eval_swap proof

examples/hoare_logic/blocking_semantics3.mlw

@@ -6,7 +6,7 @@
 theory ImpExpr
 
 use import int.Int
-use import int.MinMaxw
+use import int.MinMax
 use import bool.Bool
 use export list.List
 use export list.Append
@@ -295,23 +295,12 @@ lemma fresh_in_binop:
     fresh_in_term v (mk_tbin t op t') ->
       fresh_in_term v t  /\ fresh_in_term v t'
  
-lemma eval_msubst_term:
-  forall e:term, sigma:env, pi:stack, x:mident, v:ident.
-    fresh_in_term v e ->
-    eval_term sigma pi (msubst_term e x v) =
-    eval_term (IdMap.set sigma x (get_stack v pi)) pi e
-
 (* lemma eval_subst_term: *)
 (*   forall sigma:env, pi:stack, 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 *)
 
-lemma eval_term_change_free :
-  forall t:term, sigma:env, pi:stack, id:ident, v:value.
-    fresh_in_term id t ->
-    eval_term sigma (Cons (id,v) pi) t = eval_term sigma pi t
-
 predicate fresh_in_fmla (id:ident) (f:fmla) =
   match f with
   | Fterm e -> fresh_in_term id e
@@ -342,6 +331,10 @@ function msubst (f:fmla) (x:mident) (v:ident) : fmla =
   | Fforall y ty f -> Fforall y ty (msubst f x v)
   end
 
+lemma subst_fresh_term :
+  forall t:term, x:ident, v:ident.
+   fresh_in_term x t -> subst_term t x v = t
+
 lemma subst_fresh :
   forall f:fmla, x:ident, v:ident.
    fresh_in_fmla x f -> subst f x v = f
@@ -351,6 +344,12 @@ lemma subst_fresh :
 (*     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') *)
 
+lemma eval_msubst_term:
+  forall e:term, sigma:env, pi:stack, x:mident, v:ident.
+    fresh_in_term v e ->
+    eval_term sigma pi (msubst_term e x v) =
+    eval_term (IdMap.set sigma x (get_stack v pi)) pi e
+
 (* Need it for monotonicity and wp_reduction *)
 lemma eval_msubst:
   forall f:fmla, sigma:env, pi:stack, x:mident, v:ident.
@@ -374,7 +373,7 @@ lemma eval_msubst:
 (*     eval_fmla sigma (Cons (id,v1) (Cons (id,v2) pi)) f <-> *)
 (*     eval_fmla sigma (Cons (id,v1) pi) f *)
 
-lemma eval_swap_term_any:
+lemma eval_swap_term:
 forall t:term, sigma:env, pi l:stack, id1 id2:ident, v1 v2:value.
 id1 <> id2 ->
 (eval_term sigma (l++(Cons (id1,v1) (Cons (id2,v2) pi))) t =
@@ -386,17 +385,22 @@ eval_term sigma (l++(Cons (id2,v2) (Cons (id1,v1) pi))) t)
 (*     (eval_term sigma (Cons (id1,v1) (Cons (id2,v2) pi)) t = *)
 (*     eval_term sigma (Cons (id2,v2) (Cons (id1,v1) pi)) t) *)
 
-lemma eval_swap_any:
+lemma eval_swap:
   forall f:fmla, sigma:env, pi l:stack, id1 id2:ident, v1 v2:value.
     id1 <> id2 ->
     (eval_fmla sigma (l++(Cons (id1,v1) (Cons (id2,v2) pi))) f <->
     eval_fmla sigma (l++(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_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_term_change_free :
+  forall t:term, sigma:env, pi:stack, id:ident, v:value.
+    fresh_in_term id t ->
+    eval_term sigma (Cons (id,v) pi) t = eval_term sigma pi t
 
  (* Need it for monotonicity*)
 lemma eval_change_free :

