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prog.ml
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(* ========================================================================= *)
(* Simple WHILE-language with relational semantics. *)
(* ========================================================================= *)
prioritize_num();;
parse_as_infix("refined",(12,"right"));;
(* ------------------------------------------------------------------------- *)
(* Logical operations "lifted" to predicates, for readability. *)
(* ------------------------------------------------------------------------- *)
parse_as_infix("AND",(20,"right"));;
parse_as_infix("OR",(16,"right"));;
parse_as_infix("IMP",(13,"right"));;
parse_as_infix("IMPLIES",(12,"right"));;
let FALSE = new_definition
`FALSE = \x:S. F`;;
let TRUE = new_definition
`TRUE = \x:S. T`;;
let NOT = new_definition
`NOT p = \x:S. ~(p x)`;;
let AND = new_definition
`p AND q = \x:S. p x /\ q x`;;
let OR = new_definition
`p OR q = \x:S. p x \/ q x`;;
let ANDS = new_definition
`ANDS P = \x:S. !p. P p ==> p x`;;
let ORS = new_definition
`ORS P = \x:S. ?p. P p /\ p x`;;
let IMP = new_definition
`p IMP q = \x:S. p x ==> q x`;;
(* ------------------------------------------------------------------------- *)
(* This one is different, corresponding to "subset". *)
(* ------------------------------------------------------------------------- *)
let IMPLIES = new_definition
`p IMPLIES q <=> !x:S. p x ==> q x`;;
(* ------------------------------------------------------------------------- *)
(* Simple procedure to prove tautologies at the predicate level. *)
(* ------------------------------------------------------------------------- *)
let PRED_TAUT =
let tac =
REWRITE_TAC[FALSE; TRUE; NOT; AND; OR; ANDS; ORS; IMP;
IMPLIES; FUN_EQ_THM] THEN MESON_TAC[] in
fun tm -> prove(tm,tac);;
(* ------------------------------------------------------------------------- *)
(* Some applications. *)
(* ------------------------------------------------------------------------- *)
let IMPLIES_TRANS = PRED_TAUT
`!p q r. p IMPLIES q /\ q IMPLIES r ==> p IMPLIES r`;;
(* ------------------------------------------------------------------------- *)
(* Enumerated type of basic commands, and other derived commands. *)
(* ------------------------------------------------------------------------- *)
parse_as_infix("Seq",(26,"right"));;
let command_INDUCTION,command_RECURSION = define_type
"command = Assign (S->S)
| Seq command command
| Ite (S->bool) command command
| While (S->bool) command";;
let SKIP = new_definition
`SKIP = Assign I`;;
let ABORT = new_definition
`ABORT = While TRUE SKIP`;;
let IF = new_definition
`IF e c = Ite e c SKIP`;;
let DO = new_definition
`DO c e = c Seq (While e c)`;;
let ASSERT = new_definition
`ASSERT g = Ite g SKIP ABORT`;;
(* ------------------------------------------------------------------------- *)
(* Annotation commands, to allow insertion of loop (in)variants. *)
(* ------------------------------------------------------------------------- *)
let AWHILE = new_definition
`AWHILE (i:S->bool) (v:S->S->bool) (e:S->bool) c = While e c`;;
let ADO = new_definition
`ADO (i:S->bool) (v:S->S->bool) c (e:S->bool) = DO c e`;;
(* ------------------------------------------------------------------------- *)
(* Useful properties of type constructors for commands. *)
(* ------------------------------------------------------------------------- *)
let command_DISTINCT =
distinctness "command";;
