Section Permutation.
Context {A : Type} (eqA : relation A) (e : Equivalence eqA).
Inductive PermutationA : list A -> list A -> Prop :=
| permA_nil: PermutationA nil nil
| permA_skip x₁ x₂ l₁ l₂ :
eqA x₁ x₂ -> PermutationA l₁ l₂ -> PermutationA (x₁ :: l₁) (x₂ :: l₂)
| permA_swap x y l : PermutationA (y :: x :: l) (x :: y :: l)
| permA_trans l₁ l₂ l₃ :
PermutationA l₁ l₂ -> PermutationA l₂ l₃ -> PermutationA l₁ l₃.
Local Hint Constructors PermutationA : core.
Global Instance: Equivalence PermutationA.
Global Instance PermutationA_cons :
Proper (eqA ==> PermutationA ==> PermutationA) (@cons A).
Global Instance PermutationA_app :
Proper (PermutationA ==> PermutationA ==> PermutationA) (@app A).

Lemma PermutationA_decompose l₁ l₂ :
PermutationA l₁ l₂ ->
exists l, Permutation l₁ l /\ eqlistA eqA l l₂.