eval_type_term proof

examples/hoare_logic/blocking_semantics3.mlw

@@ -24,13 +24,13 @@ type operator = Oplus | Ominus | Omult | Ole
 (** ident for mutable variables *)
 type mident
 
-axiom mident_decide :
+lemma mident_decide :
   forall m1 m2: mident. m1 = m2 \/ m1 <> m2
 
 (** ident for immutable variables *)
 type ident = {| ident_index : int |}
  
-axiom ident_decide :
+lemma ident_decide :
   forall m1 m2: ident. m1 = m2 \/ m1 <> m2
 
 (** Terms *)
@@ -108,7 +108,7 @@ function get_vartype (i:ident) (pi:type_stack) : datatype =
   | Cons (x,ty) r -> if x=i then ty else get_vartype i r
   end
 
-    
+
 type type_env = IdMap.map mident datatype  (* map global mutable variables to their type *)
 function get_reftype (i:mident) (e:type_env) : datatype = IdMap.get e i
 
@@ -242,12 +242,29 @@ function eval_term (sigma:env) (pi:stack) (t:term) : value =
      eval_bin (eval_term sigma pi t1) op (eval_term sigma pi t2)
 end
 
-lemma eval_bool_term:
-  forall sigma:env, pi:stack, sigmat:type_env, pit:type_stack, t:term.
-    type_term sigmat pit t TYbool ->
-    (* TODO: compatibility sigma, sigmat and pi,pit *)
-    exists b:bool.
-      eval_term sigma pi t = Vbool b
+inductive compatible datatype value =
+    | Compatible_bool : 
+	forall b: bool. compatible TYbool (Vbool b)
+    | Compatible_int : 
+	forall n: int. compatible TYint (Vint n)
+    | Compatible_void : 
+	compatible TYunit Vvoid
+
+predicate existe_compatible (ty:datatype) (v:value) =
+   match ty with
+    | TYbool -> exists b: bool. v = Vbool b
+    | TYint -> exists n: int. v = Vint n
+    | TYunit -> v = Vvoid
+end 
+
+predicate compatible_env (sigma:env) (sigmat:type_env) (pi:stack) (pit: type_stack) = 
+ (forall id: mident. compatible (get_reftype id sigmat) (IdMap.get sigma id)) /\
+   (forall id: ident. compatible (get_vartype id pit) (get_stack id pi))
+
+lemma eval_type_term:
+  forall t:term, sigma:env, pi:stack, sigmat:type_env, pit:type_stack, ty:datatype.
+    compatible_env sigma sigmat pi pit ->
+    type_term sigmat pit t ty -> existe_compatible ty (eval_term sigma pi t)
 
 predicate eval_fmla (sigma:env) (pi:stack) (f:fmla) =
   match f with
@@ -723,7 +740,6 @@ predicate stmt_writes (s:stmt) (w:Set.set mident) =
         eval_fmla sigma pi (abstract_effects s q) ->
         eval_fmla sigma pi (wp s (abstract_effects s q))
 
-
   (* lemma wp_subst: *)
   (*   forall e:expr, q:fmla, id :mident, id':ident. *)
   (*   fresh_in_stmt id e -> *)
@@ -748,9 +764,10 @@ predicate stmt_writes (s:stmt) (w:Set.set mident) =
       eval_fmla sigma' pi' (wp s' q)
 
   lemma progress:
-    forall s:stmt, sigma:env, pi:stack,
+	forall s:stmt, sigma:env, pi:stack,
       sigmat: type_env, pit: type_stack, q:fmla.
-      type_stmt sigmat pit s ->
+	compatible_env sigma sigmat pi pit ->
+	  type_stmt sigmat pit s ->
 (* useful ?
       type_fmla sigmat pit q ->
 *)

examples/hoare_logic/blocking_semantics3/blocking_semantics3_ImpExpr_eval_bool_term_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.
+
+(* Why3 assumption *)
+Inductive compatible : datatype -> value -> Prop :=
+  | Compatible_bool : forall (b:bool), (compatible TYbool (Vbool b))
+  | Compatible_int : forall (n:Z), (compatible TYint (Vint n))
+  | Compatible_void : (compatible TYunit Vvoid).
+
+Require Import Why3.
+
+Ltac ae := why3 "alt-ergo" timelimit 3.
+
+(* Why3 goal *)
+Theorem eval_bool_term : forall (t:term),
+  match t with
+  | (Tvalue v) => forall (sigma:(map mident value)) (pi:(list (ident*
+      value)%type)) (sigmat:(map mident datatype)) (pit:(list (ident*
+      datatype)%type)), (type_term sigmat pit t TYbool) ->
+      ((forall (id:mident), (compatible (get sigmat id) (get sigma id))) ->
+      ((forall (id:ident), (compatible (get_vartype id pit) (get_stack id
+      pi))) -> exists b:bool, ((eval_term sigma pi t) = (Vbool b))))
+  | (Tvar i) => True
+  | (Tderef m) => True
+  | (Tbin t1 o t2) => True
+  end.
+destruct t; auto.
+simpl.
+intros.
+inversion H; subst.
+destruct v ;ae.
+Qed.
+
+

examples/hoare_logic/blocking_semantics3/blocking_semantics3_ImpExpr_eval_bool_term_3.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.