src/HOL/Product_Type.thy
author haftmann
Sun Jan 01 15:44:15 2012 +0100 (2012-01-01)
changeset 46128 53e7cc599f58
parent 46028 9f113cdf3d66
child 46556 2848e36e0348
permissions -rw-r--r--
interaction of set operations for execution and membership predicate
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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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uses
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  ("Tools/split_rule.ML")
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  ("Tools/inductive_set.ML")
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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 (open) 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 (open) ('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 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))"
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  by (simp add: fun_eq_iff split: prod.split)
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lemma fst_eqD: "fst (x, y) = a ==> x = a"
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  by simp
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   334
wenzelm@11838
   335
lemma snd_eqD: "snd (x, y) = a ==> y = a"
wenzelm@11838
   336
  by simp
wenzelm@11838
   337
haftmann@26358
   338
lemma pair_collapse [simp]: "(fst p, snd p) = p"
wenzelm@11838
   339
  by (cases p) simp
wenzelm@11838
   340
haftmann@26358
   341
lemmas surjective_pairing = pair_collapse [symmetric]
wenzelm@11838
   342
huffman@44066
   343
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
haftmann@37166
   344
  by (cases s, cases t) simp
haftmann@37166
   345
haftmann@37166
   346
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
huffman@44066
   347
  by (simp add: prod_eq_iff)
haftmann@37166
   348
haftmann@37166
   349
lemma split_conv [simp, code]: "split f (a, b) = f a b"
haftmann@37591
   350
  by (fact prod.cases)
haftmann@37166
   351
haftmann@37166
   352
lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
haftmann@37166
   353
  by (rule split_conv [THEN iffD2])
haftmann@37166
   354
haftmann@37166
   355
lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
haftmann@37166
   356
  by (rule split_conv [THEN iffD1])
haftmann@37166
   357
haftmann@37166
   358
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
nipkow@39302
   359
  by (simp add: fun_eq_iff split: prod.split)
haftmann@37166
   360
haftmann@37166
   361
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
haftmann@37166
   362
  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
nipkow@39302
   363
  by (simp add: fun_eq_iff split: prod.split)
haftmann@37166
   364
haftmann@37166
   365
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
haftmann@37166
   366
  by (cases x) simp
haftmann@37166
   367
haftmann@37166
   368
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
haftmann@37166
   369
  by (cases p) simp
haftmann@37166
   370
haftmann@37166
   371
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
haftmann@37591
   372
  by (simp add: prod_case_unfold)
haftmann@37166
   373
haftmann@37166
   374
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
haftmann@37166
   375
  -- {* Prevents simplification of @{term c}: much faster *}
huffman@40929
   376
  by (fact prod.weak_case_cong)
haftmann@37166
   377
haftmann@37166
   378
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
haftmann@37166
   379
  by (simp add: split_eta)
haftmann@37166
   380
wenzelm@11838
   381
lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
wenzelm@11820
   382
proof
wenzelm@11820
   383
  fix a b
wenzelm@11820
   384
  assume "!!x. PROP P x"
wenzelm@19535
   385
  then show "PROP P (a, b)" .
wenzelm@11820
   386
next
wenzelm@11820
   387
  fix x
wenzelm@11820
   388
  assume "!!a b. PROP P (a, b)"
wenzelm@19535
   389
  from `PROP P (fst x, snd x)` show "PROP P x" by simp
wenzelm@11820
   390
qed
wenzelm@11820
   391
wenzelm@11838
   392
text {*
wenzelm@11838
   393
  The rule @{thm [source] split_paired_all} does not work with the
wenzelm@11838
   394
  Simplifier because it also affects premises in congrence rules,
wenzelm@11838
   395
  where this can lead to premises of the form @{text "!!a b. ... =
wenzelm@11838
   396
  ?P(a, b)"} which cannot be solved by reflexivity.
wenzelm@11838
   397
*}
wenzelm@11838
   398
haftmann@26358
   399
lemmas split_tupled_all = split_paired_all unit_all_eq2
haftmann@26358
   400
wenzelm@26480
   401
ML {*
wenzelm@11838
   402
  (* replace parameters of product type by individual component parameters *)
wenzelm@11838
   403
  val safe_full_simp_tac = generic_simp_tac true (true, false, false);
wenzelm@11838
   404
  local (* filtering with exists_paired_all is an essential optimization *)
wenzelm@16121
   405
    fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
wenzelm@11838
   406
          can HOLogic.dest_prodT T orelse exists_paired_all t
wenzelm@11838
   407
      | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
wenzelm@11838
   408
      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
wenzelm@11838
   409
      | exists_paired_all _ = false;
wenzelm@11838
   410
    val ss = HOL_basic_ss
wenzelm@26340
   411
      addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
wenzelm@43594
   412
      addsimprocs [@{simproc unit_eq}];
wenzelm@11838
   413
  in
wenzelm@11838
   414
    val split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   415
      if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838
   416
    val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   417
      if exists_paired_all t then full_simp_tac ss i else no_tac);
wenzelm@11838
   418
    fun split_all th =
wenzelm@26340
   419
   if exists_paired_all (Thm.prop_of th) then full_simplify ss th else th;
wenzelm@11838
   420
  end;
wenzelm@26340
   421
*}
wenzelm@11838
   422
wenzelm@26340
   423
declaration {* fn _ =>
wenzelm@26340
   424
  Classical.map_cs (fn cs => cs addSbefore ("split_all_tac", split_all_tac))
wenzelm@16121
   425
*}
wenzelm@11838
   426
wenzelm@11838
   427
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@11838
   428
  -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
wenzelm@11838
   429
  by fast
wenzelm@11838
   430
haftmann@26358
   431
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
haftmann@26358
   432
  by fast
haftmann@26358
   433
wenzelm@11838
   434
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@11838
   435
  -- {* Can't be added to simpset: loops! *}
haftmann@26358
   436
  by (simp add: split_eta)
wenzelm@11838
   437
wenzelm@11838
   438
text {*
wenzelm@11838
   439
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
wenzelm@11838
   440
  @{thm [source] split_eta} as a rewrite rule is not general enough,
wenzelm@11838
   441
  and using @{thm [source] cond_split_eta} directly would render some
wenzelm@11838
   442
  existing proofs very inefficient; similarly for @{text
haftmann@26358
   443
  split_beta}.
