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wp2.mlw 12.07 KiB
(** {1 A certified WP calculus} *)
(** {2 A simple imperative language, syntax and semantics} *)
theory Imp
(** terms and formulas *)
type datatype = Tint | Tbool
type operator = Oplus | Ominus | Omult | Ole
type ident = int
type term =
| Tconst int
| Tvar ident
| Tderef ident
| Tbin term operator term
type fmla =
| Fterm term
| Fand fmla fmla
| Fnot fmla
| Fimplies fmla fmla
| Flet ident term fmla
| Fforall ident datatype fmla
use import int.Int
use import bool.Bool
type value =
| Vint int
| Vbool bool
use map.Map as IdMap
type env = IdMap.map ident value
(** 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
| _,_ -> Vbool False
end
function get_env (i:ident) (e:env) : value = IdMap.get e i
function eval_term (sigma:env) (pi:env) (t:term) : value =
match t with
| Tconst n -> Vint n
| Tvar id -> get_env id pi
| Tderef id -> get_env id sigma
| Tbin t1 op t2 ->
eval_bin (eval_term sigma pi t1) op (eval_term sigma pi t2)
end
predicate eval_fmla (sigma:env) (pi:env) (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 (IdMap.set pi x (eval_term sigma pi t)) f
| Fforall x Tint f ->
forall n:int. eval_fmla sigma (IdMap.set pi x (Vint n)) f
| Fforall x Tbool f ->
forall b:bool.
eval_fmla sigma (IdMap.set pi x (Vbool b)) f
end
(** substitution of a reference [r] by a logic variable [v]
warning: proper behavior only guaranted if [v] is fresh *)
function subst_term (e:term) (r:ident) (v:ident) : term =
match e with
| Tconst _ -> e
| Tvar _ -> e
| Tderef x -> if r=x then Tvar v else e
| Tbin e1 op e2 -> Tbin (subst_term e1 r v) op (subst_term e2 r v)
end
predicate fresh_in_term (id:ident) (t:term) =
match t with
| Tconst _ -> true
| Tvar v -> id <> v
| Tderef _ -> true
| Tbin t1 _ t2 -> fresh_in_term id t1 /\ fresh_in_term id t2
end
lemma eval_subst_term:
forall sigma pi:env, e:term, x:ident, v:ident.
fresh_in_term v e ->
eval_term sigma pi (subst_term e x v) =
eval_term (IdMap.set sigma x (IdMap.get pi v)) pi e
lemma eval_term_change_free :
forall t:term, sigma pi:env, id:ident, v:value.
fresh_in_term id t ->
eval_term sigma (IdMap.set pi id v) 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
lemma eval_subst:
forall f:fmla, sigma pi:env, x:ident, v:ident.
fresh_in_fmla v f ->
(eval_fmla sigma pi (subst f x v) <->
eval_fmla (IdMap.set sigma x (IdMap.get pi v)) pi f)
lemma eval_swap:
forall f:fmla, sigma pi:env, id1 id2:ident, v1 v2:value.
id1 <> id2 ->
(eval_fmla sigma (IdMap.set (IdMap.set pi id1 v1) id2 v2) f <-> eval_fmla sigma (IdMap.set (IdMap.set pi id2 v2) id1 v1) f)
lemma eval_change_free :
forall f:fmla, sigma pi:env, id:ident, v:value.
fresh_in_fmla id f ->
(eval_fmla sigma (IdMap.set pi id v) f <-> eval_fmla sigma pi f)
(* statements *)
type stmt =
| Sskip
| Sassign ident term
| Sseq stmt stmt
| Sif term stmt stmt
| Sassert fmla
| Swhile term fmla stmt
lemma check_skip:
forall s:stmt. s=Sskip \/s<>Sskip
(** small-steps semantics for statements *)
inductive one_step env env stmt env env stmt =
| one_step_assign:
forall sigma pi:env, x:ident, e:term.
one_step sigma pi (Sassign x e)
(IdMap.set sigma x (eval_term sigma pi e)) pi
Sskip
| one_step_seq:
forall sigma pi sigma' pi':env, i1 i1' i2:stmt.
one_step sigma pi i1 sigma' pi' i1' ->
one_step sigma pi (Sseq i1 i2) sigma' pi' (Sseq i1' i2)
| one_step_seq_skip:
forall sigma pi:env, i:stmt.
one_step sigma pi (Sseq Sskip i) sigma pi i
| one_step_if_true:
forall sigma pi:env, e:term, i1 i2:stmt.
