DecoratedTraversableFunctor.v

From Tealeaves Require Export
  Classes.Kleisli.DecoratedTraversableFunctor
  Classes.Kleisli.Theory.TraversableFunctor
  Classes.Kleisli.DecoratedContainerFunctor
  Classes.Kleisli.DecoratedShapelyFunctor
  Classes.Kleisli.Theory.DecoratedContainerFunctor
  Functors.Early.Environment.

From Coq.Logic Require Import Decidable.

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

Import DecoratedContainerFunctor.Notations.
Import ContainerFunctor.Notations.
Import Monoid.Notations.
Import Subset.Notations.
Import Product.Notations.

Import Kleisli.DecoratedTraversableFunctor.DerivedInstances.

(** * Theory of Decorated Traversable Functors *)
(**********************************************************************)

(** ** <<mapdt>> with constant applicative functors *)
(**********************************************************************)
Section mapdt_constant_applicatives.

  Context
    {E: Type}
    {T: Type -> Type}
    `{Mapdt_inst: Mapdt E T}
    `{Map_inst: Map T}
    `{! Compat_Map_Mapdt E T}
    `{! DecoratedTraversableFunctor E T}
    `{Monoid M}.

  Lemma mapdt_constant_applicative1 {A B: Type}
    `(f: E * A -> const M B):
    mapdt (G := const M) (A := A) (B := B) f=
      mapdt (G := const M) (B := False) f.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
f: E * A -> const M B
mapdt f = mapdt f
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
f: E * A -> const M B
mapdt f = mapdt f
    change_right
      (map (F := const M) (A := T False) (B := T B)
         (map (F := T) (@exfalso B))
         ∘ (mapdt (G := const M) (B := False) f)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
f: E * A -> const M B
mapdt f = map (map exfalso) ∘ mapdt f
    rewrite map_mapdt.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A, B: Type
f: E * A -> const M B
mapdt f = mapdt (map exfalso ∘ f)
    reflexivity.
  Qed.

  Lemma mapdt_constant_applicative2 (B fake1 fake2: Type)
    `(f: E * A -> const M B):
    mapdt (G := const M) (B := fake1) f =
      mapdt (G := const M) (B := fake2) f.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B, fake1, fake2, A: Type
f: E * A -> const M B
mapdt f = mapdt f
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B, fake1, fake2, A: Type
f: E * A -> const M B
mapdt f = mapdt f
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B, fake1, fake2, A: Type
f: E * A -> const M B
mapdt f = mapdt f
    rewrite (mapdt_constant_applicative1 (B := fake1)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B, fake1, fake2, A: Type
f: E * A -> const M B
mapdt f = mapdt f
    rewrite (mapdt_constant_applicative1 (B := fake2)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B, fake1, fake2, A: Type
f: E * A -> const M B
mapdt f = mapdt f
    easy.
  Qed.

End mapdt_constant_applicatives.

(** * Derived Operation <<mapdReduce>> *)
(**********************************************************************)
Definition mapdReduce {T: Type -> Type} `{Mapdt E T}
  `{op: Monoid_op M} `{unit: Monoid_unit M}
  {A: Type} (f: E * A -> M): T A -> M :=
  mapdt (G := const M) (B := False) f.

Section mapdt_mapdReduce.

  Context
    {E: Type}
    {T: Type -> Type}
    `{Mapdt_inst: Mapdt E T}
    `{Mapd_inst: Mapd E T}
    `{Traverse_inst: Traverse T}
    `{Map_inst: Map T}
    `{! Compat_Map_Mapdt E T}
    `{! Compat_Mapd_Mapdt E T}
    `{! Compat_Traverse_Mapdt E T}
    `{! DecoratedTraversableFunctor E T}.

  (** ** Rewriting Laws *)
  (********************************************************************)
  Lemma mapdReduce_to_mapdt1 `{Monoid M} `(f: E * A -> M):
    mapdReduce (T := T) (M := M) (A := A) f =
      mapdt (G := const M) (B := False) f.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: E * A -> M
mapdReduce f = mapdt f
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: E * A -> M
mapdReduce f = mapdt f
    reflexivity.
  Qed.

  Lemma mapdReduce_to_mapdt2 `{Monoid M} `(f: E * A -> M):
    forall (fake: Type),
      mapdReduce (T := T) (M := M) (A := A) f =
        mapdt (G := const M) (B := fake) f.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: E * A -> M
forall fake : Type, mapdReduce f = mapdt f
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: E * A -> M
forall fake : Type, mapdReduce f = mapdt f
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: E * A -> M
fake: Type
mapdReduce f = mapdt f
    rewrite mapdReduce_to_mapdt1.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: E * A -> M
fake: Type
mapdt f = mapdt f
    rewrite (mapdt_constant_applicative1 (B := fake)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: E * A -> M
fake: Type
mapdt f = mapdt f
    reflexivity.
  Qed.

  (** ** Composition Laws with <<mapd>> and <<map>> *)
  (********************************************************************)
  Lemma mapdReduce_mapd `{Monoid M} {B: Type}:
    forall `(g: E * B -> M) `(f: E * A -> B),
      mapdReduce g ∘ mapd f = mapdReduce (T := T) (g ∘ cobind f).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
forall (g : E * B -> M) (A : Type) (f : E * A -> B),
mapdReduce g ∘ mapd f = mapdReduce (g ∘ cobind f)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
forall (g : E * B -> M) (A : Type) (f : E * A -> B),
mapdReduce g ∘ mapd f = mapdReduce (g ∘ cobind f)
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: E * B -> M
A: Type
f: E * A -> B
mapdReduce g ∘ mapd f = mapdReduce (g ∘ cobind f)
    rewrite mapdReduce_to_mapdt1.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: E * B -> M
A: Type
f: E * A -> B
mapdt g ∘ mapd f = mapdReduce (g ∘ cobind f)
    rewrite (mapdt_mapd g f).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: E * B -> M
A: Type
f: E * A -> B
mapdt (kc1 g f) = mapdReduce (g ∘ cobind f)
    reflexivity.
  Qed.

  Corollary mapdReduce_map `{Monoid M}:
    forall `(g: E * B -> M) `(f: A -> B),
      mapdReduce g ∘ map f = mapdReduce (g ∘ map (F := prod E) f).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
forall (B : Type) (g : E * B -> M) (A : Type)
  (f : A -> B),
mapdReduce g ∘ map f = mapdReduce (g ∘ map f)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
forall (B : Type) (g : E * B -> M) (A : Type)
  (f : A -> B),
mapdReduce g ∘ map f = mapdReduce (g ∘ map f)
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: E * B -> M
A: Type
f: A -> B
mapdReduce g ∘ map f = mapdReduce (g ∘ map f)
    rewrite map_to_mapdt.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: E * B -> M
A: Type
f: A -> B
mapdReduce g ∘ mapdt (f ∘ extract) =
mapdReduce (g ∘ map f)
    replace (mapdt (G := fun A => A) (f ∘ extract))
      with (mapd (f ∘ extract)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: E * B -> M
A: Type
f: A -> B
mapdReduce g ∘ mapd (f ∘ extract) =
mapdReduce (g ∘ map f)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: E * B -> M
A: Type
f: A -> B
mapd (f ∘ extract) = mapdt (f ∘ extract)
    -E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: E * B -> M
A: Type
f: A -> B
mapdReduce g ∘ mapd (f ∘ extract) =
mapdReduce (g ∘ map f) rewrite mapdReduce_mapd.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: E * B -> M
A: Type
f: A -> B
mapdReduce (g ∘ cobind (f ∘ extract)) =
mapdReduce (g ∘ map f)
      reflexivity.
    -E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: E * B -> M
A: Type
f: A -> B
mapd (f ∘ extract) = mapdt (f ∘ extract) rewrite mapd_to_mapdt.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
B: Type
g: E * B -> M
A: Type
f: A -> B
mapdt (f ∘ extract) = mapdt (f ∘ extract)
      reflexivity.
  Qed.

  (** ** Composition with Monoid Homomorphisms *)
  (********************************************************************)
  Lemma mapdReduce_morphism
    `{morphism: Monoid_Morphism M1 M2 ϕ}: forall `(f: E * A -> M1),
      ϕ ∘ mapdReduce f = mapdReduce (ϕ ∘ f).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M1, M2: Type
src_op: Monoid_op M1
src_unit: Monoid_unit M1
tgt_op: Monoid_op M2
tgt_unit: Monoid_unit M2
ϕ: M1 -> M2
morphism: Monoid_Morphism M1 M2 ϕ
forall (A : Type) (f : E * A -> M1),
ϕ ∘ mapdReduce f = mapdReduce (ϕ ∘ f)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M1, M2: Type
src_op: Monoid_op M1
src_unit: Monoid_unit M1
tgt_op: Monoid_op M2
tgt_unit: Monoid_unit M2
ϕ: M1 -> M2
morphism: Monoid_Morphism M1 M2 ϕ
forall (A : Type) (f : E * A -> M1),
ϕ ∘ mapdReduce f = mapdReduce (ϕ ∘ f)
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M1, M2: Type
src_op: Monoid_op M1
src_unit: Monoid_unit M1
tgt_op: Monoid_op M2
tgt_unit: Monoid_unit M2
ϕ: M1 -> M2
morphism: Monoid_Morphism M1 M2 ϕ
A: Type
f: E * A -> M1
ϕ ∘ mapdReduce f = mapdReduce (ϕ ∘ f)
    inversion morphism.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M1, M2: Type
src_op: Monoid_op M1
src_unit: Monoid_unit M1
tgt_op: Monoid_op M2
tgt_unit: Monoid_unit M2
ϕ: M1 -> M2
morphism: Monoid_Morphism M1 M2 ϕ
A: Type
f: E * A -> M1
monmor_src: Monoid M1
monmor_tgt: Monoid M2
monmor_unit: ϕ Ƶ = Ƶ
monmor_op: forall a1 a2 : M1,
ϕ (a1 ● a2) = ϕ a1 ● ϕ a2
ϕ ∘ mapdReduce f = mapdReduce (ϕ ∘ f)
    rewrite mapdReduce_to_mapdt1.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M1, M2: Type
src_op: Monoid_op M1
src_unit: Monoid_unit M1
tgt_op: Monoid_op M2
tgt_unit: Monoid_unit M2
ϕ: M1 -> M2
morphism: Monoid_Morphism M1 M2 ϕ
A: Type
f: E * A -> M1
monmor_src: Monoid M1
monmor_tgt: Monoid M2
monmor_unit: ϕ Ƶ = Ƶ
monmor_op: forall a1 a2 : M1,
ϕ (a1 ● a2) = ϕ a1 ● ϕ a2
ϕ ∘ mapdt f = mapdReduce (ϕ ∘ f)
    change ϕ with (const ϕ (T False)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M1, M2: Type
src_op: Monoid_op M1
src_unit: Monoid_unit M1
tgt_op: Monoid_op M2
tgt_unit: Monoid_unit M2
ϕ: M1 -> M2
morphism: Monoid_Morphism M1 M2 ϕ
A: Type
f: E * A -> M1
monmor_src: Monoid M1
monmor_tgt: Monoid M2
monmor_unit: ϕ Ƶ = Ƶ
monmor_op: forall a1 a2 : M1,
ϕ (a1 ● a2) = ϕ a1 ● ϕ a2
const ϕ (T False) ∘ mapdt f =
mapdReduce (const ϕ (T False) ∘ f)
    rewrite (kdtf_morph (G1 := const M1) (G2 := const M2)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M1, M2: Type
src_op: Monoid_op M1
src_unit: Monoid_unit M1
tgt_op: Monoid_op M2
tgt_unit: Monoid_unit M2
ϕ: M1 -> M2
morphism: Monoid_Morphism M1 M2 ϕ
A: Type
f: E * A -> M1
monmor_src: Monoid M1
monmor_tgt: Monoid M2
monmor_unit: ϕ Ƶ = Ƶ
monmor_op: forall a1 a2 : M1,
ϕ (a1 ● a2) = ϕ a1 ● ϕ a2
mapdt (const ϕ False ∘ f) =
mapdReduce (const ϕ (T False) ∘ f)
    reflexivity.
  Qed.

  (** *** <<mapdReduce>> as a generalization of <<mapReduce>> *)
  (********************************************************************)
  Lemma mapReduce_to_mapdReduce: forall `{Monoid M} `(f: A -> M),
      mapReduce (T := T) f = mapdReduce (T := T) (f ∘ extract).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (M : Type) (op : Monoid_op M)
  (unit0 : Monoid_unit M),
Monoid M ->
forall (A : Type) (f : A -> M),
mapReduce f = mapdReduce (f ∘ extract)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (M : Type) (op : Monoid_op M)
  (unit0 : Monoid_unit M),
Monoid M ->
forall (A : Type) (f : A -> M),
mapReduce f = mapdReduce (f ∘ extract)
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: A -> M
mapReduce f = mapdReduce (f ∘ extract)
    rewrite mapReduce_to_traverse1.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: A -> M
traverse f = mapdReduce (f ∘ extract)
    rewrite traverse_to_mapdt.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: A -> M
mapdt (f ∘ extract) = mapdReduce (f ∘ extract)
    reflexivity.
  Qed.

End mapdt_mapdReduce.

(** * The <<toctxlist>> operation *)
(**********************************************************************)
#[local] Instance ToCtxlist_Mapdt
  `{Mapdt E T}: ToCtxlist E T :=
  fun A => mapdReduce (ret (T := list)).

