(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below    *)
Require Import BuiltIn.
Require BuiltIn.
Require HighOrd.
Require int.Int.
Require map.Map.
Require bool.Bool.

(* Why3 assumption *)
Inductive datatype :=
  | Tint : datatype
  | Tbool : datatype.
Axiom datatype_WhyType : WhyType datatype.
Existing Instance datatype_WhyType.

(* Why3 assumption *)
Inductive operator :=
  | Oplus : operator
  | Ominus : operator
  | Omult : operator
  | Ole : operator.
Axiom operator_WhyType : WhyType operator.
Existing Instance operator_WhyType.

(* Why3 assumption *)
Definition ident := Numbers.BinNums.Z.

(* Why3 assumption *)
Inductive term :=
  | Tconst : Numbers.BinNums.Z -> term
  | Tvar : Numbers.BinNums.Z -> term
  | Tderef : Numbers.BinNums.Z -> term
  | Tbin : term -> operator -> term -> term.
Axiom term_WhyType : WhyType term.
Existing Instance term_WhyType.

(* Why3 assumption *)
Inductive fmla :=
  | Fterm : term -> fmla
  | Fand : fmla -> fmla -> fmla
  | Fnot : fmla -> fmla
  | Fimplies : fmla -> fmla -> fmla
  | Flet : Numbers.BinNums.Z -> term -> fmla -> fmla
  | Fforall : Numbers.BinNums.Z -> datatype -> fmla -> fmla.
Axiom fmla_WhyType : WhyType fmla.
Existing Instance fmla_WhyType.

(* Why3 assumption *)
Inductive value :=
  | Vint : Numbers.BinNums.Z -> value
  | Vbool : Init.Datatypes.bool -> value.
Axiom value_WhyType : WhyType value.
Existing Instance value_WhyType.

(* Why3 assumption *)
Definition env := Numbers.BinNums.Z -> value.

Parameter eval_bin: value -> operator -> value -> value.

Axiom eval_bin_def :
  forall (x:value) (op:operator) (y:value),
  match (x, y) with
  | (Vint x1, Vint y1) =>
      match op with
      | Oplus => ((eval_bin x op y) = (Vint (x1 + y1)%Z))
      | Ominus => ((eval_bin x op y) = (Vint (x1 - y1)%Z))
      | Omult => ((eval_bin x op y) = (Vint (x1 * y1)%Z))
      | Ole =>
          ((x1 <= y1)%Z -> ((eval_bin x op y) = (Vbool Init.Datatypes.true))) /\
          (~ (x1 <= y1)%Z ->
           ((eval_bin x op y) = (Vbool Init.Datatypes.false)))
      end
  | (_, _) => ((eval_bin x op y) = (Vbool Init.Datatypes.false))
  end.

(* Why3 assumption *)
Fixpoint eval_term (sigma:Numbers.BinNums.Z -> value)
  (pi:Numbers.BinNums.Z -> value) (t:term) {struct t}: value :=
  match t with
  | Tconst n => Vint n
  | Tvar id => pi id
  | Tderef id => sigma id
  | Tbin t1 op t2 =>
      eval_bin (eval_term sigma pi t1) op (eval_term sigma pi t2)
  end.

(* Why3 assumption *)
Fixpoint eval_fmla (sigma:Numbers.BinNums.Z -> value)
  (pi:Numbers.BinNums.Z -> value) (f:fmla) {struct f}: Prop :=
  match f with
  | Fterm t => ((eval_term sigma pi t) = (Vbool Init.Datatypes.true))
  | Fand f1 f2 => eval_fmla sigma pi f1 /\ eval_fmla sigma pi f2
  | Fnot f1 => ~ eval_fmla sigma pi f1
  | Fimplies f1 f2 => eval_fmla sigma pi f1 -> eval_fmla sigma pi f2
  | Flet x t f1 =>
      eval_fmla sigma (map.Map.set pi x (eval_term sigma pi t)) f1
  | Fforall x Tint f1 =>
      forall (n:Numbers.BinNums.Z),
      eval_fmla sigma (map.Map.set pi x (Vint n)) f1
  | Fforall x Tbool f1 =>
      forall (b:Init.Datatypes.bool),
      eval_fmla sigma (map.Map.set pi x (Vbool b)) f1
  end.

