src/HOL/Finite.thy
 author oheimb Wed Jan 31 10:15:55 2001 +0100 (2001-01-31) changeset 11008 f7333f055ef6 parent 9087 12db178a78df child 11092 69c1abb9a129 permissions -rw-r--r--
improved theory reference in comment
```     1 (*  Title:      HOL/Finite.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Lawrence C Paulson & Tobias Nipkow
```
```     4     Copyright   1995  University of Cambridge & TU Muenchen
```
```     5
```
```     6 Finite sets, their cardinality, and a fold functional.
```
```     7 *)
```
```     8
```
```     9 Finite = Divides + Power + Inductive + SetInterval +
```
```    10
```
```    11 consts Finites :: 'a set set
```
```    12
```
```    13 inductive "Finites"
```
```    14   intrs
```
```    15     emptyI  "{} : Finites"
```
```    16     insertI "A : Finites ==> insert a A : Finites"
```
```    17
```
```    18 syntax finite :: 'a set => bool
```
```    19 translations  "finite A"  ==  "A : Finites"
```
```    20
```
```    21 axclass	finite<term
```
```    22   finite "finite UNIV"
```
```    23
```
```    24 (* This definition, although traditional, is ugly to work with
```
```    25 constdefs
```
```    26   card :: 'a set => nat
```
```    27   "card A == LEAST n. ? f. A = {f i |i. i<n}"
```
```    28 Therefore we have switched to an inductive one:
```
```    29 *)
```
```    30
```
```    31 consts cardR :: "('a set * nat) set"
```
```    32
```
```    33 inductive cardR
```
```    34 intrs
```
```    35   EmptyI  "({},0) : cardR"
```
```    36   InsertI "[| (A,n) : cardR; a ~: A |] ==> (insert a A, Suc n) : cardR"
```
```    37
```
```    38 constdefs
```
```    39   card :: 'a set => nat
```
```    40  "card A == @n. (A,n) : cardR"
```
```    41
```
```    42 (*
```
```    43 A "fold" functional for finite sets.  For n non-negative we have
```
```    44     fold f e {x1,...,xn} = f x1 (... (f xn e))
```
```    45 where f is at least left-commutative.
```
```    46 *)
```
```    47
```
```    48 consts foldSet :: "[['b,'a] => 'a, 'a] => ('b set * 'a) set"
```
```    49
```
```    50 inductive "foldSet f e"
```
```    51   intrs
```
```    52     emptyI   "({}, e) : foldSet f e"
```
```    53
```
```    54     insertI  "[| x ~: A;  (A,y) : foldSet f e |]
```
```    55 	      ==> (insert x A, f x y) : foldSet f e"
```
```    56
```
```    57 constdefs
```
```    58    fold :: "[['b,'a] => 'a, 'a, 'b set] => 'a"
```
```    59   "fold f e A == @x. (A,x) : foldSet f e"
```
```    60
```
```    61    setsum :: "('a => 'b) => 'a set => ('b::plus_ac0)"
```
```    62   "setsum f A == if finite A then fold (op+ o f) 0 A else 0"
```
```    63
```
```    64 locale LC =
```
```    65   fixes
```
```    66     f    :: ['b,'a] => 'a
```
```    67   assumes
```
```    68     lcomm    "f x (f y z) = f y (f x z)"
```
```    69
```
```    70 locale ACe =
```
```    71   fixes
```
```    72     f    :: ['a,'a] => 'a
```
```    73     e    :: 'a
```
```    74   assumes
```
```    75     ident    "f x e = x"
```
```    76     commute  "f x y = f y x"
```
```    77     assoc    "f (f x y) z = f x (f y z)"
```
```    78
```
```    79 end
```