Class Compat_ToCtxlist_Mapdt
  (E: Type)
  (T: Type -> Type)
  `{ToCtxlist_inst: ToCtxlist E T}
  `{Mapdt_inst: Mapdt E T}: Prop :=
  compat_toctxlist_mapdt:
    ToCtxlist_inst = @ToCtxlist_Mapdt E T Mapdt_inst.

#[export] Instance Compat_ToCtxlist_Mapdt_Self
  `{Mapdt_ET: Mapdt E T}:
  @Compat_ToCtxlist_Mapdt E T ToCtxlist_Mapdt Mapdt_ET
  := ltac:(reflexivity).

Lemma toctxlist_to_mapdt
  `{ToCtxlist_inst: ToCtxlist E T}
  `{Mapdt_ET: Mapdt E T}
  `{! Compat_ToCtxlist_Mapdt E T}:
  forall (A: Type),
    toctxlist = mapdReduce (ret (T := list) (A := E * A)).E: Type
T: Type -> Type
ToCtxlist_inst: ToCtxlist E T
Mapdt_ET: Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
forall A : Type, toctxlist = mapdReduce ret
Proof.E: Type
T: Type -> Type
ToCtxlist_inst: ToCtxlist E T
Mapdt_ET: Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
forall A : Type, toctxlist = mapdReduce ret
  intros.E: Type
T: Type -> Type
ToCtxlist_inst: ToCtxlist E T
Mapdt_ET: Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
A: Type
toctxlist = mapdReduce ret
  rewrite compat_toctxlist_mapdt.E: Type
T: Type -> Type
ToCtxlist_inst: ToCtxlist E T
Mapdt_ET: Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
A: Type
ToCtxlist_Mapdt A = mapdReduce ret
  reflexivity.
Qed.

Section mapdt_toctxlist.

  Context
    {E: Type}
    {T: Type -> Type}
    `{Mapdt_inst: Mapdt E T}
    `{Mapd_inst: Mapd E T}
    `{Traverse_inst: Traverse T}
    `{Map_inst: Map T}
    `{Toctxlist_inst: ToCtxlist E T}
    `{! Compat_Map_Mapdt E T}
    `{! Compat_Mapd_Mapdt E T}
    `{! Compat_Traverse_Mapdt E T}
    `{! Compat_ToCtxlist_Mapdt E T}
    `{! DecoratedTraversableFunctor E T}.

  (** ** Rewriting to <<mapdReduce>>/<<mapdt>>*)
  (********************************************************************)
  Lemma toctxlist_to_mapdReduce: forall (A: Type),
      toctxlist (F := T) = mapdReduce (ret (T := list) (A := E * A)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, toctxlist = mapdReduce ret
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, toctxlist = mapdReduce ret
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
toctxlist = mapdReduce ret
    rewrite toctxlist_to_mapdt.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapdReduce ret = mapdReduce ret
    reflexivity.
  Qed.

  Corollary toctxlist_to_mapdt1: forall (A: Type),
      toctxlist =
        mapdt (G := const (list (E * A))) (B := False) (ret (T := list)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, toctxlist = mapdt ret
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, toctxlist = mapdt ret
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
toctxlist = mapdt ret
    rewrite toctxlist_to_mapdt.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapdReduce ret = mapdt ret
    reflexivity.
  Qed.

  Corollary toctxlist_to_mapdt2: forall (A fake: Type),
      toctxlist =
        mapdt (G := const (list (E * A))) (B := fake) (ret (T := list)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A fake : Type, toctxlist = mapdt ret
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A fake : Type, toctxlist = mapdt ret
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, fake: Type
toctxlist = mapdt ret
    rewrite toctxlist_to_mapdt1.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, fake: Type
mapdt ret = mapdt ret
    rewrite (mapdt_constant_applicative1 (B := fake)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, fake: Type
mapdt ret = mapdt ret
    reflexivity.
  Qed.

  (** ** Naturality *)
  (********************************************************************)
  #[export] Instance Natural_ToCtxlist_Mapdt: Natural (@toctxlist E T _).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
Natural (@toctxlist E T Toctxlist_inst)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
Natural (@toctxlist E T Toctxlist_inst)
    constructor.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
Functor T
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
Functor (env E)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : A -> B),
map f ∘ toctxlist = toctxlist ∘ map f
    -E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
Functor T typeclasses eauto.
    -E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
Functor (env E) typeclasses eauto.
    -E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : A -> B),
map f ∘ toctxlist = toctxlist ∘ map f intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
map f ∘ toctxlist = toctxlist ∘ map f
      (* LHS *)
      change (list ○ prod E) with (env E). (* hidden *)E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
map f ∘ toctxlist = toctxlist ∘ map f
      rewrite toctxlist_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
map f ∘ mapdReduce ret = toctxlist ∘ map f
      assert (Monoid_Morphism (list (E * A)) (list (E * B)) (map f)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
Monoid_Morphism (list (E * A)) (list (E * B)) (map f)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
H: Monoid_Morphism (list (E * A)) 
  (list (E * B)) (map f)
map f ∘ mapdReduce ret = toctxlist ∘ map f
      {E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
Monoid_Morphism (list (E * A)) (list (E * B)) (map f) rewrite env_map_spec.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
Monoid_Morphism (list (E * A)) (list (E * B))
  (map (map f))
        apply Monmor_list_map. }E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
H: Monoid_Morphism (list (E * A)) 
  (list (E * B)) (map f)
map f ∘ mapdReduce ret = toctxlist ∘ map f
      rewrite (mapdReduce_morphism
                 (M1 := list (E * A)) (M2 := list (E * B))).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
H: Monoid_Morphism (list (E * A)) 
  (list (E * B)) (map f)
mapdReduce (map f ∘ ret) = toctxlist ∘ map f
      rewrite env_map_spec.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
H: Monoid_Morphism (list (E * A)) 
  (list (E * B)) (map f)
mapdReduce (map (map f) ∘ ret) = toctxlist ∘ map f
      rewrite (natural (ϕ := @ret list _)); unfold_ops @Map_I.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
H: Monoid_Morphism (list (E * A)) 
  (list (E * B)) (map f)
mapdReduce (ret ∘ map f) = toctxlist ∘ map f
      (* RHS *)
      rewrite toctxlist_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
H: Monoid_Morphism (list (E * A)) 
  (list (E * B)) (map f)
mapdReduce (ret ∘ map f) = mapdReduce ret ∘ map f
      rewrite mapdReduce_map.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
H: Monoid_Morphism (list (E * A)) 
  (list (E * B)) (map f)
mapdReduce (ret ∘ map f) = mapdReduce (ret ∘ map f)
      reflexivity.
  Qed.

  (** ** Composing <<toctxlist>> with <<mapd>> *)
  (********************************************************************)
  Lemma toctxlist_mapd: forall `(f: E * A -> B),
      toctxlist (F := T) ∘ mapd f =
        mapdReduce (ret (T := list) ∘ cobind f).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : E * A -> B),
toctxlist ∘ mapd f = mapdReduce (ret ∘ cobind f)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : E * A -> B),
toctxlist ∘ mapd f = mapdReduce (ret ∘ cobind f)
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
toctxlist ∘ mapd f = mapdReduce (ret ∘ cobind f)
    rewrite toctxlist_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce ret ∘ mapd f = mapdReduce (ret ∘ cobind f)
    rewrite mapdReduce_mapd.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce (ret ∘ cobind f) =
mapdReduce (ret ∘ cobind f)
    reflexivity.
  Qed.

  Lemma toctxlist_map: forall `(f: A -> B),
      toctxlist (F := T) ∘ map f =
        mapdReduce (ret (T := list) ∘ map (F := (E ×)) f).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : A -> B),
toctxlist ∘ map f = mapdReduce (ret ∘ map f)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : A -> B),
toctxlist ∘ map f = mapdReduce (ret ∘ map f)
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
toctxlist ∘ map f = mapdReduce (ret ∘ map f)
    rewrite toctxlist_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
mapdReduce ret ∘ map f = mapdReduce (ret ∘ map f)
    rewrite mapdReduce_map.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
mapdReduce (ret ∘ map f) = mapdReduce (ret ∘ map f)
    reflexivity.
  Qed.

  Lemma tolist_mapd: forall `(f: E * A -> B),
      tolist ∘ mapd f = mapdReduce (ret (T := list) ∘ f).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : E * A -> B),
tolist ∘ mapd f = mapdReduce (ret ∘ f)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : E * A -> B),
tolist ∘ mapd f = mapdReduce (ret ∘ f)
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
tolist ∘ mapd f = mapdReduce (ret ∘ f)
    rewrite tolist_to_mapReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapReduce ret ∘ mapd f = mapdReduce (ret ∘ f)
    rewrite mapReduce_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce (ret ∘ extract) ∘ mapd f =
mapdReduce (ret ∘ f)
    rewrite mapdReduce_mapd.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce (ret ∘ extract ∘ cobind f) =
mapdReduce (ret ∘ f)
    reassociate -> on left.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce (ret ∘ (extract ∘ cobind f)) =
mapdReduce (ret ∘ f)
    rewrite kcom_cobind0.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce (ret ∘ f) = mapdReduce (ret ∘ f)
    reflexivity.
  Qed.

  (** ** Naturality for <<toctxlist>> *)
  (********************************************************************)
  Lemma mapd_toctxlist: forall `(f: E * A -> B),
      mapd f ∘ toctxlist (F := T) = toctxlist ∘ mapd f.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : E * A -> B),
mapd f ∘ toctxlist = toctxlist ∘ mapd f
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : E * A -> B),
mapd f ∘ toctxlist = toctxlist ∘ mapd f
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapd f ∘ toctxlist = toctxlist ∘ mapd f
    rewrite toctxlist_mapd.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapd f ∘ toctxlist = mapdReduce (ret ∘ cobind f)
    rewrite toctxlist_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapd f ∘ mapdReduce ret = mapdReduce (ret ∘ cobind f)
    assert (Monoid_Morphism (env E A) (env E B) (mapd f)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
Monoid_Morphism (env E A) (env E B) (mapd f)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
H: Monoid_Morphism (env E A) (env E B) (mapd f)
mapd f ∘ mapdReduce ret = mapdReduce (ret ∘ cobind f)
    {E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
Monoid_Morphism (env E A) (env E B) (mapd f) unfold env.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
Monoid_Morphism (list (E * A)) (list (E * B)) (mapd f) rewrite env_mapd_spec.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
Monoid_Morphism (list (E * A)) (list (E * B))
  (map (cobind f))
      typeclasses eauto. }E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
H: Monoid_Morphism (env E A) (env E B) (mapd f)
mapd f ∘ mapdReduce ret = mapdReduce (ret ∘ cobind f)
    rewrite (mapdReduce_morphism).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
H: Monoid_Morphism (env E A) (env E B) (mapd f)
mapdReduce (mapd f ∘ ret) =
mapdReduce (ret ∘ cobind f)
    fequal.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
H: Monoid_Morphism (env E A) (env E B) (mapd f)
mapd f ∘ ret = ret ∘ cobind f now ext [e a].
    (* TODO ^ generalize this part *)
  Qed.

  Lemma map_toctxlist: forall `(f: A -> B),
      map f ∘ toctxlist (F := T) =
        toctxlist (F := T) ∘ map f.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : A -> B),
map f ∘ toctxlist = toctxlist ∘ map f
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : A -> B),
map f ∘ toctxlist = toctxlist ∘ map f
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
map f ∘ toctxlist = toctxlist ∘ map f
    rewrite toctxlist_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
map f ∘ mapdReduce ret = toctxlist ∘ map f
    rewrite toctxlist_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
map f ∘ mapdReduce ret = mapdReduce ret ∘ map f
    rewrite mapdReduce_map.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
map f ∘ mapdReduce ret = mapdReduce (ret ∘ map f)
    assert (Monoid_Morphism (env E A) (env E B) (map f)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
Monoid_Morphism (env E A) (env E B) (map f)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
H: Monoid_Morphism (env E A) (env E B) (map f)
map f ∘ mapdReduce ret = mapdReduce (ret ∘ map f)
    {E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
Monoid_Morphism (env E A) (env E B) (map f) unfold env at 1 2.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
Monoid_Morphism (list (E * A)) (list (E * B)) (map f) rewrite env_map_spec.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
Monoid_Morphism (list (E * A)) (list (E * B))
  (map (map f))
      typeclasses eauto. }E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
H: Monoid_Morphism (env E A) (env E B) (map f)
map f ∘ mapdReduce ret = mapdReduce (ret ∘ map f)
    rewrite (mapdReduce_morphism).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
H: Monoid_Morphism (env E A) (env E B) (map f)
mapdReduce (map f ∘ ret) = mapdReduce (ret ∘ map f)
    fequal.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
H: Monoid_Morphism (env E A) (env E B) (map f)
map f ∘ ret = ret ∘ map f
    rewrite env_map_spec.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
H: Monoid_Morphism (env E A) (env E B) (map f)
map (map f) ∘ ret = ret ∘ map f
    now rewrite (natural (ϕ := @ret list _) (A := E * A) (B := E * B)).
  Qed.