Parameter subst_term: term -> Numbers.BinNums.Z -> Numbers.BinNums.Z -> term.

Axiom subst_term_def :
  forall (e:term) (r:Numbers.BinNums.Z) (v:Numbers.BinNums.Z),
  match e with
  | Tconst _ => ((subst_term e r v) = e)
  | Tvar _ => ((subst_term e r v) = e)
  | Tderef x =>
      ((r = x) -> ((subst_term e r v) = (Tvar v))) /\
      (~ (r = x) -> ((subst_term e r v) = e))
  | Tbin e1 op e2 =>
      ((subst_term e r v) =
       (Tbin (subst_term e1 r v) op (subst_term e2 r v)))
  end.

(* Why3 assumption *)
Fixpoint fresh_in_term (id:Numbers.BinNums.Z) (t:term) {struct t}: Prop :=
  match t with
  | Tconst _ => True
  | Tvar v => ~ (id = v)
  | Tderef _ => True
  | Tbin t1 _ t2 => fresh_in_term id t1 /\ fresh_in_term id t2
  end.

Axiom eval_subst_term :
  forall (sigma:Numbers.BinNums.Z -> value) (pi:Numbers.BinNums.Z -> value)
    (e:term) (x:Numbers.BinNums.Z) (v:Numbers.BinNums.Z),
  fresh_in_term v e ->
  ((eval_term sigma pi (subst_term e x v)) =
   (eval_term (map.Map.set sigma x (pi v)) pi e)).

Axiom eval_term_change_free :
  forall (t:term) (sigma:Numbers.BinNums.Z -> value)
    (pi:Numbers.BinNums.Z -> value) (id:Numbers.BinNums.Z) (v:value),
  fresh_in_term id t ->
  ((eval_term sigma (map.Map.set pi id v) t) = (eval_term sigma pi t)).

(* Why3 assumption *)
Fixpoint fresh_in_fmla (id:Numbers.BinNums.Z) (f:fmla) {struct f}: Prop :=
  match f with
  | Fterm e => fresh_in_term id e
  | (Fand f1 f2)|(Fimplies f1 f2) =>
      fresh_in_fmla id f1 /\ fresh_in_fmla id f2
  | Fnot f1 => fresh_in_fmla id f1
  | Flet y t f1 => ~ (id = y) /\ fresh_in_term id t /\ fresh_in_fmla id f1
  | Fforall y _ f1 => ~ (id = y) /\ fresh_in_fmla id f1
  end.

(* Why3 assumption *)
Fixpoint subst (f:fmla) (x:Numbers.BinNums.Z)
  (v:Numbers.BinNums.Z) {struct f}: fmla :=
  match f with
  | Fterm e => Fterm (subst_term e x v)
  | Fand f1 f2 => Fand (subst f1 x v) (subst f2 x v)
  | Fnot f1 => Fnot (subst f1 x v)
  | Fimplies f1 f2 => Fimplies (subst f1 x v) (subst f2 x v)
  | Flet y t f1 => Flet y (subst_term t x v) (subst f1 x v)
  | Fforall y ty f1 => Fforall y ty (subst f1 x v)
  end.

Axiom eval_subst :
  forall (f:fmla) (sigma:Numbers.BinNums.Z -> value)
    (pi:Numbers.BinNums.Z -> value) (x:Numbers.BinNums.Z)
    (v:Numbers.BinNums.Z),
  fresh_in_fmla v f ->
  eval_fmla sigma pi (subst f x v) <->
  eval_fmla (map.Map.set sigma x (pi v)) pi f.

Axiom eval_swap :
  forall (f:fmla) (sigma:Numbers.BinNums.Z -> value)
    (pi:Numbers.BinNums.Z -> value) (id1:Numbers.BinNums.Z)
    (id2:Numbers.BinNums.Z) (v1:value) (v2:value),
  ~ (id1 = id2) ->
  eval_fmla sigma (map.Map.set (map.Map.set pi id1 v1) id2 v2) f <->
  eval_fmla sigma (map.Map.set (map.Map.set pi id2 v2) id1 v1) f.

