monotonicity proof

examples/hoare_logic/blocking_semantics4/blocking_semantics4_WP_monotonicity_10.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.
+Require set.Set.
+
+(* 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.
+
+Parameter result: ident.
+
+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 expr  :=
+  | Evalue : value -> expr 
+  | Ebin : expr -> operator -> expr -> expr 
+  | Evar : ident -> expr 
+  | Ederef : mident -> expr 
+  | Eassign : mident -> expr -> expr 
+  | Eseq : expr -> expr -> expr 
+  | Elet : ident -> expr -> expr -> expr 
+  | Eif : expr -> expr -> expr -> expr 
+  | Eassert : fmla -> expr 
+  | Ewhile : expr -> fmla -> expr -> expr .
+Axiom expr_WhyType : WhyType expr.
+Existing Instance expr_WhyType.
+
+(* 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_expr : (map mident datatype) -> (list (ident* datatype)%type)
+  -> expr -> datatype -> Prop :=
+  | Type_Evalue : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (v:value), (type_expr sigma pi (Evalue v)
+      (type_value v))
+  | Type_Evar : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (v:ident) (ty:datatype), ((get_vartype v pi) = ty) ->
+      (type_expr sigma pi (Evar v) ty)
+  | Type_Ederef : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (v:mident) (ty:datatype), ((get sigma v) = ty) ->
+      (type_expr sigma pi (Ederef v) ty)
+  | Type_Ebinop : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (e1:expr) (e2:expr) (op:operator) (ty1:datatype)
+      (ty2:datatype) (ty:datatype), (type_expr sigma pi e1 ty1) ->
+      ((type_expr sigma pi e2 ty2) -> ((type_operator op ty1 ty2 ty) ->
+      (type_expr sigma pi (Ebin e1 op e2) ty)))
+  | Type_seq : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (e1:expr) (e2:expr) (ty:datatype), (type_expr sigma pi
+      e1 TYunit) -> ((type_expr sigma pi e2 ty) -> (type_expr sigma pi
+      (Eseq e1 e2) ty))
+  | Type_assigns : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (x:mident) (e:expr) (ty:datatype), ((get sigma
+      x) = ty) -> ((type_expr sigma pi e ty) -> (type_expr sigma pi
+      (Eassign x e) TYunit))
+  | Type_if : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (t:expr) (e1:expr) (e2:expr) (ty:datatype),
+      (type_expr sigma pi t TYbool) -> ((type_expr sigma pi e1 ty) ->
+      ((type_expr sigma pi e2 ty) -> (type_expr sigma pi (Eif t e1 e2) ty)))
+  | Type_assert : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (p:fmla), (type_fmla sigma pi p) -> (type_expr sigma
+      pi (Eassert p) TYbool)
+  | Type_while : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (guard:expr) (body:expr) (inv:fmla) (ty:datatype),
+      (type_fmla sigma pi inv) -> ((type_expr sigma pi guard TYbool) ->
+      ((type_expr sigma pi body ty) -> (type_expr sigma pi (Ewhile guard inv
+      body) ty)))
+  | Type_Elet : forall (sigma:(map mident datatype)) (pi:(list (ident*
+      datatype)%type)) (x:ident) (e1:expr) (e2:expr) (ty1:datatype)
+      (ty2:datatype), (type_expr sigma pi e1 ty1) -> ((type_expr sigma
+      (Cons (x, ty1) pi) e2 ty2) -> (type_expr sigma pi (Elet x e1 e2) ty2)).
+
+(* 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')).
+
+(* 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_term : forall (t:term) (x:ident) (v:ident),
+  (fresh_in_term x t) -> ((subst_term t x v) = t).
+
+Axiom subst_fresh : forall (f:fmla) (x:ident) (v:ident), (fresh_in_fmla x
+  f) -> ((subst f x v) = f).
+
+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_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 : 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_term_2 : forall (t:term) (sigma:(map mident value)) (pi:(list
+  (ident* value)%type)) (id1:ident) (id2:ident) (v1:value) (v2:value),
+  (~ (id1 = id2)) -> ((eval_term sigma (Cons (id1, v1) (Cons (id2, v2) pi))
+  t) = (eval_term sigma (Cons (id2, v2) (Cons (id1, v1) pi)) t)).
+
+Axiom eval_swap : 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)).
