Verifikation reaktiver Systeme - Universität Kaiserslautern

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(* neutral elements *)
val rat_ainv_def = Define ‘rat_ainv r1 = abs_rat( ainv (rep_rat r1))‘;
val rat_minv_def = Define ‘rat_minv r1 = abs_rat( minv (rep_rat r1))‘;
(* less (absolute value) *)
val rat_les_abs_def = Define
‘rat_les_abs r1 r2 = les_abs (rep_rat r1) (rep_rat r2)‘;
(* basic arithmetics *)
val rat_add_def = Define
‘rat_add r1 r2 = abs_rat( add (rep_rat r1) (rep_rat r2) )‘;
val rat_sub_def = Define
‘rat_sub r1 r2 = abs_rat( sub (rep_rat r1) (rep_rat r2) )‘;
val rat_mul_def = Define
‘rat_mul r1 r2 = abs_rat( mul (rep_rat r1) (rep_rat r2) )‘;
val rat_div_def = Define
‘rat_div r1 r2 = abs_rat( div (rep_rat r1) (rep_rat r2) )‘;
(*--------------------------------------------------------------------------
* REP_ABS_EQUIV
*--------------------------------------------------------------------------*)
val REP_ABS_EQUIV = prove(
‘‘!a. rat_equiv a (rep_rat (abs_rat a))‘‘,
GEN_TAC THEN
REWRITE_TAC[CONJUNCT2 rat_def] THEN
REWRITE_TAC[CONJUNCT1 rat_def]);
val REP_ABS_DFN_EQUIV = prove(
‘‘!x. nmr x * dnm (rep_rat(abs_rat x))
= nmr (rep_rat(abs_rat x)) * dnm x‘‘,
GEN_TAC THEN
REWRITE_TAC[GSYM rat_equiv_def] THEN
ARW_TAC[REP_ABS_EQUIV] );
(* |- !r’ r. (abs_rat r = abs_rat r’) = rat_equiv r r’ *)
val RAT_ABS_EQUIV = GEN_ALL (SYM (SPEC_ALL (CONJUNCT2 rat_def)));
val RAT_ABS_EQUIV_TAC = REWRITE_TAC[RAT_ABS_EQUIV];
val RAT_IMP_EQUIV = prove(‘‘!r1 r2. (r1 = r2) ==> rat_equiv r1 r2‘‘,
REPEAT STRIP_TAC THEN
REWRITE_TAC[rat_equiv_def]
THEN ARW_TAC[]);
val RAT_EQUIV_TRANS_IMP = prove(
‘‘!a b c. rat_equiv a b ==> (rat_equiv a c = rat_equiv b c)‘‘,
REPEAT STRIP_TAC THEN