Axiom eval_change_free :
  forall (f:fmla) (sigma:Numbers.BinNums.Z -> value)
    (pi:Numbers.BinNums.Z -> value) (id:Numbers.BinNums.Z) (v:value),
  fresh_in_fmla id f ->
  eval_fmla sigma (map.Map.set pi id v) f <-> eval_fmla sigma pi f.

(* Why3 assumption *)
Inductive stmt :=
  | Sskip : stmt
  | Sassign : Numbers.BinNums.Z -> term -> stmt
  | Sseq : stmt -> stmt -> stmt
  | Sif : term -> stmt -> stmt -> stmt
  | Sassert : fmla -> stmt
  | Swhile : term -> fmla -> stmt -> stmt.
Axiom stmt_WhyType : WhyType stmt.
Existing Instance stmt_WhyType.

Axiom check_skip : forall (s:stmt), (s = Sskip) \/ ~ (s = Sskip).

(* Why3 assumption *)
Inductive one_step: (Numbers.BinNums.Z -> value) ->
  (Numbers.BinNums.Z -> value) -> stmt -> (Numbers.BinNums.Z -> value) ->
  (Numbers.BinNums.Z -> value) -> stmt -> Prop :=
  | one_step_assign :
      forall (sigma:Numbers.BinNums.Z -> value)
        (pi:Numbers.BinNums.Z -> value) (x:Numbers.BinNums.Z) (e:term),
      one_step sigma pi (Sassign x e)
      (map.Map.set sigma x (eval_term sigma pi e)) pi Sskip
  | one_step_seq :
      forall (sigma:Numbers.BinNums.Z -> value)
        (pi:Numbers.BinNums.Z -> value) (sigma':Numbers.BinNums.Z -> value)
        (pi':Numbers.BinNums.Z -> value) (i1:stmt) (i1':stmt) (i2:stmt),
      one_step sigma pi i1 sigma' pi' i1' ->
      one_step sigma pi (Sseq i1 i2) sigma' pi' (Sseq i1' i2)
  | one_step_seq_skip :
      forall (sigma:Numbers.BinNums.Z -> value)
        (pi:Numbers.BinNums.Z -> value) (i:stmt),
      one_step sigma pi (Sseq Sskip i) sigma pi i
  | one_step_if_true :
      forall (sigma:Numbers.BinNums.Z -> value)
        (pi:Numbers.BinNums.Z -> value) (e:term) (i1:stmt) (i2:stmt),
      ((eval_term sigma pi e) = (Vbool Init.Datatypes.true)) ->
      one_step sigma pi (Sif e i1 i2) sigma pi i1
  | one_step_if_false :
      forall (sigma:Numbers.BinNums.Z -> value)
        (pi:Numbers.BinNums.Z -> value) (e:term) (i1:stmt) (i2:stmt),
      ((eval_term sigma pi e) = (Vbool Init.Datatypes.false)) ->
      one_step sigma pi (Sif e i1 i2) sigma pi i2
  | one_step_assert :
      forall (sigma:Numbers.BinNums.Z -> value)
        (pi:Numbers.BinNums.Z -> value) (f:fmla),
      eval_fmla sigma pi f -> one_step sigma pi (Sassert f) sigma pi Sskip
  | one_step_while_true :
      forall (sigma:Numbers.BinNums.Z -> value)
        (pi:Numbers.BinNums.Z -> value) (e:term) (inv:fmla) (i:stmt),
      eval_fmla sigma pi inv ->
      ((eval_term sigma pi e) = (Vbool Init.Datatypes.true)) ->
      one_step sigma pi (Swhile e inv i) sigma pi (Sseq i (Swhile e inv i))
  | one_step_while_false :
      forall (sigma:Numbers.BinNums.Z -> value)
        (pi:Numbers.BinNums.Z -> value) (e:term) (inv:fmla) (i:stmt),
      eval_fmla sigma pi inv ->
      ((eval_term sigma pi e) = (Vbool Init.Datatypes.false)) ->
      one_step sigma pi (Swhile e inv i) sigma pi Sskip.

