Verifikation reaktiver Systeme - Universität Kaiserslautern

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end;
STRIP_TAC THEN
ASSUME_TAC (UNDISCH_ALL (SPEC ‘‘nmr (rep_rat a)‘‘
ABS_NOT_0_POSITIVE)) THEN
ARW_TAC[NMR,DNM] THEN
RAT_ABS_EQUIV_TAC THEN
REWRITE_TAC[rat_equiv_def] THEN
ASSUME_TAC (UNDISCH_ALL (SPEC ‘‘ABS (nmr (rep_rat a))‘‘
(SPEC ‘‘rep_rat a‘‘ lemmaY))) THEN
ARW_TAC[NMR,DNM] THEN
ARW_TAC[INT_ABS,sgn_def] THEN
INT_RING_TAC )
val RAT_MULT_LINV = store_thm("RAT_MULT_LINV", ‘‘!a. ~(a = rat_0) ==>
(rat_mul (rat_minv a) a = rat_1)‘‘,
ONCE_REWRITE_TAC[RAT_MULT_COMM] THEN
ARW_TAC[RAT_MULT_RINV] );
(*--------------------------------------------------------------------------
* RAT_RDISTRIB: thm
* |- !a b c. rat_mul (rat_add a b) c = rat_add (rat_mul a c) (rat_mul b c)
*
* RAT_LDISTRIB: thm
* |- !a b c. rat_mul c (rat_add a b) = rat_add (rat_mul c a) (rat_mul c b)
*--------------------------------------------------------------------------*)
val RAT_RDISTRIB = store_thm("RAT_RDISTRIB",
‘‘!a b c. rat_mul (rat_add a b) c = rat_add (rat_mul a c) (rat_mul b c)‘‘,
REPEAT GEN_TAC THEN
REWRITE_TAC[rat_mul_def,rat_add_def] THEN
ARW_TAC[RAT_MULT_CONG, RAT_ADD_CONG] THEN
REWRITE_TAC[mul_def,add_def] THEN
DNM_POS_ASM_TAC ‘‘dnm (rep_rat a) * dnm (rep_rat b)‘‘ THEN
DNM_POS_ASM_TAC ‘‘dnm (rep_rat b) * dnm (rep_rat c)‘‘ THEN
DNM_POS_ASM_TAC ‘‘dnm (rep_rat a) * dnm (rep_rat c)‘‘ THEN
ARW_TAC[NMR,DNM] THEN
RAT_ABS_EQUIV_TAC THEN REWRITE_TAC[rat_equiv_def] THEN
DNM_POS_ASM_TAC ‘‘dnm (rep_rat a) * dnm (rep_rat b) * dnm (rep_rat c)‘‘ THEN
DNM_POS_ASM_TAC ‘‘dnm (rep_rat a) * dnm (rep_rat c)
* (dnm (rep_rat b) * dnm (rep_rat c))‘‘ THEN
ARW_TAC[NMR,DNM] THEN
INT_RING_TAC );
val RAT_LDISTRIB = store_thm("RAT_LDISTRIB",
‘‘!a b c. rat_mul c (rat_add a b) = rat_add (rat_mul c a) (rat_mul c b)‘‘,
ONCE_REWRITE_TAC[RAT_MULT_COMM] THEN
REWRITE_TAC[RAT_RDISTRIB] );
(*--------------------------------------------------------------------------
* RAT_IS_RING: thm