src/HOL/Product_Type.thy
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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_printing
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  constant "HOL.equal :: bool \<Rightarrow> bool \<Rightarrow> bool" \<rightharpoonup> (Haskell) infix 4 "=="
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| class_instance "bool" :: "equal" \<rightharpoonup> (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_printing
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  type_constructor unit \<rightharpoonup>
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    (SML) "unit"
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    and (OCaml) "unit"
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    and (Haskell) "()"
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    and (Scala) "Unit"
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| constant Unity \<rightharpoonup>
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    (SML) "()"
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    and (OCaml) "()"
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    and (Haskell) "()"
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    and (Scala) "()"
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| class_instance unit :: equal \<rightharpoonup>
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    (Haskell) -
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| constant "HOL.equal :: unit \<Rightarrow> unit \<Rightarrow> bool" \<rightharpoonup>
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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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11777
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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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  "_pattern x y" => "CONST Pair x y"
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  "_patterns 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}, K 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}, K 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 tr ctxt 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 tr ctxt ts
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          else Syntax.const Q $ A $ s
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      | _ => tr ctxt ts);
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  in
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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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    |> map (fn (Q, tr) => (Q, contract Q tr))
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  end
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*}
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subsubsection {* Code generator setup *}
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code_printing
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  type_constructor prod \<rightharpoonup>
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    (SML) infix 2 "*"
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    and (OCaml) infix 2 "*"
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    and (Haskell) "!((_),/ (_))"
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    and (Scala) "((_),/ (_))"
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| constant Pair \<rightharpoonup>
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    (SML) "!((_),/ (_))"
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    and (OCaml) "!((_),/ (_))"
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    and (Haskell) "!((_),/ (_))"
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    and (Scala) "!((_),/ (_))"
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| class_instance  prod :: equal \<rightharpoonup>
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    (Haskell) -
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| constant "HOL.equal :: 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" \<rightharpoonup>
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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_printing
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  constant fst \<rightharpoonup> (Haskell) "fst"
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| constant snd \<rightharpoonup> (Haskell) "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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lemma snd_eqD: "snd (x, y) = a ==> y = a"
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  by simp
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lemma pair_collapse [simp]: "(fst p, snd p) = p"
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  by (cases p) simp
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lemmas surjective_pairing = pair_collapse [symmetric]
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lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
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  by (cases s, cases t) simp
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lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
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  by (simp add: prod_eq_iff)
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lemma split_conv [simp, code]: "split f (a, b) = f a b"
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  by (fact prod.cases)
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lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
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  by (rule split_conv [THEN iffD2])
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lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
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  by (rule split_conv [THEN iffD1])
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lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
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  by (simp add: fun_eq_iff split: prod.split)
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lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
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  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
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  by (simp add: fun_eq_iff split: prod.split)
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lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
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  by (cases x) simp
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lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
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  by (cases p) simp
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lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
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  by (simp add: prod_case_unfold)
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lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
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  -- {* Prevents simplification of @{term c}: much faster *}
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  by (fact prod.weak_case_cong)
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lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
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  by (simp add: split_eta)
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lemma split_paired_all [no_atp]: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
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proof
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  fix a b
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  assume "!!x. PROP P x"
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  then show "PROP P (a, b)" .
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next
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  fix x
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  assume "!!a b. PROP P (a, b)"
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  from `PROP P (fst x, snd x)` show "PROP P x" by simp
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qed
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   393
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lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))"
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  by (cases x) simp
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text {*
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  The rule @{thm [source] split_paired_all} does not work with the
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  Simplifier because it also affects premises in congrence rules,
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  where this can lead to premises of the form @{text "!!a b. ... =
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  ?P(a, b)"} which cannot be solved by reflexivity.
