Update examples due to the new Coq realization for map.Map.

examples/hoare_logic/imp/imp_Imp_eval_subst_expr_1.v

 (* This file is generated by Why3's Coq driver *)
 (* Beware! Only edit allowed sections below    *)
-Require Import ZArith.
-Require Import Rbase.
-Parameter ident : Type.
+Require Import BuiltIn.
+Require BuiltIn.
+Require map.Map.
+Require int.Int.
+
+Axiom ident : Type.
+Parameter ident_WhyType : WhyType ident.
+Existing Instance ident_WhyType.
 
 Axiom ident_eq_dec : forall (i1:ident) (i2:ident), (i1 = i2) \/ ~ (i1 = i2).
 
 Parameter mk_ident: Z -> ident.
 
-
 Axiom mk_ident_inj : forall (i:Z) (j:Z), ((mk_ident i) = (mk_ident j)) ->
   (i = j).
 
+(* Why3 assumption *)
 Inductive operator  :=
   | Oplus : operator 
   | Ominus : operator 
   | Omult : operator .
+Axiom operator_WhyType : WhyType operator.
+Existing Instance operator_WhyType.
 
+(* Why3 assumption *)
 Inductive expr  :=
   | Econst : Z -> expr 
   | Evar : ident -> expr 
   | Ebin : expr -> operator -> expr -> expr .
+Axiom expr_WhyType : WhyType expr.
+Existing Instance expr_WhyType.
 
+(* Why3 assumption *)
 Inductive stmt  :=
   | Sskip : stmt 
   | Sassign : ident -> expr -> stmt 
   | Sseq : stmt -> stmt -> stmt 
   | Sif : expr -> stmt -> stmt -> stmt 
   | Swhile : expr -> stmt -> stmt .
+Axiom stmt_WhyType : WhyType stmt.
+Existing Instance stmt_WhyType.
 
 Axiom check_skip : forall (s:stmt), (s = Sskip) \/ ~ (s = Sskip).
 
-Parameter map : forall (a:Type) (b:Type), Type.
-
-Parameter get: forall (a:Type) (b:Type), (map a b) -> a -> b.
-
-Implicit Arguments get.
-
-Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b -> (map a b).
-
-Implicit Arguments set.
-
-Axiom Select_eq : forall (a:Type) (b:Type), 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) (b:Type), 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 (b:Type) (a:Type), b -> (map a b).
-
-Set Contextual Implicit.
-Implicit Arguments const.
-Unset Contextual Implicit.
-
-Axiom Const : forall (b:Type) (a:Type), forall (b1:b) (a1:a), ((get (const(
-  b1):(map a b)) a1) = b1).
-
-Definition state  := (map ident Z).
+(* Why3 assumption *)
+Definition state  := (map.Map.map ident Z).
 
+(* Why3 assumption *)
 Definition eval_bin(x:Z) (op:operator) (y:Z): Z :=
   match op with
   | Oplus => (x + y)%Z
@@ -67,64 +55,77 @@ Definition eval_bin(x:Z) (op:operator) (y:Z): Z :=
   | Omult => (x * y)%Z
   end.
 
-Set Implicit Arguments.
-Fixpoint eval_expr(s:(map ident Z)) (e:expr) {struct e}: Z :=
+(* Why3 assumption *)
+Fixpoint eval_expr(s:(map.Map.map ident Z)) (e:expr) {struct e}: Z :=
   match e with
   | (Econst n) => n
-  | (Evar x) => (get s x)
+  | (Evar x) => (map.Map.get s x)
   | (Ebin e1 op e2) => (eval_bin (eval_expr s e1) op (eval_expr s e2))
   end.
-Unset Implicit Arguments.
 
