src/HOL/Finite.thy
 author paulson Thu Sep 23 13:06:31 1999 +0200 (1999-09-23) changeset 7584 5be4bb8e4e3f parent 6015 d1d5dd2f121c child 7958 f531589c9fc1 permissions -rw-r--r--
tidied; added lemma restrict_to_left
```     1 (*  Title:      HOL/Finite.thy
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```     2     ID:         \$Id\$
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```     3     Author:     Lawrence C Paulson & Tobias Nipkow
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```     4     Copyright   1995  University of Cambridge & TU Muenchen
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```     5
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```     6 Finite sets, their cardinality, and a fold functional.
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```     7 *)
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```     8
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```     9 Finite = Divides + Power + Inductive +
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```    10
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```    11 consts Finites :: 'a set set
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```    12
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```    13 inductive "Finites"
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```    14   intrs
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```    15     emptyI  "{} : Finites"
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```    16     insertI "A : Finites ==> insert a A : Finites"
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```    17
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```    18 syntax finite :: 'a set => bool
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```    19 translations  "finite A"  ==  "A : Finites"
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```    20
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```    21 (* This definition, although traditional, is ugly to work with
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```    22 constdefs
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```    23   card :: 'a set => nat
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```    24   "card A == LEAST n. ? f. A = {f i |i. i<n}"
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```    25 Therefore we have switched to an inductive one:
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```    26 *)
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```    27
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```    28 consts cardR :: "('a set * nat) set"
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```    29
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```    30 inductive cardR
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```    31 intrs
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```    32   EmptyI  "({},0) : cardR"
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```    33   InsertI "[| (A,n) : cardR; a ~: A |] ==> (insert a A, Suc n) : cardR"
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```    34
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```    35 constdefs
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```    36   card :: 'a set => nat
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```    37  "card A == @n. (A,n) : cardR"
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```    38
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```    39 (*
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```    40 A "fold" functional for finite sets.  For n non-negative we have
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```    41     fold f e {x1,...,xn} = f x1 (... (f xn e))
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```    42 where f is at least left-commutative.
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```    43 *)
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```    44
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```    45 consts foldSet :: "[['b,'a] => 'a, 'a] => ('b set * 'a) set"
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```    46
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```    47 inductive "foldSet f e"
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```    48   intrs
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```    49     emptyI   "({}, e) : foldSet f e"
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```    50
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```    51     insertI  "[| x ~: A;  (A,y) : foldSet f e |]
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```    52 	      ==> (insert x A, f x y) : foldSet f e"
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```    53
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```    54 constdefs
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```    55    fold :: "[['b,'a] => 'a, 'a, 'b set] => 'a"
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```    56   "fold f e A == @x. (A,x) : foldSet f e"
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```    57   (* A frequent instance: *)
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```    58    setsum :: ('a => nat) => 'a set => nat
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```    59   "setsum f == fold (op+ o f) 0"
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```    60
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```    61 locale LC =
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```    62   fixes
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```    63     f    :: ['b,'a] => 'a
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```    64   assumes
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```    65     lcomm    "f x (f y z) = f y (f x z)"
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```    66
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```    67 locale ACe =
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```    68   fixes
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```    69     f    :: ['a,'a] => 'a
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```    70     e    :: 'a
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```    71   assumes
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```    72     ident    "f x e = x"
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```    73     commute  "f x y = f y x"
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```    74     assoc    "f (f x y) z = f x (f y z)"
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```    75
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```    76 end
```