(* 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.
Require list.List.
Require list.Length.
Require list.Mem.
Require list.Append.

(* Why3 assumption *)
Inductive datatype :=
  | TYunit : datatype
  | TYint : datatype
  | TYbool : datatype.
Axiom datatype_WhyType : WhyType datatype.
Existing Instance datatype_WhyType.

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

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

Axiom mident : Type.
Parameter mident_WhyType : WhyType mident.
Existing Instance mident_WhyType.

Axiom mident_decide :
  forall (m1:mident) (m2:mident), (m1 = m2) \/ ~ (m1 = m2).

Axiom ident : Type.
Parameter ident_WhyType : WhyType ident.
Existing Instance ident_WhyType.

Axiom ident_decide : forall (m1:ident) (m2:ident), (m1 = m2) \/ ~ (m1 = m2).

(* Why3 assumption *)
Inductive term :=
  | Tvalue : value -> term
  | Tvar : ident -> term
  | Tderef : mident -> 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 : ident -> term -> fmla -> fmla
  | Fforall : ident -> datatype -> fmla -> fmla.
Axiom fmla_WhyType : WhyType fmla.
Existing Instance fmla_WhyType.

(* Why3 assumption *)
Inductive stmt :=
  | Sskip : stmt
  | Sassign : mident -> 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 decide_is_skip : forall (s:stmt), (s = Sskip) \/ ~ (s = Sskip).

(* Why3 assumption *)
Definition env := mident -> value.

(* Why3 assumption *)
Definition stack := (list (ident* value)%type).

Parameter get_stack: ident -> (list (ident* value)%type) -> value.

Axiom get_stack_def :
  forall (i:ident) (pi:(list (ident* value)%type)),
  match pi with
  | Init.Datatypes.nil => ((get_stack i pi) = Vvoid)
  | (Init.Datatypes.cons (x, v) r) =>
      ((x = i) -> ((get_stack i pi) = v)) /\
      (~ (x = i) -> ((get_stack i pi) = (get_stack i r)))
  end.

Axiom get_stack_eq :
  forall (x:ident) (v:value) (r:(list (ident* value)%type)),
  ((get_stack x (Init.Datatypes.cons (x, v) r)) = v).

Axiom get_stack_neq :
  forall (x:ident) (i:ident) (v:value) (r:(list (ident* value)%type)),
  ~ (x = i) ->
  ((get_stack i (Init.Datatypes.cons (x, v) r)) = (get_stack i r)).

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 true))) /\
          (~ (x1 <= y1)%Z -> ((eval_bin x op y) = (Vbool false)))
      end
  | (_, _) => ((eval_bin x op y) = Vvoid)
  end.

(* Why3 assumption *)
Fixpoint eval_term (sigma:mident -> value) (pi:(list (ident* value)%type))
  (t:term) {struct t}: value :=
  match t with
  | Tvalue v => v
  | Tvar id => get_stack id pi
  | 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:mident -> value) (pi:(list (ident* value)%type))
  (f:fmla) {struct f}: Prop :=
  match f with
  | Fterm t => ((eval_term sigma pi t) = (Vbool 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 (Init.Datatypes.cons (x, eval_term sigma pi t) pi) f1
  | Fforall x TYint f1 =>
      forall (n:Z), eval_fmla sigma (Init.Datatypes.cons (x, Vint n) pi) f1
  | Fforall x TYbool f1 =>
      forall (b:bool),
      eval_fmla sigma (Init.Datatypes.cons (x, Vbool b) pi) f1
  | Fforall x TYunit f1 =>
      eval_fmla sigma (Init.Datatypes.cons (x, Vvoid) pi) f1
  end.

(* Why3 assumption *)
Definition valid_fmla (p:fmla) : Prop :=
  forall (sigma:mident -> value) (pi:(list (ident* value)%type)),
  eval_fmla sigma pi p.

