(** {1 A certified WP calculus} *) (** {2 A simple imperative language with expressions, syntax and semantics} *) theory ImpExpr use import int.Int use import int.MinMax use import bool.Bool use export list.List use map.Map as IdMap (** types and values *) type datatype = TYunit | TYint | TYbool type value = Vvoid | Vint int | Vbool bool (** terms and formulas *) type operator = Oplus | Ominus | Omult | Ole (* TODO: introduce two distinct types mident and ident *) (** ident for mutable variable*) type mident (** ident for imutable variable*) type ident = {| ident_index : int |} constant result : ident (** Terms *) type term_node = | Tvalue value | Tvar ident | Tderef mident | Tbin term operator term with term = {| term_node : term_node; term_maxvar : int; |} predicate var_occurs_in_term (x:ident) (t:term) = match t with | {| term_node = Tvalue _ |} -> false | {| term_node = Tvar i |} -> x=i | {| term_node = Tderef _ |} -> false | {| term_node = Tbin t1 _ t2 |} -> var_occurs_in_term x t1 \/ var_occurs_in_term x t2 end predicate term_inv (t:term) = forall x:ident. var_occurs_in_term x t -> x.ident_index <= t.term_maxvar function mk_tvalue (v:value) : term = {| term_node = Tvalue v; term_maxvar = -1 |} lemma mk_tvalue_inv : forall v:value. term_inv (mk_tvalue v) function mk_tvar (i:ident) : term = {| term_node = Tvar i; term_maxvar = i.ident_index |} lemma mk_tvar_inv : forall i:ident. term_inv (mk_tvar i) function mk_tderef (r:mident) : term = {| term_node = Tderef r; term_maxvar = -1 |} lemma mk_tderef_inv : forall r:mident. term_inv (mk_tderef r) function mk_tbin (t1:term) (o:operator) (t2:term) : term = {| term_node = Tbin t1 o t2; term_maxvar = max t1.term_maxvar t2.term_maxvar |} lemma mk_tbin_inv : forall t1 t2:term, o:operator. term_inv t1 /\ term_inv t2 -> term_inv (mk_tbin t1 o t2) (** Formulas *) type fmla = | Fterm term | Fand fmla fmla | Fnot fmla | Fimplies fmla fmla | Flet ident term fmla (* let id = term in fmla *) | Fforall ident datatype fmla (* forall id : ty, fmla *) (** Expressions *) type expr = | Evalue value | Ebin expr operator expr | Evar ident | Ederef mident | Eassign mident expr | Eseq expr expr | Elet ident expr expr | Eif expr expr expr | Eassert fmla | Ewhile expr fmla expr (* while cond invariant inv body *) (** Typing *) function type_value (v:value) : datatype = match v with | Vvoid -> TYunit | Vint int -> TYint | Vbool bool -> TYbool end inductive type_operator (op:operator) (ty1 ty2 ty: datatype) = | 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 type type_stack = list (ident, datatype) (* map local immutable variables to their type *) function get_vartype (i:ident) (pi:type_stack) : datatype = match pi with | Nil -> TYunit | 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 inductive type_term type_env type_stack term datatype = | Type_value : forall sigma: type_env, pi:type_stack, v:value, m:int. type_term sigma pi {| term_node = Tvalue v; term_maxvar = m |} (type_value v) | Type_var : forall sigma: type_env, pi:type_stack, v: ident, m:int, ty:datatype. (get_vartype v pi = ty) -> type_term sigma pi {| term_node = Tvar v ; term_maxvar = m |} ty | Type_deref : forall sigma: type_env, pi:type_stack, v: mident, m:int, ty:datatype. (get_reftype v sigma = ty) -> type_term sigma pi {| term_node = Tderef