haftmann@26358
   444
*}
wenzelm@11838
   445
wenzelm@26480
   446
ML {*
wenzelm@11838
   447
local
wenzelm@35364
   448
  val cond_split_eta_ss = HOL_basic_ss addsimps @{thms cond_split_eta};
wenzelm@35364
   449
  fun Pair_pat k 0 (Bound m) = (m = k)
wenzelm@35364
   450
    | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =
wenzelm@35364
   451
        i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t
wenzelm@35364
   452
    | Pair_pat _ _ _ = false;
wenzelm@35364
   453
  fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t
wenzelm@35364
   454
    | no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@35364
   455
    | no_args k i (Bound m) = m < k orelse m > k + i
wenzelm@35364
   456
    | no_args _ _ _ = true;
wenzelm@35364
   457
  fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE
haftmann@37591
   458
    | split_pat tp i (Const (@{const_name prod_case}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
wenzelm@35364
   459
    | split_pat tp i _ = NONE;
wenzelm@20044
   460
  fun metaeq ss lhs rhs = mk_meta_eq (Goal.prove (Simplifier.the_context ss) [] []
wenzelm@35364
   461
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
wenzelm@18328
   462
        (K (simp_tac (Simplifier.inherit_context ss cond_split_eta_ss) 1)));
wenzelm@11838
   463
wenzelm@35364
   464
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t
wenzelm@35364
   465
    | beta_term_pat k i (t $ u) =
wenzelm@35364
   466
        Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@35364
   467
    | beta_term_pat k i t = no_args k i t;
wenzelm@35364
   468
  fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@35364
   469
    | eta_term_pat _ _ _ = false;
wenzelm@11838
   470
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@35364
   471
    | subst arg k i (t $ u) =
wenzelm@35364
   472
        if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@35364
   473
        else (subst arg k i t $ subst arg k i u)
wenzelm@35364
   474
    | subst arg k i t = t;
wenzelm@43595
   475
in
haftmann@37591
   476
  fun beta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t) $ arg) =
wenzelm@11838
   477
        (case split_pat beta_term_pat 1 t of
wenzelm@35364
   478
          SOME (i, f) => SOME (metaeq ss s (subst arg 0 i f))
skalberg@15531
   479
        | NONE => NONE)
wenzelm@35364
   480
    | beta_proc _ _ = NONE;
haftmann@37591
   481
  fun eta_proc ss (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t)) =
wenzelm@11838
   482
        (case split_pat eta_term_pat 1 t of
wenzelm@35364
   483
          SOME (_, ft) => SOME (metaeq ss s (let val (f $ arg) = ft in f end))
skalberg@15531
   484
        | NONE => NONE)
wenzelm@35364
   485
    | eta_proc _ _ = NONE;
wenzelm@11838
   486
end;
wenzelm@11838
   487
*}
wenzelm@43595
   488
simproc_setup split_beta ("split f z") = {* fn _ => fn ss => fn ct => beta_proc ss (term_of ct) *}
wenzelm@43595
   489
simproc_setup split_eta ("split f") = {* fn _ => fn ss => fn ct => eta_proc ss (term_of ct) *}
wenzelm@11838
   490
berghofe@26798
   491
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)"
wenzelm@11838
   492
  by (subst surjective_pairing, rule split_conv)
wenzelm@11838
   493
blanchet@35828
   494
lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
wenzelm@11838
   495
  -- {* For use with @{text split} and the Simplifier. *}
paulson@15481
   496
  by (insert surj_pair [of p], clarify, simp)
wenzelm@11838
   497
wenzelm@11838
   498
text {*
wenzelm@11838
   499
  @{thm [source] split_split} could be declared as @{text "[split]"}
wenzelm@11838
   500
  done after the Splitter has been speeded up significantly;
wenzelm@11838
   501
  precompute the constants involved and don't do anything unless the
wenzelm@11838
   502
  current goal contains one of those constants.
wenzelm@11838
   503
*}
wenzelm@11838
   504
blanchet@35828
   505
lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
paulson@14208
   506
by (subst split_split, simp)
wenzelm@11838
   507
wenzelm@11838
   508
text {*
wenzelm@11838
   509
  \medskip @{term split} used as a logical connective or set former.
