(* Definition of permutations as sequences of adjacent transpositions *) 
Inductive permutation : list A -> list A -> Prop :=
  | permutation_nil : permutation nil nil
  | permutation_skip :
      forall (a : A) (l1 l2 : list A),
      permutation l2 l1 -> permutation (a :: l2) (a :: l1)
  | permutation_swap :
      forall (a b : A) (l : list A), permutation (a :: b :: l) (b :: a :: l)
  | permutation_trans :
      forall l1 l2 l3 : list A,
      permutation l1 l2 -> permutation l2 l3 -> permutation l1 l3.(* Definition of permutations as sequences of adjacent transpositions *) 
Inductive permutation : list A -> list A -> Prop :=
  | permutation_nil : permutation nil nil
  | permutation_skip :
      forall (a : A) (l1 l2 : list A),
      permutation l2 l1 -> permutation (a :: l2) (a :: l1)
  | permutation_swap :
      forall (a b : A) (l : list A), permutation (a :: b :: l) (b :: a :: l)
  | permutation_trans :
      forall l1 l2 l3 : list A,
      permutation l1 l2 -> permutation l2 l3 -> permutation l1 l3.(* Definition of permutations as sequences of adjacent transpositions *) 
Inductive permutation : list A -> list A -> Prop :=
  | permutation_nil : permutation nil nil
  | permutation_skip :
      forall (a : A) (l1 l2 : list A),
      permutation l2 l1 -> permutation (a :: l2) (a :: l1)
  | permutation_swap :
      forall (a b : A) (l : list A), permutation (a :: b :: l) (b :: a :: l)
  | permutation_trans :
      forall l1 l2 l3 : list A,
      permutation l1 l2 -> permutation l2 l3 -> permutation l1 l3.(* Definition of permutations as sequences of adjacent transpositions *) 
Inductive permutation : list A -> list A -> Prop :=
  | permutation_nil : permutation nil nil
  | permutation_skip :
      forall (a : A) (l1 l2 : list A),
      permutation l2 l1 -> permutation (a :: l2) (a :: l1)
  | permutation_swap :
      forall (a b : A) (l : list A), permutation (a :: b :: l) (b :: a :: l)
  | permutation_trans :
      forall l1 l2 l3 : list A,
      permutation l1 l2 -> permutation l2 l3 -> permutation l1 l3.(* Definition of permutations as sequences of adjacent transpositions *) 
Inductive permutation : list A -> list A -> Prop :=
  | permutation_nil : permutation nil nil
  | permutation_skip :
      forall (a : A) (l1 l2 : list A),
      permutation l2 l1 -> permutation (a :: l2) (a :: l1)
  | permutation_swap :
      forall (a b : A) (l : list A), permutation (a :: b :: l) (b :: a :: l)
  | permutation_trans :
      forall l1 l2 l3 : list A,
      permutation l1 l2 -> permutation l2 l3 -> permutation l1 l3.

Lemma permutation_cons_ex_aux : forall (a : A) (l1 l2 : list A), permutation l1 l2 -> forall l11 l12 : list A, l1 = l11 ++ a :: l12 -> exists l3 : list A, (exists l4 : list A, l2 = l3 ++ a :: l4 /\ permutation (l11 ++ l12) (l3 ++ l4)).