src/HOL/Product_Type.thy
author hoelzl
Fri Nov 16 18:45:57 2012 +0100 (2012-11-16)
changeset 50104 de19856feb54
parent 49897 cc69be3c8f87
child 50107 289181e3e524
permissions -rw-r--r--
move theorems to be more generally useable
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(*  Title:      HOL/Product_Type.thy
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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 {* Cartesian products *}
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theory Product_Type
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imports Typedef Inductive Fun
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keywords "inductive_set" "coinductive_set" :: thy_decl
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begin
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subsection {* @{typ bool} is a datatype *}
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rep_datatype True False by (auto intro: bool_induct)
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declare case_split [cases type: bool]
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  -- "prefer plain propositional version"
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lemma
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  shows [code]: "HOL.equal False P \<longleftrightarrow> \<not> P"
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    and [code]: "HOL.equal True P \<longleftrightarrow> P" 
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    and [code]: "HOL.equal P False \<longleftrightarrow> \<not> P"
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    and [code]: "HOL.equal P True \<longleftrightarrow> P"
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    and [code nbe]: "HOL.equal P P \<longleftrightarrow> True"
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  by (simp_all add: equal)
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lemma If_case_cert:
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  assumes "CASE \<equiv> (\<lambda>b. If b f g)"
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  shows "(CASE True \<equiv> f) &&& (CASE False \<equiv> g)"
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  using assms by simp_all
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setup {*
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  Code.add_case @{thm If_case_cert}
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*}
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code_const "HOL.equal \<Colon> bool \<Rightarrow> bool \<Rightarrow> bool"
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  (Haskell infix 4 "==")
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code_instance bool :: equal
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  (Haskell -)
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subsection {* The @{text unit} type *}
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typedef unit = "{True}"
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  by auto
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definition Unity :: unit  ("'(')")
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  where "() = Abs_unit True"
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lemma unit_eq [no_atp]: "u = ()"
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  by (induct u) (simp add: Unity_def)
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text {*
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  Simplification procedure for @{thm [source] unit_eq}.  Cannot use
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  this rule directly --- it loops!
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*}
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simproc_setup unit_eq ("x::unit") = {*
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  fn _ => fn _ => fn ct =>
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    if HOLogic.is_unit (term_of ct) then NONE
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    else SOME (mk_meta_eq @{thm unit_eq})
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*}
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rep_datatype "()" by simp
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lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
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  by simp
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lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
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  by (rule triv_forall_equality)
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text {*
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  This rewrite counters the effect of simproc @{text unit_eq} on @{term
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  [source] "%u::unit. f u"}, replacing it by @{term [source]
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  f} rather than by @{term [source] "%u. f ()"}.
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*}
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lemma unit_abs_eta_conv [simp, no_atp]: "(%u::unit. f ()) = f"
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  by (rule ext) simp
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lemma UNIV_unit [no_atp]:
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  "UNIV = {()}" by auto
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instantiation unit :: default
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begin
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definition "default = ()"
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instance ..
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end
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lemma [code]:
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  "HOL.equal (u\<Colon>unit) v \<longleftrightarrow> True" unfolding equal unit_eq [of u] unit_eq [of v] by rule+
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code_type unit
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  (SML "unit")
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  (OCaml "unit")
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  (Haskell "()")
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  (Scala "Unit")
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code_const Unity
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  (SML "()")
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  (OCaml "()")
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  (Haskell "()")
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  (Scala "()")
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code_instance unit :: equal
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  (Haskell -)
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code_const "HOL.equal \<Colon> unit \<Rightarrow> unit \<Rightarrow> bool"
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  (Haskell infix 4 "==")
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code_reserved SML
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  unit
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code_reserved OCaml
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  unit
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code_reserved Scala
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  Unit
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subsection {* The product type *}
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subsubsection {* Type definition *}
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definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" where
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  "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
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definition "prod = {f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}"
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typedef ('a, 'b) prod (infixr "*" 20) = "prod :: ('a \<Rightarrow> 'b \<Rightarrow> bool) set"
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  unfolding prod_def by auto
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type_notation (xsymbols)
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  "prod"  ("(_ \<times>/ _)" [21, 20] 20)
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type_notation (HTML output)
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  "prod"  ("(_ \<times>/ _)" [21, 20] 20)
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definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" where
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  "Pair a b = Abs_prod (Pair_Rep a b)"
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rep_datatype Pair proof -
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  fix P :: "'a \<times> 'b \<Rightarrow> bool" and p
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  assume "\<And>a b. P (Pair a b)"
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  then show "P p" by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def)
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next
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  fix a c :: 'a and b d :: 'b
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  have "Pair_Rep a b = Pair_Rep c d \<longleftrightarrow> a = c \<and> b = d"
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    by (auto simp add: Pair_Rep_def fun_eq_iff)
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  moreover have "Pair_Rep a b \<in> prod" and "Pair_Rep c d \<in> prod"
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    by (auto simp add: prod_def)
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  ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d"
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    by (simp add: Pair_def Abs_prod_inject)
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qed
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declare prod.simps(2) [nitpick_simp del]
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declare prod.weak_case_cong [cong del]
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subsubsection {* Tuple syntax *}
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abbreviation (input) split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
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  "split \<equiv> prod_case"
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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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nonterminal tuple_args and 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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translations
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  "(x, y)" == "CONST 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" == "CONST prod_case (%x (y, zs). b)"
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  "%(x, y). b" == "CONST prod_case (%x y. b)"
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  "_abs (CONST 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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(*reconstruct pattern from (nested) splits, avoiding eta-contraction of body;
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  works best with enclosing "let", if "let" does not avoid eta-contraction*)
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print_translation {*
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let
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  fun split_tr' [Abs (x, T, t as (Abs abs))] =
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        (* split (%x y. t) => %(x,y) t *)
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        let
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          val (y, t') = Syntax_Trans.atomic_abs_tr' abs;
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          val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t');
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        in
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          Syntax.const @{syntax_const "_abs"} $
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            (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
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        end
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    | split_tr' [Abs (x, T, (s as Const (@{const_syntax prod_case}, _) $ t))] =
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        (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
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        let
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          val Const (@{syntax_const "_abs"}, _) $
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            (Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t];
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          val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t');
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        in
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          Syntax.const @{syntax_const "_abs"} $
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            (Syntax.const @{syntax_const "_pattern"} $ x' $
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              (Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t''
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        end
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    | split_tr' [Const (@{const_syntax prod_case}, _) $ t] =
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        (* split (split (%x y z. t)) => %((x, y), z). t *)
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        split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
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    | split_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] =
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        (* split (%pttrn z. t) => %(pttrn,z). t *)
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        let val (z, t) = Syntax_Trans.atomic_abs_tr' abs in
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          Syntax.const @{syntax_const "_abs"} $
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            (Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t
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        end
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    | split_tr' _ = raise Match;
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in [(@{const_syntax prod_case}, split_tr')] end
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*}
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(* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
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typed_print_translation {*
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let
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  fun split_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match
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    | split_guess_names_tr' T [Abs (x, xT, t)] =
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        (case (head_of t) of
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          Const (@{const_syntax prod_case}, _) => raise Match
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        | _ =>
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          let 
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            val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
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            val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0);
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            val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t');
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          in
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            Syntax.const @{syntax_const "_abs"} $
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              (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
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          end)
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    | split_guess_names_tr' T [t] =
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        (case head_of t of
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          Const (@{const_syntax prod_case}, _) => raise Match
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        | _ =>
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          let
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            val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
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            val (y, t') =
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              Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0);
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            val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t');
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          in
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            Syntax.const @{syntax_const "_abs"} $
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              (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
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          end)
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    | split_guess_names_tr' _ _ = raise Match;
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in [(@{const_syntax prod_case}, split_guess_names_tr')] end
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*}
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(* Force eta-contraction for terms of the form "Q A (%p. prod_case P p)"
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   where Q is some bounded quantifier or set operator.