  (** ** Factoring <<mapdReduce>> through <<toctxlist>> *)
  (********************************************************************)
  Corollary mapdReduce_through_toctxlist `{Monoid M}:
    forall (A: Type) (f: E * A -> M),
      mapdReduce f = mapReduce (T := list) f ∘ toctxlist.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
forall (A : Type) (f : E * A -> M),
mapdReduce f = mapReduce f ∘ toctxlist
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
forall (A : Type) (f : E * A -> M),
mapdReduce f = mapReduce f ∘ toctxlist
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: E * A -> M
mapdReduce f = mapReduce f ∘ toctxlist
    rewrite toctxlist_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: E * A -> M
mapdReduce f = mapReduce f ∘ mapdReduce ret
    rewrite mapReduce_eq_mapReduce_list.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: E * A -> M
mapdReduce f = mapReduce_list f ∘ mapdReduce ret
    rewrite (mapdReduce_morphism (M1 := list (E * A)) (M2 := M)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: E * A -> M
mapdReduce f = mapdReduce (mapReduce_list f ∘ ret)
    rewrite mapReduce_list_ret.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
op: Monoid_op M
unit0: Monoid_unit M
H: Monoid M
A: Type
f: E * A -> M
mapdReduce f = mapdReduce f
    reflexivity.
  Qed.

  (** ** Relating <<tolist>> and <<toctxlist>> *)
  (********************************************************************)
  Lemma tolist_to_toctxlist: forall (A: Type),
      tolist (F := T) (Tolist := Tolist_Traverse) (A := A) =
        map (F := list) extract ∘ toctxlist.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, tolist = map extract ∘ toctxlist
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, tolist = map extract ∘ toctxlist
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
tolist = map extract ∘ toctxlist
    rewrite tolist_to_mapReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapReduce ret = map extract ∘ toctxlist
    rewrite mapReduce_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapdReduce (ret ∘ extract) = map extract ∘ toctxlist
    rewrite toctxlist_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapdReduce (ret ∘ extract) =
map extract ∘ mapdReduce ret
    rewrite (mapdReduce_morphism).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapdReduce (ret ∘ extract) =
mapdReduce (map extract ∘ ret)
    rewrite (natural (ϕ := @ret list _)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
Toctxlist_inst: ToCtxlist E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxlist_Mapdt0: Compat_ToCtxlist_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapdReduce (ret ∘ extract) =
mapdReduce (ret ∘ map extract)
    reflexivity.
  Qed.

End mapdt_toctxlist.

(** * <<toctxset>> and <<∈d>> *)
(**********************************************************************)
#[local] Instance ToCtxset_Mapdt
  `{Mapdt E T}: ToCtxset E T :=
  fun A => mapdReduce (ret (T := subset) (A := E * A)).

Class Compat_ToCtxset_Mapdt
  (E: Type)
  (T: Type -> Type)
  `{ToCtxset_inst: ToCtxset E T}
  `{Mapdt_inst: Mapdt E T}: Prop :=
  compat_toctxset_mapdt:
    ToCtxset_inst = @ToCtxset_Mapdt E T Mapdt_inst.

#[export] Instance Compat_ToCtxset_Mapdt_Self
  `{Mapdt_ET: Mapdt E T}:
  @Compat_ToCtxset_Mapdt E T ToCtxset_Mapdt Mapdt_ET
  := ltac:(reflexivity).

Lemma toctxset_to_mapdt
  `{ToCtxset_inst: ToCtxset E T}
  `{Mapdt_ET: Mapdt E T}
  `{! Compat_ToCtxset_Mapdt E T}:
  forall (A: Type),
    toctxset = mapdReduce (ret (T := subset) (A := E * A)).E: Type
T: Type -> Type
ToCtxset_inst: ToCtxset E T
Mapdt_ET: Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
forall A : Type, toctxset = mapdReduce ret
Proof.E: Type
T: Type -> Type
ToCtxset_inst: ToCtxset E T
Mapdt_ET: Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
forall A : Type, toctxset = mapdReduce ret
  intros.E: Type
T: Type -> Type
ToCtxset_inst: ToCtxset E T
Mapdt_ET: Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
toctxset = mapdReduce ret
  rewrite compat_toctxset_mapdt.E: Type
T: Type -> Type
ToCtxset_inst: ToCtxset E T
Mapdt_ET: Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
ToCtxset_Mapdt A = mapdReduce ret
  reflexivity.
Qed.

(** A <<tosubset>> that is compatible with <<traverse>>
    is compatible with the <<toctxset>> that is compatible with <<mapdt>>,
    if <<traverse>> is compatible with <<mapdt>> *)
#[export] Instance Compat_ToSubset_ToCtxset_Traverse
  `{Mapdt E T}
  `{Traverse T}
  `{ToSubset_T: ToSubset T}
  `{! Compat_Traverse_Mapdt E T}
  `{! Compat_ToSubset_Traverse T}
  `{! DecoratedTraversableFunctor E T}:
  Compat_ToSubset_ToCtxset E T (ToSubset_T := ToSubset_T).E: Type
T: Type -> Type
H: Mapdt E T
H0: Traverse T
ToSubset_T: ToSubset T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
Compat_ToSubset_ToCtxset E T
Proof.E: Type
T: Type -> Type
H: Mapdt E T
H0: Traverse T
ToSubset_T: ToSubset T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
Compat_ToSubset_ToCtxset E T
  hnf.E: Type
T: Type -> Type
H: Mapdt E T
H0: Traverse T
ToSubset_T: ToSubset T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
ToSubset_T = ToSubset_ToCtxset
  rewrite compat_tosubset_traverse.E: Type
T: Type -> Type
H: Mapdt E T
H0: Traverse T
ToSubset_T: ToSubset T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
ToSubset_Traverse = ToSubset_ToCtxset
  unfold_ops @ToSubset_Traverse.E: Type
T: Type -> Type
H: Mapdt E T
H0: Traverse T
ToSubset_T: ToSubset T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
(fun A : Type => mapReduce ret) = ToSubset_ToCtxset
  unfold ToSubset_ToCtxset.E: Type
T: Type -> Type
H: Mapdt E T
H0: Traverse T
ToSubset_T: ToSubset T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
(fun A : Type => mapReduce ret) =
(fun A : Type => map extract ∘ toctxset)
  unfold_ops @ToCtxset_Mapdt.E: Type
T: Type -> Type
H: Mapdt E T
H0: Traverse T
ToSubset_T: ToSubset T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
(fun A : Type => mapReduce ret) =
(fun A : Type => map extract ∘ mapdReduce ret)
  ext A.E: Type
T: Type -> Type
H: Mapdt E T
H0: Traverse T
ToSubset_T: ToSubset T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapReduce ret = map extract ∘ mapdReduce ret
  rewrite mapReduce_to_mapdReduce.E: Type
T: Type -> Type
H: Mapdt E T
H0: Traverse T
ToSubset_T: ToSubset T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapdReduce (ret ∘ extract) =
map extract ∘ mapdReduce ret
  rewrite mapdReduce_morphism.E: Type
T: Type -> Type
H: Mapdt E T
H0: Traverse T
ToSubset_T: ToSubset T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapdReduce (ret ∘ extract) =
mapdReduce (map extract ∘ ret)
  rewrite (natural (ϕ := @ret subset _)).E: Type
T: Type -> Type
H: Mapdt E T
H0: Traverse T
ToSubset_T: ToSubset T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapdReduce (ret ∘ extract) =
mapdReduce (ret ∘ map extract)
  reflexivity.
Qed.

Section mapdt_toctxset.

  Context
    {E: Type}
    {T: Type -> Type}
    `{Mapdt_inst: Mapdt E T}
    `{Mapd_inst: Mapd E T}
    `{Traverse_inst: Traverse T}
    `{Map_inst: Map T}
    `{ToCtxset_inst: ToCtxset E T}
    `{! Compat_Map_Mapdt E T}
    `{! Compat_Mapd_Mapdt E T}
    `{! Compat_Traverse_Mapdt E T}
    `{! Compat_ToCtxset_Mapdt E T}
    `{! DecoratedTraversableFunctor E T}.

  (** ** Rewriting <<toctxset_of>> and <<∈d>> to <<mapdReduce>> *)
  (********************************************************************)
  Lemma toctxset_to_mapdReduce: forall (A: Type),
      toctxset (F := T) (A := A) = mapdReduce (ret (T := subset)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, toctxset = mapdReduce ret
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, toctxset = mapdReduce ret
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
toctxset = mapdReduce ret
    rewrite toctxset_to_mapdt.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapdReduce ret = mapdReduce ret
    reflexivity.
  Qed.

  Corollary toctxset_to_mapdt1: forall (A: Type),
      toctxset (F := T) =
        mapdt (G := const (subset (E * A)))
          (B := False) (ret (T := subset)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, toctxset = mapdt ret
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, toctxset = mapdt ret
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
toctxset = mapdt ret
    rewrite toctxset_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapdReduce ret = mapdt ret
    reflexivity.
  Qed.

  Corollary toctxset_to_mapdt2: forall (A fake: Type),
      toctxset (F := T) =
        mapdt (G := const (subset (E * A)))
          (B := fake) (ret (T := subset)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A fake : Type, toctxset = mapdt ret
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A fake : Type, toctxset = mapdt ret
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, fake: Type
toctxset = mapdt ret
    rewrite toctxset_to_mapdt1.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, fake: Type
mapdt ret = mapdt ret
    rewrite (mapdt_constant_applicative1 (B := fake)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, fake: Type
mapdt ret = mapdt ret
    reflexivity.
  Qed.

  Lemma element_ctx_of_to_mapdReduce
    `{ToSubset T} `{! Compat_ToSubset_Traverse T}
    : forall (A: Type) (p: E * A),
      element_ctx_of (T := T) (A := A) p =
        mapdReduce (op := Monoid_op_or)
          (unit := Monoid_unit_false) {{p}}.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
forall (A : Type) (p : E * A),
element_ctx_of p = mapdReduce {{p}}
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
forall (A : Type) (p : E * A),
element_ctx_of p = mapdReduce {{p}}
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
p: E * A
element_ctx_of p = mapdReduce {{p}}
    rewrite element_ctx_of_toctxset.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
p: E * A
evalAt p ∘ toctxset = mapdReduce {{p}}
    rewrite toctxset_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
p: E * A
evalAt p ∘ mapdReduce ret = mapdReduce {{p}}
    rewrite mapdReduce_morphism.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
p: E * A
mapdReduce (evalAt p ∘ ret) = mapdReduce {{p}}
    unfold evalAt, compose.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
p: E * A
mapdReduce (fun a : E * A => ret a p) =
mapdReduce {{p}}
    now (fequal; ext [e' a']; propext; intuition).
  Qed.

  Lemma element_ctx_of_to_mapdReduce2
    `{ToSubset T} `{! Compat_ToSubset_Traverse T}
   : forall (A: Type),
      element_ctx_of (T := T) (A := A) =
        mapdReduce (op := Monoid_op_or)
          (unit := Monoid_unit_false) ∘ ret (T := subset).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
forall A : Type, element_ctx_of = mapdReduce ∘ ret
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
forall A : Type, element_ctx_of = mapdReduce ∘ ret
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
element_ctx_of = mapdReduce ∘ ret ext p.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
p: E * A
element_ctx_of p = (mapdReduce ∘ ret) p
    apply element_ctx_of_to_mapdReduce.
  Qed.

  (** ** Factoring <<toctxset_of>> through <<toctxlist>>/<<mapdReduce>> *)
  (********************************************************************)
  Lemma toctxset_through_toctxlist: forall (A: Type),
      toctxset (F := T) (A := A) =
        tosubset (F := list) ∘ toctxlist (F := T).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, toctxset = tosubset ∘ toctxlist
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, toctxset = tosubset ∘ toctxlist
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
toctxset = tosubset ∘ toctxlist
    rewrite toctxlist_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
toctxset = tosubset ∘ mapdReduce ret
    rewrite mapdReduce_morphism.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
toctxset = mapdReduce (tosubset ∘ ret)
    rewrite toctxset_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapdReduce ret = mapdReduce (tosubset ∘ ret)
    rewrite (Monad.kmon_hom_ret (ϕ := @tosubset list _)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
mapdReduce ret = mapdReduce ret
    reflexivity.
  Qed.

  Lemma tosubset_eq_toctxset_env: forall (A: Type),
      tosubset (F := list) (A := E * A) =
        toctxset (F := env E).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, tosubset = toctxset
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, tosubset = toctxset
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
tosubset = toctxset ext l.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
l: list (E * A)
tosubset l = toctxset l
    induction l.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
tosubset nil = toctxset nil
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
a: E * A
l: list (E * A)
IHl: tosubset l = toctxset l
tosubset (a :: l) = toctxset (a :: l)
    -E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
tosubset nil = toctxset nil reflexivity.
    -E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
a: E * A
l: list (E * A)
IHl: tosubset l = toctxset l
tosubset (a :: l) = toctxset (a :: l) simpl_list.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
a: E * A
l: list (E * A)
IHl: tosubset l = toctxset l
{{a}} ∪ tosubset l = toctxset (a :: l)
      destruct a as [e a].E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
e: E
a: A
l: list (E * A)
IHl: tosubset l = toctxset l
{{(e, a)}} ∪ tosubset l = toctxset ((e, a) :: l)
      cbn.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
e: E
a: A
l: list (E * A)
IHl: tosubset l = toctxset l
{{(e, a)}} ∪ tosubset l =
(pure cons ● map (pair e) (ret (e, a))) ● toctxset l
      unfold_ops @Pure_const.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
e: E
a: A
l: list (E * A)
IHl: tosubset l = toctxset l
{{(e, a)}} ∪ tosubset l =
(Ƶ ● map (pair e) (ret (e, a))) ● toctxset l
      rewrite monoid_id_l.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
e: E
a: A
l: list (E * A)
IHl: tosubset l = toctxset l
{{(e, a)}} ∪ tosubset l =
map (pair e) (ret (e, a)) ● toctxset l
      rewrite <- IHl.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
e: E
a: A
l: list (E * A)
IHl: tosubset l = toctxset l
{{(e, a)}} ∪ tosubset l =
map (pair e) (ret (e, a)) ● tosubset l
      reflexivity.
  Qed.

