From Tealeaves Require Import
  Adapters.KleisliToCategorical.TraversableFunctor
  Adapters.KleisliToCoalgebraic.TraversableFunctor
  Adapters.KleisliToCategorical.DecoratedTraversableFunctor
  Adapters.KleisliToCoalgebraic.DecoratedTraversableFunctor
  Classes.Coalgebraic.TraversableFunctor
  Classes.Coalgebraic.DecoratedTraversableFunctor.

From Tealeaves Require Export
  Functors.Batch
  Functors.Environment
  Functors.Option
  Theory.TraversableFunctor
  Kleisli.Theory.DecoratedTraversableFunctor.

Import Product.Notations.
Import Monoid.Notations.
Import Batch.Notations.
Import List.ListNotations.
Import Subset.Notations.
Import ContainerFunctor.Notations.
Import DecoratedContainerFunctor.Notations.
Import VectorRefinement.Notations.

#[local] Generalizable Variables F M E T G A B C ϕ.

(** * Properties of <<toBatch3>> *)
(**********************************************************************)
Section theory.

  Context
    `{Mapdt_inst: Mapdt E T}
    `{Traverse_inst: Traverse T}
    `{Mapd_inst: Mapd E T}
    `{Map_inst: Map T}
    `{ToBatch_inst: ToBatch T}
    `{ToBatch3_inst: ToBatch3 E T}
    `{ToCtxset_inst: ToCtxset E T}
    `{ToCtxlist_inst: ToCtxlist E T}
    `{! Compat_Traverse_Mapdt E T}
    `{! Compat_Mapd_Mapdt E T}
    `{! Compat_Map_Mapdt E T}
    `{! Compat_ToBatch_Traverse T}
    `{! Compat_ToBatch3_Mapdt E T}
    `{! Compat_ToCtxset_Mapdt E T}
    `{! Compat_ToCtxlist_Mapdt E T}
    `{! Kleisli.DecoratedTraversableFunctor.DecoratedTraversableFunctor E T}.

  (** ** Relating <<toBatch3>> with <<toBatch>> *)
  (********************************************************************)
  Lemma toBatch_to_toBatch3 {A B: Type}:
    toBatch (A := A) (A' := B) = mapfst_Batch extract ∘ toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
toBatch = mapfst_Batch extract ∘ toBatch3
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
toBatch = mapfst_Batch extract ∘ toBatch3
    intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
toBatch = mapfst_Batch extract ∘ toBatch3
    rewrite toBatch_to_traverse.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
traverse (batch A B) = mapfst_Batch extract ∘ toBatch3
    rewrite traverse_to_mapdt.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
mapdt (batch A B ∘ extract) =
mapfst_Batch extract ∘ toBatch3
    rewrite toBatch3_to_mapdt.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
mapdt (batch A B ∘ extract) =
mapfst_Batch extract ∘ mapdt (batch (E * A) B)
    rewrite (kdtf_morph
               (G1 := Batch (E * A) B)
               (G2 := Batch A B)
               (ϕ := fun C => mapfst_Batch extract)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
mapdt (batch A B ∘ extract) =
mapdt (mapfst_Batch extract ∘ batch (E * A) B)
    rewrite ret_natural.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
mapdt (batch A B ∘ extract) =
mapdt (batch A B ∘ extract)
    reflexivity.
  Qed.

  (** ** Factoring <<mapdt>> via <<toBatch3>>  *)
  (********************************************************************)
  Lemma mapdt_through_toBatch3
    `{Applicative G} `(f: E * A -> G B):
    mapdt (E := E) (T := T) f = runBatch f ∘ toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: E * A -> G B
mapdt f = runBatch f ∘ toBatch3
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: E * A -> G B
mapdt f = runBatch f ∘ toBatch3
    intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: E * A -> G B
mapdt f = runBatch f ∘ toBatch3
    rewrite toBatch3_to_mapdt.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: E * A -> G B
mapdt f = runBatch f ∘ mapdt (batch (E * A) B)
    rewrite kdtf_morph.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: E * A -> G B
mapdt f = mapdt (runBatch f ∘ batch (E * A) B)
    rewrite (runBatch_batch G).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: E * A -> G B
mapdt f = mapdt f
    reflexivity.
  Qed.

  (** ** Naturality of <<toBatch3>> *)
  (********************************************************************)
  Lemma toBatch3_mapdt
    `{Applicative G}
    {A A' B: Type} (f: E * A -> G A'):
    map (F := G) (toBatch3 (A := A') (B := B)) ∘ mapdt (T := T) f =
      traverse (T := BATCH1 B (T B)) (strength ∘ cobind f) ∘ toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, A', B: Type
f: E * A -> G A'
map toBatch3 ∘ mapdt f =
traverse (strength ∘ cobind f) ∘ toBatch3
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, A', B: Type
f: E * A -> G A'
map toBatch3 ∘ mapdt f =
traverse (strength ∘ cobind f) ∘ toBatch3
    rewrite toBatch3_to_mapdt.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, A', B: Type
f: E * A -> G A'
map (mapdt (batch (E * A') B)) ∘ mapdt f =
traverse (strength ∘ cobind f) ∘ toBatch3
    rewrite kdtf_mapdt2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, A', B: Type
f: E * A -> G A'
mapdt (kc3 (batch (E * A') B) f) =
traverse (strength ∘ cobind f) ∘ toBatch3
    rewrite (traverse_via_runBatch G).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, A', B: Type
f: E * A -> G A'
mapdt (kc3 (batch (E * A') B) f) =
runBatch
  (map (batch (E * A') B) ∘ (strength ∘ cobind f))
∘ toBatch3
    rewrite <- (mapdt_through_toBatch3
                 (A := A) (B := B)
                 (G := G ∘ Batch (E * A') B)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, A', B: Type
f: E * A -> G A'
mapdt (kc3 (batch (E * A') B) f) =
mapdt (map (batch (E * A') B) ∘ (strength ∘ cobind f))
    reflexivity.
  Qed.

  Lemma toBatch3_mapd
    {A A' B: Type} (f: E * A -> A'):
    toBatch3 ∘ mapd (T := T) f =
      mapfst_Batch (cobind f) ∘ toBatch3 (A := A) (B := B).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: E * A -> A'
toBatch3 ∘ mapd f = mapfst_Batch (cobind f) ∘ toBatch3
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: E * A -> A'
toBatch3 ∘ mapd f = mapfst_Batch (cobind f) ∘ toBatch3
    rewrite toBatch3_to_mapdt.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: E * A -> A'
mapdt (batch (E * A') B) ∘ mapd f =
mapfst_Batch (cobind f) ∘ toBatch3
    rewrite mapdt_mapd.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: E * A -> A'
mapdt (kc1 (batch (E * A') B) f) =
mapfst_Batch (cobind f) ∘ toBatch3
    rewrite toBatch3_to_mapdt.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: E * A -> A'
mapdt (kc1 (batch (E * A') B) f) =
mapfst_Batch (cobind f) ∘ mapdt (batch (E * A) B)
    rewrite (kdtf_morph
               (G1 := Batch (E * A) B)
               (G2 := Batch (E * A') B)
               (ϕ := fun C => mapfst_Batch (cobind f))).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: E * A -> A'
mapdt (kc1 (batch (E * A') B) f) =
mapdt (mapfst_Batch (cobind f) ∘ batch (E * A) B)
    reflexivity.
  Qed.

  Lemma toBatch3_map
    {A A' B: Type} (f: A -> A') {C: Type}:
    toBatch3 ∘ map (F := T) f =
      mapfst_Batch (map f) ∘ toBatch3 (A := A) (B := B).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: A -> A'
C: Type
toBatch3 ∘ map f = mapfst_Batch (map f) ∘ toBatch3
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: A -> A'
C: Type
toBatch3 ∘ map f = mapfst_Batch (map f) ∘ toBatch3
    rewrite toBatch3_to_mapdt.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: A -> A'
C: Type
mapdt (batch (E * A') B) ∘ map f =
mapfst_Batch (map f) ∘ toBatch3
    rewrite mapdt_map.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: A -> A'
C: Type
mapdt (batch (E * A') B ∘ map f) =
mapfst_Batch (map f) ∘ toBatch3
    rewrite toBatch3_to_mapdt.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: A -> A'
C: Type
mapdt (batch (E * A') B ∘ map f) =
mapfst_Batch (map f) ∘ mapdt (batch (E * A) B)
    rewrite (kdtf_morph
               (morphism := ApplicativeMorphism_mapfst_Batch (map f))

               (ϕ := fun C => mapfst_Batch (map f))).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: A -> A'
C: Type
mapdt (batch (E * A') B ∘ map f) =
mapdt (mapfst_Batch (map f) ∘ batch (E * A) B)
    rewrite ret_natural.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: A -> A'
C: Type
mapdt (batch (E * A') B ∘ map f) =
mapdt (batch (E * A') B ∘ map f)
    reflexivity.
  Qed.

  Lemma toBatch_mapd
    {A A' B: Type} (f: E * A -> A'):
    toBatch (A := A') (A' := B) ∘ mapd (T := T) f =
      mapfst_Batch f ∘ toBatch3 (T := T) (A := A) (B := B).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: E * A -> A'
toBatch ∘ mapd f = mapfst_Batch f ∘ toBatch3
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: E * A -> A'
toBatch ∘ mapd f = mapfst_Batch f ∘ toBatch3
    rewrite toBatch_to_toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: E * A -> A'
mapfst_Batch extract ∘ toBatch3 ∘ mapd f =
mapfst_Batch f ∘ toBatch3
    reassociate -> on left.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: E * A -> A'
mapfst_Batch extract ∘ (toBatch3 ∘ mapd f) =
mapfst_Batch f ∘ toBatch3
    rewrite toBatch3_mapd.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: E * A -> A'
mapfst_Batch extract
∘ (mapfst_Batch (cobind f) ∘ toBatch3) =
mapfst_Batch f ∘ toBatch3
    reassociate <-.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: E * A -> A'
mapfst_Batch extract ∘ mapfst_Batch (cobind f)
∘ toBatch3 = mapfst_Batch f ∘ toBatch3
    rewrite (mapfst_mapfst_Batch).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: E * A -> A'
mapfst_Batch (extract ∘ cobind f) ∘ toBatch3 =
mapfst_Batch f ∘ toBatch3
    rewrite (kcom_cobind0).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
f: E * A -> A'
mapfst_Batch f ∘ toBatch3 = mapfst_Batch f ∘ toBatch3
    reflexivity.
  Qed.

  (** * Factoring Operations Through <<toBatch3>> *)
  (********************************************************************)
  Section runBatch.

    (** ** Core Operations *)
    (******************************************************************)
    Theorem mapdt_through_runBatch `{Applicative G} `(f: E * A -> G B):
      mapdt f = runBatch f ∘ toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: E * A -> G B
mapdt f = runBatch f ∘ toBatch3
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: E * A -> G B
mapdt f = runBatch f ∘ toBatch3
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: E * A -> G B
mapdt f = runBatch f ∘ toBatch3
      rewrite toBatch3_to_mapdt.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: E * A -> G B
mapdt f = runBatch f ∘ mapdt (batch (E * A) B)
      rewrite kdtf_morph.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: E * A -> G B
mapdt f = mapdt (runBatch f ∘ batch (E * A) B)
      rewrite (runBatch_batch G).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: E * A -> G B
mapdt f = mapdt f
      reflexivity.
    Qed.

    Corollary traverse_through_runBatch `{Applicative G} `(f: A -> G B):
      traverse f = runBatch (f ∘ extract (W := (E ×))) ∘ toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: A -> G B
traverse f = runBatch (f ∘ extract) ∘ toBatch3
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: A -> G B
traverse f = runBatch (f ∘ extract) ∘ toBatch3
      rewrite traverse_to_mapdt.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: A -> G B
mapdt (f ∘ extract) =
runBatch (f ∘ extract) ∘ toBatch3
      rewrite mapdt_through_runBatch.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
f: A -> G B
runBatch (f ∘ extract) ∘ toBatch3 =
runBatch (f ∘ extract) ∘ toBatch3
      reflexivity.
    Qed.

    Corollary mapd_through_runBatch `(f: E * A -> B):
      mapd f = runBatch (G := fun A => A) f ∘ toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
mapd f = runBatch f ∘ toBatch3
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
mapd f = runBatch f ∘ toBatch3
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
mapd f = runBatch f ∘ toBatch3
      rewrite mapd_to_mapdt.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
mapdt f = runBatch f ∘ toBatch3
      rewrite mapdt_through_runBatch.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
runBatch f ∘ toBatch3 = runBatch f ∘ toBatch3
      reflexivity.
    Qed.

    Corollary map_through_runBatch `(f: A -> B):
      map f = runBatch (G := fun A => A) (f ∘ extract) ∘ toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> B
map f = runBatch (f ∘ extract) ∘ toBatch3
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> B
map f = runBatch (f ∘ extract) ∘ toBatch3
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> B
map f = runBatch (f ∘ extract) ∘ toBatch3
      rewrite map_to_mapdt.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> B
mapdt (f ∘ extract) =
runBatch (f ∘ extract) ∘ toBatch3
      rewrite mapdt_through_runBatch.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> B
runBatch (f ∘ extract) ∘ toBatch3 =
runBatch (f ∘ extract) ∘ toBatch3
      reflexivity.
    Qed.

    (** ** <<mapdReduce>> Through <<toBatch3>> *)
    (******************************************************************)
    Lemma mapdReduce_through_runBatch1 {A} `{Monoid M}: forall `(f: E * A -> M),
        mapdReduce f = runBatch (G := const M) f (C := T False) ∘ toBatch3 (B := False).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, M: Type
op: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
forall f : E * A -> M,
mapdReduce f = runBatch f ∘ toBatch3
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, M: Type
op: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
forall f : E * A -> M,
mapdReduce f = runBatch f ∘ toBatch3
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, M: Type
op: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
f: E * A -> M
mapdReduce f = runBatch f ∘ toBatch3
      rewrite mapdReduce_to_mapdt1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, M: Type
op: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
f: E * A -> M
mapdt f = runBatch f ∘ toBatch3
      rewrite (mapdt_through_runBatch (G := const M)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, M: Type
op: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
f: E * A -> M
runBatch f ∘ toBatch3 = runBatch f ∘ toBatch3
      reflexivity.
    Qed.

    Lemma mapdReduce_through_runBatch2 `{Monoid M}: forall (A fake: Type) `(f: E * A -> M),
        mapdReduce f = runBatch (G := const M) f (C := T fake) ∘ toBatch3 (B := fake).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
forall (A fake : Type) (f : E * A -> M),
mapdReduce f = runBatch f ∘ toBatch3
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
forall (A fake : Type) (f : E * A -> M),
mapdReduce f = runBatch f ∘ toBatch3
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
A, fake: Type
f: E * A -> M
mapdReduce f = runBatch f ∘ toBatch3
      rewrite mapdReduce_to_mapdt1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
A, fake: Type
f: E * A -> M
mapdt f = runBatch f ∘ toBatch3
      rewrite (mapdt_constant_applicative2 False False fake).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
A, fake: Type
f: E * A -> M
mapdt f = runBatch f ∘ toBatch3
      rewrite mapdt_through_runBatch.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H0: Monoid M
A, fake: Type
f: E * A -> M
runBatch f ∘ toBatch3 = runBatch f ∘ toBatch3
      reflexivity.
    Qed.

    (** ** <<toctxlist>> Through <<toBatch3>> *)
    (******************************************************************)
    Corollary toctxlist_through_runBatch3 {A: Type} (tag: Type):
      toctxlist = runBatch (B := tag)
                    (G := const (list (E * A)))
                    (ret (T := list))
                    ∘ toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, tag: Type
toctxlist = runBatch ret ∘ toBatch3
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, tag: Type
toctxlist = runBatch ret ∘ toBatch3
      rewrite (toctxlist_to_mapdt2 A tag).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, tag: Type
mapdt ret = runBatch ret ∘ toBatch3
      now rewrite mapdt_through_runBatch.
    Qed.


    Corollary toctxset_through_runBatch1 {A: Type}:
      toctxset (F := T) = runBatch (B := False)
                            (G := const (subset (E * A)))
                            (ret (T := subset)) ∘ toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
toctxset = runBatch ret ∘ toBatch3
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
toctxset = runBatch ret ∘ toBatch3
      rewrite (toctxset_to_mapdt1 A).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
mapdt ret = runBatch ret ∘ toBatch3
      now rewrite (mapdt_through_runBatch).
    Qed.

    Corollary toctxset_through_runBatch2 {A tag: Type}:
      toctxset (F := T) = runBatch (B := tag)
                            (G := const (subset (E * A)))
                            (ret (T := subset)) ∘ toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, tag: Type
toctxset = runBatch ret ∘ toBatch3
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, tag: Type
toctxset = runBatch ret ∘ toBatch3
      rewrite (toctxset_to_mapdt2 A tag).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, tag: Type
mapdt ret = runBatch ret ∘ toBatch3
      now rewrite (mapdt_through_runBatch).
    Qed.

    Corollary element_ctx_of_through_runBatch1
      `{ToSubset T}
      `{! Compat_ToSubset_Traverse T}
      {A: Type} {p: E * A}:
      element_ctx_of (T := T) p =
        runBatch (B := False) (G := const Prop)
          (H0 := @Mult_const _ Monoid_op_or)
          (H1 := @Pure_const _ Monoid_unit_false)
          {{p}} ∘ toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
H0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
p: E * A
element_ctx_of p = runBatch {{p}} ∘ toBatch3
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
H0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
p: E * A
element_ctx_of p = runBatch {{p}} ∘ toBatch3
      rewrite element_ctx_of_to_mapdReduce.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
H0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
p: E * A
mapdReduce {{p}} = runBatch {{p}} ∘ toBatch3
      rewrite mapdReduce_through_runBatch1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
H0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
p: E * A
runBatch {{p}} ∘ toBatch3 = runBatch {{p}} ∘ toBatch3
      reflexivity.
    Qed.

  End runBatch.

  (** * Respectfulness Properties *)
  (********************************************************************)
  Section decorated_traversable_functor_theory.

    (** ** Purity *)
    (******************************************************************)
    Theorem mapdt_purity1:
      forall `{Applicative G},
        `(mapdt (G := G) (pure ∘ extract) = @pure G _ (T A)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall A : Type, mapdt (pure ∘ extract) = pure
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall A : Type, mapdt (pure ∘ extract) = pure
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A: Type
mapdt (pure ∘ extract) = pure
      rewrite <- (kdtf_morph (G1 := fun A => A) (G2 := G)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A: Type
pure ∘ mapdt extract = pure
      rewrite kdtf_mapdt1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A: Type
pure ∘ id = pure
      reflexivity.
    Qed.

    (** ** Characterizing <<∈d>> and <<mapd>> *)
    (******************************************************************)
    Lemma mapdt_respectful:
      forall A B `{Applicative G} (t: T A) (f g: E * A -> G B),
        (forall (e: E) (a: A), (e, a) ∈d t -> f (e, a) = g (e, a))
        -> mapdt f t = mapdt g t.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (t : T A) (f g : E * A -> G B),
(forall (e : E) (a : A),
 (e, a) ∈d t -> f (e, a) = g (e, a)) ->
mapdt f t = mapdt g t
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (t : T A) (f g : E * A -> G B),
(forall (e : E) (a : A),
 (e, a) ∈d t -> f (e, a) = g (e, a)) ->
mapdt f t = mapdt g t
      introv Happl hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
hyp: forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = g (e, a)
mapdt f t = mapdt g t
      do 2 rewrite mapdt_through_runBatch.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
hyp: forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = g (e, a)
(runBatch f ∘ toBatch3) t = (runBatch g ∘ toBatch3) t
      unfold element_ctx_of in hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
hyp: forall (e : E) (a : A),
toctxset t (e, a) -> f (e, a) = g (e, a)
(runBatch f ∘ toBatch3) t = (runBatch g ∘ toBatch3) t
      rewrite (toctxset_through_runBatch2 (tag := B)) in hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
hyp: forall (e : E) (a : A),
(runBatch ret ∘ toBatch3) t (e, a) ->
f (e, a) = g (e, a)
(runBatch f ∘ toBatch3) t = (runBatch g ∘ toBatch3) t
      unfold compose in *.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
hyp: forall (e : E) (a : A),
runBatch ret (toBatch3 t) (e, a) ->
f (e, a) = g (e, a)
runBatch f (toBatch3 t) = runBatch g (toBatch3 t)
      induction (toBatch3 (B := B) t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
c: C
hyp: forall (e : E) (a : A),
runBatch ret (Done c) (e, a) ->
f (e, a) = g (e, a)
runBatch f (Done c) = runBatch g (Done c)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
hyp: forall (e : E) (a0 : A),
runBatch ret (b ⧆ a) (e, a0) ->
f (e, a0) = g (e, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch f (b ⧆ a) = runBatch g (b ⧆ a)
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
c: C
hyp: forall (e : E) (a : A),
runBatch ret (Done c) (e, a) ->
f (e, a) = g (e, a)
runBatch f (Done c) = runBatch g (Done c) reflexivity.
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
hyp: forall (e : E) (a0 : A),
runBatch ret (b ⧆ a) (e, a0) ->
f (e, a0) = g (e, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch f (b ⧆ a) = runBatch g (b ⧆ a) destruct a as [e a].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch f (b ⧆ (e, a)) = runBatch g (b ⧆ (e, a)) cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
ap G (runBatch f b) (f (e, a)) =
ap G (runBatch g b) (g (e, a))
        rewrite IHb.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
ap G (runBatch g b) (f (e, a)) =
ap G (runBatch g b) (g (e, a))
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
forall (e : E) (a : A),
runBatch ret b (e, a) -> f (e, a) = g (e, a)
        rewrite hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
ap G (runBatch g b) (g (e, a)) =
ap G (runBatch g b) (g (e, a))
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch ret (b ⧆ (e, a)) (e, a)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
forall (e : E) (a : A),
runBatch ret b (e, a) -> f (e, a) = g (e, a)
        reflexivity.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch ret (b ⧆ (e, a)) (e, a)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
forall (e : E) (a : A),
runBatch ret b (e, a) -> f (e, a) = g (e, a)
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch ret (b ⧆ (e, a)) (e, a) cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
(runBatch ret b ● ret (e, a)) (e, a) now right.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
forall (e : E) (a : A),
runBatch ret b (e, a) -> f (e, a) = g (e, a) introv hyp2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
e0: E
a0: A
hyp2: runBatch ret b (e0, a0)
f (e0, a0) = g (e0, a0)
          apply hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
Happl: Applicative G
t: T A
f, g: E * A -> G B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
e0: E
a0: A
hyp2: runBatch ret b (e0, a0)
runBatch ret (b ⧆ (e, a)) (e0, a0)
          now left.
    Qed.