let command_INJECTIVE =
injectivity "command";;
(* ------------------------------------------------------------------------- *)
(* Relational semantics of commands. *)
(* ------------------------------------------------------------------------- *)
let sem_RULES,sem_INDUCT,sem_CASES = new_inductive_definition
`(!f s. sem(Assign f) s (f s)) /\
(!c1 c2 s s' s''. sem(c1) s s' /\ sem(c2) s' s''
==> sem(c1 Seq c2) s s'') /\
(!e c1 c2 s s'. e s /\ sem(c1) s s' ==> sem(Ite e c1 c2) s s') /\
(!e c1 c2 s s'. ~(e s) /\ sem(c2) s s' ==> sem(Ite e c1 c2) s s') /\
(!e c s. ~(e s) ==> sem(While e c) s s) /\
(!e c s s' s''. e s /\ sem(c) s s' /\ sem(While e c) s' s''
==> sem(While e c) s s'')`;;
(* ------------------------------------------------------------------------- *)
(* A more "denotational" view of the semantics. *)
(* ------------------------------------------------------------------------- *)
let SEM_ASSIGN = prove
(`sem(Assign f) s s' <=> (s' = f s)`,
GEN_REWRITE_TAC LAND_CONV [sem_CASES] THEN
REWRITE_TAC[command_DISTINCT; command_INJECTIVE] THEN MESON_TAC[]);;
let SEM_SEQ = prove
(`sem(c1 Seq c2) s s' <=> ?s''. sem c1 s s'' /\ sem c2 s'' s'`,
GEN_REWRITE_TAC LAND_CONV [sem_CASES] THEN
REWRITE_TAC[command_DISTINCT; command_INJECTIVE] THEN MESON_TAC[]);;
let SEM_ITE = prove
(`sem(Ite e c1 c2) s s' <=> e s /\ sem c1 s s' \/
~(e s) /\ sem c2 s s'`,
GEN_REWRITE_TAC LAND_CONV [sem_CASES] THEN
REWRITE_TAC[command_DISTINCT; command_INJECTIVE] THEN MESON_TAC[]);;
let SEM_SKIP = prove
(`sem(SKIP) s s' <=> (s' = s)`,
REWRITE_TAC[SKIP; SEM_ASSIGN; I_THM]);;
let SEM_IF = prove
(`sem(IF e c) s s' <=> e s /\ sem c s s' \/ ~(e s) /\ (s = s')`,
REWRITE_TAC[IF; SEM_ITE; SEM_SKIP; EQ_SYM_EQ]);;
let SEM_WHILE = prove
(`sem(While e c) s s' <=> sem(IF e (c Seq While e c)) s s'`,
GEN_REWRITE_TAC LAND_CONV [sem_CASES] THEN
REWRITE_TAC[FUN_EQ_THM; SEM_IF; SEM_SEQ] THEN REPEAT GEN_TAC THEN
REWRITE_TAC[command_DISTINCT; command_INJECTIVE] THEN MESON_TAC[]);;
let SEM_ABORT = prove
(`sem(ABORT) s s' <=> F`,
let lemma = prove
(`!c s s'. sem c s s' ==> ~(c = ABORT)`,
MATCH_MP_TAC sem_INDUCT THEN
REWRITE_TAC[command_DISTINCT; command_INJECTIVE; ABORT] THEN
REWRITE_TAC[FUN_EQ_THM; TRUE] THEN MESON_TAC[]) in
MESON_TAC[lemma]);;
let SEM_DO = prove
(`sem(DO c e) s s' <=> sem(c Seq IF e (DO c e)) s s'`,
REWRITE_TAC[DO; SEM_SEQ; GSYM SEM_WHILE]);;
let SEM_ASSERT = prove
(`sem(ASSERT g) s s' <=> g s /\ (s' = s)`,
REWRITE_TAC[ASSERT; SEM_ITE; SEM_SKIP; SEM_ABORT]);;
(* ------------------------------------------------------------------------- *)
(* Proofs that all commands are deterministic. *)
(* ------------------------------------------------------------------------- *)
let deterministic = new_definition
`deterministic r <=> !s s1 s2. r s s1 /\ r s s2 ==> (s1 = s2)`;;
let DETERMINISM = prove
(`!c:(S)command. deterministic(sem c)`,
REWRITE_TAC[deterministic] THEN SUBGOAL_THEN
`!c s s1. sem c s s1 ==> !s2:S. sem c s s2 ==> (s1 = s2)`
(fun th -> MESON_TAC[th]) THEN
MATCH_MP_TAC sem_INDUCT THEN CONJ_TAC THENL
[ALL_TAC; REPEAT CONJ_TAC THEN REPEAT GEN_TAC THEN DISCH_TAC] THEN
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[sem_CASES] THEN
REWRITE_TAC[command_DISTINCT; command_INJECTIVE] THEN
ASM_MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Termination, weakest liberal precondition and weakest precondition. *)
(* ------------------------------------------------------------------------- *)
let terminates = new_definition
`terminates c s <=> ?s'. sem c s s'`;;
let wlp = new_definition
