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
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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) == nat_rec m 0 (%n r.n)"
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```    20   add_def   "m+n == nat_rec m n (%u v. Suc(v))"
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```    21   diff_def  "m-n == nat_rec n m (%u v. pred(v))"
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```    22   mult_def  "m*n == nat_rec m 0 (%u v. n + v)"
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```    23   mod_def   "m mod n == wfrec (trancl pred_nat) m (%j f.(if (j<n) j (f (j-n))))"
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```    24   div_def   "m div n == wfrec (trancl pred_nat) m (%j f.(if (j<n) 0 (Suc (f (j-n)))))"
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```    25 end
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```    26
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```    27 (*"Difference" is subtraction of natural numbers.
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```    28   There are no negative numbers; we have
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```    29      m - n = 0  iff  m<=n   and     m - n = Suc(k) iff m>n.
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```    30   Also, nat_rec(m, 0, %z w.z) is pred(m).   *)
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```    31
```