(* Why3 assumption *)
Inductive many_steps: (Numbers.BinNums.Z -> value) ->
  (Numbers.BinNums.Z -> value) -> stmt -> (Numbers.BinNums.Z -> value) ->
  (Numbers.BinNums.Z -> value) -> stmt -> Numbers.BinNums.Z -> Prop :=
  | many_steps_refl :
      forall (sigma:Numbers.BinNums.Z -> value)
        (pi:Numbers.BinNums.Z -> value) (i:stmt),
      many_steps sigma pi i sigma pi i 0%Z
  | many_steps_trans :
      forall (sigma1:Numbers.BinNums.Z -> value)
        (pi1:Numbers.BinNums.Z -> value) (sigma2:Numbers.BinNums.Z -> value)
        (pi2:Numbers.BinNums.Z -> value) (sigma3:Numbers.BinNums.Z -> value)
        (pi3:Numbers.BinNums.Z -> value) (i1:stmt) (i2:stmt) (i3:stmt)
        (n:Numbers.BinNums.Z),
      one_step sigma1 pi1 i1 sigma2 pi2 i2 ->
      many_steps sigma2 pi2 i2 sigma3 pi3 i3 n ->
      many_steps sigma1 pi1 i1 sigma3 pi3 i3 (n + 1%Z)%Z.

Axiom steps_non_neg :
  forall (sigma1:Numbers.BinNums.Z -> value) (pi1:Numbers.BinNums.Z -> value)
    (sigma2:Numbers.BinNums.Z -> value) (pi2:Numbers.BinNums.Z -> value)
    (i1:stmt) (i2:stmt) (n:Numbers.BinNums.Z),
  many_steps sigma1 pi1 i1 sigma2 pi2 i2 n -> (0%Z <= n)%Z.

Axiom many_steps_seq :
  forall (sigma1:Numbers.BinNums.Z -> value) (pi1:Numbers.BinNums.Z -> value)
    (sigma3:Numbers.BinNums.Z -> value) (pi3:Numbers.BinNums.Z -> value)
    (i1:stmt) (i2:stmt) (n:Numbers.BinNums.Z),
  many_steps sigma1 pi1 (Sseq i1 i2) sigma3 pi3 Sskip n ->
  exists sigma2:Numbers.BinNums.Z -> value, exists pi2:
  Numbers.BinNums.Z -> value, exists n1:Numbers.BinNums.Z, exists n2:
  Numbers.BinNums.Z,
  many_steps sigma1 pi1 i1 sigma2 pi2 Sskip n1 /\
  many_steps sigma2 pi2 i2 sigma3 pi3 Sskip n2 /\ (n = ((1%Z + n1)%Z + n2)%Z).

(* Why3 assumption *)
Definition valid_fmla (p:fmla) : Prop :=
  forall (sigma:Numbers.BinNums.Z -> value) (pi:Numbers.BinNums.Z -> value),
  eval_fmla sigma pi p.

(* Why3 assumption *)
Definition valid_triple (p:fmla) (i:stmt) (q:fmla) : Prop :=
  forall (sigma:Numbers.BinNums.Z -> value) (pi:Numbers.BinNums.Z -> value),
  eval_fmla sigma pi p ->
  forall (sigma':Numbers.BinNums.Z -> value) (pi':Numbers.BinNums.Z -> value)
    (n:Numbers.BinNums.Z),
  many_steps sigma pi i sigma' pi' Sskip n -> eval_fmla sigma' pi' q.

Axiom fset : forall (a:Type), Type.
Parameter fset_WhyType : forall (a:Type) {a_WT:WhyType a}, WhyType (fset a).
Existing Instance fset_WhyType.

Parameter mem: forall {a:Type} {a_WT:WhyType a}, a -> fset a -> Prop.

(* Why3 assumption *)
Definition infix_eqeq {a:Type} {a_WT:WhyType a} (s1:fset a) (s2:fset a) :
    Prop :=
  forall (x:a), mem x s1 <-> mem x s2.

