Commit 96c5d906 authored by Asma Tafat-Bouzid's avatar Asma Tafat-Bouzid

Hoare logic example : monotonicity proof

parent 685c7122
......@@ -23,6 +23,9 @@ type operator = Oplus | Ominus | Omult | Ole
(** ident for mutable variables *)
type mident
axiom mident_decide :
forall m1 m2: mident. m1 = m2 \/ m1 <> m2
(** ident for immutable variables *)
type ident = {| ident_index : int |}
......@@ -301,7 +304,7 @@ predicate fresh_in_term (id:ident) (t:term) =
id.ident_index > t.term_maxvar
lemma eval_msubst_term:
forall sigma:env, pi:stack, e:term, x:mident, v:ident.
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
......@@ -355,6 +358,7 @@ lemma let_subst:
forall t:term, f:fmla, x id':ident, id :mident.
msubst (Flet x t f) id id' = Flet x (msubst_term t id id') (msubst f id id')
(* Need it for monotonicity*)
lemma eval_msubst:
forall f:fmla, sigma:env, pi:stack, x:mident, v:ident.
fresh_in_fmla v f ->
......@@ -378,6 +382,7 @@ lemma eval_same_var:
eval_fmla sigma (Cons (id,v1) (Cons (id,v2) pi)) f <->
eval_fmla sigma (Cons (id,v1) pi) f
(* Need it for monotonicity*)
lemma eval_change_free :
forall f:fmla, sigma:env, pi:stack, id:ident, v:value.
fresh_in_fmla id f ->
......@@ -386,6 +391,7 @@ lemma eval_change_free :
(** [valid_fmla f] is true when [f] is valid in any environment *)
predicate valid_fmla (p:fmla) = forall sigma:env, pi:stack. eval_fmla sigma pi p
(* Not needed *)
axiom msubst_implies :
forall p q:fmla.
valid_fmla (Fimplies p q) ->
......@@ -395,13 +401,6 @@ forall p q:fmla.
(** let id' = t in f[id <- id'] <=> let id = t in f*)
lemma let_equiv :
forall id:ident, id':ident, t:term, f:fmla.
forall sigma:env, pi:stack.
fresh_in_fmla id' f ->
(eval_fmla sigma pi (Flet id' t (subst f id id'))
-> eval_fmla sigma pi (Flet id t f))
lemma let_equiv2 :
forall id:ident, id':ident, t:term, f:fmla.
forall sigma:env, pi:stack.
fresh_in_fmla id' f ->
......@@ -666,6 +665,7 @@ predicate stmt_writes (s:stmt) (w:Set.set mident) =
function fresh_from (f:fmla) (s:stmt) : ident
(* Need it for monotonicity*)
axiom fresh_from_fmla: forall s:stmt, f:fmla.
fresh_in_fmla (fresh_from f s) f
......
......@@ -68,6 +68,9 @@ 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 .
......
(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below *)
Require Import BuiltIn.
Require BuiltIn.
Require int.Int.
Require int.MinMax.
(* Why3 assumption *)
Inductive list (a:Type) {a_WT:WhyType a} :=
| Nil : list a
| Cons : a -> (list a) -> list a.
Axiom list_WhyType : forall (a:Type) {a_WT:WhyType a}, WhyType (list a).
Existing Instance list_WhyType.
Implicit Arguments Nil [[a] [a_WT]].
Implicit Arguments Cons [[a] [a_WT]].
Axiom map : forall (a:Type) {a_WT:WhyType a} (b:Type) {b_WT:WhyType b}, Type.
Parameter map_WhyType : forall (a:Type) {a_WT:WhyType a}
(b:Type) {b_WT:WhyType b}, WhyType (map a b).
Existing Instance map_WhyType.
Parameter get: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
(map a b) -> a -> b.
Parameter set: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
(map a b) -> a -> b -> (map a b).
Axiom Select_eq : forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
forall (m:(map a b)), forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) ->
((get (set m a1 b1) a2) = b1).
Axiom Select_neq : forall {a:Type} {a_WT:WhyType a}
{b:Type} {b_WT:WhyType b}, forall (m:(map a b)), forall (a1:a) (a2:a),
forall (b1:b), (~ (a1 = a2)) -> ((get (set m a1 b1) a2) = (get m a2)).
Parameter const: forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
b -> (map a b).
Axiom Const : forall {a:Type} {a_WT:WhyType a} {b:Type} {b_WT:WhyType b},
forall (b1:b) (a1:a), ((get (const b1:(map a b)) a1) = b1).
(* Why3 assumption *)
Inductive datatype :=
| TYunit : datatype
| TYint : datatype
| TYbool : datatype .