v; term_maxvar = m |} ty | Type_bin : forall sigma: type_env, pi:type_stack, t1 t2 : term, op:operator, m:int, ty1 ty2 ty:datatype. type_term sigma pi t1 ty1 -> type_term sigma pi t2 ty2 -> type_operator op ty1 ty2 ty -> type_term sigma pi {| term_node = Tbin t1 op t2; term_maxvar = m |} ty inductive type_fmla type_env type_stack fmla = | Type_term : forall sigma: type_env, pi:type_stack, t:term. type_term sigma pi t TYbool -> type_fmla sigma pi (Fterm t) | Type_conj : forall sigma: type_env, pi:type_stack, f1 f2:fmla. type_fmla sigma pi f1 -> type_fmla sigma pi f2 -> type_fmla sigma pi (Fand f1 f2) | Type_neg : forall sigma: type_env, pi:type_stack, f:fmla. type_fmla sigma pi f -> type_fmla sigma pi (Fnot f) | Type_implies : forall sigma: type_env, pi:type_stack, f1 f2:fmla. type_fmla sigma pi f1 -> type_fmla sigma pi f2 -> type_fmla sigma pi (Fimplies f1 f2) | Type_let : forall sigma: type_env, pi:type_stack, 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: type_env, pi:type_stack, x:ident, f:fmla. type_fmla sigma (Cons (x,TYint) pi) f -> type_fmla sigma pi (Fforall x TYint f) | Type_forall2 : forall sigma: type_env, pi:type_stack, x:ident, f:fmla. type_fmla sigma (Cons (x,TYbool) pi) f -> type_fmla sigma pi (Fforall x TYbool f) | Type_forall3: forall sigma: type_env, pi:type_stack, x:ident, f:fmla. type_fmla sigma (Cons (x,TYunit) pi) f -> type_fmla sigma pi (Fforall x TYunit f) inductive type_expr type_env type_stack expr datatype = | Type_evalue : forall sigma: type_env, pi:type_stack, v:value. type_expr sigma pi (Evalue v) (type_value v) | Type_evar : forall sigma: type_env, pi:type_stack, v:ident, ty:datatype. (get_vartype v pi = ty) -> type_expr sigma pi (Evar v) ty | Type_ederef : forall sigma: type_env, pi:type_stack, v:mident, ty:datatype. (get_reftype v sigma = ty) -> type_expr sigma pi (Ederef v) ty | Type_ebin : forall sigma: type_env, pi:type_stack, e1 e2:expr, op:operator, ty1 ty2 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_eseq : forall sigma: type_env, pi:type_stack, e1 e2:expr, op:operator, ty:datatype. type_expr sigma pi e1 TYunit -> type_expr sigma pi e2 ty -> type_expr sigma pi (Eseq e1 e2) ty | Type_eassigns : forall sigma: type_env, pi:type_stack, x:mident, e:expr, ty:datatype. (get_reftype x sigma = ty) -> type_expr sigma pi e ty -> type_expr sigma pi (Eassign x e) TYunit | Type_elet : forall sigma: type_env, pi:type_stack, x:ident, e1 e2:expr, ty1 ty2 ty:datatype. type_expr sigma pi e1 ty1 -> type_expr sigma (Cons (x,ty2) pi) e2 ty2 -> type_expr sigma pi (Elet x e1 e2) ty2 | Type_eif : forall sigma: type_env, pi:type_stack, e1 e2 e3:expr, ty:datatype. type_expr sigma pi e1 TYbool -> type_expr sigma pi e2 ty -> type_expr sigma pi e3 ty -> type_expr sigma pi (Eif e1 e2 e3) ty | Type_eassert : forall sigma: type_env, pi:type_stack, p:fmla, ty:datatype. type_fmla sigma pi p -> type_expr sigma pi (Eassert p) TYbool | Type_ewhile : forall sigma: type_env, pi:type_stack, guard body:expr, inv:fmla, ty:datatype. type_fmla sigma pi inv -> type_expr sigma pi guard TYbool -> type_expr sigma pi body TYunit -> type_expr sigma pi (Ewhile guard inv body) TYunit (** Operational semantic *) type env = IdMap.map mident value (* map global mutable variables to their value *) function get_env (i:mident) (e:env) : value = IdMap.get e i type stack = list (ident, value) (* map local immutable variables to their value *) function get_stack (i:ident) (pi:stack) : value = match pi with | Nil -> Vvoid | Cons (x,v) r -> if x=i then v else get_stack i r end lemma get_stack_eq: forall x:ident, v:value, r:stack. get_stack x (Cons (x,v) r) = v lemma get_stack_neq: forall x i:ident, v:value, r:stack. x <> i -> get_stack i (Cons (x,v) r) = get_stack i r (** semantics of formulas *) function eval_bin (x:value) (op:operator) (y:value) : value = match x,y with | Vint x,Vint y -> match op with | Oplus -> Vint (x+y) | Ominus -> Vint (x-y) | Omult -> Vint (x*y) | Ole -> Vbool (if x <= y then True else False) end | _,_ -> Vvoid end function eval_term (sigma:env) (pi:stack) (t:term) : value = match t with | {| term_node = Tvalue v |} -> v | {| term_node = Tvar id |} -> get_stack id pi | {| term_node = Tderef id |} -> get_env id sigma | {| term_node = Tbin t1 op t2 |} -> eval_bin (eval_term sigma pi t1) op (eval_term sigma pi t2) end predicate eval_fmla (sigma:env) (pi:stack) (f:fmla) = 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 f -> not (eval_fmla sigma pi f) | Fimplies f1 f2 -> eval_fmla sigma pi f1 -> eval_fmla sigma pi f2 | Flet x t f -> eval_fmla sigma (Cons (x,eval_term sigma pi t) pi) f | Fforall x TYint f -> forall n:int. eval_fmla sigma (Cons (x,Vint n) pi) f | Fforall x TYbool f -> forall b:bool. eval_fmla sigma (Cons (x,Vbool b) pi) f | Fforall x TYunit f -> eval_fmla sigma (Cons (x,Vvoid) pi) f end (** substitution of a reference [r] by a logic variable [v] warning: proper behavior only guaranted if [v] is "fresh", i.e index(v) > term_maxvar(t) *) function msubst_term (t:term) (r:mident) (v:ident) : term = match t with | {| term_node = Tvalue _ | Tvar _ |} -> t | {| term_node = Tderef x |} -> if r = x then mk_tvar v else t | {| term_node = Tbin t1 op t2 |} -> mk_tbin (msubst_term t1 r v) op (msubst_term t2 r v) end function subst_term (t:term) (r:ident) (v:ident) : term = match t with | {| term_node = Tvalue _ | Tderef _ |} -> t | {| term_node = Tvar x |} -> if r = x then mk_tvar v else t | {| term_node = Tbin t1 op t2 |} -> mk_tbin (subst_term t1 r v) op (subst_term t2 r v) end (** [fresh_in_term id t] is true when [id] does not occur in [t] *) 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. 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 lemma eval_subst_term: forall sigma:env, pi:stack, e:term, x:ident, v:ident. fresh_in_term v e -> eval_term sigma pi (subst_term e x v) = eval_term sigma (Cons (x, (get_stack v pi)) pi) e lemma eval_term_change_free : forall t:term, sigma:env, pi:stack, id:ident, v:value. fresh_in_term id t -> eval_term sigma (Cons (id,v) pi) t = eval_term sigma pi t predicate fresh_in_fmla (id:ident) (f:fmla) = 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 f -> fresh_in_fmla id f | Flet y t f -> id <> y /\ fresh_in_term id t /\ fresh_in_fmla id f | Fforall y ty f -> id <> y /\ fresh_in_fmla id f end function subst (f:fmla) (x:ident) (v:ident) : 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 f -> Fnot (subst f x v) | Fimplies f1 f2 -> Fimplies (subst f1 x v) (subst f2 x v) | Flet y t f -> Flet y (subst_term t x v) (subst f x v) | Fforall y ty f -> Fforall y ty (subst f x v) end function msubst (f:fmla) (x:mident) (v:ident) : 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 f -> Fnot (msubst f x v) | Fimplies f1 f2 -> Fimplies (msubst f1 x v) (msubst f2 x v) | Flet y t f -> Flet y (msubst_term t x v) (msubst f x v) | Fforall y ty f -> Fforall y ty (msubst f x v) end lemma subst_fresh : forall f:fmla, x:ident, v:ident. fresh_in_fmla x f -> subst f x v = f 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') lemma eval_msubst: forall f:fmla, sigma:env, pi:stack, x:mident, v:ident. fresh_in_fmla v f -> (eval_fmla sigma pi (msubst f x v) <-> eval_fmla (IdMap.set sigma x (get_stack v pi)) pi f) lemma eval_subst: forall f:fmla, sigma:env, pi:stack, x:ident, v:ident. fresh_in_fmla v f -> (eval_fmla sigma pi (subst f x v) <-> eval_fmla sigma (Cons(x, (get_stack v pi)) pi) f) lemma eval_swap: forall f:fmla, sigma:env, pi:stack, id1 id2:ident, v1 v2:value. id1 <> id2 -> (eval_fmla sigma (Cons (id1,v1) (Cons (id2,v2) pi)) f <-> eval_fmla sigma (Cons (id2,v2) (Cons (id1,v1) pi)) f) lemma eval_same_var: forall f:fmla, sigma:env, pi:stack, id:ident, v1 v2:value. eval_fmla sigma (Cons (id,v1) (Cons (id,v2) pi)) f <-> eval_fmla sigma (Cons (id,v1) pi) f lemma eval_change_free : forall f:fmla, sigma:env, pi:stack, id:ident, v:value. fresh_in_fmla id f -> (eval_fmla sigma (Cons (id,v) pi) f <-> eval_fmla sigma pi f) (** [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 (** 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 -> eval_fmla sigma pi (Flet id' t (subst f id id')) -> eval_fmla sigma pi (Flet id t f) lemma let_implies : forall id:ident, t:term, p q:fmla. valid_fmla (Fimplies p q) -> valid_fmla (Fimplies (Flet id t p) (Flet id t q)) predicate fresh_in_expr (id:ident) (e:expr) = match e with | Evalue _ -> true | Eseq e1 e2 | Ebin e1 _ e2 -> fresh_in_expr id e1 /\ fresh_in_expr id e2 | Evar v -> id <> v | Ederef _ -> true | Eassign _ e -> fresh_in_expr id e | 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 constant void : expr = Evalue Vvoid (** small-steps semantics for expressions *) inductive one_step env stack expr env stack expr = | one_step_var: forall sigma:env, pi:stack, v:ident. one_step sigma pi (Evar v) sigma pi (Evalue (get_stack v pi)) | one_step_deref: forall sigma:env, pi:stack, v:mident. one_step sigma pi (Ederef v) sigma pi (Evalue (get_env v sigma)) | one_step_bin_ctxt1: forall sigma sigma':env, pi pi':stack, op:operator, e1 e1' 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 sigma':env, pi pi':stack, op:operator, v1:value, e2 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 sigma':env, pi pi':stack, op:operator, v1 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 sigma':env, pi pi':stack, x:mident, e 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:env, pi:stack, x:mident, v:value, e:term. one_step sigma pi (Eassign x (Evalue v)) (IdMap.set sigma x v) pi void | one_step_seq_ctxt: forall sigma sigma':env, pi pi':stack, e1 e1' 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:env, pi:stack, id:ident, e:expr. one_step sigma pi (Eseq void e) sigma pi e | one_step_let_ctxt: forall sigma sigma':env, pi pi':stack, id:ident, e1 e1' 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:env, pi:stack, 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 sigma':env, pi pi':stack, id:ident, e1 e1' e2 