(* Why3 assumption *)
Inductive one_step: (mident -> value) -> (list (ident* value)%type) ->
  stmt -> (mident -> value) -> (list (ident* value)%type) -> stmt -> Prop :=
  | one_step_assign :
      forall (sigma:mident -> value) (sigma':mident -> value)
        (pi:(list (ident* value)%type)) (x:mident) (t:term),
      (sigma' = (map.Map.set sigma x (eval_term sigma pi t))) ->
      one_step sigma pi (Sassign x t) sigma' pi Sskip
  | one_step_seq_noskip :
      forall (sigma:mident -> value) (sigma':mident -> value)
        (pi:(list (ident* value)%type)) (pi':(list (ident* value)%type))
        (s1:stmt) (s1':stmt) (s2:stmt),
      (one_step sigma pi s1 sigma' pi' s1') ->
      one_step sigma pi (Sseq s1 s2) sigma' pi' (Sseq s1' s2)
  | one_step_seq_skip :
      forall (sigma:mident -> value) (pi:(list (ident* value)%type)) (s:stmt),
      one_step sigma pi (Sseq Sskip s) sigma pi s
  | one_step_if_true :
      forall (sigma:mident -> value) (pi:(list (ident* value)%type)) 
        (t:term) (s1:stmt) (s2:stmt),
      ((eval_term sigma pi t) = (Vbool true)) ->
      one_step sigma pi (Sif t s1 s2) sigma pi s1
  | one_step_if_false :
      forall (sigma:mident -> value) (pi:(list (ident* value)%type)) 
        (t:term) (s1:stmt) (s2:stmt),
      ((eval_term sigma pi t) = (Vbool false)) ->
      one_step sigma pi (Sif t s1 s2) sigma pi s2
  | one_step_assert :
      forall (sigma:mident -> value) (pi:(list (ident* value)%type)) (f:fmla),
      (eval_fmla sigma pi f) -> one_step sigma pi (Sassert f) sigma pi Sskip
  | one_step_while_true :
      forall (sigma:mident -> value) (pi:(list (ident* value)%type))
        (cond:term) (inv:fmla) (body:stmt),
      ((eval_fmla sigma pi inv) /\
       ((eval_term sigma pi cond) = (Vbool true))) ->
      one_step sigma pi (Swhile cond inv body) sigma pi
      (Sseq body (Swhile cond inv body))
  | one_step_while_false :
      forall (sigma:mident -> value) (pi:(list (ident* value)%type))
        (cond:term) (inv:fmla) (body:stmt),
      ((eval_fmla sigma pi inv) /\
       ((eval_term sigma pi cond) = (Vbool false))) ->
      one_step sigma pi (Swhile cond inv body) sigma pi Sskip.

(* Why3 assumption *)
Inductive many_steps: (mident -> value) -> (list (ident* value)%type) ->
  stmt -> (mident -> value) -> (list (ident* value)%type) -> stmt -> 
  Z -> Prop :=
  | many_steps_refl :
      forall (sigma:mident -> value) (pi:(list (ident* value)%type)) (s:stmt),
      many_steps sigma pi s sigma pi s 0%Z
  | many_steps_trans :
      forall (sigma1:mident -> value) (sigma2:mident -> value)
        (sigma3:mident -> value) (pi1:(list (ident* value)%type))
        (pi2:(list (ident* value)%type)) (pi3:(list (ident* value)%type))
        (s1:stmt) (s2:stmt) (s3:stmt) (n:Z),
      (one_step sigma1 pi1 s1 sigma2 pi2 s2) ->
      (many_steps sigma2 pi2 s2 sigma3 pi3 s3 n) ->
      many_steps sigma1 pi1 s1 sigma3 pi3 s3 (n + 1%Z)%Z.

Axiom steps_non_neg :
  forall (sigma1:mident -> value) (sigma2:mident -> value)
    (pi1:(list (ident* value)%type)) (pi2:(list (ident* value)%type))
    (s1:stmt) (s2:stmt) (n:Z),
  (many_steps sigma1 pi1 s1 sigma2 pi2 s2 n) -> (0%Z <= n)%Z.