Axiom datatype_WhyType : WhyType datatype.
Existing Instance datatype_WhyType.
(* Why3 assumption *)
Inductive value :=
| Vvoid : value
| Vint : Z -> value
| Vbool : bool -> value .
Axiom value_WhyType : WhyType value.
Existing Instance value_WhyType.
(* Why3 assumption *)
Inductive operator :=
| Oplus : operator
| Ominus : operator
| Omult : operator
| Ole : operator .
Axiom operator_WhyType : WhyType operator.
Existing Instance operator_WhyType.
Axiom mident : Type.
Parameter mident_WhyType : WhyType mident.
Existing Instance mident_WhyType.
Axiom mident_decide : forall (m1:mident) (m2:mident), (m1 = m2) \/
~ (m1 = m2).
(* Why3 assumption *)
Inductive ident :=
| mk_ident : Z -> ident .
Axiom ident_WhyType : WhyType ident.
Existing Instance ident_WhyType.
(* Why3 assumption *)
Definition ident_index(v:ident): Z := match v with
| (mk_ident x) => x
end.
(* Why3 assumption *)
Inductive term_node :=
| Tvalue : value -> term_node
| Tvar : ident -> term_node
| Tderef : mident -> term_node
| Tbin : term -> operator -> term -> term_node
with term :=
| mk_term : term_node -> Z -> term .
Axiom term_WhyType : WhyType term.
Existing Instance term_WhyType.
Axiom term_node_WhyType : WhyType term_node.
Existing Instance term_node_WhyType.
(* Why3 assumption *)
Definition term_maxvar(v:term): Z := match v with
| (mk_term x x1) => x1
end.
(* Why3 assumption *)
Definition term_node1(v:term): term_node :=
match v with
| (mk_term x x1) => x
end.
(* Why3 assumption *)
Fixpoint var_occurs_in_term(x:ident) (t:term) {struct t}: Prop :=
match t with
| (mk_term (Tvalue _) _) => False
| (mk_term (Tvar i) _) => (x = i)
| (mk_term (Tderef _) _) => False
| (mk_term (Tbin t1 _ t2) _) => (var_occurs_in_term x t1) \/
(var_occurs_in_term x t2)
end.
(* Why3 assumption *)
Definition term_inv(t:term): Prop := forall (x:ident), (var_occurs_in_term x
t) -> ((ident_index x) <= (term_maxvar t))%Z.
(* Why3 assumption *)
Definition mk_tvalue(v:value): term := (mk_term (Tvalue v) (-1%Z)%Z).
Axiom mk_tvalue_inv : forall (v:value), (term_inv (mk_tvalue v)).
(* Why3 assumption *)
Definition mk_tvar(i:ident): term := (mk_term (Tvar i) (ident_index i)).
Axiom mk_tvar_inv : forall (i:ident), (term_inv (mk_tvar i)).
(* Why3 assumption *)
Definition mk_tderef(r:mident): term := (mk_term (Tderef r) (-1%Z)%Z).
Axiom mk_tderef_inv : forall (r:mident), (term_inv (mk_tderef r)).
(* Why3 assumption *)
Definition mk_tbin(t1:term) (o:operator) (t2:term): term := (mk_term (Tbin t1
o t2) (Zmax (term_maxvar t1) (term_maxvar t2))).
Axiom mk_tbin_inv : forall (t1:term) (t2:term) (o:operator),
((term_inv t1) /\ (term_inv t2)) -> (term_inv (mk_tbin t1 o t2)).
(* Why3 assumption *)
Inductive fmla :=
| Fterm : term -> fmla
| Fand : fmla -> fmla -> fmla
| Fnot : fmla -> fmla
| Fimplies : fmla -> fmla -> fmla
| Flet : ident -> term -> fmla -> fmla
| Fforall : ident -> datatype -> fmla -> fmla .
Axiom fmla_WhyType : WhyType fmla.
Existing Instance fmla_WhyType.
(* Why3 assumption *)
Inductive stmt :=
| Sskip : stmt
| Sassign : mident -> term -> stmt
| Sseq : stmt -> stmt -> stmt
| Sif : term -> stmt -> stmt -> stmt
| Sassert : fmla -> stmt
| Swhile : term -> fmla -> stmt -> stmt .
Axiom stmt_WhyType : WhyType stmt.
Existing Instance stmt_WhyType.
Axiom decide_is_skip : forall (s:stmt), (s = Sskip) \/ ~ (s = Sskip).
