(* ========================================================================= *) (* All you wanted to know about reflexive symmetric and transitive closures. *) (* ========================================================================= *) prioritize_num();; let RULE_INDUCT_TAC = MATCH_MP_TAC o DISCH_ALL o SPEC_ALL o UNDISCH o SPEC_ALL;; (* ------------------------------------------------------------------------- *) (* Little lemmas about equivalent forms of symmetry and transitivity. *) (* ------------------------------------------------------------------------- *) let SYM_ALT = prove (`!R:A->A->bool. (!x y. R x y ==> R y x) <=> (!x y. R x y <=> R y x)`, GEN_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL [EQ_TAC THEN DISCH_TAC THEN FIRST_ASSUM MATCH_MP_TAC; FIRST_ASSUM(fun th -> GEN_REWRITE_TAC I [th])] THEN FIRST_ASSUM MATCH_ACCEPT_TAC);; let TRANS_ALT = prove (`!(R:A->A->bool) (S:A->A->bool) U. (!x z. (?y. R x y /\ S y z) ==> U x z) <=> (!x y z. R x y /\ S y z ==> U x z)`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN EQ_TAC THEN DISCH_TAC THEN ASM_REWRITE_TAC[]);; (* ------------------------------------------------------------------------- *) (* Reflexive closure *) (* ------------------------------------------------------------------------- *) let RC_RULES,RC_INDUCT,RC_CASES = new_inductive_definition `(!x y. R x y ==> RC R x y) /\ (!x:A. RC R x x)`;; let RC_INC = prove (`!(R:A->A->bool) x y. R x y ==> RC R x y`, REWRITE_TAC[RC_RULES]);; let RC_REFL = prove (`!(R:A->A->bool) x. RC R x x`, REWRITE_TAC[RC_RULES]);; let RC_EXPLICIT = prove (`!(R:A->A->bool) x y. RC R x y <=> R x y \/ (x = y)`, REWRITE_TAC[RC_CASES; EQ_SYM_EQ]);; let RC_MONO = prove (`!(R:A->A->bool) S. (!x y. R x y ==> S x y) ==> (!x y. RC R x y ==> RC S x y)`, MESON_TAC[RC_CASES]);; let RC_CLOSED = prove (`!R:A->A->bool. (RC R = R) <=> !x. R x x`, REWRITE_TAC[FUN_EQ_THM; RC_EXPLICIT] THEN MESON_TAC[]);; let RC_IDEMP = prove (`!R:A->A->bool. RC(RC R) = RC R`, REWRITE_TAC[RC_CLOSED; RC_REFL]);; let RC_SYM = prove (`!R:A->A->bool. (!x y. R x y ==> R y x) ==> (!x y. RC R x y ==> RC R y x)`, MESON_TAC[RC_CASES]);; let RC_TRANS = prove (`!R:A->A->bool. (!x y z. R x y /\ R y z ==> R x z) ==> (!x y z. RC R x y /\ RC R y z ==> RC R x z)`, REWRITE_TAC[RC_CASES] THEN MESON_TAC[]);; (* ------------------------------------------------------------------------- *) (* Symmetric closure *) (* ------------------------------------------------------------------------- *) let SC_RULES,SC_INDUCT,SC_CASES = new_inductive_definition `(!x y. R x y ==> SC R x y) /\ (!x:A y. SC R x y ==> SC R y x)`;; let SC_INC = prove (`!(R:A->A->bool) x y. R x y ==> SC R x y`, REWRITE_TAC[SC_RULES]);; let SC_SYM = prove (`!(R:A->A->bool) x y. SC R x y ==> SC R y x`, REWRITE_TAC[SC_RULES]);; let SC_EXPLICIT = prove (`!R:A->A->bool. SC(R) x y <=> R x y \/ R y x`, GEN_TAC THEN EQ_TAC THENL [RULE_INDUCT_TAC SC_INDUCT THEN MESON_TAC[]; MESON_TAC[SC_CASES]]);; let SC_MONO = prove (`!