-Inductive one_step : (map ident Z) -> stmt -> (map ident Z)
+(* Why3 assumption *)
+Inductive one_step : (map.Map.map ident Z) -> stmt -> (map.Map.map ident Z)
   -> stmt -> Prop :=
-  | one_step_assign : forall (s:(map ident Z)) (x:ident) (e:expr),
-      (one_step s (Sassign x e) (set s x (eval_expr s e)) Sskip)
-  | one_step_seq : forall (s:(map ident Z)) (sqt:(map ident Z)) (i1:stmt)
-      (i1qt:stmt) (i2:stmt), (one_step s i1 sqt i1qt) -> (one_step s (Sseq i1
-      i2) sqt (Sseq i1qt i2))
-  | one_step_seq_skip : forall (s:(map ident Z)) (i:stmt), (one_step s
-      (Sseq Sskip i) s i)
-  | one_step_if_true : forall (s:(map ident Z)) (e:expr) (i1:stmt) (i2:stmt),
-      (~ ((eval_expr s e) = 0%Z)) -> (one_step s (Sif e i1 i2) s i1)
-  | one_step_if_false : forall (s:(map ident Z)) (e:expr) (i1:stmt)
+  | one_step_assign : forall (s:(map.Map.map ident Z)) (x:ident) (e:expr),
+      (one_step s (Sassign x e) (map.Map.set s x (eval_expr s e)) Sskip)
+  | one_step_seq : forall (s:(map.Map.map ident Z)) (s':(map.Map.map ident
+      Z)) (i1:stmt) (i1':stmt) (i2:stmt), (one_step s i1 s' i1') ->
+      (one_step s (Sseq i1 i2) s' (Sseq i1' i2))
+  | one_step_seq_skip : forall (s:(map.Map.map ident Z)) (i:stmt),
+      (one_step s (Sseq Sskip i) s i)
+  | one_step_if_true : forall (s:(map.Map.map ident Z)) (e:expr) (i1:stmt)
+      (i2:stmt), (~ ((eval_expr s e) = 0%Z)) -> (one_step s (Sif e i1 i2) s
+      i1)
+  | one_step_if_false : forall (s:(map.Map.map ident Z)) (e:expr) (i1:stmt)
       (i2:stmt), ((eval_expr s e) = 0%Z) -> (one_step s (Sif e i1 i2) s i2)
-  | one_step_while_true : forall (s:(map ident Z)) (e:expr) (i:stmt),
+  | one_step_while_true : forall (s:(map.Map.map ident Z)) (e:expr) (i:stmt),
       (~ ((eval_expr s e) = 0%Z)) -> (one_step s (Swhile e i) s (Sseq i
       (Swhile e i)))
-  | one_step_while_false : forall (s:(map ident Z)) (e:expr) (i:stmt),
-      ((eval_expr s e) = 0%Z) -> (one_step s (Swhile e i) s Sskip).
+  | one_step_while_false : forall (s:(map.Map.map ident Z)) (e:expr)
+      (i:stmt), ((eval_expr s e) = 0%Z) -> (one_step s (Swhile e i) s Sskip).
 
-Axiom progress : forall (s:(map ident Z)) (i:stmt), (~ (i = Sskip)) ->
-  exists sqt:(map ident Z), exists iqt:stmt, (one_step s i sqt iqt).
+Axiom progress : forall (s:(map.Map.map ident Z)) (i:stmt),
+  (~ (i = Sskip)) -> exists s':(map.Map.map ident Z), exists i':stmt,
+  (one_step s i s' i').
 