  Lemma toctxset_through_toctxlist2: forall (A: Type),
      toctxset (F := T) (A := A) =
        toctxset (F := env E) ∘ toctxlist (F := T).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, toctxset = toctxset ∘ toctxlist
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, toctxset = toctxset ∘ toctxlist
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
toctxset = toctxset ∘ toctxlist
    rewrite toctxset_through_toctxlist.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
tosubset ∘ toctxlist = toctxset ∘ toctxlist
    rewrite tosubset_eq_toctxset_env.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
toctxset ∘ toctxlist = toctxset ∘ toctxlist
    reflexivity.
  Qed.

  Lemma toctxset_through_mapdReduce: forall (A: Type),
      toctxset (F := T) (A := A) =
        tosubset ∘ mapdReduce (ret (T := list)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, toctxset = tosubset ∘ mapdReduce ret
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall A : Type, toctxset = tosubset ∘ mapdReduce ret
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
toctxset = tosubset ∘ mapdReduce ret
    apply toctxset_through_toctxlist.
  Qed.

  (** ** Fusion Laws for <<toctxset>> *)
  (********************************************************************)
  Lemma toctxset_mapd_fusion: forall `(f: E * A -> B),
      toctxset (F := T) ∘ mapd f =
        mapdReduce (ret (T := subset) ∘ cobind f).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : E * A -> B),
toctxset ∘ mapd f = mapdReduce (ret ∘ cobind f)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : E * A -> B),
toctxset ∘ mapd f = mapdReduce (ret ∘ cobind f)
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
toctxset ∘ mapd f = mapdReduce (ret ∘ cobind f)
    rewrite toctxset_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce ret ∘ mapd f = mapdReduce (ret ∘ cobind f)
    rewrite mapdReduce_mapd.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce (ret ∘ cobind f) =
mapdReduce (ret ∘ cobind f)
    reflexivity.
  Qed.

  Lemma toctxset_map_fusion: forall `(f: A -> B),
      toctxset (F := T) ∘ map f =
        mapdReduce (ret (T := subset) ∘ map (F := (E ×)) f).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : A -> B),
toctxset ∘ map f = mapdReduce (ret ∘ map f)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : A -> B),
toctxset ∘ map f = mapdReduce (ret ∘ map f)
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
toctxset ∘ map f = mapdReduce (ret ∘ map f)
    rewrite toctxset_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
mapdReduce ret ∘ map f = mapdReduce (ret ∘ map f)
    rewrite mapdReduce_map.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
mapdReduce (ret ∘ map f) = mapdReduce (ret ∘ map f)
    reflexivity.
  Qed.

  Lemma tosubset_mapd_fusion
    `{ToSubset T} `{! Compat_ToSubset_Traverse T}: forall `(f: E * A -> B),
      tosubset ∘ mapd f = mapdReduce (ret (T := subset) ∘ f).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
forall (A B : Type) (f : E * A -> B),
tosubset ∘ mapd f = mapdReduce (ret ∘ f)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
forall (A B : Type) (f : E * A -> B),
tosubset ∘ mapd f = mapdReduce (ret ∘ f)
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
f: E * A -> B
tosubset ∘ mapd f = mapdReduce (ret ∘ f)
    rewrite tosubset_to_mapReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
f: E * A -> B
mapReduce ret ∘ mapd f = mapdReduce (ret ∘ f)
    rewrite mapReduce_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
f: E * A -> B
mapdReduce (ret ∘ extract) ∘ mapd f =
mapdReduce (ret ∘ f)
    rewrite mapdReduce_mapd.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
f: E * A -> B
mapdReduce (ret ∘ extract ∘ cobind f) =
mapdReduce (ret ∘ f)
    reassociate -> on left.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
f: E * A -> B
mapdReduce (ret ∘ (extract ∘ cobind f)) =
mapdReduce (ret ∘ f)
    rewrite kcom_cobind0.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A, B: Type
f: E * A -> B
mapdReduce (ret ∘ f) = mapdReduce (ret ∘ f)
    reflexivity.
  Qed.

  (** ** Naturality for <<toctxset>> *)
  (********************************************************************)
  Instance DecoratedHom_ret_subst: (*TODO Move me *)
    DecoratedHom E (E ×) (ctxset E) (@ret subset _ ○ (E ×)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
DecoratedHom E (prod E) (ctxset E)
  (@ret subset Return_subset ○ prod E)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
DecoratedHom E (prod E) (ctxset E)
  (@ret subset Return_subset ○ prod E)
    constructor.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : E * A -> B),
mapd f ∘ ret = ret ∘ mapd f
    intros A B f.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapd f ∘ ret = ret ∘ mapd f
    ext [e a].E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
(mapd f ∘ ret) (e, a) = (ret ∘ mapd f) (e, a)
    unfold compose.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
mapd f (ret (e, a)) = ret (mapd f (e, a))
    unfold_ops @Return_subset.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
mapd f (fun b : E * A => (e, a) = b) =
(fun b : E * B => mapd f (e, a) = b)
    unfold_ops @Mapd_ctxset.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
(fun '(e0, b) =>
 exists a0 : A, (e, a) = (e0, a0) /\ f (e0, a0) = b) =
(fun b : E * B => mapd f (e, a) = b)
    unfold mapd, Mapd_Reader.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
(fun '(e0, b) =>
 exists a0 : A, (e, a) = (e0, a0) /\ f (e0, a0) = b) =
(fun b : E * B => cobind f (e, a) = b)
    ext [e' b].E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
e': E
b: B
(exists a0 : A, (e, a) = (e', a0) /\ f (e', a0) = b) =
(cobind f (e, a) = (e', b)) cbn.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
e': E
b: B
(exists a0 : A, (e, a) = (e', a0) /\ f (e', a0) = b) =
((e, f (e, a)) = (e', b)) propext.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
e': E
b: B
(exists a0 : A, (e, a) = (e', a0) /\ f (e', a0) = b) ->
(e, f (e, a)) = (e', b)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
e': E
b: B
(e, f (e, a)) = (e', b) ->
exists a0 : A, (e, a) = (e', a0) /\ f (e', a0) = b
    -E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
e': E
b: B
(exists a0 : A, (e, a) = (e', a0) /\ f (e', a0) = b) ->
(e, f (e, a)) = (e', b) intros [a'' [Heq Hf]].E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
e': E
b: B
a'': A
Heq: (e, a) = (e', a'')
Hf: f (e', a'') = b
(e, f (e, a)) = (e', b)
      inversion Heq.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
e': E
b: B
a'': A
Heq: (e, a) = (e', a'')
Hf: f (e', a'') = b
H0: e = e'
H1: a = a''
(e', f (e', a'')) = (e', b) rewrite Hf.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
e': E
b: B
a'': A
Heq: (e, a) = (e', a'')
Hf: f (e', a'') = b
H0: e = e'
H1: a = a''
(e', b) = (e', b)
      reflexivity.
    -E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
e': E
b: B
(e, f (e, a)) = (e', b) ->
exists a0 : A, (e, a) = (e', a0) /\ f (e', a0) = b intro H.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
e': E
b: B
H: (e, f (e, a)) = (e', b)
exists a0 : A, (e, a) = (e', a0) /\ f (e', a0) = b
      exists a.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
e: E
a: A
e': E
b: B
H: (e, f (e, a)) = (e', b)
(e, a) = (e', a) /\ f (e', a) = b now inversion H.
  Qed.

  Lemma toctxset_mapd: forall `(f: E * A -> B),
      toctxset (F := T) ∘ mapd f = mapd f ∘ toctxset.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : E * A -> B),
toctxset ∘ mapd f = mapd f ∘ toctxset
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : E * A -> B),
toctxset ∘ mapd f = mapd f ∘ toctxset
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
toctxset ∘ mapd f = mapd f ∘ toctxset
    rewrite toctxset_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce ret ∘ mapd f = mapd f ∘ toctxset
    rewrite toctxset_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce ret ∘ mapd f = mapd f ∘ mapdReduce ret
    rewrite mapdReduce_mapd.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce (ret ∘ cobind f) = mapd f ∘ mapdReduce ret
    rewrite mapdReduce_morphism.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce (ret ∘ cobind f) =
mapdReduce (mapd f ∘ ret)
    change (cobind f) with (mapd (T := (E ×)) f).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce (ret ∘ mapd f) = mapdReduce (mapd f ∘ ret)
    change (@ret subset _ (E * B))
      with ((@ret subset _ ○ (E ×)) B).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce
  ((@ret subset Return_subset ○ prod E) B ∘ mapd f) =
mapdReduce (mapd f ∘ ret)
    rewrite <- dhom_natural.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: E * A -> B
mapdReduce (mapd f ∘ ret) = mapdReduce (mapd f ∘ ret)
    reflexivity.
  Qed.

  Lemma toctxset_map: forall `(f: A -> B),
      toctxset (F := T) ∘ map f = map f ∘ toctxset.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : A -> B),
toctxset ∘ map f = map f ∘ toctxset
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : A -> B),
toctxset ∘ map f = map f ∘ toctxset
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
toctxset ∘ map f = map f ∘ toctxset
    rewrite ctxset_map_spec.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
toctxset ∘ map f = map (map f) ∘ toctxset
    rewrite toctxset_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
mapdReduce ret ∘ map f = map (map f) ∘ toctxset
    rewrite mapdReduce_map.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
mapdReduce (ret ∘ map f) = map (map f) ∘ toctxset
    rewrite toctxset_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
mapdReduce (ret ∘ map f) =
map (map f) ∘ mapdReduce ret
    rewrite mapdReduce_morphism.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
mapdReduce (ret ∘ map f) =
mapdReduce (map (map f) ∘ ret)
    rewrite (natural (ϕ := @ret subset _) (A := E * A) (B := E * B)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
mapdReduce (ret ∘ map f) =
mapdReduce (ret ∘ map (map f))
    reflexivity.
  Qed.

  (*
    Theorem ind_mapd_iff_core:
    forall `(f: E * A -> B),
    mapd f ∘ toctxset = toctxset ∘ mapd (T := T) f.
    Proof.
    intros.
    rewrite toctxset_through_toctxlist.
    rewrite toctxset_through_toctxlist.
    reassociate -> on right.
    change (list (prod ?E ?X)) with (env E X). (* hidden *)
    rewrite <- (mapd_toctxlist f).
    rewrite env_mapd_spec.
    reassociate <- on right.
    rewrite ctxset_mapd_spec.
    change (env ?E ?X) with (list (prod E X)). (* hidden *)
    unfold ctxset.
    rewrite <- (natural (ϕ := @tosubset list _)).
    reflexivity.
    Qed.
   *)

  #[export] Instance Natural_Elementd_Mapdt: Natural (@toctxset E T _).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
Natural (@toctxset E T ToCtxset_inst)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
Natural (@toctxset E T ToCtxset_inst)
    constructor;
      try typeclasses eauto.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A B : Type) (f : A -> B),
map f ∘ toctxset = toctxset ∘ map f
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
map f ∘ toctxset = toctxset ∘ map f rewrite toctxset_map.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A, B: Type
f: A -> B
map f ∘ toctxset = map f ∘ toctxset
    reflexivity.
  Qed.

  (** ** Relating <<tosubset>> and <<toctxset>> *)
  (********************************************************************)
  Lemma tosubset_to_toctxset
    `{ToSubset T} `{! Compat_ToSubset_Traverse T}: forall (A: Type),
      tosubset (F := T) (A := A) =
        map (F := subset) extract ∘ toctxset.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
forall A : Type, tosubset = map extract ∘ toctxset
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
forall A : Type, tosubset = map extract ∘ toctxset
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
tosubset = map extract ∘ toctxset
    rewrite tosubset_to_mapReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
mapReduce ret = map extract ∘ toctxset
    rewrite mapReduce_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
mapdReduce (ret ∘ extract) = map extract ∘ toctxset
    rewrite toctxset_to_mapdReduce.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
mapdReduce (ret ∘ extract) =
map extract ∘ mapdReduce ret
    rewrite mapdReduce_morphism.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
mapdReduce (ret ∘ extract) =
mapdReduce (map extract ∘ ret)
    rewrite (natural (ϕ := @ret subset _)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
H: ToSubset T
Compat_ToSubset_Traverse0: Compat_ToSubset_Traverse T
A: Type
mapdReduce (ret ∘ extract) =
mapdReduce (ret ∘ map extract)
    reflexivity.
  Qed.

