author | wenzelm |
Mon, 15 Oct 2001 20:33:42 +0200 | |
changeset 11777 | b03c8a3fcc6d |
parent 11602 | bf6700f4c010 |
child 11820 | 015a82d4ee96 |
permissions | -rw-r--r-- |
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(* Title: HOL/Product_Type.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1992 University of Cambridge |
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*) |
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header {* Finite products (including unit) *} |
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theory Product_Type = Fun |
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files ("Product_Type_lemmas.ML") ("Tools/split_rule.ML"): |
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subsection {* Products *} |
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subsubsection {* Type definition *} |
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constdefs |
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Pair_Rep :: "['a, 'b] => ['a, 'b] => bool" |
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"Pair_Rep == (%a b. %x y. x=a & y=b)" |
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global |
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typedef (Prod) |
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('a, 'b) "*" (infixr 20) |
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= "{f. EX a b. f = Pair_Rep (a::'a) (b::'b)}" |
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proof |
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fix a b show "Pair_Rep a b : ?Prod" |
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by blast |
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qed |
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syntax (symbols) |
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"*" :: "[type, type] => type" ("(_ \<times>/ _)" [21, 20] 20) |
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syntax (HTML output) |
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"*" :: "[type, type] => type" ("(_ \<times>/ _)" [21, 20] 20) |
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local |
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subsubsection {* Abstract constants and syntax *} |
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global |
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consts |
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fst :: "'a * 'b => 'a" |
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snd :: "'a * 'b => 'b" |
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split :: "[['a, 'b] => 'c, 'a * 'b] => 'c" |
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prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd" |
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Pair :: "['a, 'b] => 'a * 'b" |
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Sigma :: "['a set, 'a => 'b set] => ('a * 'b) set" |
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local |
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text {* |
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Patterns -- extends pre-defined type @{typ pttrn} used in |
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abstractions. |
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*} |
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nonterminals |
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tuple_args patterns |
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syntax |
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"_tuple" :: "'a => tuple_args => 'a * 'b" ("(1'(_,/ _'))") |
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"_tuple_arg" :: "'a => tuple_args" ("_") |
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"_tuple_args" :: "'a => tuple_args => tuple_args" ("_,/ _") |
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"_pattern" :: "[pttrn, patterns] => pttrn" ("'(_,/ _')") |
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"" :: "pttrn => patterns" ("_") |
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"_patterns" :: "[pttrn, patterns] => patterns" ("_,/ _") |
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"@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" 10) |
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"@Times" ::"['a set, 'a => 'b set] => ('a * 'b) set" (infixr "<*>" 80) |
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translations |
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"(x, y)" == "Pair x y" |
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"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" |
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"%(x,y,zs).b" == "split(%x (y,zs).b)" |
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"%(x,y).b" == "split(%x y. b)" |
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"_abs (Pair x y) t" => "%(x,y).t" |
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(* The last rule accommodates tuples in `case C ... (x,y) ... => ...' |
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The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *) |
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"SIGMA x:A. B" => "Sigma A (%x. B)" |
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"A <*> B" => "Sigma A (_K B)" |
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syntax (symbols) |
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"@Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3\<Sigma> _\<in>_./ _)" 10) |
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"@Times" :: "['a set, 'a => 'b set] => ('a * 'b) set" ("_ \<times> _" [81, 80] 80) |
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print_translation {* [("Sigma", dependent_tr' ("@Sigma", "@Times"))] *} |
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subsubsection {* Definitions *} |
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defs |
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Pair_def: "Pair a b == Abs_Prod(Pair_Rep a b)" |
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fst_def: "fst p == THE a. EX b. p = (a, b)" |
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snd_def: "snd p == THE b. EX a. p = (a, b)" |
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split_def: "split == (%c p. c (fst p) (snd p))" |
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prod_fun_def: "prod_fun f g == split(%x y.(f(x), g(y)))" |
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Sigma_def: "Sigma A B == UN x:A. UN y:B(x). {(x, y)}" |
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subsection {* Unit *} |
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typedef unit = "{True}" |
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proof |
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show "True : ?unit" |
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by blast |
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qed |
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constdefs |
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Unity :: unit ("'(')") |
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"() == Abs_unit True" |
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subsection {* Lemmas and tool setup *} |
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use "Product_Type_lemmas.ML" |
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lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A" |
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apply (rule_tac x = "(a, b)" in image_eqI) |
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apply auto |
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done |
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constdefs |
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internal_split :: "('a => 'b => 'c) => 'a * 'b => 'c" |
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"internal_split == split" |
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lemma internal_split_conv: "internal_split c (a, b) = c a b" |
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by (simp only: internal_split_def split_conv) |
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hide const internal_split |
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use "Tools/split_rule.ML" |
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setup SplitRule.setup |
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end |