wp3.mlw 13.8 KB
 MARCHE Claude committed Jun 08, 2012 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 `````` (** {1 A certified WP calculus} *) (** {2 A simple imperative language with expressions, syntax and semantics} *) theory ImpExpr use import int.Int use import bool.Bool (** types and values *) type datatype = TYunit | TYint | TYbool type value = Vvoid | Vint int | Vbool bool (** terms and formulas *) type operator = Oplus | Ominus | Omult | Ole type ident = int constant result : ident = (-1) type term = | Tvalue value | 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 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 | Tvalue v -> v | 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 TYint f -> forall n:int. eval_fmla sigma (IdMap.set pi x (Vint n)) f | Fforall x TYbool f -> forall b:bool. eval_fmla sigma (IdMap.set pi x (Vbool b)) f | Fforall x TYunit f -> eval_fmla sigma (IdMap.set pi x Vvoid) 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 | Tvalue _ | 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 | Tvalue _ -> 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) (* expressions *) type expr = | Evalue value | Ebin expr operator expr | Evar ident | Ederef ident | Eassign ident expr | Eseq expr expr | Elet ident expr expr | Eif expr expr expr | Eassert fmla | Ewhile expr fmla expr constant void : expr = Evalue Vvoid (* lemma check_skip: forall s:stmt. s=Sskip \/s<>Sskip *) (** small-steps semantics for statements *) inductive one_step env env expr env env expr = | one_step_assign_ctxt: forall sigma pi sigma' pi':env, x:ident, 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: `````` MARCHE Claude committed Sep 03, 2012 182 `````` forall sigma pi:env, x:ident, v:value. `````` MARCHE Claude committed Jun 08, 2012 183 184 185 186 187 188 189 190 191 `````` one_step sigma pi (Eassign x (Evalue v)) (IdMap.set sigma x v) pi void | one_step_seq_ctxt: forall sigma pi sigma' pi':env, 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: `````` MARCHE Claude committed Oct 11, 2012 192 `````` forall sigma pi:env, e:expr. `````` MARCHE Claude committed Jun 08, 2012 193 194 195 196 197 198 199 200 201 202 203 204 `````` one_step sigma pi (Eseq void e) sigma pi e | one_step_let_ctxt: forall sigma pi sigma' pi':env, 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 pi:env, id:ident, v:value, e:expr. one_step sigma pi (Elet id (Evalue v) e) sigma (IdMap.set pi id v) e | one_step_if_ctxt: `````` MARCHE Claude committed Oct 11, 2012 205 `````` forall sigma pi sigma' pi':env, e1 e1' e2 e3:expr. `````` MARCHE Claude committed Jun 08, 2012 206 207 208 209 `````` 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: `````` MARCHE Claude committed Oct 11, 2012 210 `````` forall sigma pi:env, e1 e2:expr. `````` MARCHE Claude committed Jun 08, 2012 211 212 213 `````` one_step sigma pi (Eif (Evalue (Vbool True)) e1 e2) sigma pi e1 | one_step_if_false: `````` MARCHE Claude committed Oct 11, 2012 214 `````` forall sigma pi:env, e1 e2:expr. `````` MARCHE Claude committed Jun 08, 2012 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 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506 507 508 509 510 511 512 513 514 515 516 517 518 `````` one_step sigma pi (Eif (Evalue (Vbool False)) e1 e2) sigma pi e2 | one_step_assert: forall sigma pi:env, f:fmla. eval_fmla sigma pi f -> one_step sigma pi (Eassert f) sigma pi void | one_step_while: forall sigma pi:env, e:expr, inv:fmla, e':expr. one_step sigma pi (Ewhile e inv e') sigma pi (Eif e (Eseq e' (Ewhile e inv e')) void) (*** lemma progress: forall s:state, i:expr. i <> Sskip -> exists