src/HOL/Arith.thy
 author nipkow Tue Apr 08 10:48:42 1997 +0200 (1997-04-08) changeset 2919 953a47dc0519 parent 2887 00b8ee790d89 child 3235 351565b7321b permissions -rw-r--r--
Dep. on Provers/nat_transitive
```     1 (*  Title:      HOL/Arith.thy
```
```     2     ID:         \$Id\$
```
```     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 +
```
```    10
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```    11 instance
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```    12   nat :: {plus, minus, times}
```
```    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)
```
```    17
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```    18 defs
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```    19   pred_def  "pred(m) == case m of 0 => 0 | Suc n => n"
```
```    20
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```    21   mod_def   "m mod n == wfrec (trancl pred_nat)
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```    22                           (%f j. if j<n then j else f (j-n)) m"
```
```    23   div_def   "m div n == wfrec (trancl pred_nat)
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```    24                           (%f j. if j<n then 0 else Suc (f (j-n))) m"
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```    25
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```    26 primrec "op +" nat
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```    27   "0 + n = n"
```
```    28   "Suc m + n = Suc(m + n)"
```
```    29
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```    30 primrec "op -" nat
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```    31   "m - 0 = m"
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```    32   "m - Suc n = pred(m - n)"
```
```    33
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```    34 primrec "op *"  nat
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```    35   "0 * n = 0"
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```    36   "Suc m * n = n + (m * n)"
```
```    37
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```    38 end
```