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*}
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lemmas split_tupled_all = split_paired_all unit_all_eq2
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ML {*
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  (* replace parameters of product type by individual component parameters *)
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  local (* filtering with exists_paired_all is an essential optimization *)
16121
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    fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
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          can HOLogic.dest_prodT T orelse exists_paired_all t
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      | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
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      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
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      | exists_paired_all _ = false;
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    val ss =
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      simpset_of
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       (put_simpset HOL_basic_ss @{context}
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        addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
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        addsimprocs [@{simproc unit_eq}]);
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  in
51717
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    fun split_all_tac ctxt = SUBGOAL (fn (t, i) =>
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      if exists_paired_all t then safe_full_simp_tac (put_simpset ss ctxt) i else no_tac);
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    fun unsafe_split_all_tac ctxt = SUBGOAL (fn (t, i) =>
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      if exists_paired_all t then full_simp_tac (put_simpset ss ctxt) i else no_tac);
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   426
    fun split_all ctxt th =
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   427
      if exists_paired_all (Thm.prop_of th)
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   428
      then full_simplify (put_simpset ss ctxt) th else th;
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   429
  end;
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*}
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setup {* map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac)) *}
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lemma split_paired_All [simp, no_atp]: "(ALL x. P x) = (ALL a b. P (a, b))"
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   435
  -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
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   436
  by fast
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   437
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diff changeset
   438
lemma split_paired_Ex [simp, no_atp]: "(EX x. P x) = (EX a b. P (a, b))"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   439
  by fast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   440
47740
a8989fe9a3a5 added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents: 46950
diff changeset
   441
lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))"
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   442
  -- {* Can't be added to simpset: loops! *}
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   443
  by (simp add: split_eta)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   444
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   445
text {*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   446
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   447
  @{thm [source] split_eta} as a rewrite rule is not general enough,
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   448
  and using @{thm [source] cond_split_eta} directly would render some
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   449
  existing proofs very inefficient; similarly for @{text
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   450
  split_beta}.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   451
*}
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   452
26480
544cef16045b replaced 'ML_setup' by 'ML';
wenzelm
parents: 26358
diff changeset
   453
ML {*
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   454
local
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   455
  val cond_split_eta_ss =
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   456
    simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms cond_split_eta});
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   457
  fun Pair_pat k 0 (Bound m) = (m = k)
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   458
    | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   459
        i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   460
    | Pair_pat _ _ _ = false;
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   461
  fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   462
    | no_args k i (t $ u) = no_args k i t andalso no_args k i u
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   463
    | no_args k i (Bound m) = m < k orelse m > k + i
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   464
    | no_args _ _ _ = true;
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   465
  fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   466
    | split_pat tp i (Const (@{const_name prod_case}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   467
    | split_pat tp i _ = NONE;
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   468
  fun metaeq ctxt lhs rhs = mk_meta_eq (Goal.prove ctxt [] []
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   469
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   470
        (K (simp_tac (put_simpset cond_split_eta_ss ctxt) 1)));
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   471
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   472
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   473
    | beta_term_pat k i (t $ u) =
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   474
        Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   475
    | beta_term_pat k i t = no_args k i t;
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   476
  fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   477
    | eta_term_pat _ _ _ = false;
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   478
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   479
    | subst arg k i (t $ u) =
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   480
        if Pair_pat k i (t $ u) then incr_boundvars k arg
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   481
        else (subst arg k i t $ subst arg k i u)
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   482
    | subst arg k i t = t;
43595
7ae4a23b5be6 modernized some simproc setup;
wenzelm
parents: 43594
diff changeset
   483
in
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   484
  fun beta_proc ctxt (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t) $ arg) =
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   485
        (case split_pat beta_term_pat 1 t of
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   486
          SOME (i, f) => SOME (metaeq ctxt s (subst arg 0 i f))
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15481
diff changeset
   487
        | NONE => NONE)
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   488
    | beta_proc _ _ = NONE;
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   489
  fun eta_proc ctxt (s as Const (@{const_name prod_case}, _) $ Abs (_, _, t)) =
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   490
        (case split_pat eta_term_pat 1 t of
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   491
          SOME (_, ft) => SOME (metaeq ctxt s (let val (f $ arg) = ft in f end))
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15481
diff changeset
   492
        | NONE => NONE)
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   493
    | eta_proc _ _ = NONE;
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   494
end;
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   495
*}
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   496
