blocking semantic

examples/hoare_logic/blocking_semantics3.mlw

@@ -359,35 +359,36 @@ lemma subst_fresh :
   forall f:fmla, x:ident, v:ident.
    fresh_in_fmla x f -> subst f x v = f
 
-lemma let_subst:
-    forall t:term, f:fmla, x id':ident, id :mident.
-    msubst (Flet x t f) id id' = Flet x (msubst_term t id id') (msubst f id id')
+(* Not needed *)
+(* lemma let_subst: *)
+(*     forall t:term, f:fmla, x id':ident, id :mident. *)
+(*     msubst (Flet x t f) id id' = Flet x (msubst_term t id id') (msubst f id id') *)
 
-(* Need it for monotonicity*)
+(* Need it for monotonicity and wp_reduction *)
 lemma eval_msubst:
   forall f:fmla, sigma:env, pi:stack, x:mident, v:ident.
     fresh_in_fmla v f ->
     (eval_fmla sigma pi (msubst f x v) <->
      eval_fmla (IdMap.set sigma x (get_stack v pi)) pi f)
 
-lemma eval_subst:
-  forall f:fmla, sigma:env, pi:stack, x:ident, v:ident.
-    fresh_in_fmla v f ->
-    (eval_fmla sigma pi (subst f x v) <->
-     eval_fmla sigma (Cons(x, (get_stack v pi)) pi) f)
+(* lemma eval_subst: *)
+(*   forall f:fmla, sigma:env, pi:stack, x:ident, v:ident. *)
+(*     fresh_in_fmla v f -> *)
+(*     (eval_fmla sigma pi (subst f x v) <-> *)
+(*      eval_fmla sigma (Cons(x, (get_stack v pi)) pi) f) *)
 
-lemma eval_swap:
-  forall f:fmla, sigma:env, pi:stack, id1 id2:ident, v1 v2:value.
-    id1 <> id2 ->
-    (eval_fmla sigma (Cons (id1,v1) (Cons (id2,v2) pi)) f <->
-    eval_fmla sigma (Cons (id2,v2) (Cons (id1,v1) pi)) f)
+(* lemma eval_swap: *)
+(*   forall f:fmla, sigma:env, pi:stack, id1 id2:ident, v1 v2:value. *)
+(*     id1 <> id2 -> *)
+(*     (eval_fmla sigma (Cons (id1,v1) (Cons (id2,v2) pi)) f <-> *)
+(*     eval_fmla sigma (Cons (id2,v2) (Cons (id1,v1) pi)) f) *)
 
-lemma eval_same_var:
-  forall f:fmla, sigma:env, pi:stack, id:ident, v1 v2:value.
-    eval_fmla sigma (Cons (id,v1) (Cons (id,v2) pi)) f <->
-    eval_fmla sigma (Cons (id,v1) pi) f
+(* lemma eval_same_var: *)
+(*   forall f:fmla, sigma:env, pi:stack, id:ident, v1 v2:value. *)
+(*     eval_fmla sigma (Cons (id,v1) (Cons (id,v2) pi)) f <-> *)
+(*     eval_fmla sigma (Cons (id,v1) pi) f *)
 
-  (* Need it for monotonicity*)
+ (* Need it for monotonicity*)
 lemma eval_change_free :
   forall f:fmla, sigma:env, pi:stack, id:ident, v:value.
     fresh_in_fmla id f ->
@@ -397,25 +398,26 @@ lemma eval_change_free :
   predicate valid_fmla (p:fmla) = forall sigma:env, pi:stack. eval_fmla sigma pi p
 
 (* Not needed *)
-axiom msubst_implies :
-forall p q:fmla.
-  valid_fmla (Fimplies p q) ->
-  forall sigma:env, pi:stack, x:mident, id:ident.
-    fresh_in_fmla id (Fand p q) -> 
-    eval_fmla sigma (Cons (id, (get_env x sigma)) pi) (Fimplies (msubst p x id) (msubst q x id)) 
+(* axiom msubst_implies : *)
+(* forall p q:fmla. *)
+(*   valid_fmla (Fimplies p q) -> *)
+(*   forall sigma:env, pi:stack, x:mident, id:ident. *)
+(*     fresh_in_fmla id (Fand p q) ->  *)
+(*     eval_fmla sigma (Cons (id, (get_env x sigma)) pi) (Fimplies (msubst p x id) (msubst q x id))  *)
 