-Inductive many_steps : (map ident Z) -> stmt -> (map ident Z)
+(* Why3 assumption *)
+Inductive many_steps : (map.Map.map ident Z) -> stmt -> (map.Map.map ident Z)
   -> stmt -> Prop :=
-  | many_steps_refl : forall (s:(map ident Z)) (i:stmt), (many_steps s i s i)
-  | many_steps_trans : forall (s1:(map ident Z)) (s2:(map ident Z)) (s3:(map
-      ident Z)) (i1:stmt) (i2:stmt) (i3:stmt), (one_step s1 i1 s2 i2) ->
-      ((many_steps s2 i2 s3 i3) -> (many_steps s1 i1 s3 i3)).
-
-Axiom many_steps_seq_rec : forall (s1:(map ident Z)) (s3:(map ident Z))
-  (i:stmt) (i3:stmt), (many_steps s1 i s3 i3) -> ((i3 = Sskip) ->
-  forall (i1:stmt) (i2:stmt), (i = (Sseq i1 i2)) -> exists s2:(map ident Z),
-  (many_steps s1 i1 s2 Sskip) /\ (many_steps s2 i2 s3 Sskip)).
-
-Axiom many_steps_seq : forall (s1:(map ident Z)) (s3:(map ident Z)) (i1:stmt)
-  (i2:stmt), (many_steps s1 (Sseq i1 i2) s3 Sskip) -> exists s2:(map ident
-  Z), (many_steps s1 i1 s2 Sskip) /\ (many_steps s2 i2 s3 Sskip).
-
+  | many_steps_refl : forall (s:(map.Map.map ident Z)) (i:stmt),
+      (many_steps s i s i)
+  | many_steps_trans : forall (s1:(map.Map.map ident Z)) (s2:(map.Map.map
+      ident Z)) (s3:(map.Map.map ident Z)) (i1:stmt) (i2:stmt) (i3:stmt),
+      (one_step s1 i1 s2 i2) -> ((many_steps s2 i2 s3 i3) -> (many_steps s1
+      i1 s3 i3)).
+
+Axiom many_steps_seq_rec : forall (s1:(map.Map.map ident Z)) (s3:(map.Map.map
+  ident Z)) (i:stmt) (i3:stmt), (many_steps s1 i s3 i3) -> ((i3 = Sskip) ->
+  forall (i1:stmt) (i2:stmt), (i = (Sseq i1 i2)) -> exists s2:(map.Map.map
+  ident Z), (many_steps s1 i1 s2 Sskip) /\ (many_steps s2 i2 s3 Sskip)).
+
+Axiom many_steps_seq : forall (s1:(map.Map.map ident Z)) (s3:(map.Map.map
+  ident Z)) (i1:stmt) (i2:stmt), (many_steps s1 (Sseq i1 i2) s3 Sskip) ->
+  exists s2:(map.Map.map ident Z), (many_steps s1 i1 s2 Sskip) /\
+  (many_steps s2 i2 s3 Sskip).
+
+(* Why3 assumption *)
 Inductive fmla  :=
-  | Fterm : expr -> fmla .
-
-Definition eval_fmla(s:(map ident Z)) (f:fmla): Prop :=
+  | Fterm : expr -> fmla 
+  | Fand : fmla -> fmla -> fmla 
+  | Fnot : fmla -> fmla .
+Axiom fmla_WhyType : WhyType fmla.
+Existing Instance fmla_WhyType.
+
+(* Why3 assumption *)
+Fixpoint eval_fmla(s:(map.Map.map ident Z)) (f:fmla) {struct f}: Prop :=
   match f with
   | (Fterm e) => ~ ((eval_expr s e) = 0%Z)
+  | (Fand f1 f2) => (eval_fmla s f1) /\ (eval_fmla s f2)
+  | (Fnot f1) => ~ (eval_fmla s f1)
   end.
 
 Parameter subst_expr: expr -> ident -> expr -> expr.
 
-
 Axiom subst_expr_def : forall (e:expr) (x:ident) (t:expr),
   match e with
   | (Econst _) => ((subst_expr e x t) = e)
@@ -134,14 +135,13 @@ Axiom subst_expr_def : forall (e:expr) (x:ident) (t:expr),
       (subst_expr e2 x t)))
   end.
 