simproc_setup split_beta ("split f z") = {* fn _ => fn ctxt => fn ct => beta_proc ctxt (term_of ct) *}
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   497
simproc_setup split_eta ("split f") = {* fn _ => fn ctxt => fn ct => eta_proc ctxt (term_of ct) *}
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   498
26798
a9134a089106 split_beta is now declared as monotonicity rule, to allow bounded
berghofe
parents: 26588
diff changeset
   499
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)"
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   500
  by (subst surjective_pairing, rule split_conv)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   501
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
   502
lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
   503
  by (auto simp: fun_eq_iff)
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
   504
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
   505
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35427
diff changeset
   506
lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   507
  -- {* For use with @{text split} and the Simplifier. *}
15481
fc075ae929e4 the new subst tactic, by Lucas Dixon
paulson
parents: 15422
diff changeset
   508
  by (insert surj_pair [of p], clarify, simp)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   509
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   510
text {*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   511
  @{thm [source] split_split} could be declared as @{text "[split]"}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   512
  done after the Splitter has been speeded up significantly;
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   513
  precompute the constants involved and don't do anything unless the
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   514
  current goal contains one of those constants.
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   515
*}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   516
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35427
diff changeset
   517
lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
14208
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   518
by (subst split_split, simp)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   519
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   520
text {*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   521
  \medskip @{term split} used as a logical connective or set former.
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   522
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   523
  \medskip These rules are for use with @{text blast}; could instead
40929
7ff03a5e044f theorem names generated by the (rep_)datatype command now have mandatory qualifiers
huffman
parents: 40702
diff changeset
   524
  call @{text simp} using @{thm [source] prod.split} as rewrite. *}
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   525
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   526
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   527
  apply (simp only: split_tupled_all)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   528
  apply (simp (no_asm_simp))
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   529
  done
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   530
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   531
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   532
  apply (simp only: split_tupled_all)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   533
  apply (simp (no_asm_simp))
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   534
  done
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   535
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   536
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   537
  by (induct p) auto
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   538
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   539
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   540
  by (induct p) auto
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   541
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   542
lemma splitE2:
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   543
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   544
proof -
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   545
  assume q: "Q (split P z)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   546
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   547
  show R
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   548
    apply (rule r surjective_pairing)+
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   549
    apply (rule split_beta [THEN subst], rule q)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   550
    done
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   551
qed
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   552
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   553
lemma splitD': "split R (a,b) c ==> R a b c"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   554
  by simp
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   555
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   556
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   557
  by simp
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   558
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   559
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
14208
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   560
by (simp only: split_tupled_all, simp)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   561
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   562
lemma mem_splitE:
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   563
  assumes major: "z \<in> split c p"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   564
    and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   565
  shows Q
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   566
  by (rule major [unfolded prod_case_unfold] cases surjective_pairing)+
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   567
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   568
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   569
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   570
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   571
ML {*
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   572
local (* filtering with exists_p_split is an essential optimization *)
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   573
  fun exists_p_split (Const (@{const_name prod_case},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   574
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   575
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   576
    | exists_p_split _ = false;
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   577
in
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   578
fun split_conv_tac ctxt = SUBGOAL (fn (t, i) =>
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   579
  if exists_p_split t
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   580
  then safe_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms split_conv}) i
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   581
  else no_tac);
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   582
end;
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   583
*}
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   584
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   585
(* This prevents applications of splitE for already splitted arguments leading
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   586
   to quite time-consuming computations (in particular for nested tuples) *)
51703
f2e92fc0c8aa modifiers for classical wrappers operate on Proof.context instead of claset;
wenzelm
parents: 51392
diff changeset
   587
setup {* map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac)) *}
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   588
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35427
diff changeset
   589
lemma split_eta_SetCompr [simp,no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   590
  by (rule ext) fast
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   591
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35427
diff changeset
   592
lemma split_eta_SetCompr2 [simp,no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   593