`wlp c q s <=> !s'. sem c s s' ==> q s'`;;
let wp = new_definition
`wp c q s <=> terminates c s /\ wlp c q s`;;
(* ------------------------------------------------------------------------- *)
(* Dijkstra's healthiness conditions (the last because of determinism). *)
(* ------------------------------------------------------------------------- *)
let WP_TOTAL = prove
(`!c. (wp c FALSE = FALSE)`,
REWRITE_TAC[FUN_EQ_THM; wp; wlp; terminates; FALSE] THEN MESON_TAC[]);;
let WP_MONOTONIC = prove
(`q IMPLIES r ==> wp c q IMPLIES wp c r`,
REWRITE_TAC[IMPLIES; wp; wlp; terminates] THEN MESON_TAC[]);;
let WP_CONJUNCTIVE = prove
(`(wp c q) AND (wp c r) = wp c (q AND r)`,
REWRITE_TAC[FUN_EQ_THM; wp; wlp; terminates; AND] THEN MESON_TAC[]);;
let WP_DISJUNCTIVE = prove
(`(wp c p) OR (wp c q) = wp c (p OR q)`,
REWRITE_TAC[FUN_EQ_THM; wp; wlp; OR; terminates] THEN
MESON_TAC[REWRITE_RULE[deterministic] DETERMINISM]);;
(* ------------------------------------------------------------------------- *)
(* Weakest preconditions for the primitive and derived commands. *)
(* ------------------------------------------------------------------------- *)
let WP_ASSIGN = prove
(`!f q. wp (Assign f) q = q o f`,
REWRITE_TAC[wp; wlp; terminates; o_THM; FUN_EQ_THM; SEM_ASSIGN] THEN
MESON_TAC[]);;
let WP_SEQ = prove
(`!c1 c2 q. wp (c1 Seq c2) q = wp c1 (wp c2 q)`,
REWRITE_TAC[wp; wlp; terminates; SEM_SEQ; FUN_EQ_THM] THEN
MESON_TAC[REWRITE_RULE[deterministic] DETERMINISM]);;
let WP_ITE = prove
(`!e c1 c2 q. wp (Ite e c1 c2) q = (e AND wp c1 q) OR (NOT e AND wp c2 q)`,
REWRITE_TAC[wp; wlp; terminates; SEM_ITE; FUN_EQ_THM; AND; OR; NOT] THEN
MESON_TAC[]);;
let WP_WHILE = prove
(`!e c. wp (IF e (c Seq While e c)) q = wp (While e c) q`,
REWRITE_TAC[FUN_EQ_THM; wp; wlp; terminates; GSYM SEM_WHILE]);;
let WP_SKIP = prove
(`!q. wp SKIP q = q`,
REWRITE_TAC[FUN_EQ_THM; SKIP; WP_ASSIGN; I_THM; o_THM]);;
let WP_ABORT = prove
(`!q. wp ABORT q = FALSE`,
REWRITE_TAC[FUN_EQ_THM; wp; wlp; terminates; SEM_ABORT; FALSE]);;
let WP_IF = prove
(`!e c q. wp (IF e c) q = (e AND wp c q) OR (NOT e AND q)`,
REWRITE_TAC[IF; WP_ITE; WP_SKIP]);;
let WP_DO = prove
(`!e c. wp (c Seq IF e (DO c e)) q = wp (DO c e) q`,
REWRITE_TAC[FUN_EQ_THM; wp; wlp; terminates; GSYM SEM_DO]);;
let WP_ASSERT = prove
(`!g q. wp (ASSERT g) q = g AND q`,
REWRITE_TAC[wp; wlp; terminates; SEM_ASSERT; FUN_EQ_THM; AND] THEN
MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Rules for total correctness. *)
(* ------------------------------------------------------------------------- *)
let correct = new_definition
`correct p c q <=> p IMPLIES (wp c q)`;;
let CORRECT_PRESTRENGTH = prove
(`!p p' c q. p IMPLIES p' /\ correct p' c q ==> correct p c q`,
REWRITE_TAC[correct; IMPLIES_TRANS]);;
let CORRECT_POSTWEAK = prove
(`!p c q q'. correct p c q' /\ q' IMPLIES q ==> correct p c q`,
REWRITE_TAC[correct] THEN MESON_TAC[WP_MONOTONIC; IMPLIES_TRANS]);;
let CORRECT_ASSIGN = prove
(`!p f q. (p IMPLIES (\s. q(f s))) ==> correct p (Assign f) q`,
REWRITE_TAC[correct; WP_ASSIGN; IMPLIES; o_THM]);;
let CORRECT_SEQ = prove
(`!p q r c1 c2.
correct p c1 r /\ correct r c2 q ==> correct p (c1 Seq c2) q`,
REWRITE_TAC[correct; WP_SEQ; o_THM] THEN
MESON_TAC[WP_MONOTONIC; IMPLIES_TRANS]);;
let CORRECT_ITE = prove
(`!p e c1 c2 q.
correct (p AND e) c1 q /\ correct (p AND (NOT e)) c2 q
==> correct p (Ite e c1 c2) q`,
REWRITE_TAC[correct; WP_ITE; AND; NOT; IMPLIES; OR] THEN MESON_TAC[]);;
let CORRECT_WHILE = prove
(`! (<<) p c q e invariant.