 (** let id' = t in f[id <- id'] <=> let id = t in f*)
-lemma let_equiv :
-  forall id:ident, id':ident, t:term, f:fmla.
-    forall sigma:env, pi:stack.
-      fresh_in_fmla id' f ->
-	eval_fmla sigma pi (Flet id' t (subst f id id'))
-	 -> eval_fmla sigma pi (Flet id t f)
-
-lemma let_implies :
-  forall id:ident, t:term, p q:fmla.
-    valid_fmla (Fimplies p q) ->
-    valid_fmla (Fimplies (Flet id t p) (Flet id t q))
+(* Not needed *)
+(* lemma let_equiv : *)
+(*   forall id:ident, id':ident, t:term, f:fmla. *)
+(*     forall sigma:env, pi:stack. *)
+(*       fresh_in_fmla id' f -> *)
+(* 	eval_fmla sigma pi (Flet id' t (subst f id id')) *)
+(* 	 -> eval_fmla sigma pi (Flet id t f) *)
+
+(* lemma let_implies : *)
+(*   forall id:ident, t:term, p q:fmla. *)
+(*     valid_fmla (Fimplies p q) -> *)
+(*     valid_fmla (Fimplies (Flet id t p) (Flet id t q)) *)
 
 predicate fresh_in_stmt (id:ident) (s:stmt) =
   match s with
@@ -488,10 +490,12 @@ inductive one_step env stack stmt env stack stmt =
        many_steps sigma2 pi2 s2 sigma3 pi3 s3 n ->
        many_steps sigma1 pi1 s1 sigma3 pi3 s3 (n+1)
 
-  lemma steps_non_neg:
-    forall sigma1 sigma2:env, pi1 pi2:stack, s1 s2:stmt, n:int.
-      many_steps sigma1 pi1 s1 sigma2 pi2 s2 n -> n >= 0
+(* Not neede *)
+(* lemma steps_non_neg: *)
+(*   forall sigma1 sigma2:env, pi1 pi2:stack, s1 s2:stmt, n:int. *)
+(*     many_steps sigma1 pi1 s1 sigma2 pi2 s2 n -> n >= 0 *)
 
+(* Used by Hoare_logic/seq_rule*)
   lemma many_steps_seq:
     forall sigma1 sigma3:env, pi1 pi3:stack, s1 s2:stmt, n:int.
       many_steps sigma1 pi1 (Sseq s1 s2) sigma3 pi3 Sskip n ->
@@ -500,12 +504,11 @@ inductive one_step env stack stmt env stack stmt =
         many_steps sigma2 pi2 s2 sigma3 pi3 Sskip n2 /\
         n = 1 + n1 + n2
 
- lemma one_step_change_free :
-  forall s s':stmt, sigma sigma':env, pi pi':stack, id:ident, v:value.
-    fresh_in_stmt id s ->
-    one_step sigma (Cons (id,v) pi) s sigma' pi' s' ->
-    one_step sigma pi s sigma' pi' s'
-
+ (* lemma one_step_change_free : *)
+ (*  forall s s':stmt, sigma sigma':env, pi pi':stack, id:ident, v:value. *)
+ (*    fresh_in_stmt id s -> *)
+ (*    one_step sigma (Cons (id,v) pi) s sigma' pi' s' -> *)
+ (*    one_step sigma pi s sigma' pi' s' *)
 