  by (rule ext) fast
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   594
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   595
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   596
  -- {* Allows simplifications of nested splits in case of independent predicates. *}
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   597
  by (rule ext) blast
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   598
14337
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   599
(* Do NOT make this a simp rule as it
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   600
   a) only helps in special situations
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   601
   b) can lead to nontermination in the presence of split_def
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   602
*)
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   603
lemma split_comp_eq: 
20415
e3d2d7b01279 explicit type variables prevent empty sorts
paulson
parents: 20380
diff changeset
   604
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
e3d2d7b01279 explicit type variables prevent empty sorts
paulson
parents: 20380
diff changeset
   605
  shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   606
  by (rule ext) auto
14101
d25c23e46173 added upd_fst, upd_snd, some thms
oheimb
parents: 13480
diff changeset
   607
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   608
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   609
  apply (rule_tac x = "(a, b)" in image_eqI)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   610
   apply auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   611
  done
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   612
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   613
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   614
  by blast
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   615
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   616
(*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   617
the following  would be slightly more general,
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   618
but cannot be used as rewrite rule:
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   619
### Cannot add premise as rewrite rule because it contains (type) unknowns:
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   620
### ?y = .x
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   621
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
14208
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   622
by (rtac some_equality 1)
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   623
by ( Simp_tac 1)
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   624
by (split_all_tac 1)
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   625
by (Asm_full_simp_tac 1)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   626
qed "The_split_eq";
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   627
*)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   628
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   629
text {*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   630
  Setup of internal @{text split_rule}.
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   631
*}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   632
45607
16b4f5774621 eliminated obsolete "standard";
wenzelm
parents: 45205
diff changeset
   633
lemmas prod_caseI = prod.cases [THEN iffD2]
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   634
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   635
lemma prod_caseI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> prod_case c p"
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   636
  by (fact splitI2)
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   637
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   638
lemma prod_caseI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> prod_case c p x"
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   639
  by (fact splitI2')
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   640
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   641
lemma prod_caseE: "prod_case c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   642
  by (fact splitE)
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   643
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   644
lemma prod_caseE': "prod_case c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   645
  by (fact splitE')
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   646
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   647
declare prod_caseI [intro!]
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   648
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 25885
diff changeset
   649
lemma prod_case_beta:
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 25885
diff changeset
   650
  "prod_case f p = f (fst p) (snd p)"
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   651
  by (fact split_beta)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 25885
diff changeset
   652
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   653
lemma prod_cases3 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   654
  obtains (fields) a b c where "y = (a, b, c)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   655
  by (cases y, case_tac b) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   656
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   657
lemma prod_induct3 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   658
    "(!!a b c. P (a, b, c)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   659
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   660
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   661
lemma prod_cases4 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   662
  obtains (fields) a b c d where "y = (a, b, c, d)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   663
  by (cases y, case_tac c) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   664
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   665
lemma prod_induct4 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   666
    "(!!a b c d. P (a, b, c, d)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   667
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   668
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   669
lemma prod_cases5 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   670
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   671
  by (cases y, case_tac d) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   672
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   673
lemma prod_induct5 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   674
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   675
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   676
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   677
lemma prod_cases6 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   678
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   679
  by (cases y, case_tac e) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   680
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   681
lemma prod_induct6 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   682
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   683
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   684
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   685
lemma prod_cases7 [cases type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   686
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   687
  by (cases y, case_tac f) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   688
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   689
lemma prod_induct7 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   690
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   691
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   692
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   693
lemma split_def:
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   694
  "split = (\<lambda>c p. c (fst p) (snd p))"
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   695
  by (fact prod_case_unfold)
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   696
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   697
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   698
  "internal_split == split"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   699
