src/HOL/Arith.thy
author clasohm
Fri Mar 03 12:02:25 1995 +0100 (1995-03-03)
changeset 923 ff1574a81019
child 965 24eef3860714
permissions -rw-r--r--
new version of HOL with curried function application
     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) m (%j f.(if (j<n) j (f (j-n))))"
    24   div_def   "m div n == wfrec (trancl pred_nat) m (%j f.(if (j<n) 0 (Suc (f (j-n)))))"
    25 end
    26 
    27 (*"Difference" is subtraction of natural numbers.
    28   There are no negative numbers; we have
    29      m - n = 0  iff  m<=n   and     m - n = Suc(k) iff m>n.
    30   Also, nat_rec(m, 0, %z w.z) is pred(m).   *)
    31