-(* YOU MAY EDIT THE CONTEXT BELOW *)
 
-(* DO NOT EDIT BELOW *)
 
-Theorem eval_subst_expr : forall (s:(map ident Z)) (e:expr) (x:ident)
-  (t:expr), ((eval_expr s (subst_expr e x t)) = (eval_expr (set s x
+(* Why3 goal *)
+Theorem eval_subst_expr : forall (s:(map.Map.map ident Z)) (e:expr) (x:ident)
+  (t:expr), ((eval_expr s (subst_expr e x t)) = (eval_expr (map.Map.set s x
   (eval_expr s t)) e)).
-(* YOU MAY EDIT THE PROOF BELOW *)
+Proof.
 intros s e x t.
 induction e.
 (* case Econst *)
@@ -153,11 +153,11 @@ intros (H1,H2).
 case (ident_eq_dec x i).
 (* subcase x=i *)
 simpl; intro; subst x.
-rewrite Select_eq; auto.
+rewrite Map.Select_eq; auto.
 now rewrite H1.
 (* subcase x<>i *)
 simpl; intro.
-rewrite Select_neq; auto.
+rewrite Map.Select_neq; auto.
 now rewrite H2.
 (* case Ebin *)
 rewrite (subst_expr_def (Ebin e1 o e2) x t).
@@ -165,6 +165,5 @@ simpl.
 rewrite IHe2.
 now rewrite IHe1.
 Qed.
-(* DO NOT EDIT BELOW *)
 
 

examples/hoare_logic/imp/imp_Imp_progress_1.v

 (* This file is generated by Why3's Coq driver *)
 (* Beware! Only edit allowed sections below    *)
-Require Import ZArith.
-Require Import Rbase.
-Parameter ident : Type.
+Require Import BuiltIn.
+Require BuiltIn.
+Require map.Map.
+Require int.Int.
+
+Axiom ident : Type.
+Parameter ident_WhyType : WhyType ident.
+Existing Instance ident_WhyType.
+
+Axiom ident_eq_dec : forall (i1:ident) (i2:ident), (i1 = i2) \/ ~ (i1 = i2).
 
 Parameter mk_ident: Z -> ident.
 
+Axiom mk_ident_inj : forall (i:Z) (j:Z), ((mk_ident i) = (mk_ident j)) ->
+  (i = j).
 
+(* Why3 assumption *)
 Inductive operator  :=
   | Oplus : operator 
   | Ominus : operator 
   | Omult : operator .
+Axiom operator_WhyType : WhyType operator.
+Existing Instance operator_WhyType.
 
+(* Why3 assumption *)
 Inductive expr  :=
   | Econst : Z -> expr 
   | Evar : ident -> expr 
   | Ebin : expr -> operator -> expr -> expr .
+Axiom expr_WhyType : WhyType expr.
+Existing Instance expr_WhyType.
 
+(* Why3 assumption *)
 Inductive stmt  :=
   | Sskip : stmt 
   | Sassign : ident -> expr -> stmt 
   | Sseq : stmt -> stmt -> stmt 
   | Sif : expr -> stmt -> stmt -> stmt 
   | Swhile : expr -> stmt -> stmt .
+Axiom stmt_WhyType : WhyType stmt.
+Existing Instance stmt_WhyType.
 
 Axiom check_skip : forall (s:stmt), (s = Sskip) \/ ~ (s = Sskip).
 
-Parameter map : forall (a:Type) (b:Type), Type.
-
-Parameter get: forall (a:Type) (b:Type), (map a b) -> a -> b.
-
-Implicit Arguments get.
-
-Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b -> (map a b).
-
-Implicit Arguments set.
-
-Axiom Select_eq : forall (a:Type) (b:Type), 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) (b:Type), 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 (b:Type) (a:Type), b -> (map a b).
-
-Set Contextual Implicit.
-Implicit Arguments const.
-Unset Contextual Implicit.
-
-Axiom Const : forall (b:Type) (a:Type), forall (b1:b) (a1:a), ((get (const(
-  b1):(map a b)) a1) = b1).
-
-Definition state  := (map ident Z).
+(* Why3 assumption *)
+Definition state  := (map.Map.map ident Z).
 