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   700
lemma internal_split_conv: "internal_split c (a, b) = c a b"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   701
  by (simp only: internal_split_def split_conv)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   702
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47988
diff changeset
   703
ML_file "Tools/split_rule.ML"
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   704
setup Split_Rule.setup
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   705
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   706
hide_const internal_split
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   707
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   708
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   709
subsubsection {* Derived operations *}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   710
37387
3581483cca6c qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents: 37278
diff changeset
   711
definition curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where
3581483cca6c qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents: 37278
diff changeset
   712
  "curry = (\<lambda>c x y. c (x, y))"
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   713
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   714
lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   715
  by (simp add: curry_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   716
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   717
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   718
  by (simp add: curry_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   719
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   720
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   721
  by (simp add: curry_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   722
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   723
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   724
  by (simp add: curry_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   725
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   726
lemma curry_split [simp]: "curry (split f) = f"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   727
  by (simp add: curry_def split_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   728
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   729
lemma split_curry [simp]: "split (curry f) = f"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   730
  by (simp add: curry_def split_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   731
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   732
text {*
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   733
  The composition-uncurry combinator.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   734
*}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   735
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   736
notation fcomp (infixl "\<circ>>" 60)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   737
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   738
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   739
  "f \<circ>\<rightarrow> g = (\<lambda>x. prod_case g (f x))"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   740
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   741
lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   742
  by (simp add: fun_eq_iff scomp_def prod_case_unfold)
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   743
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   744
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = prod_case g (f x)"
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   745
  by (simp add: scomp_unfold prod_case_unfold)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   746
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   747
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
   748
  by (simp add: fun_eq_iff)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   749
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   750
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
   751
  by (simp add: fun_eq_iff)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   752
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   753
lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   754
  by (simp add: fun_eq_iff scomp_unfold)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   755
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   756
lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   757
  by (simp add: fun_eq_iff scomp_unfold fcomp_def)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   758
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   759
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
   760
  by (simp add: fun_eq_iff scomp_unfold)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   761
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 52143
diff changeset
   762
code_printing
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 52143
diff changeset
   763
  constant scomp \<rightharpoonup> (Eval) infixl 3 "#->"
31202
52d332f8f909 pretty printing of functional combinators for evaluation code
haftmann
parents: 30924
diff changeset
   764
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   765
no_notation fcomp (infixl "\<circ>>" 60)
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   766
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   767
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   768
text {*
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   769
  @{term map_pair} --- action of the product functor upon
36664
6302f9ad7047 repaired comments where SOMEthing went utterly wrong (cf. 2b04504fcb69)
krauss
parents: 36622
diff changeset
   770
  functions.
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   771
*}
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   772
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   773
definition map_pair :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   774
  "map_pair f g = (\<lambda>(x, y). (f x, g y))"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   775
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   776
lemma map_pair_simp [simp, code]:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   777
  "map_pair f g (a, b) = (f a, g b)"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   778
  by (simp add: map_pair_def)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   779
41505
6d19301074cf "enriched_type" replaces less specific "type_lifting"
haftmann
parents: 41372
diff changeset
   780
enriched_type map_pair: map_pair
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
   781
  by (auto simp add: split_paired_all)
37278
307845cc7f51 added lemmas
nipkow
parents: 37166
diff changeset
   782
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   783
lemma fst_map_pair [simp]:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   784
  "fst (map_pair f g x) = f (fst x)"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   785
  by (cases x) simp_all
37278
307845cc7f51 added lemmas
nipkow
parents: 37166
diff changeset
   786
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   787
lemma snd_prod_fun [simp]:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   788
  "snd (map_pair f g x) = g (snd x)"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   789
  by (cases x) simp_all
37278
307845cc7f51 added lemmas
nipkow
parents: 37166
diff changeset
   790
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   791
lemma fst_comp_map_pair [simp]:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   792
  "fst \<circ> map_pair f g = f \<circ> fst"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   793
  by (rule ext) simp_all
37278
307845cc7f51 added lemmas
nipkow
parents: 37166
diff changeset
   794
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   795
lemma snd_comp_map_pair [simp]:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   796
  "snd \<circ> map_pair f g = g \<circ> snd"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   797
  by (rule ext) simp_all
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   798
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   799
lemma map_pair_compose:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   800