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:env, pi:stack, e:term, e1 e2:expr. one_step sigma pi (Eif (Evalue (Vbool True)) e1 e2) sigma pi e1 | one_step_if_false: forall sigma:env, pi:stack, e:term, e1 e2:expr. one_step sigma pi (Eif (Evalue (Vbool False)) e1 e2) sigma pi e2 | one_step_assert: forall sigma:env, pi:stack, f:fmla. (* blocking semantics *) eval_fmla sigma pi f -> one_step sigma pi (Eassert f) sigma pi void | one_step_while: forall sigma:env, pi:stack, cond:expr, inv:fmla, body:expr. (* blocking semantics *) eval_fmla sigma pi inv -> one_step sigma pi (Ewhile cond inv body) sigma pi (Eif cond (Eseq body (Ewhile cond inv body)) void) (** many steps of execution *) inductive many_steps env stack expr env stack expr int = | many_steps_refl: forall sigma:env, pi:stack, e:expr. many_steps sigma pi e sigma pi e 0 | many_steps_trans: forall sigma1 sigma2 sigma3:env, pi1 pi2 pi3:stack, e1 e2 e3:expr, n:int. one_step sigma1 pi1 e1 sigma2 pi2 e2 -> many_steps sigma2 pi2 e2 sigma3 pi3 e3 n -> many_steps sigma1 pi1 e1 sigma3 pi3 e3 (n+1) lemma steps_non_neg: forall sigma1 sigma2:env, pi1 pi2:stack, e1 e2:expr, n:int. many_steps sigma1 pi1 e1 sigma2 pi2 e2 n -> n >= 0 lemma many_steps_seq: forall sigma1 sigma3:env, pi1 pi3:stack, e1 e2:expr, n:int. many_steps sigma1 pi1 (Eseq e1 e2) sigma3 pi3 void n -> exists sigma2:env, pi2:stack, n1 n2:int. many_steps sigma1 pi1 e1 sigma2 pi2 void n1 /\ many_steps sigma2 pi2 e2 sigma3 pi3 void n2 /\ n = 1 + n1 + n2 lemma many_steps_let: forall sigma1 sigma3:env, pi1 pi3:stack, id:ident, e1 e2:expr, v2:value, n:int. many_steps sigma1 pi1 (Elet id e1 e2) sigma3 pi3 (Evalue v2) n -> exists sigma2:env, pi2:stack, v1:value, n1 n2:int. many_steps sigma1 pi1 e1 sigma2 pi2 (Evalue v1) n1 /\ many_steps sigma2 (Cons (id,v1) pi2) e2 sigma3 pi3 (Evalue v2) n2 /\ n = 1 + n1 + n2 lemma one_step_change_free : forall e e':expr, sigma sigma':env, pi pi':stack, id:ident, v:value. fresh_in_expr id e -> one_step sigma (Cons (id,v) pi) e sigma' pi' e' -> one_step sigma pi e sigma' pi' e' (** {3 Hoare triples} *) (** partial correctness *) predicate valid_triple (p:fmla) (e:expr) (q:fmla) = forall sigma:env, pi:stack. eval_fmla sigma pi p -> forall sigma':env, pi':stack, v:value, n:int. many_steps sigma pi e sigma' pi' (Evalue v) n -> eval_fmla sigma' (Cons (result,v) pi') q (*** total correctness *) predicate total_valid_triple (p:fmla) (e:expr) (q:fmla) = forall sigma:env, pi:stack. eval_fmla sigma pi p -> exists sigma':env, pi':stack, v:value, n:int. many_steps sigma pi e sigma' pi' (Evalue v) n /\ eval_fmla sigma' (Cons (result,v) pi') q end theory TestSemantics use import ImpExpr function my_sigma : env = IdMap.const (Vint 0) constant x : ident constant y : mident function my_pi : stack = Cons (x, Vint 42) Nil goal Test13 : eval_term my_sigma my_pi (mk_tvalue (Vint 13)) = Vint 13 goal Test13expr : many_steps my_sigma my_pi (Evalue (Vint 13)) my_sigma my_pi (Evalue (Vint 13)) 0 goal Test42 : eval_term my_sigma my_pi (mk_tvar x) = Vint 42 goal Test42expr : many_steps my_sigma my_pi (Evar x) my_sigma my_pi (Evalue (Vint 42)) 1 goal Test0 : eval_term my_sigma my_pi (mk_tderef y) = Vint 0 goal Test0expr : many_steps my_sigma my_pi (Ederef y) my_sigma my_pi (Evalue (Vint 0)) 1 goal Test55 : eval_term my_sigma my_pi (mk_tbin (mk_tvar x) Oplus (mk_tvalue (Vint 13))) = Vint 55 goal Test55expr : many_steps my_sigma my_pi (Ebin (Evar x) Oplus (Evalue (Vint 13))) my_sigma my_pi (Evalue (Vint 55)) 2 goal Ass42 : forall sigma':env, pi':stack. one_step my_sigma my_pi (Eassign y (Evalue (Vint 42))) sigma' pi' void -> IdMap.get sigma' y = Vint 42 goal If42 : forall sigma1 sigma2:env, pi1 pi2:stack, e:expr. one_step my_sigma my_pi (Eif (Ebin (Ederef y) Ole (Evalue (Vint 10))) (Eassign y (Evalue (Vint 13))) (Eassign y (Evalue (Vint 42)))) sigma1 pi1 e -> one_step sigma1 pi1 e sigma2 pi2 void -> IdMap.get sigma2 y = Vint 13 end (** {2 Hoare logic} *) theory HoareLogic use import ImpExpr (** Hoare logic rules (partial correctness) *) lemma consequence_rule: forall p p' q q':fmla, e:expr. valid_fmla (Fimplies p' p) -> valid_triple p e q -> valid_fmla (Fimplies q q') -> valid_triple p' e q' lemma value_rule: forall q:fmla, v:value. fresh_in_fmla result q -> valid_triple q (Evalue v) q lemma assign_rule: forall p q:fmla, x:mident, e:expr. valid_triple p e (msubst q x result) -> valid_triple p (Eassign x e) q lemma seq_rule: forall p q r:fmla, e1 e2:expr. valid_triple p e1 r /\ valid_triple r e2 q -> valid_triple p (Eseq e1 e2) q lemma let_rule: forall p q r:fmla, id:ident, e1 e2:expr. id <> result -> fresh_in_fmla id r -> valid_triple p e1 r /\ valid_triple (Flet result (mk_tvar id) r) e2 q -> valid_triple p (Elet id e1 e2) q (* lemma if_rule: forall e:expr, p q:fmla, i1 i2:expr. valid_triple (Fand p (Fterm e)) i1 q /\ valid_triple (Fand p (Fnot (Fterm e))) i2 q -> valid_triple p (Eif e e1 e2) q *) lemma assert_rule: forall f p:fmla. valid_fmla (Fimplies p f) -> valid_triple p (Eassert f) p lemma assert_rule_ext: forall f p:fmla. valid_triple (Fimplies f p) (Eassert f) p (* lemma while_rule: forall e:term, inv:fmla, i:expr. valid_triple (Fand (Fterm e) inv) i inv -> valid_triple inv (Swhile e inv i) (Fand (Fnot (Fterm e)) inv) lemma while_rule_ext: forall e:term, inv inv':fmla, i:expr. valid_fmla (Fimplies inv' inv) -> valid_triple (Fand (Fterm e) inv') i inv' -> valid_triple inv' (Swhile e inv i) (Fand (Fnot (Fterm e)) inv') *) (*** frame rule ? *) end theory Simpl_tautology predicate p predicate q lemma simpl_tautology : (p -> q) <-> (p /\ q <-> p) end (** {2 WP calculus} *) theory WP use import ImpExpr use import bool.Bool use set.Set (** [assigns sigma W sigma'] is true when the only differences between [sigma] and [sigma'] are the value of references in [W] *) predicate assigns (sigma:env) (a:Set.set mident) (sigma':env) = forall i:mident. not (Set.mem i a) -> IdMap.get sigma i = IdMap.get sigma' i lemma assigns_refl: forall sigma:env, a:Set.set mident. assigns sigma a sigma lemma assigns_trans: forall sigma1 sigma2 sigma3:env, a:Set.set mident. assigns sigma1 a sigma2 /\ assigns sigma2 a sigma3 -> assigns sigma1 a sigma3 lemma assigns_union_left: forall sigma sigma':env, s1 s2:Set.set mident. assigns sigma s1 sigma' -> assigns sigma (Set.union s1 s2) sigma' lemma assigns_union_right: forall sigma sigma':env, s1 s2:Set.set mident. assigns sigma s2 sigma' -> assigns sigma (Set.union s1 s2) sigma' (** [expr_writes e W] is true when the only references modified by [e] are in [W] *) predicate expr_writes (e:expr) (w:Set.set mident) = match e with | Evalue _ | Evar _ | Ederef _ | Eassert _ -> true | Ebin e1 _ e2 -> expr_writes e1 w /\ expr_writes e2 w | Eassign id _ -> Set.mem id w | Eseq e1 e2 -> expr_writes e1 w /\ expr_writes e2 w | Elet id 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 end function fresh_from (f:fmla) (e:expr) : ident axiom fresh_from_fmla: forall e:expr, f:fmla. fresh_in_fmla (fresh_from f e) f axiom fresh_from_expr: forall e:expr, f:fmla. fresh_in_expr (fresh_from f e) e function abstract_effects (e:expr) (f:fmla) : fmla function wp (e:expr) (q:fmla) : fmla = match e with | Evalue v -> (* let result = v in q *) Flet result (mk_tvalue v) q | Evar v -> Flet result (mk_tvar v) q | Ederef x -> Flet result (mk_tderef x) q | Eassert f -> (* asymmetric and *) Fand f (Fimplies f (Flet result (mk_tvalue Vvoid) q)) | Eseq e1 e2 -> wp e1 (wp e2 q) | Elet id e1 e2 -> wp e1 (Flet id (mk_tvar result) (wp e2 q)) | Ebin e1 op e2 -> let t1 = fresh_from q e in let t2 = fresh_from (Fand (Fterm (mk_tvar t1)) q) e in let q' = Flet result (mk_tbin (mk_tvar t1) op (mk_tvar t2)) q in let f = wp e2 (Flet t2 (mk_tvar result) q') in wp e1 (Flet t1 (mk_tvar result) f) (* let wp_op = subst q (Tvar result) (Tbin (Tvar t1) op (Tvar t2)) in *) (* let wp2 = (subst (wp e2 wp_op) t2 (Tvar result)) in *) (* (subst (wp e1 wp2) t1 (Tvar result)) *) | Eassign x e -> let id = fresh_from q e in let q' = Flet result (mk_tvalue Vvoid) q in wp e (Flet id (mk_tvar result) (msubst q' x id)) | Eif e1 e2 e3 -> let f = Fand (Fimplies (Fterm (mk_tvar result)) (wp e2 q)) (Fimplies (Fnot (Fterm (mk_tvar result))) (wp e3 q)) in wp e1 f | Ewhile cond inv body -> Fand inv (abstract_effects body (wp cond (Fand (Fimplies (Fand (Fterm (mk_tvar result)) inv) (wp body inv)) (Fimplies (Fand (Fnot (Fterm (mk_tvar result))) inv) q)))) end lemma result_always_fresh_in_wp: forall e:expr, q:fmla. fresh_in_fmla result (wp e q) (* lemma wp_subst: *) (* forall e:expr, q:fmla, id :mident, id':ident. *) (* fresh_in_expr id e -> *) (* subst (wp e q) id id' = wp e (subst q id id') *) lemma distrib_conj: forall sigma:env, pi:stack, e:expr, p q:fmla. eval_fmla sigma pi (wp e (Fand p q)) <-> (eval_fmla sigma pi (wp e p)) /\ (eval_fmla sigma pi (wp e q)) lemma monotonicity: forall e:expr, p q:fmla. valid_fmla (Fimplies p q) -> valid_fmla (Fimplies (wp e p) (wp e q) ) lemma wp_reduction: forall sigma sigma':env, pi pi':stack, e e':expr. one_step sigma pi e sigma' pi' e' -> forall q:fmla. eval_fmla sigma pi (wp e q) -> eval_fmla sigma' pi' (wp e' q) predicate is_value (e:expr) = match e with | Evalue _ -> true | _ -> false end lemma decide_value : forall e:expr. not (is_value e) \/ exists v:value. e = Evalue v lemma bool_value: forall v:value, sigmat: type_env, pit:type_stack. type_expr sigmat pit (Evalue v) TYbool -> (v = Vbool False) \/ (v = Vbool True) lemma unit_value: forall v:value, sigmat: type_env, pit:type_stack. type_expr sigmat pit (Evalue v) TYunit -> v = Vvoid lemma progress: forall e:expr, sigma:env, pi:stack, sigmat: type_env, pit: type_stack, ty: datatype, q:fmla. type_expr sigmat pit e ty -> type_fmla sigmat (Cons(result, ty) pit) q -> eval_fmla sigma pi (wp e q) -> not is_value e -> exists sigma':env, pi':stack, e':expr. one_step sigma pi e sigma' pi' e' end (*** Local Variables: compile-command: "why3ide blocking_semantics3.mlw" End: *)