+(* Why3 assumption *)
 Definition eval_bin(x:Z) (op:operator) (y:Z): Z :=
   match op with
   | Oplus => (x + y)%Z
@@ -62,41 +55,42 @@ Definition eval_bin(x:Z) (op:operator) (y:Z): Z :=
   | Omult => (x * y)%Z
   end.
 
-Set Implicit Arguments.
-Fixpoint eval_expr(s:(map ident Z)) (e:expr) {struct e}: Z :=
+(* Why3 assumption *)
+Fixpoint eval_expr(s:(map.Map.map ident Z)) (e:expr) {struct e}: Z :=
   match e with
   | (Econst n) => n
-  | (Evar x) => (get s x)
+  | (Evar x) => (map.Map.get s x)
   | (Ebin e1 op e2) => (eval_bin (eval_expr s e1) op (eval_expr s e2))
   end.
-Unset Implicit Arguments.
 
-Inductive one_step : (map ident Z) -> stmt -> (map ident Z)
+(* Why3 assumption *)
+Inductive one_step : (map.Map.map ident Z) -> stmt -> (map.Map.map ident Z)
   -> stmt -> Prop :=
-  | one_step_assign : forall (s:(map ident Z)) (x:ident) (e:expr),
-      (one_step s (Sassign x e) (set s x (eval_expr s e)) Sskip)
-  | one_step_seq : forall (s:(map ident Z)) (sqt:(map ident Z)) (i1:stmt)
-      (i1qt:stmt) (i2:stmt), (one_step s i1 sqt i1qt) -> (one_step s (Sseq i1
-      i2) sqt (Sseq i1qt i2))
-  | one_step_seq_skip : forall (s:(map ident Z)) (i:stmt), (one_step s
-      (Sseq Sskip i) s i)
-  | one_step_if_true : forall (s:(map ident Z)) (e:expr) (i1:stmt) (i2:stmt),
-      (~ ((eval_expr s e) = 0%Z)) -> (one_step s (Sif e i1 i2) s i1)
-  | one_step_if_false : forall (s:(map ident Z)) (e:expr) (i1:stmt)
+  | one_step_assign : forall (s:(map.Map.map ident Z)) (x:ident) (e:expr),
+      (one_step s (Sassign x e) (map.Map.set s x (eval_expr s e)) Sskip)
+  | one_step_seq : forall (s:(map.Map.map ident Z)) (s':(map.Map.map ident
+      Z)) (i1:stmt) (i1':stmt) (i2:stmt), (one_step s i1 s' i1') ->
+      (one_step s (Sseq i1 i2) s' (Sseq i1' i2))
+  | one_step_seq_skip : forall (s:(map.Map.map ident Z)) (i:stmt),
+      (one_step s (Sseq Sskip i) s i)
+  | one_step_if_true : forall (s:(map.Map.map ident Z)) (e:expr) (i1:stmt)
+      (i2:stmt), (~ ((eval_expr s e) = 0%Z)) -> (one_step s (Sif e i1 i2) s
+      i1)
+  | one_step_if_false : forall (s:(map.Map.map ident Z)) (e:expr) (i1:stmt)
       (i2:stmt), ((eval_expr s e) = 0%Z) -> (one_step s (Sif e i1 i2) s i2)
-  | one_step_while_true : forall (s:(map ident Z)) (e:expr) (i:stmt),
+  | one_step_while_true : forall (s:(map.Map.map ident Z)) (e:expr) (i:stmt),
       (~ ((eval_expr s e) = 0%Z)) -> (one_step s (Swhile e i) s (Sseq i
       (Swhile e i)))
-  | one_step_while_false : forall (s:(map ident Z)) (e:expr) (i:stmt),
-      ((eval_expr s e) = 0%Z) -> (one_step s (Swhile e i) s Sskip).
+  | one_step_while_false : forall (s:(map.Map.map ident Z)) (e:expr)
+      (i:stmt), ((eval_expr s e) = 0%Z) -> (one_step s (Swhile e i) s Sskip).
 