wenzelm@11838
   510
wenzelm@11838
   511
  \medskip These rules are for use with @{text blast}; could instead
huffman@40929
   512
  call @{text simp} using @{thm [source] prod.split} as rewrite. *}
wenzelm@11838
   513
wenzelm@11838
   514
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
wenzelm@11838
   515
  apply (simp only: split_tupled_all)
wenzelm@11838
   516
  apply (simp (no_asm_simp))
wenzelm@11838
   517
  done
wenzelm@11838
   518
wenzelm@11838
   519
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
wenzelm@11838
   520
  apply (simp only: split_tupled_all)
wenzelm@11838
   521
  apply (simp (no_asm_simp))
wenzelm@11838
   522
  done
wenzelm@11838
   523
wenzelm@11838
   524
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@37591
   525
  by (induct p) auto
wenzelm@11838
   526
wenzelm@11838
   527
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@37591
   528
  by (induct p) auto
wenzelm@11838
   529
wenzelm@11838
   530
lemma splitE2:
wenzelm@11838
   531
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
wenzelm@11838
   532
proof -
wenzelm@11838
   533
  assume q: "Q (split P z)"
wenzelm@11838
   534
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
wenzelm@11838
   535
  show R
wenzelm@11838
   536
    apply (rule r surjective_pairing)+
wenzelm@11838
   537
    apply (rule split_beta [THEN subst], rule q)
wenzelm@11838
   538
    done
wenzelm@11838
   539
qed
wenzelm@11838
   540
wenzelm@11838
   541
lemma splitD': "split R (a,b) c ==> R a b c"
wenzelm@11838
   542
  by simp
wenzelm@11838
   543
wenzelm@11838
   544
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
wenzelm@11838
   545
  by simp
wenzelm@11838
   546
wenzelm@11838
   547
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
paulson@14208
   548
by (simp only: split_tupled_all, simp)
wenzelm@11838
   549
wenzelm@18372
   550
lemma mem_splitE:
haftmann@37166
   551
  assumes major: "z \<in> split c p"
haftmann@37166
   552
    and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q"
wenzelm@18372
   553
  shows Q
haftmann@37591
   554
  by (rule major [unfolded prod_case_unfold] cases surjective_pairing)+
wenzelm@11838
   555
wenzelm@11838
   556
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
wenzelm@11838
   557
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
wenzelm@11838
   558
wenzelm@26340
   559
ML {*
wenzelm@11838
   560
local (* filtering with exists_p_split is an essential optimization *)
haftmann@37591
   561
  fun exists_p_split (Const (@{const_name prod_case},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true
wenzelm@11838
   562
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838
   563
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838
   564
    | exists_p_split _ = false;
wenzelm@35364
   565
  val ss = HOL_basic_ss addsimps @{thms split_conv};
wenzelm@11838
   566
in
wenzelm@11838
   567
val split_conv_tac = SUBGOAL (fn (t, i) =>
wenzelm@11838
   568
    if exists_p_split t then safe_full_simp_tac ss i else no_tac);
wenzelm@11838
   569
end;
wenzelm@26340
   570
*}
wenzelm@26340
   571
wenzelm@11838
   572
(* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838
   573
   to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@26340
   574
declaration {* fn _ =>
wenzelm@26340
   575
  Classical.map_cs (fn cs => cs addSbefore ("split_conv_tac", split_conv_tac))
wenzelm@16121
   576
*}
wenzelm@11838
   577
blanchet@35828
   578
lemma split_eta_SetCompr [simp,no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
wenzelm@18372
   579
  by (rule ext) fast
wenzelm@11838
   580
blanchet@35828
   581
lemma split_eta_SetCompr2 [simp,no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
wenzelm@18372
   582
  by (rule ext) fast
wenzelm@11838
   583
wenzelm@11838
   584
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
wenzelm@11838
   585
  -- {* Allows simplifications of nested splits in case of independent predicates. *}
wenzelm@18372
   586
  by (rule ext) blast
wenzelm@11838
   587
nipkow@14337
   588
(* Do NOT make this a simp rule as it
nipkow@14337
   589
   a) only helps in special situations
nipkow@14337
   590
   b) can lead to nontermination in the presence of split_def
nipkow@14337
   591
*)
nipkow@14337
   592
lemma split_comp_eq: 
paulson@20415
   593
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
paulson@20415
   594
  shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
wenzelm@18372
   595
  by (rule ext) auto
oheimb@14101
   596
haftmann@26358
   597
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
haftmann@26358
   598
  apply (rule_tac x = "(a, b)" in image_eqI)
haftmann@26358
   599
   apply auto
haftmann@26358
   600
  done
haftmann@26358
   601
wenzelm@11838
   602
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838
   603
  by blast
wenzelm@11838
   604
wenzelm@11838
   605
(*
wenzelm@11838
   606
the following  would be slightly more general,
wenzelm@11838
   607
but cannot be used as rewrite rule:
wenzelm@11838
   608
### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838
   609
### ?y = .x
wenzelm@11838
   610
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208
   611
by (rtac some_equality 1)
paulson@14208
   612
by ( Simp_tac 1)
paulson@14208
   613
by (split_all_tac 1)
paulson@14208
   614
by (Asm_full_simp_tac 1)
wenzelm@11838
   615
qed "The_split_eq";
wenzelm@11838
   616
*)
wenzelm@11838
   617
wenzelm@11838
   618
text {*
wenzelm@11838
   619
  Setup of internal @{text split_rule}.