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   Reason: the above prints as "Q p : A. case p of (x,y) \<Rightarrow> P x y"
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   whereas we want "Q (x,y):A. P x y".
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   Otherwise prevent eta-contraction.
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*)
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print_translation {*
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let
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  fun contract Q f ts =
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    case ts of
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      [A, Abs(_, _, (s as Const (@{const_syntax prod_case},_) $ t) $ Bound 0)]
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      => if Term.is_dependent t then f ts else Syntax.const Q $ A $ s
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    | _ => f ts;
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  fun contract2 (Q,f) = (Q, contract Q f);
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  val pairs =
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    [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"},
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     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Bex} @{syntax_const "_Bex"},
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     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFI} @{syntax_const "_INF"},
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     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPR} @{syntax_const "_SUP"}]
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in map contract2 pairs end
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*}
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subsubsection {* Code generator setup *}
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code_type prod
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  (SML infix 2 "*")
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  (OCaml infix 2 "*")
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  (Haskell "!((_),/ (_))")
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  (Scala "((_),/ (_))")
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code_const Pair
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  (SML "!((_),/ (_))")
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  (OCaml "!((_),/ (_))")
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  (Haskell "!((_),/ (_))")
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  (Scala "!((_),/ (_))")
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code_instance prod :: equal
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  (Haskell -)
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code_const "HOL.equal \<Colon> 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"
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  (Haskell infix 4 "==")
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subsubsection {* Fundamental operations and properties *}
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lemma Pair_inject:
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  assumes "(a, b) = (a', b')"
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    and "a = a' ==> b = b' ==> R"
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  shows R
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  using assms by simp
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lemma surj_pair [simp]: "EX x y. p = (x, y)"
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  by (cases p) simp
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definition fst :: "'a \<times> 'b \<Rightarrow> 'a" where
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  "fst p = (case p of (a, b) \<Rightarrow> a)"
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definition snd :: "'a \<times> 'b \<Rightarrow> 'b" where
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  "snd p = (case p of (a, b) \<Rightarrow> b)"
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lemma fst_conv [simp, code]: "fst (a, b) = a"
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  unfolding fst_def by simp
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lemma snd_conv [simp, code]: "snd (a, b) = b"
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  unfolding snd_def by simp
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code_const fst and snd
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  (Haskell "fst" and "snd")
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lemma prod_case_unfold [nitpick_unfold]: "prod_case = (%c p. c (fst p) (snd p))"
nipkow@39302
   334
  by (simp add: fun_eq_iff split: prod.split)
haftmann@26358
   335
wenzelm@11838
   336
lemma fst_eqD: "fst (x, y) = a ==> x = a"
wenzelm@11838
   337
  by simp
wenzelm@11838
   338
wenzelm@11838
   339
lemma snd_eqD: "snd (x, y) = a ==> y = a"
wenzelm@11838
   340
  by simp
wenzelm@11838
   341
haftmann@26358
   342
lemma pair_collapse [simp]: "(fst p, snd p) = p"
wenzelm@11838
   343
  by (cases p) simp
wenzelm@11838
   344
haftmann@26358
   345
lemmas surjective_pairing = pair_collapse [symmetric]
wenzelm@11838
   346
huffman@44066
   347
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
haftmann@37166
   348
  by (cases s, cases t) simp
haftmann@37166
   349
haftmann@37166
   350
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
huffman@44066
   351
  by (simp add: prod_eq_iff)
haftmann@37166
   352
haftmann@37166
   353
lemma split_conv [simp, code]: "split f (a, b) = f a b"
haftmann@37591
   354
  by (fact prod.cases)
haftmann@37166
   355
haftmann@37166
   356
lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
haftmann@37166
   357
  by (rule split_conv [THEN iffD2])
haftmann@37166
   358
haftmann@37166
   359
lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
haftmann@37166
   360
  by (rule split_conv [THEN iffD1])
haftmann@37166
   361
haftmann@37166
   362
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
nipkow@39302
   363
  by (simp add: fun_eq_iff split: prod.split)
haftmann@37166
   364
haftmann@37166
   365
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
haftmann@37166
   366
  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
nipkow@39302
   367
  by (simp add: fun_eq_iff split: prod.split)
haftmann@37166
   368
haftmann@37166
   369
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
haftmann@37166
   370
  by (cases x) simp
haftmann@37166
   371
haftmann@37166
   372
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
haftmann@37166
   373
  by (cases p) simp
haftmann@37166
   374
haftmann@37166
   375
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
haftmann@37591
   376
  by (simp add: prod_case_unfold)
haftmann@37166
   377
haftmann@37166
   378
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
haftmann@37166
   379
  -- {* Prevents simplification of @{term c}: much faster *}
huffman@40929
   380
  by (fact prod.weak_case_cong)
haftmann@37166
   381
haftmann@37166
   382
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
haftmann@37166
   383
  by (simp add: split_eta)
haftmann@37166
   384
blanchet@47740
   385
lemma split_paired_all [no_atp]: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
wenzelm@11820
   386
proof
wenzelm@11820
   387
  fix a b
wenzelm@11820
   388
  assume "!!x. PROP P x"
wenzelm@19535
   389
  then show "PROP P (a, b)" .