  "map_pair (f1 o f2) (g1 o g2) = (map_pair f1 g1 o map_pair f2 g2)"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   801
  by (rule ext) (simp add: map_pair.compositionality comp_def)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   802
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   803
lemma map_pair_ident [simp]:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   804
  "map_pair (%x. x) (%y. y) = (%z. z)"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   805
  by (rule ext) (simp add: map_pair.identity)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   806
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   807
lemma map_pair_imageI [intro]:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   808
  "(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_pair f g ` R"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   809
  by (rule image_eqI) simp_all
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   810
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   811
lemma prod_fun_imageE [elim!]:
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   812
  assumes major: "c \<in> map_pair f g ` R"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   813
    and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   814
  shows P
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   815
  apply (rule major [THEN imageE])
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   816
  apply (case_tac x)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   817
  apply (rule cases)
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   818
  apply simp_all
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   819
  done
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   820
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   821
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   822
  "apfst f = map_pair f id"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   823
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   824
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   825
  "apsnd f = map_pair id f"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   826
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   827
lemma apfst_conv [simp, code]:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   828
  "apfst f (x, y) = (f x, y)" 
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   829
  by (simp add: apfst_def)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   830
33638
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   831
lemma apsnd_conv [simp, code]:
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   832
  "apsnd f (x, y) = (x, f y)" 
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   833
  by (simp add: apsnd_def)
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   834
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   835
lemma fst_apfst [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   836
  "fst (apfst f x) = f (fst x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   837
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   838
51173
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   839
lemma fst_comp_apfst [simp]:
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   840
  "fst \<circ> apfst f = f \<circ> fst"
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   841
  by (simp add: fun_eq_iff)
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   842
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   843
lemma fst_apsnd [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   844
  "fst (apsnd f x) = fst x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   845
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   846
51173
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   847
lemma fst_comp_apsnd [simp]:
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   848
  "fst \<circ> apsnd f = fst"
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   849
  by (simp add: fun_eq_iff)
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   850
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   851
lemma snd_apfst [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   852
  "snd (apfst f x) = snd x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   853
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   854
51173
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   855
lemma snd_comp_apfst [simp]:
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   856
  "snd \<circ> apfst f = snd"
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   857
  by (simp add: fun_eq_iff)
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   858
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   859
lemma snd_apsnd [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   860
  "snd (apsnd f x) = f (snd x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   861
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   862
51173
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   863
lemma snd_comp_apsnd [simp]:
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   864
  "snd \<circ> apsnd f = f \<circ> snd"
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   865
  by (simp add: fun_eq_iff)
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   866
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   867
lemma apfst_compose:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   868
  "apfst f (apfst g x) = apfst (f \<circ> g) x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   869
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   870
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   871
lemma apsnd_compose:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   872
  "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   873
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   874
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   875
lemma apfst_apsnd [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   876
  "apfst f (apsnd g x) = (f (fst x), g (snd x))"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   877
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   878
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   879
lemma apsnd_apfst [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   880
  "apsnd f (apfst g x) = (g (fst x), f (snd x))"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   881
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   882
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   883
lemma apfst_id [simp] :
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   884
  "apfst id = id"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   885
  by (simp add: fun_eq_iff)
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   886
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   887
lemma apsnd_id [simp] :
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   888
  "apsnd id = id"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   889
  by (simp add: fun_eq_iff)
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   890
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   891
lemma apfst_eq_conv [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   892
  "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   893
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   894
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   895
lemma apsnd_eq_conv [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   896
  "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   897
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   898
33638
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   899
lemma apsnd_apfst_commute:
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   900
  "apsnd f (apfst g p) = apfst g (apsnd f p)"
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   901
  by simp
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   902
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   903
text {*
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   904
  Disjoint union of a family of sets -- Sigma.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   905
*}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   906
45986
c9e50153e5ae moved various set operations to theory Set (resp. Product_Type)
haftmann
parents: 45696
diff changeset
   907
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   908
  Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   909