(R:A->A->bool) S. (!x y. R x y ==> S x y) ==> (!x y. SC R x y ==> SC S x y)`, MESON_TAC[SC_EXPLICIT]);; let SC_CLOSED = prove (`!R:A->A->bool. (SC R = R) <=> !x y. R x y ==> R y x`, REWRITE_TAC[FUN_EQ_THM; SC_EXPLICIT] THEN MESON_TAC[]);; let SC_IDEMP = prove (`!R:A->A->bool. SC(SC R) = SC R`, REWRITE_TAC[SC_CLOSED; SC_SYM]);; let SC_REFL = prove (`!R:A->A->bool. (!x. R x x) ==> (!x. SC R x x)`, MESON_TAC[SC_EXPLICIT]);; (* ------------------------------------------------------------------------- *) (* Transitive closure *) (* ------------------------------------------------------------------------- *) let TC_RULES,TC_INDUCT,TC_CASES = new_inductive_definition `(!x y. R x y ==> TC R x y) /\ (!(x:A) y z. TC R x y /\ TC R y z ==> TC R x z)`;; let TC_INC = prove (`!(R:A->A->bool) x y. R x y ==> TC R x y`, REWRITE_TAC[TC_RULES]);; let TC_TRANS = prove (`!(R:A->A->bool) x y z. TC R x y /\ TC R y z ==> TC R x z`, REWRITE_TAC[TC_RULES]);; let TC_MONO = prove (`!(R:A->A->bool) S. (!x y. R x y ==> S x y) ==> (!x y. TC R x y ==> TC S x y)`, REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC TC_INDUCT THEN ASM_MESON_TAC[TC_RULES]);; let TC_CLOSED = prove (`!R:A->A->bool. (TC R = R) <=> !x y z. R x y /\ R y z ==> R x z`, GEN_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN EQ_TAC THENL [MESON_TAC[TC_RULES]; REPEAT STRIP_TAC] THEN EQ_TAC THENL [RULE_INDUCT_TAC TC_INDUCT; ALL_TAC] THEN ASM_MESON_TAC[TC_RULES]);; let TC_IDEMP = prove (`!R:A->A->bool. TC(TC R) = TC R`, REWRITE_TAC[TC_CLOSED; TC_TRANS]);; let TC_REFL = prove (`!R:A->A->bool. (!x. R x x) ==> (!x. TC R x x)`, MESON_TAC[TC_INC]);; let TC_SYM = prove (`!R:A->A->bool. (!x y. R x y ==> R y x) ==> (!x y. TC R x y ==> TC R y x)`, GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC TC_INDUCT THEN ASM_MESON_TAC[TC_RULES]);; (* ------------------------------------------------------------------------- *) (* Commutativity properties of the three basic closure operations *) (* ------------------------------------------------------------------------- *) let RC_SC = prove (`!R:A->A->bool. RC(SC R) = SC(RC R)`, REWRITE_TAC[FUN_EQ_THM; RC_EXPLICIT; SC_EXPLICIT] THEN MESON_TAC[]);; let SC_RC = prove (`!R:A->A->bool. SC(RC R) = RC(SC R)`, REWRITE_TAC[RC_SC]);; let RC_TC = prove (`!R:A->A->bool. RC(TC R) = TC(RC R)`, REWRITE_TAC[FUN_EQ_THM] THEN REPEAT GEN_TAC THEN EQ_TAC THENL [RULE_INDUCT_TAC RC_INDUCT THEN MESON_TAC[TC_RULES; RC_RULES; TC_MONO]; RULE_INDUCT_TAC TC_INDUCT THEN MESON_TAC[RC_TRANS; TC_RULES; RC_MONO]]);; let TC_RC = prove (`!R:A->A->bool. TC(RC R) = RC(TC R)`, REWRITE_TAC[RC_TC]);; let TC_SC = prove (`!(R:A->A->bool) x y. SC(TC R) x y ==> TC(SC R) x y`, GEN_TAC THEN MATCH_MP_TAC SC_INDUCT THEN MESON_TAC[TC_MONO; TC_SYM; SC_RULES]);; let SC_TC = prove (`!