    Corollary mapdt_respectful_pure:
      forall A `{Applicative G} (t: T A) (f: E * A -> G A),
        (forall (e: E) (a: A), (e, a) ∈d t -> f (e, a) = pure (F := G) a)
        -> mapdt f t = pure t.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A : Type) (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (t : T A) (f : E * A -> G A),
(forall (e : E) (a : A),
 (e, a) ∈d t -> f (e, a) = pure a) ->
mapdt f t = pure t
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A : Type) (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (t : T A) (f : E * A -> G A),
(forall (e : E) (a : A),
 (e, a) ∈d t -> f (e, a) = pure a) ->
mapdt f t = pure t
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
t: T A
f: E * A -> G A
H1: forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = pure a
mapdt f t = pure t
      rewrite <- mapdt_purity1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
t: T A
f: E * A -> G A
H1: forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = pure a
mapdt f t = mapdt (pure ∘ extract) t
      now apply mapdt_respectful.
    Qed.

    Corollary mapdt_respectful_id:
      forall A (t: T A) (f: E * A -> A),
        (forall (e: E) (a: A), (e, a) ∈d t -> f (e, a) = a)
        -> mapdt (G := fun A => A) f t = t.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A : Type) (t : T A) (f : E * A -> A),
(forall (e : E) (a : A), (e, a) ∈d t -> f (e, a) = a) ->
mapdt f t = t
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A : Type) (t : T A) (f : E * A -> A),
(forall (e : E) (a : A), (e, a) ∈d t -> f (e, a) = a) ->
mapdt f t = t
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
f: E * A -> A
H0: forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = a
mapdt f t = t
      change t with (id t) at 2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
f: E * A -> A
H0: forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = a
mapdt f t = id t
      rewrite <- kdtf_mapdt1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
f: E * A -> A
H0: forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = a
mapdt f t = mapdt extract t
      apply (mapdt_respectful A A (G := fun A => A)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
f: E * A -> A
H0: forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = a
forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = extract (e, a)
      assumption.
    Qed.

    Lemma mapd_respectful:
      forall A B (t: T A) (f g: E * A -> B),
        (forall (e: E) (a: A), (e, a) ∈d t -> f (e, a) = g (e, a))
        -> mapd f t = mapd g t.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (t : T A) (f g : E * A -> B),
(forall (e : E) (a : A),
 (e, a) ∈d t -> f (e, a) = g (e, a)) ->
mapd f t = mapd g t
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (t : T A) (f g : E * A -> B),
(forall (e : E) (a : A),
 (e, a) ∈d t -> f (e, a) = g (e, a)) ->
mapd f t = mapd g t
      introv hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
hyp: forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = g (e, a)
mapd f t = mapd g t
      rewrite mapd_through_runBatch.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
hyp: forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = g (e, a)
(runBatch f ∘ toBatch3) t = mapd g t
      rewrite mapd_through_runBatch.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
hyp: forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = g (e, a)
(runBatch f ∘ toBatch3) t = (runBatch g ∘ toBatch3) t
      unfold element_ctx_of in hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
hyp: forall (e : E) (a : A),
toctxset t (e, a) -> f (e, a) = g (e, a)
(runBatch f ∘ toBatch3) t = (runBatch g ∘ toBatch3) t
      rewrite (toctxset_through_runBatch2 (tag := B)) in hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
hyp: forall (e : E) (a : A),
(runBatch ret ∘ toBatch3) t (e, a) ->
f (e, a) = g (e, a)
(runBatch f ∘ toBatch3) t = (runBatch g ∘ toBatch3) t
      unfold compose in *.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
hyp: forall (e : E) (a : A),
runBatch ret (toBatch3 t) (e, a) ->
f (e, a) = g (e, a)
runBatch f (toBatch3 t) = runBatch g (toBatch3 t)
      induction (toBatch3 (B := B) t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
c: C
hyp: forall (e : E) (a : A),
runBatch ret (Done c) (e, a) ->
f (e, a) = g (e, a)
runBatch f (Done c) = runBatch g (Done c)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
hyp: forall (e : E) (a0 : A),
runBatch ret (b ⧆ a) (e, a0) ->
f (e, a0) = g (e, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch f (b ⧆ a) = runBatch g (b ⧆ a)
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
c: C
hyp: forall (e : E) (a : A),
runBatch ret (Done c) (e, a) ->
f (e, a) = g (e, a)
runBatch f (Done c) = runBatch g (Done c) reflexivity.
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
hyp: forall (e : E) (a0 : A),
runBatch ret (b ⧆ a) (e, a0) ->
f (e, a0) = g (e, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch f (b ⧆ a) = runBatch g (b ⧆ a) destruct a as [e a].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch f (b ⧆ (e, a)) = runBatch g (b ⧆ (e, a)) cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch f b (f (e, a)) = runBatch g b (g (e, a))
        rewrite IHb.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch g b (f (e, a)) = runBatch g b (g (e, a))
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
forall (e : E) (a : A),
runBatch ret b (e, a) -> f (e, a) = g (e, a)
        rewrite hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch g b (g (e, a)) = runBatch g b (g (e, a))
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch ret (b ⧆ (e, a)) (e, a)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
forall (e : E) (a : A),
runBatch ret b (e, a) -> f (e, a) = g (e, a)
        reflexivity.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch ret (b ⧆ (e, a)) (e, a)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
forall (e : E) (a : A),
runBatch ret b (e, a) -> f (e, a) = g (e, a)
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
runBatch ret (b ⧆ (e, a)) (e, a) cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
(runBatch ret b ● ret (e, a)) (e, a) now right.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
forall (e : E) (a : A),
runBatch ret b (e, a) -> f (e, a) = g (e, a) introv hyp2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
e0: E
a0: A
hyp2: runBatch ret b (e0, a0)
f (e0, a0) = g (e0, a0)
          apply hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
hyp: forall (e0 : E) (a0 : A),
runBatch ret (b ⧆ (e, a)) (e0, a0) ->
f (e0, a0) = g (e0, a0)
IHb: (forall (e : E) (a : A),
 runBatch ret b (e, a) -> f (e, a) = g (e, a)) ->
runBatch f b = runBatch g b
e0: E
a0: A
hyp2: runBatch ret b (e0, a0)
runBatch ret (b ⧆ (e, a)) (e0, a0)
          now left.
    Qed.

    Lemma mapd_eq_iff:
      forall A B (t: T A) (f g: E * A -> B),
        mapd f t = mapd g t ->
        mapfst_Batch f (toBatch3 (B := B) t) = mapfst_Batch g (toBatch3 t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (t : T A) (f g : E * A -> B),
mapd f t = mapd g t ->
mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (t : T A) (f g : E * A -> B),
mapd f t = mapd g t ->
mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
      assert (cut: mapfst_Batch (cobind f) (toBatch3 (B := B) t) = mapfst_Batch (cobind g) (toBatch3 t)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
mapfst_Batch (cobind f) (toBatch3 t) =
mapfst_Batch (cobind g) (toBatch3 t)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
cut: mapfst_Batch (cobind f) (toBatch3 t) =
mapfst_Batch (cobind g) (toBatch3 t)
mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
mapfst_Batch (cobind f) (toBatch3 t) =
mapfst_Batch (cobind g) (toBatch3 t) compose near t.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
(mapfst_Batch (cobind f) ∘ toBatch3) t =
(mapfst_Batch (cobind g) ∘ toBatch3) t
        rewrite <- toBatch3_mapd.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
(toBatch3 ∘ mapd f) t =
(mapfst_Batch (cobind g) ∘ toBatch3) t
        rewrite <- toBatch3_mapd.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
(toBatch3 ∘ mapd f) t = (toBatch3 ∘ mapd g) t
        unfold compose.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
toBatch3 (mapd f t) = toBatch3 (mapd g t) rewrite H0.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
toBatch3 (mapd g t) = toBatch3 (mapd g t)
        reflexivity.
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
cut: mapfst_Batch (cobind f) (toBatch3 t) =
mapfst_Batch (cobind g) (toBatch3 t)
mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t) induction (toBatch3 t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
c: C
cut: mapfst_Batch (cobind f) (Done c) =
mapfst_Batch (cobind g) (Done c)
mapfst_Batch f (Done c) = mapfst_Batch g (Done c)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
cut: mapfst_Batch (cobind f) (b ⧆ a) =
mapfst_Batch (cobind g) (b ⧆ a)
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
mapfst_Batch f (b ⧆ a) = mapfst_Batch g (b ⧆ a)
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
c: C
cut: mapfst_Batch (cobind f) (Done c) =
mapfst_Batch (cobind g) (Done c)
mapfst_Batch f (Done c) = mapfst_Batch g (Done c) reflexivity.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
cut: mapfst_Batch (cobind f) (b ⧆ a) =
mapfst_Batch (cobind g) (b ⧆ a)
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
mapfst_Batch f (b ⧆ a) = mapfst_Batch g (b ⧆ a) cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
cut: mapfst_Batch (cobind f) (b ⧆ a) =
mapfst_Batch (cobind g) (b ⧆ a)
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
mapfst_Batch f b ⧆ f a = mapfst_Batch g b ⧆ g a fequal.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
cut: mapfst_Batch (cobind f) (b ⧆ a) =
mapfst_Batch (cobind g) (b ⧆ a)
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
mapfst_Batch f b = mapfst_Batch g b
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
cut: mapfst_Batch (cobind f) (b ⧆ a) =
mapfst_Batch (cobind g) (b ⧆ a)
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
f a = g a
          {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
cut: mapfst_Batch (cobind f) (b ⧆ a) =
mapfst_Batch (cobind g) (b ⧆ a)
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
mapfst_Batch f b = mapfst_Batch g b apply IHb.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
cut: mapfst_Batch (cobind f) (b ⧆ a) =
mapfst_Batch (cobind g) (b ⧆ a)
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
mapd f t = mapd g t
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
cut: mapfst_Batch (cobind f) (b ⧆ a) =
mapfst_Batch (cobind g) (b ⧆ a)
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
mapfst_Batch (cobind f) b = mapfst_Batch (cobind g) b
            assumption.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
cut: mapfst_Batch (cobind f) (b ⧆ a) =
mapfst_Batch (cobind g) (b ⧆ a)
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
mapfst_Batch (cobind f) b = mapfst_Batch (cobind g) b
            inversion cut.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
cut: mapfst_Batch (cobind f) (b ⧆ a) =
mapfst_Batch (cobind g) (b ⧆ a)
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
H2: mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b
H3: cobind f a = cobind g a
mapfst_Batch (cobind f) b = mapfst_Batch (cobind f) b
            eauto. }
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
cut: mapfst_Batch (cobind f) (b ⧆ a) =
mapfst_Batch (cobind g) (b ⧆ a)
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
f a = g a
          {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
cut: mapfst_Batch (cobind f) (b ⧆ a) =
mapfst_Batch (cobind g) (b ⧆ a)
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
f a = g a inversion cut.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
cut: mapfst_Batch (cobind f) (b ⧆ a) =
mapfst_Batch (cobind g) (b ⧆ a)
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
H2: mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b
H3: cobind f a = cobind g a
f a = g a
            destruct a as [e a].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
cut: mapfst_Batch (cobind f) (b ⧆ (e, a)) =
mapfst_Batch (cobind g) (b ⧆ (e, a))
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
H2: mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b
H3: cobind f (e, a) = cobind g (e, a)
f (e, a) = g (e, a)
            cbn in H3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: mapd f t = mapd g t
C: Type
b: Batch (E * A) B (B -> C)
e: E
a: A
cut: mapfst_Batch (cobind f) (b ⧆ (e, a)) =
mapfst_Batch (cobind g) (b ⧆ (e, a))
IHb: mapd f t = mapd g t ->
mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b ->
mapfst_Batch f b = mapfst_Batch g b
H2: mapfst_Batch (cobind f) b =
mapfst_Batch (cobind g) b
H3: (e, f (e, a)) = (e, g (e, a))
f (e, a) = g (e, a)
            inversion H3; auto.
          }
    Qed.

    Lemma mapd_respectful_iff
      `{! ToSubset T} `{! Compat_ToSubset_Traverse T}:
      forall A B (t: T A) (f g: E * A -> B),
        (forall (e: E) (a: A), (e, a) ∈d t -> f (e, a) = g (e, a))
        <-> mapd f t = mapd g t.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
forall (A B : Type) (t : T A) (f g : E * A -> B),
(forall (e : E) (a : A),
 (e, a) ∈d t -> f (e, a) = g (e, a)) <->
mapd f t = mapd g t
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
forall (A B : Type) (t : T A) (f g : E * A -> B),
(forall (e : E) (a : A),
 (e, a) ∈d t -> f (e, a) = g (e, a)) <->
mapd f t = mapd g t
      split.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
(forall (e : E) (a : A),
 (e, a) ∈d t -> f (e, a) = g (e, a)) ->
mapd f t = mapd g t
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
mapd f t = mapd g t ->
forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = g (e, a)
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
(forall (e : E) (a : A),
 (e, a) ∈d t -> f (e, a) = g (e, a)) ->
mapd f t = mapd g t apply mapd_respectful.
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
mapd f t = mapd g t ->
forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = g (e, a) introv Heq Hf.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapd f t = mapd g t
e: E
a: A
Hf: (e, a) ∈d t
f (e, a) = g (e, a)
        apply mapd_eq_iff in Heq.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: (e, a) ∈d t
f (e, a) = g (e, a)
        rewrite element_ctx_of_to_mapdReduce in Hf.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: mapdReduce {{(e, a)}} t
f (e, a) = g (e, a)
        rewrite (mapdReduce_through_runBatch2 A B) in Hf.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: (runBatch {{(e, a)}} ∘ toBatch3) t
f (e, a) = g (e, a)
        unfold compose in *.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
f (e, a) = g (e, a)
        pose (Batch_ind (E * A) B
                (fun (C: Type) (b: Batch (E * A) B C) =>
                   mapfst_Batch f b = mapfst_Batch g b ->
                   runBatch
                     (H := Map_const)
                     (H0 := (@Mult_const Prop Monoid_op_or))
                     (H1 := (@Pure_const Prop Monoid_unit_false))
                     (C := C) (G := const Prop) {{(e, a)}} b
                   -> f (e, a) = g (e, a))).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
f (e, a) = g (e, a)
        eapply e0.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
forall (C : Type) (c : C),
mapfst_Batch f (Done c) = mapfst_Batch g (Done c) ->
runBatch {{(e, a)}} (Done c) -> f (e, a) = g (e, a)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
forall (C : Type) (b : Batch (E * A) B (B -> C)),
(mapfst_Batch f b = mapfst_Batch g b ->
 runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) ->
forall a0 : E * A,
mapfst_Batch f (b ⧆ a0) = mapfst_Batch g (b ⧆ a0) ->
runBatch {{(e, a)}} (b ⧆ a0) -> f (e, a) = g (e, a)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
mapfst_Batch f ?b = mapfst_Batch g ?b
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
runBatch {{(e, a)}} ?b
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
forall (C : Type) (c : C),
mapfst_Batch f (Done c) = mapfst_Batch g (Done c) ->
runBatch {{(e, a)}} (Done c) -> f (e, a) = g (e, a) introv X Y.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
c: C
X: mapfst_Batch f (Done c) = mapfst_Batch g (Done c)
Y: runBatch {{(e, a)}} (Done c)
f (e, a) = g (e, a)
          inversion Y.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
forall (C : Type) (b : Batch (E * A) B (B -> C)),
(mapfst_Batch f b = mapfst_Batch g b ->
 runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) ->
forall a0 : E * A,
mapfst_Batch f (b ⧆ a0) = mapfst_Batch g (b ⧆ a0) ->
runBatch {{(e, a)}} (b ⧆ a0) -> f (e, a) = g (e, a) intros C b Hb [e' a'].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e, a)}} b -> f (e, a) = g (e, a)
e': E
a': A
mapfst_Batch f (b ⧆ (e', a')) =
mapfst_Batch g (b ⧆ (e', a')) ->
runBatch {{(e, a)}} (b ⧆ (e', a')) ->
f (e, a) = g (e, a)
          rewrite mapfst_Batch_rw2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e, a)}} b -> f (e, a) = g (e, a)
e': E
a': A
mapfst_Batch f b ⧆ f (e', a') =
mapfst_Batch g (b ⧆ (e', a')) ->
runBatch {{(e, a)}} (b ⧆ (e', a')) ->
f (e, a) = g (e, a)
          rewrite mapfst_Batch_rw2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e, a)}} b -> f (e, a) = g (e, a)
e': E
a': A
mapfst_Batch f b ⧆ f (e', a') =
mapfst_Batch g b ⧆ g (e', a') ->
runBatch {{(e, a)}} (b ⧆ (e', a')) ->
f (e, a) = g (e, a)
          rewrite runBatch_rw2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e, a)}} b -> f (e, a) = g (e, a)
e': E
a': A
mapfst_Batch f b ⧆ f (e', a') =
mapfst_Batch g b ⧆ g (e', a') ->
ap (const Prop) (runBatch {{(e, a)}} b)
  ({{(e, a)}} (e', a')) -> f (e, a) = g (e, a)
          unfold ap.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e, a)}} b -> f (e, a) = g (e, a)
e': E
a': A
mapfst_Batch f b ⧆ f (e', a') =
mapfst_Batch g b ⧆ g (e', a') ->
map (fun '(f, a) => f a)
  (mult (runBatch {{(e, a)}} b, {{(e, a)}} (e', a'))) ->
f (e, a) = g (e, a)
          unfold_ops @Map_const.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e, a)}} b -> f (e, a) = g (e, a)
e': E
a': A
mapfst_Batch f b ⧆ f (e', a') =
mapfst_Batch g b ⧆ g (e', a') ->
mult (runBatch {{(e, a)}} b, {{(e, a)}} (e', a')) ->
f (e, a) = g (e, a)
          unfold_ops @Mult_const.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e, a)}} b -> f (e, a) = g (e, a)
e': E
a': A
mapfst_Batch f b ⧆ f (e', a') =
mapfst_Batch g b ⧆ g (e', a') ->
runBatch {{(e, a)}} b ● {{(e, a)}} (e', a') ->
f (e, a) = g (e, a)
          unfold transparent tcs.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e, a)}} b -> f (e, a) = g (e, a)
e': E
a': A
mapfst_Batch f b ⧆ f (e', a') =
mapfst_Batch g b ⧆ g (e', a') ->
runBatch {{(e, a)}} b \/ {{(e, a)}} (e', a') ->
f (e, a) = g (e, a)
          intros X [Case1 | Case2].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e, a)}} b -> f (e, a) = g (e, a)
e': E
a': A
X: mapfst_Batch f b ⧆ f (e', a') =
mapfst_Batch g b ⧆ g (e', a')
Case1: runBatch {{(e, a)}} b
f (e, a) = g (e, a)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e, a)}} b -> f (e, a) = g (e, a)
e': E
a': A
X: mapfst_Batch f b ⧆ f (e', a') =
mapfst_Batch g b ⧆ g (e', a')
Case2: {{(e, a)}} (e', a')
f (e, a) = g (e, a)
          {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e, a)}} b -> f (e, a) = g (e, a)
e': E
a': A
X: mapfst_Batch f b ⧆ f (e', a') =
mapfst_Batch g b ⧆ g (e', a')
Case1: runBatch {{(e, a)}} b
f (e, a) = g (e, a) apply Hb; inversion X; auto. }
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e, a)}} b -> f (e, a) = g (e, a)
e': E
a': A
X: mapfst_Batch f b ⧆ f (e', a') =
mapfst_Batch g b ⧆ g (e', a')
Case2: {{(e, a)}} (e', a')
f (e, a) = g (e, a)
          {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e, a)}} b -> f (e, a) = g (e, a)
e': E
a': A
X: mapfst_Batch f b ⧆ f (e', a') =
mapfst_Batch g b ⧆ g (e', a')
Case2: {{(e, a)}} (e', a')
f (e, a) = g (e, a) inversion X.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e, a)}} b -> f (e, a) = g (e, a)
e': E
a': A
X: mapfst_Batch f b ⧆ f (e', a') =
mapfst_Batch g b ⧆ g (e', a')
Case2: {{(e, a)}} (e', a')
H1: mapfst_Batch f b = mapfst_Batch g b
H2: f (e', a') = g (e', a')
f (e, a) = g (e, a)
            inversion Case2; subst.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e': E
a': A
Hf: runBatch {{(e', a')}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e', a')}} b ->
   f (e', a') = g (e', a')): (forall (C : Type) (c : C),
 mapfst_Batch f (Done c) =
 mapfst_Batch g (Done c) ->
 runBatch {{(e', a')}} (Done c) ->
 f (e', a') = g (e', a')) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e', a')}} b ->
  f (e', a') = g (e', a')) ->
 forall a : E * A,
 mapfst_Batch f (b ⧆ a) = mapfst_Batch g (b ⧆ a) ->
 runBatch {{(e', a')}} (b ⧆ a) ->
 f (e', a') = g (e', a')) ->
forall (C : Type) (b : Batch (E * A) B C),
mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e', a')}} b ->
f (e', a') = g (e', a')
C: Type
b: Batch (E * A) B (B -> C)
Hb: mapfst_Batch f b = mapfst_Batch g b ->
runBatch {{(e', a')}} b ->
f (e', a') = g (e', a')
X: mapfst_Batch f b ⧆ f (e', a') =
mapfst_Batch g b ⧆ g (e', a')
Case2: {{(e', a')}} (e', a')
H1: mapfst_Batch f b = mapfst_Batch g b
H2: f (e', a') = g (e', a')
f (e', a') = g (e', a')
            assumption. }
          Unshelve.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
Type
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
Batch (E * A) B ?C
          exact (T B).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
Batch (E * A) B (T B)
          exact (toBatch3 t).
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t) assumption.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
ToSubset0: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
t: T A
f, g: E * A -> B
Heq: mapfst_Batch f (toBatch3 t) =
mapfst_Batch g (toBatch3 t)
e: E
a: A
Hf: runBatch {{(e, a)}} (toBatch3 t)
e0:= Batch_ind (E * A) B
  (fun (C : Type) (b : Batch (E * A) B C) =>
   mapfst_Batch f b = mapfst_Batch g b ->
   runBatch {{(e, a)}} b -> f (e, a) = g (e, a)): (forall (C : Type) (c : C),
 (fun (C0 : Type) (b : Batch (E * A) B C0) =>
  mapfst_Batch f b = mapfst_Batch g b ->
  runBatch {{(e, a)}} b -> f (e, a) = g (e, a)) C
   (Done c)) ->
(forall (C : Type) (b : Batch (E * A) B (B -> C)),
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   (B -> C) b ->
 forall a0 : E * A,
 (fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
  mapfst_Batch f b0 = mapfst_Batch g b0 ->
  runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a))
   C (b ⧆ a0)) ->
forall (C : Type) (b : Batch (E * A) B C),
(fun (C0 : Type) (b0 : Batch (E * A) B C0) =>
 mapfst_Batch f b0 = mapfst_Batch g b0 ->
 runBatch {{(e, a)}} b0 -> f (e, a) = g (e, a)) C
  b
runBatch {{(e, a)}} (toBatch3 t) assumption.
    Qed.

