Verifikation reaktiver Systeme - Universität Kaiserslautern

241
ARW_TAC[] THEN
ARW_TAC[ODD_DOUBLE] );
val ODD_EVEN_PRODUCT = prove(‘‘!x y. ODD x /\ EVEN (y*x) ==> EVEN y‘‘,
REPEAT GEN_TAC THEN
REWRITE_TAC[EVEN_MULT] THEN
ONCE_REWRITE_TAC[LEFT_AND_OVER_OR] THEN
ARW_TAC[ONCE_REWRITE_RULE[CONJ_COMM] EVEN_AND_ODD] );
val ASM_IMP_CONJ = prove(‘‘!a b c. (a /\ b ==> c) = (a ==> b ==> c)‘‘,
PROVE_TAC[] );
val DIVIDES_ODD_EVEN = store_thm("DIVIDES_ODD_EVEN",
‘‘!x y. ODD x /\ divides x (2 * y) ==> divides x y‘‘,
REPEAT GEN_TAC THEN
REWRITE_TAC[ODD_EXISTS] THEN
REWRITE_TAC[divides_def] THEN
STRIP_TAC THEN
ONCE_REWRITE_TAC[GSYM (SPEC ‘‘1‘‘ (SPEC ‘‘q * x‘‘
(SPEC ‘‘y:num‘‘ MULT_MONO_EQ)))] THEN
REWRITE_TAC[SUC1_IS_2] THEN
ASM_REWRITE_TAC[] THEN
REWRITE_TAC[EQT_ELIM (AC_CONV(MULT_ASSOC,MULT_COMM)
‘‘2 * (q’ * SUC (2 * m)) = 2 * q’ * SUC (2 * m)‘‘)] THEN
REWRITE_TAC[ONCE_REWRITE_RULE[MULT_COMM] MULT_MONO_EQ] THEN
REWRITE_TAC[GSYM EVEN_EXISTS] THEN
ASSUME_TAC (UNDISCH_ALL (REWRITE_RULE[SUC1_IS_2] (SPEC ‘‘m:num‘‘
(SPEC ‘‘x:num‘‘ ODD_EXISTS_IMP)))) THEN
ASSUME_TAC (UNDISCH_ALL
(SPEC ‘‘y:num‘‘ (SPEC ‘‘q*x‘‘ EVEN_DOUBLE_EQ))) THEN
ASSUME_TAC (UNDISCH_ALL (REWRITE_RULE [ASM_IMP_CONJ] (SPEC ‘‘q:num‘‘
(SPEC ‘‘x:num‘‘ ODD_EVEN_PRODUCT)))) THEN
PROVE_TAC[] );
val ODD_DIVIDES_ODD = store_thm("ODD_DIVIDES_ODD",
‘‘!x y. ODD x /\ divides y x ==> ODD y‘‘,
REPEAT GEN_TAC THEN
REWRITE_TAC[CONTRAPOS_CONV ‘‘ODD x /\ divides y x ==> ODD y‘‘] THEN
REWRITE_TAC[DE_MORGAN_THM] THEN
REWRITE_TAC[GSYM EVEN_ODD] THEN
REWRITE_TAC[EVEN_EXISTS, divides_def] THEN
STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN
LEFT_DISJ_TAC THEN
ARW_TAC[] THEN
EXISTS_TAC ‘‘m*q‘‘ THEN
ARW_TAC[AC_CONV(MULT_ASSOC,MULT_COMM) ‘‘q * (2 * m) = 2 * (m * q)‘‘] );
val IS_GCD_DOUBLE = store_thm("IS_GCD_DOUBLE",
‘‘is_gcd (2*a) (2*b) (2*c) = is_gcd a b c‘‘,