Verifikation reaktiver Systeme - Universität Kaiserslautern

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* |- !x y. ~(nmr y = 0) ==>
* (abs_rat (div (rep_rat (abs_rat x)) y) = abs_rat (div x y))
*
* RAT_DIV_CONG2: thm
* |- !x y. ~(nmr y = 0) ==>
(abs_rat (div x (rep_rat (abs_rat y))) = abs_rat (div x y))
*
* RAT_DIV_CONG: thm
* |- !x y. ~(nmr y = 0) ==>
* (abs_rat (div (rep_rat (abs_rat x)) y) = abs_rat (div x y)) /\
* !x y. ~(nmr y = 0) ==>
(abs_rat (div x (rep_rat (abs_rat y))) = abs_rat (div x y))
*--------------------------------------------------------------------------*)
val RAT_DIV_CONG1 = store_thm("RAT_DIV_CONG1", ‘‘!x y. ~(nmr y = 0) ==>
(abs_rat (div (rep_rat (abs_rat x)) y) = abs_rat (div x y))‘‘,
REPEAT STRIP_TAC THEN
REWRITE_TAC[div_def] THEN
ARW_TAC[RAT_MULT_CONG] );
val RAT_DIV_CONG2 = store_thm("RAT_DIV_CONG2", ‘‘!x y. ~(nmr y = 0) ==>
(abs_rat (div x (rep_rat (abs_rat y))) = abs_rat (div x y))‘‘,
REPEAT STRIP_TAC THEN
REWRITE_TAC[div_def, mul_def] THEN
DNM_POS_ASM_TAC ‘‘dnm x * dnm (minv y)‘‘ THEN
DNM_POS_ASM_TAC ‘‘dnm x * dnm (minv (rep_rat (abs_rat y)))‘‘ THEN
RAT_ABS_EQUIV_TAC THEN
REWRITE_TAC [rat_equiv_def] THEN
ARW_TAC[NMR,DNM] THEN
REWRITE_TAC[EQT_ELIM (AC_CONV (INT_MUL_ASSOC,INT_MUL_SYM) ‘‘nmr x
* nmr (minv (rep_rat (abs_rat y))) * (dnm x * dnm (minv y))
= (nmr x * dnm x) * (nmr (minv (rep_rat (abs_rat y)))
* dnm (minv y))‘‘)] THEN
REWRITE_TAC[EQT_ELIM (AC_CONV (INT_MUL_ASSOC,INT_MUL_SYM)
‘‘nmr x * nmr (minv y) * (dnm x * dnm (minv (rep_rat (abs_rat y))))
= (nmr x * dnm x) *(nmr (minv y) * dnm (minv (rep_rat
(abs_rat y))))‘‘)] THEN
REWRITE_TAC[INT_EQ_LMUL] THEN
DISJ2_TAC THEN
REWRITE_TAC[GSYM rat_equiv_def] THEN
REWRITE_TAC[GSYM RAT_ABS_EQUIV] THEN
ARW_TAC[RAT_MINV_CONG] );
val RAT_DIV_CONG =
save_thm("RAT_DIV_CONG", CONJ RAT_DIV_CONG1 RAT_DIV_CONG2 );
(*--------------------------------------------------------------------------
* RAT_ADD_ASSOC: thm
* |- !a b c. rat_add a (rat_add b c) = rat_add (rat_add a b) c