WF(<<) /\
p IMPLIES invariant /\
(NOT e) AND invariant IMPLIES q /\
(!X:S. correct
(invariant AND e AND (\s. X = s)) c (invariant AND (\s. s << X)))
==> correct p (While e c) q`,
REWRITE_TAC[correct; IMPLIES; IN; AND; NOT] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN `!s:S. invariant s ==> wp (While e c) q s` MP_TAC THENL
[ALL_TAC; ASM_MESON_TAC[]] THEN
FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[WF_IND]) THEN
X_GEN_TAC `s:S` THEN REPEAT DISCH_TAC THEN
ONCE_REWRITE_TAC[GSYM WP_WHILE] THEN
REWRITE_TAC[WP_IF; WP_SEQ; AND; OR; NOT; o_THM] THEN
ASM_CASES_TAC `(e:S->bool) s` THEN ASM_REWRITE_TAC[] THENL
[ALL_TAC; ASM_MESON_TAC[]] THEN
SUBGOAL_THEN `wp c (\x:S. invariant x /\ x << s) (s:S) :bool` MP_TAC THENL
[FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]; ALL_TAC] THEN
SUBGOAL_THEN `(\x:S. invariant x /\ x << (s:S)) IMPLIES wp (While e c) q`
MP_TAC THENL [REWRITE_TAC[IMPLIES] THEN ASM_MESON_TAC[]; ALL_TAC] THEN
MESON_TAC[WP_MONOTONIC; IMPLIES]);;
let CORRECT_SKIP = prove
(`!p q. (p IMPLIES q) ==> correct p SKIP q`,
REWRITE_TAC[correct; WP_SKIP]);;
let CORRECT_ABORT = prove
(`!p q. F ==> correct p ABORT q`,
REWRITE_TAC[]);;
let CORRECT_IF = prove
(`!p e c q.
correct (p AND e) c q /\ (p AND (NOT e)) IMPLIES q
==> correct p (IF e c) q`,
REWRITE_TAC[correct; WP_IF; AND; NOT; IMPLIES; OR] THEN MESON_TAC[]);;
let CORRECT_DO = prove
(`! (<<) p q c invariant.
WF(<<) /\
(e AND invariant) IMPLIES p /\
((NOT e) AND invariant) IMPLIES q /\
(!X:S. correct
(p AND (\s. X = s)) c (invariant AND (\s. s << X)))
==> correct p (DO c e) q`,
REPEAT STRIP_TAC THEN REWRITE_TAC[DO] THEN
MATCH_MP_TAC CORRECT_SEQ THEN EXISTS_TAC `invariant:S->bool` THEN
ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[POP_ASSUM_LIST(MP_TAC o end_itlist CONJ) THEN
REWRITE_TAC[correct; GSYM WP_CONJUNCTIVE] THEN
REWRITE_TAC[AND; IMPLIES] THEN MESON_TAC[];
MATCH_MP_TAC CORRECT_WHILE THEN
MAP_EVERY EXISTS_TAC [`(<<) :S->S->bool`; `invariant:S->bool`] THEN
ASM_REWRITE_TAC[IMPLIES] THEN X_GEN_TAC `X:S` THEN
MATCH_MP_TAC CORRECT_PRESTRENGTH THEN
EXISTS_TAC `p AND (\s:S. X = s)` THEN ASM_REWRITE_TAC[] THEN
UNDISCH_TAC `(e:S->bool) AND invariant IMPLIES p` THEN
REWRITE_TAC[AND; IMPLIES] THEN MESON_TAC[]]);;
let CORRECT_ASSERT = prove
(`!p g q. p IMPLIES (g AND q) ==> correct p (ASSERT g) q`,
REWRITE_TAC[correct; WP_ASSERT]);;
(* ------------------------------------------------------------------------- *)
(* VCs for the basic commands (in fact only assign should be needed). *)
(* ------------------------------------------------------------------------- *)
let VC_ASSIGN = prove
(`p IMPLIES (q o f) ==> correct p (Assign f) q`,
REWRITE_TAC[o_DEF; CORRECT_ASSIGN]);;
let VC_SKIP = prove
(`p IMPLIES q ==> correct p SKIP q`,
REWRITE_TAC[CORRECT_SKIP]);;
let VC_ABORT = prove
(`F ==> correct p ABORT q`,
MATCH_ACCEPT_TAC CORRECT_ABORT);;
let VC_ASSERT = prove
(`p IMPLIES (b AND q) ==> correct p (ASSERT b) q`,
REWRITE_TAC[CORRECT_ASSERT]);;
(* ------------------------------------------------------------------------- *)
(* VCs for composite commands other than sequences. *)
(* ------------------------------------------------------------------------- *)
let VC_ITE = prove
(`correct (p AND e) c1 q /\ correct (p AND NOT e) c2 q
==> correct p (Ite e c1 c2) q`,
REWRITE_TAC[CORRECT_ITE]);;
let VC_IF = prove
(`correct (p AND e) c q /\ p AND NOT e IMPLIES q
==> correct p (IF e c) q`,