wenzelm@11838
   620
*}
wenzelm@11838
   621
wenzelm@45607
   622
lemmas prod_caseI = prod.cases [THEN iffD2]
haftmann@24699
   623
haftmann@24699
   624
lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
haftmann@37678
   625
  by (fact splitI2)
haftmann@24699
   626
haftmann@24699
   627
lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
haftmann@37678
   628
  by (fact splitI2')
haftmann@24699
   629
haftmann@24699
   630
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@37678
   631
  by (fact splitE)
haftmann@24699
   632
haftmann@24699
   633
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@37678
   634
  by (fact splitE')
haftmann@24699
   635
haftmann@37678
   636
declare prod_caseI [intro!]
haftmann@24699
   637
bulwahn@26143
   638
lemma prod_case_beta:
bulwahn@26143
   639
  "prod_case f p = f (fst p) (snd p)"
haftmann@37591
   640
  by (fact split_beta)
bulwahn@26143
   641
haftmann@24699
   642
lemma prod_cases3 [cases type]:
haftmann@24699
   643
  obtains (fields) a b c where "y = (a, b, c)"
haftmann@24699
   644
  by (cases y, case_tac b) blast
haftmann@24699
   645
haftmann@24699
   646
lemma prod_induct3 [case_names fields, induct type]:
haftmann@24699
   647
    "(!!a b c. P (a, b, c)) ==> P x"
haftmann@24699
   648
  by (cases x) blast
haftmann@24699
   649
haftmann@24699
   650
lemma prod_cases4 [cases type]:
haftmann@24699
   651
  obtains (fields) a b c d where "y = (a, b, c, d)"
haftmann@24699
   652
  by (cases y, case_tac c) blast
haftmann@24699
   653
haftmann@24699
   654
lemma prod_induct4 [case_names fields, induct type]:
haftmann@24699
   655
    "(!!a b c d. P (a, b, c, d)) ==> P x"
haftmann@24699
   656
  by (cases x) blast
haftmann@24699
   657
haftmann@24699
   658
lemma prod_cases5 [cases type]:
haftmann@24699
   659
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
haftmann@24699
   660
  by (cases y, case_tac d) blast
haftmann@24699
   661
haftmann@24699
   662
lemma prod_induct5 [case_names fields, induct type]:
haftmann@24699
   663
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
haftmann@24699
   664
  by (cases x) blast
haftmann@24699
   665
haftmann@24699
   666
lemma prod_cases6 [cases type]:
haftmann@24699
   667
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
haftmann@24699
   668
  by (cases y, case_tac e) blast
haftmann@24699
   669
haftmann@24699
   670
lemma prod_induct6 [case_names fields, induct type]:
haftmann@24699
   671
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
haftmann@24699
   672
  by (cases x) blast
haftmann@24699
   673
haftmann@24699
   674
lemma prod_cases7 [cases type]:
haftmann@24699
   675
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
haftmann@24699
   676
  by (cases y, case_tac f) blast
haftmann@24699
   677
haftmann@24699
   678
lemma prod_induct7 [case_names fields, induct type]:
haftmann@24699
   679
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
haftmann@24699
   680
  by (cases x) blast
haftmann@24699
   681
haftmann@37166
   682
lemma split_def:
haftmann@37166
   683
  "split = (\<lambda>c p. c (fst p) (snd p))"
haftmann@37591
   684
  by (fact prod_case_unfold)
haftmann@37166
   685
haftmann@37166
   686
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
haftmann@37166
   687
  "internal_split == split"
haftmann@37166
   688
haftmann@37166
   689
lemma internal_split_conv: "internal_split c (a, b) = c a b"
haftmann@37166
   690
  by (simp only: internal_split_def split_conv)
haftmann@37166
   691
haftmann@37166
   692
use "Tools/split_rule.ML"
haftmann@37166
   693
setup Split_Rule.setup
haftmann@37166
   694
haftmann@37166
   695
hide_const internal_split
haftmann@37166
   696
haftmann@24699
   697
haftmann@26358
   698
subsubsection {* Derived operations *}
haftmann@26358
   699
haftmann@37387
   700
definition curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where
haftmann@37387
   701
  "curry = (\<lambda>c x y. c (x, y))"
haftmann@37166
   702
haftmann@37166
   703
lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
haftmann@37166
   704
  by (simp add: curry_def)
haftmann@37166
   705
haftmann@37166
   706
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
haftmann@37166
   707
  by (simp add: curry_def)
haftmann@37166
   708
haftmann@37166
   709
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
haftmann@37166
   710
  by (simp add: curry_def)
haftmann@37166
   711
haftmann@37166
   712
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@37166
   713
  by (simp add: curry_def)
haftmann@37166
   714
haftmann@37166
   715
lemma curry_split [simp]: "curry (split f) = f"
haftmann@37166
   716
  by (simp add: curry_def split_def)
haftmann@37166
   717
haftmann@37166
   718
lemma split_curry [simp]: "split (curry f) = f"
haftmann@37166
   719
  by (simp add: curry_def split_def)
haftmann@37166
   720
haftmann@26358
   721
text {*
haftmann@26358
   722
  The composition-uncurry combinator.
