(* 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.

Theorem permutation_one_inv : forall (a : A) (l : list A), permutation (a :: nil) l -> l = a :: nil.