REWRITE_TAC[CORRECT_IF]);;
let VC_AWHILE_VARIANT = prove
(`WF(<<) /\
p IMPLIES invariant /\
(NOT e) AND invariant IMPLIES q /\
(!X. correct
(invariant AND e AND (\s. X = s)) c (invariant AND (\s. s << X)))
==> correct p (AWHILE invariant (<<) e c) q`,
REWRITE_TAC[AWHILE; CORRECT_WHILE]);;
let VC_AWHILE_MEASURE = prove
(`p IMPLIES invariant /\
(NOT e) AND invariant IMPLIES q /\
(!X. correct
(invariant AND e AND (\s:S. X = m(s)))
c
(invariant AND (\s. m(s) < X)))
==> correct p (AWHILE invariant (MEASURE m) e c) q`,
STRIP_TAC THEN MATCH_MP_TAC VC_AWHILE_VARIANT THEN
ASM_REWRITE_TAC[WF_MEASURE] THEN
X_GEN_TAC `X:S` THEN FIRST_ASSUM(MP_TAC o SPEC `(m:S->num) X`) THEN
REWRITE_TAC[correct; AND; IMPLIES; MEASURE] THEN MESON_TAC[]);;
let VC_ADO_VARIANT = prove
(`WF(<<) /\
(e AND invariant) IMPLIES p /\
((NOT e) AND invariant) IMPLIES q /\
(!X. correct
(p AND (\s. X = s)) c (invariant AND (\s. s << X)))
==> correct p (ADO invariant (<<) c e) q`,
REWRITE_TAC[ADO; CORRECT_DO]);;
let VC_ADO_MEASURE = prove
(`(e AND invariant) IMPLIES p /\
((NOT e) AND invariant) IMPLIES q /\
(!X. correct
(p AND (\s:S. X = m(s))) c (invariant AND (\s. m(s) < X)))
==> correct p (ADO invariant (MEASURE m) c e) q`,
STRIP_TAC THEN MATCH_MP_TAC VC_ADO_VARIANT THEN
ASM_REWRITE_TAC[WF_MEASURE] THEN
X_GEN_TAC `X:S` THEN FIRST_ASSUM(MP_TAC o SPEC `(m:S->num) X`) THEN
REWRITE_TAC[correct; AND; IMPLIES; MEASURE] THEN MESON_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* VCs for sequences of commands, using intelligence where possible. *)
(* ------------------------------------------------------------------------- *)
let VC_SEQ_ASSERT_LEFT = prove
(`p IMPLIES b /\ correct b c q ==> correct p (ASSERT b Seq c) q`,
MESON_TAC[CORRECT_SEQ; CORRECT_ASSERT; CORRECT_PRESTRENGTH;
PRED_TAUT `(p IMPLIES b) ==> (p IMPLIES b AND p)`]);;
let VC_SEQ_ASSERT_RIGHT = prove
(`correct p c b /\ b IMPLIES q ==> correct p (c Seq (ASSERT b)) q`,
MESON_TAC[CORRECT_SEQ; CORRECT_ASSERT;
PRED_TAUT `(p IMPLIES b) ==> (p IMPLIES p AND b)`]);;
let VC_SEQ_ASSERT_MIDDLE = prove
(`correct p c b /\ correct b c' q
==> correct p (c Seq (ASSERT b) Seq c') q`,
MESON_TAC[CORRECT_SEQ; CORRECT_ASSERT; PRED_TAUT `b IMPLIES b AND b`]);;
let VC_SEQ_ASSIGN_LEFT = prove
(`(p o f = p) /\ (f o f = f) /\
correct (p AND (\s:S. s = f s)) c q
==> correct p ((Assign f) Seq c) q`,
REWRITE_TAC[FUN_EQ_THM; o_THM] THEN STRIP_TAC THEN
MATCH_MP_TAC CORRECT_SEQ THEN EXISTS_TAC `p AND (\s:S. s = f s)` THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC VC_ASSIGN THEN
ASM_REWRITE_TAC[IMPLIES; AND; o_THM]);;
let VC_SEQ_ASSIGN_RIGHT = prove
(`correct p c (q o f) ==> correct p (c Seq (Assign f)) q`,
MESON_TAC[CORRECT_SEQ; VC_ASSIGN; PRED_TAUT `(p:S->bool) IMPLIES p`]);;
(* ------------------------------------------------------------------------- *)
(* Parser for correctness assertions. *)
(* ------------------------------------------------------------------------- *)
let rec dive_to_var ptm =
match ptm with
Varp(_,_) as vp -> vp | Typing(t,_) -> dive_to_var t | _ -> fail();;
let reserve_program_words,unreserve_program_words =
let words = ["var"; "end"; "skip"; "abort";
":="; "if"; "then"; "else"; "while"; "do"] in
(fun () -> reserve_words words),
(fun () -> unreserve_words words);;
reserve_program_words();;
let parse_program,parse_program_assertion =
let assign_ptm = Varp("Assign",dpty)
and seq_ptm = Varp("Seq",dpty)
and ite_ptm = Varp("Ite",dpty)