(* Why3 assumption *)
Definition reductible (sigma:mident -> value) (pi:(list (ident* value)%type))
    (s:stmt) : Prop :=
  exists sigma':mident -> value, exists pi':(list (ident* value)%type),
  exists s':stmt, one_step sigma pi s sigma' pi' s'.

(* Why3 assumption *)
Definition type_value (v:value) : datatype :=
  match v with
  | Vvoid => TYunit
  | Vint _ => TYint
  | Vbool _ => TYbool
  end.

(* Why3 assumption *)
Inductive type_operator: operator -> datatype -> datatype -> datatype ->
  Prop :=
  | Type_plus : type_operator Oplus TYint TYint TYint
  | Type_minus : type_operator Ominus TYint TYint TYint
  | Type_mult : type_operator Omult TYint TYint TYint
  | Type_le : type_operator Ole TYint TYint TYbool.

(* Why3 assumption *)
Definition type_stack := (list (ident* datatype)%type).

Parameter get_vartype: ident -> (list (ident* datatype)%type) -> datatype.

Axiom get_vartype_def :
  forall (i:ident) (pi:(list (ident* datatype)%type)),
  match pi with
  | Init.Datatypes.nil => ((get_vartype i pi) = TYunit)
  | (Init.Datatypes.cons (x, ty) r) =>
      ((x = i) -> ((get_vartype i pi) = ty)) /\
      (~ (x = i) -> ((get_vartype i pi) = (get_vartype i r)))
  end.

(* Why3 assumption *)
Definition type_env := mident -> datatype.

(* Why3 assumption *)
Inductive type_term: (mident -> datatype) -> (list (ident* datatype)%type) ->
  term -> datatype -> Prop :=
  | Type_value :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (v:value),
      type_term sigma pi (Tvalue v) (type_value v)
  | Type_var :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (v:ident) (ty:datatype),
      ((get_vartype v pi) = ty) -> type_term sigma pi (Tvar v) ty
  | Type_deref :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (v:mident) (ty:datatype),
      ((sigma v) = ty) -> type_term sigma pi (Tderef v) ty
  | Type_bin :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (t1:term) (t2:term) (op:operator) (ty1:datatype) (ty2:datatype)
        (ty:datatype),
      ((type_term sigma pi t1 ty1) /\
       ((type_term sigma pi t2 ty2) /\ (type_operator op ty1 ty2 ty))) ->
      type_term sigma pi (Tbin t1 op t2) ty.

(* Why3 assumption *)
Inductive type_fmla: (mident -> datatype) -> (list (ident* datatype)%type) ->
  fmla -> Prop :=
  | Type_term :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (t:term),
      (type_term sigma pi t TYbool) -> type_fmla sigma pi (Fterm t)
  | Type_conj :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (f1:fmla) (f2:fmla),
      ((type_fmla sigma pi f1) /\ (type_fmla sigma pi f2)) ->
      type_fmla sigma pi (Fand f1 f2)
  | Type_neg :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (f:fmla),
      (type_fmla sigma pi f) -> type_fmla sigma pi (Fnot f)
  | Type_implies :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (f1:fmla) (f2:fmla),
      (type_fmla sigma pi f1) -> (type_fmla sigma pi f2) ->
      type_fmla sigma pi (Fimplies f1 f2)
  | Type_let :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (x:ident) (t:term) (f:fmla) (ty:datatype),
      (type_term sigma pi t ty) ->
      (type_fmla sigma (Init.Datatypes.cons (x, ty) pi) f) ->
      type_fmla sigma pi (Flet x t f)
  | Type_forall :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (x:ident) (f:fmla) (ty:datatype),
      (type_fmla sigma (Init.Datatypes.cons (x, ty) pi) f) ->
      type_fmla sigma pi (Fforall x ty f).

