(* QuickChick update_permute. *)

Definition total_map := (partial_map * nat)%type.Definition t_empty v : total_map := (empty, v).Definition t_update (m : total_map) (x : id) (v : nat) :=
  let (pm,d) := m in
  (update (x, v) pm, d).Inductive t_bind : partial_map -> nat -> id -> nat -> Prop :=
  | T_BindDef   : forall x v, t_bind nil v x v
  | T_BindNow   : forall x a m d, t_bind (cons (x, a) m) d x a
  | T_BindLater : forall x x' a a' m' d,
                  ~ (x = x') ->
                  t_bind m' d x a -> 
                  t_bind (cons (x',a') m') d x a.Inductive t_bind : partial_map -> nat -> id -> nat -> Prop :=
  | T_BindDef   : forall x v, t_bind nil v x v
  | T_BindNow   : forall x a m d, t_bind (cons (x, a) m) d x a
  | T_BindLater : forall x x' a a' m' d,
                  ~ (x = x') ->
                  t_bind m' d x a -> 
                  t_bind (cons (x',a') m') d x a.Inductive t_bind : partial_map -> nat -> id -> nat -> Prop :=
  | T_BindDef   : forall x v, t_bind nil v x v
  | T_BindNow   : forall x a m d, t_bind (cons (x, a) m) d x a
  | T_BindLater : forall x x' a a' m' d,
                  ~ (x = x') ->
                  t_bind m' d x a -> 
                  t_bind (cons (x',a') m') d x a.Inductive t_bind : partial_map -> nat -> id -> nat -> Prop :=
  | T_BindDef   : forall x v, t_bind nil v x v
  | T_BindNow   : forall x a m d, t_bind (cons (x, a) m) d x a
  | T_BindLater : forall x x' a a' m' d,
                  ~ (x = x') ->
                  t_bind m' d x a -> 
                  t_bind (cons (x',a') m') d x a.Instance adm_st' d m a : SuchThatCorrect (fun x => t_bind m d x a) (genST (fun x => t_bind m d x a)).

(* Specifications: *)
Lemma t_apply_empty: forall x v v', t_bind empty v x v' -> v = v'.