and while_ptm = Varp("While",dpty)
and skip_ptm = Varp("SKIP",dpty)
and abort_ptm = Varp("ABORT",dpty)
and if_ptm = Varp("IF",dpty)
and do_ptm = Varp("DO",dpty)
and assert_ptm = Varp("ASSERT",dpty)
and awhile_ptm = Varp("AWHILE",dpty)
and ado_ptm = Varp("ADO",dpty) in
let pmk_pair(ptm1,ptm2) = Combp(Combp(Varp(",",dpty),ptm1),ptm2) in
let varname ptm =
match dive_to_var ptm with Varp(n,_) -> n | _ -> fail() in
let rec assign s v e =
match s with
Combp(Combp(pop,lptm),rptm) ->
if varname pop = "," then
Combp(Combp(pop,assign lptm v e),assign rptm v e)
else fail()
| _ -> if varname s = v then e else s in
let lmk_assign s v e = Combp(assign_ptm,Absp(s,assign s v e))
and lmk_seq c cs =
if cs = [] then c else Combp(Combp(seq_ptm,c),hd cs)
and lmk_ite e c1 c2 = Combp(Combp(Combp(ite_ptm,e),c1),c2)
and lmk_while e c = Combp(Combp(while_ptm,e),c)
and lmk_skip _ = skip_ptm
and lmk_abort _ = abort_ptm
and lmk_if e c = Combp(Combp(if_ptm,e),c)
and lmk_do c e = Combp(Combp(do_ptm,c),e)
and lmk_assert e = Combp(assert_ptm,e)
and lmk_awhile i v e c = Combp(Combp(Combp(Combp(awhile_ptm,i),v),e),c)
and lmk_ado i v c e = Combp(Combp(Combp(Combp(ado_ptm,i),v),c),e) in
let lmk_gwhile al e c =
if al = [] then lmk_while e c
else lmk_awhile (fst(hd al)) (snd(hd al)) e c
and lmk_gdo al c e =
if al = [] then lmk_do c e
else lmk_ado (fst(hd al)) (snd(hd al)) c e in
let expression s = parse_preterm >> (fun p -> Absp(s,p)) in
let identifier =
function ((Ident n)::rest) -> n,rest
| _ -> raise Noparse in
let variant s =
(a (Ident "variant") ++ parse_preterm
>> snd)
||| (a (Ident "measure") ++ expression s
>> fun (_,m) -> Combp(Varp("MEASURE",dpty),m)) in
let annotation s =
a (Resword "[") ++ a (Ident "invariant") ++ expression s ++
a (Resword ";") ++ variant s ++ a (Resword "]")
>> fun (((((_,_),i),_),v),_) -> (i,v) in
let rec command s i =
( (a (Resword "(") ++ commands s ++ a (Resword ")")
>> (fun ((_,c),_) -> c))
||| (a (Resword "skip")
>> lmk_skip)
||| (a (Resword "abort")
>> lmk_abort)
||| (a (Resword "if") ++ expression s ++ a (Resword "then") ++ command s ++
possibly (a (Resword "else") ++ command s >> snd)
>> (fun ((((_,e),_),c),cs) -> if cs = [] then lmk_if e c
else lmk_ite e c (hd cs)))
||| (a (Resword "while") ++ expression s ++ a (Resword "do") ++
possibly (annotation s) ++ command s
>> (fun ((((_,e),_),al),c) -> lmk_gwhile al e c))
||| (a (Resword "do") ++ possibly (annotation s) ++
command s ++ a (Resword "while") ++ expression s
>> (fun ((((_,al),c),_),e) -> lmk_gdo al c e))
||| (a (Resword "{") ++ expression s ++ a (Resword "}")
>> (fun ((_,e),_) -> lmk_assert e))
||| (identifier ++ a (Resword ":=") ++ parse_preterm
>> (fun ((v,_),e) -> lmk_assign s v e))) i
and commands s i =
(command s ++ possibly (a (Resword ";") ++ commands s >> snd)
>> (fun (c,cs) -> lmk_seq c cs)) i in
let program i =
let ((_,s),_),r =
(a (Resword "var") ++ parse_preterm ++ a (Resword ";")) i in
let c,r' = (commands s ++ a (Resword "end") >> fst) r in
(s,c),r' in
let assertion =
a (Ident "correct") ++ parse_preterm ++ program ++ parse_preterm
>> fun (((_,p),(s,c)),q) ->
Combp(Combp(Combp(Varp("correct",dpty),Absp(s,p)),c),Absp(s,q)) in
(program >> snd),assertion;;
(* ------------------------------------------------------------------------- *)
(* Introduce the variables in the VCs. *)
(* ------------------------------------------------------------------------- *)
let STATE_GEN_TAC =
