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.
```     1 (*  Title:      HOL/Arith.thy
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```     2     ID:         \$Id\$
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```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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```     4     Copyright   1993  University of Cambridge
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```     5
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```     6 Arithmetic operators and their definitions
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```     7 *)
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```     8
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```     9 Arith = Nat +
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```    10
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```    11 instance
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```    12   nat :: {plus, minus, times}
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```    13
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```    14 consts
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```    15   pred      :: nat => nat
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```    16   div, mod  :: [nat, nat] => nat  (infixl 70)
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```    17
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```    18 defs
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```    19   pred_def  "pred(m) == case m of 0 => 0 | Suc n => n"
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```    20   add_def   "m+n == nat_rec n (%u v. Suc(v)) m"
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```    21   diff_def  "m-n == nat_rec m (%u v. pred(v)) n"
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```    22   mult_def  "m*n == nat_rec 0 (%u v. n + v) m"
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```    23
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```    24   mod_def   "m mod n == wfrec (trancl pred_nat)
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```    25                           (%f j. if j<n then j else f (j-n)) m"
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```    26   div_def   "m div n == wfrec (trancl pred_nat)
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```    27                           (%f j. if j<n then 0 else Suc (f (j-n))) m"
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```    28 end
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```    29
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```    30 (*"Difference" is subtraction of natural numbers.
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```    31   There are no negative numbers; we have
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```    32      m - n = 0  iff  m<=n   and     m - n = Suc(k) iff m)n.
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```    33   Also, nat_rec(0, %z w.z, m) is pred(m).   *)
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```    34
```