  (** ** Characterizing <<∈d>> *)
  (********************************************************************)
  Lemma ind_iff_in_toctxlist1: forall (A: Type) (e: E) (a: A) (t: T A),
      (e, a) ∈d t <-> (e, a) ∈ (toctxlist t: list (E * A)).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A : Type) (e : E) (a : A) (t : T A),
(e, a) ∈d t <-> (e, a) ∈ (toctxlist t : list (E * A))
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A : Type) (e : E) (a : A) (t : T A),
(e, a) ∈d t <-> (e, a) ∈ (toctxlist t : list (E * A))
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
e: E
a: A
t: T A
(e, a) ∈d t <-> (e, a) ∈ (toctxlist t : list (E * A))
    unfold element_ctx_of.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
e: E
a: A
t: T A
toctxset t (e, a) <-> (e, a) ∈ toctxlist t
    rewrite toctxset_through_toctxlist.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
e: E
a: A
t: T A
(tosubset ∘ toctxlist) t (e, a) <->
(e, a) ∈ toctxlist t
    reflexivity.
  Qed.

  Lemma ind_iff_in_toctxlist2: forall (A: Type) (e: E) (a: A) (t: T A),
      (e, a) ∈d t <-> (e, a) ∈d toctxlist t.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A : Type) (e : E) (a : A) (t : T A),
(e, a) ∈d t <-> (e, a) ∈d toctxlist t
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
forall (A : Type) (e : E) (a : A) (t : T A),
(e, a) ∈d t <-> (e, a) ∈d toctxlist t
    intros.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
e: E
a: A
t: T A
(e, a) ∈d t <-> (e, a) ∈d toctxlist t
    unfold element_ctx_of.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
e: E
a: A
t: T A
toctxset t (e, a) <-> toctxset (toctxlist t) (e, a)
    rewrite <- tosubset_eq_toctxset_env.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
e: E
a: A
t: T A
toctxset t (e, a) <-> tosubset (toctxlist t) (e, a)
    rewrite toctxset_through_toctxlist.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
A: Type
e: E
a: A
t: T A
(tosubset ∘ toctxlist) t (e, a) <->
tosubset (toctxlist t) (e, a)
    reflexivity.
  Qed.

  (** ** Folding by Preordered Monoids *)
  (********************************************************************)
  Lemma mapdReduce_mono {M R op unit}
    `{@PreOrderedMonoid M R op unit}:
    forall `(f: E * A -> M) (g: E * A -> M)
      (t: T A),
      (forall e a, (e, a) ∈d t ->
              R (f (e, a)) (g (e, a))) ->
      R (mapdReduce f t) (mapdReduce g t).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
forall (A : Type) (f g : E * A -> M) (t : T A),
(forall (e : E) (a : A),
 (e, a) ∈d t -> R (f (e, a)) (g (e, a))) ->
R (mapdReduce f t) (mapdReduce g t)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
forall (A : Type) (f g : E * A -> M) (t : T A),
(forall (e : E) (a : A),
 (e, a) ∈d t -> R (f (e, a)) (g (e, a))) ->
R (mapdReduce f t) (mapdReduce g t)
    introv Hin.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
Hin: forall (e : E) (a : A),
(e, a) ∈d t -> R (f (e, a)) (g (e, a))
R (mapdReduce f t) (mapdReduce g t)
    setoid_rewrite ind_iff_in_toctxlist1 in Hin.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
Hin: forall (e : E) (a : A),
(e, a) ∈ toctxlist t -> R (f (e, a)) (g (e, a))
R (mapdReduce f t) (mapdReduce g t)
    rewrite mapdReduce_through_toctxlist.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
Hin: forall (e : E) (a : A),
(e, a) ∈ toctxlist t -> R (f (e, a)) (g (e, a))
R ((mapReduce f ∘ toctxlist) t) (mapdReduce g t)
    rewrite mapdReduce_through_toctxlist.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
Hin: forall (e : E) (a : A),
(e, a) ∈ toctxlist t -> R (f (e, a)) (g (e, a))
R ((mapReduce f ∘ toctxlist) t)
  ((mapReduce g ∘ toctxlist) t)
    unfold compose.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
Hin: forall (e : E) (a : A),
(e, a) ∈ toctxlist t -> R (f (e, a)) (g (e, a))
R (mapReduce f (toctxlist t))
  (mapReduce g (toctxlist t))
    induction (toctxlist t).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
Hin: forall (e : E) (a : A),
(e, a) ∈ nil -> R (f (e, a)) (g (e, a))
R (mapReduce f nil) (mapReduce g nil)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
a: E * A
e: list (E * A)
Hin: forall (e0 : E) (a0 : A),
(e0, a0) ∈ (a :: e) ->
R (f (e0, a0)) (g (e0, a0))
IHe: (forall (e0 : E) (a : A),
 (e0, a) ∈ e -> R (f (e0, a)) (g (e0, a))) ->
R (mapReduce f e) (mapReduce g e)
R (mapReduce f (a :: e)) (mapReduce g (a :: e))
    -E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
Hin: forall (e : E) (a : A),
(e, a) ∈ nil -> R (f (e, a)) (g (e, a))
R (mapReduce f nil) (mapReduce g nil) cbv.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
Hin: forall (e : E) (a : A),
(e, a) ∈ nil -> R (f (e, a)) (g (e, a))
R unit unit reflexivity.
    -E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
a: E * A
e: list (E * A)
Hin: forall (e0 : E) (a0 : A),
(e0, a0) ∈ (a :: e) ->
R (f (e0, a0)) (g (e0, a0))
IHe: (forall (e0 : E) (a : A),
 (e0, a) ∈ e -> R (f (e0, a)) (g (e0, a))) ->
R (mapReduce f e) (mapReduce g e)
R (mapReduce f (a :: e)) (mapReduce g (a :: e)) rename a into hd.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
hd: E * A
e: list (E * A)
Hin: forall (e0 : E) (a : A),
(e0, a) ∈ (hd :: e) -> R (f (e0, a)) (g (e0, a))
IHe: (forall (e0 : E) (a : A),
 (e0, a) ∈ e -> R (f (e0, a)) (g (e0, a))) ->
R (mapReduce f e) (mapReduce g e)
R (mapReduce f (hd :: e)) (mapReduce g (hd :: e))
      rename e into tl.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
hd: E * A
tl: list (E * A)
Hin: forall (e : E) (a : A),
(e, a) ∈ (hd :: tl) -> R (f (e, a)) (g (e, a))
IHe: (forall (e : E) (a : A),
 (e, a) ∈ tl -> R (f (e, a)) (g (e, a))) ->
R (mapReduce f tl) (mapReduce g tl)
R (mapReduce f (hd :: tl)) (mapReduce g (hd :: tl))
      destruct hd as [e a].E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
e: E
a: A
tl: list (E * A)
Hin: forall (e0 : E) (a0 : A),
(e0, a0) ∈ ((e, a) :: tl) ->
R (f (e0, a0)) (g (e0, a0))
IHe: (forall (e : E) (a : A),
 (e, a) ∈ tl -> R (f (e, a)) (g (e, a))) ->
R (mapReduce f tl) (mapReduce g tl)
R (mapReduce f ((e, a) :: tl))
  (mapReduce g ((e, a) :: tl))
      setoid_rewrite element_of_list_cons in Hin.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
e: E
a: A
tl: list (E * A)
Hin: forall (e0 : E) (a0 : A),
(e0, a0) = (e, a) \/ (e0, a0) ∈ tl ->
R (f (e0, a0)) (g (e0, a0))
IHe: (forall (e : E) (a : A),
 (e, a) ∈ tl -> R (f (e, a)) (g (e, a))) ->
R (mapReduce f tl) (mapReduce g tl)
R (mapReduce f ((e, a) :: tl))
  (mapReduce g ((e, a) :: tl))
      do 2 rewrite mapReduce_eq_mapReduce_list.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
e: E
a: A
tl: list (E * A)
Hin: forall (e0 : E) (a0 : A),
(e0, a0) = (e, a) \/ (e0, a0) ∈ tl ->
R (f (e0, a0)) (g (e0, a0))
IHe: (forall (e : E) (a : A),
 (e, a) ∈ tl -> R (f (e, a)) (g (e, a))) ->
R (mapReduce f tl) (mapReduce g tl)
R (mapReduce_list f ((e, a) :: tl))
  (mapReduce_list g ((e, a) :: tl))
      do 2 rewrite mapReduce_list_cons.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
e: E
a: A
tl: list (E * A)
Hin: forall (e0 : E) (a0 : A),
(e0, a0) = (e, a) \/ (e0, a0) ∈ tl ->
R (f (e0, a0)) (g (e0, a0))
IHe: (forall (e : E) (a : A),
 (e, a) ∈ tl -> R (f (e, a)) (g (e, a))) ->
R (mapReduce f tl) (mapReduce g tl)
R (f (e, a) ● mapReduce_list f tl)
  (g (e, a) ● mapReduce_list g tl)
      apply pompos_both.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
e: E
a: A
tl: list (E * A)
Hin: forall (e0 : E) (a0 : A),
(e0, a0) = (e, a) \/ (e0, a0) ∈ tl ->
R (f (e0, a0)) (g (e0, a0))
IHe: (forall (e : E) (a : A),
 (e, a) ∈ tl -> R (f (e, a)) (g (e, a))) ->
R (mapReduce f tl) (mapReduce g tl)
R (f (e, a)) (g (e, a))
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
e: E
a: A
tl: list (E * A)
Hin: forall (e0 : E) (a0 : A),
(e0, a0) = (e, a) \/ (e0, a0) ∈ tl ->
R (f (e0, a0)) (g (e0, a0))
IHe: (forall (e : E) (a : A),
 (e, a) ∈ tl -> R (f (e, a)) (g (e, a))) ->
R (mapReduce f tl) (mapReduce g tl)
R (mapReduce_list f tl) (mapReduce_list g tl)
      +E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
e: E
a: A
tl: list (E * A)
Hin: forall (e0 : E) (a0 : A),
(e0, a0) = (e, a) \/ (e0, a0) ∈ tl ->
R (f (e0, a0)) (g (e0, a0))
IHe: (forall (e : E) (a : A),
 (e, a) ∈ tl -> R (f (e, a)) (g (e, a))) ->
R (mapReduce f tl) (mapReduce g tl)
R (f (e, a)) (g (e, a)) auto.
      +E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
e: E
a: A
tl: list (E * A)
Hin: forall (e0 : E) (a0 : A),
(e0, a0) = (e, a) \/ (e0, a0) ∈ tl ->
R (f (e0, a0)) (g (e0, a0))
IHe: (forall (e : E) (a : A),
 (e, a) ∈ tl -> R (f (e, a)) (g (e, a))) ->
R (mapReduce f tl) (mapReduce g tl)
R (mapReduce_list f tl) (mapReduce_list g tl) do 2 rewrite <- mapReduce_eq_mapReduce_list.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
e: E
a: A
tl: list (E * A)
Hin: forall (e0 : E) (a0 : A),
(e0, a0) = (e, a) \/ (e0, a0) ∈ tl ->
R (f (e0, a0)) (g (e0, a0))
IHe: (forall (e : E) (a : A),
 (e, a) ∈ tl -> R (f (e, a)) (g (e, a))) ->
R (mapReduce f tl) (mapReduce g tl)
R (mapReduce f tl) (mapReduce g tl)
        apply IHe.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoid M R
A: Type
f, g: E * A -> M
t: T A
e: E
a: A
tl: list (E * A)
Hin: forall (e0 : E) (a0 : A),
(e0, a0) = (e, a) \/ (e0, a0) ∈ tl ->
R (f (e0, a0)) (g (e0, a0))
IHe: (forall (e : E) (a : A),
 (e, a) ∈ tl -> R (f (e, a)) (g (e, a))) ->
R (mapReduce f tl) (mapReduce g tl)
forall (e : E) (a : A),
(e, a) ∈ tl -> R (f (e, a)) (g (e, a)) intuition.
  Qed.