    Import Kleisli.DecoratedTraversableFunctor.DerivedInstances.

    #[export] Instance
      DecoratedContainerFunctor_DecoratedTraversableFunctor:
      DecoratedContainerFunctor E T.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
DecoratedContainerFunctor E T
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
DecoratedContainerFunctor E T
      constructor.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
DecoratedFunctor E T
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
DecoratedHom E T (ctxset E)
  (@toctxset E T ToCtxset_inst)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (t : T A) (f g : E * A -> B),
(forall (e : E) (a : A),
 (e, a) ∈d t -> f (e, a) = g (e, a)) ->
mapd f t = mapd g t
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
DecoratedFunctor E T typeclasses eauto.
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
DecoratedHom E T (ctxset E)
  (@toctxset E T ToCtxset_inst) constructor.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (f : E * A -> B),
mapd f ∘ toctxset = toctxset ∘ mapd f
        intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
mapd f ∘ toctxset = toctxset ∘ mapd f
        rewrite toctxset_mapd.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
mapd f ∘ toctxset = mapd f ∘ toctxset
        reflexivity.
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (t : T A) (f g : E * A -> B),
(forall (e : E) (a : A),
 (e, a) ∈d t -> f (e, a) = g (e, a)) ->
mapd f t = mapd g t intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = g (e, a)
mapd f t = mapd g t
        apply mapd_respectful.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f, g: E * A -> B
H0: forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = g (e, a)
forall (e : E) (a : A),
(e, a) ∈d t -> f (e, a) = g (e, a)
        assumption.
    Qed.

  End decorated_traversable_functor_theory.

  (** * Shapeliness *)
  (********************************************************************)
  Section shapeliness.

    Import Kleisli.DecoratedTraversableFunctor.DerivedInstances.

    Lemma mapd_shape {A B}: forall (f: E * A -> B) t,
        shape (mapd f t) = shape t.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
forall (f : E * A -> B) (t : T A),
shape (mapd f t) = shape t
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
forall (f : E * A -> B) (t : T A),
shape (mapd f t) = shape t
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t: T A
shape (mapd f t) = shape t
      unfold shape.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t: T A
map (const tt) (mapd f t) = map (const tt) t
      compose near t on left.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t: T A
(map (const tt) ∘ mapd f) t = map (const tt) t
      rewrite map_mapd.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t: T A
mapd (const tt ∘ f) t = map (const tt) t
      rewrite map_to_mapd.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t: T A
mapd (const tt ∘ f) t = mapd (const tt ∘ extract) t
      reflexivity.
    Qed.

    Lemma mapd_ctxlist_injective {A B}:
      forall (f: E * A -> B) (t1 t2: T A),
        (forall p q: E * A, f p = f q -> p = q) ->
        mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
        tolist t1 = tolist t2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
forall (f : E * A -> B) (t1 t2 : T A),
(forall p q : E * A, f p = f q -> p = q) ->
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
tolist t1 = tolist t2
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
forall (f : E * A -> B) (t1 t2 : T A),
(forall p q : E * A, f p = f q -> p = q) ->
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
tolist t1 = tolist t2
      introv Hinj.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
tolist t1 = tolist t2
      rewrite tolist_to_toctxlist.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
(map extract ∘ toctxlist) t1 =
(map extract ∘ toctxlist) t2
      unfold compose.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
map extract (toctxlist t1) =
map extract (toctxlist t2)
      generalize dependent (toctxlist t2).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
forall e : env E A,
mapd f (toctxlist t1) = mapd f e ->
map extract (toctxlist t1) = map extract e
      induction (toctxlist t1) as [| [e a] rest IHrest];
        intros toctxlist_t2 premise.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
toctxlist_t2: env E A
premise: mapd f [] = mapd f toctxlist_t2
map extract [] = map extract toctxlist_t2
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
premise: mapd f ((e, a) :: rest) =
mapd f toctxlist_t2
map extract ((e, a) :: rest) =
map extract toctxlist_t2
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
toctxlist_t2: env E A
premise: mapd f [] = mapd f toctxlist_t2
map extract [] = map extract toctxlist_t2 destruct (toctxlist_t2) as [| [e' a'] rest'].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
toctxlist_t2: env E A
premise: mapd f [] = mapd f []
map extract [] = map extract []
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
premise: mapd f [] = mapd f ((e', a') :: rest')
map extract [] = map extract ((e', a') :: rest')
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
toctxlist_t2: env E A
premise: mapd f [] = mapd f []
map extract [] = map extract [] reflexivity.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
premise: mapd f [] = mapd f ((e', a') :: rest')
map extract [] = map extract ((e', a') :: rest') exfalso.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
premise: mapd f [] = mapd f ((e', a') :: rest')
False inversion premise.
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
premise: mapd f ((e, a) :: rest) =
mapd f toctxlist_t2
map extract ((e, a) :: rest) =
map extract toctxlist_t2 destruct (toctxlist_t2) as [| [e' a'] rest'].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
premise: mapd f ((e, a) :: rest) = mapd f []
map extract ((e, a) :: rest) = map extract []
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
premise: mapd f ((e, a) :: rest) =
mapd f ((e', a') :: rest')
map extract ((e, a) :: rest) =
map extract ((e', a') :: rest')
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
premise: mapd f ((e, a) :: rest) = mapd f []
map extract ((e, a) :: rest) = map extract [] exfalso.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
premise: mapd f ((e, a) :: rest) = mapd f []
False inversion premise.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
premise: mapd f ((e, a) :: rest) =
mapd f ((e', a') :: rest')
map extract ((e, a) :: rest) =
map extract ((e', a') :: rest') cbn in *.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
a :: map extract rest = a' :: map extract rest'
          fequal.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
a = a'
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
map extract rest = map extract rest'
          *
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
a = a' assert (Heq: f (e, a) = f (e', a')) by
              now inversion premise.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
Heq: f (e, a) = f (e', a')
a = a'
            specialize (Hinj (e, a) (e', a') Heq).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
e': E
a': A
Hinj: (e, a) = (e', a')
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
rest': list (E * A)
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
Heq: f (e, a) = f (e', a')
a = a'
            now inversion Hinj.
          *
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
map extract rest = map extract rest' apply IHrest.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
mapd f rest = mapd f rest'
            now inversion premise.
    Qed.

    Lemma mapd_ctxlist_injective_restricted1 {A B}:
      forall (f: E * A -> B) (t1 t2: T A),
        (forall p q: E * A,
            p ∈d (toctxlist t1) ->
            q ∈d (toctxlist t2) ->
            f p = f q -> p = q) ->
        mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
        tolist t1 = tolist t2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
forall (f : E * A -> B) (t1 t2 : T A),
(forall p q : E * A,
 p ∈d toctxlist t1 ->
 q ∈d toctxlist t2 -> f p = f q -> p = q) ->
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
tolist t1 = tolist t2
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
forall (f : E * A -> B) (t1 t2 : T A),
(forall p q : E * A,
 p ∈d toctxlist t1 ->
 q ∈d toctxlist t2 -> f p = f q -> p = q) ->
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
tolist t1 = tolist t2
      introv Hinj.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A,
p ∈d toctxlist t1 ->
q ∈d toctxlist t2 -> f p = f q -> p = q
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
tolist t1 = tolist t2
      rewrite tolist_to_toctxlist.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A,
p ∈d toctxlist t1 ->
q ∈d toctxlist t2 -> f p = f q -> p = q
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
(map extract ∘ toctxlist) t1 =
(map extract ∘ toctxlist) t2
      unfold compose.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A,
p ∈d toctxlist t1 ->
q ∈d toctxlist t2 -> f p = f q -> p = q
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
map extract (toctxlist t1) =
map extract (toctxlist t2)
      generalize dependent (toctxlist t2).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
forall e : env E A,
(forall p q : E * A,
 p ∈d toctxlist t1 -> q ∈d e -> f p = f q -> p = q) ->
mapd f (toctxlist t1) = mapd f e ->
map extract (toctxlist t1) = map extract e
      induction (toctxlist t1) as [| [e a] rest IHrest];
        intros toctxlist_t2 Hinj premise.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
toctxlist_t2: env E A
Hinj: forall p q : E * A,
p ∈d [] ->
q ∈d toctxlist_t2 -> f p = f q -> p = q
premise: mapd f [] = mapd f toctxlist_t2
map extract [] = map extract toctxlist_t2
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
Hinj: forall p q : E * A,
p ∈d ((e, a) :: rest) ->
q ∈d toctxlist_t2 -> f p = f q -> p = q
premise: mapd f ((e, a) :: rest) =
mapd f toctxlist_t2
map extract ((e, a) :: rest) =
map extract toctxlist_t2
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
toctxlist_t2: env E A
Hinj: forall p q : E * A,
p ∈d [] ->
q ∈d toctxlist_t2 -> f p = f q -> p = q
premise: mapd f [] = mapd f toctxlist_t2
map extract [] = map extract toctxlist_t2 destruct (toctxlist_t2) as [| [e' a'] rest'].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
toctxlist_t2: env E A
Hinj: forall p q : E * A,
p ∈d [] -> q ∈d [] -> f p = f q -> p = q
premise: mapd f [] = mapd f []
map extract [] = map extract []
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
p ∈d [] ->
q ∈d ((e', a') :: rest') -> f p = f q -> p = q
premise: mapd f [] = mapd f ((e', a') :: rest')
map extract [] = map extract ((e', a') :: rest')
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
toctxlist_t2: env E A
Hinj: forall p q : E * A,
p ∈d [] -> q ∈d [] -> f p = f q -> p = q
premise: mapd f [] = mapd f []
map extract [] = map extract [] reflexivity.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
p ∈d [] ->
q ∈d ((e', a') :: rest') -> f p = f q -> p = q
premise: mapd f [] = mapd f ((e', a') :: rest')
map extract [] = map extract ((e', a') :: rest') exfalso.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
p ∈d [] ->
q ∈d ((e', a') :: rest') -> f p = f q -> p = q
premise: mapd f [] = mapd f ((e', a') :: rest')
False inversion premise.
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
Hinj: forall p q : E * A,
p ∈d ((e, a) :: rest) ->
q ∈d toctxlist_t2 -> f p = f q -> p = q
premise: mapd f ((e, a) :: rest) =
mapd f toctxlist_t2
map extract ((e, a) :: rest) =
map extract toctxlist_t2 destruct (toctxlist_t2) as [| [e' a'] rest'].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
Hinj: forall p q : E * A,
p ∈d ((e, a) :: rest) ->
q ∈d [] -> f p = f q -> p = q
premise: mapd f ((e, a) :: rest) = mapd f []
map extract ((e, a) :: rest) = map extract []
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
p ∈d ((e, a) :: rest) ->
q ∈d ((e', a') :: rest') -> f p = f q -> p = q
premise: mapd f ((e, a) :: rest) =
mapd f ((e', a') :: rest')
map extract ((e, a) :: rest) =
map extract ((e', a') :: rest')
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
Hinj: forall p q : E * A,
p ∈d ((e, a) :: rest) ->
q ∈d [] -> f p = f q -> p = q
premise: mapd f ((e, a) :: rest) = mapd f []
map extract ((e, a) :: rest) = map extract [] exfalso.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
Hinj: forall p q : E * A,
p ∈d ((e, a) :: rest) ->
q ∈d [] -> f p = f q -> p = q
premise: mapd f ((e, a) :: rest) = mapd f []
False inversion premise.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
p ∈d ((e, a) :: rest) ->
q ∈d ((e', a') :: rest') -> f p = f q -> p = q
premise: mapd f ((e, a) :: rest) =
mapd f ((e', a') :: rest')
map extract ((e, a) :: rest) =
map extract ((e', a') :: rest') cbn in *.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
(e, a) = p \/ List.In p rest ->
(e', a') = q \/ List.In q rest' ->
f p = f q -> p = q
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
a :: map extract rest = a' :: map extract rest'
          fequal.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
(e, a) = p \/ List.In p rest ->
(e', a') = q \/ List.In q rest' ->
f p = f q -> p = q
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
a = a'
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
(e, a) = p \/ List.In p rest ->
(e', a') = q \/ List.In q rest' ->
f p = f q -> p = q
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
map extract rest = map extract rest'
          *
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
(e, a) = p \/ List.In p rest ->
(e', a') = q \/ List.In q rest' ->
f p = f q -> p = q
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
a = a' assert (Heq: f (e, a) = f (e', a')) by
              now inversion premise.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
(e, a) = p \/ List.In p rest ->
(e', a') = q \/ List.In q rest' ->
f p = f q -> p = q
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
Heq: f (e, a) = f (e', a')
a = a'
            enough (X: (e, a) = (e', a'))
              by now inversion X.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
(e, a) = p \/ List.In p rest ->
(e', a') = q \/ List.In q rest' ->
f p = f q -> p = q
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
Heq: f (e, a) = f (e', a')
(e, a) = (e', a')
            apply Hinj; auto.
          *
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
(e, a) = p \/ List.In p rest ->
(e', a') = q \/ List.In q rest' ->
f p = f q -> p = q
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
map extract rest = map extract rest' apply IHrest.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
(e, a) = p \/ List.In p rest ->
(e', a') = q \/ List.In q rest' ->
f p = f q -> p = q
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
forall p q : E * A,
p ∈d rest -> q ∈d rest' -> f p = f q -> p = q
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
(e, a) = p \/ List.In p rest ->
(e', a') = q \/ List.In q rest' ->
f p = f q -> p = q
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
mapd f rest = mapd f rest'
            {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
(e, a) = p \/ List.In p rest ->
(e', a') = q \/ List.In q rest' ->
f p = f q -> p = q
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
forall p q : E * A,
p ∈d rest -> q ∈d rest' -> f p = f q -> p = q intros p q Hinp Hinq Heq.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
(e, a) = p \/ List.In p rest ->
(e', a') = q \/ List.In q rest' ->
f p = f q -> p = q
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
p, q: E * A
Hinp: p ∈d rest
Hinq: q ∈d rest'
Heq: f p = f q
p = q
              apply Hinj; eauto. }
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
(e, a) = p \/ List.In p rest ->
(e', a') = q \/ List.In q rest' ->
f p = f q -> p = q
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
mapd f rest = mapd f rest'
            {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall p q : E * A,
 p ∈d rest -> q ∈d e -> f p = f q -> p = q) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall p q : E * A,
(e, a) = p \/ List.In p rest ->
(e', a') = q \/ List.In q rest' ->
f p = f q -> p = q
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
mapd f rest = mapd f rest' now inversion premise. }
    Qed.

    Lemma mapd_ctxlist_injective_restricted2 {A B}:
      forall (f: E * A -> B) (t1 t2: T A),
        (forall (e: E) (a1: A) (a2: A),
            (e, a1) ∈d (toctxlist t1) ->
            (e, a2) ∈d (toctxlist t2) ->
            f (e, a1) = f (e, a2) -> a1 = a2) ->
        mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
        tolist t1 = tolist t2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
forall (f : E * A -> B) (t1 t2 : T A),
(forall (e : E) (a1 a2 : A),
 (e, a1) ∈d toctxlist t1 ->
 (e, a2) ∈d toctxlist t2 ->
 f (e, a1) = f (e, a2) -> a1 = a2) ->
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
tolist t1 = tolist t2
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
forall (f : E * A -> B) (t1 t2 : T A),
(forall (e : E) (a1 a2 : A),
 (e, a1) ∈d toctxlist t1 ->
 (e, a2) ∈d toctxlist t2 ->
 f (e, a1) = f (e, a2) -> a1 = a2) ->
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
tolist t1 = tolist t2
      introv Hinj.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈d toctxlist t1 ->
(e, a2) ∈d toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
tolist t1 = tolist t2
      rewrite tolist_to_toctxlist.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈d toctxlist t1 ->
(e, a2) ∈d toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
(map extract ∘ toctxlist) t1 =
(map extract ∘ toctxlist) t2
      unfold compose.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈d toctxlist t1 ->
(e, a2) ∈d toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
mapd f (toctxlist t1) = mapd f (toctxlist t2) ->
map extract (toctxlist t1) =
map extract (toctxlist t2)
      generalize dependent (toctxlist t2).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d toctxlist t1 ->
 (e0, a2) ∈d e -> f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f (toctxlist t1) = mapd f e ->
map extract (toctxlist t1) = map extract e
      induction (toctxlist t1) as [| [e a] rest IHrest];
        intros toctxlist_t2 Hinj premise.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
toctxlist_t2: env E A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈d [] ->
(e, a2) ∈d toctxlist_t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
premise: mapd f [] = mapd f toctxlist_t2
map extract [] = map extract toctxlist_t2
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
Hinj: forall (e0 : E) (a1 a2 : A),
(e0, a1) ∈d ((e, a) :: rest) ->
(e0, a2) ∈d toctxlist_t2 ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: mapd f ((e, a) :: rest) =
mapd f toctxlist_t2
map extract ((e, a) :: rest) =
map extract toctxlist_t2
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
toctxlist_t2: env E A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈d [] ->
(e, a2) ∈d toctxlist_t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
premise: mapd f [] = mapd f toctxlist_t2
map extract [] = map extract toctxlist_t2 destruct (toctxlist_t2) as [| [e' a'] rest'].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
toctxlist_t2: env E A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈d [] ->
(e, a2) ∈d [] ->
f (e, a1) = f (e, a2) -> a1 = a2
premise: mapd f [] = mapd f []
map extract [] = map extract []
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈d [] ->
(e, a2) ∈d ((e', a') :: rest') ->
f (e, a1) = f (e, a2) -> a1 = a2
premise: mapd f [] = mapd f ((e', a') :: rest')
map extract [] = map extract ((e', a') :: rest')
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
toctxlist_t2: env E A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈d [] ->
(e, a2) ∈d [] ->
f (e, a1) = f (e, a2) -> a1 = a2
premise: mapd f [] = mapd f []
map extract [] = map extract [] reflexivity.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈d [] ->
(e, a2) ∈d ((e', a') :: rest') ->
f (e, a1) = f (e, a2) -> a1 = a2
premise: mapd f [] = mapd f ((e', a') :: rest')
map extract [] = map extract ((e', a') :: rest') exfalso.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈d [] ->
(e, a2) ∈d ((e', a') :: rest') ->
f (e, a1) = f (e, a2) -> a1 = a2
premise: mapd f [] = mapd f ((e', a') :: rest')
False inversion premise.
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
Hinj: forall (e0 : E) (a1 a2 : A),
(e0, a1) ∈d ((e, a) :: rest) ->
(e0, a2) ∈d toctxlist_t2 ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: mapd f ((e, a) :: rest) =
mapd f toctxlist_t2
map extract ((e, a) :: rest) =
map extract toctxlist_t2 destruct (toctxlist_t2) as [| [e' a'] rest'].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
Hinj: forall (e0 : E) (a1 a2 : A),
(e0, a1) ∈d ((e, a) :: rest) ->
(e0, a2) ∈d [] ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: mapd f ((e, a) :: rest) = mapd f []
map extract ((e, a) :: rest) = map extract []
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e0, a1) ∈d ((e, a) :: rest) ->
(e0, a2) ∈d ((e', a') :: rest') ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: mapd f ((e, a) :: rest) =
mapd f ((e', a') :: rest')
map extract ((e, a) :: rest) =
map extract ((e', a') :: rest')
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
Hinj: forall (e0 : E) (a1 a2 : A),
(e0, a1) ∈d ((e, a) :: rest) ->
(e0, a2) ∈d [] ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: mapd f ((e, a) :: rest) = mapd f []
map extract ((e, a) :: rest) = map extract [] exfalso.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
Hinj: forall (e0 : E) (a1 a2 : A),
(e0, a1) ∈d ((e, a) :: rest) ->
(e0, a2) ∈d [] ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: mapd f ((e, a) :: rest) = mapd f []
False inversion premise.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e0, a1) ∈d ((e, a) :: rest) ->
(e0, a2) ∈d ((e', a') :: rest') ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: mapd f ((e, a) :: rest) =
mapd f ((e', a') :: rest')
map extract ((e, a) :: rest) =
map extract ((e', a') :: rest') cbn in *.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e, a) = (e0, a1) \/ List.In (e0, a1) rest ->
(e', a') = (e0, a2) \/ List.In (e0, a2) rest' ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
a :: map extract rest = a' :: map extract rest'
          fequal.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e, a) = (e0, a1) \/ List.In (e0, a1) rest ->
(e', a') = (e0, a2) \/ List.In (e0, a2) rest' ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
a = a'
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e, a) = (e0, a1) \/ List.In (e0, a1) rest ->
(e', a') = (e0, a2) \/ List.In (e0, a2) rest' ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
map extract rest = map extract rest'
          *
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e, a) = (e0, a1) \/ List.In (e0, a1) rest ->
(e', a') = (e0, a2) \/ List.In (e0, a2) rest' ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
a = a' assert (e = e') by now inversion premise; subst.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e, a) = (e0, a1) \/ List.In (e0, a1) rest ->
(e', a') = (e0, a2) \/ List.In (e0, a2) rest' ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
H0: e = e'
a = a'
            assert (Heq: f (e, a) = f (e, a')) by
              now inversion premise.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e, a) = (e0, a1) \/ List.In (e0, a1) rest ->
(e', a') = (e0, a2) \/ List.In (e0, a2) rest' ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
H0: e = e'
Heq: f (e, a) = f (e, a')
a = a'
            eapply Hinj; auto.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e, a) = (e0, a1) \/ List.In (e0, a1) rest ->
(e', a') = (e0, a2) \/ List.In (e0, a2) rest' ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
H0: e = e'
Heq: f (e, a) = f (e, a')
(e', a') = (e, a') \/ List.In (e, a') rest'
            subst; now left.
          *
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e, a) = (e0, a1) \/ List.In (e0, a1) rest ->
(e', a') = (e0, a2) \/ List.In (e0, a2) rest' ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
map extract rest = map extract rest' apply IHrest.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e, a) = (e0, a1) \/ List.In (e0, a1) rest ->
(e', a') = (e0, a2) \/ List.In (e0, a2) rest' ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
forall (e : E) (a1 a2 : A),
(e, a1) ∈d rest ->
(e, a2) ∈d rest' -> f (e, a1) = f (e, a2) -> a1 = a2
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e, a) = (e0, a1) \/ List.In (e0, a1) rest ->
(e', a') = (e0, a2) \/ List.In (e0, a2) rest' ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
mapd f rest = mapd f rest'
            {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e, a) = (e0, a1) \/ List.In (e0, a1) rest ->
(e', a') = (e0, a2) \/ List.In (e0, a2) rest' ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
forall (e : E) (a1 a2 : A),
(e, a1) ∈d rest ->
(e, a2) ∈d rest' -> f (e, a1) = f (e, a2) -> a1 = a2 intros e'' a1 a2 Hinp Hinq Heq.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e, a) = (e0, a1) \/ List.In (e0, a1) rest ->
(e', a') = (e0, a2) \/ List.In (e0, a2) rest' ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
e'': E
a1, a2: A
Hinp: (e'', a1) ∈d rest
Hinq: (e'', a2) ∈d rest'
Heq: f (e'', a1) = f (e'', a2)
a1 = a2
              eapply Hinj; eauto. }
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e, a) = (e0, a1) \/ List.In (e0, a1) rest ->
(e', a') = (e0, a2) \/ List.In (e0, a2) rest' ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
mapd f rest = mapd f rest'
            {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
e: E
a: A
rest: list (E * A)
IHrest: forall e : env E A,
(forall (e0 : E) (a1 a2 : A),
 (e0, a1) ∈d rest ->
 (e0, a2) ∈d e ->
 f (e0, a1) = f (e0, a2) -> a1 = a2) ->
mapd f rest = mapd f e ->
map extract rest = map extract e
toctxlist_t2: env E A
e': E
a': A
rest': list (E * A)
Hinj: forall (e0 : E) (a1 a2 : A),
(e, a) = (e0, a1) \/ List.In (e0, a1) rest ->
(e', a') = (e0, a2) \/ List.In (e0, a2) rest' ->
f (e0, a1) = f (e0, a2) -> a1 = a2
premise: (e, f (e, a)) :: mapd f rest =
(e', f (e', a')) :: mapd f rest'
mapd f rest = mapd f rest' now inversion premise. }
    Qed.

