src/HOL/Arith.thy
author nipkow
Wed Oct 16 10:37:17 1996 +0200 (1996-10-16)
changeset 2099 c5f004bfcbab
parent 1824 44254696843a
child 2681 93ed51a91622
permissions -rw-r--r--
Defined pred using nat_case rather than nat_rec.
Added expand_nat_case
     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   add_def   "m+n == nat_rec n (%u v. Suc(v)) m"
    21   diff_def  "m-n == nat_rec m (%u v. pred(v)) n"
    22   mult_def  "m*n == nat_rec 0 (%u v. n + v) m"
    23 
    24   mod_def   "m mod n == wfrec (trancl pred_nat)
    25                           (%f j. if j<n then j else f (j-n)) m"
    26   div_def   "m div n == wfrec (trancl pred_nat) 
    27                           (%f j. if j<n then 0 else Suc (f (j-n))) m"
    28 end
    29 
    30 (*"Difference" is subtraction of natural numbers.
    31   There are no negative numbers; we have
    32      m - n = 0  iff  m<=n   and     m - n = Suc(k) iff m)n.
    33   Also, nat_rec(0, %z w.z, m) is pred(m).   *)
    34