src/HOL/Arith.thy
author clasohm
Wed Mar 13 11:55:25 1996 +0100 (1996-03-13 ago)
changeset 1574 5a63ab90ee8a
parent 1475 7f5a4cd08209
child 1796 c42db9ab8728
permissions -rw-r--r--
modified primrec so it can be used in MiniML/Type.thy
     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) == nat_rec m 0 (%n r.n)"
    20   add_def   "m+n == nat_rec m n (%u v. Suc(v))"
    21   diff_def  "m-n == nat_rec n m (%u v. pred(v))"
    22   mult_def  "m*n == nat_rec m 0 (%u v. n + v)"
    23 mod_def "m mod n == wfrec (trancl pred_nat)
    24                           (%f j. if j<n then j else f (j-n)) m"
    25 div_def "m div n == wfrec (trancl pred_nat) 
    26                           (%f j. if j<n then 0 else Suc (f (j-n))) m"
    27 end
    28 
    29 (*"Difference" is subtraction of natural numbers.
    30   There are no negative numbers; we have
    31      m - n = 0  iff  m<=n   and     m - n = Suc(k) iff m)n.
    32   Also, nat_rec(m, 0, %z w.z) is pred(m).   *)
    33