src/CTT/arith.thy
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     1 (*  Title: 	CTT/arith
       
     2     ID:         $Id$
       
     3     Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
       
     4     Copyright   1991  University of Cambridge
       
     5 
       
     6 Arithmetic operators and their definitions
       
     7 
       
     8 Proves about elementary arithmetic: addition, multiplication, etc.
       
     9 Tests definitions and simplifier.
       
    10 *)
       
    11 
       
    12 Arith = CTT +
       
    13 
       
    14 consts "#+","-","|-|"	:: "[i,i]=>i"	(infixr 65)
       
    15        "#*",div,mod     :: "[i,i]=>i"	(infixr 70)
       
    16 
       
    17 rules
       
    18   add_def     "a#+b == rec(a, b, %u v.succ(v))"  
       
    19   diff_def    "a-b == rec(b, a, %u v.rec(v, 0, %x y.x))"  
       
    20   absdiff_def "a|-|b == (a-b) #+ (b-a)"  
       
    21   mult_def    "a#*b == rec(a, 0, %u v. b #+ v)"  
       
    22   mod_def     "a mod b == rec(a, 0, %u v. rec(succ(v) |-| b, 0, %x y.succ(v)))"
       
    23   div_def     "a div b == rec(a, 0, %u v. rec(succ(u) mod b, succ(v), %x y.v))"
       
    24 end