| author | paulson |
| Tue, 17 Oct 1995 12:09:46 +0100 | |
| changeset 1283 | ea8b657a9c92 |
| parent 972 | e61b058d58d2 |
| child 1370 | 7361ac9b024d |
| permissions | -rw-r--r-- |
| 923 | 1 |
(* 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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instance |
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nat :: {plus, minus, 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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defs |
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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. pred(v))" |
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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 then j else f (j-n))" |
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div_def "m div n == wfrec (trancl pred_nat) m (%j f.if j<n then 0 else 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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972
e61b058d58d2
changed syntax of tuples from <..., ...> to (..., ...)
clasohm
parents:
965
diff
changeset
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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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