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(* Title: HOL/arith.thy
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1993 University of Cambridge
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Arithmetic operators and their definitions
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*)
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Arith = Nat +
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arities nat::plus
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nat::minus
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nat::times
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consts
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pred :: "nat => nat"
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div,mod :: "[nat,nat]=>nat" (infixl 70)
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rules
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pred_def "pred(m) == nat_rec(m, 0, %n r.n)"
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add_def "m+n == nat_rec(m, n, %u v.Suc(v))"
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diff_def "m-n == nat_rec(n, m, %u v. nat_rec(v, 0, %x y.x))"
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mult_def "m*n == nat_rec(m, 0, %u v. n + v)"
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mod_def "m mod n == wfrec(trancl(pred_nat), m, %j f. if(j<n, j, f(j-n)))"
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div_def "m div n == wfrec(trancl(pred_nat), m, %j f. if(j<n, 0, Suc(f(j-n))))"
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end
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(*"Difference" is subtraction of natural numbers.
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There are no negative numbers; we have
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m - n = 0 iff m<=n and m - n = Suc(k) iff m>n.
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Also, nat_rec(m, 0, %z w.z) is pred(m). *)
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