let PAIR_CONV = REWR_CONV(GSYM PAIR) in
let rec repair vs v acc =
try let l,r = dest_pair vs in
let th = PAIR_CONV v in
let tm = rand(concl th) in
let rtm = rator tm in
let lth,acc1 = repair l (rand rtm) acc in
let rth,acc2 = repair r (rand tm) acc1 in
TRANS th (MK_COMB(AP_TERM (rator rtm) lth,rth)),acc2
with Failure _ -> REFL v,((v,vs)::acc) in
fun (asl,w) ->
let abstm = find_term (fun t -> not (is_abs t) && is_gabs t) w in
let vs = fst(dest_gabs abstm) in
let v = genvar(type_of(fst(dest_forall w))) in
let th,gens = repair vs v [] in
(X_GEN_TAC v THEN SUBST1_TAC th THEN
MAP_EVERY SPEC_TAC gens THEN REPEAT GEN_TAC) (asl,w);;
let STATE_GEN_TAC' =
let PAIR_CONV = REWR_CONV(GSYM PAIR) in
let rec repair vs v acc =
try let l,r = dest_pair vs in
let th = PAIR_CONV v in
let tm = rand(concl th) in
let rtm = rator tm in
let lth,acc1 = repair l (rand rtm) acc in
let rth,acc2 = repair r (rand tm) acc1 in
TRANS th (MK_COMB(AP_TERM (rator rtm) lth,rth)),acc2
with Failure _ -> REFL v,((v,vs)::acc) in
fun (asl,w) ->
let abstm = find_term (fun t -> not (is_abs t) && is_gabs t) w in
let vs0 = fst(dest_gabs abstm) in
let vl0 = striplist dest_pair vs0 in
let vl = map (variant (variables (list_mk_conj(w::map (concl o snd) asl))))
vl0 in
let vs = end_itlist (curry mk_pair) vl in
let v = genvar(type_of(fst(dest_forall w))) in
let th,gens = repair vs v [] in
(X_GEN_TAC v THEN SUBST1_TAC th THEN
MAP_EVERY SPEC_TAC gens THEN REPEAT GEN_TAC) (asl,w);;
(* ------------------------------------------------------------------------- *)
(* Tidy up a verification condition. *)
(* ------------------------------------------------------------------------- *)
let VC_UNPACK_TAC =
REWRITE_TAC[IMPLIES; o_THM; FALSE; TRUE; AND; OR; NOT; IMP] THEN
STATE_GEN_TAC THEN CONV_TAC(REDEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[PAIR_EQ; GSYM CONJ_ASSOC];;
(* ------------------------------------------------------------------------- *)
(* Calculate a (pseudo-) weakest precondition for command. *)
(* ------------------------------------------------------------------------- *)
let find_pwp =
let wptms =
(map (snd o strip_forall o concl)
[WP_ASSIGN; WP_ITE; WP_SKIP; WP_ABORT; WP_IF; WP_ASSERT]) @
[`wp (AWHILE i v e c) q = i`; `wp (ADO i v c e) q = i`] in
let conv tm =
tryfind (fun t -> rand (instantiate (term_match [] (lhand t) tm) t))
wptms in
fun tm q -> conv(mk_comb(list_mk_icomb "wp" [tm],q));;
(* ------------------------------------------------------------------------- *)
(* Tools for automatic VC generation from annotated program. *)
(* ------------------------------------------------------------------------- *)
let VC_SEQ_TAC =
let is_seq = is_binary "Seq"
and strip_seq = striplist (dest_binary "Seq")
and is_assert tm =
try fst(dest_const(rator tm)) = "ASSERT" with Failure _ -> false
and is_assign tm =
try fst(dest_const(rator tm)) = "Assign" with Failure _ -> false
and SIDE_TAC =
GEN_REWRITE_TAC I [FUN_EQ_THM] THEN STATE_GEN_TAC THEN
PURE_REWRITE_TAC[IMPLIES; o_THM; FALSE; TRUE; AND; OR; NOT; IMP] THEN
CONV_TAC(REDEPTH_CONV GEN_BETA_CONV) THEN
REWRITE_TAC[PAIR_EQ] THEN NO_TAC in
let ADJUST_TAC ptm ptm' ((_,w) as gl) =
let w' = subst [ptm',ptm] w in
let th = EQT_ELIM(REWRITE_CONV[correct; WP_SEQ] (mk_eq(w,w'))) in
GEN_REWRITE_TAC I [th] gl in
fun (asl,w) ->
let cptm,q = dest_comb w in
let cpt,ptm = dest_comb cptm in
let ctm,p = dest_comb cpt in
let ptms = strip_seq ptm in
let seq = rator(rator ptm) in
