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(* Title: CTT/arith

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ID: $Id$

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory

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Copyright 1991 University of Cambridge


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Arithmetic operators and their definitions


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Proves about elementary arithmetic: addition, multiplication, etc.


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Tests definitions and simplifier.


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*)


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Arith = CTT +


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consts "#+","","" :: "[i,i]=>i" (infixr 65)


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"#*",div,mod :: "[i,i]=>i" (infixr 70)

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rules

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add_def "a#+b == rec(a, b, %u v. succ(v))"


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diff_def "ab == rec(b, a, %u v. rec(v, 0, %x y. x))"

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absdiff_def "ab == (ab) #+ (ba)"


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mult_def "a#*b == rec(a, 0, %u v. b #+ v)"

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mod_def "a mod b == rec(a, 0, %u v. rec(succ(v)  b, 0, %x y. succ(v)))"


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div_def "a div b == rec(a, 0, %u v. rec(succ(u) mod b, succ(v), %x y. v))"

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end
