Verifikation reaktiver Systeme - Universität Kaiserslautern

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val add_def = Define ‘add f1 f2 =
abs_frac(nmr f1 * dnm f2 + nmr f2 * dnm f1, dnm f1 * dnm f2)‘;
val mul_def = Define ‘mul f1 f2 =
abs_frac(nmr f1 * nmr f2, dnm f1 * dnm f2)‘;
val sub_def = Define ‘sub f1 f2 = add f1 (ainv f2)‘;
val div_def = Define ‘div f1 f2 = mul f1 (minv f2)‘;
Addition and multiplication are defined intuitively. Substraction and division
use them and the inverse elements.
Like in many other areas, good tools are the key to success. A well considered
set of tactics and lemmas makes proofs much simpler.
val FRAC_EQ_TAC:tactic = fn (asl,w) =>
let
val (lhs,rhs) = dest_eq w
in
let
val (lhc, lha) = dest_comb lhs;
val (rhc, rha ) = dest_comb rhs;
in
let
val [a1,b1] = strip_pair lha;
val [a2,b2] = strip_pair rha;
in
let
val sg1 = mk_eq(a1,a2);
val sg2 = mk_eq(b1,b2);
in
(
[(asl,sg1), (asl,sg2)],
fn [thm1,thm2] => MP
(SPEC b2 (SPEC a2 (SPEC b1 (SPEC a1 (
prove(‘‘!a1 b1 a2 b2. (a1=a2) /\ (b1=b2) ==>
(abs_frac(a1,b1)=abs_frac(a2,b2))‘‘, ARW_TAC[])
)))))
(CONJ thm1 thm2)
)
end
end
end
end
handle HOL_ERR _ => raise ERR "FRAC_EQ_TAC" "";
FRAC EQ TAC is a tactic which is used in almost every proof involving fractions:
To show that two fractions are equal, it is sufficient to show that respectively
their numerators and denominators are equal. In the first three let parts,
the numerator and denominator of the two fractions are extracted. This is accomplished
by successively applying the appropriate destructor functions. The
strategy assumes a certain syntactical structure of the goal. If this is not given,
the tactic will fail (i.e. the goal will remain unchanged).