  Lemma mapdReduce_pompos {M R op unit}
    `{@PreOrderedMonoidPos M R op unit}:
    forall `(f: E * A -> M) (t: T A),
    forall e a, (e, a) ∈d t -> R (f (e, a)) (mapdReduce f t).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
forall (A : Type) (f : E * A -> M) (t : T A) (e : E)
  (a : A),
(e, a) ∈d t -> R (f (e, a)) (mapdReduce f t)
  Proof.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
forall (A : Type) (f : E * A -> M) (t : T A) (e : E)
  (a : A),
(e, a) ∈d t -> R (f (e, a)) (mapdReduce f t)
    introv Hin.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
Hin: (e, a) ∈d t
R (f (e, a)) (mapdReduce f t)
    rewrite ind_iff_in_toctxlist1 in Hin.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
Hin: (e, a) ∈ toctxlist t
R (f (e, a)) (mapdReduce f t)
    rewrite mapdReduce_through_toctxlist.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
Hin: (e, a) ∈ toctxlist t
R (f (e, a)) ((mapReduce f ∘ toctxlist) t)
    unfold compose.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
Hin: (e, a) ∈ toctxlist t
R (f (e, a)) (mapReduce f (toctxlist t))
    induction (toctxlist t).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
Hin: (e, a) ∈ nil
R (f (e, a)) (mapReduce f nil)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
a0: E * A
e0: list (E * A)
Hin: (e, a) ∈ (a0 :: e0)
IHe0: (e, a) ∈ e0 -> R (f (e, a)) (mapReduce f e0)
R (f (e, a)) (mapReduce f (a0 :: e0))
    -E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
Hin: (e, a) ∈ nil
R (f (e, a)) (mapReduce f nil) inversion Hin.
    -E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
a0: E * A
e0: list (E * A)
Hin: (e, a) ∈ (a0 :: e0)
IHe0: (e, a) ∈ e0 -> R (f (e, a)) (mapReduce f e0)
R (f (e, a)) (mapReduce f (a0 :: e0)) rename a0 into hd.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
hd: E * A
e0: list (E * A)
Hin: (e, a) ∈ (hd :: e0)
IHe0: (e, a) ∈ e0 -> R (f (e, a)) (mapReduce f e0)
R (f (e, a)) (mapReduce f (hd :: e0))
      rename e0 into tl.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
hd: E * A
tl: list (E * A)
Hin: (e, a) ∈ (hd :: tl)
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
R (f (e, a)) (mapReduce f (hd :: tl))
      destruct hd as [e' a'].E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
e': E
a': A
tl: list (E * A)
Hin: (e, a) ∈ ((e', a') :: tl)
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
R (f (e, a)) (mapReduce f ((e', a') :: tl))
      rewrite element_of_list_cons in Hin.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
e': E
a': A
tl: list (E * A)
Hin: (e, a) = (e', a') \/ (e, a) ∈ tl
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
R (f (e, a)) (mapReduce f ((e', a') :: tl))
      rewrite mapReduce_eq_mapReduce_list.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
e': E
a': A
tl: list (E * A)
Hin: (e, a) = (e', a') \/ (e, a) ∈ tl
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
R (f (e, a)) (mapReduce_list f ((e', a') :: tl))
      rewrite mapReduce_list_cons.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
e': E
a': A
tl: list (E * A)
Hin: (e, a) = (e', a') \/ (e, a) ∈ tl
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
R (f (e, a)) (f (e', a') ● mapReduce_list f tl)
      rewrite <- mapReduce_eq_mapReduce_list.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
e': E
a': A
tl: list (E * A)
Hin: (e, a) = (e', a') \/ (e, a) ∈ tl
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
R (f (e, a)) (f (e', a') ● mapReduce f tl)
      destruct Hin as [Hin | Hin].E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
e': E
a': A
tl: list (E * A)
Hin: (e, a) = (e', a')
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
R (f (e, a)) (f (e', a') ● mapReduce f tl)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
e': E
a': A
tl: list (E * A)
Hin: (e, a) ∈ tl
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
R (f (e, a)) (f (e', a') ● mapReduce f tl)
      +E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
e': E
a': A
tl: list (E * A)
Hin: (e, a) = (e', a')
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
R (f (e, a)) (f (e', a') ● mapReduce f tl) inversion Hin.E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
e': E
a': A
tl: list (E * A)
Hin: (e, a) = (e', a')
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
H1: e = e'
H2: a = a'
R (f (e', a')) (f (e', a') ● mapReduce f tl)
        apply pompos_incr_r.
      +E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
e': E
a': A
tl: list (E * A)
Hin: (e, a) ∈ tl
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
R (f (e, a)) (f (e', a') ● mapReduce f tl) transitivity (mapReduce f tl).E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
e': E
a': A
tl: list (E * A)
Hin: (e, a) ∈ tl
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
R (f (e, a)) (mapReduce f tl)
E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
e': E
a': A
tl: list (E * A)
Hin: (e, a) ∈ tl
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
R (mapReduce f tl) (f (e', a') ● mapReduce f tl)
        *E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
e': E
a': A
tl: list (E * A)
Hin: (e, a) ∈ tl
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
R (f (e, a)) (mapReduce f tl) auto.
        *E: Type
T: Type -> Type
Mapdt_inst: Mapdt E T
Mapd_inst: Mapd E T
Traverse_inst: Traverse T
Map_inst: Map T
ToCtxset_inst: ToCtxset E T
Compat_Map_Mapdt0: Compat_Map_Mapdt E T
Compat_Mapd_Mapdt0: Compat_Mapd_Mapdt E T
Compat_Traverse_Mapdt0: Compat_Traverse_Mapdt E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
DecoratedTraversableFunctor0: DecoratedTraversableFunctor
  E T
M: Type
R: relation M
op: Monoid_op M
unit: Monoid_unit M
H: PreOrderedMonoidPos M R
A: Type
f: E * A -> M
t: T A
e: E
a: A
e': E
a': A
tl: list (E * A)
Hin: (e, a) ∈ tl
IHe0: (e, a) ∈ tl -> R (f (e, a)) (mapReduce f tl)
R (mapReduce f tl) (f (e', a') ● mapReduce f tl) apply pompos_incr_l.
  Qed.

End mapdt_toctxset.

(** * Quantification with Context: <<Forall_ctx>> and <<Forany_ctx>> *)
(**********************************************************************)
Section quantification.

  Context
    `{DecoratedTraversableFunctor E T}
    `{ToCtxset E T}
    `{! Compat_ToCtxset_Mapdt E T}.

  Definition Forall_ctx `(P: E * A -> Prop): T A -> Prop :=
    @mapdReduce T E _ Prop Monoid_op_and Monoid_unit_true A P.

  Definition Forany_ctx `(P: E * A -> Prop): T A -> Prop :=
    @mapdReduce T E _ Prop Monoid_op_or Monoid_unit_false A P.

  Lemma forall_ctx_iff `(P: E * A -> Prop) (t: T A):
    Forall_ctx P t <-> forall (e: E) (a: A), (e, a) ∈d t -> P (e, a).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
Forall_ctx P t <->
(forall (e : E) (a : A), (e, a) ∈d t -> P (e, a))
  Proof.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
Forall_ctx P t <->
(forall (e : E) (a : A), (e, a) ∈d t -> P (e, a))
    unfold Forall_ctx.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
mapdReduce P t <->
(forall (e : E) (a : A), (e, a) ∈d t -> P (e, a))
    rewrite mapdReduce_through_toctxlist.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
(mapReduce P ∘ toctxlist) t <->
(forall (e : E) (a : A), (e, a) ∈d t -> P (e, a))
    setoid_rewrite ind_iff_in_toctxlist2.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
(mapReduce P ∘ toctxlist) t <->
(forall (e : E) (a : A),
 (e, a) ∈d toctxlist t -> P (e, a))
    unfold compose at 1.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
mapReduce P (toctxlist t) <->
(forall (e : E) (a : A),
 (e, a) ∈d toctxlist t -> P (e, a))
    induction (toctxlist t) as [|[e a] rest IHrest].E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
mapReduce P nil <->
(forall (e : E) (a : A), (e, a) ∈d nil -> P (e, a))
E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
IHrest: mapReduce P rest <->
(forall (e : E) (a : A),
 (e, a) ∈d rest -> P (e, a))
mapReduce P ((e, a) :: rest) <->
(forall (e0 : E) (a0 : A),
 (e0, a0) ∈d ((e, a) :: rest) -> P (e0, a0))
    -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
mapReduce P nil <->
(forall (e : E) (a : A), (e, a) ∈d nil -> P (e, a)) cbv.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
(True -> forall (e : E) (a : A), False -> P (e, a)) /\
((forall (e : E) (a : A), False -> P (e, a)) -> True) intuition.
    -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
IHrest: mapReduce P rest <->
(forall (e : E) (a : A),
 (e, a) ∈d rest -> P (e, a))
mapReduce P ((e, a) :: rest) <->
(forall (e0 : E) (a0 : A),
 (e0, a0) ∈d ((e, a) :: rest) -> P (e0, a0)) rewrite mapReduce_eq_mapReduce_list;
        simpl_list;
        rewrite <- mapReduce_eq_mapReduce_list.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
IHrest: mapReduce P rest <->
(forall (e : E) (a : A),
 (e, a) ∈d rest -> P (e, a))
P (e, a) ● mapReduce P rest <->
(forall (e0 : E) (a0 : A),
 (e0, a0) ∈d ((e, a) :: rest) -> P (e0, a0))
      rewrite IHrest; clear IHrest.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a)
● (forall (e : E) (a : A), (e, a) ∈d rest -> P (e, a)) <->
(forall (e0 : E) (a0 : A),
 (e0, a0) ∈d ((e, a) :: rest) -> P (e0, a0))
      unfold element_ctx_of.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a)
● (forall (e : E) (a : A),
   toctxset rest (e, a) -> P (e, a)) <->
(forall (e0 : E) (a0 : A),
 toctxset ((e, a) :: rest) (e0, a0) -> P (e0, a0))
      rewrite <- tosubset_eq_toctxset_env.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a)
● (forall (e : E) (a : A),
   tosubset rest (e, a) -> P (e, a)) <->
(forall (e0 : E) (a0 : A),
 tosubset ((e, a) :: rest) (e0, a0) -> P (e0, a0))
      rewrite tosubset_list_cons.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a)
● (forall (e : E) (a : A),
   tosubset rest (e, a) -> P (e, a)) <->
(forall (e0 : E) (a0 : A),
 ({{(e, a)}} ∪ tosubset rest) (e0, a0) -> P (e0, a0))
      change (tosubset ?t ?p) with (p ∈ t).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a)
● (forall (e : E) (a : A), (e, a) ∈ rest -> P (e, a)) <->
(forall (e0 : E) (a0 : A),
 ({{(e, a)}} ∪ tosubset rest) (e0, a0) -> P (e0, a0))
      setoid_rewrite element_of_list_cons.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a)
● (forall (e : E) (a : A), (e, a) ∈ rest -> P (e, a)) <->
(forall (e0 : E) (a0 : A),
 (e0, a0) = (e, a) \/ (e0, a0) ∈ rest -> P (e0, a0))
      setoid_rewrite pair_equal_spec.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a)
● (forall (e : E) (a : A), (e, a) ∈ rest -> P (e, a)) <->
(forall (e0 : E) (a0 : A),
 e0 = e /\ a0 = a \/ (e0, a0) ∈ rest -> P (e0, a0))
      unfold_all_transparent_tcs.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a) /\
(forall (e : E) (a : A), (e, a) ∈ rest -> P (e, a)) <->
(forall (e0 : E) (a0 : A),
 e0 = e /\ a0 = a \/ (e0, a0) ∈ rest -> P (e0, a0))
      intuition (subst; auto).
  Qed.

  Corollary forall_ctx_iff_eq `(P: E * A -> Prop) (t: T A):
    Forall_ctx P t = forall (e: E) (a: A), (e, a) ∈d t -> P (e, a).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
Forall_ctx P t =
(forall (e : E) (a : A), (e, a) ∈d t -> P (e, a))
  Proof.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
Forall_ctx P t =
(forall (e : E) (a : A), (e, a) ∈d t -> P (e, a))
    apply propositional_extensionality.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
Forall_ctx P t <->
(forall (e : E) (a : A), (e, a) ∈d t -> P (e, a))
    apply forall_ctx_iff.
  Qed.