+
+Axiom eval_swap_2 : forall (f:fmla) (id1:ident) (id2:ident) (v1:value)
+  (v2:value), (~ (id1 = id2)) -> forall (sigma:(map mident value)) (pi:(list
+  (ident* value)%type)), (eval_fmla sigma (Cons (id1, v1) (Cons (id2, v2)
+  pi)) f) <-> (eval_fmla sigma (Cons (id2, v2) (Cons (id1, v1) pi)) f).
+
+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)).
+
+Axiom eval_change_free : forall (f:fmla) (id:ident) (v:value),
+  (fresh_in_fmla id f) -> forall (sigma:(map mident value)) (pi:(list (ident*
+  value)%type)), (eval_fmla sigma (Cons (id, v) pi) f) <-> (eval_fmla sigma
+  pi f).
+
+(* Why3 assumption *)
+Definition valid_fmla(p:fmla): Prop := forall (sigma:(map mident value))
+  (pi:(list (ident* value)%type)), (eval_fmla sigma pi p).
+
+(* Why3 assumption *)
+Fixpoint fresh_in_expr(id:ident) (e:expr) {struct e}: Prop :=
+  match e with
+  | (Evalue _) => True
+  | (Ebin e1 op e2) => (fresh_in_expr id e1) /\ (fresh_in_expr id e2)
+  | (Evar v) => ~ (id = v)
+  | (Ederef _) => True
+  | (Eassign x e1) => (fresh_in_expr id e1)
+  | (Eseq e1 e2) => (fresh_in_expr id e1) /\ (fresh_in_expr id e2)
+  | (Elet v e1 e2) => (~ (id = v)) /\ ((fresh_in_expr id e1) /\
+      (fresh_in_expr id e2))
+  | (Eif e1 e2 e3) => (fresh_in_expr id e1) /\ ((fresh_in_expr id e2) /\
+      (fresh_in_expr id e3))
+  | (Eassert f) => (fresh_in_fmla id f)
+  | (Ewhile cond inv body) => (fresh_in_expr id cond) /\ ((fresh_in_fmla id
+      inv) /\ (fresh_in_expr id body))
+  end.
+
+(* Why3 assumption *)
+Inductive one_step : (map mident value) -> (list (ident* value)%type) -> expr
+  -> (map mident value) -> (list (ident* value)%type) -> expr -> Prop :=
+  | one_step_var : forall (sigma:(map mident value)) (pi:(list (ident*
+      value)%type)) (v:ident), (one_step sigma pi (Evar v) sigma pi
+      (Evalue (get_stack v pi)))
+  | one_step_deref : forall (sigma:(map mident value)) (pi:(list (ident*
+      value)%type)) (v:mident), (one_step sigma pi (Ederef v) sigma pi
+      (Evalue (get sigma v)))
+  | one_step_bin_ctxt1 : forall (sigma:(map mident value)) (sigma':(map
+      mident value)) (pi:(list (ident* value)%type)) (pi':(list (ident*
+      value)%type)) (op:operator) (e1:expr) (e1':expr) (e2:expr),
+      (one_step sigma pi e1 sigma' pi' e1') -> (one_step sigma pi (Ebin e1 op
+      e2) sigma' pi' (Ebin e1' op e2))
+  | one_step_bin_ctxt2 : forall (sigma:(map mident value)) (sigma':(map
+      mident value)) (pi:(list (ident* value)%type)) (pi':(list (ident*
+      value)%type)) (op:operator) (v1:value) (e2:expr) (e2':expr),
+      (one_step sigma pi e2 sigma' pi' e2') -> (one_step sigma pi
+      (Ebin (Evalue v1) op e2) sigma' pi' (Ebin (Evalue v1) op e2'))
+  | one_step_bin_value : forall (sigma:(map mident value)) (pi:(list (ident*
+      value)%type)) (op:operator) (v1:value) (v2:value), (one_step sigma pi
+      (Ebin (Evalue v1) op (Evalue v2)) sigma pi (Evalue (eval_bin v1 op
+      v2)))
+  | one_step_assign_ctxt : forall (sigma:(map mident value)) (sigma':(map
+      mident value)) (pi:(list (ident* value)%type)) (pi':(list (ident*
+      value)%type)) (x:mident) (e:expr) (e':expr), (one_step sigma pi e
+      sigma' pi' e') -> (one_step sigma pi (Eassign x e) sigma' pi'
+      (Eassign x e'))
+  | one_step_assign_value : forall (sigma:(map mident value)) (sigma':(map
+      mident value)) (pi:(list (ident* value)%type)) (x:mident) (v:value),