s':state, i':expr. one_step s i s' i' *) (** many steps of execution *) inductive many_steps env env expr env env expr int = | many_steps_refl: forall sigma pi:env, i:expr. many_steps sigma pi i sigma pi i 0 | many_steps_trans: forall sigma1 pi1 sigma2 pi2 sigma3 pi3:env, i1 i2 i3:expr, 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:expr, n:int. many_steps sigma1 pi1 i1 sigma2 pi2 i2 n -> n >= 0 lemma many_steps_seq: forall sigma1 pi1 sigma3 pi3:env, e1 e2:expr, n:int. many_steps sigma1 pi1 (Eseq e1 e2) sigma3 pi3 void n -> exists sigma2 pi2:env, 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 pi1 sigma3 pi3:env, id:ident, e1 e2:expr, v2:value, n:int. many_steps sigma1 pi1 (Elet id e1 e2) sigma3 pi3 (Evalue v2) n -> exists sigma2 pi2:env, v1:value, n1 n2:int. many_steps sigma1 pi1 e1 sigma2 pi2 (Evalue v1) n1 /\ many_steps sigma2 (IdMap.set pi2 id v1) e2 sigma3 pi3 (Evalue v2) 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) (e:expr) (q:fmla) = forall sigma pi:env. eval_fmla sigma pi p -> forall sigma' pi':env, v:value, n:int. many_steps sigma pi e sigma' pi' (Evalue v) n -> eval_fmla sigma' (IdMap.set pi' result v) q (*** total correctness *) (*** predicate total_valid_triple (p:fmla) (i:expr) (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 ImpExpr function my_sigma : env = IdMap.const (Vint 0) function my_pi : env = IdMap.const (Vint 42) goal Test13 : eval_term my_sigma my_pi (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 (Tvar 0) = Vint 42 goal Test42expr : many_steps my_sigma my_pi (Evar 0) my_sigma my_pi (Evalue (Vint 42)) 1 goal Test0 : eval_term my_sigma my_pi (Tderef 0) = Vint 0 goal Test0expr : many_steps my_sigma my_pi (Ederef 0) my_sigma my_pi (Evalue (Vint 0)) 1 goal Test55 : eval_term my_sigma my_pi (Tbin (Tvar 0) Oplus (Tvalue (Vint 13))) = Vint 55 goal Test55expr : many_steps my_sigma my_pi (Ebin (Evar 0) Oplus (Evalue (Vint 13))) my_sigma my_pi (Evalue (Vint 55)) 3 goal Ass42 : let x = 0 in forall sigma' pi':env. one_step my_sigma my_pi (Eassign x (Evalue (Vint 42))) sigma' pi' void -> IdMap.get sigma' x = Vint 42 goal If42 : let x = 0 in forall sigma1 pi1 sigma2 pi2:env, e:expr. one_step my_sigma my_pi (Eif (Ebin (Ederef x) Ole (Evalue (Vint 10))) (Eassign x (Evalue (Vint 13))) (Eassign x (Evalue (Vint 42)))) sigma1 pi1 e -> one_step sigma1 pi1 e sigma2 pi2 void -> IdMap.get sigma2 x = 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:ident, e:expr. valid_triple p e (subst 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. fresh_in_fmla result p -> valid_triple p e1 r /\ valid_triple (Flet result (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 (** {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 expr_writes (i:expr) (w:Set.set ident) = match i with | Sskip | Sassert _ -> true | Sassign id _ -> Set.mem id w | Sseq s1 s2 | Sif _ s1 s2 -> expr_writes s1 w /\ expr_writes s2 w | Swhile _ _ s -> expr_writes s w end let rec compute_writes (s:expr) : Set.set ident = { } 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 { forall sigma pi sigma' pi':env, n:int. many_steps sigma pi s sigma' pi' Sskip n -> assigns sigma result sigma' } val fresh_from_fmla (q:fmla) : { } ident { fresh_in_fmla result q } val abstract_effects (i:expr) (f:fmla) : { } fmla { forall sigma pi:env. eval_fmla sigma pi result -> eval_fmla sigma pi f /\ (*** forall sigma':env, w:Set.set ident. expr_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:expr) (q:fmla) = { true } 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 { valid_triple result i q } end *) (*** Local Variables: compile-command: "why3ide wp3.mlw" End: *) ``````