(* Why3 assumption *)
Inductive type_stmt: (mident -> datatype) -> (list (ident* datatype)%type) ->
  stmt -> Prop :=
  | Type_skip :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type)),
      type_stmt sigma pi Sskip
  | Type_seq :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (s1:stmt) (s2:stmt),
      (type_stmt sigma pi s1) -> (type_stmt sigma pi s2) ->
      type_stmt sigma pi (Sseq s1 s2)
  | Type_assigns :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (x:mident) (t:term) (ty:datatype),
      ((sigma x) = ty) -> (type_term sigma pi t ty) ->
      type_stmt sigma pi (Sassign x t)
  | Type_if :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (t:term) (s1:stmt) (s2:stmt),
      (type_term sigma pi t TYbool) -> (type_stmt sigma pi s1) ->
      (type_stmt sigma pi s2) -> type_stmt sigma pi (Sif t s1 s2)
  | Type_assert :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (p:fmla),
      (type_fmla sigma pi p) -> type_stmt sigma pi (Sassert p)
  | Type_while :
      forall (sigma:mident -> datatype) (pi:(list (ident* datatype)%type))
        (cond:term) (body:stmt) (inv:fmla),
      (type_fmla sigma pi inv) -> (type_term sigma pi cond TYbool) ->
      (type_stmt sigma pi body) -> type_stmt sigma pi (Swhile cond inv body).

(* Why3 assumption *)
Definition compatible_env (sigma:mident -> value) (sigmat:mident -> datatype)
    (pi:(list (ident* value)%type)) (pit:(list (ident* datatype)%type)) :
    Prop :=
  (forall (id:mident), ((type_value (sigma id)) = (sigmat id))) /\
  forall (id:ident), ((type_value (get_stack id pi)) = (get_vartype id pit)).

Axiom type_inversion :
  forall (v:value),
  match type_value v with
  | TYbool => exists b:bool, (v = (Vbool b))
  | TYint => exists n:Z, (v = (Vint n))
  | TYunit => (v = Vvoid)
  end.

Axiom eval_type_term :
  forall (t:term) (sigma:mident -> value) (pi:(list (ident* value)%type))
    (sigmat:mident -> datatype) (pit:(list (ident* datatype)%type))
    (ty:datatype),
  (compatible_env sigma sigmat pi pit) -> (type_term sigmat pit t ty) ->
  ((type_value (eval_term sigma pi t)) = ty).

Axiom type_preservation :
  forall (s1:stmt) (s2:stmt) (sigma1:mident -> value)
    (sigma2:mident -> value) (pi1:(list (ident* value)%type))
    (pi2:(list (ident* value)%type)) (sigmat:mident -> datatype)
    (pit:(list (ident* datatype)%type)),
  (type_stmt sigmat pit s1) -> (compatible_env sigma1 sigmat pi1 pit) ->
  (one_step sigma1 pi1 s1 sigma2 pi2 s2) ->
  (type_stmt sigmat pit s2) /\ (compatible_env sigma2 sigmat pi2 pit).

Axiom Cons_append :
  forall {a:Type} {a_WT:WhyType a},
  forall (a1:a) (l1:(list a)) (l2:(list a)),
  ((Init.Datatypes.cons a1 (Init.Datatypes.app l1 l2)) =
   (Init.Datatypes.app (Init.Datatypes.cons a1 l1) l2)).

Axiom Append_nil_l :
  forall {a:Type} {a_WT:WhyType a},
  forall (l:(list a)), ((Init.Datatypes.app Init.Datatypes.nil l) = l).

Parameter msubst_term: term -> mident -> ident -> term.

Axiom msubst_term_def :
  forall (t:term) (x:mident) (v:ident),
  match t with
  | (Tvalue _)|(Tvar _) => ((msubst_term t x v) = t)
  | Tderef y =>
      ((x = y) -> ((msubst_term t x v) = (Tvar v))) /\
      (~ (x = y) -> ((msubst_term t x v) = t))
  | Tbin t1 op t2 =>
      ((msubst_term t x v) =
       (Tbin (msubst_term t1 x v) op (msubst_term t2 x v)))
  end.

