src/ZF/OrderArith.thy
 author paulson Thu Sep 07 17:36:37 2000 +0200 (2000-09-07 ago) changeset 9883 c1c8647af477 parent 1478 2b8c2a7547ab child 13140 6d97dbb189a9 permissions -rw-r--r--
a number of new theorems
```     1 (*  Title:      ZF/OrderArith.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   1994  University of Cambridge
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```     5
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```     6 Towards ordinal arithmetic.  Also useful with wfrec.
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```     7 *)
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```     8
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```     9 OrderArith = Order + Sum + Ordinal +
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```    10 consts
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```    11   radd, rmult      :: [i,i,i,i]=>i
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```    12   rvimage          :: [i,i,i]=>i
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```    13
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```    14 defs
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```    15   (*disjoint sum of two relations; underlies ordinal addition*)
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```    16   radd_def "radd(A,r,B,s) ==
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```    17                 {z: (A+B) * (A+B).
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```    18                     (EX x y. z = <Inl(x), Inr(y)>)   |
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```    19                     (EX x' x. z = <Inl(x'), Inl(x)> & <x',x>:r)   |
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```    20                     (EX y' y. z = <Inr(y'), Inr(y)> & <y',y>:s)}"
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```    21
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```    22   (*lexicographic product of two relations; underlies ordinal multiplication*)
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```    23   rmult_def "rmult(A,r,B,s) ==
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```    24                 {z: (A*B) * (A*B).
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```    25                     EX x' y' x y. z = <<x',y'>, <x,y>> &
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```    26                        (<x',x>: r | (x'=x & <y',y>: s))}"
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```    27
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```    28   (*inverse image of a relation*)
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```    29   rvimage_def "rvimage(A,f,r) == {z: A*A. EX x y. z = <x,y> & <f`x,f`y>: r}"
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```    30
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```    31 constdefs
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```    32    measure :: "[i, i\\<Rightarrow>i] \\<Rightarrow> i"
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```    33    "measure(A,f) == {<x,y>: A*A. f(x) < f(y)}"
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```    34
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```    35
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```    36 end
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