    Theorem mapd_injective1 {A B}: forall (f: E * A -> B) (t1 t2: T A),
        (forall p q, f p = f q -> p = q) ->
        mapd f t1 = mapd f t2 ->
        t1 = t2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
forall (f : E * A -> B) (t1 t2 : T A),
(forall p q : E * A, f p = f q -> p = q) ->
mapd f t1 = mapd f t2 -> t1 = t2
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
forall (f : E * A -> B) (t1 t2 : T A),
(forall p q : E * A, f p = f q -> p = q) ->
mapd f t1 = mapd f t2 -> t1 = t2
      introv Hinj Heq.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
t1 = t2
      assert (cut: (toctxlist ∘ mapd f) t1 = (toctxlist ∘ mapd f) t2).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
(toctxlist ∘ mapd f) t1 = (toctxlist ∘ mapd f) t2
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
cut: (toctxlist ∘ mapd f) t1 =
(toctxlist ∘ mapd f) t2
t1 = t2
      {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
(toctxlist ∘ mapd f) t1 = (toctxlist ∘ mapd f) t2 unfold compose.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
toctxlist (mapd f t1) = toctxlist (mapd f t2) fequal.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
mapd f t1 = mapd f t2 auto. }
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
cut: (toctxlist ∘ mapd f) t1 =
(toctxlist ∘ mapd f) t2
t1 = t2
      rewrite <- mapd_toctxlist in cut.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
cut: (mapd f ∘ toctxlist) t1 =
(mapd f ∘ toctxlist) t2
t1 = t2
      unfold compose in cut.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
cut: mapd f (toctxlist t1) = mapd f (toctxlist t2)
t1 = t2
      apply Traversable_shapeliness.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
cut: mapd f (toctxlist t1) = mapd f (toctxlist t2)
shape t1 = shape t2 /\ tolist t1 = tolist t2
      split.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
cut: mapd f (toctxlist t1) = mapd f (toctxlist t2)
shape t1 = shape t2
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
cut: mapd f (toctxlist t1) = mapd f (toctxlist t2)
tolist t1 = tolist t2
      - (* Same shape *)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
cut: mapd f (toctxlist t1) = mapd f (toctxlist t2)
shape t1 = shape t2
        enough (cut2: shape (mapd f t1) = shape (mapd f t2)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
cut: mapd f (toctxlist t1) = mapd f (toctxlist t2)
cut2: shape (mapd f t1) = shape (mapd f t2)
shape t1 = shape t2
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
cut: mapd f (toctxlist t1) = mapd f (toctxlist t2)
shape (mapd f t1) = shape (mapd f t2)
        do 2 rewrite mapd_shape in cut2; auto.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
cut: mapd f (toctxlist t1) = mapd f (toctxlist t2)
shape (mapd f t1) = shape (mapd f t2)
        now rewrite Heq.
      - (* Same contents *)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall p q : E * A, f p = f q -> p = q
Heq: mapd f t1 = mapd f t2
cut: mapd f (toctxlist t1) = mapd f (toctxlist t2)
tolist t1 = tolist t2
        eapply mapd_ctxlist_injective; eassumption.
    Qed.

    Theorem mapd_injective2 {A B}:
      forall (f: E * A -> B) (t1 t2: T A),
        (forall e a1 a2,
            (e, a1) ∈d t1 ->
            (e, a2) ∈d t2 ->
            f (e, a1) = f (e, a2) -> a1 = a2) ->
        mapd f t1 = mapd f t2 ->
        t1 = t2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
forall (f : E * A -> B) (t1 t2 : T A),
(forall (e : E) (a1 a2 : A),
 (e, a1) ∈d t1 ->
 (e, a2) ∈d t2 -> f (e, a1) = f (e, a2) -> a1 = a2) ->
mapd f t1 = mapd f t2 -> t1 = t2
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
forall (f : E * A -> B) (t1 t2 : T A),
(forall (e : E) (a1 a2 : A),
 (e, a1) ∈d t1 ->
 (e, a2) ∈d t2 -> f (e, a1) = f (e, a2) -> a1 = a2) ->
mapd f t1 = mapd f t2 -> t1 = t2
      introv Hinj Heq.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈d t1 ->
(e, a2) ∈d t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
t1 = t2
      setoid_rewrite ind_iff_in_toctxlist1 in Hinj.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
t1 = t2
      assert (Heq2: (toctxlist ∘ mapd f) t1 = (toctxlist ∘ mapd f) t2).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
(toctxlist ∘ mapd f) t1 = (toctxlist ∘ mapd f) t2
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: (toctxlist ∘ mapd f) t1 =
(toctxlist ∘ mapd f) t2
t1 = t2
      {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
(toctxlist ∘ mapd f) t1 = (toctxlist ∘ mapd f) t2 unfold compose.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
toctxlist (mapd f t1) = toctxlist (mapd f t2) fequal.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
mapd f t1 = mapd f t2 auto. }
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: (toctxlist ∘ mapd f) t1 =
(toctxlist ∘ mapd f) t2
t1 = t2
      rewrite <- mapd_toctxlist in Heq2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: (mapd f ∘ toctxlist) t1 =
(mapd f ∘ toctxlist) t2
t1 = t2
      unfold compose in Heq2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
t1 = t2
      apply Traversable_shapeliness.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
shape t1 = shape t2 /\ tolist t1 = tolist t2
      split.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
shape t1 = shape t2
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
tolist t1 = tolist t2
      - (* Same shape *)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
shape t1 = shape t2
        enough (cut2: shape (mapd f t1) = shape (mapd f t2)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
cut2: shape (mapd f t1) = shape (mapd f t2)
shape t1 = shape t2
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
shape (mapd f t1) = shape (mapd f t2)
        do 2 rewrite mapd_shape in cut2; auto.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
shape (mapd f t1) = shape (mapd f t2)
        now rewrite Heq.
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
tolist t1 = tolist t2 eapply mapd_ctxlist_injective_restricted2; eauto.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
forall (e : E) (a1 a2 : A),
(e, a1) ∈d toctxlist t1 ->
(e, a2) ∈d toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
        intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
e: E
a1, a2: A
H0: (e, a1) ∈d toctxlist t1
H1: (e, a2) ∈d toctxlist t2
H2: f (e, a1) = f (e, a2)
a1 = a2
        eapply Hinj; eauto.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
e: E
a1, a2: A
H0: (e, a1) ∈d toctxlist t1
H1: (e, a2) ∈d toctxlist t2
H2: f (e, a1) = f (e, a2)
(e, a1) ∈ toctxlist t1
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
e: E
a1, a2: A
H0: (e, a1) ∈d toctxlist t1
H1: (e, a2) ∈d toctxlist t2
H2: f (e, a1) = f (e, a2)
(e, a2) ∈ toctxlist t2
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
e: E
a1, a2: A
H0: (e, a1) ∈d toctxlist t1
H1: (e, a2) ∈d toctxlist t2
H2: f (e, a1) = f (e, a2)
(e, a1) ∈ toctxlist t1 repeat rewrite toctxlist_to_mapdt in *.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ mapdReduce ret t1 ->
(e, a2) ∈ mapdReduce ret t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (mapdReduce ret t1) =
mapd f (mapdReduce ret t2)
e: E
a1, a2: A
H0: (e, a1) ∈d mapdReduce ret t1
H1: (e, a2) ∈d mapdReduce ret t2
H2: f (e, a1) = f (e, a2)
(e, a1) ∈ mapdReduce ret t1
          assumption.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ toctxlist t1 ->
(e, a2) ∈ toctxlist t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (toctxlist t1) = mapd f (toctxlist t2)
e: E
a1, a2: A
H0: (e, a1) ∈d toctxlist t1
H1: (e, a2) ∈d toctxlist t2
H2: f (e, a1) = f (e, a2)
(e, a2) ∈ toctxlist t2 repeat rewrite toctxlist_to_mapdt in *.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> B
t1, t2: T A
Hinj: forall (e : E) (a1 a2 : A),
(e, a1) ∈ mapdReduce ret t1 ->
(e, a2) ∈ mapdReduce ret t2 ->
f (e, a1) = f (e, a2) -> a1 = a2
Heq: mapd f t1 = mapd f t2
Heq2: mapd f (mapdReduce ret t1) =
mapd f (mapdReduce ret t2)
e: E
a1, a2: A
H0: (e, a1) ∈d mapdReduce ret t1
H1: (e, a2) ∈d mapdReduce ret t2
H2: f (e, a1) = f (e, a2)
(e, a2) ∈ mapdReduce ret t2
          assumption.
    Qed.

  End shapeliness.

  (** * Deconstructing with refinement-type vectors *)
  (********************************************************************)
  Section deconstruction.

    Import Kleisli.DecoratedTraversableFunctor.DerivedInstances.

    #[local] Generalizable Variables v.

    (** ** Relating <<plength>> and <<toBatch3>> *)
    (******************************************************************)
    Lemma plength_toBatch3:
      forall {A} {B} (t: T A),
        plength t = length_Batch (toBatch3 (A := A) (B := B) t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (t : T A),
plength t = length_Batch (toBatch3 t)
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (t : T A),
plength t = length_Batch (toBatch3 t)
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
plength t = length_Batch (toBatch3 t)
      unfold plength.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
mapReduce (fun _ : A => 1) t =
length_Batch (toBatch3 t)
      rewrite (mapReduce_through_runBatch2 A B).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
(runBatch (fun _ : A => 1) ∘ toBatch) t =
length_Batch (toBatch3 t)
      rewrite toBatch_to_toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
(runBatch (fun _ : A => 1)
 ∘ (mapfst_Batch extract ∘ toBatch3)) t =
length_Batch (toBatch3 t)
      unfold compose.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
runBatch (fun _ : A => 1)
  (mapfst_Batch extract (toBatch3 t)) =
length_Batch (toBatch3 t)
      induction (toBatch3 t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
c: C
runBatch (fun _ : A => 1)
  (mapfst_Batch extract (Done c)) =
length_Batch (Done c)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: runBatch (fun _ : A => 1)
  (mapfst_Batch extract b) = 
length_Batch b
runBatch (fun _ : A => 1)
  (mapfst_Batch extract (b ⧆ a)) =
length_Batch (b ⧆ a)
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
c: C
runBatch (fun _ : A => 1)
  (mapfst_Batch extract (Done c)) =
length_Batch (Done c) reflexivity.
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: runBatch (fun _ : A => 1)
  (mapfst_Batch extract b) = 
length_Batch b
runBatch (fun _ : A => 1)
  (mapfst_Batch extract (b ⧆ a)) =
length_Batch (b ⧆ a) cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: runBatch (fun _ : A => 1)
  (mapfst_Batch extract b) = 
length_Batch b
runBatch (fun _ : A => 1) (mapfst_Batch extract b) ● 1 =
S (length_Batch b)
        rewrite IHb.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: runBatch (fun _ : A => 1)
  (mapfst_Batch extract b) = 
length_Batch b
length_Batch b ● 1 = S (length_Batch b)
        unfold_ops @NaturalNumbers.Monoid_op_plus.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: runBatch (fun _ : A => 1)
  (mapfst_Batch extract b) = 
length_Batch b
(length_Batch b + 1)%nat = S (length_Batch b)
        lia.
    Qed.

    Lemma length_toBatch3_toBatch:
      forall {A B} (t: T A),
        length_Batch (toBatch3 (A := A) (B := B) t) =
          length_Batch (toBatch (A := A) (A' := B) t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (t : T A),
length_Batch (toBatch3 t) = length_Batch (toBatch t)
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (t : T A),
length_Batch (toBatch3 t) = length_Batch (toBatch t)
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
length_Batch (toBatch3 t) = length_Batch (toBatch t)
      rewrite toBatch_to_toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
length_Batch (toBatch3 t) =
length_Batch ((mapfst_Batch extract ∘ toBatch3) t)
      unfold compose.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
length_Batch (toBatch3 t) =
length_Batch (mapfst_Batch extract (toBatch3 t)) induction (toBatch3 t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
c: C
length_Batch (Done c) =
length_Batch (mapfst_Batch extract (Done c))
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: length_Batch b =
length_Batch (mapfst_Batch extract b)
length_Batch (b ⧆ a) =
length_Batch (mapfst_Batch extract (b ⧆ a))
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
c: C
length_Batch (Done c) =
length_Batch (mapfst_Batch extract (Done c)) reflexivity.
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: length_Batch b =
length_Batch (mapfst_Batch extract b)
length_Batch (b ⧆ a) =
length_Batch (mapfst_Batch extract (b ⧆ a)) cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: length_Batch b =
length_Batch (mapfst_Batch extract b)
S (length_Batch b) =
S (length_Batch (mapfst_Batch extract b)) now rewrite IHb.
    Qed.

    (** ** <<mapdt_contents>> and <<mapdt_make>> *)
    (******************************************************************)
    Definition mapdt_contents {A} (t: T A):
      Vector (plength t) (E * A) :=
      let v: Vector
               (length_Batch (toBatch3 (B := False) (A := A) t))
               (E * A)
        := Batch_contents (toBatch3 t)
      in coerce_Vector_length (eq_sym (plength_toBatch3 t)) v.

    Definition mapdt_make {A B} (t: T A):
      Vector (plength t) B -> T B := trav_make t.

    (** ** Relation <<toctxlist>> and <<mapdt_contents>> *)
    (******************************************************************)
    Lemma Batch3_contents_toctxlist:
      forall {A B} (t: T A),
        Vector_to_list (E * A) (Batch_contents (toBatch3 (B := B) t)) =
          List.rev (toctxlist t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (t : T A),
Vector_to_list (E * A) (Batch_contents (toBatch3 t)) =
List.rev (toctxlist t)
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (t : T A),
Vector_to_list (E * A) (Batch_contents (toBatch3 t)) =
List.rev (toctxlist t)
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
Vector_to_list (E * A) (Batch_contents (toBatch3 t)) =
List.rev (toctxlist t)
      rewrite toctxlist_to_mapdReduce.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
Vector_to_list (E * A) (Batch_contents (toBatch3 t)) =
List.rev (mapdReduce ret t)
      rewrite (mapdReduce_through_runBatch2 A B).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
Vector_to_list (E * A) (Batch_contents (toBatch3 t)) =
List.rev ((runBatch ret ∘ toBatch3) t)
      unfold compose.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
Vector_to_list (E * A) (Batch_contents (toBatch3 t)) =
List.rev (runBatch ret (toBatch3 t))
      induction (toBatch3 t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
c: C
Vector_to_list (E * A) (Batch_contents (Done c)) =
List.rev (runBatch ret (Done c))
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: Vector_to_list (E * A) (Batch_contents b) =
List.rev (runBatch ret b)
Vector_to_list (E * A) (Batch_contents (b ⧆ a)) =
List.rev (runBatch ret (b ⧆ a))
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
c: C
Vector_to_list (E * A) (Batch_contents (Done c)) =
List.rev (runBatch ret (Done c)) cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
c: C
[] = [] reflexivity.
      -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: Vector_to_list (E * A) (Batch_contents b) =
List.rev (runBatch ret b)
Vector_to_list (E * A) (Batch_contents (b ⧆ a)) =
List.rev (runBatch ret (b ⧆ a)) cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: Vector_to_list (E * A) (Batch_contents b) =
List.rev (runBatch ret b)
Vector_to_list (E * A)
  (vcons (length_Batch b) a (Batch_contents b)) =
List.rev (runBatch ret b ● ret a)
        rewrite Vector_to_list_vcons.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: Vector_to_list (E * A) (Batch_contents b) =
List.rev (runBatch ret b)
a :: Vector_to_list (E * A) (Batch_contents b) =
List.rev (runBatch ret b ● ret a)
        rewrite IHb.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: Vector_to_list (E * A) (Batch_contents b) =
List.rev (runBatch ret b)
a :: List.rev (runBatch ret b) =
List.rev (runBatch ret b ● ret a)
        unfold_ops @Monoid_op_list @Return_list.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: Vector_to_list (E * A) (Batch_contents b) =
List.rev (runBatch ret b)
a :: List.rev (runBatch (fun a : E * A => [a]) b) =
List.rev (runBatch (fun a : E * A => [a]) b ++ [a])
        rewrite List.rev_unit.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: Vector_to_list (E * A) (Batch_contents b) =
List.rev (runBatch ret b)
a :: List.rev (runBatch (fun a : E * A => [a]) b) =
a :: List.rev (runBatch (fun a : E * A => [a]) b)
        reflexivity.
    Qed.

    Corollary mapdt_contents_toctxlist:
      forall {A} (t: T A),
        Vector_to_list (E * A) (mapdt_contents t) =
          List.rev (toctxlist t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A : Type) (t : T A),
Vector_to_list (E * A) (mapdt_contents t) =
List.rev (toctxlist t)
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A : Type) (t : T A),
Vector_to_list (E * A) (mapdt_contents t) =
List.rev (toctxlist t)
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
Vector_to_list (E * A) (mapdt_contents t) =
List.rev (toctxlist t)
      unfold mapdt_contents.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
Vector_to_list (E * A)
  (coerce eq_sym (plength_toBatch3 t)
   in Batch_contents (toBatch3 t)) =
List.rev (toctxlist t)
      unfold Vector_to_list.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
proj1_sig
  (coerce eq_sym (plength_toBatch3 t)
   in Batch_contents (toBatch3 t)) =
List.rev (toctxlist t)
      rewrite <- coerce_Vector_contents.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
proj1_sig (Batch_contents (toBatch3 t)) =
List.rev (toctxlist t)
      apply Batch3_contents_toctxlist.
    Qed.

    (** ** <<_sim>> Properties *)
    (******************************************************************)
    Lemma Batch_contents_toBatch3_sim:
      forall {A B B'} (t: T A),
        Batch_contents (toBatch3 (B := B) t) ~~
          Batch_contents (toBatch3 (B := B') t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B B' : Type) (t : T A),
Batch_contents (toBatch3 t) ~~
Batch_contents (toBatch3 t)
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B B' : Type) (t : T A),
Batch_contents (toBatch3 t) ~~
Batch_contents (toBatch3 t)
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B, B': Type
t: T A
Batch_contents (toBatch3 t) ~~
Batch_contents (toBatch3 t)
      unfold Vector_sim.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B, B': Type
t: T A
proj1_sig (Batch_contents (toBatch3 t)) =
proj1_sig (Batch_contents (toBatch3 t))
      change (proj1_sig ?v) with (Vector_to_list _ v).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B, B': Type
t: T A
Vector_to_list (E * A) (Batch_contents (toBatch3 t)) =
Vector_to_list (E * A) (Batch_contents (toBatch3 t))
      rewrite Batch3_contents_toctxlist.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B, B': Type
t: T A
List.rev (toctxlist t) =
Vector_to_list (E * A) (Batch_contents (toBatch3 t))
      rewrite Batch3_contents_toctxlist.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B, B': Type
t: T A
List.rev (toctxlist t) = List.rev (toctxlist t)
      reflexivity.
    Qed.