(R:A->A->bool) x y. SC(TC R) x y ==> TC(SC R) x y`, REWRITE_TAC[TC_SC]);; (* ------------------------------------------------------------------------- *) (* Left and right variants of TC. *) (* ------------------------------------------------------------------------- *) let TC_TRANS_L = prove (`!(R:A->A->bool) x y z. TC R x y /\ R y z ==> TC R x z`, MESON_TAC[TC_RULES]);; let TC_TRANS_R = prove (`!(R:A->A->bool) x y z. R x y /\ TC R y z ==> TC R x z`, MESON_TAC[TC_RULES]);; let TC_CASES_L = prove (`!(R:A->A->bool) x z. TC R x z <=> R x z \/ (?y. TC R x y /\ R y z)`, REPEAT GEN_TAC THEN EQ_TAC THENL [RULE_INDUCT_TAC TC_INDUCT THEN MESON_TAC[TC_RULES]; MESON_TAC[TC_RULES]]);; let TC_CASES_R = prove (`!(R:A->A->bool) x z. TC R x z <=> R x z \/ (?y. R x y /\ TC R y z)`, REPEAT GEN_TAC THEN EQ_TAC THENL [RULE_INDUCT_TAC TC_INDUCT THEN MESON_TAC[TC_RULES]; MESON_TAC[TC_RULES]]);; let TC_INDUCT_L = prove (`!(R:A->A->bool) P. (!x y. R x y ==> P x y) /\ (!x y z. P x y /\ R y z ==> P x z) ==> (!x y. TC R x y ==> P x y)`, REPEAT GEN_TAC THEN STRIP_TAC THEN SUBGOAL_THEN `!y:A z. TC(R) y z ==> !x:A. P x y ==> P x z` MP_TAC THENL [MATCH_MP_TAC TC_INDUCT THEN ASM_MESON_TAC[]; ASM_MESON_TAC[TC_CASES_R]]);; let TC_INDUCT_R = prove (`!(R:A->A->bool) P. (!x y. R x y ==> P x y) /\ (!x z. (?y. R x y /\ P y z) ==> P x z) ==> (!x y. TC R x y ==> P x y)`, REPEAT GEN_TAC THEN STRIP_TAC THEN SUBGOAL_THEN `!x:A y. TC(R) x y ==> !z:A. P y z ==> P x z` MP_TAC THENL [MATCH_MP_TAC TC_INDUCT THEN ASM_MESON_TAC[]; ASM_MESON_TAC[TC_CASES_L]]);; (* ------------------------------------------------------------------------- *) (* Reflexive symmetric closure *) (* ------------------------------------------------------------------------- *) let RSC = new_definition `RSC(R:A->A->bool) = RC(SC R)`;; let RSC_INC = prove (`!(R:A->A->bool) x y. R x y ==> RSC R x y`, REWRITE_TAC[RSC] THEN MESON_TAC[RC_INC; SC_INC]);; let RSC_REFL = prove (`!(R:A->A->bool) x. RSC R x x`, REWRITE_TAC[RSC; RC_REFL]);; let RSC_SYM = prove (`!(R:A->A->bool) x y. RSC R x y ==> RSC R y x`, REWRITE_TAC[RSC; RC_SC; SC_SYM]);; let RSC_CASES = prove (`!(R:A->A->bool) x y. RSC R x y <=> (x = y) \/ R x y \/ R y x`, REWRITE_TAC[RSC; RC_EXPLICIT; SC_EXPLICIT; DISJ_ACI]);; let RSC_INDUCT = prove (`!(R:A->A->bool) P. (!x y. R x y ==> P x y) /\ (!x. P x x) /\ (!x y. P x y ==> P y x) ==> !x y. RSC R x y ==> P x y`, REWRITE_TAC[RSC; RC_EXPLICIT; SC_EXPLICIT] THEN MESON_TAC[]);; let RSC_MONO = prove (`!(R:A->A->bool) S. (!x y. R x y ==> S x y) ==> (!x y. RSC R x y ==> RSC S x y)`, REWRITE_TAC[RSC] THEN MESON_TAC[SC_MONO; RC_MONO]);; let RSC_CLOSED = prove (`!R:A->A->bool. (RSC R = R) <=> (!x. R x x) /\ (!x y. R x y ==> R y x)`, REWRITE_TAC[FUN_EQ_THM; RSC; RC_EXPLICIT; SC_EXPLICIT] THEN MESON_TAC[]);; let RSC_IDEMP = prove (`!R:A->A->bool. RSC(RSC R) = RSC R`, REWRITE_TAC[RSC_CLOSED; RSC_REFL; RSC_SYM]);; (* ------------------------------------------------------------------------- *) (* Reflexive transitive closure *) (* ------------------------------------------------------------------------- *) let RTC = new_definition `RTC(R:A->A->bool) = RC(TC R)`;; let RTC_INC = prove (`!