+      (sigma' = (set sigma x v)) -> (one_step sigma pi (Eassign x (Evalue v))
+      sigma' pi (Evalue Vvoid))
+  | one_step_seq_ctxt : forall (sigma:(map mident value)) (sigma':(map mident
+      value)) (pi:(list (ident* value)%type)) (pi':(list (ident*
+      value)%type)) (e1:expr) (e1':expr) (e2:expr), (one_step sigma pi e1
+      sigma' pi' e1') -> (one_step sigma pi (Eseq e1 e2) sigma' pi' (Eseq e1'
+      e2))
+  | one_step_seq_value : forall (sigma:(map mident value)) (pi:(list (ident*
+      value)%type)) (e:expr), (one_step sigma pi (Eseq (Evalue Vvoid) e)
+      sigma pi e)
+  | one_step_let_ctxt : forall (sigma:(map mident value)) (sigma':(map mident
+      value)) (pi:(list (ident* value)%type)) (pi':(list (ident*
+      value)%type)) (id:ident) (e1:expr) (e1':expr) (e2:expr),
+      (one_step sigma pi e1 sigma' pi' e1') -> (one_step sigma pi (Elet id e1
+      e2) sigma' pi' (Elet id e1' e2))
+  | one_step_let_value : forall (sigma:(map mident value)) (pi:(list (ident*
+      value)%type)) (id:ident) (v:value) (e:expr), (one_step sigma pi
+      (Elet id (Evalue v) e) sigma (Cons (id, v) pi) e)
+  | one_step_if_ctxt : forall (sigma:(map mident value)) (sigma':(map mident
+      value)) (pi:(list (ident* value)%type)) (pi':(list (ident*
+      value)%type)) (e1:expr) (e1':expr) (e2:expr) (e3:expr), (one_step sigma
+      pi e1 sigma' pi' e1') -> (one_step sigma pi (Eif e1 e2 e3) sigma' pi'
+      (Eif e1' e2 e3))
+  | one_step_if_true : forall (sigma:(map mident value)) (pi:(list (ident*
+      value)%type)) (e1:expr) (e2:expr), (one_step sigma pi
+      (Eif (Evalue (Vbool true)) e1 e2) sigma pi e1)
+  | one_step_if_false : forall (sigma:(map mident value)) (pi:(list (ident*
+      value)%type)) (e1:expr) (e2:expr), (one_step sigma pi
+      (Eif (Evalue (Vbool false)) e1 e2) sigma pi e2)
+  | one_step_assert : forall (sigma:(map mident value)) (pi:(list (ident*
+      value)%type)) (f:fmla), (eval_fmla sigma pi f) -> (one_step sigma pi
+      (Eassert f) sigma pi (Evalue Vvoid))
+  | one_step_while_true : forall (sigma:(map mident value)) (pi:(list (ident*
+      value)%type)) (cond:expr) (body:expr) (inv:fmla), (eval_fmla sigma pi
+      inv) -> (one_step sigma pi (Ewhile (Evalue (Vbool true)) inv body)
+      sigma pi (Eseq body (Ewhile cond inv body)))
+  | one_step_while_false : forall (sigma:(map mident value)) (pi:(list
+      (ident* value)%type)) (inv:fmla) (body:expr), (eval_fmla sigma pi
+      inv) -> (one_step sigma pi (Ewhile (Evalue (Vbool false)) inv body)
+      sigma pi (Evalue Vvoid)).
+
+(* Why3 assumption *)
+Inductive many_steps : (map mident value) -> (list (ident* value)%type)
+  -> expr -> (map mident value) -> (list (ident* value)%type) -> expr
+  -> Z -> Prop :=
+  | many_steps_refl : forall (sigma:(map mident value)) (pi:(list (ident*
+      value)%type)) (s:expr), (many_steps sigma pi s sigma pi s 0%Z)
+  | many_steps_trans : forall (sigma1:(map mident value)) (sigma2:(map mident
+      value)) (sigma3:(map mident value)) (pi1:(list (ident* value)%type))
+      (pi2:(list (ident* value)%type)) (pi3:(list (ident* value)%type))
+      (s1:expr) (s2:expr) (s3:expr) (n:Z), (0%Z < n)%Z -> ((one_step sigma1
+      pi1 s1 sigma2 pi2 s2) -> ((many_steps sigma2 pi2 s2 sigma3 pi3 s3
+      (n - 1%Z)%Z) -> (many_steps sigma1 pi1 s1 sigma3 pi3 s3 n))).