 
 (** {3 Hoare triples} *)

examples/hoare_logic/blocking_semantics3/blocking_semantics3_ImpExpr_eval_change_free_1.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import BuiltIn.
+Require BuiltIn.
+Require int.Int.
+Require int.MinMax.
+
+(* Why3 assumption *)
+Inductive list (a:Type) {a_WT:WhyType a} :=
+  | Nil : list a
+  | Cons : a -> (list a) -> list a.
+Axiom list_WhyType : forall (a:Type) {a_WT:WhyType a}, WhyType (list a).
+Existing Instance list_WhyType.
+Implicit Arguments Nil [[a] [a_WT]].
+Implicit Arguments Cons [[a] [a_WT]].
+
+Axiom map : forall (a:Type) {a_WT:WhyType a} (b:Type) {b_WT:WhyType b}, Type.
+Parameter map_WhyType : forall (a:Type) {a_WT:WhyType a}
+  (b:Type) {b_WT:WhyType b}, WhyType (map a b).
+Existing Instance map_WhyType.
+
+Parameter get: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
+  (map a b) -> a -> b.
+
+Parameter set: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
+  (map a b) -> a -> b -> (map a b).
+
+Axiom Select_eq : forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
+  forall (m:(map a b)), forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) ->
+  ((get (set m a1 b1) a2) = b1).
+
+Axiom Select_neq : forall {a:Type} {a_WT:WhyType a}
+  {b:Type} {b_WT:WhyType b}, forall (m:(map a b)), forall (a1:a) (a2:a),
+  forall (b1:b), (~ (a1 = a2)) -> ((get (set m a1 b1) a2) = (get m a2)).
+
+Parameter const: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
+  b -> (map a b).
+
+Axiom Const : forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
+  forall (b1:b) (a1:a), ((get (const b1:(map a b)) a1) = b1).
+
+(* 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).
+
+(* Why3 assumption *)
+Inductive ident  :=
+  | mk_ident : Z -> ident .
+Axiom ident_WhyType : WhyType ident.
+Existing Instance ident_WhyType.
+
+(* Why3 assumption *)
+Definition ident_index(v:ident): Z := match v with
+  | (mk_ident x) => x
+  end.
+
+(* Why3 assumption *)
+Inductive term_node  :=
+  | Tvalue : value -> term_node 
+  | Tvar : ident -> term_node 
+  | Tderef : mident -> term_node 
+  | Tbin : term -> operator -> term -> term_node 
+  with term  :=
+  | mk_term : term_node -> Z -> term .
+Axiom term_WhyType : WhyType term.
+Existing Instance term_WhyType.
+
+Axiom term_node_WhyType : WhyType term_node.
+Existing Instance term_node_WhyType.
+
+(* Why3 assumption *)
+Definition term_maxvar(v:term): Z := match v with
+  | (mk_term x x1) => x1
+  end.
+
+(* Why3 assumption *)
+Definition term_node1(v:term): term_node :=
+  match v with
+  | (mk_term x x1) => x
+  end.
+
+(* Why3 assumption *)
+Fixpoint var_occurs_in_term(x:ident) (t:term) {struct t}: Prop :=
+  match t with
+  | (mk_term (Tvalue _) _) => False
+  | (mk_term (Tvar i) _) => (x = i)
+  | (mk_term (Tderef _) _) => False
+  | (mk_term (Tbin t1 _ t2) _) => (var_occurs_in_term x t1) \/
+      (var_occurs_in_term x t2)
+  end.
+
+(* Why3 assumption *)
+Definition term_inv(t:term): Prop := forall (x:ident), (var_occurs_in_term x
+  t) -> ((ident_index x) <= (term_maxvar t))%Z.
+
+(* Why3 assumption *)
+Definition mk_tvalue(v:value): term := (mk_term (Tvalue v) (-1%Z)%Z).
+
+Axiom mk_tvalue_inv : forall (v:value), (term_inv (mk_tvalue v)).
+
+(* Why3 assumption *)
+Definition mk_tvar(i:ident): term := (mk_term (Tvar i) (ident_index i)).
+
+Axiom mk_tvar_inv : forall (i:ident), (term_inv (mk_tvar i)).
+
+(* Why3 assumption *)
+Definition mk_tderef(r:mident): term := (mk_term (Tderef r) (-1%Z)%Z).
+
+Axiom mk_tderef_inv : forall (r:mident), (term_inv (mk_tderef r)).
+
+(* Why3 assumption *)