    Lemma Batch_make_toBatch3 {A B} (t: T A):
      Batch_make (toBatch3 t) =
        (fun v => Batch_make (B := B)
                 (toBatch t) (coerce (length_toBatch3_toBatch t) in v)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
Batch_make (toBatch3 t) =
Batch_make (toBatch t)
○ coerce_Vector_length (length_toBatch3_toBatch t)
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
Batch_make (toBatch3 t) =
Batch_make (toBatch t)
○ coerce_Vector_length (length_toBatch3_toBatch t)
      ext v.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
v: Vector (length_Batch (toBatch3 t)) B
Batch_make (toBatch3 t) v =
Batch_make (toBatch t)
  (coerce length_toBatch3_toBatch t in v)
      generalize dependent (length_toBatch3_toBatch (B := B) t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
v: Vector (length_Batch (toBatch3 t)) B
forall
  e : length_Batch (toBatch3 t) =
      length_Batch (toBatch t),
Batch_make (toBatch3 t) v =
Batch_make (toBatch t) (coerce e in v)
      intro e.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
v: Vector (length_Batch (toBatch3 t)) B
e: length_Batch (toBatch3 t) =
length_Batch (toBatch t)
Batch_make (toBatch3 t) v =
Batch_make (toBatch t) (coerce e in v)
      apply Batch_make_sim3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
v: Vector (length_Batch (toBatch3 t)) B
e: length_Batch (toBatch3 t) =
length_Batch (toBatch t)
shape (toBatch3 t) = shape (toBatch t)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
v: Vector (length_Batch (toBatch3 t)) B
e: length_Batch (toBatch3 t) =
length_Batch (toBatch t)
v ~~ coerce e in v
      2: {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
v: Vector (length_Batch (toBatch3 t)) B
e: length_Batch (toBatch3 t) =
length_Batch (toBatch t)
v ~~ coerce e in v vector_sim. }
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
v: Vector (length_Batch (toBatch3 t)) B
e: length_Batch (toBatch3 t) =
length_Batch (toBatch t)
shape (toBatch3 t) = shape (toBatch t)
      {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
v: Vector (length_Batch (toBatch3 t)) B
e: length_Batch (toBatch3 t) =
length_Batch (toBatch t)
shape (toBatch3 t) = shape (toBatch t) rewrite toBatch_to_toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
v: Vector (length_Batch (toBatch3 t)) B
e: length_Batch (toBatch3 t) =
length_Batch (toBatch t)
shape (toBatch3 t) =
shape ((mapfst_Batch extract ∘ toBatch3) t)
        unfold compose.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
v: Vector (length_Batch (toBatch3 t)) B
e: length_Batch (toBatch3 t) =
length_Batch (toBatch t)
shape (toBatch3 t) =
shape (mapfst_Batch extract (toBatch3 t))
        clear e v.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
shape (toBatch3 t) =
shape (mapfst_Batch extract (toBatch3 t))
        induction (toBatch3 t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
c: C
shape (Done c) = shape (mapfst_Batch extract (Done c))
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: shape b = shape (mapfst_Batch extract b)
shape (b ⧆ a) = shape (mapfst_Batch extract (b ⧆ a))
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
c: C
shape (Done c) = shape (mapfst_Batch extract (Done c)) reflexivity.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: shape b = shape (mapfst_Batch extract b)
shape (b ⧆ a) = shape (mapfst_Batch extract (b ⧆ a)) unfold shape in *.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: map (const tt) b =
map (const tt) (mapfst_Batch extract b)
map (const tt) (b ⧆ a) =
map (const tt) (mapfst_Batch extract (b ⧆ a)) cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: map (const tt) b =
map (const tt) (mapfst_Batch extract b)
mapfst_Batch (const tt) b ⧆ const tt a =
mapfst_Batch (const tt) (mapfst_Batch extract b)
⧆ const tt (extract a)
          fequal.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
C: Type
b: Batch (E * A) B (B -> C)
a: E * A
IHb: map (const tt) b =
map (const tt) (mapfst_Batch extract b)
mapfst_Batch (const tt) b =
mapfst_Batch (const tt) (mapfst_Batch extract b) apply IHb.
      }
    Qed.


    (** * Miscellaneous *)
    (******************************************************************)
    Lemma toBatch_mapdt_make {A A' B} {t: T A} {v: Vector (plength t) B}:
      toBatch (A' := A') (mapdt_make t v) =
        Batch_replace_contents
          (toBatch (A' := A') t)
          (coerce eq_sym (plength_eq_length t) in v).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
t: T A
v: Vector (plength t) B
toBatch (mapdt_make t v) =
Batch_replace_contents (toBatch t)
  (coerce eq_sym (plength_eq_length t) in v)
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, A', B: Type
t: T A
v: Vector (plength t) B
toBatch (mapdt_make t v) =
Batch_replace_contents (toBatch t)
  (coerce eq_sym (plength_eq_length t) in v)
      apply toBatch_trav_make_to_replace_contents.
    Qed.

    Lemma mapdt_same_shape
      `(t1: T A) `(t2: T A'):
      shape t1 = shape t2 ->
      forall B, mapdt_make (B := B) t1 ~!~ mapdt_make t2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t1: T A
A': Type
t2: T A'
shape t1 = shape t2 ->
forall B : Type, mapdt_make t1 ~!~ mapdt_make t2
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t1: T A
A': Type
t2: T A'
shape t1 = shape t2 ->
forall B : Type, mapdt_make t1 ~!~ mapdt_make t2
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t1: T A
A': Type
t2: T A'
H0: shape t1 = shape t2
B: Type
mapdt_make t1 ~!~ mapdt_make t2
      now apply trav_same_shape.
    Qed.

    Lemma plength_mapdt_make: forall `(t: T A) `(v: Vector _ B),
        plength (mapdt_make t v) = plength t.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A : Type) (t : T A) (B : Type)
  (v : Vector (plength t) B),
plength (mapdt_make t v) = plength t
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A : Type) (t : T A) (B : Type)
  (v : Vector (plength t) B),
plength (mapdt_make t v) = plength t
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
B: Type
v: Vector (plength t) B
plength (mapdt_make t v) = plength t
      unfold mapdt_make.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
B: Type
v: Vector (plength t) B
plength (trav_make t v) = plength t
      symmetry.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
B: Type
v: Vector (plength t) B
plength t = plength (trav_make t v) apply plength_trav_make.
    Qed.

    (** * Lens Laws *)
    (******************************************************************)
    Section lens_laws.

      Import KleisliToCoalgebraic.DecoratedTraversableFunctor.DerivedInstances.
      Import KleisliToCoalgebraic.TraversableFunctor.DerivedInstances.

      (** ** get-put *)
      (****************************************************************)
      Lemma mapdt_get_put `{t: T A}:
        mapdt_make t (map extract (mapdt_contents t)) = t.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
mapdt_make t (map extract (mapdt_contents t)) = t
      Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
mapdt_make t (map extract (mapdt_contents t)) = t
        unfold mapdt_make, trav_make, mapdt_contents.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
Batch_make (toBatch t)
  (coerce eq_sym (plength_eq_length t)
   in map extract
        (coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t))) = t
        hide_lhs;
          change t with (id t);
          rewrite Heqlhs; clear Heqlhs lhs.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
Batch_make (toBatch t)
  (coerce eq_sym (plength_eq_length t)
   in map extract
        (coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t))) = id t
        rewrite <- trf_extract.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
Batch_make (toBatch t)
  (coerce eq_sym (plength_eq_length t)
   in map extract
        (coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t))) =
(extract_Batch ∘ toBatch) t
        unfold compose.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
Batch_make (toBatch t)
  (coerce eq_sym (plength_eq_length t)
   in map extract
        (coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t))) =
extract_Batch (toBatch t)
        rewrite <- Batch_make_contents.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
Batch_make (toBatch t)
  (coerce eq_sym (plength_eq_length t)
   in map extract
        (coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t))) =
Batch_make (toBatch t) (Batch_contents (toBatch t))
        apply Batch_make_sim1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
coerce eq_sym (plength_eq_length t)
in map extract
     (coerce eq_sym (plength_toBatch3 t)
      in Batch_contents (toBatch3 t)) ~~
Batch_contents (toBatch t)
        vector_sim.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
map extract (Batch_contents (toBatch3 t)) ~~
Batch_contents (toBatch t)
        unfold Vector_sim.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
proj1_sig (map extract (Batch_contents (toBatch3 t))) =
proj1_sig (Batch_contents (toBatch t))
        rewrite Batch_contents_natural.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
proj1_sig
  (Batch_contents (mapfst_Batch extract (toBatch3 t))) =
proj1_sig (Batch_contents (toBatch t))
        compose near t on left.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
proj1_sig
  (Batch_contents
     ((mapfst_Batch extract ∘ toBatch3) t)) =
proj1_sig (Batch_contents (toBatch t))
        rewrite <- toBatch_to_toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
proj1_sig (Batch_contents (toBatch t)) =
proj1_sig (Batch_contents (toBatch t))
        apply Batch_contents_toBatch_sim.
      Qed.

      (** ** put-get *)
      (****************************************************************)
      (* Ill-typed, needs clarification of statement *)
      (*
        Lemma mapdt_contents_make {A} {t: T A}
        {v: Vector (plength t) A}:
        mapdt_contents (trav_make t v) ~~ v.
        Proof.
        unfold trav_contents.
        vector_sim.
        rewrite toBatch_trav_make.
        rewrite Batch_put_get.
        vector_sim.
        Qed.
       *)

      (** ** put-put *)
      (****************************************************************)
      Lemma mapdt_make_make
        `(t: T A) `(v: Vector (plength t) B)
        `(v1: Vector _ B')
        (v2: Vector _ B')
        (pf: v1 ~~ v2):
        mapdt_make (mapdt_make t v) v1 =
          mapdt_make t v2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
B: Type
v: Vector (plength t) B
B': Type
v1: Vector (plength (mapdt_make t v)) B'
v2: Vector (plength t) B'
pf: v1 ~~ v2
mapdt_make (mapdt_make t v) v1 = mapdt_make t v2
      Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
B: Type
v: Vector (plength t) B
B': Type
v1: Vector (plength (mapdt_make t v)) B'
v2: Vector (plength t) B'
pf: v1 ~~ v2
mapdt_make (mapdt_make t v) v1 = mapdt_make t v2
        unfold mapdt_make at 1 2 3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
B: Type
v: Vector (plength t) B
B': Type
v1: Vector (plength (mapdt_make t v)) B'
v2: Vector (plength t) B'
pf: v1 ~~ v2
trav_make (trav_make t v) v1 = trav_make t v2
        apply trav_make_make.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
t: T A
B: Type
v: Vector (plength t) B
B': Type
v1: Vector (plength (mapdt_make t v)) B'
v2: Vector (plength t) B'
pf: v1 ~~ v2
v1 ~~ v2
        assumption.
      Qed.

    End lens_laws.

    (** * Representation theorems *)
    (******************************************************************)
    Lemma mapdt_repr:
      forall `{Applicative G} (A B: Type) (t: T A) (f: E * A -> G B),
        mapdt f t =
          map (mapdt_make t)
            (forwards (traverse (mkBackwards ∘ f) (mapdt_contents t))).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (A B : Type) (t : T A) (f : E * A -> G B),
mapdt f t =
map (mapdt_make t)
  (forwards
     (traverse (mkBackwards ∘ f) (mapdt_contents t)))
    Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (G : Type -> Type) (Map_G : Map G)
  (Pure_G : Pure G) (Mult_G : Mult G),
Applicative G ->
forall (A B : Type) (t : T A) (f : E * A -> G B),
mapdt f t =
map (mapdt_make t)
  (forwards
     (traverse (mkBackwards ∘ f) (mapdt_contents t)))
      intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
mapdt f t =
map (mapdt_make t)
  (forwards
     (traverse (mkBackwards ∘ f) (mapdt_contents t)))
      unfold mapdt_make.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
mapdt f t =
map (trav_make t)
  (forwards
     (traverse (mkBackwards ∘ f) (mapdt_contents t)))
      unfold mapdt_contents.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
mapdt f t =
map (trav_make t)
  (forwards
     (traverse (mkBackwards ∘ f)
        (coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t))))
      unfold trav_make.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
mapdt f t =
map
  (Batch_make (toBatch t)
   ○ coerce_Vector_length
       (eq_sym (plength_eq_length t)))
  (forwards
     (traverse (mkBackwards ∘ f)
        (coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t))))
      change  (map (?f ○ ?g)) with (map (f ∘ g)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
mapdt f t =
map
  (Batch_make (toBatch t)
   ∘ coerce_Vector_length
       (eq_sym (plength_eq_length t)))
  (forwards
     (traverse (mkBackwards ∘ f)
        (coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t))))
      rewrite <- (fun_map_map (F := G)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
mapdt f t =
(map (Batch_make (toBatch t))
 ∘ map
     (coerce_Vector_length
        (eq_sym (plength_eq_length t))))
  (forwards
     (traverse (mkBackwards ∘ f)
        (coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t))))
      unfold compose at 1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
mapdt f t =
map (Batch_make (toBatch t))
  (map
     (coerce_Vector_length
        (eq_sym (plength_eq_length t)))
     (forwards
        (traverse (mkBackwards ∘ f)
           (coerce eq_sym (plength_toBatch3 t)
            in Batch_contents (toBatch3 t)))))
      do 2 change (map ?f (forwards ?x)) with (forwards (map f x)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
mapdt f t =
forwards
  (map (Batch_make (toBatch t))
     (map
        (coerce_Vector_length
           (eq_sym (plength_eq_length t)))
        (traverse (mkBackwards ∘ f)
           (coerce eq_sym (plength_toBatch3 t)
            in Batch_contents (toBatch3 t)))))
      rewrite traverse_Vector_coerce_natural;[|typeclasses eauto].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
mapdt f t =
forwards
  (map (Batch_make (toBatch t))
     (traverse (mkBackwards ∘ f)
        (coerce eq_sym (plength_eq_length t)
         in coerce eq_sym (plength_toBatch3 t)
            in Batch_contents (toBatch3 t))))

      rewrite mapdt_through_runBatch.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
(runBatch f ∘ toBatch3) t =
forwards
  (map (Batch_make (toBatch t))
     (traverse (mkBackwards ∘ f)
        (coerce eq_sym (plength_eq_length t)
         in coerce eq_sym (plength_toBatch3 t)
            in Batch_contents (toBatch3 t))))
      unfold compose at 1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
runBatch f (toBatch3 t) =
forwards
  (map (Batch_make (toBatch t))
     (traverse (mkBackwards ∘ f)
        (coerce eq_sym (plength_eq_length t)
         in coerce eq_sym (plength_toBatch3 t)
            in Batch_contents (toBatch3 t))))
      rewrite runBatch_repr2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
map (Batch_make (toBatch3 t))
  (forwards
     (traverse (mkBackwards ∘ f)
        (Batch_contents (toBatch3 t)))) =
forwards
  (map (Batch_make (toBatch t))
     (traverse (mkBackwards ∘ f)
        (coerce eq_sym (plength_eq_length t)
         in coerce eq_sym (plength_toBatch3 t)
            in Batch_contents (toBatch3 t))))
      change (map ?f (forwards ?x)) with (forwards (map f x)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
forwards
  (map (Batch_make (toBatch3 t))
     (traverse (mkBackwards ∘ f)
        (Batch_contents (toBatch3 t)))) =
forwards
  (map (Batch_make (toBatch t))
     (traverse (mkBackwards ∘ f)
        (coerce eq_sym (plength_eq_length t)
         in coerce eq_sym (plength_toBatch3 t)
            in Batch_contents (toBatch3 t))))
      fequal.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
map (Batch_make (toBatch3 t))
  (traverse (mkBackwards ∘ f)
     (Batch_contents (toBatch3 t))) =
map (Batch_make (toBatch t))
  (traverse (mkBackwards ∘ f)
     (coerce eq_sym (plength_eq_length t)
      in coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t)))
      rewrite Batch_make_toBatch3.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
map
  (Batch_make (toBatch t)
   ○ coerce_Vector_length (length_toBatch3_toBatch t))
  (traverse (mkBackwards ∘ f)
     (Batch_contents (toBatch3 t))) =
map (Batch_make (toBatch t))
  (traverse (mkBackwards ∘ f)
     (coerce eq_sym (plength_eq_length t)
      in coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t)))
      change  (map (?f ○ ?g) ?t) with (map (f ∘ g) t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
map
  (Batch_make (toBatch t)
   ∘ coerce_Vector_length (length_toBatch3_toBatch t))
  (traverse (mkBackwards ∘ f)
     (Batch_contents (toBatch3 t))) =
map (Batch_make (toBatch t))
  (traverse (mkBackwards ∘ f)
     (coerce eq_sym (plength_eq_length t)
      in coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t)))
      rewrite <- (fun_map_map (F := Backwards G)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
(map (Batch_make (toBatch t))
 ∘ map
     (coerce_Vector_length (length_toBatch3_toBatch t)))
  (traverse (mkBackwards ∘ f)
     (Batch_contents (toBatch3 t))) =
map (Batch_make (toBatch t))
  (traverse (mkBackwards ∘ f)
     (coerce eq_sym (plength_eq_length t)
      in coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t)))
      unfold compose at 1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
map (Batch_make (toBatch t))
  (map
     (coerce_Vector_length (length_toBatch3_toBatch t))
     (traverse (mkBackwards ∘ f)
        (Batch_contents (toBatch3 t)))) =
map (Batch_make (toBatch t))
  (traverse (mkBackwards ∘ f)
     (coerce eq_sym (plength_eq_length t)
      in coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t)))
      rewrite traverse_Vector_coerce_natural;
        [| typeclasses eauto ].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
map (Batch_make (toBatch t))
  (traverse (mkBackwards ∘ f)
     (coerce length_toBatch3_toBatch t
      in Batch_contents (toBatch3 t))) =
map (Batch_make (toBatch t))
  (traverse (mkBackwards ∘ f)
     (coerce eq_sym (plength_eq_length t)
      in coerce eq_sym (plength_toBatch3 t)
         in Batch_contents (toBatch3 t)))
      fequal.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
traverse (mkBackwards ∘ f)
  (coerce length_toBatch3_toBatch t
   in Batch_contents (toBatch3 t)) =
traverse (mkBackwards ∘ f)
  (coerce eq_sym (plength_eq_length t)
   in coerce eq_sym (plength_toBatch3 t)
      in Batch_contents (toBatch3 t))
      fequal.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
coerce length_toBatch3_toBatch t
in Batch_contents (toBatch3 t) =
coerce eq_sym (plength_eq_length t)
in coerce eq_sym (plength_toBatch3 t)
   in Batch_contents (toBatch3 t)
      apply Vector_sim_eq.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
coerce length_toBatch3_toBatch t
in Batch_contents (toBatch3 t) ~~
coerce eq_sym (plength_eq_length t)
in coerce eq_sym (plength_toBatch3 t)
   in Batch_contents (toBatch3 t)
      vector_sim.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
G: Type -> Type
Map_G: Map G
Pure_G: Pure G
Mult_G: Mult G
H0: Applicative G
A, B: Type
t: T A
f: E * A -> G B
Batch_contents (toBatch3 t) ~~
Batch_contents (toBatch3 t)
      apply Batch_contents_toBatch3_sim.
    Qed.

  End deconstruction.

  (** ** Specialized to option *)
  (**********************************************************************)
  Lemma map_None_eq_iff: forall (A B: Type) (f: A -> B) (o: option A),
      map f o = None <-> o = None.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (f : A -> B) (o : option A),
map f o = None <-> o = None
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (f : A -> B) (o : option A),
map f o = None <-> o = None
    intros; destruct o; easy.
  Qed.

  Import Applicative.Notations.

  Lemma ap_None_eq_None1: forall (A B: Type) (o: option A),
      (None: option (A -> B)) <⋆> o = None.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (o : option A),
(None : option (A -> B)) <⋆> o = None
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (o : option A),
(None : option (A -> B)) <⋆> o = None
    now destruct o.
  Qed.

  Lemma ap_None_eq_None1_iff: forall (A B: Type) (f: option (A -> B)) (a: A),
      f <⋆> Some a = None <-> f = None.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (f : option (A -> B)) (a : A),
f <⋆> Some a = None <-> f = None
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (f : option (A -> B)) (a : A),
f <⋆> Some a = None <-> f = None
    intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: option (A -> B)
a: A
f <⋆> Some a = None <-> f = None
    now destruct f.
  Qed.

  Lemma ap_None_eq_None2: forall (A B: Type) (f: option (A -> B)),
      f <⋆> None = None.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (f : option (A -> B)),
f <⋆> None = None
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (f : option (A -> B)),
f <⋆> None = None
    now destruct f.
  Qed.

  Lemma ap_None_eq_None2_iff: forall (A B: Type) (f: A -> B) (o: option A),
      Some f <⋆> o = None <-> o = None.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (f : A -> B) (o : option A),
Some f <⋆> o = None <-> o = None
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (f : A -> B) (o : option A),
Some f <⋆> o = None <-> o = None
    now destruct o.
  Qed.

  Lemma list_rev_cons: forall (A: Type) (a: A) (l: list A),
      List.rev (a :: l) = List.rev l ++ [a].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A : Type) (a : A) (l : list A),
List.rev (a :: l) = List.rev l ++ [a]
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A : Type) (a : A) (l : list A),
List.rev (a :: l) = List.rev l ++ [a]
    reflexivity.
  Qed.

  Lemma in_list_rev_spec: forall (A: Type) (l: list A) (a: A),
      a ∈ l <-> a ∈ List.rev l.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A : Type) (l : list A) (a : A),
a ∈ l <-> a ∈ List.rev l
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A : Type) (l : list A) (a : A),
a ∈ l <-> a ∈ List.rev l
    induction l.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
forall a : A, a ∈ [] <-> a ∈ List.rev []
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
a: A
l: list A
IHl: forall a : A, a ∈ l <-> a ∈ List.rev l
forall a0 : A,
a0 ∈ (a :: l) <-> a0 ∈ List.rev (a :: l)
    -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
forall a : A, a ∈ [] <-> a ∈ List.rev [] reflexivity.
    -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
a: A
l: list A
IHl: forall a : A, a ∈ l <-> a ∈ List.rev l
forall a0 : A,
a0 ∈ (a :: l) <-> a0 ∈ List.rev (a :: l) setoid_rewrite element_of_list_cons.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
a: A
l: list A
IHl: forall a : A, a ∈ l <-> a ∈ List.rev l
forall a0 : A,
a0 = a \/ a0 ∈ l <-> a0 ∈ List.rev (a :: l)
      setoid_rewrite IHl.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
a: A
l: list A
IHl: forall a : A, a ∈ l <-> a ∈ List.rev l
forall a0 : A,
a0 = a \/ a0 ∈ List.rev l <-> a0 ∈ List.rev (a :: l)
      rewrite list_rev_cons.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
a: A
l: list A
IHl: forall a : A, a ∈ l <-> a ∈ List.rev l
forall a0 : A,
a0 = a \/ a0 ∈ List.rev l <-> a0 ∈ (List.rev l ++ [a])
      setoid_rewrite element_of_list_app.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
a: A
l: list A
IHl: forall a : A, a ∈ l <-> a ∈ List.rev l
forall a0 : A,
a0 = a \/ a0 ∈ List.rev l <->
a0 ∈ List.rev l \/ a0 ∈ [a]
      setoid_rewrite element_of_list_one.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A: Type
a: A
l: list A
IHl: forall a : A, a ∈ l <-> a ∈ List.rev l
forall a0 : A,
a0 = a \/ a0 ∈ List.rev l <->
a0 ∈ List.rev l \/ a0 = a
      tauto.
  Qed.

