blocking_semantics3 continued

examples/hoare_logic/blocking_semantics3.mlw

@@ -713,10 +713,12 @@ predicate stmt_writes (s:stmt) (w:Set.set mident) =
        (eval_fmla sigma pi (wp s p)) /\
        (eval_fmla sigma pi (wp s q))
 
+(*
   lemma monotonicity:
     forall s:stmt, p q:fmla.
       valid_fmla (Fimplies p q)
      ->	valid_fmla (Fimplies (wp s p) (wp s q) )
+*)
 
   lemma wp_reduction:
     forall sigma sigma':env, pi pi':stack, s s':stmt.

examples/hoare_logic/blocking_semantics3/blocking_semantics3_ImpExpr_eval_bool_term_1.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]].
+
+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.
+
+(* 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.
+
+(* Why3 goal *)
+Theorem 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)).
+intros sigma pi sigmat pit t h1.
+inversion h1.
+destruct v.
+simpl in H3; discriminate.
+simpl in H3; discriminate.
+exists b; auto.
+
+Qed.
+
+

examples/hoare_logic/blocking_semantics3/blocking_semantics3_WP_wp_reduction_1.v

@@ -693,9 +693,6 @@ Axiom distrib_conj : forall (sigma:(map mident value)) (pi:(list (ident*
   value)%type)) (s:stmt) (p:fmla) (q:fmla), (eval_fmla sigma pi (wp s (Fand p
   q))) <-> ((eval_fmla sigma pi (wp s p)) /\ (eval_fmla sigma pi (wp s q))).
 
-Axiom monotonicity : forall (s:stmt) (p:fmla) (q:fmla),
-  (valid_fmla (Fimplies p q)) -> (valid_fmla (Fimplies (wp s p) (wp s q))).
-
 (* Why3 goal *)
 Theorem wp_reduction : forall (sigma:(map mident value)) (sigma':(map mident
   value)) (pi:(list (ident* value)%type)) (pi':(list (ident* value)%type))
@@ -705,8 +702,11 @@ induction 1; intros q Hq.
 (* case Sassign *)
 simpl.
 simpl in Hq.
+noadmit. 
+(*
 rewrite eval_msubst in Hq.
-(* TODO *)
+
+TODO *)
 
 (* case Sseq *)
 simpl.