haftmann@26358
   723
*}
haftmann@26358
   724
haftmann@37751
   725
notation fcomp (infixl "\<circ>>" 60)
haftmann@26358
   726
haftmann@37751
   727
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where
haftmann@37751
   728
  "f \<circ>\<rightarrow> g = (\<lambda>x. prod_case g (f x))"
haftmann@26358
   729
haftmann@37678
   730
lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))"
nipkow@39302
   731
  by (simp add: fun_eq_iff scomp_def prod_case_unfold)
haftmann@37678
   732
haftmann@37751
   733
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = prod_case g (f x)"
haftmann@37751
   734
  by (simp add: scomp_unfold prod_case_unfold)
haftmann@26358
   735
haftmann@37751
   736
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x"
huffman@44921
   737
  by (simp add: fun_eq_iff)
haftmann@26358
   738
haftmann@37751
   739
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x"
huffman@44921
   740
  by (simp add: fun_eq_iff)
haftmann@26358
   741
haftmann@37751
   742
lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
nipkow@39302
   743
  by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358
   744
haftmann@37751
   745
lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
nipkow@39302
   746
  by (simp add: fun_eq_iff scomp_unfold fcomp_def)
haftmann@26358
   747
haftmann@37751
   748
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
huffman@44921
   749
  by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358
   750
haftmann@31202
   751
code_const scomp
haftmann@31202
   752
  (Eval infixl 3 "#->")
haftmann@31202
   753
haftmann@37751
   754
no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
   755
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@26358
   756
haftmann@26358
   757
text {*
haftmann@40607
   758
  @{term map_pair} --- action of the product functor upon
krauss@36664
   759
  functions.
haftmann@26358
   760
*}
haftmann@21195
   761
haftmann@40607
   762
definition map_pair :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
haftmann@40607
   763
  "map_pair f g = (\<lambda>(x, y). (f x, g y))"
haftmann@26358
   764
haftmann@40607
   765
lemma map_pair_simp [simp, code]:
haftmann@40607
   766
  "map_pair f g (a, b) = (f a, g b)"
haftmann@40607
   767
  by (simp add: map_pair_def)
haftmann@26358
   768
haftmann@41505
   769
enriched_type map_pair: map_pair
huffman@44921
   770
  by (auto simp add: split_paired_all)
nipkow@37278
   771
haftmann@40607
   772
lemma fst_map_pair [simp]:
haftmann@40607
   773
  "fst (map_pair f g x) = f (fst x)"
haftmann@40607
   774
  by (cases x) simp_all
nipkow@37278
   775
haftmann@40607
   776
lemma snd_prod_fun [simp]:
haftmann@40607
   777
  "snd (map_pair f g x) = g (snd x)"
haftmann@40607
   778
  by (cases x) simp_all
nipkow@37278
   779
haftmann@40607
   780
lemma fst_comp_map_pair [simp]:
haftmann@40607
   781
  "fst \<circ> map_pair f g = f \<circ> fst"
haftmann@40607
   782
  by (rule ext) simp_all
nipkow@37278
   783
haftmann@40607
   784
lemma snd_comp_map_pair [simp]:
haftmann@40607
   785
  "snd \<circ> map_pair f g = g \<circ> snd"
haftmann@40607
   786
  by (rule ext) simp_all
haftmann@26358
   787
haftmann@40607
   788
lemma map_pair_compose:
haftmann@40607
   789
  "map_pair (f1 o f2) (g1 o g2) = (map_pair f1 g1 o map_pair f2 g2)"
haftmann@40607
   790
  by (rule ext) (simp add: map_pair.compositionality comp_def)
haftmann@26358
   791
haftmann@40607
   792
lemma map_pair_ident [simp]:
haftmann@40607
   793
  "map_pair (%x. x) (%y. y) = (%z. z)"
haftmann@40607
   794
  by (rule ext) (simp add: map_pair.identity)
haftmann@40607
   795
haftmann@40607
   796
lemma map_pair_imageI [intro]:
haftmann@40607
   797
  "(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_pair f g ` R"
haftmann@40607
   798
  by (rule image_eqI) simp_all
haftmann@21195
   799
haftmann@26358
   800
lemma prod_fun_imageE [elim!]:
haftmann@40607
   801
  assumes major: "c \<in> map_pair f g ` R"
haftmann@40607
   802
    and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P"
haftmann@26358
   803
  shows P
haftmann@26358
   804
  apply (rule major [THEN imageE])
haftmann@37166
   805
  apply (case_tac x)
haftmann@26358
   806
  apply (rule cases)
haftmann@40607
   807
  apply simp_all
haftmann@26358
   808
  done
haftmann@26358
   809
haftmann@37166
   810
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where
haftmann@40607
   811
  "apfst f = map_pair f id"
haftmann@26358
   812
haftmann@37166