(R:A->A->bool) x y. R x y ==> RTC R x y`, REWRITE_TAC[RTC] THEN MESON_TAC[RC_INC; TC_INC]);; let RTC_REFL = prove (`!(R:A->A->bool) x. RTC R x x`, REWRITE_TAC[RTC; RC_REFL]);; let RTC_TRANS = prove (`!(R:A->A->bool) x y z. RTC R x y /\ RTC R y z ==> RTC R x z`, REWRITE_TAC[RTC; RC_TC; TC_TRANS]);; let RTC_RULES = prove (`!(R:A->A->bool). (!x y. R x y ==> RTC R x y) /\ (!x. RTC R x x) /\ (!x y z. RTC R x y /\ RTC R y z ==> RTC R x z)`, REWRITE_TAC[RTC_INC; RTC_REFL; RTC_TRANS]);; let RTC_TRANS_L = prove (`!(R:A->A->bool) x y z. RTC R x y /\ R y z ==> RTC R x z`, REWRITE_TAC[RTC; RC_TC] THEN MESON_TAC[TC_TRANS_L; RC_INC]);; let RTC_TRANS_R = prove (`!(R:A->A->bool) x y z. R x y /\ RTC R y z ==> RTC R x z`, REWRITE_TAC[RTC; RC_TC] THEN MESON_TAC[TC_TRANS_R; RC_INC]);; let RTC_CASES = prove (`!(R:A->A->bool) x z. RTC R x z <=> (x = z) \/ ?y. RTC R x y /\ RTC R y z`, REWRITE_TAC[RTC; RC_EXPLICIT] THEN MESON_TAC[TC_TRANS]);; let RTC_CASES_L = prove (`!(R:A->A->bool) x z. RTC R x z <=> (x = z) \/ ?y. RTC R x y /\ R y z`, REWRITE_TAC[RTC; RC_EXPLICIT] THEN MESON_TAC[TC_CASES_L; TC_TRANS_L]);; let RTC_CASES_R = prove (`!(R:A->A->bool) x z. RTC R x z <=> (x = z) \/ ?y. R x y /\ RTC R y z`, REWRITE_TAC[RTC; RC_EXPLICIT] THEN MESON_TAC[TC_CASES_R; TC_TRANS_R]);; let RTC_INDUCT = prove (`!(R:A->A->bool) P. (!x y. R x y ==> P x y) /\ (!x. P x x) /\ (!x y z. P x y /\ P y z ==> P x z) ==> !x y. RTC R x y ==> P x y`, REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[RTC; RC_TC] THEN MATCH_MP_TAC TC_INDUCT THEN REWRITE_TAC[RC_EXPLICIT] THEN ASM_MESON_TAC[]);; let RTC_INDUCT_L = prove (`!(R:A->A->bool) P. (!x. P x x) /\ (!x y z. P x y /\ R y z ==> P x z) ==> !x y. RTC R x y ==> P x y`, REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[RTC; RC_TC] THEN MATCH_MP_TAC TC_INDUCT_L THEN REWRITE_TAC[RC_EXPLICIT] THEN ASM_MESON_TAC[]);; let RTC_INDUCT_R = prove (`!(R:A->A->bool) P. (!x. P x x) /\ (!x y z. R x y /\ P y z ==> P x z) ==> !x y. RTC R x y ==> P x y`, REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[RTC; RC_TC] THEN MATCH_MP_TAC TC_INDUCT_R THEN REWRITE_TAC[RC_EXPLICIT] THEN ASM_MESON_TAC[]);; let RTC_MONO = prove (`!(R:A->A->bool) S. (!x y. R x y ==> S x y) ==> (!x y. RTC R x y ==> RTC S x y)`, REWRITE_TAC[RTC] THEN MESON_TAC[RC_MONO; TC_MONO]);; let RTC_CLOSED = prove (`!R:A->A->bool. (RTC R = R) <=> (!x. R x x) /\ (!x y z. R x y /\ R y z ==> R x z)`, REWRITE_TAC[FUN_EQ_THM; RTC; RC_EXPLICIT] THEN MESON_TAC[TC_CLOSED; TC_RULES]);; let RTC_IDEMP = prove (`!R:A->A->bool. RTC(RTC R) = RTC R`, REWRITE_TAC[RTC_CLOSED; RTC_REFL; RTC_TRANS]);; let RTC_SYM = prove (`!R:A->A->bool. (!x y. R x y ==> R y x) ==> (!x y. RTC R x y ==> RTC R y