+Definition mk_tbin(t1:term) (o:operator) (t2:term): term := (mk_term (Tbin t1
+  o t2) (Zmax (term_maxvar t1) (term_maxvar t2))).
+
+Axiom mk_tbin_inv : forall (t1:term) (t2:term) (o:operator),
+  ((term_inv t1) /\ (term_inv t2)) -> (term_inv (mk_tbin t1 o t2)).
+
+(* 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 type_value(v:value): datatype :=
+  match v with
+  | Vvoid => TYunit
+  | (Vint int) => TYint
+  | (Vbool bool1) => 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
+  | Nil => ((get_vartype i pi) = TYunit)
+  | (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  := (map mident datatype).
+
+(* Why3 assumption *)
+Inductive type_term : (map mident datatype) -> (list (ident* datatype)%type)
+  -> term -> datatype -> Prop :=
+  | Type_value : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (v:value) (m:Z), (type_term sigma pi
+      (mk_term (Tvalue v) m) (type_value v))
+  | Type_var : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (v:ident) (m:Z) (ty:datatype), ((get_vartype v
+      pi) = ty) -> (type_term sigma pi (mk_term (Tvar v) m) ty)
+  | Type_deref : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (v:mident) (m:Z) (ty:datatype), ((get sigma
+      v) = ty) -> (type_term sigma pi (mk_term (Tderef v) m) ty)
+  | Type_bin : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (t1:term) (t2:term) (op:operator) (m:Z) (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 (mk_term (Tbin t1 op t2) m) ty))).
+
+(* Why3 assumption *)
+Inductive type_fmla : (map mident datatype) -> (list (ident* datatype)%type)
+  -> fmla -> Prop :=
+  | Type_term : forall (sigma:(map 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:(map 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:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (f:fmla), (type_fmla sigma pi f) -> (type_fmla sigma
+      pi (Fnot f))
+  | Type_implies : forall (sigma:(map 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:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (x:ident) (t:term) (f:fmla) (ty:datatype),
+      (type_term sigma pi t ty) -> ((type_fmla sigma (Cons (x, ty) pi) f) ->
+      (type_fmla sigma pi (Flet x t f)))
+  | Type_forall1 : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (x:ident) (f:fmla), (type_fmla sigma (Cons (x, TYint)
+      pi) f) -> (type_fmla sigma pi (Fforall x TYint f))
+  | Type_forall2 : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (x:ident) (f:fmla), (type_fmla sigma (Cons (x, TYbool)
+      pi) f) -> (type_fmla sigma pi (Fforall x TYbool f))
+  | Type_forall3 : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (x:ident) (f:fmla), (type_fmla sigma (Cons (x, TYunit)
+      pi) f) -> (type_fmla sigma pi (Fforall x TYunit f)).
+
+(* Why3 assumption *)
+Inductive type_stmt : (map mident datatype) -> (list (ident* datatype)%type)
+  -> stmt -> Prop :=
+  | Type_skip : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)), (type_stmt sigma pi Sskip)
+  | Type_seq : forall (sigma:(map 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:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (x:mident) (t:term) (ty:datatype), ((get sigma
+      x) = ty) -> ((type_term sigma pi t ty) -> (type_stmt sigma pi
+      (Sassign x t)))
+  | Type_if : forall (sigma:(map 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:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (p:fmla), (type_fmla sigma pi p) -> (type_stmt sigma
+      pi (Sassert p))
+  | Type_while : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (guard:term) (body:stmt) (inv:fmla), (type_fmla sigma