Axiom extensionality :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a), infix_eqeq s1 s2 -> (s1 = s2).

(* Why3 assumption *)
Definition subset {a:Type} {a_WT:WhyType a} (s1:fset a) (s2:fset a) : Prop :=
  forall (x:a), mem x s1 -> mem x s2.

Axiom subset_refl :
  forall {a:Type} {a_WT:WhyType a}, forall (s:fset a), subset s s.

Axiom subset_trans :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a) (s3:fset a), subset s1 s2 -> subset s2 s3 ->
  subset s1 s3.

(* Why3 assumption *)
Definition is_empty {a:Type} {a_WT:WhyType a} (s:fset a) : Prop :=
  forall (x:a), ~ mem x s.

Parameter empty: forall {a:Type} {a_WT:WhyType a}, fset a.

Axiom is_empty_empty :
  forall {a:Type} {a_WT:WhyType a}, is_empty (empty : fset a).

Axiom empty_is_empty :
  forall {a:Type} {a_WT:WhyType a},
  forall (s:fset a), is_empty s -> (s = (empty : fset a)).

Parameter add: forall {a:Type} {a_WT:WhyType a}, a -> fset a -> fset a.

Axiom add_def :
  forall {a:Type} {a_WT:WhyType a},
  forall (x:a) (s:fset a) (y:a), mem y (add x s) <-> mem y s \/ (y = x).

Axiom mem_singleton :
  forall {a:Type} {a_WT:WhyType a},
  forall (x:a) (y:a), mem y (add x (empty : fset a)) -> (y = x).

Parameter remove: forall {a:Type} {a_WT:WhyType a}, a -> fset a -> fset a.

Axiom remove_def :
  forall {a:Type} {a_WT:WhyType a},
  forall (x:a) (s:fset a) (y:a), mem y (remove x s) <-> mem y s /\ ~ (y = x).

Axiom add_remove :
  forall {a:Type} {a_WT:WhyType a},
  forall (x:a) (s:fset a), mem x s -> ((add x (remove x s)) = s).

Axiom remove_add :
  forall {a:Type} {a_WT:WhyType a},
  forall (x:a) (s:fset a), ((remove x (add x s)) = (remove x s)).

Axiom subset_remove :
  forall {a:Type} {a_WT:WhyType a},
  forall (x:a) (s:fset a), subset (remove x s) s.

Parameter union:
  forall {a:Type} {a_WT:WhyType a}, fset a -> fset a -> fset a.

Axiom union_def :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a) (x:a),
  mem x (union s1 s2) <-> mem x s1 \/ mem x s2.

Axiom subset_union_1 :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a), subset s1 (union s1 s2).

Axiom subset_union_2 :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a), subset s2 (union s1 s2).

Parameter inter:
  forall {a:Type} {a_WT:WhyType a}, fset a -> fset a -> fset a.

Axiom inter_def :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a) (x:a),
  mem x (inter s1 s2) <-> mem x s1 /\ mem x s2.

Axiom subset_inter_1 :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a), subset (inter s1 s2) s1.

Axiom subset_inter_2 :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a), subset (inter s1 s2) s2.

Parameter diff: forall {a:Type} {a_WT:WhyType a}, fset a -> fset a -> fset a.

Axiom diff_def :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a) (x:a),
  mem x (diff s1 s2) <-> mem x s1 /\ ~ mem x s2.

Axiom subset_diff :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a), subset (diff s1 s2) s1.

Parameter pick: forall {a:Type} {a_WT:WhyType a}, fset a -> a.

Axiom pick_def :
  forall {a:Type} {a_WT:WhyType a},
  forall (s:fset a), ~ is_empty s -> mem (pick s) s.

(* Why3 assumption *)
Definition disjoint {a:Type} {a_WT:WhyType a} (s1:fset a) (s2:fset a) : Prop :=
  forall (x:a), ~ mem x s1 \/ ~ mem x s2.

Axiom disjoint_inter_empty :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a), disjoint s1 s2 <-> is_empty (inter s1 s2).