x)`, REWRITE_TAC[RTC] THEN MESON_TAC[RC_SYM; TC_SYM]);; let RTC_STUTTER = prove (`RTC R = RTC (\x y. R x y /\ ~(x = y))`, REWRITE_TAC[RC_TC; RTC] THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN REWRITE_TAC[RC_CASES] THEN MESON_TAC[]);; let TC_RTC_CASES_L = prove (`TC R x z <=> ?y. RTC R x y /\ R y z`, REWRITE_TAC[RTC; RC_CASES] THEN MESON_TAC[TC_CASES_L; TC_INC]);; let TC_RTC_CASES_R = prove (`!R x z. TC R x z <=> ?y. R x y /\ RTC R y z`, REWRITE_TAC[RTC; RC_CASES] THEN MESON_TAC[TC_CASES_R; TC_INC]);; let TC_TC_RTC_CASES = prove (`!R x z. TC R x z <=> ?y. TC R x y /\ RTC R y z`, REWRITE_TAC[RTC; RC_CASES] THEN MESON_TAC[TC_TRANS]);; let TC_RTC_TC_CASES = prove (`!R x z. TC R x z <=> ?y. RTC R x y /\ TC R y z`, REWRITE_TAC[RTC; RC_CASES] THEN MESON_TAC[TC_TRANS]);; let RTC_NE_IMP_TC = prove (`!R x y. RTC R x y /\ ~(x = y) ==> TC R x y`, GEN_TAC THEN ONCE_REWRITE_TAC[GSYM IMP_IMP] THEN MATCH_MP_TAC RTC_INDUCT THEN REWRITE_TAC[] THEN MESON_TAC[TC_INC; TC_CASES]);; (* ------------------------------------------------------------------------- *) (* Symmetric transitive closure *) (* ------------------------------------------------------------------------- *) let STC = new_definition `STC(R:A->A->bool) = TC(SC R)`;; let STC_INC = prove (`!(R:A->A->bool) x y. R x y ==> STC R x y`, REWRITE_TAC[STC] THEN MESON_TAC[SC_INC; TC_INC]);; let STC_SYM = prove (`!(R:A->A->bool) x y. STC R x y ==> STC R y x`, REWRITE_TAC[STC] THEN MESON_TAC[TC_SYM; SC_SYM]);; let STC_TRANS = prove (`!(R:A->A->bool) x y z. STC R x y /\ STC R y z ==> STC R x z`, REWRITE_TAC[STC; TC_TRANS]);; let STC_TRANS_L = prove (`!(R:A->A->bool) x y z. STC R x y /\ R y z ==> STC R x z`, REWRITE_TAC[STC] THEN MESON_TAC[TC_TRANS_L; SC_INC]);; let STC_TRANS_R = prove (`!(R:A->A->bool) x y z. R x y /\ STC R y z ==> STC R x z`, REWRITE_TAC[STC] THEN MESON_TAC[TC_TRANS_R; SC_INC]);; let STC_CASES = prove (`!(R:A->A->bool) x z. STC R x z <=> R x z \/ STC R z x \/ ?y. STC R x y /\ STC R y z`, REWRITE_TAC[STC] THEN MESON_TAC[SC_SYM; TC_SYM; TC_INC; TC_TRANS; SC_INC]);; let STC_CASES_L = prove (`!(R:A->A->bool) x z. STC R x z <=> R x z \/ STC R z x \/ ?y. STC R x y /\ R y z`, REWRITE_TAC[STC] THEN MESON_TAC[SC_SYM; TC_SYM; TC_INC; TC_TRANS; SC_INC]);; let STC_CASES_R = prove (`!(R:A->A->bool) x z. STC R x z <=> R x z \/ STC R z x \/ ?y. R x y /\ STC R y z`, REWRITE_TAC[STC] THEN MESON_TAC[SC_SYM; TC_SYM; TC_INC; TC_TRANS; SC_INC]);; let STC_INDUCT = prove (`!(R:A->A->bool) P. (!x y. R x y ==> P x y) /\ (!x y. P x y ==> P y x) /\ (!x y z. P x y /\ P y z ==> P x z) ==> !x y. STC R x y ==> P x y`, REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[STC] THEN MATCH_MP_TAC TC_INDUCT THEN ASM_MESON_TAC[SC_EXPLICIT]);; let STC_MONO = prove (`!