eval_term sigma pi e = (Vbool True) ->
one_step sigma pi (Sif e i1 i2) sigma pi i1
| one_step_if_false:
forall sigma pi:env, e:term, i1 i2:stmt.
eval_term sigma pi e = (Vbool False) ->
one_step sigma pi (Sif e i1 i2) sigma pi i2
| one_step_assert:
forall sigma pi:env, f:fmla.
eval_fmla sigma pi f ->
one_step sigma pi (Sassert f) sigma pi Sskip
| one_step_while_true:
forall sigma pi:env, e:term, inv:fmla, i:stmt.
eval_fmla sigma pi inv ->
eval_term sigma pi e = (Vbool True) ->
one_step sigma pi (Swhile e inv i) sigma pi (Sseq i (Swhile e inv i))
| one_step_while_false:
forall sigma pi:env, e:term, inv:fmla, i:stmt.
eval_fmla sigma pi inv ->
eval_term sigma pi e = (Vbool False) ->
one_step sigma pi (Swhile e inv i) sigma pi Sskip
(***
lemma progress:
forall s:state, i:stmt. i <> Sskip ->
exists s':state, i':stmt. one_step s i s' i'
*)
(** many steps of execution *)
inductive many_steps env env stmt env env stmt int =
| many_steps_refl:
forall sigma pi:env, i:stmt. many_steps sigma pi i sigma pi i 0
| many_steps_trans:
forall sigma1 pi1 sigma2 pi2 sigma3 pi3:env, i1 i2 i3:stmt, n:int.
one_step sigma1 pi1 i1 sigma2 pi2 i2 ->
many_steps sigma2 pi2 i2 sigma3 pi3 i3 n ->
many_steps sigma1 pi1 i1 sigma3 pi3 i3 (n+1)
lemma steps_non_neg:
forall sigma1 pi1 sigma2 pi2:env, i1 i2:stmt, n:int.
many_steps sigma1 pi1 i1 sigma2 pi2 i2 n -> n >= 0
lemma many_steps_seq:
forall sigma1 pi1 sigma3 pi3:env, i1 i2:stmt, n:int.
many_steps sigma1 pi1 (Sseq i1 i2) sigma3 pi3 Sskip n ->
exists sigma2 pi2:env, n1 n2:int.
many_steps sigma1 pi1 i1 sigma2 pi2 Sskip n1 /\
many_steps sigma2 pi2 i2 sigma3 pi3 Sskip n2 /\
n = 1 + n1 + n2
predicate valid_fmla (p:fmla) = forall sigma pi:env. eval_fmla sigma pi p
(** {3 Hoare triples} *)
(** partial correctness *)
predicate valid_triple (p:fmla) (i:stmt) (q:fmla) =
forall sigma pi:env. eval_fmla sigma pi p ->
forall sigma' pi':env, n:int. many_steps sigma pi i sigma' pi' Sskip n ->
eval_fmla sigma' pi' q
(*** total correctness *)
(***
predicate total_valid_triple (p:fmla) (i:stmt) (q:fmla) =
forall s:state. eval_fmla s p ->
exists s':state, n:int. many_steps s i s' Sskip n /\
eval_fmla s' q
*)
end
theory TestSemantics
use import Imp
function my_sigma : env = IdMap.const (Vint 0)
function my_pi : env = IdMap.const (Vint 42)
goal Test13 :
eval_term my_sigma my_pi (Tconst 13) = Vint 13
goal Test42 :
eval_term my_sigma my_pi (Tvar 0) = Vint 42
goal Test0 :
eval_term my_sigma my_pi (Tderef 0) = Vint 0
goal Test55 :
eval_term my_sigma my_pi (Tbin (Tvar 0) Oplus (Tconst 13)) = Vint 55
goal Ass42 :
let x = 0 in
forall sigma' pi':env.
one_step my_sigma my_pi (Sassign x (Tconst 42)) sigma' pi' Sskip ->
IdMap.get sigma' x = Vint 42
goal If42 :
let x = 0 in
forall sigma1 pi1 sigma2 pi2:env, i:stmt.
one_step my_sigma my_pi
(Sif (Tbin (Tderef x) Ole (Tconst 10))
(Sassign x (Tconst 13))
(Sassign x (Tconst 42)))
sigma1 pi1 i ->
one_step sigma1 pi1 i sigma2 pi2 Sskip ->
IdMap.get sigma2 x = Vint 13
end
(** {2 Hoare logic} *)
theory HoareLogic
use import Imp
(** Hoare logic rules (partial correctness) *)
lemma consequence_rule:
forall p p' q q':fmla, i:stmt.
