Verifikation reaktiver Systeme - Universität Kaiserslautern

255
(* interactive mode
app load ["fractionTheory", "fractionLib", "quotient"];
*)
open
pairTheory arithmeticTheory ringTheory ringLib
integerTheory intLib integerRingTheory integerRingLib
fractionTheory fractionLib quotient;
val _ = new_theory "rational";
val ARW_TAC = RW_TAC int_ss;
val frac_bij = DB.fetch "fraction" "frac_tybij";
(*--------------------------------------------------------------------------*
* some lemmas
*--------------------------------------------------------------------------*)
(* LESS_IMP_NOT_0: |- !n. 0i ~(n=0i) *)
val LESS_IMP_NOT_0 = store_thm("LESS_IMP_NOT_0",
‘‘!n:int. 0i ~(n=0i)‘‘,
GEN_TAC THEN
ASM_CASES_TAC ‘‘n=0i‘‘
THEN ARW_TAC[] );
(* INT_EQ_RMUL_EXP: |- !a b n. 0 ((a=b) = (a*n=b*n)) *)
val INT_EQ_RMUL_EXP = store_thm("INT_EQ_RMUL_EXP",
‘‘!a:int b:int n:int. 0 ((a=b) = (a*n=b*n))‘‘,
REPEAT STRIP_TAC
THEN EQ_TAC
THEN ASSUME_TAC (prove(‘‘0i ~(n=0i)‘‘,
ASM_CASES_TAC ‘‘n=0i‘‘ THEN ARW_TAC[]))
THEN ASSUME_TAC (SPEC ‘‘n:int‘‘ (SPEC ‘‘b:int‘‘
(SPEC ‘‘a:int‘‘ INT_EQ_RMUL_IMP)))
THEN ARW_TAC[] );
(* INT_LT_RMUL_EXP: |- !a b n. 0 ((ab*n
REPEAT STRIP_TAC THEN
REWRITE_TAC[int_gt] THEN
ASSUME_TAC (UNDISCH_ALL (GSYM (SPEC ‘‘a:int‘‘ (SPEC ‘‘b:int‘‘
(SPEC ‘‘n:int‘‘ INT_LT_MONO))))) THEN
ARW_TAC[INT_MUL_SYM] );