   813
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where
haftmann@40607
   814
  "apsnd f = map_pair id f"
haftmann@26358
   815
haftmann@26358
   816
lemma apfst_conv [simp, code]:
haftmann@26358
   817
  "apfst f (x, y) = (f x, y)" 
haftmann@26358
   818
  by (simp add: apfst_def)
haftmann@26358
   819
hoelzl@33638
   820
lemma apsnd_conv [simp, code]:
haftmann@26358
   821
  "apsnd f (x, y) = (x, f y)" 
haftmann@26358
   822
  by (simp add: apsnd_def)
haftmann@21195
   823
haftmann@33594
   824
lemma fst_apfst [simp]:
haftmann@33594
   825
  "fst (apfst f x) = f (fst x)"
haftmann@33594
   826
  by (cases x) simp
haftmann@33594
   827
haftmann@33594
   828
lemma fst_apsnd [simp]:
haftmann@33594
   829
  "fst (apsnd f x) = fst x"
haftmann@33594
   830
  by (cases x) simp
haftmann@33594
   831
haftmann@33594
   832
lemma snd_apfst [simp]:
haftmann@33594
   833
  "snd (apfst f x) = snd x"
haftmann@33594
   834
  by (cases x) simp
haftmann@33594
   835
haftmann@33594
   836
lemma snd_apsnd [simp]:
haftmann@33594
   837
  "snd (apsnd f x) = f (snd x)"
haftmann@33594
   838
  by (cases x) simp
haftmann@33594
   839
haftmann@33594
   840
lemma apfst_compose:
haftmann@33594
   841
  "apfst f (apfst g x) = apfst (f \<circ> g) x"
haftmann@33594
   842
  by (cases x) simp
haftmann@33594
   843
haftmann@33594
   844
lemma apsnd_compose:
haftmann@33594
   845
  "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
haftmann@33594
   846
  by (cases x) simp
haftmann@33594
   847
haftmann@33594
   848
lemma apfst_apsnd [simp]:
haftmann@33594
   849
  "apfst f (apsnd g x) = (f (fst x), g (snd x))"
haftmann@33594
   850
  by (cases x) simp
haftmann@33594
   851
haftmann@33594
   852
lemma apsnd_apfst [simp]:
haftmann@33594
   853
  "apsnd f (apfst g x) = (g (fst x), f (snd x))"
haftmann@33594
   854
  by (cases x) simp
haftmann@33594
   855
haftmann@33594
   856
lemma apfst_id [simp] :
haftmann@33594
   857
  "apfst id = id"
nipkow@39302
   858
  by (simp add: fun_eq_iff)
haftmann@33594
   859
haftmann@33594
   860
lemma apsnd_id [simp] :
haftmann@33594
   861
  "apsnd id = id"
nipkow@39302
   862
  by (simp add: fun_eq_iff)
haftmann@33594
   863
haftmann@33594
   864
lemma apfst_eq_conv [simp]:
haftmann@33594
   865
  "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
haftmann@33594
   866
  by (cases x) simp
haftmann@33594
   867
haftmann@33594
   868
lemma apsnd_eq_conv [simp]:
haftmann@33594
   869
  "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"
haftmann@33594
   870
  by (cases x) simp
haftmann@33594
   871
hoelzl@33638
   872
lemma apsnd_apfst_commute:
hoelzl@33638
   873
  "apsnd f (apfst g p) = apfst g (apsnd f p)"
hoelzl@33638
   874
  by simp
haftmann@21195
   875
haftmann@26358
   876
text {*
haftmann@26358
   877
  Disjoint union of a family of sets -- Sigma.
haftmann@26358
   878
*}
haftmann@26358
   879
haftmann@45986
   880
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where
haftmann@26358
   881
  Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
haftmann@26358
   882
haftmann@26358
   883
abbreviation
haftmann@45986
   884
  Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set"
haftmann@26358
   885
    (infixr "<*>" 80) where
haftmann@26358
   886
  "A <*> B == Sigma A (%_. B)"
haftmann@26358
   887
haftmann@26358
   888
notation (xsymbols)
haftmann@26358
   889
  Times  (infixr "\<times>" 80)
berghofe@15394
   890
haftmann@26358
   891
notation (HTML output)
haftmann@26358
   892
  Times  (infixr "\<times>" 80)
haftmann@26358
   893
nipkow@45662
   894
hide_const (open) Times
nipkow@45662
   895
haftmann@26358
   896
syntax
wenzelm@35115
   897
  "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
haftmann@26358
   898
translations
wenzelm@35115
   899
  "SIGMA x:A. B" == "CONST Sigma A (%x. B)"
haftmann@26358
   900
haftmann@26358
   901
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
haftmann@26358
   902
  by (unfold Sigma_def) blast
haftmann@26358
   903
haftmann@26358
   904
lemma SigmaE [elim!]:
haftmann@26358
   905
    "[| c: Sigma A B;
haftmann@26358
   906
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
haftmann@26358
   907
     |] ==> P"
haftmann@26358
   908
  -- {* The general elimination rule. *}
haftmann@26358
   909
  by (unfold Sigma_def) blast
haftmann@20588
   910
haftmann@26358
   911
text {*
haftmann@26358
   912
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
haftmann@26358
   913
  eigenvariables.