-(* YOU MAY EDIT THE CONTEXT BELOW *)
 
-(* DO NOT EDIT BELOW *)
 
-Theorem progress : forall (s:(map ident Z)) (i:stmt), (~ (i = Sskip)) ->
-  exists sqt:(map ident Z), exists iqt:stmt, (one_step s i sqt iqt).
-(* YOU MAY EDIT THE PROOF BELOW *)
+(* Why3 goal *)
+Theorem progress : forall (s:(map.Map.map ident Z)) (i:stmt),
+  (~ (i = Sskip)) -> exists s':(map.Map.map ident Z), exists i':stmt,
+  (one_step s i s' i').
+Proof.
 intros s i Hskip.
 induction i.
 
@@ -104,7 +98,7 @@ induction i.
 intuition.
 
 (* case i=assign *)
-exists (set s i (eval_expr s e)).
+exists (Map.set s i (eval_expr s e)).
 exists Sskip.
 constructor.
 
@@ -129,6 +123,5 @@ exists s. exists Sskip. constructor. auto.
 exists s. eexists. econstructor. auto.
 
 Qed.
-(* DO NOT EDIT BELOW *)
 
 

examples/hoare_logic/imp_n/imp_n_Imp_eval_subst_expr_1.v

 (* This file is generated by Why3's Coq driver *)
 (* Beware! Only edit allowed sections below    *)
-Require Import ZArith.
-Require Import Rbase.
-Parameter ident : Type.
+Require Import BuiltIn.
+Require BuiltIn.
+Require map.Map.
+Require int.Int.
+
+Axiom ident : Type.
+Parameter ident_WhyType : WhyType ident.
+Existing Instance ident_WhyType.
 
 Axiom ident_eq_dec : forall (i1:ident) (i2:ident), (i1 = i2) \/ ~ (i1 = i2).
 
 Parameter mk_ident: Z -> ident.
 
-
 Axiom mk_ident_inj : forall (i:Z) (j:Z), ((mk_ident i) = (mk_ident j)) ->
   (i = j).
 
+(* Why3 assumption *)
 Inductive operator  :=
   | Oplus : operator 
   | Ominus : operator 
   | Omult : operator .
+Axiom operator_WhyType : WhyType operator.
+Existing Instance operator_WhyType.
 
+(* Why3 assumption *)
 Inductive expr  :=
   | Econst : Z -> expr 
   | Evar : ident -> expr 
   | Ebin : expr -> operator -> expr -> expr .
+Axiom expr_WhyType : WhyType expr.
+Existing Instance expr_WhyType.
 
+(* Why3 assumption *)
 Inductive stmt  :=
   | Sskip : stmt 
   | Sassign : ident -> expr -> stmt 
   | Sseq : stmt -> stmt -> stmt 
   | Sif : expr -> stmt -> stmt -> stmt 
   | Swhile : expr -> stmt -> stmt .
+Axiom stmt_WhyType : WhyType stmt.
+Existing Instance stmt_WhyType.
 
 Axiom check_skip : forall (s:stmt), (s = Sskip) \/ ~ (s = Sskip).
 
-Parameter map : forall (a:Type) (b:Type), Type.
-
-Parameter get: forall (a:Type) (b:Type), (map a b) -> a -> b.
-
-Implicit Arguments get.
-
-Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b -> (map a b).
-
-Implicit Arguments set.
-
-Axiom Select_eq : forall (a:Type) (b:Type), 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) (b:Type), 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 (b:Type) (a:Type), b -> (map a b).
-
-Set Contextual Implicit.
-Implicit Arguments const.
-Unset Contextual Implicit.
-
-Axiom Const : forall (b:Type) (a:Type), forall (b1:b) (a1:a), ((get (const(
-  b1):(map a b)) a1) = b1).
-
-Definition state  := (map ident Z).
+(* Why3 assumption *)
+Definition state  := (map.Map.map ident Z).
 