(* Why3 assumption *)
Definition type_value(v:value): datatype :=
match v with
| Vvoid => TYunit
| (Vint int) => TYint
| (Vbool bool1) => TYbool
end.
(* Why3 assumption *)
Inductive type_operator : operator -> datatype -> datatype
-> datatype -> Prop :=
| Type_plus : (type_operator Oplus TYint TYint TYint)
| Type_minus : (type_operator Ominus TYint TYint TYint)
| Type_mult : (type_operator Omult TYint TYint TYint)
| Type_le : (type_operator Ole TYint TYint TYbool).
(* Why3 assumption *)
Definition type_stack := (list (ident* datatype)%type).
Parameter get_vartype: ident -> (list (ident* datatype)%type) -> datatype.
Axiom get_vartype_def : forall (i:ident) (pi:(list (ident* datatype)%type)),
match pi with
| Nil => ((get_vartype i pi) = TYunit)
| (Cons (x, ty) r) => ((x = i) -> ((get_vartype i pi) = ty)) /\
((~ (x = i)) -> ((get_vartype i pi) = (get_vartype i r)))
end.
(* Why3 assumption *)
Definition type_env := (map mident datatype).
(* Why3 assumption *)
Inductive type_term : (map mident datatype) -> (list (ident* datatype)%type)
-> term -> datatype -> Prop :=
| Type_value : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (v:value) (m:Z), (type_term sigma pi
(mk_term (Tvalue v) m) (type_value v))
| Type_var : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (v:ident) (m:Z) (ty:datatype), ((get_vartype v
pi) = ty) -> (type_term sigma pi (mk_term (Tvar v) m) ty)
| Type_deref : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (v:mident) (m:Z) (ty:datatype), ((get sigma
v) = ty) -> (type_term sigma pi (mk_term (Tderef v) m) ty)
| Type_bin : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (t1:term) (t2:term) (op:operator) (m:Z) (ty1:datatype)
(ty2:datatype) (ty:datatype), (type_term sigma pi t1 ty1) ->
((type_term sigma pi t2 ty2) -> ((type_operator op ty1 ty2 ty) ->
(type_term sigma pi (mk_term (Tbin t1 op t2) m) ty))).
(* Why3 assumption *)
Inductive type_fmla : (map mident datatype) -> (list (ident* datatype)%type)
-> fmla -> Prop :=
| Type_term : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (t:term), (type_term sigma pi t TYbool) ->
(type_fmla sigma pi (Fterm t))
| Type_conj : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (f1:fmla) (f2:fmla), (type_fmla sigma pi f1) ->
((type_fmla sigma pi f2) -> (type_fmla sigma pi (Fand f1 f2)))
| Type_neg : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (f:fmla), (type_fmla sigma pi f) -> (type_fmla sigma
pi (Fnot f))
| Type_implies : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (f1:fmla) (f2:fmla), (type_fmla sigma pi f1) ->
((type_fmla sigma pi f2) -> (type_fmla sigma pi (Fimplies f1 f2)))
| Type_let : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (x:ident) (t:term) (f:fmla) (ty:datatype),
(type_term sigma pi t ty) -> ((type_fmla sigma (Cons (x, ty) pi) f) ->
(type_fmla sigma pi (Flet x t f)))
| Type_forall1 : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (x:ident) (f:fmla), (type_fmla sigma (Cons (x, TYint)
pi) f) -> (type_fmla sigma pi (Fforall x TYint f))
| Type_forall2 : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (x:ident) (f:fmla), (type_fmla sigma (Cons (x, TYbool)
pi) f) -> (type_fmla sigma pi (Fforall x TYbool f))
| Type_forall3 : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (x:ident) (f:fmla), (type_fmla sigma (Cons (x, TYunit)
pi) f) -> (type_fmla sigma pi (Fforall x TYunit f)).
(* Why3 assumption *)
Inductive type_stmt : (map mident datatype) -> (list (ident* datatype)%type)
-> stmt -> Prop :=
| Type_skip : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)), (type_stmt sigma pi Sskip)
| Type_seq : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (s1:stmt) (s2:stmt), (type_stmt sigma pi s1) ->
((type_stmt sigma pi s2) -> (type_stmt sigma pi (Sseq s1 s2)))
| Type_assigns : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (x:mident) (t:term) (ty:datatype), ((get sigma
x) = ty) -> ((type_term sigma pi t ty) -> (type_stmt sigma pi
(Sassign x t)))
| Type_if : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (t:term) (s1:stmt) (s2:stmt), (type_term sigma pi t
TYbool) -> ((type_stmt sigma pi s1) -> ((type_stmt sigma pi s2) ->
(type_stmt sigma pi (Sif t s1 s2))))
| Type_assert : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (p:fmla), (type_fmla sigma pi p) -> (type_stmt sigma
pi (Sassert p))
| Type_while : forall (sigma:(map mident datatype)) (pi:(list (ident*
datatype)%type)) (guard:term) (body:stmt) (inv:fmla), (type_fmla sigma
pi inv) -> ((type_term sigma pi guard TYbool) -> ((type_stmt sigma pi
body) -> (type_stmt sigma pi (Swhile guard inv body)))).
