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(* ========================================================================= *) | |
(* Formal definition of "safety" and "liveness" for properties of traces. *) | |
(* Proof that any property of traces can be decomposed into the conjunction *) | |
(* of a safety and a liveness property, following Fred Schneider's paper *) | |
(* "Decomposing Properties into Safety and Liveness using Predicate Logic". *) | |
(* *) | |
(* https://apps.dtic.mil/dtic/tr/fulltext/u2/a187556.pdf *) | |
(* ========================================================================= *) | |
let safety = new_definition | |
`safety (p:(num->S)->bool) <=> | |
!s. ~(p s) ==> ?n. !s'. (!m. m <= n ==> s(m) = s'(m)) ==> ~(p s')`;; | |
let liveness = new_definition | |
`liveness (p:(num->S)->bool) <=> | |
!s n. ?s'. (!m. m <= n ==> s(m) = s'(m)) /\ p s'`;; | |
(* ------------------------------------------------------------------------- *) | |
(* It doesn't matter whether we take strict or non-strict subsequences. *) | |
(* ------------------------------------------------------------------------- *) | |
let SAFETY = prove | |
(`!p:(num->S)->bool. | |
safety p <=> | |
!s. ~(p s) ==> ?n. !s'. (!m. m < n ==> s(m) = s'(m)) ==> ~(p s')`, | |
GEN_TAC THEN REWRITE_TAC[safety; GSYM LT_SUC_LE] THEN | |
MESON_TAC[ARITH_RULE `m < n ==> m < SUC n`]);; | |
let LIVENESS = prove | |
(`!p:(num->S)->bool. | |
liveness p <=> | |
!s n. ?s'. (!m. m < n ==> s(m) = s'(m)) /\ p s'`, | |
GEN_TAC THEN REWRITE_TAC[liveness; GSYM LT_SUC_LE] THEN | |
MESON_TAC[ARITH_RULE `m < n ==> m < SUC n`]);; | |
(* ------------------------------------------------------------------------- *) | |
(* Decomposition theorem (following top level of Schneider's proof). *) | |
(* ------------------------------------------------------------------------- *) | |
let SAFETY_LIVENESS_DECOMPOSITION = prove | |
(`!P:(num->S)->bool. | |
?S L. safety S /\ liveness L /\ (!x. S x /\ L x <=> P x)`, | |
REPEAT GEN_TAC THEN | |
ABBREV_TAC `Q = \s:num->S. !i. ?b. (!j. j <= i ==> b j = s j) /\ P b` THEN | |
MAP_EVERY EXISTS_TAC [`\s:num->S. P s \/ Q s`; `\s:num->S. P s \/ ~Q s`] THEN | |
REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL | |
[EXPAND_TAC "Q" THEN REWRITE_TAC[safety; liveness]; MESON_TAC[]] THEN | |
REWRITE_TAC[DE_MORGAN_THM; IMP_CONJ; NOT_FORALL_THM] THEN CONJ_TAC THENL | |
[GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC MONO_EXISTS THEN ASM_MESON_TAC[]; | |
MESON_TAC[]]);; | |