wenzelm@11820
   390
next
wenzelm@11820
   391
  fix x
wenzelm@11820
   392
  assume "!!a b. PROP P (a, b)"
wenzelm@19535
   393
  from `PROP P (fst x, snd x)` show "PROP P x" by simp
wenzelm@11820
   394
qed
wenzelm@11820
   395
hoelzl@50104
   396
lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))"
hoelzl@50104
   397
  by (cases x) simp
hoelzl@50104
   398
wenzelm@11838
   399
text {*
wenzelm@11838
   400
  The rule @{thm [source] split_paired_all} does not work with the
wenzelm@11838
   401
  Simplifier because it also affects premises in congrence rules,
wenzelm@11838
   402
  where this can lead to premises of the form @{text "!!a b. ... =
wenzelm@11838
   403
  ?P(a, b)"} which cannot be solved by reflexivity.
wenzelm@11838
   404
*}
wenzelm@11838
   405
haftmann@26358
   406
lemmas split_tupled_all = split_paired_all unit_all_eq2
haftmann@26358
   407
wenzelm@26480
   408
ML {*
wenzelm@11838
   409
  (* replace parameters of product type by individual component parameters *)
wenzelm@11838
   410
  val safe_full_simp_tac = generic_simp_tac true (true, false, false);
wenzelm@11838
   411
  local (* filtering with exists_paired_all is an essential optimization *)
wenzelm@16121
   412
    fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
wenzelm@11838
   413
          can HOLogic.dest_prodT T orelse exists_paired_all t
wenzelm@11838
   414
      | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
wenzelm@11838
   415
      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
wenzelm@11838
   416
      | exists_paired_all _ = false;
wenzelm@11838
   417
    val ss = HOL_basic_ss
wenzelm@26340
   418
      addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
wenzelm@43594
   419
      addsimprocs [@{simproc unit_eq}];
wenzelm@11838
   420
  in
wenzelm@11838
   421
    val split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   422
      if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838
   423
    val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   424
      if exists_paired_all t then full_simp_tac ss i else no_tac);
wenzelm@11838
   425
    fun split_all th =
wenzelm@26340
   426
   if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th;
wenzelm@11838
   427
  end;
wenzelm@26340
   428
*}
wenzelm@11838
   429
wenzelm@26340
   430
declaration {* fn _ =>
wenzelm@26340
   431
  Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac))
wenzelm@16121
   432
*}
wenzelm@11838
   433
blanchet@47740
   434
lemma split_paired_All [simp, no_atp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@11838
   435
  -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
wenzelm@11838
   436
  by fast
wenzelm@11838
   437
blanchet@47740
   438
lemma split_paired_Ex [simp, no_atp]: "(EX x. P x) = (EX a b. P (a, b))"
haftmann@26358
   439
  by fast
haftmann@26358
   440
blanchet@47740
   441
lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@11838
   442
  -- {* Can't be added to simpset: loops! *}
haftmann@26358
   443
  by (simp add: split_eta)
wenzelm@11838
   444
wenzelm@11838
   445
text {*
wenzelm@11838
   446
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
wenzelm@11838
   447
  @{thm [source] split_eta} as a rewrite rule is not general enough,
wenzelm@11838
   448
  and using @{thm [source] cond_split_eta} directly would render some
wenzelm@11838
   449
  existing proofs very inefficient; similarly for @{text
haftmann@26358
   450
  split_beta}.
haftmann@26358
   451
*}
wenzelm@11838
   452
wenzelm@26480
   453
ML {*
wenzelm@11838
   454
local
wenzelm@35364
   455
  val cond_split_eta_ss = HOL_basic_ss addsimps @{thms cond_split_eta};
wenzelm@35364
   456
  fun Pair_pat k 0 (Bound m) = (m = k)
wenzelm@35364
   457
    | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =
wenzelm@35364
   458
        i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t
wenzelm@35364
   459
    | Pair_pat _ _ _ = false;
wenzelm@35364
   460
  fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t
wenzelm@35364
   461
    | no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@35364
   462
    | no_args k i (Bound m) = m < k orelse m > k + i
wenzelm@35364
   463
    | no_args _ _ _ = true;
wenzelm@35364
   464
  fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE
haftmann@37591
   465
    | split_pat tp i (Const (@{const_name prod_case}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
wenzelm@35364
   466
    | split_pat tp i _ = NONE;
wenzelm@20044
   467
  fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
wenzelm@35364
   468
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
wenzelm@18328
   469
        (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1)));
wenzelm@11838
   470
wenzelm@35364
   471
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t
wenzelm@35364
   472
    | beta_term_pat k i (t $ u) =
wenzelm@35364
   473
        Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@35364
   474
    | beta_term_pat k i t = no_args k i t;
wenzelm@35364
   475
  fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@35364
   476
    | eta_term_pat _ _ _ = false;
wenzelm@11838
   477
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@35364
   478
    | subst arg k i (t $ u) =
wenzelm@35364
   479
        if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@35364
   480
        else (subst arg k i t $ subst arg k i u)
wenzelm@35364
   481
    | subst arg k i t = t;
wenzelm@43595
   482
in
haftmann@37591
   483
  fun beta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t) $ arg) =
wenzelm@11838
   484
        (case split_pat beta_term_pat 1 t of
wenzelm@35364
   485
          SOME (i, f) => SOME (metaeq ss s (subst arg 0 i f))
skalberg@15531
   486
        | NONE => NONE)
wenzelm@35364
   487
    | beta_proc _ _ = NONE;
haftmann@37591
   488
  fun eta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t)) =
wenzelm@11838
   489
        (case split_pat eta_term_pat 1 t of
wenzelm@35364
   490
          SOME (_, ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))
skalberg@15531
   491
        | NONE => NONE)
wenzelm@35364
   492
    | eta_proc _ _ = NONE;
wenzelm@11838
   493
end;
wenzelm@11838
   494
*}
wenzelm@43595
   495
simproc_setup split_beta ("split f z") = {* fn _ => fn ss => fn ct => beta_proc ss (term_of ct) *}
wenzelm@43595
   496
simproc_setup split_eta ("split f") = {* fn _ => fn ss => fn ct => eta_proc ss (term_of ct) *}
wenzelm@11838
   497
berghofe@26798
   498
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)"
wenzelm@11838
   499
  by (subst surjective_pairing, rule split_conv)
wenzelm@11838
   500
hoelzl@50104
   501
lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
hoelzl@50104
   502
  by (auto simp: fun_eq_iff)
hoelzl@50104
   503
hoelzl@50104
   504
blanchet@35828
   505
lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
wenzelm@11838
   506
  -- {* For use with @{text split} and the Simplifier. *}
paulson@15481
   507
  by (insert surj_pair [of p], clarify, simp)
wenzelm@11838
   508
wenzelm@11838
   509
text {*
wenzelm@11838
   510
  @{thm [source] split_split} could be declared as @{text "[split]"}
wenzelm@11838
   511
  done after the Splitter has been speeded up significantly;
wenzelm@11838
   512
  precompute the constants involved and don't do anything unless the
wenzelm@11838
   513
  current goal contains one of those constants.