(R:A->A->bool) S. (!x y. R x y ==> S x y) ==> (!x y. STC R x y ==> STC S x y)`, REWRITE_TAC[STC] THEN MESON_TAC[SC_MONO; TC_MONO]);; let STC_CLOSED = prove (`!R:A->A->bool. (STC R = R) <=> (!x y. R x y ==> R y x) /\ (!x y z. R x y /\ R y z ==> R x z)`, GEN_TAC THEN REWRITE_TAC[STC; SC_EXPLICIT] THEN EQ_TAC THENL [DISCH_THEN(SUBST1_TAC o SYM) THEN MESON_TAC[TC_TRANS; TC_SYM; SC_SYM]; REWRITE_TAC[GSYM SC_CLOSED; GSYM TC_CLOSED] THEN MESON_TAC[]]);; let STC_IDEMP = prove (`!R:A->A->bool. STC(STC R) = STC R`, REWRITE_TAC[STC_CLOSED; STC_SYM; STC_TRANS]);; let STC_REFL = prove (`!R:A->A->bool. (!x. R x x) ==> !x. STC R x x`, MESON_TAC[STC_INC]);; (* ------------------------------------------------------------------------- *) (* Reflexive symmetric transitive closure (smallest equivalence relation) *) (* ------------------------------------------------------------------------- *) let RSTC = new_definition `RSTC(R:A->A->bool) = RC(TC(SC R))`;; let RSTC_INC = prove (`!(R:A->A->bool) x y. R x y ==> RSTC R x y`, REWRITE_TAC[RSTC] THEN MESON_TAC[RC_INC; TC_INC; SC_INC]);; let RSTC_REFL = prove (`!(R:A->A->bool) x. RSTC R x x`, REWRITE_TAC[RSTC; RC_REFL]);; let RSTC_SYM = prove (`!(R:A->A->bool) x y. RSTC R x y ==> RSTC R y x`, REWRITE_TAC[RSTC] THEN MESON_TAC[SC_SYM; TC_SYM; RC_SYM]);; let RSTC_TRANS = prove (`!(R:A->A->bool) x y z. RSTC R x y /\ RSTC R y z ==> RSTC R x z`, REWRITE_TAC[RSTC; RC_TC; TC_TRANS]);; let RSTC_RULES = prove (`!(R:A->A->bool). (!x y. R x y ==> RSTC R x y) /\ (!x. RSTC R x x) /\ (!x y. RSTC R x y ==> RSTC R y x) /\ (!x y z. RSTC R x y /\ RSTC R y z ==> RSTC R x z)`, REWRITE_TAC[RSTC_INC; RSTC_REFL; RSTC_SYM; RSTC_TRANS]);; let RSTC_TRANS_L = prove (`!(R:A->A->bool) x y z. RSTC R x y /\ R y z ==> RSTC R x z`, REWRITE_TAC[RSTC; RC_TC] THEN MESON_TAC[TC_TRANS_L; RC_INC; SC_INC]);; let RSTC_TRANS_R = prove (`!(R:A->A->bool) x y z. R x y /\ RSTC R y z ==> RSTC R x z`, REWRITE_TAC[RSTC; RC_TC] THEN MESON_TAC[TC_TRANS_R; RC_INC; SC_INC]);; let RSTC_CASES = prove (`!(R:A->A->bool) x z. RSTC R x z <=> (x = z) \/ R x z \/ RSTC R z x \/ ?y. RSTC R x y /\ RSTC R y z`, REWRITE_TAC[RSTC; RC_TC; RC_SC] THEN REWRITE_TAC[GSYM STC] THEN MESON_TAC[STC_CASES; RC_CASES]);; let RSTC_CASES_L = prove (`!(R:A->A->bool) x z. RSTC R x z <=> (x = z) \/ R x z \/ RSTC R z x \/ ?y. RSTC R x y /\ R y z`, REWRITE_TAC[RSTC; RC_TC; RC_SC] THEN REWRITE_TAC[GSYM STC] THEN MESON_TAC[STC_CASES_L; RC_CASES]);; let RSTC_CASES_R = prove (`!(R:A->A->bool) x z. RSTC R x z <=> (x = z) \/ R x z \/ RSTC R z x \/ ?y. R x y /\ RSTC R y z`, REWRITE_TAC[RSTC; RC_TC; RC_SC] THEN REWRITE_TAC[GSYM STC] THEN MESON_TAC[STC_CASES_R; RC_CASES]);; let RSTC_INDUCT = prove (`!(R:A->A->bool) P. (!x y. R x y ==> P x y) /\ (!x. P x x) /\ (!x y. P x y ==> P y x) /\ (!x y z. P x y /\ P y z ==> P x z) ==> !x y. RSTC R x y ==> P x y`, REPEAT GEN_TAC THEN STRIP_TAC THEN REWRITE_TAC[RSTC; RC_TC; RC_SC] THEN REWRITE_TAC[GSYM STC] THEN MATCH_MP_TAC STC_INDUCT THEN REWRITE_TAC[RC_EXPLICIT] THEN ASM_MESON_TAC[]);; let RSTC_MONO = prove (`!