(* Why3 assumption *)
Definition env := (map mident value).
(* Why3 assumption *)
Definition stack := (list (ident* value)%type).
Parameter get_stack: ident -> (list (ident* value)%type) -> value.
Axiom get_stack_def : forall (i:ident) (pi:(list (ident* value)%type)),
match pi with
| Nil => ((get_stack i pi) = Vvoid)
| (Cons (x, v) r) => ((x = i) -> ((get_stack i pi) = v)) /\ ((~ (x = i)) ->
((get_stack i pi) = (get_stack i r)))
end.
Axiom get_stack_eq : forall (x:ident) (v:value) (r:(list (ident*
value)%type)), ((get_stack x (Cons (x, v) r)) = v).
Axiom get_stack_neq : forall (x:ident) (i:ident) (v:value) (r:(list (ident*
value)%type)), (~ (x = i)) -> ((get_stack i (Cons (x, v) r)) = (get_stack i
r)).
Parameter eval_bin: value -> operator -> value -> value.
Axiom eval_bin_def : forall (x:value) (op:operator) (y:value), match (x,
y) with
| ((Vint x1), (Vint y1)) =>
match op with
| Oplus => ((eval_bin x op y) = (Vint (x1 + y1)%Z))
| Ominus => ((eval_bin x op y) = (Vint (x1 - y1)%Z))
| Omult => ((eval_bin x op y) = (Vint (x1 * y1)%Z))
| Ole => ((x1 <= y1)%Z -> ((eval_bin x op y) = (Vbool true))) /\
((~ (x1 <= y1)%Z) -> ((eval_bin x op y) = (Vbool false)))
end
| (_, _) => ((eval_bin x op y) = Vvoid)
end.
(* Why3 assumption *)
Fixpoint eval_term(sigma:(map mident value)) (pi:(list (ident* value)%type))
(t:term) {struct t}: value :=
match t with
| (mk_term (Tvalue v) _) => v
| (mk_term (Tvar id) _) => (get_stack id pi)
| (mk_term (Tderef id) _) => (get sigma id)
| (mk_term (Tbin t1 op t2) _) => (eval_bin (eval_term sigma pi t1) op
(eval_term sigma pi t2))
end.
Axiom eval_bool_term : forall (sigma:(map mident value)) (pi:(list (ident*
value)%type)) (sigmat:(map mident datatype)) (pit:(list (ident*
datatype)%type)) (t:term), (type_term sigmat pit t TYbool) ->
exists b:bool, ((eval_term sigma pi t) = (Vbool b)).
(* Why3 assumption *)
Fixpoint eval_fmla(sigma:(map mident value)) (pi:(list (ident* value)%type))
(f:fmla) {struct f}: Prop :=
match f with
| (Fterm t) => ((eval_term sigma pi t) = (Vbool true))
| (Fand f1 f2) => (eval_fmla sigma pi f1) /\ (eval_fmla sigma pi f2)
| (Fnot f1) => ~ (eval_fmla sigma pi f1)
| (Fimplies f1 f2) => (eval_fmla sigma pi f1) -> (eval_fmla sigma pi f2)
| (Flet x t f1) => (eval_fmla sigma (Cons (x, (eval_term sigma pi t)) pi)
f1)
| (Fforall x TYint f1) => forall (n:Z), (eval_fmla sigma (Cons (x,
(Vint n)) pi) f1)
| (Fforall x TYbool f1) => forall (b:bool), (eval_fmla sigma (Cons (x,
(Vbool b)) pi) f1)
| (Fforall x TYunit f1) => (eval_fmla sigma (Cons (x, Vvoid) pi) f1)
end.
Parameter msubst_term: term -> mident -> ident -> term.
Axiom msubst_term_def : forall (t:term) (r:mident) (v:ident),
match t with
| (mk_term ((Tvalue _)|(Tvar _)) _) => ((msubst_term t r v) = t)
| (mk_term (Tderef x) _) => ((r = x) -> ((msubst_term t r
v) = (mk_tvar v))) /\ ((~ (r = x)) -> ((msubst_term t r v) = t))
| (mk_term (Tbin t1 op t2) _) => ((msubst_term t r
v) = (mk_tbin (msubst_term t1 r v) op (msubst_term t2 r v)))
end.