wenzelm@11838
   514
*}
wenzelm@11838
   515
blanchet@35828
   516
lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
paulson@14208
   517
by (subst split_split, simp)
wenzelm@11838
   518
wenzelm@11838
   519
text {*
wenzelm@11838
   520
  \medskip @{term split} used as a logical connective or set former.
wenzelm@11838
   521
wenzelm@11838
   522
  \medskip These rules are for use with @{text blast}; could instead
huffman@40929
   523
  call @{text simp} using @{thm [source] prod.split} as rewrite. *}
wenzelm@11838
   524
wenzelm@11838
   525
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
wenzelm@11838
   526
  apply (simp only: split_tupled_all)
wenzelm@11838
   527
  apply (simp (no_asm_simp))
wenzelm@11838
   528
  done
wenzelm@11838
   529
wenzelm@11838
   530
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
wenzelm@11838
   531
  apply (simp only: split_tupled_all)
wenzelm@11838
   532
  apply (simp (no_asm_simp))
wenzelm@11838
   533
  done
wenzelm@11838
   534
wenzelm@11838
   535
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@37591
   536
  by (induct p) auto
wenzelm@11838
   537
wenzelm@11838
   538
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@37591
   539
  by (induct p) auto
wenzelm@11838
   540
wenzelm@11838
   541
lemma splitE2:
wenzelm@11838
   542
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
wenzelm@11838
   543
proof -
wenzelm@11838
   544
  assume q: "Q (split P z)"
wenzelm@11838
   545
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
wenzelm@11838
   546
  show R
wenzelm@11838
   547
    apply (rule r surjective_pairing)+
wenzelm@11838
   548
    apply (rule split_beta [THEN subst], rule q)
wenzelm@11838
   549
    done
wenzelm@11838
   550
qed
wenzelm@11838
   551
wenzelm@11838
   552
lemma splitD': "split R (a,b) c ==> R a b c"
wenzelm@11838
   553
  by simp
wenzelm@11838
   554
wenzelm@11838
   555
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
wenzelm@11838
   556
  by simp
wenzelm@11838
   557
wenzelm@11838
   558
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
paulson@14208
   559
by (simp only: split_tupled_all, simp)
wenzelm@11838
   560
wenzelm@18372
   561
lemma mem_splitE:
haftmann@37166
   562
  assumes major: "z \<in> split c p"
haftmann@37166
   563
    and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q"
wenzelm@18372
   564
  shows Q
haftmann@37591
   565
  by (rule major [unfolded prod_case_unfold] cases surjective_pairing)+
wenzelm@11838
   566
wenzelm@11838
   567
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
wenzelm@11838
   568
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
wenzelm@11838
   569
wenzelm@26340
   570
ML {*
wenzelm@11838
   571
local (* filtering with exists_p_split is an essential optimization *)
haftmann@37591
   572
  fun exists_p_split (Const (@{const_name prod_case},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true
wenzelm@11838
   573
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838
   574
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838
   575
    | exists_p_split _ = false;
wenzelm@35364
   576
  val ss = HOL_basic_ss addsimps @{thms split_conv};
wenzelm@11838
   577
in
wenzelm@11838
   578
val split_conv_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   579
    if exists_p_split t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838
   580
end;
wenzelm@26340
   581
*}
wenzelm@26340
   582
wenzelm@11838
   583
(* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838
   584
   to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@26340
   585
declaration {* fn _ =>
wenzelm@26340
   586
  Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac))
wenzelm@16121
   587
*}
wenzelm@11838
   588
blanchet@35828
   589
lemma split_eta_SetCompr [simp,no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
wenzelm@18372
   590
  by (rule ext) fast
wenzelm@11838
   591
blanchet@35828
   592
lemma split_eta_SetCompr2 [simp,no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
wenzelm@18372
   593
  by (rule ext) fast
wenzelm@11838
   594
wenzelm@11838
   595
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
wenzelm@11838
   596
  -- {* Allows simplifications of nested splits in case of independent predicates. *}
wenzelm@18372
   597
  by (rule ext) blast
wenzelm@11838
   598
nipkow@14337
   599
(* Do NOT make this a simp rule as it
nipkow@14337
   600
   a) only helps in special situations
nipkow@14337
   601
   b) can lead to nontermination in the presence of split_def
nipkow@14337
   602
*)
nipkow@14337
   603
lemma split_comp_eq: 
paulson@20415
   604
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
paulson@20415
   605
  shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
wenzelm@18372
   606
  by (rule ext) auto
oheimb@14101
   607
haftmann@26358
   608
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
haftmann@26358
   609
  apply (rule_tac x = "(a, b)" in image_eqI)
haftmann@26358
   610
   apply auto
haftmann@26358
   611
  done
haftmann@26358
   612
wenzelm@11838
   613
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838
   614
  by blast
wenzelm@11838
   615
wenzelm@11838
   616
(*
wenzelm@11838
   617
the following  would be slightly more general,
wenzelm@11838
   618
but cannot be used as rewrite rule:
wenzelm@11838
   619
### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838
   620
### ?y = .x
wenzelm@11838
   621
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208
   622
by (rtac some_equality 1)
paulson@14208
   623
by ( Simp_tac 1)
paulson@14208
   624
by (split_all_tac 1)
paulson@14208
   625
by (Asm_full_simp_tac 1)
wenzelm@11838
   626
qed "The_split_eq";
wenzelm@11838
   627
*)
wenzelm@11838
   628
wenzelm@11838
   629
text {*
wenzelm@11838
   630
  Setup of internal @{text split_rule}.
wenzelm@11838
   631
*}
wenzelm@11838
   632
wenzelm@45607
   633
lemmas prod_caseI = prod.cases [THEN iffD2]
haftmann@24699
   634
haftmann@24699
   635
lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
haftmann@37678
   636
  by (fact splitI2)
haftmann@24699
   637
haftmann@24699
   638
lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
haftmann@37678
   639
  by (fact splitI2')
haftmann@24699
   640
haftmann@24699
   641
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@37678
   642
  by (fact splitE)
haftmann@24699
   643
haftmann@24699
   644
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@37678
   645
  by (fact splitE')
haftmann@24699
   646
haftmann@37678
   647
declare prod_caseI [intro!]
