Verifikation reaktiver Systeme - Universität Kaiserslautern

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REWRITE_TAC[nmr_def] THEN
REWRITE_TAC[FRAC_REP_ABS_SUBST] );
(* DNM: !a b. 0 < b ==> (nmr (abs_frac (a,b)) = a) *)
val DNM = store_thm("DNM", ‘‘!a b. 0 < b ==> (dnm (abs_frac (a,b)) = b)‘‘,
REPEAT STRIP_TAC THEN
REWRITE_TAC[dnm_def] THEN
REWRITE_TAC[FRAC_REP_ABS_SUBST] );
(*--------------------------------------------------------------------------
* FRAC_ID: thm
* |- !f. abs_frac (nmr f,dnm f) = f
*--------------------------------------------------------------------------*)
val FRAC_ID = store_thm("FRAC_ID", ‘‘!f. abs_frac (nmr f,dnm f) = f‘‘,
STRIP_TAC THEN
REWRITE_TAC[nmr_def,dnm_def]
THEN ARW_TAC[FRAC_ABS_REP_SUBST]);
(*--------------------------------------------------------------------------
* FRAC_EXPLODE_THM: thm
* |- !f a b. 0 < b ==> ((f = abs_frac(a,b)) = (nmr f = a) /\ (dnm f = b))
*--------------------------------------------------------------------------*)
val FRAC_EXPLODE_THM = store_thm("FRAC_EXPLODE_THM",
‘‘!f a b. 0 < b ==> ((f = abs_frac (a,b)) = (nmr f = a) /\ (dnm f = b))‘‘,
REPEAT STRIP_TAC THEN
EQ_TAC THEN
STRIP_TAC THEN
ARW_TAC[NMR,DNM,FRAC_ID] );
(*--------------------------------------------------------------------------
* FRAC_ADD_ASSOC: thm
* |- !a b c. add a (add b c) = add (add a b) c
*
* FRAC_MULT_ASSOC: thm
* |- !a b c. mul a (mul b c) = mul (mul a b) c
*--------------------------------------------------------------------------*)
val FRAC_ADD_ASSOC = store_thm("FRAC_ADD_ASSOC",
‘‘!a b c. add a (add b c) = add (add a b) c‘‘,
REPEAT STRIP_TAC THEN REWRITE_TAC[add_def]
THEN DNM_POS_ASM_TAC ‘‘dnm a * dnm b‘‘
THEN DNM_POS_ASM_TAC ‘‘dnm b * dnm c‘‘
THEN ARW_TAC[NMR,DNM]
THEN FRAC_EQ_TAC THEN INT_RING_TAC );
val FRAC_MULT_ASSOC = store_thm("FRAC_MULT_ASSOC",
‘‘!a b c. mul a (mul b c) = mul (mul a b) c‘‘,
REPEAT STRIP_TAC THEN REWRITE_TAC[mul_def]