Verifikation reaktiver Systeme - Universität Kaiserslautern

235
The fourth let part constructs the two subgoals (numerators are equal, denominators
are equal) using the constructor function mk_eq. These subgoals prove
the initial goal; the argument for this is !a1 b1 a2 b2. (a1=a2) /\ (b1=b2) ==>
(abs\_frac(a1,b1)=abs\_frac(a2,b2)).
(* FRAC_REP_ABS_SUBST: 0 < b |- (rep_frac (abs_frac (a,b)) = (a,b)) : thm *)
(* NMR: !a b. 0 < b ==> (nmr (abs_frac (a,b)) = a) *)
val NMR = store_thm("NMR", ‘‘!a b. 0 < b ==> (nmr (abs_frac (a,b)) = a)‘‘,
REPEAT STRIP_TAC THEN
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] );
The pair library offers theorems to simplify terms of the form FST(a,b)=a or
SND(a,b)=b. Similar theorems are provided by the fractions library. NMR and DNM
reduce the terms nmr(abs frac(a,b)) and dnm(abs frac(a,b)). Both of them
require that the denominator of the included fraction is positive. Otherwise, this
simplification step cannot be done. In most cases, this means the proof gets stuck
at this place.
To show the required precondition, another tactic comes to help:
(* INT_MUL_POS_SIGN: |- !a b. 0 0 0
ASSUME_TAC( (prove(intSyntax.mk_less(intSyntax.zero_tm,term1),
ARW_TAC[FRAC_DNM_POSITIVE,INT_MUL_POS_SIGN])) )
handle HOL_ERR _ => raise ERR "FRAC_NOT_NAN_ASM_TAC" "";
For a given term, the tactic tries to prove that it is positive, using the facts
that the denominator of a fraction is positive and the product of two positive
numbers is positive.
Equipped with these utilities, some properties of fractions can be shown quite
elegantly, e.g. the associativity of the addition:
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 );