Verifikation reaktiver Systeme - Universität Kaiserslautern

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end;
* dnm (rep_rat (abs_rat x)) = nmr y * (dnm x
* dnm (rep_rat (abs_rat x)))‘‘)] THEN
REWRITE_TAC[EQT_ELIM (RING_CONV ‘‘nmr y * dnm (rep_rat (abs_rat x))
* dnm x = nmr y * (dnm (rep_rat (abs_rat x)) * dnm x)‘‘)] THEN
ARW_TAC[lemmaX] THEN
REWRITE_TAC[GSYM INT_SUB_LDISTRIB, GSYM INT_SUB_RDISTRIB] THEN
REWRITE_TAC[EQT_ELIM (RING_CONV
‘‘dnm x * dnm (rep_rat (abs_rat x))
= dnm (rep_rat (abs_rat x)) * dnm x‘‘)] THEN
ARW_TAC[INT_MUL_RZERO,INT_SUB_REFL] THEN
ARW_TAC[lemmaY] THEN
ARW_TAC[INT_EQ_SUB_LADD, INT_ADD_LID] THEN
REWRITE_TAC[GSYM rat_equiv_def] THEN
ARW_TAC[GSYM REP_ABS_EQUIV] )
val RAT_ADD_CONG2 = store_thm("RAT_ADD_CONG",
‘‘!x y. abs_rat (add x (rep_rat (abs_rat y))) = abs_rat (add x y)‘‘,
ONCE_REWRITE_TAC[FRAC_ADD_COMM] THEN
ARW_TAC[RAT_ADD_CONG1]);
val RAT_ADD_CONG = save_thm
("RAT_ADD_CONG", CONJ RAT_ADD_CONG1 RAT_ADD_CONG2);
(*--------------------------------------------------------------------------
* RAT_MULT_CONG1: thm
* |- !x y. abs_rat (mul (rep_rat (abs_rat x)) y) = abs_rat (mul x y)
*
* RAT_MULT_CONG2: thm
* |- !x y. abs_rat (mul x (rep_rat (abs_rat y))) = abs_rat (mul x y)
*
* RAT_MULT_CONG: thm
* |- !x y. abs_rat (mul (rep_rat (abs_rat x)) y) = abs_rat (mul x y) /\
* !x y. abs_rat (mul x (rep_rat (abs_rat y))) = abs_rat (mul x y)
*--------------------------------------------------------------------------*)
val RAT_MULT_CONG1 = store_thm("RAT_MULT_CONG1",
‘‘!x y. abs_rat (mul (rep_rat (abs_rat x)) y) = abs_rat (mul x y)‘‘,
REPEAT GEN_TAC THEN
REWRITE_TAC[RAT_ABS_EQUIV] THEN
REWRITE_TAC[rat_equiv_def,mul_def] THEN
DNM_POS_ASM_TAC ‘‘dnm x * dnm y‘‘ THEN
DNM_POS_ASM_TAC ‘‘dnm (rep_rat (abs_rat x)) * dnm y‘‘ THEN
ARW_TAC[NMR,DNM] THEN
REWRITE_TAC[EQT_ELIM (RING_CONV ‘‘nmr (rep_rat (abs_rat x))
* nmr y * (dnm x * dnm y) = nmr (rep_rat (abs_rat x)) *dnm x
* nmr y * dnm y‘‘)] THEN
REWRITE_TAC[EQT_ELIM (RING_CONV ‘‘nmr x * nmr y
* (dnm (rep_rat (abs_rat x)) * dnm y) = nmr x