  Lemma forany_ctx_iff `(P: E * A -> Prop) (t: T A):
    Forany_ctx P t <-> exists (e: E) (a: A), (e, a) ∈d t /\ P (e, a).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
Forany_ctx P t <->
(exists (e : E) (a : A), (e, a) ∈d t /\ P (e, a))
  Proof.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
Forany_ctx P t <->
(exists (e : E) (a : A), (e, a) ∈d t /\ P (e, a))
    unfold Forany_ctx.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
mapdReduce P t <->
(exists (e : E) (a : A), (e, a) ∈d t /\ P (e, a))
    rewrite mapdReduce_through_toctxlist.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
(mapReduce P ∘ toctxlist) t <->
(exists (e : E) (a : A), (e, a) ∈d t /\ P (e, a))
    setoid_rewrite ind_iff_in_toctxlist2.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
(mapReduce P ∘ toctxlist) t <->
(exists (e : E) (a : A),
   (e, a) ∈d toctxlist t /\ P (e, a))
    unfold compose at 1.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
mapReduce P (toctxlist t) <->
(exists (e : E) (a : A),
   (e, a) ∈d toctxlist t /\ P (e, a))
    induction (toctxlist t) as [|[e a] rest IHrest].E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
mapReduce P nil <->
(exists (e : E) (a : A), (e, a) ∈d nil /\ P (e, a))
E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
IHrest: mapReduce P rest <->
(exists (e : E) 
 (a : A), (e, a) ∈d rest /\ P (e, a))
mapReduce P ((e, a) :: rest) <->
(exists (e0 : E) 
 (a0 : A), (e0, a0) ∈d ((e, a) :: rest) /\ P (e0, a0))
    -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
mapReduce P nil <->
(exists (e : E) (a : A), (e, a) ∈d nil /\ P (e, a)) cbv.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
(False -> exists (e : E) (a : A), False /\ P (e, a)) /\
((exists (e : E) (a : A), False /\ P (e, a)) -> False) intuition.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
H2: exists (e : E) (a : A), False /\ P (e, a)
False
      firstorder.
    -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
IHrest: mapReduce P rest <->
(exists (e : E) 
 (a : A), (e, a) ∈d rest /\ P (e, a))
mapReduce P ((e, a) :: rest) <->
(exists (e0 : E) (a0 : A),
   (e0, a0) ∈d ((e, a) :: rest) /\ P (e0, a0)) rewrite mapReduce_eq_mapReduce_list;
        simpl_list;
        rewrite <- mapReduce_eq_mapReduce_list.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
IHrest: mapReduce P rest <->
(exists (e : E) 
 (a : A), (e, a) ∈d rest /\ P (e, a))
P (e, a) ● mapReduce P rest <->
(exists (e0 : E) (a0 : A),
   (e0, a0) ∈d ((e, a) :: rest) /\ P (e0, a0))
      rewrite IHrest; clear IHrest.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a)
● (exists (e : E) (a : A), (e, a) ∈d rest /\ P (e, a)) <->
(exists (e0 : E) (a0 : A),
   (e0, a0) ∈d ((e, a) :: rest) /\ P (e0, a0))
      unfold element_ctx_of.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a)
● (exists (e : E) (a : A),
     toctxset rest (e, a) /\ P (e, a)) <->
(exists (e0 : E) (a0 : A),
   toctxset ((e, a) :: rest) (e0, a0) /\ P (e0, a0))
      rewrite <- tosubset_eq_toctxset_env.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a)
● (exists (e : E) (a : A),
     tosubset rest (e, a) /\ P (e, a)) <->
(exists (e0 : E) (a0 : A),
   tosubset ((e, a) :: rest) (e0, a0) /\ P (e0, a0))
      rewrite tosubset_list_cons.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a)
● (exists (e : E) (a : A),
     tosubset rest (e, a) /\ P (e, a)) <->
(exists (e0 : E) (a0 : A),
   ({{(e, a)}} ∪ tosubset rest) (e0, a0) /\ P (e0, a0))
      change (tosubset ?t ?p) with (p ∈ t).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a)
● (exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a)) <->
(exists (e0 : E) (a0 : A),
   ({{(e, a)}} ∪ tosubset rest) (e0, a0) /\ P (e0, a0))
      setoid_rewrite element_of_list_cons.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a)
● (exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a)) <->
(exists (e0 : E) (a0 : A),
   ((e0, a0) = (e, a) \/ (e0, a0) ∈ rest) /\
   P (e0, a0))
      setoid_rewrite pair_equal_spec.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a)
● (exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a)) <->
(exists (e0 : E) (a0 : A),
   (e0 = e /\ a0 = a \/ (e0, a0) ∈ rest) /\ P (e0, a0))
      unfold_all_transparent_tcs.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a) \/
(exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a)) <->
(exists (e0 : E) (a0 : A),
   (e0 = e /\ a0 = a \/ (e0, a0) ∈ rest) /\ P (e0, a0))
      split.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a) \/
(exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a)) ->
exists (e0 : E) (a0 : A),
  (e0 = e /\ a0 = a \/ (e0, a0) ∈ rest) /\ P (e0, a0)
E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
(exists (e0 : E) 
 (a0 : A),
   (e0 = e /\ a0 = a \/ (e0, a0) ∈ rest) /\ P (e0, a0)) ->
P (e, a) \/
(exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a))
      {E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
P (e, a) \/
(exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a)) ->
exists (e0 : E) (a0 : A),
  (e0 = e /\ a0 = a \/ (e0, a0) ∈ rest) /\ P (e0, a0) intros [hyp | hyp].E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
hyp: P (e, a)
exists (e0 : E) (a0 : A),
  (e0 = e /\ a0 = a \/ (e0, a0) ∈ rest) /\ P (e0, a0)
E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
hyp: exists (e : E) (a : A),
  (e, a) ∈ rest /\ P (e, a)
exists (e0 : E) 
(a0 : A),
  (e0 = e /\ a0 = a \/ (e0, a0) ∈ rest) /\ P (e0, a0)
        -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
hyp: P (e, a)
exists (e0 : E) (a0 : A),
  (e0 = e /\ a0 = a \/ (e0, a0) ∈ rest) /\ P (e0, a0) exists e a.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
hyp: P (e, a)
(e = e /\ a = a \/ (e, a) ∈ rest) /\ P (e, a) split.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
hyp: P (e, a)
e = e /\ a = a \/ (e, a) ∈ rest
E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
hyp: P (e, a)
P (e, a)
          now left.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
hyp: P (e, a)
P (e, a) assumption.
        -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
hyp: exists (e : E) (a : A),
  (e, a) ∈ rest /\ P (e, a)
exists (e0 : E) (a0 : A),
  (e0 = e /\ a0 = a \/ (e0, a0) ∈ rest) /\ P (e0, a0) destruct hyp as [e' [a' [Hin HP]]].E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
e': E
a': A
Hin: (e', a') ∈ rest
HP: P (e', a')
exists (e0 : E) (a0 : A),
  (e0 = e /\ a0 = a \/ (e0, a0) ∈ rest) /\ P (e0, a0)
          exists e' a'.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
e': E
a': A
Hin: (e', a') ∈ rest
HP: P (e', a')
(e' = e /\ a' = a \/ (e', a') ∈ rest) /\ P (e', a') split.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
e': E
a': A
Hin: (e', a') ∈ rest
HP: P (e', a')
e' = e /\ a' = a \/ (e', a') ∈ rest
E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
e': E
a': A
Hin: (e', a') ∈ rest
HP: P (e', a')
P (e', a')
          now right.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
e': E
a': A
Hin: (e', a') ∈ rest
HP: P (e', a')
P (e', a') assumption.
      }E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
(exists (e0 : E) (a0 : A),
   (e0 = e /\ a0 = a \/ (e0, a0) ∈ rest) /\ P (e0, a0)) ->
P (e, a) \/
(exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a))
      {E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
(exists (e0 : E) (a0 : A),
   (e0 = e /\ a0 = a \/ (e0, a0) ∈ rest) /\ P (e0, a0)) ->
P (e, a) \/
(exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a)) intros [e' [a' [hyp1 hyp2]]].E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
e': E
a': A
hyp1: e' = e /\ a' = a \/ (e', a') ∈ rest
hyp2: P (e', a')
P (e, a) \/
(exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a))
        destruct hyp1 as [hyp1 | hyp1].E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
e': E
a': A
hyp1: e' = e /\ a' = a
hyp2: P (e', a')
P (e, a) \/
(exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a))
E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
e': E
a': A
hyp1: (e', a') ∈ rest
hyp2: P (e', a')
P (e, a) \/
(exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a))
        -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
e': E
a': A
hyp1: e' = e /\ a' = a
hyp2: P (e', a')
P (e, a) \/
(exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a)) left.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
e': E
a': A
hyp1: e' = e /\ a' = a
hyp2: P (e', a')
P (e, a) inversion hyp1; subst.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
hyp1: e = e /\ a = a
hyp2: P (e, a)
P (e, a)
          assumption.
        -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
e': E
a': A
hyp1: (e', a') ∈ rest
hyp2: P (e', a')
P (e, a) \/
(exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a)) right.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
e': E
a': A
hyp1: (e', a') ∈ rest
hyp2: P (e', a')
exists (e : E) (a : A), (e, a) ∈ rest /\ P (e, a) exists e' a'.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
e: E
a: A
rest: list (E * A)
e': E
a': A
hyp1: (e', a') ∈ rest
hyp2: P (e', a')
(e', a') ∈ rest /\ P (e', a') easy.
      }
  Qed.

  Corollary forany_ctx_iff_eq `(P: E * A -> Prop) (t: T A):
    Forany_ctx P t = exists (e: E) (a: A), (e, a) ∈d t /\ P (e, a).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
Forany_ctx P t =
(exists (e : E) (a : A), (e, a) ∈d t /\ P (e, a))
  Proof.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
Forany_ctx P t =
(exists (e : E) (a : A), (e, a) ∈d t /\ P (e, a))
    apply propositional_extensionality.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
t: T A
Forany_ctx P t <->
(exists (e : E) (a : A), (e, a) ∈d t /\ P (e, a))
    apply forany_ctx_iff.
  Qed.

  Lemma element_ctx_of_env_cons {A}: forall e a e' a' (rest: env E A),
      (e, a) ∈d ((e', a') :: rest) =
        ((e = e' /\ a = a') \/ (e, a) ∈d rest).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
forall (e : E) (a : A) (e' : E) (a' : A)
  (rest : env E A),
(e, a) ∈d ((e', a') :: rest) =
(e = e' /\ a = a' \/ (e, a) ∈d rest)
  Proof.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
forall (e : E) (a : A) (e' : E) (a' : A)
  (rest : env E A),
(e, a) ∈d ((e', a') :: rest) =
(e = e' /\ a = a' \/ (e, a) ∈d rest)
    intros.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
(e, a) ∈d ((e', a') :: rest) =
(e = e' /\ a = a' \/ (e, a) ∈d rest)
    unfold element_ctx_of.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
toctxset ((e', a') :: rest) (e, a) =
(e = e' /\ a = a' \/ toctxset rest (e, a))
    rewrite toctxset_to_mapdt.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
mapdReduce ret ((e', a') :: rest) (e, a) =
(e = e' /\ a = a' \/ mapdReduce ret rest (e, a))
    rewrite mapdReduce_to_mapdt1.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
mapdt ret ((e', a') :: rest) (e, a) =
(e = e' /\ a = a' \/ mapdt ret rest (e, a))
    cbn.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
((pure cons ● map (pair e') (ret (e', a')))
 ● mapdt ret rest) (e, a) =
(e = e' /\ a = a' \/ mapdt ret rest (e, a))
    unfold const at 1.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
((pure cons ● map (pair e') (ret (e', a')))
 ● mapdt ret rest) (e, a) =
(e = e' /\ a = a' \/ mapdt ret rest (e, a))
    simplify_applicative_const.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
((Ƶ ● map (pair e') (ret (e', a'))) ● mapdt ret rest)
  (e, a) = (e = e' /\ a = a' \/ mapdt ret rest (e, a))
    repeat simplify_monoid_subset.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
(∅ ● map (pair e') (ret (e', a')) ∪ mapdt ret rest)
  (e, a) = (e = e' /\ a = a' \/ mapdt ret rest (e, a))
    setoid_rewrite monoid_subset_rw.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
(∅ ∪ map (pair e') (ret (e', a')) ∪ mapdt ret rest)
  (e, a) = (e = e' /\ a = a' \/ mapdt ret rest (e, a))
    simplify_map_const.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
(∅ ∪ id (ret (e', a')) ∪ mapdt ret rest) (e, a) =
(e = e' /\ a = a' \/ mapdt ret rest (e, a))
    simpl_subset.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
((False \/ id (ret (e', a')) (e, a)) \/
 mapdt ret rest (e, a)) =
(e = e' /\ a = a' \/ mapdt ret rest (e, a))
    unfold_ops @Return_subset.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
((False \/ (e', a') = (e, a)) \/
 mapdt (fun a b : E * A => a = b) rest (e, a)) =
(e = e' /\ a = a' \/
 mapdt (fun a b : E * A => a = b) rest (e, a))
    propext.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
(False \/ (e', a') = (e, a)) \/
mapdt (fun a b : E * A => a = b) rest (e, a) ->
e = e' /\ a = a' \/
mapdt (fun a b : E * A => a = b) rest (e, a)
E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
e = e' /\ a = a' \/
mapdt (fun a b : E * A => a = b) rest (e, a) ->
(False \/ (e', a') = (e, a)) \/
mapdt (fun a b : E * A => a = b) rest (e, a)
    -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
(False \/ (e', a') = (e, a)) \/
mapdt (fun a b : E * A => a = b) rest (e, a) ->
e = e' /\ a = a' \/
mapdt (fun a b : E * A => a = b) rest (e, a) setoid_rewrite pair_equal_spec.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
(False \/ e' = e /\ a' = a) \/
mapdt (fun a b : E * A => a = b) rest (e, a) ->
e = e' /\ a = a' \/
mapdt (fun a b : E * A => a = b) rest (e, a) firstorder.
    -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
e = e' /\ a = a' \/
mapdt (fun a b : E * A => a = b) rest (e, a) ->
(False \/ (e', a') = (e, a)) \/
mapdt (fun a b : E * A => a = b) rest (e, a) setoid_rewrite pair_equal_spec.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
e: E
a: A
e': E
a': A
rest: env E A
e = e' /\ a = a' \/
mapdt (fun a b : E * A => a = b) rest (e, a) ->
(False \/ e' = e /\ a' = a) \/
mapdt (fun a b : E * A => a = b) rest (e, a) firstorder.
  Qed.

  (** ** Decidability of <<Forall_ctx>> and <<Forany_ctx>> *)
  (**********************************************************************)
  Lemma decidable_Forall_ctx `(P: E * A -> Prop) `(Dec: decidable_pred P):
    decidable_pred (Forall_ctx P).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
decidable_pred (Forall_ctx P)
  Proof.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
decidable_pred (Forall_ctx P)
    unfold decidable_pred.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
forall a : T A, decidable (Forall_ctx P a)
    intro t.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable (Forall_ctx P t)
    unfold Forall_ctx.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable (mapdReduce P t)
    rewrite mapdReduce_through_toctxlist.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable ((mapReduce P ∘ toctxlist) t)
    change (decidable (Forall (T := list) P (toctxlist t))).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable (Forall P (toctxlist t))
    apply decidable_Forall.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable_pred P
    assumption.
  Qed.

