Verifikation reaktiver Systeme - Universität Kaiserslautern

258
*
* alternative representation of equivalence relation
*--------------------------------------------------------------------------*)
val RAT_EQUIV_ALT =
let
val lemma1 = prove(‘‘! x y. 0 0 < dnm x * y‘‘,
REPEAT GEN_TAC THEN DNM_POS_ASM_TAC ‘‘dnm x‘‘ THEN
ARW_TAC[INT_MUL_POS_SIGN]);
val lemma2 = prove(
‘‘(abs_frac (nmr a * b,dnm a * b) = abs_frac (nmr f * c,dnm f * c))
==> (rep_frac (abs_frac (nmr a * b,dnm a * b))
= rep_frac (abs_frac (nmr f * c,dnm f * c)))‘‘, ARW_TAC[]);
(* lemma3: 0 < dnm a * b |=
rep_frac (abs_frac (nmr a * b,dnm a * b)) = (nmr a * b,dnm a * b) *)
val lemma3 = UNDISCH_ALL (REWRITE_RULE[SND]
(#1 (EQ_IMP_RULE (SPEC ‘‘(nmr a * b,dnm a * b)‘‘
(BETA_RULE (CONJUNCT2 frac_bij))))));
(* lemma4: 0 < dnm f * c |=
rep_frac (abs_frac (nmr f * c,dnm f * c)) = (nmr f * c,dnm f * c) *)
val lemma4 = UNDISCH_ALL (REWRITE_RULE[SND]
(#1 (EQ_IMP_RULE (SPEC ‘‘(nmr f * c,dnm f * c)‘‘
(BETA_RULE (CONJUNCT2 frac_bij))))));
in
val subst1 = UNDISCH_ALL (SPEC ‘‘b:int‘‘
(SPEC ‘‘nmr f * dnm a‘‘ (SPEC ‘‘nmr a * dnm f‘‘ INT_EQ_RMUL_EXP)));
val subst2 = UNDISCH_ALL (SPEC ‘‘c:int‘‘
(SPEC ‘‘nmr f * dnm a * b‘‘ (SPEC ‘‘nmr a * dnm f * b‘‘
INT_EQ_RMUL_EXP)));
store_thm("RAT_EQUIV_ALT",
‘‘!a. rat_equiv a = \x. (?b c. 0