haftmann@24699
   648
bulwahn@26143
   649
lemma prod_case_beta:
bulwahn@26143
   650
  "prod_case f p = f (fst p) (snd p)"
haftmann@37591
   651
  by (fact split_beta)
bulwahn@26143
   652
haftmann@24699
   653
lemma prod_cases3 [cases type]:
haftmann@24699
   654
  obtains (fields) a b c where "y = (a, b, c)"
haftmann@24699
   655
  by (cases y, case_tac b) blast
haftmann@24699
   656
haftmann@24699
   657
lemma prod_induct3 [case_names fields, induct type]:
haftmann@24699
   658
    "(!!a b c. P (a, b, c)) ==> P x"
haftmann@24699
   659
  by (cases x) blast
haftmann@24699
   660
haftmann@24699
   661
lemma prod_cases4 [cases type]:
haftmann@24699
   662
  obtains (fields) a b c d where "y = (a, b, c, d)"
haftmann@24699
   663
  by (cases y, case_tac c) blast
haftmann@24699
   664
haftmann@24699
   665
lemma prod_induct4 [case_names fields, induct type]:
haftmann@24699
   666
    "(!!a b c d. P (a, b, c, d)) ==> P x"
haftmann@24699
   667
  by (cases x) blast
haftmann@24699
   668
haftmann@24699
   669
lemma prod_cases5 [cases type]:
haftmann@24699
   670
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
haftmann@24699
   671
  by (cases y, case_tac d) blast
haftmann@24699
   672
haftmann@24699
   673
lemma prod_induct5 [case_names fields, induct type]:
haftmann@24699
   674
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
haftmann@24699
   675
  by (cases x) blast
haftmann@24699
   676
haftmann@24699
   677
lemma prod_cases6 [cases type]:
haftmann@24699
   678
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
haftmann@24699
   679
  by (cases y, case_tac e) blast
haftmann@24699
   680
haftmann@24699
   681
lemma prod_induct6 [case_names fields, induct type]:
haftmann@24699
   682
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
haftmann@24699
   683
  by (cases x) blast
haftmann@24699
   684
haftmann@24699
   685
lemma prod_cases7 [cases type]:
haftmann@24699
   686
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
haftmann@24699
   687
  by (cases y, case_tac f) blast
haftmann@24699
   688
haftmann@24699
   689
lemma prod_induct7 [case_names fields, induct type]:
haftmann@24699
   690
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
haftmann@24699
   691
  by (cases x) blast
haftmann@24699
   692
haftmann@37166
   693
lemma split_def:
haftmann@37166
   694
  "split = (\<lambda>c p. c (fst p) (snd p))"
haftmann@37591
   695
  by (fact prod_case_unfold)
haftmann@37166
   696
haftmann@37166
   697
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
haftmann@37166
   698
  "internal_split == split"
haftmann@37166
   699
haftmann@37166
   700
lemma internal_split_conv: "internal_split c (a, b) = c a b"
haftmann@37166
   701
  by (simp only: internal_split_def split_conv)
haftmann@37166
   702
wenzelm@48891
   703
ML_file "Tools/split_rule.ML"
haftmann@37166
   704
setup Split_Rule.setup
haftmann@37166
   705
haftmann@37166
   706
hide_const internal_split
haftmann@37166
   707
haftmann@24699
   708
haftmann@26358
   709
subsubsection {* Derived operations *}
haftmann@26358
   710
haftmann@37387
   711
definition curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where
haftmann@37387
   712
  "curry = (\<lambda>c x y. c (x, y))"
haftmann@37166
   713
haftmann@37166
   714
lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
haftmann@37166
   715
  by (simp add: curry_def)
haftmann@37166
   716
haftmann@37166
   717
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
haftmann@37166
   718
  by (simp add: curry_def)
haftmann@37166
   719
haftmann@37166
   720
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
haftmann@37166
   721
  by (simp add: curry_def)
haftmann@37166
   722
haftmann@37166
   723
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@37166
   724
  by (simp add: curry_def)
haftmann@37166
   725
haftmann@37166
   726
lemma curry_split [simp]: "curry (split f) = f"
haftmann@37166
   727
  by (simp add: curry_def split_def)
haftmann@37166
   728
haftmann@37166
   729
lemma split_curry [simp]: "split (curry f) = f"
haftmann@37166
   730
  by (simp add: curry_def split_def)
haftmann@37166
   731
haftmann@26358
   732
text {*
haftmann@26358
   733
  The composition-uncurry combinator.
haftmann@26358
   734
*}
haftmann@26358
   735
haftmann@37751
   736
notation fcomp (infixl "\<circ>>" 60)
haftmann@26358
   737
haftmann@37751
   738
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where
haftmann@37751
   739
  "f \<circ>\<rightarrow> g = (\<lambda>x. prod_case g (f x))"
haftmann@26358
   740
haftmann@37678
   741
lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))"
nipkow@39302
   742
  by (simp add: fun_eq_iff scomp_def prod_case_unfold)
haftmann@37678
   743
haftmann@37751
   744
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = prod_case g (f x)"
haftmann@37751
   745
  by (simp add: scomp_unfold prod_case_unfold)
haftmann@26358
   746
haftmann@37751
   747
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x"
huffman@44921
   748
  by (simp add: fun_eq_iff)
haftmann@26358
   749
haftmann@37751
   750
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x"
huffman@44921
   751
  by (simp add: fun_eq_iff)
haftmann@26358
   752
haftmann@37751
   753
lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
nipkow@39302
   754
  by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358
   755
haftmann@37751
   756
lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
nipkow@39302
   757
  by (simp add: fun_eq_iff scomp_unfold fcomp_def)
haftmann@26358
   758
haftmann@37751
   759
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
huffman@44921
   760
  by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358
   761
haftmann@31202
   762
code_const scomp
haftmann@31202
   763
  (Eval infixl 3 "#->")
haftmann@31202
   764
haftmann@37751
   765
no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
   766
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@26358
   767
haftmann@26358
   768
text {*
haftmann@40607
   769
  @{term map_pair} --- action of the product functor upon
krauss@36664
   770
  functions.