  Lemma list_traverse_None_iff: forall (A B: Type) (f: A -> option B) (l: list A),
      traverse f l = None <-> exists (a: A), a ∈ l /\ f a = None.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (f : A -> option B) (l : list A),
traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (f : A -> option B) (l : list A),
traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
    intros; induction l.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
traverse f [] = None <->
(exists a : A, a ∈ [] /\ f a = None)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
traverse f (a :: l) = None <->
(exists a0 : A, a0 ∈ (a :: l) /\ f a0 = None)
    -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
traverse f [] = None <->
(exists a : A, a ∈ [] /\ f a = None) cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
pure [] = None <-> (exists a : A, False /\ f a = None) split; intro Hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
Hyp: pure [] = None
exists a : A, False /\ f a = None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
Hyp: exists a : A, False /\ f a = None
pure [] = None
      +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
Hyp: pure [] = None
exists a : A, False /\ f a = None false.
      +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
Hyp: exists a : A, False /\ f a = None
pure [] = None destruct Hyp as [a [contra _]].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
contra: False
pure [] = None
        false.
    -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
traverse f (a :: l) = None <->
(exists a0 : A, a0 ∈ (a :: l) /\ f a0 = None) rewrite traverse_list_cons.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
pure cons <⋆> f a <⋆> traverse f l = None <->
(exists a0 : A, a0 ∈ (a :: l) /\ f a0 = None)
      remember (f a) as Fa.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
Fa: option B
HeqFa: Fa = f a
pure cons <⋆> Fa <⋆> traverse f l = None <->
(exists a0 : A, a0 ∈ (a :: l) /\ f a0 = None)
      destruct Fa.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
pure cons <⋆> Some b <⋆> traverse f l = None <->
(exists a0 : A, a0 ∈ (a :: l) /\ f a0 = None)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
HeqFa: None = f a
pure cons <⋆> None <⋆> traverse f l = None <->
(exists a0 : A, a0 ∈ (a :: l) /\ f a0 = None)
      +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
pure cons <⋆> Some b <⋆> traverse f l = None <->
(exists a0 : A, a0 ∈ (a :: l) /\ f a0 = None) cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
Some (cons b) <⋆> traverse f l = None <->
(exists a0 : A,
   (a = a0 \/ List.In a0 l) /\ f a0 = None) split; intro Hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
Hyp: Some (cons b) <⋆> traverse f l = None
exists a0 : A, (a = a0 \/ List.In a0 l) /\ f a0 = None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
Hyp: exists a0 : A,
  (a = a0 \/ List.In a0 l) /\ f a0 = None
Some (cons b) <⋆> traverse f l = None
        {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
Hyp: Some (cons b) <⋆> traverse f l = None
exists a0 : A, (a = a0 \/ List.In a0 l) /\ f a0 = None apply ap_None_eq_None2_iff in Hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
Hyp: traverse f l = None
exists a0 : A, (a = a0 \/ List.In a0 l) /\ f a0 = None
          rewrite IHl in Hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
Hyp: exists a : A, a ∈ l /\ f a = None
exists a0 : A, (a = a0 \/ List.In a0 l) /\ f a0 = None
          destruct Hyp as [a' [Hain Hfa]].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
a': A
Hain: a' ∈ l
Hfa: f a' = None
exists a0 : A, (a = a0 \/ List.In a0 l) /\ f a0 = None
          +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
a': A
Hain: a' ∈ l
Hfa: f a' = None
exists a0 : A, (a = a0 \/ List.In a0 l) /\ f a0 = None exists a'.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
a': A
Hain: a' ∈ l
Hfa: f a' = None
(a = a' \/ List.In a' l) /\ f a' = None split; auto.
        }
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
Hyp: exists a0 : A,
  (a = a0 \/ List.In a0 l) /\ f a0 = None
Some (cons b) <⋆> traverse f l = None
        {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
Hyp: exists a0 : A,
  (a = a0 \/ List.In a0 l) /\ f a0 = None
Some (cons b) <⋆> traverse f l = None rewrite ap_None_eq_None2_iff.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
Hyp: exists a0 : A,
  (a = a0 \/ List.In a0 l) /\ f a0 = None
traverse f l = None
          rewrite IHl.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
Hyp: exists a0 : A,
  (a = a0 \/ List.In a0 l) /\ f a0 = None
exists a : A, a ∈ l /\ f a = None
          destruct Hyp as [a' [Hain Hfa]].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
a': A
Hain: a = a' \/ List.In a' l
Hfa: f a' = None
exists a : A, a ∈ l /\ f a = None
          destruct Hain as [Case1 | Case2].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
a': A
Case1: a = a'
Hfa: f a' = None
exists a : A, a ∈ l /\ f a = None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
a': A
Case2: List.In a' l
Hfa: f a' = None
exists a : A, a ∈ l /\ f a = None
          -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
a': A
Case1: a = a'
Hfa: f a' = None
exists a : A, a ∈ l /\ f a = None false.
          -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
b: B
HeqFa: Some b = f a
a': A
Case2: List.In a' l
Hfa: f a' = None
exists a : A, a ∈ l /\ f a = None exists a'; auto.
        }
      +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
HeqFa: None = f a
pure cons <⋆> None <⋆> traverse f l = None <->
(exists a0 : A, a0 ∈ (a :: l) /\ f a0 = None) split; intro Hyp.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
HeqFa: None = f a
Hyp: pure cons <⋆> None <⋆> traverse f l = None
exists a0 : A, a0 ∈ (a :: l) /\ f a0 = None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
HeqFa: None = f a
Hyp: exists a0 : A, a0 ∈ (a :: l) /\ f a0 = None
pure cons <⋆> None <⋆> traverse f l = None
        {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
HeqFa: None = f a
Hyp: pure cons <⋆> None <⋆> traverse f l = None
exists a0 : A, a0 ∈ (a :: l) /\ f a0 = None exists a.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
HeqFa: None = f a
Hyp: pure cons <⋆> None <⋆> traverse f l = None
a ∈ (a :: l) /\ f a = None split; [now left|].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
HeqFa: None = f a
Hyp: pure cons <⋆> None <⋆> traverse f l = None
f a = None
          now rewrite HeqFa. }
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
HeqFa: None = f a
Hyp: exists a0 : A, a0 ∈ (a :: l) /\ f a0 = None
pure cons <⋆> None <⋆> traverse f l = None
        {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: A -> option B
a: A
l: list A
IHl: traverse f l = None <->
(exists a : A, a ∈ l /\ f a = None)
HeqFa: None = f a
Hyp: exists a0 : A, a0 ∈ (a :: l) /\ f a0 = None
pure cons <⋆> None <⋆> traverse f l = None apply ap_None_eq_None1. }
  Qed.

  Lemma traverse_Vector_None_spec1 {A B} (n: nat) (f: E * A -> option B):
    forall (l: list (E * A)) (e: length l = n),
      (forwards (traverse_Vector_core (mkBackwards ∘ f) l e) = None) -> traverse f l = None.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
n: nat
f: E * A -> option B
forall (l : list (E * A)) (e : length l = n),
forwards (traverse_Vector_core (mkBackwards ∘ f) l e) =
None -> traverse f l = None
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
n: nat
f: E * A -> option B
forall (l : list (E * A)) (e : length l = n),
forwards (traverse_Vector_core (mkBackwards ∘ f) l e) =
None -> traverse f l = None
    introv HeqNone.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
n: nat
f: E * A -> option B
l: list (E * A)
e: length l = n
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l e) = None
traverse f l = None
    destruct e.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
traverse f l = None
    induction l.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) [] eq_refl) = None
traverse f [] = None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
traverse f (a :: l) = None
    -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) [] eq_refl) = None
traverse f [] = None cbn in HeqNone.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
HeqNone: pure
  (exist (fun l : list B => length l = 0) []
     eq_refl) = None
traverse f [] = None false.
    -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
traverse f (a :: l) = None assert (cut: f a = None \/ forwards (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) = None).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
f a = None \/
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
cut: f a = None \/
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
traverse f (a :: l) = None
      {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
f a = None \/
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None cbn in HeqNone.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: map (fun '(f, a) => f a)
  (map swap
     (match
        forwards
          ((fix go
              (vl : list (E * A)) 
              (n : nat) {struct vl} :
                length vl = n ->
                Backwards option
                  (Vector n B) :=
              match
                vl as vl0
                return
                  (length vl0 = n ->
                   Backwards option
                     (Vector n B))
              with
              | [] =>
                  pure
                  ○ exist
                     (fun l : list B =>
                     length l = n) []
              | a :: rest =>
                  fun
                    vlen : 
                     S (length rest) = n =>
                  match
                    n as n0
                    return
                     (S ... = n0 ->
                     Backwards option (...))
                  with
                  | 0 =>
                     zero_not_S
                     ○ 
                     eq_sym (y:=0)
                  | S m =>
                     fun
                     vlen' : S (...) = S m
                     =>
                     pure (vcons m) <⋆>
                     (mkBackwards ∘ f) a <⋆>
                     go rest m
                     (S_uncons vlen')
                  end vlen
              end) l (length l) eq_refl)
      with
      | Some a =>
          fun
            o : option
                  (Vector (length l) B ->
                   Vector (S (length l)) B)
          =>
          match o with
          | Some a0 => Some (a, a0)
          | None => None
          end
      | None =>
          fun
            o : option
                  (Vector (length l) B ->
                   Vector (S (length l)) B)
          =>
          match o with
          | Some _ | _ => None
          end
      end
        (map (fun '(f, a) => f a)
           (map swap
              (match f a with
               | Some a =>
                   fun
                     o : 
                     option
                     (B ->
                     Vector (length l) B ->
                     Vector (S (length l)) B)
                   =>
                   match o with
                   | Some a0 => Some (a, a0)
                   | None => None
                   end
               | None =>
                   fun
                     o : 
                     option
                     (B ->
                     Vector (length l) B ->
                     Vector (S (length l)) B)
                   =>
                   match o with
                   | Some _ | _ => None
                   end
               end (pure (vcons (length l)))))))) =
None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
f a = None \/
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
        rewrite map_None_eq_iff in HeqNone.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: map swap
  (match
     forwards
       ((fix go
           (vl : list (E * A)) 
           (n : nat) {struct vl} :
             length vl = n ->
             Backwards option (Vector n B) :=
           match
             vl as vl0
             return
               (length vl0 = n ->
                Backwards option
                  (Vector n B))
           with
           | [] =>
               pure
               ○ exist
                   (fun l : list B =>
                    length l = n) []
           | a :: rest =>
               fun
                 vlen : S (length rest) = n
               =>
               match
                 n as n0
                 return
                   (S (length rest) = n0 ->
                    Backwards option
                     (Vector n0 B))
               with
               | 0 =>
                   zero_not_S
                   ○ eq_sym (y:=0)
               | S m =>
                   fun
                     vlen' : 
                     S (length rest) = 
                     S m =>
                   pure (vcons m) <⋆>
                   (mkBackwards ∘ f) a <⋆>
                   go rest m
                     (S_uncons vlen')
               end vlen
           end) l (length l) eq_refl)
   with
   | Some a =>
       fun
         o : option
               (Vector (length l) B ->
                Vector (S (length l)) B) =>
       match o with
       | Some a0 => Some (a, a0)
       | None => None
       end
   | None =>
       fun
         o : option
               (Vector (length l) B ->
                Vector (S (length l)) B) =>
       match o with
       | Some _ | _ => None
       end
   end
     (map (fun '(f, a) => f a)
        (map swap
           (match f a with
            | Some a =>
                fun
                  o : 
                   option
                     (B ->
                     Vector (length l) B ->
                     Vector (S (length l)) B)
                =>
                match o with
                | Some a0 => Some (a, a0)
                | None => None
                end
            | None =>
                fun
                  o : 
                   option
                     (B ->
                     Vector (length l) B ->
                     Vector (S (length l)) B)
                =>
                match o with
                | Some _ | _ => None
                end
            end (pure (vcons (length l))))))) =
None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
f a = None \/
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
        rewrite map_None_eq_iff in HeqNone.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: match
  forwards
    ((fix go
        (vl : list (E * A)) 
        (n : nat) {struct vl} :
          length vl = n ->
          Backwards option (Vector n B) :=
        match
          vl as vl0
          return
            (length vl0 = n ->
             Backwards option (Vector n B))
        with
        | [] =>
            pure
            ○ exist
                (fun l : list B =>
                 length l = n) []
        | a :: rest =>
            fun vlen : S (length rest) = n
            =>
            match
              n as n0
              return
                (S (length rest) = n0 ->
                 Backwards option
                   (Vector n0 B))
            with
            | 0 =>
                zero_not_S ○ eq_sym (y:=0)
            | S m =>
                fun
                  vlen' : 
                   S (length rest) = 
                   S m =>
                pure (vcons m) <⋆>
                (mkBackwards ∘ f) a <⋆>
                go rest m (S_uncons vlen')
            end vlen
        end) l (length l) eq_refl)
with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end
  (map (fun '(f, a) => f a)
     (map swap
        (match f a with
         | Some a =>
             fun
               o : option
                     (B ->
                     Vector (length l) B ->
                     Vector (S (length l)) B)
             =>
             match o with
             | Some a0 => Some (a, a0)
             | None => None
             end
         | None =>
             fun
               o : option
                     (B ->
                     Vector (length l) B ->
                     Vector (S (length l)) B)
             =>
             match o with
             | Some _ | _ => None
             end
         end (pure (vcons (length l)))))) =
None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
f a = None \/
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
        destruct (f a).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
HeqNone: match
  forwards
    ((fix go
        (vl : list (E * A)) 
        (n : nat) {struct vl} :
          length vl = n ->
          Backwards option (Vector n B) :=
        match
          vl as vl0
          return
            (length vl0 = n ->
             Backwards option (Vector n B))
        with
        | [] =>
            pure
            ○ exist
                (fun l : list B =>
                 length l = n) []
        | a :: rest =>
            fun vlen : S (length rest) = n
            =>
            match
              n as n0
              return
                (S (length rest) = n0 ->
                 Backwards option
                   (Vector n0 B))
            with
            | 0 =>
                zero_not_S ○ eq_sym (y:=0)
            | S m =>
                fun
                  vlen' : 
                   S (length rest) = 
                   S m =>
                pure (vcons m) <⋆>
                (mkBackwards ∘ f) a <⋆>
                go rest m (S_uncons vlen')
            end vlen
        end) l (length l) eq_refl)
with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end
  (map (fun '(f, a) => f a)
     (map swap
        match pure (vcons (length l)) with
        | Some a => Some (b, a)
        | None => None
        end)) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
Some b = None \/
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: match
  forwards
    ((fix go
        (vl : list (E * A)) 
        (n : nat) {struct vl} :
          length vl = n ->
          Backwards option (Vector n B) :=
        match
          vl as vl0
          return
            (length vl0 = n ->
             Backwards option (Vector n B))
        with
        | [] =>
            pure
            ○ exist
                (fun l : list B =>
                 length l = n) []
        | a :: rest =>
            fun vlen : S (length rest) = n
            =>
            match
              n as n0
              return
                (S (length rest) = n0 ->
                 Backwards option
                   (Vector n0 B))
            with
            | 0 =>
                zero_not_S ○ eq_sym (y:=0)
            | S m =>
                fun
                  vlen' : 
                   S (length rest) = 
                   S m =>
                pure (vcons m) <⋆>
                (mkBackwards ∘ f) a <⋆>
                go rest m (S_uncons vlen')
            end vlen
        end) l (length l) eq_refl)
with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end
  (map (fun '(f, a) => f a)
     (map swap
        match pure (vcons (length l)) with
        | Some _ | _ => None
        end)) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
None = None \/
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
        right.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
HeqNone: match
  forwards
    ((fix go
        (vl : list (E * A)) 
        (n : nat) {struct vl} :
          length vl = n ->
          Backwards option (Vector n B) :=
        match
          vl as vl0
          return
            (length vl0 = n ->
             Backwards option (Vector n B))
        with
        | [] =>
            pure
            ○ exist
                (fun l : list B =>
                 length l = n) []
        | a :: rest =>
            fun vlen : S (length rest) = n
            =>
            match
              n as n0
              return
                (S (length rest) = n0 ->
                 Backwards option
                   (Vector n0 B))
            with
            | 0 =>
                zero_not_S ○ eq_sym (y:=0)
            | S m =>
                fun
                  vlen' : 
                   S (length rest) = 
                   S m =>
                pure (vcons m) <⋆>
                (mkBackwards ∘ f) a <⋆>
                go rest m (S_uncons vlen')
            end vlen
        end) l (length l) eq_refl)
with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end
  (map (fun '(f, a) => f a)
     (map swap
        match pure (vcons (length l)) with
        | Some a => Some (b, a)
        | None => None
        end)) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: match
  forwards
    ((fix go
        (vl : list (E * A)) 
        (n : nat) {struct vl} :
          length vl = n ->
          Backwards option (Vector n B) :=
        match
          vl as vl0
          return
            (length vl0 = n ->
             Backwards option (Vector n B))
        with
        | [] =>
            pure
            ○ exist
                (fun l : list B =>
                 length l = n) []
        | a :: rest =>
            fun vlen : S (length rest) = n
            =>
            match
              n as n0
              return
                (S (length rest) = n0 ->
                 Backwards option
                   (Vector n0 B))
            with
            | 0 =>
                zero_not_S ○ eq_sym (y:=0)
            | S m =>
                fun
                  vlen' : 
                   S (length rest) = 
                   S m =>
                pure (vcons m) <⋆>
                (mkBackwards ∘ f) a <⋆>
                go rest m (S_uncons vlen')
            end vlen
        end) l (length l) eq_refl)
with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end
  (map (fun '(f, a) => f a)
     (map swap
        match pure (vcons (length l)) with
        | Some _ | _ => None
        end)) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
None = None \/
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
        cbn in HeqNone.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
HeqNone: match
  forwards
    ((fix go
        (vl : list (E * A)) 
        (n : nat) {struct vl} :
          length vl = n ->
          Backwards option (Vector n B) :=
        match
          vl as vl0
          return
            (length vl0 = n ->
             Backwards option (Vector n B))
        with
        | [] =>
            pure
            ○ exist
                (fun l : list B =>
                 length l = n) []
        | a :: rest =>
            fun vlen : S (length rest) = n
            =>
            match
              n as n0
              return
                (S (length rest) = n0 ->
                 Backwards option
                   (Vector n0 B))
            with
            | 0 =>
                zero_not_S ○ eq_sym (y:=0)
            | S m =>
                fun
                  vlen' : 
                   S (length rest) = 
                   S m =>
                pure (vcons m) <⋆>
                (mkBackwards ∘ f) a <⋆>
                go rest m (S_uncons vlen')
            end vlen
        end) l (length l) eq_refl)
with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end (Some (vcons (length l) b)) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: match
  forwards
    ((fix go
        (vl : list (E * A)) 
        (n : nat) {struct vl} :
          length vl = n ->
          Backwards option (Vector n B) :=
        match
          vl as vl0
          return
            (length vl0 = n ->
             Backwards option (Vector n B))
        with
        | [] =>
            pure
            ○ exist
                (fun l : list B =>
                 length l = n) []
        | a :: rest =>
            fun vlen : S (length rest) = n
            =>
            match
              n as n0
              return
                (S (length rest) = n0 ->
                 Backwards option
                   (Vector n0 B))
            with
            | 0 =>
                zero_not_S ○ eq_sym (y:=0)
            | S m =>
                fun
                  vlen' : 
                   S (length rest) = 
                   S m =>
                pure (vcons m) <⋆>
                (mkBackwards ∘ f) a <⋆>
                go rest m (S_uncons vlen')
            end vlen
        end) l (length l) eq_refl)
with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end
  (map (fun '(f, a) => f a)
     (map swap
        match pure (vcons (length l)) with
        | Some _ | _ => None
        end)) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
None = None \/
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
        remember (@forwards option (Vector (@length (prod E A) l) B)
                    ((fix go (vl : list (prod E A)) (n : nat) {struct vl} :
                       forall _ : @eq nat (@length (prod E A) vl) n, Backwards option (Vector n B) :=
                        match
                          vl as vl0 return (forall _ : @eq nat (@length (prod E A) vl0) n, Backwards option (Vector n B))
                        with
                        | nil =>
                            fun vlen : @eq nat O n =>
                              @pure (Backwards option) (@Pure_Backwards option Pure_option)
                                (@sig (list B) (fun l : list B => @eq nat (@length B l) n))
                                (@exist (list B) (fun l : list B => @eq nat (@length B l) n) (@nil B) vlen)
                        | cons a rest =>
                            fun vlen : @eq nat (S (@length (prod E A) rest)) n =>
                              match
                                n as n0
                                return (forall _ : @eq nat (S (@length (prod E A) rest)) n0, Backwards option (Vector n0 B))
                              with
                              | O =>
                                  fun vlen' : @eq nat (S (@length (prod E A) rest)) O =>
                                    @zero_not_S
                                      ((fix length (l : list (prod E A)) : nat :=
                                          match l return nat with
                                          | nil => O
                                          | cons _ l' => S (length l')
                                          end) rest) (Backwards option (Vector O B))
                                      (@eq_sym nat (S (@length (prod E A) rest)) O vlen')
                              | S m =>
                                  fun vlen' : @eq nat (S (@length (prod E A) rest)) (S m) =>
                                    @ap (Backwards option) (@Map_Backwards option Map_option)
                                      (@Mult_Backwards option Map_option Mult_option) (Vector m B) (Vector (S m) B)
                                      (@ap (Backwards option) (@Map_Backwards option Map_option)
                                         (@Mult_Backwards option Map_option Mult_option) B
                                         (forall _ : Vector m B, Vector (S m) B)
                                         (@pure (Backwards option) (@Pure_Backwards option Pure_option)
                                            (forall (_ : B) (_ : Vector m B), Vector (S m) B) (@vcons m B))
                                         (@compose (prod E A) (option B) (Backwards option B) (@mkBackwards option B) f a))
                                      (go rest m
                                         (@S_uncons
                                            ((fix length (l : list (prod E A)) : nat :=
                                                match l return nat with
                                                | nil => O
                                                | cons _ l' => S (length l')
                                                end) rest) m vlen'))
                              end vlen
                        end) l (@length (prod E A) l) (@eq_refl nat (@length (prod E A) l)))) as X.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
X: option (Vector (length l) B)
HeqX: X =
forwards
  ((fix go
      (vl : list (E * A)) (n : nat) {struct vl} :
        length vl = n ->
        Backwards option (Vector n B) :=
      match
        vl as vl0
        return
          (length vl0 = n ->
           Backwards option (Vector n B))
      with
      | [] =>
          pure
          ○ exist
              (fun l : list B => length l = n)
              []
      | a :: rest =>
          fun vlen : S (length rest) = n =>
          match
            n as n0
            return
              (S (length rest) = n0 ->
               Backwards option (Vector n0 B))
          with
          | 0 => zero_not_S ○ eq_sym (y:=0)
          | S m =>
              fun vlen' : S (length rest) = S m
              =>
              pure (vcons m) <⋆>
              (mkBackwards ∘ f) a <⋆>
              go rest m (S_uncons vlen')
          end vlen
      end) l (length l) eq_refl)
HeqNone: match X with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end (Some (vcons (length l) b)) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: match
  forwards
    ((fix go
        (vl : list (E * A)) 
        (n : nat) {struct vl} :
          length vl = n ->
          Backwards option (Vector n B) :=
        match
          vl as vl0
          return
            (length vl0 = n ->
             Backwards option (Vector n B))
        with
        | [] =>
            pure
            ○ exist
                (fun l : list B =>
                 length l = n) []
        | a :: rest =>
            fun vlen : S (length rest) = n
            =>
            match
              n as n0
              return
                (S (length rest) = n0 ->
                 Backwards option
                   (Vector n0 B))
            with
            | 0 =>
                zero_not_S ○ eq_sym (y:=0)
            | S m =>
                fun
                  vlen' : 
                   S (length rest) = 
                   S m =>
                pure (vcons m) <⋆>
                (mkBackwards ∘ f) a <⋆>
                go rest m (S_uncons vlen')
            end vlen
        end) l (length l) eq_refl)
with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end
  (map (fun '(f, a) => f a)
     (map swap
        match pure (vcons (length l)) with
        | Some _ | _ => None
        end)) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
None = None \/
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
        destruct X.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
v: Vector (length l) B
HeqX: Some v =
forwards
  ((fix go
      (vl : list (E * A)) (n : nat) {struct vl} :
        length vl = n ->
        Backwards option (Vector n B) :=
      match
        vl as vl0
        return
          (length vl0 = n ->
           Backwards option (Vector n B))
      with
      | [] =>
          pure
          ○ exist
              (fun l : list B => length l = n)
              []
      | a :: rest =>
          fun vlen : S (length rest) = n =>
          match
            n as n0
            return
              (S (length rest) = n0 ->
               Backwards option (Vector n0 B))
          with
          | 0 => zero_not_S ○ eq_sym (y:=0)
          | S m =>
              fun vlen' : S (length rest) = S m
              =>
              pure (vcons m) <⋆>
              (mkBackwards ∘ f) a <⋆>
              go rest m (S_uncons vlen')
          end vlen
      end) l (length l) eq_refl)
HeqNone: Some (v, vcons (length l) b) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
HeqX: None =
forwards
  ((fix go
      (vl : list (E * A)) (n : nat) {struct vl} :
        length vl = n ->
        Backwards option (Vector n B) :=
      match
        vl as vl0
        return
          (length vl0 = n ->
           Backwards option (Vector n B))
      with
      | [] =>
          pure
          ○ exist
              (fun l : list B => length l = n)
              []
      | a :: rest =>
          fun vlen : S (length rest) = n =>
          match
            n as n0
            return
              (S (length rest) = n0 ->
               Backwards option (Vector n0 B))
          with
          | 0 => zero_not_S ○ eq_sym (y:=0)
          | S m =>
              fun vlen' : S (length rest) = S m
              =>
              pure (vcons m) <⋆>
              (mkBackwards ∘ f) a <⋆>
              go rest m (S_uncons vlen')
          end vlen
      end) l (length l) eq_refl)
HeqNone: None = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: match
  forwards
    ((fix go
        (vl : list (E * A)) 
        (n : nat) {struct vl} :
          length vl = n ->
          Backwards option (Vector n B) :=
        match
          vl as vl0
          return
            (length vl0 = n ->
             Backwards option (Vector n B))
        with
        | [] =>
            pure
            ○ exist
                (fun l : list B =>
                 length l = n) []
        | a :: rest =>
            fun vlen : S (length rest) = n
            =>
            match
              n as n0
              return
                (S (length rest) = n0 ->
                 Backwards option
                   (Vector n0 B))
            with
            | 0 =>
                zero_not_S ○ eq_sym (y:=0)
            | S m =>
                fun
                  vlen' : 
                   S (length rest) = 
                   S m =>
                pure (vcons m) <⋆>
                (mkBackwards ∘ f) a <⋆>
                go rest m (S_uncons vlen')
            end vlen
        end) l (length l) eq_refl)
with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end
  (map (fun '(f, a) => f a)
     (map swap
        match pure (vcons (length l)) with
        | Some _ | _ => None
        end)) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
None = None \/
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
v: Vector (length l) B
HeqX: Some v =
forwards
  ((fix go
      (vl : list (E * A)) (n : nat) {struct vl} :
        length vl = n ->
        Backwards option (Vector n B) :=
      match
        vl as vl0
        return
          (length vl0 = n ->
           Backwards option (Vector n B))
      with
      | [] =>
          pure
          ○ exist
              (fun l : list B => length l = n)
              []
      | a :: rest =>
          fun vlen : S (length rest) = n =>
          match
            n as n0
            return
              (S (length rest) = n0 ->
               Backwards option (Vector n0 B))
          with
          | 0 => zero_not_S ○ eq_sym (y:=0)
          | S m =>
              fun vlen' : S (length rest) = S m
              =>
              pure (vcons m) <⋆>
              (mkBackwards ∘ f) a <⋆>
              go rest m (S_uncons vlen')
          end vlen
      end) l (length l) eq_refl)
HeqNone: Some (v, vcons (length l) b) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None false.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
HeqX: None =
forwards
  ((fix go
      (vl : list (E * A)) (n : nat) {struct vl} :
        length vl = n ->
        Backwards option (Vector n B) :=
      match
        vl as vl0
        return
          (length vl0 = n ->
           Backwards option (Vector n B))
      with
      | [] =>
          pure
          ○ exist
              (fun l : list B => length l = n)
              []
      | a :: rest =>
          fun vlen : S (length rest) = n =>
          match
            n as n0
            return
              (S (length rest) = n0 ->
               Backwards option (Vector n0 B))
          with
          | 0 => zero_not_S ○ eq_sym (y:=0)
          | S m =>
              fun vlen' : S (length rest) = S m
              =>
              pure (vcons m) <⋆>
              (mkBackwards ∘ f) a <⋆>
              go rest m (S_uncons vlen')
          end vlen
      end) l (length l) eq_refl)
HeqNone: None = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None symmetry in HeqX.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
HeqX: forwards
  ((fix go
      (vl : list (E * A)) (n : nat) {struct vl} :
        length vl = n ->
        Backwards option (Vector n B) :=
      match
        vl as vl0
        return
          (length vl0 = n ->
           Backwards option (Vector n B))
      with
      | [] =>
          pure
          ○ exist
              (fun l : list B => length l = n)
              []
      | a :: rest =>
          fun vlen : S (length rest) = n =>
          match
            n as n0
            return
              (S (length rest) = n0 ->
               Backwards option (Vector n0 B))
          with
          | 0 => zero_not_S ○ eq_sym (y:=0)
          | S m =>
              fun vlen' : S (length rest) = S m
              =>
              pure (vcons m) <⋆>
              (mkBackwards ∘ f) a <⋆>
              go rest m (S_uncons vlen')
          end vlen
      end) l (length l) eq_refl) = None
HeqNone: None = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
          exact HeqX.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: match
  forwards
    ((fix go
        (vl : list (E * A)) 
        (n : nat) {struct vl} :
          length vl = n ->
          Backwards option (Vector n B) :=
        match
          vl as vl0
          return
            (length vl0 = n ->
             Backwards option (Vector n B))
        with
        | [] =>
            pure
            ○ exist
                (fun l : list B =>
                 length l = n) []
        | a :: rest =>
            fun vlen : S (length rest) = n
            =>
            match
              n as n0
              return
                (S (length rest) = n0 ->
                 Backwards option
                   (Vector n0 B))
            with
            | 0 =>
                zero_not_S ○ eq_sym (y:=0)
            | S m =>
                fun
                  vlen' : 
                   S (length rest) = 
                   S m =>
                pure (vcons m) <⋆>
                (mkBackwards ∘ f) a <⋆>
                go rest m (S_uncons vlen')
            end vlen
        end) l (length l) eq_refl)
with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end
  (map (fun '(f, a) => f a)
     (map swap
        match pure (vcons (length l)) with
        | Some _ | _ => None
        end)) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
None = None \/
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None now left.
      }
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
cut: f a = None \/
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
traverse f (a :: l) = None
      rewrite traverse_list_cons.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
cut: f a = None \/
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
pure cons <⋆> f a <⋆> traverse f l = None
      destruct cut as [Case1 | Case2].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
Case1: f a = None
pure cons <⋆> f a <⋆> traverse f l = None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
Case2: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
pure cons <⋆> f a <⋆> traverse f l = None
      +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
Case1: f a = None
pure cons <⋆> f a <⋆> traverse f l = None rewrite Case1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
Case1: f a = None
pure cons <⋆> None <⋆> traverse f l = None
        cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
Case1: f a = None
None <⋆> traverse f l = None
        now rewrite ap_None_eq_None1.
      +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
Case2: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
pure cons <⋆> f a <⋆> traverse f l = None rewrite IHl; auto.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
Case2: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
pure cons <⋆> f a <⋆> None = None
        rewrite ap_None_eq_None2.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None ->
traverse f l = None
Case2: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
None = None
        reflexivity.
  Qed.