Axiom disjoint_diff_eq :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a), disjoint s1 s2 <-> ((diff s1 s2) = s1).

Axiom disjoint_diff_s2 :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a), disjoint (diff s1 s2) s2.

Parameter filter:
  forall {a:Type} {a_WT:WhyType a}, fset a -> (a -> Init.Datatypes.bool) ->
  fset a.

Axiom filter_def :
  forall {a:Type} {a_WT:WhyType a},
  forall (s:fset a) (p:a -> Init.Datatypes.bool) (x:a),
  mem x (filter s p) <-> mem x s /\ ((p x) = Init.Datatypes.true).

Axiom subset_filter :
  forall {a:Type} {a_WT:WhyType a},
  forall (s:fset a) (p:a -> Init.Datatypes.bool), subset (filter s p) s.

Parameter map:
  forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b}, (a -> b) ->
  fset a -> fset b.

Axiom map_def :
  forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
  forall (f:a -> b) (u:fset a) (y:b),
  mem y (map f u) <-> (exists x:a, mem x u /\ (y = (f x))).

Axiom mem_map :
  forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
  forall (f:a -> b) (u:fset a), forall (x:a), mem x u -> mem (f x) (map f u).

Parameter cardinal:
  forall {a:Type} {a_WT:WhyType a}, fset a -> Numbers.BinNums.Z.

Axiom cardinal_nonneg :
  forall {a:Type} {a_WT:WhyType a},
  forall (s:fset a), (0%Z <= (cardinal s))%Z.

Axiom cardinal_empty :
  forall {a:Type} {a_WT:WhyType a},
  forall (s:fset a), is_empty s <-> ((cardinal s) = 0%Z).

Axiom cardinal_add :
  forall {a:Type} {a_WT:WhyType a},
  forall (x:a), forall (s:fset a),
  (mem x s -> ((cardinal (add x s)) = (cardinal s))) /\
  (~ mem x s -> ((cardinal (add x s)) = ((cardinal s) + 1%Z)%Z)).

Axiom cardinal_remove :
  forall {a:Type} {a_WT:WhyType a},
  forall (x:a), forall (s:fset a),
  (mem x s -> ((cardinal (remove x s)) = ((cardinal s) - 1%Z)%Z)) /\
  (~ mem x s -> ((cardinal (remove x s)) = (cardinal s))).

Axiom cardinal_subset :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a), subset s1 s2 ->
  ((cardinal s1) <= (cardinal s2))%Z.

Axiom subset_eq :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a), subset s1 s2 ->
  ((cardinal s1) = (cardinal s2)) -> (s1 = s2).

Axiom cardinal1 :
  forall {a:Type} {a_WT:WhyType a},
  forall (s:fset a), ((cardinal s) = 1%Z) -> forall (x:a), mem x s ->
  (x = (pick s)).

Axiom cardinal_union :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a),
  ((cardinal (union s1 s2)) =
   (((cardinal s1) + (cardinal s2))%Z - (cardinal (inter s1 s2)))%Z).

Axiom cardinal_inter_disjoint :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a), disjoint s1 s2 ->
  ((cardinal (inter s1 s2)) = 0%Z).

Axiom cardinal_diff :
  forall {a:Type} {a_WT:WhyType a},
  forall (s1:fset a) (s2:fset a),
  ((cardinal (diff s1 s2)) = ((cardinal s1) - (cardinal (inter s1 s2)))%Z).

Axiom cardinal_filter :
  forall {a:Type} {a_WT:WhyType a},
  forall (s:fset a) (p:a -> Init.Datatypes.bool),
  ((cardinal (filter s p)) <= (cardinal s))%Z.

Axiom cardinal_map :
  forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
  forall (f:a -> b) (s:fset a), ((cardinal (map f s)) <= (cardinal s))%Z.

Axiom set : Type.
Parameter set_WhyType : WhyType set.
Existing Instance set_WhyType.

Parameter to_fset: set -> fset Numbers.BinNums.Z.

Parameter choose: set -> Numbers.BinNums.Z.