haftmann@26358
   914
*}
haftmann@26358
   915
haftmann@26358
   916
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
haftmann@26358
   917
  by blast
haftmann@26358
   918
haftmann@26358
   919
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
haftmann@26358
   920
  by blast
haftmann@26358
   921
haftmann@26358
   922
lemma SigmaE2:
haftmann@26358
   923
    "[| (a, b) : Sigma A B;
haftmann@26358
   924
        [| a:A;  b:B(a) |] ==> P
haftmann@26358
   925
     |] ==> P"
haftmann@26358
   926
  by blast
haftmann@20588
   927
haftmann@26358
   928
lemma Sigma_cong:
haftmann@26358
   929
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
haftmann@26358
   930
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
haftmann@26358
   931
  by auto
haftmann@26358
   932
haftmann@26358
   933
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
haftmann@26358
   934
  by blast
haftmann@26358
   935
haftmann@26358
   936
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
haftmann@26358
   937
  by blast
haftmann@26358
   938
haftmann@26358
   939
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
haftmann@26358
   940
  by blast
haftmann@26358
   941
haftmann@26358
   942
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
haftmann@26358
   943
  by auto
haftmann@21908
   944
haftmann@26358
   945
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
haftmann@26358
   946
  by auto
haftmann@26358
   947
haftmann@26358
   948
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
haftmann@26358
   949
  by auto
haftmann@26358
   950
haftmann@26358
   951
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
haftmann@26358
   952
  by blast
haftmann@26358
   953
haftmann@26358
   954
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
haftmann@26358
   955
  by blast
haftmann@26358
   956
haftmann@26358
   957
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
haftmann@26358
   958
  by (blast elim: equalityE)
haftmann@20588
   959
haftmann@26358
   960
lemma SetCompr_Sigma_eq:
haftmann@26358
   961
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
haftmann@26358
   962
  by blast
haftmann@26358
   963
haftmann@26358
   964
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
haftmann@26358
   965
  by blast
haftmann@26358
   966
haftmann@26358
   967
lemma UN_Times_distrib:
haftmann@26358
   968
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
haftmann@26358
   969
  -- {* Suggested by Pierre Chartier *}
haftmann@26358
   970
  by blast
haftmann@26358
   971
blanchet@35828
   972
lemma split_paired_Ball_Sigma [simp,no_atp]:
haftmann@26358
   973
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
haftmann@26358
   974
  by blast
haftmann@26358
   975
blanchet@35828
   976
lemma split_paired_Bex_Sigma [simp,no_atp]:
haftmann@26358
   977
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
haftmann@26358
   978
  by blast
haftmann@21908
   979
haftmann@26358
   980
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
haftmann@26358
   981
  by blast
haftmann@26358
   982
haftmann@26358
   983
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
   984
  by blast
haftmann@26358
   985
haftmann@26358
   986
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
haftmann@26358
   987
  by blast
haftmann@26358
   988
haftmann@26358
   989
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
   990
  by blast
haftmann@26358
   991
haftmann@26358
   992
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
haftmann@26358
   993
  by blast
haftmann@26358
   994
haftmann@26358
   995
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
haftmann@26358
   996
  by blast
haftmann@21908
   997
haftmann@26358
   998
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
haftmann@26358
   999
  by blast
haftmann@26358
  1000
haftmann@26358
  1001
text {*
haftmann@26358
  1002
  Non-dependent versions are needed to avoid the need for higher-order
haftmann@26358
  1003
  matching, especially when the rules are re-oriented.
haftmann@26358
  1004
*}
haftmann@21908
  1005
haftmann@26358
  1006
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
nipkow@28719
  1007
by blast
haftmann@26358
  1008
haftmann@26358
  1009
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
nipkow@28719
  1010
by blast
haftmann@26358
  1011
haftmann@26358
  1012
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
nipkow@28719
  1013
by blast
haftmann@26358
  1014
hoelzl@36622
  1015
lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
hoelzl@36622
  1016
  by auto
hoelzl@36622
  1017
hoelzl@36622
  1018
lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)"
huffman@44921
  1019
  by force
hoelzl@36622
  1020
hoelzl@36622
  1021
lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)"
huffman@44921
  1022
  by force
hoelzl@36622
  1023
nipkow@28719
  1024
lemma insert_times_insert[simp]:
nipkow@28719
  1025
  "insert a A \<times> insert b B =
nipkow@28719
  1026
   insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
nipkow@28719
  1027
by blast
haftmann@26358
  1028
paulson@33271
  1029
lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)"
haftmann@37166
  1030
  by (auto, case_tac "f x", auto)
paulson@33271
  1031
haftmann@35822
  1032
lemma swap_inj_on:
hoelzl@36622
  1033
  "inj_on (\<lambda>(i, j). (j, i)) A"
hoelzl@36622
  1034
  by (auto intro!: inj_onI)
haftmann@35822
  1035
haftmann@35822
  1036
lemma swap_product:
haftmann@35822
  1037
  "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
haftmann@35822
  1038
  by (simp add: split_def image_def) blast
haftmann@35822
  1039
hoelzl@36622
  1040
lemma image_split_eq_Sigma:
hoelzl@36622
  1041
  "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
haftmann@46128
  1042
proof (safe intro!: imageI)
hoelzl@36622
  1043
  fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