  Lemma traverse_Vector_None_spec2 {A B} (n: nat) (f: E * A -> option B):
    forall (l: list (E * A)) (e: length l = n),
      traverse f l = None -> (forwards (traverse_Vector_core (mkBackwards ∘ f) l e) = None).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
n: nat
f: E * A -> option B
forall (l : list (E * A)) (e : length l = n),
traverse f l = None ->
forwards (traverse_Vector_core (mkBackwards ∘ f) l e) =
None
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
n: nat
f: E * A -> option B
forall (l : list (E * A)) (e : length l = n),
traverse f l = None ->
forwards (traverse_Vector_core (mkBackwards ∘ f) l e) =
None
    introv HeqNone.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
n: nat
f: E * A -> option B
l: list (E * A)
e: length l = n
HeqNone: traverse f l = None
forwards (traverse_Vector_core (mkBackwards ∘ f) l e) =
None
    destruct e.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
l: list (E * A)
HeqNone: traverse f l = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
None
    induction l.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
HeqNone: traverse f [] = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) [] eq_refl) =
None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
    -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
HeqNone: traverse f [] = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) [] eq_refl) =
None cbn in HeqNone.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
HeqNone: pure [] = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) [] eq_refl) =
None false.
    -
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) (a :: l)
     eq_refl) = None assert (cut: f a = None \/
                     (exists b, f a = Some b /\ forwards (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) = None)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
f a = None \/
(exists b : B,
   f a = Some b /\
   forwards
     (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
   None)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
cut: f a = None \/
(exists b : B,
   f a = Some b /\
   forwards
     (traverse_Vector_core 
        (mkBackwards ∘ f) l eq_refl) = None)
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
      {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
f a = None \/
(exists b : B,
   f a = Some b /\
   forwards
     (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
   None) cbn in HeqNone.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: pure cons <⋆> f a <⋆> traverse f l = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
f a = None \/
(exists b : B,
   f a = Some b /\
   forwards
     (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
   None)
        destruct (f a).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
HeqNone: pure cons <⋆> Some b <⋆> traverse f l = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Some b = None \/
(exists b0 : B,
   Some b = Some b0 /\
   forwards
     (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
   None)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: pure cons <⋆> None <⋆> traverse f l = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
None = None \/
(exists b : B,
   None = Some b /\
   forwards
     (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
   None)
        right.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
HeqNone: pure cons <⋆> Some b <⋆> traverse f l = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
exists b0 : B,
  Some b = Some b0 /\
  forwards
    (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
  None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: pure cons <⋆> None <⋆> traverse f l = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
None = None \/
(exists b : B,
   None = Some b /\
   forwards
     (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
   None)
        cbn in HeqNone.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
HeqNone: Some (cons b) <⋆> traverse f l = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
exists b0 : B,
  Some b = Some b0 /\
  forwards
    (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
  None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: pure cons <⋆> None <⋆> traverse f l = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
None = None \/
(exists b : B,
   None = Some b /\
   forwards
     (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
   None)
        destruct (traverse f l).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
l0: list B
HeqNone: Some (cons b) <⋆> Some l0 = None
IHl: Some l0 = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
exists b0 : B,
  Some b = Some b0 /\
  forwards
    (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
  None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
HeqNone: Some (cons b) <⋆> None = None
IHl: None = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
exists b0 : B,
  Some b = Some b0 /\
  forwards
    (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
  None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: pure cons <⋆> None <⋆> traverse f l = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
None = None \/
(exists b : B,
   None = Some b /\
   forwards
     (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
   None)
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
l0: list B
HeqNone: Some (cons b) <⋆> Some l0 = None
IHl: Some l0 = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
exists b0 : B,
  Some b = Some b0 /\
  forwards
    (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
  None cbn in HeqNone.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
l0: list B
HeqNone: Some (b :: l0) = None
IHl: Some l0 = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
exists b0 : B,
  Some b = Some b0 /\
  forwards
    (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
  None false.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
HeqNone: Some (cons b) <⋆> None = None
IHl: None = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
exists b0 : B,
  Some b = Some b0 /\
  forwards
    (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
  None rewrite ap_None_eq_None2 in HeqNone.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
HeqNone: None = None
IHl: None = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
exists b0 : B,
  Some b = Some b0 /\
  forwards
    (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
  None
          specialize (IHl HeqNone).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
b: B
HeqNone: None = None
IHl: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
exists b0 : B,
  Some b = Some b0 /\
  forwards
    (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
  None
          exists b; split; auto.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: pure cons <⋆> None <⋆> traverse f l = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
None = None \/
(exists b : B,
   None = Some b /\
   forwards
     (traverse_Vector_core (mkBackwards ∘ f) l eq_refl) =
   None) now left.
      }
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
cut: f a = None \/
(exists b : B,
   f a = Some b /\
   forwards
     (traverse_Vector_core 
        (mkBackwards ∘ f) l eq_refl) = None)
forwards
  (traverse_Vector_core (mkBackwards ∘ f) (a :: l)
     eq_refl) = None
      destruct cut as [Case1 | Case2].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case1: f a = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) (a :: l)
     eq_refl) = None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case2: exists b : B,
  f a = Some b /\
  forwards
    (traverse_Vector_core 
       (mkBackwards ∘ f) l eq_refl) = None
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) 
     (a :: l) eq_refl) = None
      { (* Case 1 *)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case1: f a = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) (a :: l)
     eq_refl) = None
        cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case1: f a = None
map (fun '(f, a) => f a)
  (map swap
     (match
        forwards
          ((fix go
              (vl : list (E * A)) (n : nat) {struct
               vl} :
                length vl = n ->
                Backwards option (Vector n B) :=
              match
                vl as vl0
                return
                  (length vl0 = n ->
                   Backwards option (Vector n B))
              with
              | [] =>
                  pure
                  ○ exist
                      (fun l : list B => length l = n)
                      []
              | a :: rest =>
                  fun vlen : S (length rest) = n =>
                  match
                    n as n0
                    return
                      (S ... = n0 ->
                       Backwards option (...))
                  with
                  | 0 => zero_not_S ○ eq_sym (y:=0)
                  | S m =>
                      fun vlen' : S (...) = S m =>
                      pure (vcons m) <⋆>
                      (mkBackwards ∘ f) a <⋆>
                      go rest m (S_uncons vlen')
                  end vlen
              end) l (length l) eq_refl)
      with
      | Some a =>
          fun
            o : option
                  (Vector (length l) B ->
                   Vector (S (length l)) B) =>
          match o with
          | Some a0 => Some (a, a0)
          | None => None
          end
      | None =>
          fun
            o : option
                  (Vector (length l) B ->
                   Vector (S (length l)) B) =>
          match o with
          | Some _ | _ => None
          end
      end
        (map (fun '(f, a) => f a)
           (map swap
              (match f a with
               | Some a =>
                   fun
                     o : option
                           (B ->
                            Vector (length l) B ->
                            Vector (S (length l)) B)
                   =>
                   match o with
                   | Some a0 => Some (a, a0)
                   | None => None
                   end
               | None =>
                   fun
                     o : option
                           (B ->
                            Vector (length l) B ->
                            Vector (S (length l)) B)
                   =>
                   match o with
                   | Some _ | _ => None
                   end
               end (pure (vcons (length l)))))))) =
None
        rewrite Case1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case1: f a = None
map (fun '(f, a) => f a)
  (map swap
     (match
        forwards
          ((fix go
              (vl : list (E * A)) (n : nat) {struct
               vl} :
                length vl = n ->
                Backwards option (Vector n B) :=
              match
                vl as vl0
                return
                  (length vl0 = n ->
                   Backwards option (Vector n B))
              with
              | [] =>
                  pure
                  ○ exist
                      (fun l : list B => length l = n)
                      []
              | a :: rest =>
                  fun vlen : S (length rest) = n =>
                  match
                    n as n0
                    return
                      (S ... = n0 ->
                       Backwards option (...))
                  with
                  | 0 => zero_not_S ○ eq_sym (y:=0)
                  | S m =>
                      fun vlen' : S (...) = S m =>
                      pure (vcons m) <⋆>
                      (mkBackwards ∘ f) a <⋆>
                      go rest m (S_uncons vlen')
                  end vlen
              end) l (length l) eq_refl)
      with
      | Some a =>
          fun
            o : option
                  (Vector (length l) B ->
                   Vector (S (length l)) B) =>
          match o with
          | Some a0 => Some (a, a0)
          | None => None
          end
      | None =>
          fun
            o : option
                  (Vector (length l) B ->
                   Vector (S (length l)) B) =>
          match o with
          | Some _ | _ => None
          end
      end
        (map (fun '(f, a) => f a)
           (map swap
              match pure (vcons (length l)) with
              | Some _ | _ => None
              end)))) = None
        cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case1: f a = None
map (fun '(f, a) => f a)
  (map swap
     (match
        forwards
          ((fix go
              (vl : list (E * A)) (n : nat) {struct
               vl} :
                length vl = n ->
                Backwards option (Vector n B) :=
              match
                vl as vl0
                return
                  (length vl0 = n ->
                   Backwards option (Vector n B))
              with
              | [] =>
                  pure
                  ○ exist
                      (fun l : list B => length l = n)
                      []
              | a :: rest =>
                  fun vlen : S (length rest) = n =>
                  match
                    n as n0
                    return
                      (S ... = n0 ->
                       Backwards option (...))
                  with
                  | 0 => zero_not_S ○ eq_sym (y:=0)
                  | S m =>
                      fun vlen' : S (...) = S m =>
                      pure (vcons m) <⋆>
                      (mkBackwards ∘ f) a <⋆>
                      go rest m (S_uncons vlen')
                  end vlen
              end) l (length l) eq_refl)
      with
      | Some a =>
          fun
            o : option
                  (Vector (length l) B ->
                   Vector (S (length l)) B) =>
          match o with
          | Some a0 => Some (a, a0)
          | None => None
          end
      | None =>
          fun
            o : option
                  (Vector (length l) B ->
                   Vector (S (length l)) B) =>
          match o with
          | Some _ | _ => None
          end
      end None)) = None
        rewrite map_None_eq_iff.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case1: f a = None
map swap
  (match
     forwards
       ((fix go
           (vl : list (E * A)) (n : nat) {struct vl} :
             length vl = n ->
             Backwards option (Vector n B) :=
           match
             vl as vl0
             return
               (length vl0 = n ->
                Backwards option (Vector n B))
           with
           | [] =>
               pure
               ○ exist
                   (fun l : list B => length l = n) []
           | a :: rest =>
               fun vlen : S (length rest) = n =>
               match
                 n as n0
                 return
                   (S (length rest) = n0 ->
                    Backwards option (Vector n0 B))
               with
               | 0 => zero_not_S ○ eq_sym (y:=0)
               | S m =>
                   fun vlen' : S (length rest) = S m
                   =>
                   pure (vcons m) <⋆>
                   (mkBackwards ∘ f) a <⋆>
                   go rest m (S_uncons vlen')
               end vlen
           end) l (length l) eq_refl)
   with
   | Some a =>
       fun
         o : option
               (Vector (length l) B ->
                Vector (S (length l)) B) =>
       match o with
       | Some a0 => Some (a, a0)
       | None => None
       end
   | None =>
       fun
         o : option
               (Vector (length l) B ->
                Vector (S (length l)) B) =>
       match o with
       | Some _ | _ => None
       end
   end None) = None
        rewrite map_None_eq_iff.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case1: f a = None
match
  forwards
    ((fix go
        (vl : list (E * A)) (n : nat) {struct vl} :
          length vl = n ->
          Backwards option (Vector n B) :=
        match
          vl as vl0
          return
            (length vl0 = n ->
             Backwards option (Vector n B))
        with
        | [] =>
            pure
            ○ exist (fun l : list B => length l = n)
                []
        | a :: rest =>
            fun vlen : S (length rest) = n =>
            match
              n as n0
              return
                (S (length rest) = n0 ->
                 Backwards option (Vector n0 B))
            with
            | 0 => zero_not_S ○ eq_sym (y:=0)
            | S m =>
                fun vlen' : S (length rest) = S m =>
                pure (vcons m) <⋆> (mkBackwards ∘ f) a <⋆>
                go rest m (S_uncons vlen')
            end vlen
        end) l (length l) eq_refl)
with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end None = None
        remember (@forwards option (Vector (@length (prod E A) l) B)
                    ((fix go (vl : list (prod E A)) (n : nat) {struct vl} :
                       forall _ : @eq nat (@length (prod E A) vl) n, Backwards option (Vector n B) :=
                        match
                          vl as vl0 return (forall _ : @eq nat (@length (prod E A) vl0) n, Backwards option (Vector n B))
                        with
                        | nil =>
                            fun vlen : @eq nat O n =>
                              @pure (Backwards option) (@Pure_Backwards option Pure_option)
                                (@sig (list B) (fun l : list B => @eq nat (@length B l) n))
                                (@exist (list B) (fun l : list B => @eq nat (@length B l) n) (@nil B) vlen)
                        | cons a rest =>
                            fun vlen : @eq nat (S (@length (prod E A) rest)) n =>
                              match
                                n as n0
                                return (forall _ : @eq nat (S (@length (prod E A) rest)) n0, Backwards option (Vector n0 B))
                              with
                              | O =>
                                  fun vlen' : @eq nat (S (@length (prod E A) rest)) O =>
                                    @zero_not_S
                                      ((fix length (l : list (prod E A)) : nat :=
                                          match l return nat with
                                          | nil => O
                                          | cons _ l' => S (length l')
                                          end) rest) (Backwards option (Vector O B))
                                      (@eq_sym nat (S (@length (prod E A) rest)) O vlen')
                              | S m =>
                                  fun vlen' : @eq nat (S (@length (prod E A) rest)) (S m) =>
                                    @ap (Backwards option) (@Map_Backwards option Map_option)
                                      (@Mult_Backwards option Map_option Mult_option) (Vector m B) (Vector (S m) B)
                                      (@ap (Backwards option) (@Map_Backwards option Map_option)
                                         (@Mult_Backwards option Map_option Mult_option) B
                                         (forall _ : Vector m B, Vector (S m) B)
                                         (@pure (Backwards option) (@Pure_Backwards option Pure_option)
                                            (forall (_ : B) (_ : Vector m B), Vector (S m) B) (@vcons m B))
                                         (@compose (prod E A) (option B) (Backwards option B) (@mkBackwards option B) f a))
                                      (go rest m
                                         (@S_uncons
                                            ((fix length (l : list (prod E A)) : nat :=
                                                match l return nat with
                                                | nil => O
                                                | cons _ l' => S (length l')
                                                end) rest) m vlen'))
                              end vlen
                        end) l (@length (prod E A) l) (@eq_refl nat (@length (prod E A) l)))) as X.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case1: f a = None
X: option (Vector (length l) B)
HeqX: X =
forwards
  ((fix go
      (vl : list (E * A)) (n : nat) {struct vl} :
        length vl = n ->
        Backwards option (Vector n B) :=
      match
        vl as vl0
        return
          (length vl0 = n ->
           Backwards option (Vector n B))
      with
      | [] =>
          pure
          ○ exist
              (fun l : list B => length l = n)
              []
      | a :: rest =>
          fun vlen : S (length rest) = n =>
          match
            n as n0
            return
              (S (length rest) = n0 ->
               Backwards option (Vector n0 B))
          with
          | 0 => zero_not_S ○ eq_sym (y:=0)
          | S m =>
              fun vlen' : S (length rest) = S m
              =>
              pure (vcons m) <⋆>
              (mkBackwards ∘ f) a <⋆>
              go rest m (S_uncons vlen')
          end vlen
      end) l (length l) eq_refl)
match X with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end None = None
        destruct X.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case1: f a = None
v: Vector (length l) B
HeqX: Some v =
forwards
  ((fix go
      (vl : list (E * A)) (n : nat) {struct vl} :
        length vl = n ->
        Backwards option (Vector n B) :=
      match
        vl as vl0
        return
          (length vl0 = n ->
           Backwards option (Vector n B))
      with
      | [] =>
          pure
          ○ exist
              (fun l : list B => length l = n)
              []
      | a :: rest =>
          fun vlen : S (length rest) = n =>
          match
            n as n0
            return
              (S (length rest) = n0 ->
               Backwards option (Vector n0 B))
          with
          | 0 => zero_not_S ○ eq_sym (y:=0)
          | S m =>
              fun vlen' : S (length rest) = S m
              =>
              pure (vcons m) <⋆>
              (mkBackwards ∘ f) a <⋆>
              go rest m (S_uncons vlen')
          end vlen
      end) l (length l) eq_refl)
None = None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case1: f a = None
HeqX: None =
forwards
  ((fix go
      (vl : list (E * A)) (n : nat) {struct vl} :
        length vl = n ->
        Backwards option (Vector n B) :=
      match
        vl as vl0
        return
          (length vl0 = n ->
           Backwards option (Vector n B))
      with
      | [] =>
          pure
          ○ exist
              (fun l : list B => length l = n)
              []
      | a :: rest =>
          fun vlen : S (length rest) = n =>
          match
            n as n0
            return
              (S (length rest) = n0 ->
               Backwards option (Vector n0 B))
          with
          | 0 => zero_not_S ○ eq_sym (y:=0)
          | S m =>
              fun vlen' : S (length rest) = S m
              =>
              pure (vcons m) <⋆>
              (mkBackwards ∘ f) a <⋆>
              go rest m (S_uncons vlen')
          end vlen
      end) l (length l) eq_refl)
None = None
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case1: f a = None
v: Vector (length l) B
HeqX: Some v =
forwards
  ((fix go
      (vl : list (E * A)) (n : nat) {struct vl} :
        length vl = n ->
        Backwards option (Vector n B) :=
      match
        vl as vl0
        return
          (length vl0 = n ->
           Backwards option (Vector n B))
      with
      | [] =>
          pure
          ○ exist
              (fun l : list B => length l = n)
              []
      | a :: rest =>
          fun vlen : S (length rest) = n =>
          match
            n as n0
            return
              (S (length rest) = n0 ->
               Backwards option (Vector n0 B))
          with
          | 0 => zero_not_S ○ eq_sym (y:=0)
          | S m =>
              fun vlen' : S (length rest) = S m
              =>
              pure (vcons m) <⋆>
              (mkBackwards ∘ f) a <⋆>
              go rest m (S_uncons vlen')
          end vlen
      end) l (length l) eq_refl)
None = None reflexivity.
        +
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case1: f a = None
HeqX: None =
forwards
  ((fix go
      (vl : list (E * A)) (n : nat) {struct vl} :
        length vl = n ->
        Backwards option (Vector n B) :=
      match
        vl as vl0
        return
          (length vl0 = n ->
           Backwards option (Vector n B))
      with
      | [] =>
          pure
          ○ exist
              (fun l : list B => length l = n)
              []
      | a :: rest =>
          fun vlen : S (length rest) = n =>
          match
            n as n0
            return
              (S (length rest) = n0 ->
               Backwards option (Vector n0 B))
          with
          | 0 => zero_not_S ○ eq_sym (y:=0)
          | S m =>
              fun vlen' : S (length rest) = S m
              =>
              pure (vcons m) <⋆>
              (mkBackwards ∘ f) a <⋆>
              go rest m (S_uncons vlen')
          end vlen
      end) l (length l) eq_refl)
None = None reflexivity.
      }
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case2: exists b : B,
  f a = Some b /\
  forwards
    (traverse_Vector_core 
       (mkBackwards ∘ f) l eq_refl) = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) (a :: l)
     eq_refl) = None
      { (* Case 2 *)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
Case2: exists b : B,
  f a = Some b /\
  forwards
    (traverse_Vector_core 
       (mkBackwards ∘ f) l eq_refl) = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) (a :: l)
     eq_refl) = None
        destruct Case2 as [b [Hfa rest]].
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
b: B
Hfa: f a = Some b
rest: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
forwards
  (traverse_Vector_core (mkBackwards ∘ f) (a :: l)
     eq_refl) = None
        cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
b: B
Hfa: f a = Some b
rest: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
map (fun '(f, a) => f a)
  (map swap
     (match
        forwards
          ((fix go
              (vl : list (E * A)) (n : nat) {struct
               vl} :
                length vl = n ->
                Backwards option (Vector n B) :=
              match
                vl as vl0
                return
                  (length vl0 = n ->
                   Backwards option (Vector n B))
              with
              | [] =>
                  pure
                  ○ exist
                      (fun l : list B => length l = n)
                      []
              | a :: rest =>
                  fun vlen : S (length rest) = n =>
                  match
                    n as n0
                    return
                      (S ... = n0 ->
                       Backwards option (...))
                  with
                  | 0 => zero_not_S ○ eq_sym (y:=0)
                  | S m =>
                      fun vlen' : S (...) = S m =>
                      pure (vcons m) <⋆>
                      (mkBackwards ∘ f) a <⋆>
                      go rest m (S_uncons vlen')
                  end vlen
              end) l (length l) eq_refl)
      with
      | Some a =>
          fun
            o : option
                  (Vector (length l) B ->
                   Vector (S (length l)) B) =>
          match o with
          | Some a0 => Some (a, a0)
          | None => None
          end
      | None =>
          fun
            o : option
                  (Vector (length l) B ->
                   Vector (S (length l)) B) =>
          match o with
          | Some _ | _ => None
          end
      end
        (map (fun '(f, a) => f a)
           (map swap
              (match f a with
               | Some a =>
                   fun
                     o : option
                           (B ->
                            Vector (length l) B ->
                            Vector (S (length l)) B)
                   =>
                   match o with
                   | Some a0 => Some (a, a0)
                   | None => None
                   end
               | None =>
                   fun
                     o : option
                           (B ->
                            Vector (length l) B ->
                            Vector (S (length l)) B)
                   =>
                   match o with
                   | Some _ | _ => None
                   end
               end (pure (vcons (length l)))))))) =
None
        rewrite map_None_eq_iff.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
b: B
Hfa: f a = Some b
rest: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
map swap
  (match
     forwards
       ((fix go
           (vl : list (E * A)) (n : nat) {struct vl} :
             length vl = n ->
             Backwards option (Vector n B) :=
           match
             vl as vl0
             return
               (length vl0 = n ->
                Backwards option (Vector n B))
           with
           | [] =>
               pure
               ○ exist
                   (fun l : list B => length l = n) []
           | a :: rest =>
               fun vlen : S (length rest) = n =>
               match
                 n as n0
                 return
                   (S (length rest) = n0 ->
                    Backwards option (Vector n0 B))
               with
               | 0 => zero_not_S ○ eq_sym (y:=0)
               | S m =>
                   fun vlen' : S (length rest) = S m
                   =>
                   pure (vcons m) <⋆>
                   (mkBackwards ∘ f) a <⋆>
                   go rest m (S_uncons vlen')
               end vlen
           end) l (length l) eq_refl)
   with
   | Some a =>
       fun
         o : option
               (Vector (length l) B ->
                Vector (S (length l)) B) =>
       match o with
       | Some a0 => Some (a, a0)
       | None => None
       end
   | None =>
       fun
         o : option
               (Vector (length l) B ->
                Vector (S (length l)) B) =>
       match o with
       | Some _ | _ => None
       end
   end
     (map (fun '(f, a) => f a)
        (map swap
           (match f a with
            | Some a =>
                fun
                  o : option
                        (B ->
                         Vector (length l) B ->
                         Vector (S (length l)) B) =>
                match o with
                | Some a0 => Some (a, a0)
                | None => None
                end
            | None =>
                fun
                  o : option
                        (B ->
                         Vector (length l) B ->
                         Vector (S (length l)) B) =>
                match o with
                | Some _ | _ => None
                end
            end (pure (vcons (length l))))))) = None
        rewrite map_None_eq_iff.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
b: B
Hfa: f a = Some b
rest: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
match
  forwards
    ((fix go
        (vl : list (E * A)) (n : nat) {struct vl} :
          length vl = n ->
          Backwards option (Vector n B) :=
        match
          vl as vl0
          return
            (length vl0 = n ->
             Backwards option (Vector n B))
        with
        | [] =>
            pure
            ○ exist (fun l : list B => length l = n)
                []
        | a :: rest =>
            fun vlen : S (length rest) = n =>
            match
              n as n0
              return
                (S (length rest) = n0 ->
                 Backwards option (Vector n0 B))
            with
            | 0 => zero_not_S ○ eq_sym (y:=0)
            | S m =>
                fun vlen' : S (length rest) = S m =>
                pure (vcons m) <⋆> (mkBackwards ∘ f) a <⋆>
                go rest m (S_uncons vlen')
            end vlen
        end) l (length l) eq_refl)
with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end
  (map (fun '(f, a) => f a)
     (map swap
        (match f a with
         | Some a =>
             fun
               o : option
                     (B ->
                      Vector (length l) B ->
                      Vector (S (length l)) B) =>
             match o with
             | Some a0 => Some (a, a0)
             | None => None
             end
         | None =>
             fun
               o : option
                     (B ->
                      Vector (length l) B ->
                      Vector (S (length l)) B) =>
             match o with
             | Some _ | _ => None
             end
         end (pure (vcons (length l)))))) = None
        Set Keyed Unification.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
b: B
Hfa: f a = Some b
rest: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
match
  forwards
    ((fix go
        (vl : list (E * A)) (n : nat) {struct vl} :
          length vl = n ->
          Backwards option (Vector n B) :=
        match
          vl as vl0
          return
            (length vl0 = n ->
             Backwards option (Vector n B))
        with
        | [] =>
            pure
            ○ exist (fun l : list B => length l = n)
                []
        | a :: rest =>
            fun vlen : S (length rest) = n =>
            match
              n as n0
              return
                (S (length rest) = n0 ->
                 Backwards option (Vector n0 B))
            with
            | 0 => zero_not_S ○ eq_sym (y:=0)
            | S m =>
                fun vlen' : S (length rest) = S m =>
                pure (vcons m) <⋆> (mkBackwards ∘ f) a <⋆>
                go rest m (S_uncons vlen')
            end vlen
        end) l (length l) eq_refl)
with
| Some a =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some a0 => Some (a, a0)
    | None => None
    end
| None =>
    fun
      o : option
            (Vector (length l) B ->
             Vector (S (length l)) B) =>
    match o with
    | Some _ | _ => None
    end
end
  (map (fun '(f, a) => f a)
     (map swap
        (match f a with
         | Some a =>
             fun
               o : option
                     (B ->
                      Vector (length l) B ->
                      Vector (S (length l)) B) =>
             match o with
             | Some a0 => Some (a, a0)
             | None => None
             end
         | None =>
             fun
               o : option
                     (B ->
                      Vector (length l) B ->
                      Vector (S (length l)) B) =>
             match o with
             | Some _ | _ => None
             end
         end (pure (vcons (length l)))))) = None
        rewrite rest.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
b: B
Hfa: f a = Some b
rest: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
match
  map (fun '(f, a) => f a)
    (map swap
       (match f a with
        | Some a =>
            fun
              o : option
                    (B ->
                     Vector (length l) B ->
                     Vector (S (length l)) B) =>
            match o with
            | Some a0 => Some (a, a0)
            | None => None
            end
        | None =>
            fun
              o : option
                    (B ->
                     Vector (length l) B ->
                     Vector (S (length l)) B) =>
            match o with
            | Some _ | _ => None
            end
        end (pure (vcons (length l)))))
with
| Some _ | _ => None
end = None
        Unset Keyed Unification.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
b: B
Hfa: f a = Some b
rest: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
match
  map (fun '(f, a) => f a)
    (map swap
       (match f a with
        | Some a =>
            fun
              o : option
                    (B ->
                     Vector (length l) B ->
                     Vector (S (length l)) B) =>
            match o with
            | Some a0 => Some (a, a0)
            | None => None
            end
        | None =>
            fun
              o : option
                    (B ->
                     Vector (length l) B ->
                     Vector (S (length l)) B) =>
            match o with
            | Some _ | _ => None
            end
        end (pure (vcons (length l)))))
with
| Some _ | _ => None
end = None
        rewrite Hfa.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
b: B
Hfa: f a = Some b
rest: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
match
  map (fun '(f, a) => f a)
    (map swap
       match pure (vcons (length l)) with
       | Some a => Some (b, a)
       | None => None
       end)
with
| Some _ | _ => None
end = None
        cbn.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
f: E * A -> option B
a: E * A
l: list (E * A)
HeqNone: traverse f (a :: l) = None
IHl: traverse f l = None ->
forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
b: B
Hfa: f a = Some b
rest: forwards
  (traverse_Vector_core 
     (mkBackwards ∘ f) l eq_refl) = None
None = None
        reflexivity.
      }
  Qed.