haftmann@26358
   771
*}
haftmann@21195
   772
haftmann@40607
   773
definition map_pair :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
haftmann@40607
   774
  "map_pair f g = (\<lambda>(x, y). (f x, g y))"
haftmann@26358
   775
haftmann@40607
   776
lemma map_pair_simp [simp, code]:
haftmann@40607
   777
  "map_pair f g (a, b) = (f a, g b)"
haftmann@40607
   778
  by (simp add: map_pair_def)
haftmann@26358
   779
haftmann@41505
   780
enriched_type map_pair: map_pair
huffman@44921
   781
  by (auto simp add: split_paired_all)
nipkow@37278
   782
haftmann@40607
   783
lemma fst_map_pair [simp]:
haftmann@40607
   784
  "fst (map_pair f g x) = f (fst x)"
haftmann@40607
   785
  by (cases x) simp_all
nipkow@37278
   786
haftmann@40607
   787
lemma snd_prod_fun [simp]:
haftmann@40607
   788
  "snd (map_pair f g x) = g (snd x)"
haftmann@40607
   789
  by (cases x) simp_all
nipkow@37278
   790
haftmann@40607
   791
lemma fst_comp_map_pair [simp]:
haftmann@40607
   792
  "fst \<circ> map_pair f g = f \<circ> fst"
haftmann@40607
   793
  by (rule ext) simp_all
nipkow@37278
   794
haftmann@40607
   795
lemma snd_comp_map_pair [simp]:
haftmann@40607
   796
  "snd \<circ> map_pair f g = g \<circ> snd"
haftmann@40607
   797
  by (rule ext) simp_all
haftmann@26358
   798
haftmann@40607
   799
lemma map_pair_compose:
haftmann@40607
   800
  "map_pair (f1 o f2) (g1 o g2) = (map_pair f1 g1 o map_pair f2 g2)"
haftmann@40607
   801
  by (rule ext) (simp add: map_pair.compositionality comp_def)
haftmann@26358
   802
haftmann@40607
   803
lemma map_pair_ident [simp]:
haftmann@40607
   804
  "map_pair (%x. x) (%y. y) = (%z. z)"
haftmann@40607
   805
  by (rule ext) (simp add: map_pair.identity)
haftmann@40607
   806
haftmann@40607
   807
lemma map_pair_imageI [intro]:
haftmann@40607
   808
  "(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_pair f g ` R"
haftmann@40607
   809
  by (rule image_eqI) simp_all
haftmann@21195
   810
haftmann@26358
   811
lemma prod_fun_imageE [elim!]:
haftmann@40607
   812
  assumes major: "c \<in> map_pair f g ` R"
haftmann@40607
   813
    and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P"
haftmann@26358
   814
  shows P
haftmann@26358
   815
  apply (rule major [THEN imageE])
haftmann@37166
   816
  apply (case_tac x)
haftmann@26358
   817
  apply (rule cases)
haftmann@40607
   818
  apply simp_all
haftmann@26358
   819
  done
haftmann@26358
   820
haftmann@37166
   821
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where
haftmann@40607
   822
  "apfst f = map_pair f id"
haftmann@26358
   823
haftmann@37166
   824
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where
haftmann@40607
   825
  "apsnd f = map_pair id f"
haftmann@26358
   826
haftmann@26358
   827
lemma apfst_conv [simp, code]:
haftmann@26358
   828
  "apfst f (x, y) = (f x, y)" 
haftmann@26358
   829
  by (simp add: apfst_def)
haftmann@26358
   830
hoelzl@33638
   831
lemma apsnd_conv [simp, code]:
haftmann@26358
   832
  "apsnd f (x, y) = (x, f y)" 
haftmann@26358
   833
  by (simp add: apsnd_def)
haftmann@21195
   834
haftmann@33594
   835
lemma fst_apfst [simp]:
haftmann@33594
   836
  "fst (apfst f x) = f (fst x)"
haftmann@33594
   837
  by (cases x) simp
haftmann@33594
   838
haftmann@33594
   839
lemma fst_apsnd [simp]:
haftmann@33594
   840
  "fst (apsnd f x) = fst x"
haftmann@33594
   841
  by (cases x) simp
haftmann@33594
   842
haftmann@33594
   843
lemma snd_apfst [simp]:
haftmann@33594
   844
  "snd (apfst f x) = snd x"
haftmann@33594
   845
  by (cases x) simp
haftmann@33594
   846
haftmann@33594
   847
lemma snd_apsnd [simp]:
haftmann@33594
   848
  "snd (apsnd f x) = f (snd x)"
haftmann@33594
   849
  by (cases x) simp
haftmann@33594
   850
haftmann@33594
   851
lemma apfst_compose:
haftmann@33594
   852
  "apfst f (apfst g x) = apfst (f \<circ> g) x"
haftmann@33594
   853
  by (cases x) simp
haftmann@33594
   854
haftmann@33594
   855
lemma apsnd_compose:
haftmann@33594
   856
  "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
haftmann@33594
   857
  by (cases x) simp
haftmann@33594
   858
haftmann@33594
   859
lemma apfst_apsnd [simp]:
haftmann@33594
   860
  "apfst f (apsnd g x) = (f (fst x), g (snd x))"
haftmann@33594
   861
  by (cases x) simp
haftmann@33594
   862
haftmann@33594
   863
lemma apsnd_apfst [simp]:
haftmann@33594
   864
  "apsnd f (apfst g x) = (g (fst x), f (snd x))"
haftmann@33594
   865
  by (cases x) simp
haftmann@33594
   866
haftmann@33594
   867
lemma apfst_id [simp] :
haftmann@33594
   868
  "apfst id = id"
nipkow@39302
   869
  by (simp add: fun_eq_iff)
haftmann@33594
   870
haftmann@33594
   871
lemma apsnd_id [simp] :
haftmann@33594
   872
  "apsnd id = id"
nipkow@39302
   873
  by (simp add: fun_eq_iff)
haftmann@33594
   874
haftmann@33594
   875
lemma apfst_eq_conv [simp]:
haftmann@33594
   876
  "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
haftmann@33594
   877
  by (cases x) simp
haftmann@33594
   878
haftmann@33594
   879
lemma apsnd_eq_conv [simp]:
haftmann@33594
   880
  "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"
haftmann@33594
   881
  by (cases x) simp
haftmann@33594
   882
hoelzl@33638
   883
lemma apsnd_apfst_commute:
hoelzl@33638
   884
  "apsnd f (apfst g p) = apfst g (apsnd f p)"
hoelzl@33638
   885
  by simp
haftmann@21195
   886
haftmann@26358
   887
text {*
haftmann@26358
   888
  Disjoint union of a family of sets -- Sigma.
haftmann@26358
   889
*}
haftmann@26358
   890
haftmann@45986
   891
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where
haftmann@26358
   892
  Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
haftmann@26358
   893
haftmann@26358
   894
abbreviation
haftmann@45986
   895
  Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set"
haftmann@26358
   896
    (infixr "<*>" 80) where
haftmann@26358
   897
  "A <*> B == Sigma A (%_. B)"
haftmann@26358
   898
haftmann@26358
   899
notation (xsymbols)
haftmann@26358
   900
  Times  (infixr "\<times>" 80)
berghofe@15394
   901
haftmann@26358
   902
notation (HTML output)
haftmann@26358
   903
  Times  (infixr "\<times>" 80)
haftmann@26358
   904
nipkow@45662
   905
hide_const (open) Times
nipkow@45662
   906
haftmann@26358
   907
syntax
wenzelm@35115
   908
  "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
haftmann@26358
   909
translations
wenzelm@35115
   910
  "SIGMA x:A. B" == "CONST Sigma A (%x. B)"
haftmann@26358
   911
haftmann@26358
   912
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
haftmann@26358
   913
  by (unfold Sigma_def) blast
haftmann@26358
   914
haftmann@26358
   915
lemma SigmaE [elim!]:
haftmann@26358
   916
    "[| c: Sigma A B;
haftmann@26358
   917
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
haftmann@26358
   918
     |] ==> P"
haftmann@26358
   919
  -- {* The general elimination rule. *}
haftmann@26358
   920
  by (unfold Sigma_def) blast
haftmann@20588
   921
haftmann@26358
   922
text {*
haftmann@26358
   923
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
haftmann@26358
   924
  eigenvariables.