hoelzl@36622
  1044
  show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
hoelzl@36622
  1045
    using * eq[symmetric] by auto
hoelzl@36622
  1046
qed simp_all
haftmann@35822
  1047
haftmann@46128
  1048
definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
haftmann@46128
  1049
  [code_abbrev]: "product A B = A \<times> B"
haftmann@46128
  1050
haftmann@46128
  1051
hide_const (open) product
haftmann@46128
  1052
haftmann@46128
  1053
lemma member_product:
haftmann@46128
  1054
  "x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B"
haftmann@46128
  1055
  by (simp add: product_def)
haftmann@46128
  1056
haftmann@40607
  1057
text {* The following @{const map_pair} lemmas are due to Joachim Breitner: *}
haftmann@40607
  1058
haftmann@40607
  1059
lemma map_pair_inj_on:
haftmann@40607
  1060
  assumes "inj_on f A" and "inj_on g B"
haftmann@40607
  1061
  shows "inj_on (map_pair f g) (A \<times> B)"
haftmann@40607
  1062
proof (rule inj_onI)
haftmann@40607
  1063
  fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c"
haftmann@40607
  1064
  assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto
haftmann@40607
  1065
  assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto
haftmann@40607
  1066
  assume "map_pair f g x = map_pair f g y"
haftmann@40607
  1067
  hence "fst (map_pair f g x) = fst (map_pair f g y)" by (auto)
haftmann@40607
  1068
  hence "f (fst x) = f (fst y)" by (cases x,cases y,auto)
haftmann@40607
  1069
  with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A`
haftmann@40607
  1070
  have "fst x = fst y" by (auto dest:dest:inj_onD)
haftmann@40607
  1071
  moreover from `map_pair f g x = map_pair f g y`
haftmann@40607
  1072
  have "snd (map_pair f g x) = snd (map_pair f g y)" by (auto)
haftmann@40607
  1073
  hence "g (snd x) = g (snd y)" by (cases x,cases y,auto)
haftmann@40607
  1074
  with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B`
haftmann@40607
  1075
  have "snd x = snd y" by (auto dest:dest:inj_onD)
haftmann@40607
  1076
  ultimately show "x = y" by(rule prod_eqI)
haftmann@40607
  1077
qed
haftmann@40607
  1078
haftmann@40607
  1079
lemma map_pair_surj:
hoelzl@40702
  1080
  fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd"
haftmann@40607
  1081
  assumes "surj f" and "surj g"
haftmann@40607
  1082
  shows "surj (map_pair f g)"
haftmann@40607
  1083
unfolding surj_def
haftmann@40607
  1084
proof
haftmann@40607
  1085
  fix y :: "'b \<times> 'd"
haftmann@40607
  1086
  from `surj f` obtain a where "fst y = f a" by (auto elim:surjE)
haftmann@40607
  1087
  moreover
haftmann@40607
  1088
  from `surj g` obtain b where "snd y = g b" by (auto elim:surjE)
haftmann@40607
  1089
  ultimately have "(fst y, snd y) = map_pair f g (a,b)" by auto
haftmann@40607
  1090
  thus "\<exists>x. y = map_pair f g x" by auto
haftmann@40607
  1091
qed
haftmann@40607
  1092
haftmann@40607
  1093
lemma map_pair_surj_on:
haftmann@40607
  1094
  assumes "f ` A = A'" and "g ` B = B'"
haftmann@40607
  1095
  shows "map_pair f g ` (A \<times> B) = A' \<times> B'"
haftmann@40607
  1096
unfolding image_def
haftmann@40607
  1097
proof(rule set_eqI,rule iffI)
haftmann@40607
  1098
  fix x :: "'a \<times> 'c"
haftmann@40607
  1099
  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
  1100
  then obtain y where "y \<in> A \<times> B" and "x = map_pair f g y" by blast
haftmann@40607
  1101
  from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto
haftmann@40607
  1102
  moreover from `image g B = B'` and `y \<in> A \<times> B` have "g (snd y) \<in> B'" by auto
haftmann@40607
  1103
  ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto
haftmann@40607
  1104
  with `x = map_pair f g y` show "x \<in> A' \<times> B'" by (cases y, auto)
haftmann@40607
  1105
next
haftmann@40607
  1106
  fix x :: "'a \<times> 'c"
haftmann@40607
  1107
  assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto
haftmann@40607
  1108
  from `image f A = A'` and `fst x \<in> A'` have "fst x \<in> image f A" by auto
haftmann@40607
  1109
  then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE)
haftmann@40607
  1110
  moreover from `image g B = B'` and `snd x \<in> B'`
haftmann@40607
  1111
  obtain b where "b \<in> B" and "snd x = g b" by auto
haftmann@40607
  1112
  ultimately have "(fst x, snd x) = map_pair f g (a,b)" by auto
haftmann@40607
  1113
  moreover from `a \<in> A` and  `b \<in> B` have "(a , b) \<in> A \<times> B" by auto
haftmann@40607
  1114
  ultimately have "\<exists>y \<in> A \<times> B. x = map_pair f g y" by auto
haftmann@40607
  1115
  thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_pair f g y}" by auto
haftmann@40607
  1116
qed
haftmann@40607
  1117
haftmann@21908
  1118
haftmann@37166
  1119
subsection {* Inductively defined sets *}
berghofe@15394
  1120
haftmann@31723
  1121
use "Tools/inductive_set.ML"
haftmann@31723
  1122
setup Inductive_Set.setup
haftmann@24699
  1123
haftmann@37166
  1124
haftmann@37166
  1125
subsection {* Legacy theorem bindings and duplicates *}
haftmann@37166
  1126
haftmann@37166
  1127
lemma PairE:
haftmann@37166
  1128
  obtains x y where "p = (x, y)"
haftmann@37166
  1129
  by (fact prod.exhaust)
haftmann@37166
  1130
haftmann@37166
  1131
lemma Pair_inject:
haftmann@37166
  1132
  assumes "(a, b) = (a', b')"
haftmann@37166
  1133
    and "a = a' ==> b = b' ==> R"
haftmann@37166
  1134
  shows R
haftmann@37166
  1135
  using assms by simp
haftmann@37166
  1136
haftmann@37166
  1137
lemmas Pair_eq = prod.inject
haftmann@37166
  1138
haftmann@37166
  1139
lemmas split = split_conv  -- {* for backwards compatibility *}
haftmann@37166
  1140
huffman@44066
  1141
lemmas Pair_fst_snd_eq = prod_eq_iff
huffman@44066
  1142
huffman@45204
  1143
hide_const (open) prod
huffman@45204
  1144
nipkow@10213
  1145
end