  Lemma traverse_Vector_None_spec {A B} (n: nat) (f: E * A -> option B):
    forall (l: list (E * A)) (e: length l = n),
      (forwards (traverse_Vector_core (mkBackwards ∘ f) l e) = None) <-> traverse f l = None.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
n: nat
f: E * A -> option B
forall (l : list (E * A)) (e : length l = n),
forwards (traverse_Vector_core (mkBackwards ∘ f) l e) =
None <-> traverse f l = None
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
n: nat
f: E * A -> option B
forall (l : list (E * A)) (e : length l = n),
forwards (traverse_Vector_core (mkBackwards ∘ f) l e) =
None <-> traverse f l = None
    intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
n: nat
f: E * A -> option B
l: list (E * A)
e: length l = n
forwards (traverse_Vector_core (mkBackwards ∘ f) l e) =
None <-> traverse f l = None split.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
n: nat
f: E * A -> option B
l: list (E * A)
e: length l = n
forwards (traverse_Vector_core (mkBackwards ∘ f) l e) =
None -> traverse f l = None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
n: nat
f: E * A -> option B
l: list (E * A)
e: length l = n
traverse f l = None ->
forwards (traverse_Vector_core (mkBackwards ∘ f) l e) =
None
    apply traverse_Vector_None_spec1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
n: nat
f: E * A -> option B
l: list (E * A)
e: length l = n
traverse f l = None ->
forwards (traverse_Vector_core (mkBackwards ∘ f) l e) =
None
    apply traverse_Vector_None_spec2.
  Qed.

  Lemma mapdt_option_None_spec:
    forall (A B: Type) (t: T A) (f: E * A -> option B),
      (mapdt f t = None) <->
        exists (e: E) (a: A), (e, a) ∈d t /\ f (e, a) = None.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (t : T A) (f : E * A -> option B),
mapdt f t = None <->
(exists (e : E) (a : A),
   (e, a) ∈d t /\ f (e, a) = None)
  Proof.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
forall (A B : Type) (t : T A) (f : E * A -> option B),
mapdt f t = None <->
(exists (e : E) (a : A),
   (e, a) ∈d t /\ f (e, a) = None)
    intros.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
mapdt f t = None <->
(exists (e : E) (a : A),
   (e, a) ∈d t /\ f (e, a) = None)
    rewrite mapdt_repr.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
map (mapdt_make t)
  (forwards
     (traverse (mkBackwards ∘ f) (mapdt_contents t))) =
None <->
(exists (e : E) (a : A),
   (e, a) ∈d t /\ f (e, a) = None)
    setoid_rewrite ind_iff_in_toctxlist1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
map (mapdt_make t)
  (forwards
     (traverse (mkBackwards ∘ f) (mapdt_contents t))) =
None <->
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None)
    remember (mapdt_contents t).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
v: Vector (plength t) (E * A)
Heqv: v = mapdt_contents t
map (mapdt_make t)
  (forwards (traverse (mkBackwards ∘ f) v)) = None <->
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None)
    destruct v.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
map (mapdt_make t)
  (forwards
     (traverse (mkBackwards ∘ f)
        (exist
           (fun l : list (E * A) =>
            length l = plength t) x e))) = None <->
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None)
    assert (list_spec: List.rev (toctxlist t) = x).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
List.rev (toctxlist t) = x
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
list_spec: List.rev (toctxlist t) = x
map (mapdt_make t)
  (forwards
     (traverse (mkBackwards ∘ f)
        (exist
           (fun l : list (E * A) =>
            length l = plength t) x e))) = None <->
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None)
    {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
List.rev (toctxlist t) = x symmetry.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
x = List.rev (toctxlist t)
      enough (x = Vector_to_list _ (exist (fun l : list (E * A) => length l = plength t) x e)).
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
H0: x =
Vector_to_list (E * A)
  (exist
     (fun l : list (E * A) =>
      length l = plength t) x e)
x = List.rev (toctxlist t)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
x =
Vector_to_list (E * A)
  (exist
     (fun l : list (E * A) => length l = plength t) x
     e)
      rewrite H0.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
H0: x =
Vector_to_list (E * A)
  (exist
     (fun l : list (E * A) =>
      length l = plength t) x e)
Vector_to_list (E * A)
  (exist
     (fun l : list (E * A) => length l = plength t) x
     e) = List.rev (toctxlist t)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
x =
Vector_to_list (E * A)
  (exist
     (fun l : list (E * A) => length l = plength t) x
     e)
      rewrite Heqv.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
H0: x =
Vector_to_list (E * A)
  (exist
     (fun l : list (E * A) =>
      length l = plength t) x e)
Vector_to_list (E * A) (mapdt_contents t) =
List.rev (toctxlist t)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
x =
Vector_to_list (E * A)
  (exist
     (fun l : list (E * A) => length l = plength t) x
     e)
      apply mapdt_contents_toctxlist.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
x =
Vector_to_list (E * A)
  (exist
     (fun l : list (E * A) => length l = plength t) x
     e)
      reflexivity. }
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
list_spec: List.rev (toctxlist t) = x
map (mapdt_make t)
  (forwards
     (traverse (mkBackwards ∘ f)
        (exist
           (fun l : list (E * A) =>
            length l = plength t) x e))) = None <->
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None)
    cbn in *.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
list_spec: List.rev (toctxlist t) = x
map (mapdt_make t)
  (forwards
     (traverse_Vector_core (mkBackwards ∘ f) x e)) =
None <->
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None)
    rewrite map_None_eq_iff.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
Heqv: exist
  (fun l : list (E * A) => length l = plength t)
  x e = mapdt_contents t
list_spec: List.rev (toctxlist t) = x
forwards (traverse_Vector_core (mkBackwards ∘ f) x e) =
None <->
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None)
    clear Heqv.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
e: length x = plength t
list_spec: List.rev (toctxlist t) = x
forwards (traverse_Vector_core (mkBackwards ∘ f) x e) =
None <->
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None)
    destruct e.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
list_spec: List.rev (toctxlist t) = x
forwards
  (traverse_Vector_core (mkBackwards ∘ f) x eq_refl) =
None <->
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None)
    rewrite traverse_Vector_None_spec.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
list_spec: List.rev (toctxlist t) = x
traverse f x = None <->
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None)
    rewrite list_traverse_None_iff.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
list_spec: List.rev (toctxlist t) = x
(exists a : E * A, a ∈ x /\ f a = None) <->
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None)
    rewrite <- list_spec.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
list_spec: List.rev (toctxlist t) = x
(exists a : E * A,
   a ∈ List.rev (toctxlist t) /\ f a = None) <->
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None)
    setoid_rewrite <- in_list_rev_spec at 1.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
list_spec: List.rev (toctxlist t) = x
(exists a : E * A, a ∈ toctxlist t /\ f a = None) <->
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None)
    rewrite <- compat_toctxlist_mapdt.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
list_spec: List.rev (toctxlist t) = x
(exists a : E * A, a ∈ toctxlist t /\ f a = None) <->
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None)
    split.
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
list_spec: List.rev (toctxlist t) = x
(exists a : E * A, a ∈ toctxlist t /\ f a = None) ->
exists (e : E) (a : A),
  (e, a) ∈ toctxlist t /\ f (e, a) = None
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
list_spec: List.rev (toctxlist t) = x
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None) ->
exists a : E * A, a ∈ toctxlist t /\ f a = None
    {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
list_spec: List.rev (toctxlist t) = x
(exists a : E * A, a ∈ toctxlist t /\ f a = None) ->
exists (e : E) (a : A),
  (e, a) ∈ toctxlist t /\ f (e, a) = None intros [[e a] rest]; eauto. }
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
list_spec: List.rev (toctxlist t) = x
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None) ->
exists a : E * A, a ∈ toctxlist t /\ f a = None
    {
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Traverse_inst: Traverse T
Mapd_inst: Mapd E T
Map_inst: Map T
ToBatch_inst: ToBatch T
ToBatch3_inst: ToBatch3 E T
ToCtxset_inst: ToCtxset E T
ToCtxlist_inst: ToCtxlist E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_ToBatch_Traverse0: Compat_ToBatch_Traverse T
Compat_ToBatch3_Mapdt0: Compat_ToBatch3_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
H: DecoratedTraversableFunctor E T
A, B: Type
t: T A
f: E * A -> option B
x: list (E * A)
list_spec: List.rev (toctxlist t) = x
(exists (e : E) (a : A),
   (e, a) ∈ toctxlist t /\ f (e, a) = None) ->
exists a : E * A, a ∈ toctxlist t /\ f a = None intros [e [a rest]]; eauto. }
  Qed.

End theory.