+      pi inv) -> ((type_term sigma pi guard TYbool) -> ((type_stmt sigma pi
+      body) -> (type_stmt sigma pi (Swhile guard inv body)))).
+
+(* Why3 assumption *)
+Definition env  := (map 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
+  | Nil => ((get_stack i pi) = Vvoid)
+  | (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 (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 (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:(map mident value)) (pi:(list (ident* value)%type))
+  (t:term) {struct t}: value :=
+  match t with
+  | (mk_term (Tvalue v) _) => v
+  | (mk_term (Tvar id) _) => (get_stack id pi)
+  | (mk_term (Tderef id) _) => (get sigma id)
+  | (mk_term (Tbin t1 op t2) _) => (eval_bin (eval_term sigma pi t1) op
+      (eval_term sigma pi t2))
+  end.
+
+Axiom eval_bool_term : forall (sigma:(map mident value)) (pi:(list (ident*
+  value)%type)) (sigmat:(map mident datatype)) (pit:(list (ident*
+  datatype)%type)) (t:term), (type_term sigmat pit t TYbool) ->
+  exists b:bool, ((eval_term sigma pi t) = (Vbool b)).
+
+(* Why3 assumption *)
+Fixpoint eval_fmla(sigma:(map 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 (Cons (x, (eval_term sigma pi t)) pi)
+      f1)
+  | (Fforall x TYint f1) => forall (n:Z), (eval_fmla sigma (Cons (x,
+      (Vint n)) pi) f1)
+  | (Fforall x TYbool f1) => forall (b:bool), (eval_fmla sigma (Cons (x,
+      (Vbool b)) pi) f1)
+  | (Fforall x TYunit f1) => (eval_fmla sigma (Cons (x, Vvoid) pi) f1)
+  end.
+
+Parameter msubst_term: term -> mident -> ident -> term.
+
+Axiom msubst_term_def : forall (t:term) (r:mident) (v:ident),
+  match t with
+  | (mk_term ((Tvalue _)|(Tvar _)) _) => ((msubst_term t r v) = t)
+  | (mk_term (Tderef x) _) => ((r = x) -> ((msubst_term t r
+      v) = (mk_tvar v))) /\ ((~ (r = x)) -> ((msubst_term t r v) = t))
+  | (mk_term (Tbin t1 op t2) _) => ((msubst_term t r
+      v) = (mk_tbin (msubst_term t1 r v) op (msubst_term t2 r v)))
+  end.
+
+Parameter subst_term: term -> ident -> ident -> term.
+
+Axiom subst_term_def : forall (t:term) (r:ident) (v:ident),
+  match t with
+  | (mk_term ((Tvalue _)|(Tderef _)) _) => ((subst_term t r v) = t)
+  | (mk_term (Tvar x) _) => ((r = x) -> ((subst_term t r
+      v) = (mk_tvar v))) /\ ((~ (r = x)) -> ((subst_term t r v) = t))
+  | (mk_term (Tbin t1 op t2) _) => ((subst_term t r
+      v) = (mk_tbin (subst_term t1 r v) op (subst_term t2 r v)))
+  end.
+
+(* Why3 assumption *)
+Definition fresh_in_term(id:ident) (t:term): Prop :=
+  ((term_maxvar t) < (ident_index id))%Z.
+
+Axiom fresh_in_binop : forall (t:term) (t':term) (op:operator) (v:ident),
+  (fresh_in_term v (mk_tbin t op t')) -> ((fresh_in_term v t) /\
+  (fresh_in_term v t')).
+
+Axiom eval_msubst_term : forall (e:term) (sigma:(map 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 (set sigma x
+  (get_stack v pi)) pi e)).
+
+Axiom eval_subst_term : forall (sigma:(map mident value)) (pi:(list (ident*
+  value)%type)) (e:term) (x:ident) (v:ident), (fresh_in_term v e) ->
+  ((eval_term sigma pi (subst_term e x v)) = (eval_term sigma (Cons (x,
+  (get_stack v pi)) pi) e)).
+
+Axiom eval_term_change_free : forall (t:term) (sigma:(map mident value))
+  (pi:(list (ident* value)%type)) (id:ident) (v:value), (fresh_in_term id
+  t) -> ((eval_term sigma (Cons (id, v) pi) t) = (eval_term sigma pi t)).
+
+(* 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 ty f1) => (~ (id = y)) /\ (fresh_in_fmla id f1)
+  end.
+
+(* Why3 assumption *)
+Fixpoint subst(f:fmla) (x:ident) (v:ident) {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.
+