  Lemma decidable_Forany_ctx `(P: E * A -> Prop) (Dec: decidable_pred P):
    decidable_pred (Forany_ctx P).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
decidable_pred (Forany_ctx P)
  Proof.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
decidable_pred (Forany_ctx P)
    unfold decidable_pred.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
forall a : T A, decidable (Forany_ctx P a)
    intro t.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable (Forany_ctx P t)
    unfold Forany_ctx.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable (mapdReduce P t)
    rewrite mapdReduce_through_toctxlist.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable ((mapReduce P ∘ toctxlist) t)
    change (decidable (Forany (T := list) P (toctxlist t))).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable (Forany P (toctxlist t))
    apply decidable_Forany.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable_pred P
    assumption.
  Qed.

  Lemma decidable_Forall_element_ctx
    `(P: E * A -> Prop) `(Dec: decidable_pred P) (t: T A):
    decidable (forall (e: E) (a: A), (e, a) ∈d t -> P (e, a)).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable
  (forall (e : E) (a : A), (e, a) ∈d t -> P (e, a))
  Proof.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable
  (forall (e : E) (a : A), (e, a) ∈d t -> P (e, a))
    rewrite <- forall_ctx_iff_eq.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable (Forall_ctx P t)
    apply decidable_Forall_ctx.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable_pred P
    assumption.
  Qed.

  Lemma decidable_Forany_element_ctx
    `(P: E * A -> Prop) `(Dec: decidable_pred P) (t: T A):
    decidable (exists (e: E) (a: A), (e, a) ∈d t /\ P (e, a)).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable
  (exists (e : E) (a : A), (e, a) ∈d t /\ P (e, a))
  Proof.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable
  (exists (e : E) (a : A), (e, a) ∈d t /\ P (e, a))
    rewrite <- forany_ctx_iff_eq.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable (Forany_ctx P t)
    apply decidable_Forany_ctx.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
decidable_pred P
    assumption.
  Qed.

  Lemma not_Forall_ctx_Forany_ctx_lemma1
    `(P: E * A -> Prop) (Dec: decidable_pred P) (t: T A):
      ~ (Forall_ctx P t) -> Forany_ctx (not ∘ P) t.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
~ Forall_ctx P t -> Forany_ctx (not ∘ P) t
  Proof.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
~ Forall_ctx P t -> Forany_ctx (not ∘ P) t
    unfold Forall_ctx, Forany_ctx.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
~ mapdReduce P t -> mapdReduce (not ∘ P) t
    rewrite mapdReduce_through_toctxlist.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
~ (mapReduce P ∘ toctxlist) t ->
mapdReduce (not ∘ P) t
    rewrite mapdReduce_through_toctxlist.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
~ (mapReduce P ∘ toctxlist) t ->
(mapReduce (not ∘ P) ∘ toctxlist) t
    unfold compose.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
~ mapReduce P (toctxlist t) ->
mapReduce (not ○ P) (toctxlist t)
    induction (toctxlist t).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
~ mapReduce P nil -> mapReduce (not ○ P) nil
E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
a: E * A
e: list (E * A)
IHe: ~ mapReduce P e -> mapReduce (not ○ P) e
~ mapReduce P (a :: e) -> mapReduce (not ○ P) (a :: e)
    -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
~ mapReduce P nil -> mapReduce (not ○ P) nil cbv.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
(True -> False) -> False firstorder.
    -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
a: E * A
e: list (E * A)
IHe: ~ mapReduce P e -> mapReduce (not ○ P) e
~ mapReduce P (a :: e) -> mapReduce (not ○ P) (a :: e) do 2 rewrite mapReduce_eq_mapReduce_list in *.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
a: E * A
e: list (E * A)
IHe: ~ mapReduce_list P e ->
mapReduce_list (not ○ P) e
~ mapReduce_list P (a :: e) ->
mapReduce_list (not ○ P) (a :: e)
      simpl_list.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
a: E * A
e: list (E * A)
IHe: ~ mapReduce_list P e ->
mapReduce_list (not ○ P) e
~ P a ● mapReduce_list P e ->
(~ P a) ● mapReduce_list (not ○ P) e
      simplify_monoid_conjunction.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
a: E * A
e: list (E * A)
IHe: ~ mapReduce_list P e ->
mapReduce_list (not ○ P) e
~ (P a /\ mapReduce_list P e) ->
(~ P a) ● mapReduce_list (not ○ P) e
      simplify_monoid_disjunction.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
a: E * A
e: list (E * A)
IHe: ~ mapReduce_list P e ->
mapReduce_list (not ○ P) e
~ (P a /\ mapReduce_list P e) ->
~ P a \/ mapReduce_list (not ○ P) e
      firstorder.
  Qed.

  Lemma not_Forall_ctx_Forany_ctx
    `(P: E * A -> Prop) (Dec: decidable_pred P) (t: T A):
      ~ (Forall_ctx P t) <-> Forany_ctx (not ∘ P) t.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
~ Forall_ctx P t <-> Forany_ctx (not ∘ P) t
  Proof.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
~ Forall_ctx P t <-> Forany_ctx (not ∘ P) t
    unfold not at 2, compose at 1.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
~ Forall_ctx P t <->
Forany_ctx (fun a : E * A => P a -> False) t
    destruct (decidable_Forall_ctx P Dec t) as [YesAll | NotAll].E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
YesAll: Forall_ctx P t
~ Forall_ctx P t <->
Forany_ctx (fun a : E * A => P a -> False) t
E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
NotAll: ~ Forall_ctx P t
~ Forall_ctx P t <->
Forany_ctx (fun a : E * A => P a -> False) t
    -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
YesAll: Forall_ctx P t
~ Forall_ctx P t <->
Forany_ctx (fun a : E * A => P a -> False) t split.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
YesAll: Forall_ctx P t
~ Forall_ctx P t ->
Forany_ctx (fun a : E * A => P a -> False) t
E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
YesAll: Forall_ctx P t
Forany_ctx (fun a : E * A => P a -> False) t ->
~ Forall_ctx P t
      +E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
YesAll: Forall_ctx P t
~ Forall_ctx P t ->
Forany_ctx (fun a : E * A => P a -> False) t contradiction.
      +E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
YesAll: Forall_ctx P t
Forany_ctx (fun a : E * A => P a -> False) t ->
~ Forall_ctx P t rewrite forall_ctx_iff, forany_ctx_iff in *.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
YesAll: forall (e : E) (a : A),
(e, a) ∈d t -> P (e, a)
(exists (e : E) (a : A),
   (e, a) ∈d t /\ (P (e, a) -> False)) ->
~ (forall (e : E) (a : A), (e, a) ∈d t -> P (e, a))
        intros [e [a [Hin HP]]] _.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
YesAll: forall (e : E) (a : A),
(e, a) ∈d t -> P (e, a)
e: E
a: A
Hin: (e, a) ∈d t
HP: P (e, a) -> False
False
        intuition.
    -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
NotAll: ~ Forall_ctx P t
~ Forall_ctx P t <->
Forany_ctx (fun a : E * A => P a -> False) t split.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
NotAll: ~ Forall_ctx P t
~ Forall_ctx P t ->
Forany_ctx (fun a : E * A => P a -> False) t
E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
NotAll: ~ Forall_ctx P t
Forany_ctx (fun a : E * A => P a -> False) t ->
~ Forall_ctx P t
      +E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
NotAll: ~ Forall_ctx P t
~ Forall_ctx P t ->
Forany_ctx (fun a : E * A => P a -> False) t apply not_Forall_ctx_Forany_ctx_lemma1.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
NotAll: ~ Forall_ctx P t
decidable_pred P
        assumption.
      +E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
NotAll: ~ Forall_ctx P t
Forany_ctx (fun a : E * A => P a -> False) t ->
~ Forall_ctx P t easy.
  Qed.

  Lemma not_forall_ctx_iff `(P: E * A -> Prop) (Dec: decidable_pred P) (t: T A):
    ~ Forall_ctx P t <-> exists (e: E) (a: A), (e, a) ∈d t /\ ~ P (e, a).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
~ Forall_ctx P t <->
(exists (e : E) (a : A), (e, a) ∈d t /\ ~ P (e, a))
  Proof.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
~ Forall_ctx P t <->
(exists (e : E) (a : A), (e, a) ∈d t /\ ~ P (e, a))
    rewrite not_Forall_ctx_Forany_ctx; auto.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
Forany_ctx (not ∘ P) t <->
(exists (e : E) (a : A), (e, a) ∈d t /\ ~ P (e, a))
    rewrite forany_ctx_iff.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Dec: decidable_pred P
t: T A
(exists (e : E) (a : A),
   (e, a) ∈d t /\ (not ∘ P) (e, a)) <->
(exists (e : E) (a : A), (e, a) ∈d t /\ ~ P (e, a))
    reflexivity.
  Qed.

  (** ** Booleans *)
  (**********************************************************************)
  Definition Forall_ctx_b `(P: E * A -> bool): T A -> bool :=
    @mapdReduce T E _ bool Monoid_op_bool_and Monoid_unit_bool_true A P.

  Lemma decidable_Forall_ctx_b `(P: E * A -> Prop) `(Q: E * A -> bool)
    (Qspec: forall p, Q p = true <-> P p) (t: T A):
    Forall_ctx P t <-> Forall_ctx_b Q t = true.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
Forall_ctx P t <-> Forall_ctx_b Q t = true
  Proof.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
Forall_ctx P t <-> Forall_ctx_b Q t = true
    unfold Forall_ctx.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
mapdReduce P t <-> Forall_ctx_b Q t = true
    unfold Forall_ctx_b.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
mapdReduce P t <-> mapdReduce Q t = true
    rewrite mapdReduce_through_toctxlist.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
(mapReduce P ∘ toctxlist) t <-> mapdReduce Q t = true
    rewrite mapdReduce_through_toctxlist.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
(mapReduce P ∘ toctxlist) t <->
(mapReduce Q ∘ toctxlist) t = true
    unfold compose.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
mapReduce P (toctxlist t) <->
mapReduce Q (toctxlist t) = true
    induction (toctxlist t).E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
mapReduce P nil <-> mapReduce Q nil = true
E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce P e <-> mapReduce Q e = true
mapReduce P (a :: e) <-> mapReduce Q (a :: e) = true
    -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
mapReduce P nil <-> mapReduce Q nil = true cbv.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
(True -> true = true) /\ (true = true -> True) easy.
    -E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce P e <-> mapReduce Q e = true
mapReduce P (a :: e) <-> mapReduce Q (a :: e) = true do 2 rewrite mapReduce_eq_mapReduce_list in *.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce_list P e <-> mapReduce_list Q e = true
mapReduce_list P (a :: e) <->
mapReduce_list Q (a :: e) = true
      simpl_list.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce_list P e <-> mapReduce_list Q e = true
P a ● mapReduce_list P e <->
Q a ● mapReduce_list Q e = true
      repeat simplify_applicative_const.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce_list P e <-> mapReduce_list Q e = true
P a ● mapReduce_list P e <->
Q a ● mapReduce_list Q e = true
      repeat simplify_monoid_conjunction.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce_list P e <-> mapReduce_list Q e = true
P a /\ mapReduce_list P e <->
Q a ● mapReduce_list Q e = true
      unfold transparent tcs.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce_list P e <-> mapReduce_list Q e = true
P a /\ mapReduce_list P e <->
(Q a && mapReduce_list Q e)%bool = true
      split.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce_list P e <-> mapReduce_list Q e = true
P a /\ mapReduce_list P e ->
(Q a && mapReduce_list Q e)%bool = true
E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce_list P e <-> mapReduce_list Q e = true
(Q a && mapReduce_list Q e)%bool = true ->
P a /\ mapReduce_list P e
      +E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce_list P e <-> mapReduce_list Q e = true
P a /\ mapReduce_list P e ->
(Q a && mapReduce_list Q e)%bool = true rewrite IHe; clear IHe.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
P a /\ mapReduce_list Q e = true ->
(Q a && mapReduce_list Q e)%bool = true
        rewrite <- Qspec.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
Q a = true /\ mapReduce_list Q e = true ->
(Q a && mapReduce_list Q e)%bool = true
        intros [X Y]; rewrite X; rewrite Y.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
X: Q a = true
Y: mapReduce_list Q e = true
(true && true)%bool = true
        reflexivity.
      +E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce_list P e <-> mapReduce_list Q e = true
(Q a && mapReduce_list Q e)%bool = true ->
P a /\ mapReduce_list P e intro Hyp.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce_list P e <-> mapReduce_list Q e = true
Hyp: (Q a && mapReduce_list Q e)%bool = true
P a /\ mapReduce_list P e
        rewrite Bool.andb_true_iff in Hyp.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce_list P e <-> mapReduce_list Q e = true
Hyp: Q a = true /\ mapReduce_list Q e = true
P a /\ mapReduce_list P e
        rewrite IHe.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce_list P e <-> mapReduce_list Q e = true
Hyp: Q a = true /\ mapReduce_list Q e = true
P a /\ mapReduce_list Q e = true
        rewrite <- Qspec.E: Type
T: Type -> Type
H: Mapdt E T
H0: DecoratedTraversableFunctor E T
H1: ToCtxset E T
Compat_ToCtxset_Mapdt0: Compat_ToCtxset_Mapdt E T
A: Type
P: E * A -> Prop
Q: E * A -> bool
Qspec: forall p : E * A, Q p = true <-> P p
t: T A
a: E * A
e: list (E * A)
IHe: mapReduce_list P e <-> mapReduce_list Q e = true
Hyp: Q a = true /\ mapReduce_list Q e = true
Q a = true /\ mapReduce_list Q e = true
        assumption.
  Qed.

End quantification.