src/HOL/Arith.thy
author pusch
Tue Feb 25 15:11:12 1997 +0100 (1997-02-25)
changeset 2681 93ed51a91622
parent 2099 c5f004bfcbab
child 2887 00b8ee790d89
permissions -rw-r--r--
definitions of +,-,* replaced by primrec definitions
     1 (*  Title:      HOL/Arith.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1993  University of Cambridge
     5 
     6 Arithmetic operators and their definitions
     7 *)
     8 
     9 Arith = Nat +
    10 
    11 instance
    12   nat :: {plus, minus, times}
    13 
    14 consts
    15   pred      :: nat => nat
    16   div, mod  :: [nat, nat] => nat  (infixl 70)
    17 
    18 defs
    19   pred_def  "pred(m) == case m of 0 => 0 | Suc n => n"
    20 
    21   mod_def   "m mod n == wfrec (trancl pred_nat)
    22                           (%f j. if j<n then j else f (j-n)) m"
    23   div_def   "m div n == wfrec (trancl pred_nat) 
    24                           (%f j. if j<n then 0 else Suc (f (j-n))) m"
    25 
    26 
    27 primrec "op +" nat 
    28 "0 + n = n"
    29 "Suc m + n = Suc(m + n)"
    30 
    31 
    32 primrec "op -" nat 
    33 "m - 0 = m"
    34 "m - Suc n = pred(m - n)"
    35 
    36 primrec "op *"  nat 
    37 "0 * n = 0"
    38 "Suc m * n = n + (m * n)"
    39 
    40 
    41 end
    42 
    43 (*"Difference" is subtraction of natural numbers.
    44   There are no negative numbers; we have
    45      m - n = 0  iff  m<=n   and     m - n = Suc(k) iff m)n.
    46   Also, nat_rec(0, %z w.z, m) is pred(m).   *)
    47