+
+Axiom steps_non_neg : forall (sigma1:(map mident value)) (sigma2:(map mident
+  value)) (pi1:(list (ident* value)%type)) (pi2:(list (ident* value)%type))
+  (s1:expr) (s2:expr) (n:Z), (many_steps sigma1 pi1 s1 sigma2 pi2 s2 n) ->
+  (0%Z <= n)%Z.
+
+Axiom many_steps_seq : forall (sigma1:(map mident value)) (sigma3:(map mident
+  value)) (pi1:(list (ident* value)%type)) (pi3:(list (ident* value)%type))
+  (e1:expr) (e2:expr) (n:Z), (many_steps sigma1 pi1 (Eseq e1 e2) sigma3 pi3
+  (Evalue Vvoid) n) -> exists sigma2:(map mident value), exists pi2:(list
+  (ident* value)%type), exists n1:Z, exists n2:Z, (many_steps sigma1 pi1 e1
+  sigma2 pi2 (Evalue Vvoid) n1) /\ ((many_steps sigma2 pi2 e2 sigma3 pi3
+  (Evalue Vvoid) n2) /\ (n = ((1%Z + n1)%Z + n2)%Z)).
+
+(* Why3 assumption *)
+Definition valid_triple(p:fmla) (e:expr) (q:fmla): Prop := forall (sigma:(map
+  mident value)) (pi:(list (ident* value)%type)), (eval_fmla sigma pi p) ->
+  forall (sigma':(map mident value)) (pi':(list (ident* value)%type)) (n:Z),
+  (many_steps sigma pi e sigma' pi' (Evalue Vvoid) n) -> (eval_fmla sigma'
+  pi' q).
+
+(* Why3 assumption *)
+Definition total_valid_triple(p:fmla) (e:expr) (q:fmla): Prop :=
+  forall (sigma:(map mident value)) (pi:(list (ident* value)%type)),
+  (eval_fmla sigma pi p) -> exists sigma':(map mident value),
+  exists pi':(list (ident* value)%type), exists n:Z, (many_steps sigma pi e
+  sigma' pi' (Evalue Vvoid) n) /\ (eval_fmla sigma' pi' q).
+
+(* Why3 assumption *)
+Definition assigns(sigma:(map mident value)) (a:(set.Set.set mident))
+  (sigma':(map mident value)): Prop := forall (i:mident), (~ (set.Set.mem i
+  a)) -> ((get sigma i) = (get sigma' i)).
+
+Axiom assigns_refl : forall (sigma:(map mident value)) (a:(set.Set.set
+  mident)), (assigns sigma a sigma).
+
+Axiom assigns_trans : forall (sigma1:(map mident value)) (sigma2:(map mident
+  value)) (sigma3:(map mident value)) (a:(set.Set.set mident)),
+  ((assigns sigma1 a sigma2) /\ (assigns sigma2 a sigma3)) -> (assigns sigma1
+  a sigma3).
+
+Axiom assigns_union_left : forall (sigma:(map mident value)) (sigma':(map
+  mident value)) (s1:(set.Set.set mident)) (s2:(set.Set.set mident)),
+  (assigns sigma s1 sigma') -> (assigns sigma (set.Set.union s1 s2) sigma').
+
+Axiom assigns_union_right : forall (sigma:(map mident value)) (sigma':(map
+  mident value)) (s1:(set.Set.set mident)) (s2:(set.Set.set mident)),
+  (assigns sigma s2 sigma') -> (assigns sigma (set.Set.union s1 s2) sigma').
+
+(* Why3 assumption *)
+Fixpoint expr_writes(s:expr) (w:(set.Set.set mident)) {struct s}: Prop :=
+  match s with
+  | ((Evalue _)|((Evar _)|((Ederef _)|(Eassert _)))) => True
+  | (Eassign id _) => (set.Set.mem id w)
+  | (Eseq e1 e2) => (expr_writes e1 w) /\ (expr_writes e2 w)
+  | (Eif e1 e2 e3) => (expr_writes e1 w) /\ ((expr_writes e2 w) /\
+      (expr_writes e3 w))
+  | (Ewhile cond _ body) => (expr_writes cond w) /\ (expr_writes body w)
+  | (Ebin e1 o e2) => (expr_writes e1 w) /\ (expr_writes e2 w)
+  | (Elet id e1 e2) => (expr_writes e1 w) /\ (expr_writes e2 w)
+  end.
+
+Parameter fresh_from: fmla -> expr -> ident.
+
+Axiom fresh_from_fmla : forall (s:expr) (f:fmla),
+  (fresh_in_fmla (fresh_from f s) f).