try let atm = find is_assert ptms in
let i = index atm ptms in
if i = 0 then
let ptm' = mk_binop seq (hd ptms) (list_mk_binop seq (tl ptms)) in
(ADJUST_TAC ptm ptm' THEN
MATCH_MP_TAC VC_SEQ_ASSERT_LEFT THEN CONJ_TAC THENL
[VC_UNPACK_TAC; ALL_TAC]) (asl,w)
else if i = length ptms - 1 then
let ptm' = mk_binop seq (list_mk_binop seq (butlast ptms))
(last ptms) in
(ADJUST_TAC ptm ptm' THEN
MATCH_MP_TAC VC_SEQ_ASSERT_RIGHT THEN CONJ_TAC THENL
[ALL_TAC; VC_UNPACK_TAC]) (asl,w)
else
let l,mr = chop_list (index atm ptms) ptms in
let ptm' = mk_binop seq (list_mk_binop seq l)
(mk_binop seq (hd mr) (list_mk_binop seq (tl mr))) in
(ADJUST_TAC ptm ptm' THEN
MATCH_MP_TAC VC_SEQ_ASSERT_MIDDLE THEN CONJ_TAC) (asl,w)
with Failure "find" -> try
if is_assign (hd ptms) then
let ptm' = mk_binop seq (hd ptms) (list_mk_binop seq (tl ptms)) in
(ADJUST_TAC ptm ptm' THEN
MATCH_MP_TAC VC_SEQ_ASSIGN_LEFT THEN REPEAT CONJ_TAC THENL
[SIDE_TAC; SIDE_TAC; ALL_TAC]) (asl,w)
else fail()
with Failure _ ->
let ptm' = mk_binop seq
(list_mk_binop seq (butlast ptms)) (last ptms) in
let pwp = find_pwp (rand ptm') q in
(ADJUST_TAC ptm ptm' THEN MATCH_MP_TAC CORRECT_SEQ THEN
EXISTS_TAC pwp THEN CONJ_TAC)
(asl,w);;
(* ------------------------------------------------------------------------- *)
(* Tactic to apply a 1-step VC generation. *)
(* ------------------------------------------------------------------------- *)
let VC_STEP_TAC =
let tacnet =
itlist (enter [])
[`correct p SKIP q`,
MATCH_MP_TAC VC_SKIP THEN VC_UNPACK_TAC;
`correct p (ASSERT b) q`,
MATCH_MP_TAC VC_ASSERT THEN VC_UNPACK_TAC;
`correct p (Assign f) q`,
MATCH_MP_TAC VC_ASSIGN THEN VC_UNPACK_TAC;
`correct p (Ite e c1 c2) q`,
MATCH_MP_TAC VC_ITE THEN CONJ_TAC;
`correct p (IF e c) q`,
MATCH_MP_TAC VC_IF THEN CONJ_TAC THENL [ALL_TAC; VC_UNPACK_TAC];
`correct p (AWHILE i (MEASURE m) e c) q`,
MATCH_MP_TAC VC_AWHILE_MEASURE THEN REPEAT CONJ_TAC THENL
[VC_UNPACK_TAC; VC_UNPACK_TAC; GEN_TAC];
`correct p (AWHILE i v e c) q`,
MATCH_MP_TAC VC_AWHILE_VARIANT THEN REPEAT CONJ_TAC THENL
[ALL_TAC; VC_UNPACK_TAC; VC_UNPACK_TAC; STATE_GEN_TAC'];
`correct p (ADO i (MEASURE m) c e) q`,
MATCH_MP_TAC VC_ADO_MEASURE THEN REPEAT CONJ_TAC THENL
[VC_UNPACK_TAC; VC_UNPACK_TAC; STATE_GEN_TAC'];
`correct p (ADO i v c e) q`,
MATCH_MP_TAC VC_ADO_VARIANT THEN REPEAT CONJ_TAC THENL
[ALL_TAC; VC_UNPACK_TAC; VC_UNPACK_TAC; STATE_GEN_TAC'];
`correct p (c1 Seq c2) q`,
VC_SEQ_TAC] empty_net in
fun (asl,w) -> FIRST(lookup w tacnet) (asl,w);;
(* ------------------------------------------------------------------------- *)
(* Final packaging to strip away the program completely. *)
(* ------------------------------------------------------------------------- *)
let VC_TAC = REPEAT VC_STEP_TAC;;
(* ------------------------------------------------------------------------- *)
(* Some examples. *)
(* ------------------------------------------------------------------------- *)
install_parser ("correct",parse_program_assertion);;
let EXAMPLE_FACTORIAL = prove
(`correct
T
var x,y,n;
x := 0;
y := 1;
while x < n do [invariant x <= n /\ (y = FACT x); measure n - x]
(x := x + 1;
y := y * x)
end
y = FACT n`,
VC_TAC THENL
[STRIP_TAC THEN ASM_REWRITE_TAC[FACT; LE_0];
REWRITE_TAC[CONJ_ASSOC; NOT_LT; LE_ANTISYM] THEN MESON_TAC[];
REWRITE_TAC[GSYM ADD1; FACT] THEN STRIP_TAC THEN
ASM_REWRITE_TAC[MULT_AC] THEN UNDISCH_TAC `x < n` THEN ARITH_TAC]);;
delete_parser "correct";;