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   910
abbreviation
45986
c9e50153e5ae moved various set operations to theory Set (resp. Product_Type)
haftmann
parents: 45696
diff changeset
   911
  Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   912
    (infixr "<*>" 80) where
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   913
  "A <*> B == Sigma A (%_. B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   914
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   915
notation (xsymbols)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   916
  Times  (infixr "\<times>" 80)
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
   917
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   918
notation (HTML output)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   919
  Times  (infixr "\<times>" 80)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   920
45662
4f7c05990420 Hide Product_Type.Times - too precious an identifier
nipkow
parents: 45607
diff changeset
   921
hide_const (open) Times
4f7c05990420 Hide Product_Type.Times - too precious an identifier
nipkow
parents: 45607
diff changeset
   922
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   923
syntax
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34900
diff changeset
   924
  "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   925
translations
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34900
diff changeset
   926
  "SIGMA x:A. B" == "CONST Sigma A (%x. B)"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   927
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   928
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   929
  by (unfold Sigma_def) blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   930
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   931
lemma SigmaE [elim!]:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   932
    "[| c: Sigma A B;
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   933
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   934
     |] ==> P"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   935
  -- {* The general elimination rule. *}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   936
  by (unfold Sigma_def) blast
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
   937
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   938
text {*
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   939
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   940
  eigenvariables.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   941
*}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   942
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   943
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   944
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   945
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   946
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   947
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   948
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   949
lemma SigmaE2:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   950
    "[| (a, b) : Sigma A B;
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   951
        [| a:A;  b:B(a) |] ==> P
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   952
     |] ==> P"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   953
  by blast
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
   954
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   955
lemma Sigma_cong:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   956
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   957
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   958
  by auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   959
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   960
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   961
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   962
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   963
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   964
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   965
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   966
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   967
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   968
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   969
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   970
  by auto
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
   971
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   972
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   973
  by auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   974
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   975
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   976
  by auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   977
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   978
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   979
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   980
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   981
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   982
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   983
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   984
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   985
  by (blast elim: equalityE)
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
   986
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   987
lemma SetCompr_Sigma_eq:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   988
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   989
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   990
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   991
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   992
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   993
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   994
lemma UN_Times_distrib:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   995
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   996
  -- {* Suggested by Pierre Chartier *}
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   997
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   998
47740
a8989fe9a3a5 added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents: 46950
diff changeset
   999
lemma split_paired_Ball_Sigma [simp, no_atp]:
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1000
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1001
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1002
47740
a8989fe9a3a5 added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents: 46950
diff changeset
  1003
lemma split_paired_Bex_Sigma [simp, no_atp]:
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1004
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1005
  by blast
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1006
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1007
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1008
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1009
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1010
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1011
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1012
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1013
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1014
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1015
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1016
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1017
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1018
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1019
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1020
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1021
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1022
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1023
  by blast
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1024
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1025
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1026
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1027
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1028
text {*
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1029
  Non-dependent versions are needed to avoid the need for higher-order
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1030
  matching, especially when the rules are re-oriented.
