Verifikation reaktiver Systeme - Universität Kaiserslautern

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in
(SPEC ‘‘a:int - c:int‘‘ INT_EQ_SUB_LADD))] THEN
ONCE_REWRITE_TAC[EQT_ELIM (intLib.ARITH_CONV
‘‘(a:int) - (c:int) + (b:int) = a + b - c‘‘)] THEN
REWRITE_TAC[SPEC ‘‘d:int‘‘ (SPEC ‘‘c:int‘‘
(SPEC ‘‘(a:int) + (b:int)‘‘ INT_EQ_SUB_RADD)) ] THEN
ARW_TAC[EQT_ELIM (intLib.ARITH_CONV ‘‘(d:int) + (c:int) = c + d‘‘)] );
val lemmaY = prove(‘‘!a:int b:int. 0 ( (a*b=0)=(a=0) )‘‘,
REPEAT STRIP_TAC THEN
EQ_TAC THENL
[
SUBST_TAC[UNDISCH_ALL (SPEC ‘‘b:int‘‘ (SPEC ‘‘0i‘‘ (SPEC ‘‘a:int‘‘
INT_EQ_RMUL_EXP)))] THEN
ASM_REWRITE_TAC[] THEN
ARW_TAC[INT_MUL_LZERO]
,
ARW_TAC[INT_MUL_LZERO]
]);
val subst1 = UNDISCH_ALL (SPEC ‘‘dnm y‘‘
(SPEC ‘‘(nmr x * (dnm (rep_rat (abs_rat x)) * dnm y)
+ nmr y * dnm x * dnm (rep_rat (abs_rat x)))‘‘
(SPEC ‘‘(nmr (rep_rat (abs_rat x)) * dnm x * dnm y
+ nmr y * dnm (rep_rat (abs_rat x)) * dnm x)‘‘
(GSYM INT_EQ_RMUL_EXP))));
store_thm("RAT_ADD_CONG1",
‘‘!x y. abs_rat (add (rep_rat (abs_rat x)) y) = abs_rat (add x y)‘‘,
REPEAT GEN_TAC THEN
REWRITE_TAC[RAT_ABS_EQUIV] THEN
REWRITE_TAC[rat_equiv_def,add_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[INT_RDISTRIB] THEN
REWRITE_TAC[EQT_ELIM (
RING_CONV ‘‘ nmr (rep_rat (abs_rat x)) * dnm y
* (dnm x * dnm y) + nmr y * dnm (rep_rat (abs_rat x))
* (dnm x * dnm y) = (nmr (rep_rat (abs_rat x)) * dnm x
* dnm y + nmr y * dnm (rep_rat (abs_rat x)) * dnm x)
* dnm y‘‘)] THEN
REWRITE_TAC[EQT_ELIM (RING_CONV ‘‘ nmr x * dnm y
* (dnm (rep_rat (abs_rat x)) * dnm y) + nmr y * dnm x
* (dnm (rep_rat (abs_rat x)) * dnm y)=(nmr x
* (dnm (rep_rat (abs_rat x)) * dnm y) + nmr y
* dnm x * (dnm (rep_rat (abs_rat x)))) * dnm y‘‘)] THEN
DNM_POS_ASM_TAC ‘‘dnm y‘‘ THEN
SUBST_TAC[subst1] THEN
REWRITE_TAC[EQT_ELIM (RING_CONV ‘‘nmr x
* (dnm (rep_rat (abs_rat x)) * dnm y) = nmr x
* dnm (rep_rat (abs_rat x)) * dnm y‘‘)] THEN
REWRITE_TAC[EQT_ELIM (RING_CONV ‘‘nmr y * dnm x