(* Why3 assumption *)
Fixpoint msubst (f:fmla) (x:mident) (v:ident) {struct f}: fmla :=
  match f with
  | Fterm e => Fterm (msubst_term e x v)
  | Fand f1 f2 => Fand (msubst f1 x v) (msubst f2 x v)
  | Fnot f1 => Fnot (msubst f1 x v)
  | Fimplies f1 f2 => Fimplies (msubst f1 x v) (msubst f2 x v)
  | Flet y t f1 => Flet y (msubst_term t x v) (msubst f1 x v)
  | Fforall y ty f1 => Fforall y ty (msubst f1 x v)
  end.

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

(* Why3 assumption *)
Fixpoint fresh_in_fmla (id:ident) (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.

Axiom eval_msubst_term :
  forall (e:term) (sigma:mident -> value) (pi:(list (ident* value)%type))
    (x:mident) (v:ident),
  (fresh_in_term v e) ->
  ((eval_term sigma pi (msubst_term e x v)) =
   (eval_term (map.Map.set sigma x (get_stack v pi)) pi e)).

Axiom eval_msubst :
  forall (f:fmla) (sigma:mident -> value) (pi:(list (ident* value)%type))
    (x:mident) (v:ident),
  (fresh_in_fmla v f) ->
  (eval_fmla sigma pi (msubst f x v)) <->
  (eval_fmla (map.Map.set sigma x (get_stack v pi)) pi f).

Axiom eval_swap_term :
  forall (t:term) (sigma:mident -> value) (pi:(list (ident* value)%type))
    (l:(list (ident* value)%type)) (id1:ident) (id2:ident) (v1:value)
    (v2:value),
  ~ (id1 = id2) ->
  ((eval_term sigma
    (Init.Datatypes.app l
     (Init.Datatypes.cons (id1, v1) (Init.Datatypes.cons (id2, v2) pi)))
    t)
   =
   (eval_term sigma
    (Init.Datatypes.app l
     (Init.Datatypes.cons (id2, v2) (Init.Datatypes.cons (id1, v1) pi)))
    t)).

Axiom eval_swap_gen :
  forall (f:fmla) (sigma:mident -> value) (pi:(list (ident* value)%type))
    (l:(list (ident* value)%type)) (id1:ident) (id2:ident) (v1:value)
    (v2:value),
  ~ (id1 = id2) ->
  (eval_fmla sigma
   (Init.Datatypes.app l
    (Init.Datatypes.cons (id1, v1) (Init.Datatypes.cons (id2, v2) pi)))
   f) <->
  (eval_fmla sigma
   (Init.Datatypes.app l
    (Init.Datatypes.cons (id2, v2) (Init.Datatypes.cons (id1, v1) pi)))
   f).

Axiom eval_swap :
  forall (f:fmla) (sigma:mident -> value) (pi:(list (ident* value)%type))
    (id1:ident) (id2:ident) (v1:value) (v2:value),
  ~ (id1 = id2) ->
  (eval_fmla sigma
   (Init.Datatypes.cons (id1, v1) (Init.Datatypes.cons (id2, v2) pi)) f) <->
  (eval_fmla sigma
   (Init.Datatypes.cons (id2, v2) (Init.Datatypes.cons (id1, v1) pi)) f).

Axiom eval_term_change_free :
  forall (t:term) (sigma:mident -> value) (pi:(list (ident* value)%type))
    (id:ident) (v:value),
  (fresh_in_term id t) ->
  ((eval_term sigma (Init.Datatypes.cons (id, v) pi) t) =
   (eval_term sigma pi t)).

Axiom eval_change_free :
  forall (f:fmla) (sigma:mident -> value) (pi:(list (ident* value)%type))
    (id:ident) (v:value),
  (fresh_in_fmla id f) ->
  (eval_fmla sigma (Init.Datatypes.cons (id, v) pi) f) <->
  (eval_fmla sigma pi f).

Parameter fresh_from: fmla -> ident.

Axiom fresh_from_fmla : forall (f:fmla), fresh_in_fmla (fresh_from f) f.

Parameter abstract_effects: stmt -> fmla -> fmla.

