Verifikation reaktiver Systeme - Universität Kaiserslautern

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The steps of the proof are as follows:
– First, all quantifiers are stripped off the goal by repeatedly applying STRIP_TAC.
Thus, all variables become free.
– Then, the definition of the addition is used to rewrite the goal. The resulting
proof obligation contains subterms of the form nmr(abs_frac(x,y)) and
dnm(abs_frac(x,y)).
– Assumptions are added, stating that respectively dnm a * dnm b and dnm b * dnm c
are positive.
– With this, the goal can be simplified. NMR and DNM are used.
– The goal of the form x=y is split up into the two subgoals x==>y and y==>x.
– Both of them are solved by INT_RING_TAC, which transforms all terms of the
goal to a normal form (using ring properties), and then compares them.
In a similar way, some more properties of fractions can be shown, e.g. the following
lemmas:
(* FRAC_MULT_ASSOC: |- !a b c. mul a (mul b c) = mul (mul a b) c *)
(* FRAC_ADD_COMM: |- !a b c. add a b = add b a *)
(* FRAC_MULT_COMM: |- !a b c. mul a b = mul b a *)
(* FRAC_ADD_RID: |- !a. add a frac_0 = a *)
(* FRAC_MUL_RID: |- !a. mul a frac_1 = a *)
(* FRAC_SUB_PLUS: |- !a b c. sub a (add b c) = sub (sub a b) c *)
(* FRAC_SUB_MINUS: |- !a b c. sub a (sub b c) = add (sub a b) c *)
(* SUB_AINV_THM: |- !a b. sub a b = ainv (sub b a) *)
Rational numbers Rational numbers are constructed as equivalence classes of
fractions. Two fractions f1 = f1n
f1 d
and f2 = f2n
f2 d
are equivalent, if the fractions
f1‘ and f2 ′ obtained by reduction of f1 andf2 are equal. This can be defined
as follows:
f1 n
∼ f2 n
⇔ f1 n · f2 d = f2 n · f1 d
f1 d f2 d
In HOL, this definition can be written as:
val rat_equiv_def =
Define ‘rat_equiv f1 f2 = (nmr f1 * dnm f2 = nmr f2 * dnm f1)‘;
The following theorem asserts that the equivalence relation has the intended
meaning:
(* RAT_EQUIV_ALT |- !a. rat_equiv a = \x. (?b c. 0