nipkow@10213: (*  Title:      HOL/Product_Type.thy
nipkow@10213:     ID:         $Id$
nipkow@10213:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
nipkow@10213:     Copyright   1992  University of Cambridge
nipkow@10213: 
nipkow@10213: Ordered Pairs and the Cartesian product type.
nipkow@10213: The unit type.
nipkow@10213: *)
nipkow@10213: 
oheimb@11025: theory Product_Type = Fun
wenzelm@11032: files ("Product_Type_lemmas.ML") ("Tools/split_rule.ML"):
nipkow@10213: 
nipkow@10213: 
nipkow@10213: (** products **)
nipkow@10213: 
nipkow@10213: (* type definition *)
nipkow@10213: 
nipkow@10213: constdefs
oheimb@11025:   Pair_Rep :: "['a, 'b] => ['a, 'b] => bool"
wenzelm@11032:   "Pair_Rep == (%a b. %x y. x=a & y=b)"
nipkow@10213: 
nipkow@10213: global
nipkow@10213: 
nipkow@10213: typedef (Prod)
nipkow@10213:   ('a, 'b) "*"          (infixr 20)
wenzelm@11032:     = "{f. EX a b. f = Pair_Rep (a::'a) (b::'b)}"
oheimb@11025: proof
oheimb@11025:   fix a b show "Pair_Rep a b : ?Prod"
oheimb@11025:     by blast
oheimb@11025: qed
nipkow@10213: 
nipkow@10213: syntax (symbols)
paulson@11451:   "*"      :: "[type, type] => type"         ("(_ \\<times>/ _)" [21, 20] 20)
nipkow@10213: syntax (HTML output)
paulson@11451:   "*"      :: "[type, type] => type"         ("(_ \\<times>/ _)" [21, 20] 20)
nipkow@10213: 
nipkow@10213: 
nipkow@10213: (* abstract constants and syntax *)
nipkow@10213: 
nipkow@10213: consts
oheimb@11025:   fst      :: "'a * 'b => 'a"
oheimb@11025:   snd      :: "'a * 'b => 'b"
oheimb@11025:   split    :: "[['a, 'b] => 'c, 'a * 'b] => 'c"
oheimb@11025:   prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd"
oheimb@11025:   Pair     :: "['a, 'b] => 'a * 'b"
oheimb@11025:   Sigma    :: "['a set, 'a => 'b set] => ('a * 'b) set"
nipkow@10213: 
nipkow@10213: 
nipkow@10213: (* patterns -- extends pre-defined type "pttrn" used in abstractions *)
nipkow@10213: 
nipkow@10213: nonterminals
nipkow@10213:   tuple_args patterns
nipkow@10213: 
nipkow@10213: syntax
nipkow@10213:   "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
nipkow@10213:   "_tuple_arg"  :: "'a => tuple_args"                   ("_")
nipkow@10213:   "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
oheimb@11025:   "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
oheimb@11025:   ""            :: "pttrn => patterns"                  ("_")
oheimb@11025:   "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
oheimb@11025:   "@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" 10)
oheimb@11025:   "@Times" ::"['a set,  'a => 'b set] => ('a * 'b) set" (infixr "<*>" 80)
nipkow@10213: 
nipkow@10213: translations
nipkow@10213:   "(x, y)"       == "Pair x y"
nipkow@10213:   "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
nipkow@10213:   "%(x,y,zs).b"  == "split(%x (y,zs).b)"
nipkow@10213:   "%(x,y).b"     == "split(%x y. b)"
nipkow@10213:   "_abs (Pair x y) t" => "%(x,y).t"
nipkow@10213:   (* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
nipkow@10213:      The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *)
nipkow@10213: 
nipkow@10213:   "SIGMA x:A. B" => "Sigma A (%x. B)"
nipkow@10213:   "A <*> B"      => "Sigma A (_K B)"
nipkow@10213: 
nipkow@10213: syntax (symbols)
paulson@11451:   "@Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3\\<Sigma> _\\<in>_./ _)"   10)
paulson@11451:   "@Times" :: "['a set,  'a => 'b set] => ('a * 'b) set"  ("_ \\<times> _" [81, 80] 80)
oheimb@11025: 
wenzelm@11032: print_translation {* [("Sigma", dependent_tr' ("@Sigma", "@Times"))] *}
nipkow@10213: 
nipkow@10213: 
nipkow@10213: (* definitions *)
nipkow@10213: 
nipkow@10213: local
nipkow@10213: 
nipkow@10213: defs
oheimb@11025:   Pair_def:     "Pair a b == Abs_Prod(Pair_Rep a b)"
paulson@11451:   fst_def:      "fst p == THE a. EX b. p = (a, b)"
paulson@11451:   snd_def:      "snd p == THE b. EX a. p = (a, b)"
oheimb@11025:   split_def:    "split == (%c p. c (fst p) (snd p))"
oheimb@11025:   prod_fun_def: "prod_fun f g == split(%x y.(f(x), g(y)))"
oheimb@11025:   Sigma_def:    "Sigma A B == UN x:A. UN y:B(x). {(x, y)}"
nipkow@10213: 
nipkow@10213: 
nipkow@10213: 
nipkow@10213: (** unit **)
nipkow@10213: 
nipkow@10213: global
nipkow@10213: 
wenzelm@11032: typedef unit = "{True}"
oheimb@11025: proof
oheimb@11025:   show "True : ?unit"
oheimb@11025:     by blast
oheimb@11025: qed
nipkow@10213: 
nipkow@10213: consts
nipkow@10213:   "()"          :: unit                           ("'(')")
nipkow@10213: 
nipkow@10213: local
nipkow@10213: 
nipkow@10213: defs
oheimb@11025:   Unity_def:    "() == Abs_unit True"
oheimb@11025: 
wenzelm@11032: 
wenzelm@11032: 
wenzelm@11032: (** lemmas and tool setup **)
wenzelm@11032: 
oheimb@11025: use "Product_Type_lemmas.ML"
oheimb@11025: 
wenzelm@11032: constdefs
wenzelm@11425:   internal_split :: "('a => 'b => 'c) => 'a * 'b => 'c"
wenzelm@11032:   "internal_split == split"
wenzelm@11032: 
wenzelm@11032: lemma internal_split_conv: "internal_split c (a, b) = c a b"
wenzelm@11032:   by (simp only: internal_split_def split_conv)
wenzelm@11032: 
wenzelm@11032: hide const internal_split
wenzelm@11032: 
oheimb@11025: use "Tools/split_rule.ML"
wenzelm@11032: setup SplitRule.setup
nipkow@10213: 
nipkow@10213: end