Axiom abstract_effects_specialize :
  forall (sigma:mident -> value) (pi:(list (ident* value)%type)) (s:stmt)
    (f:fmla),
  (eval_fmla sigma pi (abstract_effects s f)) -> eval_fmla sigma pi f.

Axiom abstract_effects_distrib_conj :
  forall (s:stmt) (p:fmla) (q:fmla) (sigma:mident -> value)
    (pi:(list (ident* value)%type)),
  ((eval_fmla sigma pi (abstract_effects s p)) /\
   (eval_fmla sigma pi (abstract_effects s q))) ->
  eval_fmla sigma pi (abstract_effects s (Fand p q)).

Axiom abstract_effects_monotonic :
  forall (s:stmt) (p:fmla) (q:fmla), (valid_fmla (Fimplies p q)) ->
  forall (sigma:mident -> value) (pi:(list (ident* value)%type)),
  (eval_fmla sigma pi (abstract_effects s p)) ->
  eval_fmla sigma pi (abstract_effects s q).

(* Why3 assumption *)
Fixpoint wp (s:stmt) (q:fmla) {struct s}: fmla :=
  match s with
  | Sskip => q
  | Sassert f => Fand f (Fimplies f q)
  | Sseq s1 s2 => wp s1 (wp s2 q)
  | Sassign x t => let id := fresh_from q in Flet id t (msubst q x id)
  | Sif t s1 s2 =>
      Fand (Fimplies (Fterm t) (wp s1 q))
      (Fimplies (Fnot (Fterm t)) (wp s2 q))
  | Swhile cond inv body =>
      Fand inv
      (abstract_effects body
       (Fand (Fimplies (Fand (Fterm cond) inv) (wp body inv))
        (Fimplies (Fand (Fnot (Fterm cond)) inv) q)))
  end.

Axiom abstract_effects_writes :
  forall (sigma:mident -> value) (pi:(list (ident* value)%type)) (body:stmt)
    (cond:term) (inv:fmla) (q:fmla),
  let f :=
  abstract_effects body
  (Fand (Fimplies (Fand (Fterm cond) inv) (wp body inv))
   (Fimplies (Fand (Fnot (Fterm cond)) inv) q)) in
  (eval_fmla sigma pi f) -> eval_fmla sigma pi (wp body f).

Axiom monotonicity :
  forall (s:stmt) (p:fmla) (q:fmla), (valid_fmla (Fimplies p q)) ->
  valid_fmla (Fimplies (wp s p) (wp s q)).

(* Why3 goal *)
Theorem distrib_conj :
  forall (s:stmt), forall (x:term) (x1:fmla) (x2:stmt),
  (s = (Swhile x x1 x2)) ->
  (forall (sigma:mident -> value) (pi:(list (ident* value)%type)) (p:fmla)
     (q:fmla),
   ((eval_fmla sigma pi (wp x2 p)) /\ (eval_fmla sigma pi (wp x2 q))) ->
   eval_fmla sigma pi (wp x2 (Fand p q))) ->
  forall (sigma:mident -> value) (pi:(list (ident* value)%type)) (p:fmla)
    (q:fmla),
  ((eval_fmla sigma pi (wp s p)) /\ (eval_fmla sigma pi (wp s q))) ->
  eval_fmla sigma pi (wp s (Fand p q)).
(* Why3 intros s x x1 x2 h1 h2 sigma pi p q (h3,h4). *)
intros s x x1 x2 H_;rewrite H_ in *.
simpl.
intros H sigma pi p q ((h1 & h2) & (h3 & h4)).
split; auto.
apply abstract_effects_monotonic with (p:=
(Fand (Fand (Fimplies (Fand (Fterm x) x1) (wp x2 x1))
             (Fimplies (Fand (Fnot (Fterm x)) x1) p))
  (Fand (Fimplies (Fand (Fterm x) x1) (wp x2 x1))
             (Fimplies (Fand (Fnot (Fterm x)) x1) q)))).
unfold valid_fmla; simpl.
intuition.
apply abstract_effects_distrib_conj; auto.
Qed.