haftmann@26358
   925
*}
haftmann@26358
   926
haftmann@26358
   927
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
haftmann@26358
   928
  by blast
haftmann@26358
   929
haftmann@26358
   930
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
haftmann@26358
   931
  by blast
haftmann@26358
   932
haftmann@26358
   933
lemma SigmaE2:
haftmann@26358
   934
    "[| (a, b) : Sigma A B;
haftmann@26358
   935
        [| a:A;  b:B(a) |] ==> P
haftmann@26358
   936
     |] ==> P"
haftmann@26358
   937
  by blast
haftmann@20588
   938
haftmann@26358
   939
lemma Sigma_cong:
haftmann@26358
   940
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
haftmann@26358
   941
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
haftmann@26358
   942
  by auto
haftmann@26358
   943
haftmann@26358
   944
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
haftmann@26358
   945
  by blast
haftmann@26358
   946
haftmann@26358
   947
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
haftmann@26358
   948
  by blast
haftmann@26358
   949
haftmann@26358
   950
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
haftmann@26358
   951
  by blast
haftmann@26358
   952
haftmann@26358
   953
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
haftmann@26358
   954
  by auto
haftmann@21908
   955
haftmann@26358
   956
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
haftmann@26358
   957
  by auto
haftmann@26358
   958
haftmann@26358
   959
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
haftmann@26358
   960
  by auto
haftmann@26358
   961
haftmann@26358
   962
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
haftmann@26358
   963
  by blast
haftmann@26358
   964
haftmann@26358
   965
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
haftmann@26358
   966
  by blast
haftmann@26358
   967
haftmann@26358
   968
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
haftmann@26358
   969
  by (blast elim: equalityE)
haftmann@20588
   970
haftmann@26358
   971
lemma SetCompr_Sigma_eq:
haftmann@26358
   972
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
haftmann@26358
   973
  by blast
haftmann@26358
   974
haftmann@26358
   975
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
haftmann@26358
   976
  by blast
haftmann@26358
   977
haftmann@26358
   978
lemma UN_Times_distrib:
haftmann@26358
   979
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
haftmann@26358
   980
  -- {* Suggested by Pierre Chartier *}
haftmann@26358
   981
  by blast
haftmann@26358
   982
blanchet@47740
   983
lemma split_paired_Ball_Sigma [simp, no_atp]:
haftmann@26358
   984
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
haftmann@26358
   985
  by blast
haftmann@26358
   986
blanchet@47740
   987
lemma split_paired_Bex_Sigma [simp, no_atp]:
haftmann@26358
   988
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
haftmann@26358
   989
  by blast
haftmann@21908
   990
haftmann@26358
   991
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
haftmann@26358
   992
  by blast
haftmann@26358
   993
haftmann@26358
   994
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
haftmann@26358
   995
  by blast
haftmann@26358
   996
haftmann@26358
   997
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
haftmann@26358
   998
  by blast
haftmann@26358
   999
haftmann@26358
  1000
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
haftmann@26358
  1001
  by blast
haftmann@26358
  1002
haftmann@26358
  1003
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
haftmann@26358
  1004
  by blast
haftmann@26358
  1005
haftmann@26358
  1006
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
haftmann@26358
  1007
  by blast
haftmann@21908
  1008
haftmann@26358
  1009
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
haftmann@26358
  1010
  by blast
haftmann@26358
  1011
haftmann@26358
  1012
text {*
haftmann@26358
  1013
  Non-dependent versions are needed to avoid the need for higher-order
haftmann@26358
  1014
  matching, especially when the rules are re-oriented.
haftmann@26358
  1015
*}
haftmann@21908
  1016
haftmann@26358
  1017
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
nipkow@28719
  1018
by blast
haftmann@26358
  1019
haftmann@26358
  1020
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
nipkow@28719
  1021
by blast
haftmann@26358
  1022
haftmann@26358
  1023
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
nipkow@28719
  1024
by blast
haftmann@26358
  1025
hoelzl@36622
  1026
lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
hoelzl@36622
  1027
  by auto
hoelzl@36622
  1028
hoelzl@50104
  1029
lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
hoelzl@50104
  1030
  by auto
hoelzl@50104
  1031
hoelzl@36622
  1032
lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)"
huffman@44921
  1033
  by force
hoelzl@36622
  1034
hoelzl@36622
  1035
lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)"
huffman@44921
  1036
  by force
hoelzl@36622
  1037
nipkow@28719
  1038
lemma insert_times_insert[simp]:
nipkow@28719
  1039
  "insert a A \<times> insert b B =
nipkow@28719
  1040
   insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
nipkow@28719
  1041
by blast
haftmann@26358
  1042
paulson@33271
  1043
lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)"
wenzelm@47988
  1044
  apply auto
wenzelm@47988
  1045
  apply (case_tac "f x")
wenzelm@47988
  1046
  apply auto
wenzelm@47988
  1047
  done
paulson@33271
  1048
hoelzl@50104
  1049
lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
hoelzl@50104
  1050
  by auto
hoelzl@50104
  1051
haftmann@35822
  1052
lemma swap_inj_on:
hoelzl@36622
  1053
  "inj_on (\<lambda>(i, j). (j, i)) A"
hoelzl@36622
  1054
  by (auto intro!: inj_onI)
haftmann@35822
  1055
haftmann@35822
  1056
lemma swap_product:
haftmann@35822
  1057
  "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
haftmann@35822
  1058
  by (simp add: split_def image_def) blast
haftmann@35822
  1059
hoelzl@36622
  1060
lemma image_split_eq_Sigma:
hoelzl@36622
  1061
  "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
haftmann@46128
  1062
proof (safe intro!: imageI)
hoelzl@36622
  1063
  fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