Parameter subst_term: term -> ident -> ident -> term.
Axiom subst_term_def : forall (t:term) (r:ident) (v:ident),
match t with
| (mk_term ((Tvalue _)|(Tderef _)) _) => ((subst_term t r v) = t)
| (mk_term (Tvar x) _) => ((r = x) -> ((subst_term t r
v) = (mk_tvar v))) /\ ((~ (r = x)) -> ((subst_term t r v) = t))
| (mk_term (Tbin t1 op t2) _) => ((subst_term t r
v) = (mk_tbin (subst_term t1 r v) op (subst_term t2 r v)))
end.
(* Why3 assumption *)
Definition fresh_in_term(id:ident) (t:term): Prop :=
((term_maxvar t) < (ident_index id))%Z.
Require Import Why3.
Ltac ae := why3 "alt-ergo" timelimit 3.
(* Why3 goal *)
Theorem 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)).
induction e.
destruct t.
(* Value *)
ae.
(* Var *)
intros sigma pi x v H.
simpl.
rewrite (msubst_term_def (mk_term (Tvar i) z)).
easy.
(* Ref *)
intros sigma pi x v H.
simpl.
destruct (mident_decide m x).
(* m = x*)
ae.
(* m <> x*)
ae.
(* Bin *)
(*Il manque le bon schema d'induction*)
Qed.
......@@ -717,7 +717,17 @@ pose (id1 := fresh_from p (Sassign m t)).
fold id1 in H.
pose (id2 := fresh_from q (Sassign m t)).
fold id2.
rewrite eval_msubst.
rewrite get_stack_eq.
rewrite eval_change_free.
apply H2.
rewrite eval_msubst in H.
rewrite get_stack_eq in H.
rewrite eval_change_free in H; auto.
apply fresh_from_fmla; auto.
apply fresh_from_fmla; auto.
apply fresh_from_fmla; auto.
apply fresh_from_fmla; auto.
Qed.
......@@ -162,6 +162,8 @@ Inductive 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
......@@ -312,6 +314,11 @@ Fixpoint eval_term(sigma:(map mident value)) (pi:(list (ident* value)%type))
(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 :=
......@@ -438,6 +445,11 @@ Axiom eval_change_free : forall (f:fmla) (sigma:(map mident value)) (pi:(list
Definition valid_fmla(p:fmla): Prop := forall (sigma:(map mident value))
(pi:(list (ident* value)%type)), (eval_fmla sigma pi p).
Axiom msubst_implies : forall (p:fmla) (q:fmla), (valid_fmla (Fimplies p
q)) -> forall (sigma:(map mident value)) (pi:(list (ident* value)%type))
(x:mident) (id:ident), (fresh_in_fmla id (Fand p q)) -> (eval_fmla sigma
(Cons (id, (get sigma x)) pi) (Fimplies (msubst p x id) (msubst q x id))).
Axiom let_equiv : forall (id:ident) (id':ident) (t:term) (f:fmla),
forall (sigma:(map mident value)) (pi:(list (ident* value)%type)),
(fresh_in_fmla id' f) -> ((eval_fmla sigma pi (Flet id' t (subst f id
......@@ -492,7 +504,7 @@ Inductive one_step : (map mident value) -> (list (ident* value)%type) -> stmt
value)%type)) (cond:term) (inv:fmla) (body:stmt), (eval_fmla sigma pi
inv) -> (((eval_term sigma pi cond) = (Vbool true)) -> (one_step sigma
pi (Swhile cond inv body) sigma pi (Sseq body (Swhile cond inv body))))
| one_step_while_falsee : forall (sigma:(map mident value)) (pi:(list
| one_step_while_false : forall (sigma:(map mident value)) (pi:(list
(ident* value)%type)) (cond:term) (inv:fmla) (body:stmt),
(eval_fmla sigma pi inv) -> (((eval_term sigma pi
cond) = (Vbool false)) -> (one_step sigma pi (Swhile cond inv body)
......@@ -658,6 +670,15 @@ Axiom fresh_from_stmt : forall (s:stmt) (f:fmla),
Parameter abstract_effects: stmt -> fmla -> fmla.
Axiom abstract_effects_generalize : forall (sigma:(map mident value))
(pi:(list (ident* value)%type)) (s:stmt) (f:fmla), (eval_fmla sigma pi
(abstract_effects s f)) -> (eval_fmla sigma pi f).
Axiom abstract_effects_monotonic : forall (s:stmt) (f:fmla),