Axiom choose_spec :
  forall (s:set), ~ is_empty (to_fset s) -> mem (choose s) (to_fset s).

(* Why3 assumption *)
Definition assigns (sigma:Numbers.BinNums.Z -> value)
    (a:fset Numbers.BinNums.Z) (sigma':Numbers.BinNums.Z -> value) : Prop :=
  forall (i:Numbers.BinNums.Z), ~ mem i a -> ((sigma i) = (sigma' i)).

Axiom assigns_refl :
  forall (sigma:Numbers.BinNums.Z -> value) (a:fset Numbers.BinNums.Z),
  assigns sigma a sigma.

Axiom assigns_trans :
  forall (sigma1:Numbers.BinNums.Z -> value)
    (sigma2:Numbers.BinNums.Z -> value) (sigma3:Numbers.BinNums.Z -> value)
    (a:fset Numbers.BinNums.Z),
  assigns sigma1 a sigma2 /\ assigns sigma2 a sigma3 ->
  assigns sigma1 a sigma3.

Axiom assigns_union_left :
  forall (sigma:Numbers.BinNums.Z -> value)
    (sigma':Numbers.BinNums.Z -> value) (s1:fset Numbers.BinNums.Z)
    (s2:fset Numbers.BinNums.Z),
  assigns sigma s1 sigma' -> assigns sigma (union s1 s2) sigma'.

Axiom assigns_union_right :
  forall (sigma:Numbers.BinNums.Z -> value)
    (sigma':Numbers.BinNums.Z -> value) (s1:fset Numbers.BinNums.Z)
    (s2:fset Numbers.BinNums.Z),
  assigns sigma s2 sigma' -> assigns sigma (union s1 s2) sigma'.

(* Why3 assumption *)
Fixpoint stmt_writes (i:stmt) (w:fset Numbers.BinNums.Z) {struct i}: Prop :=
  match i with
  | Sskip|(Sassert _) => True
  | Sassign id _ => mem id w
  | (Sseq s1 s2)|(Sif _ s1 s2) => stmt_writes s1 w /\ stmt_writes s2 w
  | Swhile _ _ s => stmt_writes s w
  end.

(* Why3 goal *)
Theorem VC_compute_writes :
  forall (s:stmt), forall (result:set),
  (exists x:term, exists x1:fmla, exists x2:stmt,
   (s = (Swhile x x1 x2)) /\
   (forall (sigma:Numbers.BinNums.Z -> value) (pi:Numbers.BinNums.Z -> value)
      (sigma':Numbers.BinNums.Z -> value) (pi':Numbers.BinNums.Z -> value)
      (n:Numbers.BinNums.Z),
    many_steps sigma pi x2 sigma' pi' Sskip n ->
    assigns sigma (to_fset result) sigma')) ->
  forall (sigma:Numbers.BinNums.Z -> value) (pi:Numbers.BinNums.Z -> value)
    (sigma':Numbers.BinNums.Z -> value) (pi':Numbers.BinNums.Z -> value)
    (n:Numbers.BinNums.Z),
  many_steps sigma pi s sigma' pi' Sskip n ->
  assigns sigma (to_fset result) sigma'.
Proof.
intros s result (x,(x1,(x2,(h1,h2)))); rewrite h1 in *. intros.
generalize sigma pi sigma' pi' H.
generalize (steps_non_neg _ _ _ _ _ _ _ H).
clear sigma pi sigma' pi' H.
intro H_n_pos.
pattern n.
apply Z_lt_induction; auto.
intros x3 Hind sigma pi sigma' qt' Hsteps.
inversion Hsteps; subst; clear Hsteps.
inversion H; subst; clear H.

generalize (many_steps_seq _ _ _ _ _ _ _ H0).
intros (sigma3&pi3&n1&n2&h3&h4&h5).
apply assigns_trans with sigma3; split.
eapply h2; eauto.
eapply Hind; eauto.
generalize (steps_non_neg _ _ _ _ _ _ _ h3).
generalize (steps_non_neg _ _ _ _ _ _ _ h4).
omega.

inversion H0; subst; clear H0.
apply assigns_refl.
inversion H.
Qed.