+
+Axiom fresh_from_expr : forall (s:expr) (f:fmla),
+  (fresh_in_expr (fresh_from f s) s).
+
+Parameter abstract_effects: expr -> fmla -> fmla.
+
+Axiom abstract_effects_generalize : forall (sigma:(map mident value))
+  (pi:(list (ident* value)%type)) (s:expr) (f:fmla), (eval_fmla sigma pi
+  (abstract_effects s f)) -> (eval_fmla sigma pi f).
+
+Axiom abstract_effects_monotonic : forall (s:expr) (f:fmla),
+  forall (sigma:(map mident value)) (pi:(list (ident* value)%type)),
+  (eval_fmla sigma pi f) -> forall (sigma1:(map mident value)) (pi1:(list
+  (ident* value)%type)), (eval_fmla sigma1 pi1 (abstract_effects s f)).
+
+(* Why3 assumption *)
+Fixpoint wp(e:expr) (q:fmla) {struct e}: fmla :=
+  match e with
+  | (Evalue v) => (Flet result (Tvalue v) q)
+  | (Evar v) => (Flet result (Tvar v) q)
+  | (Ederef v) => (Flet result (Tderef v) q)
+  | (Eassert f) => (Fand f (Fimplies f q))
+  | (Eseq e1 e2) => (wp e1 (wp e2 q))
+  | (Elet id e1 e2) => (wp e1 (Flet id (Tvar result) (wp e2 q)))
+  | (Ebin e1 op e2) => let t1 := (fresh_from q e) in let t2 :=
+      (fresh_from (Fand (Fterm (Tvar t1)) q) e) in let q' := (Flet result
+      (Tbin (Tvar t1) op (Tvar t2)) q) in let f := (wp e2 (Flet t2
+      (Tvar result) q')) in (wp e1 (Flet t1 (Tvar result) f))
+  | (Eassign x e1) => let id := (fresh_from q e1) in let q' := (Flet result
+      (Tvalue Vvoid) q) in (wp e1 (Flet id (Tvar result) (msubst q' x id)))
+  | (Eif e1 e2 e3) => let f := (Fand (Fimplies (Fterm (Tvar result)) (wp e2
+      q)) (Fimplies (Fnot (Fterm (Tvar result))) (wp e3 q))) in (wp e1 f)
+  | (Ewhile cond inv body) => (Fand inv (abstract_effects body (wp cond
+      (Fand (Fimplies (Fand (Fterm (Tvar result)) inv) (wp body inv))
+      (Fimplies (Fand (Fnot (Fterm (Tvar result))) inv) q)))))
+  end.
+
+Axiom abstract_effects_writes : forall (sigma:(map mident value)) (pi:(list
+  (ident* value)%type)) (s:expr) (q:fmla), (eval_fmla sigma pi
+  (abstract_effects s q)) -> (eval_fmla sigma pi (wp s (abstract_effects s
+  q))).
+
+(* Why3 goal *)
+Theorem monotonicity : forall (s:expr),
+  match s with
+  | (Evalue v) => True
+  | (Ebin e o e1) => True
+  | (Evar i) => True
+  | (Ederef m) => True
+  | (Eassign m e) => True
+  | (Eseq e e1) => True
+  | (Elet i e e1) => True
+  | (Eif e e1 e2) => True
+  | (Eassert f) => True
+  | (Ewhile e f e1) => (forall (p:fmla) (q:fmla), (valid_fmla (Fimplies p
+      q)) -> (valid_fmla (Fimplies (wp e1 p) (wp e1 q)))) ->
+      ((forall (p:fmla) (q:fmla), (valid_fmla (Fimplies p q)) ->
+      (valid_fmla (Fimplies (wp e p) (wp e q)))) -> forall (p:fmla) (q:fmla),
+      (valid_fmla (Fimplies p q)) -> (valid_fmla (Fimplies (wp s p) (wp s
+      q))))
+  end.
+destruct s; auto.
+unfold valid_fmla.
+simpl.
+intros H2 H1 p q H sigma pi (H3 & H4).
+split; auto.
+apply abstract_effects_generalize in H4.
+apply abstract_effects_monotonic with (sigma := sigma) (pi := pi).
+apply H1 with (p := (Fand (Fimplies (Fand (Fterm (Tvar result)) f) (wp s2 f))
+             (Fimplies (Fand (Fnot (Fterm (Tvar result))) f) p))); auto.
+intros.
+simpl in *.
+destruct H0.
+split; intros.
+apply H0; auto.
+apply H; auto.
+Qed.
+
+