hoelzl@36622
  1064
  show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
hoelzl@36622
  1065
    using * eq[symmetric] by auto
hoelzl@36622
  1066
qed simp_all
haftmann@35822
  1067
haftmann@46128
  1068
definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
haftmann@46128
  1069
  [code_abbrev]: "product A B = A \<times> B"
haftmann@46128
  1070
haftmann@46128
  1071
hide_const (open) product
haftmann@46128
  1072
haftmann@46128
  1073
lemma member_product:
haftmann@46128
  1074
  "x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B"
haftmann@46128
  1075
  by (simp add: product_def)
haftmann@46128
  1076
haftmann@40607
  1077
text {* The following @{const map_pair} lemmas are due to Joachim Breitner: *}
haftmann@40607
  1078
haftmann@40607
  1079
lemma map_pair_inj_on:
haftmann@40607
  1080
  assumes "inj_on f A" and "inj_on g B"
haftmann@40607
  1081
  shows "inj_on (map_pair f g) (A \<times> B)"
haftmann@40607
  1082
proof (rule inj_onI)
haftmann@40607
  1083
  fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c"
haftmann@40607
  1084
  assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto
haftmann@40607
  1085
  assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto
haftmann@40607
  1086
  assume "map_pair f g x = map_pair f g y"
haftmann@40607
  1087
  hence "fst (map_pair f g x) = fst (map_pair f g y)" by (auto)
haftmann@40607
  1088
  hence "f (fst x) = f (fst y)" by (cases x,cases y,auto)
haftmann@40607
  1089
  with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A`
haftmann@40607
  1090
  have "fst x = fst y" by (auto dest:dest:inj_onD)
haftmann@40607
  1091
  moreover from `map_pair f g x = map_pair f g y`
haftmann@40607
  1092
  have "snd (map_pair f g x) = snd (map_pair f g y)" by (auto)
haftmann@40607
  1093
  hence "g (snd x) = g (snd y)" by (cases x,cases y,auto)
haftmann@40607
  1094
  with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B`
haftmann@40607
  1095
  have "snd x = snd y" by (auto dest:dest:inj_onD)
haftmann@40607
  1096
  ultimately show "x = y" by(rule prod_eqI)
haftmann@40607
  1097
qed
haftmann@40607
  1098
haftmann@40607
  1099
lemma map_pair_surj:
hoelzl@40702
  1100
  fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd"
haftmann@40607
  1101
  assumes "surj f" and "surj g"
haftmann@40607
  1102
  shows "surj (map_pair f g)"
haftmann@40607
  1103
unfolding surj_def
haftmann@40607
  1104
proof
haftmann@40607
  1105
  fix y :: "'b \<times> 'd"
haftmann@40607
  1106
  from `surj f` obtain a where "fst y = f a" by (auto elim:surjE)
haftmann@40607
  1107
  moreover
haftmann@40607
  1108
  from `surj g` obtain b where "snd y = g b" by (auto elim:surjE)
haftmann@40607
  1109
  ultimately have "(fst y, snd y) = map_pair f g (a,b)" by auto
haftmann@40607
  1110
  thus "\<exists>x. y = map_pair f g x" by auto
haftmann@40607
  1111
qed
haftmann@40607
  1112
haftmann@40607
  1113
lemma map_pair_surj_on:
haftmann@40607
  1114
  assumes "f ` A = A'" and "g ` B = B'"
haftmann@40607
  1115
  shows "map_pair f g ` (A \<times> B) = A' \<times> B'"
haftmann@40607
  1116
unfolding image_def
haftmann@40607
  1117
proof(rule set_eqI,rule iffI)
haftmann@40607
  1118
  fix x :: "'a \<times> 'c"
haftmann@40607
  1119
  assume "x \<in> {y\<Colon>'a \<times> 'c. \<exists>x\<Colon>'b \<times> 'd\<in>A \<times> B. y = map_pair f g x}"
haftmann@40607
  1120
  then obtain y where "y \<in> A \<times> B" and "x = map_pair f g y" by blast
haftmann@40607
  1121
  from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto
haftmann@40607
  1122
  moreover from `image g B = B'` and `y \<in> A \<times> B` have "g (snd y) \<in> B'" by auto
haftmann@40607
  1123
  ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto
haftmann@40607
  1124
  with `x = map_pair f g y` show "x \<in> A' \<times> B'" by (cases y, auto)
haftmann@40607
  1125
next
haftmann@40607
  1126
  fix x :: "'a \<times> 'c"
haftmann@40607
  1127
  assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto
haftmann@40607
  1128
  from `image f A = A'` and `fst x \<in> A'` have "fst x \<in> image f A" by auto
haftmann@40607
  1129
  then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE)
haftmann@40607
  1130
  moreover from `image g B = B'` and `snd x \<in> B'`
haftmann@40607
  1131
  obtain b where "b \<in> B" and "snd x = g b" by auto
haftmann@40607
  1132
  ultimately have "(fst x, snd x) = map_pair f g (a,b)" by auto
haftmann@40607
  1133
  moreover from `a \<in> A` and  `b \<in> B` have "(a , b) \<in> A \<times> B" by auto
haftmann@40607
  1134
  ultimately have "\<exists>y \<in> A \<times> B. x = map_pair f g y" by auto
haftmann@40607
  1135
  thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_pair f g y}" by auto
haftmann@40607
  1136
qed
haftmann@40607
  1137
haftmann@21908
  1138
bulwahn@49764
  1139
subsection {* Simproc for rewriting a set comprehension into a pointfree expression *}
bulwahn@49764
  1140
bulwahn@49764
  1141
ML_file "Tools/set_comprehension_pointfree.ML"
bulwahn@49764
  1142
bulwahn@49764
  1143
setup {*
bulwahn@49764
  1144
  Code_Preproc.map_pre (fn ss => ss addsimprocs
bulwahn@49764
  1145
    [Raw_Simplifier.make_simproc {name = "set comprehension", lhss = [@{cpat "Collect ?P"}],
bulwahn@49764
  1146
    proc = K Set_Comprehension_Pointfree.code_simproc, identifier = []}])
bulwahn@49764
  1147
*}
bulwahn@49764
  1148
bulwahn@49764
  1149
haftmann@37166
  1150
subsection {* Inductively defined sets *}
berghofe@15394
  1151
wenzelm@48891
  1152
ML_file "Tools/inductive_set.ML"
haftmann@31723
  1153
setup Inductive_Set.setup
haftmann@24699
  1154
haftmann@37166
  1155
haftmann@37166
  1156
subsection {* Legacy theorem bindings and duplicates *}
haftmann@37166
  1157
haftmann@37166
  1158
lemma PairE:
haftmann@37166
  1159
  obtains x y where "p = (x, y)"
haftmann@37166
  1160
  by (fact prod.exhaust)
haftmann@37166
  1161
haftmann@37166
  1162
lemmas Pair_eq = prod.inject
haftmann@37166
  1163
haftmann@37166
  1164
lemmas split = split_conv  -- {* for backwards compatibility *}
haftmann@37166
  1165
huffman@44066
  1166
lemmas Pair_fst_snd_eq = prod_eq_iff
huffman@44066
  1167
huffman@45204
  1168
hide_const (open) prod
huffman@45204
  1169
nipkow@10213
  1170
end