+(* Why3 assumption *)
 Definition eval_bin(x:Z) (op:operator) (y:Z): Z :=
   match op with
   | Oplus => (x + y)%Z
@@ -67,71 +55,79 @@ Definition eval_bin(x:Z) (op:operator) (y:Z): Z :=
   | Omult => (x * y)%Z
   end.
 
-Set Implicit Arguments.
-Fixpoint eval_expr(s:(map ident Z)) (e:expr) {struct e}: Z :=
+(* Why3 assumption *)
+Fixpoint eval_expr(s:(map.Map.map ident Z)) (e:expr) {struct e}: Z :=
   match e with
   | (Econst n) => n
-  | (Evar x) => (get s x)
+  | (Evar x) => (map.Map.get s x)
   | (Ebin e1 op e2) => (eval_bin (eval_expr s e1) op (eval_expr s e2))
   end.
-Unset Implicit Arguments.
 
-Inductive one_step : (map ident Z) -> stmt -> (map ident Z)
+(* Why3 assumption *)
+Inductive one_step : (map.Map.map ident Z) -> stmt -> (map.Map.map ident Z)
   -> stmt -> Prop :=
-  | one_step_assign : forall (s:(map ident Z)) (x:ident) (e:expr),
-      (one_step s (Sassign x e) (set s x (eval_expr s e)) Sskip)
-  | one_step_seq : forall (s:(map ident Z)) (sqt:(map ident Z)) (i1:stmt)
-      (i1qt:stmt) (i2:stmt), (one_step s i1 sqt i1qt) -> (one_step s (Sseq i1
-      i2) sqt (Sseq i1qt i2))
-  | one_step_seq_skip : forall (s:(map ident Z)) (i:stmt), (one_step s
-      (Sseq Sskip i) s i)
-  | one_step_if_true : forall (s:(map ident Z)) (e:expr) (i1:stmt) (i2:stmt),
-      (~ ((eval_expr s e) = 0%Z)) -> (one_step s (Sif e i1 i2) s i1)
-  | one_step_if_false : forall (s:(map ident Z)) (e:expr) (i1:stmt)
+  | one_step_assign : forall (s:(map.Map.map ident Z)) (x:ident) (e:expr),
+      (one_step s (Sassign x e) (map.Map.set s x (eval_expr s e)) Sskip)
+  | one_step_seq : forall (s:(map.Map.map ident Z)) (s':(map.Map.map ident
+      Z)) (i1:stmt) (i1':stmt) (i2:stmt), (one_step s i1 s' i1') ->
+      (one_step s (Sseq i1 i2) s' (Sseq i1' i2))
+  | one_step_seq_skip : forall (s:(map.Map.map ident Z)) (i:stmt),
+      (one_step s (Sseq Sskip i) s i)
+  | one_step_if_true : forall (s:(map.Map.map ident Z)) (e:expr) (i1:stmt)
+      (i2:stmt), (~ ((eval_expr s e) = 0%Z)) -> (one_step s (Sif e i1 i2) s
+      i1)
+  | one_step_if_false : forall (s:(map.Map.map ident Z)) (e:expr) (i1:stmt)
       (i2:stmt), ((eval_expr s e) = 0%Z) -> (one_step s (Sif e i1 i2) s i2)
-  | one_step_while_true : forall (s:(map ident Z)) (e:expr) (i:stmt),
+  | one_step_while_true : forall (s:(map.Map.map ident Z)) (e:expr) (i:stmt),
       (~ ((eval_expr s e) = 0%Z)) -> (one_step s (Swhile e i) s (Sseq i
       (Swhile e i)))