valid_fmla (Fimplies p' p) ->
valid_triple p i q ->
valid_fmla (Fimplies q q') ->
valid_triple p' i q'
lemma skip_rule:
forall q:fmla. valid_triple q Sskip q
lemma assign_rule:
forall q:fmla, x id:ident, e:term.
fresh_in_fmla id q ->
valid_triple (Flet id e (subst q x id)) (Sassign x e) q
lemma seq_rule:
forall p q r:fmla, i1 i2:stmt.
valid_triple p i1 r /\ valid_triple r i2 q ->
valid_triple p (Sseq i1 i2) q
lemma if_rule:
forall e:term, p q:fmla, i1 i2:stmt.
valid_triple (Fand p (Fterm e)) i1 q /\
valid_triple (Fand p (Fnot (Fterm e))) i2 q ->
valid_triple p (Sif e i1 i2) q
lemma assert_rule:
forall f p:fmla. valid_fmla (Fimplies p f) ->
valid_triple p (Sassert f) p
lemma assert_rule_ext:
forall f p:fmla.
valid_triple (Fimplies f p) (Sassert f) p
lemma while_rule:
forall e:term, inv:fmla, i:stmt.
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:stmt.
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
(** {2 WP calculus} *)
module WP
use import Imp
use set.Set
predicate assigns (sigma:env) (a:Set.set ident) (sigma':env) =
forall i:ident. not (Set.mem i a) ->
IdMap.get sigma i = IdMap.get sigma' i
lemma assigns_refl:
forall sigma:env, a:Set.set ident. assigns sigma a sigma
lemma assigns_trans:
forall sigma1 sigma2 sigma3:env, a:Set.set ident.
assigns sigma1 a sigma2 /\ assigns sigma2 a sigma3 ->
assigns sigma1 a sigma3
lemma assigns_union_left:
forall sigma sigma':env, s1 s2:Set.set ident.
assigns sigma s1 sigma' -> assigns sigma (Set.union s1 s2) sigma'
lemma assigns_union_right:
forall sigma sigma':env, s1 s2:Set.set ident.
assigns sigma s2 sigma' -> assigns sigma (Set.union s1 s2) sigma'
predicate stmt_writes (i:stmt) (w:Set.set ident) =
match i with
| Sskip | Sassert _ -> true
| Sassign id _ -> Set.mem id w
| Sseq s1 s2 | Sif _ s1 s2 -> stmt_writes s1 w /\ stmt_writes s2 w
| Swhile _ _ s -> stmt_writes s w
end
let rec compute_writes (s:stmt) : Set.set ident
ensures {
forall sigma pi sigma' pi':env, n:int.
many_steps sigma pi s sigma' pi' Sskip n ->
assigns sigma result sigma' }
= match s with
| Sskip -> Set.empty
| Sassign i _ -> Set.singleton i
| Sseq s1 s2 -> Set.union (compute_writes s1) (compute_writes s2)
| Sif _ s1 s2 -> Set.union (compute_writes s1) (compute_writes s2)
| Swhile _ _ s -> compute_writes s
| Sassert _ -> Set.empty
end
val fresh_from_fmla (q:fmla) : ident
ensures { fresh_in_fmla result q }
val abstract_effects (i:stmt) (f:fmla) : fmla
ensures { forall sigma pi:env. eval_fmla sigma pi result ->
eval_fmla sigma pi f /\
(***
forall sigma':env, w:Set.set ident.
stmt_writes i w /\ assigns sigma w sigma' ->
eval_fmla sigma' pi result
*) forall sigma' pi':env, n:int.
many_steps sigma pi i sigma' pi' Sskip n ->
eval_fmla sigma' pi' result
}
use HoareLogic
let rec wp (i:stmt) (q:fmla)
ensures { valid_triple result i q }
= match i with
| Sskip -> q
| Sseq i1 i2 -> wp i1 (wp i2 q)
| Sassign x e ->
let id = fresh_from_fmla q in Flet id e (subst q x id)
| Sif e i1 i2 ->
Fand (Fimplies (Fterm e) (wp i1 q))
(Fimplies (Fnot (Fterm e)) (wp i2 q))
| Sassert f ->
Fimplies f q (* liberal wp, no termination required *)
(* Fand f q *) (* strict wp, termination required *)
| Swhile e inv i ->
Fand inv
(abstract_effects i
(Fand
(Fimplies (Fand (Fterm e) inv) (wp i inv))
(Fimplies (Fand (Fnot (Fterm e)) inv) q)))
end
end
(***
Local Variables:
compile-command: "why3ide wp2.mlw"
End:
*)