(R:A->A->bool) S. (!x y. R x y ==> S x y) ==> (!x y. RSTC R x y ==> RSTC S x y)`, REWRITE_TAC[RSTC] THEN MESON_TAC[RC_MONO; SC_MONO; TC_MONO]);; let RSTC_CLOSED = prove (`!R:A->A->bool. (RSTC R = R) <=> (!x. R x x) /\ (!x y. R x y ==> R y x) /\ (!x y z. R x y /\ R y z ==> R x z)`, REWRITE_TAC[RSTC] THEN REWRITE_TAC[GSYM STC; GSYM STC_CLOSED] THEN REWRITE_TAC[RC_EXPLICIT; FUN_EQ_THM] THEN MESON_TAC[STC_INC]);; let RSTC_IDEMP = prove (`!R:A->A->bool. RSTC(RSTC R) = RSTC R`, REWRITE_TAC[RSTC_CLOSED; RSTC_REFL; RSTC_SYM; RSTC_TRANS]);; (* ------------------------------------------------------------------------- *) (* Finally, we prove the inclusion properties for composite closures *) (* ------------------------------------------------------------------------- *) let RSC_INC_RC = prove (`!R:A->A->bool. !x y. RC R x y ==> RSC R x y`, REWRITE_TAC[RSC; RC_SC; SC_INC]);; let RSC_INC_SC = prove (`!R:A->A->bool. !x y. SC R x y ==> RSC R x y`, REWRITE_TAC[RSC; RC_INC]);; let RTC_INC_RC = prove (`!R:A->A->bool. !x y. RC R x y ==> RTC R x y`, REWRITE_TAC[RTC; RC_TC; TC_INC]);; let RTC_INC_TC = prove (`!R:A->A->bool. !x y. TC R x y ==> RTC R x y`, REWRITE_TAC[RTC; RC_INC]);; let STC_INC_SC = prove (`!R:A->A->bool. !x y. SC R x y ==> STC R x y`, REWRITE_TAC[STC; TC_INC]);; let STC_INC_TC = prove (`!R:A->A->bool. !x y. TC R x y ==> STC R x y`, REWRITE_TAC[STC] THEN MESON_TAC[TC_MONO; SC_INC]);; let RSTC_INC_RC = prove (`!R:A->A->bool. !x y. RC R x y ==> RSTC R x y`, REWRITE_TAC[RSTC; RC_TC; RC_SC; GSYM STC; STC_INC]);; let RSTC_INC_SC = prove (`!R:A->A->bool. !x y. SC R x y ==> RSTC R x y`, REWRITE_TAC[RSTC; GSYM RTC; RTC_INC]);; let RSTC_INC_TC = prove (`!R:A->A->bool. !x y. TC R x y ==> RSTC R x y`, REWRITE_TAC[RSTC; RC_TC; GSYM RSC] THEN MESON_TAC[TC_MONO; RSC_INC]);; let RSTC_INC_RSC = prove (`!R:A->A->bool. !x y. RSC R x y ==> RSTC R x y`, REWRITE_TAC[RSC; RSTC; RC_TC; TC_INC]);; let RSTC_INC_RTC = prove (`!R:A->A->bool. !x y. RTC R x y ==> RSTC R x y`, REWRITE_TAC[GSYM RTC; RSTC] THEN MESON_TAC[RTC_MONO; SC_INC]);; let RSTC_INC_STC = prove (`!R:A->A->bool. !x y. STC R x y ==> RSTC R x y`, REWRITE_TAC[GSYM STC; RSTC; RC_INC]);; (* ------------------------------------------------------------------------- *) (* Handy things about reverse relations. *) (* ------------------------------------------------------------------------- *) let INV = new_definition `INV R (x:A) (y:B) <=> R y x`;; let RC_INV = prove (`RC(INV R) = INV(RC R)`, REWRITE_TAC[FUN_EQ_THM; RC_EXPLICIT; INV; EQ_SYM_EQ]);; let SC_INV = prove (`SC(INV R) = INV(SC R)`, REWRITE_TAC[FUN_EQ_THM; SC_EXPLICIT; INV; DISJ_SYM]);; let SC_INV_STRONG = prove (`SC(INV R) = SC R`, REWRITE_TAC[FUN_EQ_THM; SC_EXPLICIT; INV; DISJ_SYM]);; let TC_INV = prove (`TC(INV R) = INV(TC R)`, REWRITE_TAC[FUN_EQ_THM; INV] THEN REPEAT GEN_TAC THEN EQ_TAC THEN RULE_INDUCT_TAC TC_INDUCT THEN MESON_TAC[INV; TC_RULES]);; let RSC_INV = prove (`RSC(INV R) = INV(RSC R)`, REWRITE_TAC[RSC; RC_INV; SC_INV]);; let RTC_INV = prove (`RTC(INV R) = INV(RTC R)`, REWRITE_TAC[RTC; RC_INV; TC_INV]);; let STC_INV = prove (`STC(INV R) = INV(STC R)`, REWRITE_TAC[STC; SC_INV; TC_INV]);; let RSTC_INV = prove (`RSTC(INV R) = INV(RSTC R)`, REWRITE_TAC[RSTC; RC_INV; SC_INV; TC_INV]);; (* ------------------------------------------------------------------------- *) (* An iterative version of (R)TC. *) (* ------------------------------------------------------------------------- *) let RELPOW = new_recursive_definition num_RECURSION `(RELPOW 0 (R:A->A->bool) x y <=> (x = y)) /\ (RELPOW (SUC n) R x y <=> ?z. RELPOW n R x z /\ R z y)`;; let RELPOW_R = prove (`(RELPOW 0 (R:A->A->bool) x y <=> (x = y)) /\ (RELPOW (SUC n) R x y <=> ?z. R x z /\ RELPOW n R z y)`, CONJ_TAC THENL [REWRITE_TAC[RELPOW]; ALL_TAC] THEN MAP_EVERY (fun t -> SPEC_TAC(t,t)) [`x:A`; `y:A`; `n:num`] THEN INDUCT_TAC THEN ASM_MESON_TAC[RELPOW]);; let RELPOW_M = prove (`!m n x:A y. RELPOW (m + n) R x y <=> ?z. RELPOW m R x z /\ RELPOW n R z y`, INDUCT_TAC THEN ASM_REWRITE_TAC[ADD_CLAUSES; RELPOW_R; UNWIND_THM1] THEN MESON_TAC[]);; let RTC_RELPOW = prove (`!R (x:A) y. RTC R x y <=> ?n. RELPOW n R x y`, REPEAT GEN_TAC THEN EQ_TAC THENL [RULE_INDUCT_TAC RTC_INDUCT_L THEN MESON_TAC[RELPOW]; REWRITE_TAC[LEFT_IMP_EXISTS_THM] THEN SPEC_TAC(`y:A`,`y:A`) THEN ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN INDUCT_TAC THEN REWRITE_TAC[RELPOW] THEN ASM_MESON_TAC[RTC_REFL; RTC_TRANS_L]]);; let TC_RELPOW = prove (`!R (x:A) y. TC R x y <=> ?n. RELPOW (SUC n) R x y`, REWRITE_TAC[RELPOW] THEN ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN REWRITE_TAC[LEFT_EXISTS_AND_THM; GSYM RTC_RELPOW] THEN ONCE_REWRITE_TAC[TC_CASES_L] THEN REWRITE_TAC[RTC; RC_EXPLICIT] THEN MESON_TAC[]);; let RELPOW_SEQUENCE = prove (`!R n x y. RELPOW n R x y <=> ?f. (f(0) = x:A) /\ (f(n) = y) /\ !i. i < n ==> R (f i) (f(SUC i))`, GEN_TAC THEN INDUCT_TAC THEN ASM_REWRITE_TAC[LT; RELPOW] THENL [REPEAT GEN_TAC THEN EQ_TAC THENL [DISCH_THEN SUBST1_TAC THEN EXISTS_TAC `\n:num. y:A` THEN REWRITE_TAC[]; MESON_TAC[]]; REPEAT GEN_TAC THEN EQ_TAC THEN REPEAT STRIP_TAC THENL [DISJ_CASES_TAC(ARITH_RULE `(n = 0) \/ 0 < n`) THENL [EXISTS_TAC `\i. if i = 0 then x else y:A` THEN ASM_REWRITE_TAC[ARITH; LT] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[NOT_SUC] THEN ASM_MESON_TAC[]; EXISTS_TAC `\i. if i <= n then f(i) else (y:A)` THEN ASM_REWRITE_TAC[LE_0; ARITH_RULE `~(SUC n <= n)`] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[LE_REFL; ARITH_RULE `~(SUC n <= n)`] THEN ASM_REWRITE_TAC[LE_SUC_LT] THEN ASM_REWRITE_TAC[LE_LT] THEN FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]]; EXISTS_TAC `(f:num->A) n` THEN CONJ_TAC THENL [EXISTS_TAC `f:num->A` THEN ASM_REWRITE_TAC[] THEN REPEAT STRIP_TAC THEN FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]; ASM_MESON_TAC[]]]]);;