examples/hoare_logic/blocking_semantics3/blocking_semantics3_ImpExpr_eval_change_free_2.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]].
+
+(* Why3 assumption *)
+Fixpoint infix_plpl {a:Type} {a_WT:WhyType a}(l1:(list a)) (l2:(list
+  a)) {struct l1}: (list a) :=
+  match l1 with
+  | Nil => l2
+  | (Cons x1 r1) => (Cons x1 (infix_plpl r1 l2))
+  end.
+
+Axiom Append_assoc : forall {a:Type} {a_WT:WhyType a}, forall (l1:(list a))
+  (l2:(list a)) (l3:(list a)), ((infix_plpl l1 (infix_plpl l2
+  l3)) = (infix_plpl (infix_plpl l1 l2) l3)).
+
+Axiom Append_l_nil : forall {a:Type} {a_WT:WhyType a}, forall (l:(list a)),
+  ((infix_plpl l (Nil :(list a))) = l).
+
+(* Why3 assumption *)
+Fixpoint length {a:Type} {a_WT:WhyType a}(l:(list a)) {struct l}: Z :=
+  match l with
+  | Nil => 0%Z
+  | (Cons _ r) => (1%Z + (length r))%Z
+  end.
+
+Axiom Length_nonnegative : forall {a:Type} {a_WT:WhyType a}, forall (l:(list
+  a)), (0%Z <= (length l))%Z.
+
+Axiom Length_nil : forall {a:Type} {a_WT:WhyType a}, forall (l:(list a)),
+  ((length l) = 0%Z) <-> (l = (Nil :(list a))).
+
+Axiom Append_length : forall {a:Type} {a_WT:WhyType a}, forall (l1:(list a))
+  (l2:(list a)), ((length (infix_plpl l1
+  l2)) = ((length l1) + (length l2))%Z).
+
+(* Why3 assumption *)
+Fixpoint mem {a:Type} {a_WT:WhyType a}(x:a) (l:(list a)) {struct l}: Prop :=
+  match l with
+  | Nil => False
+  | (Cons y r) => (x = y) \/ (mem x r)
+  end.
+
+Axiom mem_append : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (l1:(list
+  a)) (l2:(list a)), (mem x (infix_plpl l1 l2)) <-> ((mem x l1) \/ (mem x
+  l2)).
+
+Axiom mem_decomp : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (l:(list
+  a)), (mem x l) -> exists l1:(list a), exists l2:(list a),
+  (l = (infix_plpl l1 (Cons x l2))).
+
+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.
+
+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 *)
+Fixpoint var_occurs_in_term(x:ident) (t:term) {struct t}: Prop :=
+  match t with
+  | (Tvalue _) => False
+  | (Tvar i) => (x = i)
+  | (Tderef _) => False
+  | (Tbin t1 _ t2) => (var_occurs_in_term x t1) \/ (var_occurs_in_term x t2)
+  end.
+
+(* 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), (type_term sigma pi (Tvalue v)
+      (type_value v))
+  | Type_var : forall (sigma:(map 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:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (v:mident) (ty:datatype), ((get sigma v) = ty) ->
+      (type_term sigma pi (Tderef v) ty)
+  | Type_bin : forall (sigma:(map 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 : (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
+  | (Tvalue v) => v
+  | (Tvar id) => (get_stack id pi)
+  | (Tderef id) => (get sigma id)
+  | (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
+  | ((Tvalue _)|(Tvar _)) => ((msubst_term t r v) = t)
+  | (Tderef x) => ((r = x) -> ((msubst_term t r v) = (Tvar v))) /\
+      ((~ (r = x)) -> ((msubst_term t r v) = t))
+  | (Tbin t1 op t2) => ((msubst_term t r v) = (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
+  | ((Tvalue _)|(Tderef _)) => ((subst_term t r v) = t)
+  | (Tvar x) => ((r = x) -> ((subst_term t r v) = (Tvar v))) /\
+      ((~ (r = x)) -> ((subst_term t r v) = t))
+  | (Tbin t1 op t2) => ((subst_term t r v) = (Tbin (subst_term t1 r v) op
+      (subst_term t2 r v)))
+  end.
+
+(* Why3 assumption *)
+Definition fresh_in_term(id:ident) (t:term): Prop := ~ (var_occurs_in_term id
+  t).
+
+Axiom fresh_in_binop : forall (t:term) (t':term) (op:operator) (v:ident),
+  (fresh_in_term v (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_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.
+
+(* 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.
+
+Axiom subst_fresh : forall (f:fmla) (x:ident) (v:ident), (fresh_in_fmla x
+  f) -> ((subst f x v) = f).
+
+Axiom eval_msubst : forall (f:fmla) (sigma:(map 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 (set sigma x
+  (get_stack v pi)) pi f)).
+
+Axiom eval_swap_term_any : forall (t:term) (sigma:(map 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
+  (infix_plpl l (Cons (id1, v1) (Cons (id2, v2) pi))) t) = (eval_term sigma
+  (infix_plpl l (Cons (id2, v2) (Cons (id1, v1) pi))) t)).
+
+Axiom eval_swap_any : forall (f:fmla) (sigma:(map 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
+  (infix_plpl l (Cons (id1, v1) (Cons (id2, v2) pi))) f) <-> (eval_fmla sigma
+  (infix_plpl l (Cons (id2, v2) (Cons (id1, v1) pi))) f)).
+
+(* Why3 goal *)
+Theorem eval_change_free : forall (sigma:(map mident value)) (pi:(list