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1031
*}
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1032
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1033
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
28719
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1034
by blast
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1035
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1036
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
28719
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1037
by blast
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1038
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1039
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
28719
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1040
by blast
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1041
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1042
lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1043
  by auto
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1044
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
  1045
lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
  1046
  by auto
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
  1047
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1048
lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
  1049
  by force
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1050
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1051
lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
  1052
  by force
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1053
28719
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1054
lemma insert_times_insert[simp]:
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1055
  "insert a A \<times> insert b B =
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1056
   insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1057
by blast
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1058
33271
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 33089
diff changeset
  1059
lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)"
47988
e4b69e10b990 tuned proofs;
wenzelm
parents: 47740
diff changeset
  1060
  apply auto
e4b69e10b990 tuned proofs;
wenzelm
parents: 47740
diff changeset
  1061
  apply (case_tac "f x")
e4b69e10b990 tuned proofs;
wenzelm
parents: 47740
diff changeset
  1062
  apply auto
e4b69e10b990 tuned proofs;
wenzelm
parents: 47740
diff changeset
  1063
  done
33271
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 33089
diff changeset
  1064
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
  1065
lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
  1066
  by auto
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
  1067
35822
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1068
lemma swap_inj_on:
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1069
  "inj_on (\<lambda>(i, j). (j, i)) A"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1070
  by (auto intro!: inj_onI)
35822
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1071
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1072
lemma swap_product:
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1073
  "(%(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1074
  by (simp add: split_def image_def) blast
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1075
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1076
lemma image_split_eq_Sigma:
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1077
  "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
46128
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1078
proof (safe intro!: imageI)
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1079
  fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1080
  show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1081
    using * eq[symmetric] by auto
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1082
qed simp_all
35822
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1083
46128
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1084
definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1085
  [code_abbrev]: "product A B = A \<times> B"
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1086
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1087
hide_const (open) product
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1088
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1089
lemma member_product:
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1090
  "x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B"
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1091
  by (simp add: product_def)
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1092
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1093
text {* The following @{const map_pair} lemmas are due to Joachim Breitner: *}
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1094
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1095
lemma map_pair_inj_on:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1096
  assumes "inj_on f A" and "inj_on g B"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1097
  shows "inj_on (map_pair f g) (A \<times> B)"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1098
proof (rule inj_onI)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1099
  fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1100
  assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1101
  assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1102
  assume "map_pair f g x = map_pair f g y"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1103
  hence "fst (map_pair f g x) = fst (map_pair f g y)" by (auto)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1104
  hence "f (fst x) = f (fst y)" by (cases x,cases y,auto)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1105
  with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A`
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1106
  have "fst x = fst y" by (auto dest:dest:inj_onD)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1107
  moreover from `map_pair f g x = map_pair f g y`
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1108
  have "snd (map_pair f g x) = snd (map_pair f g y)" by (auto)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1109
  hence "g (snd x) = g (snd y)" by (cases x,cases y,auto)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1110
  with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B`
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1111
  have "snd x = snd y" by (auto dest:dest:inj_onD)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1112
  ultimately show "x = y" by(rule prod_eqI)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1113
qed
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1114
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1115
lemma map_pair_surj:
40702
cf26dd7395e4 Replace surj by abbreviation; remove surj_on.
hoelzl
parents: 40607
diff changeset
  1116
  fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd"
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1117
  assumes "surj f" and "surj g"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1118
  shows "surj (map_pair f g)"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1119
unfolding surj_def
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1120
proof
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1121
  fix y :: "'b \<times> 'd"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1122
  from `surj f` obtain a where "fst y = f a" by (auto elim:surjE)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1123
  moreover
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1124
  from `surj g` obtain b where "snd y = g b" by (auto elim:surjE)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1125
  ultimately have "(fst y, snd y) = map_pair f g (a,b)" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1126
  thus "\<exists>x. y = map_pair f g x" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1127
qed
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1128
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1129
lemma map_pair_surj_on:
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1130
  assumes "f ` A = A'" and "g ` B = B'"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1131
  shows "map_pair f g ` (A \<times> B) = A' \<times> B'"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1132
unfolding image_def
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1133
proof(rule set_eqI,rule iffI)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1134
  fix x :: "'a \<times> 'c"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1135
  assume "x \<in> {y\<Colon>'a \<times> 'c. \<exists>x\<Colon>'b \<times> 'd\<in>A \<times> B. y = map_pair f g x}"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1136
  then obtain y where "y \<in> A \<times> B" and "x = map_pair f g y" by blast
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1137
  from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1138
  moreover from `image g B = B'` and `y \<in> A \<times> B` have "g (snd y) \<in> B'" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1139
  ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1140
  with `x = map_pair f g y` show "x \<in> A' \<times> B'" by (cases y, auto)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1141
next
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1142
  fix x :: "'a \<times> 'c"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1143
  assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1144
  from `image f A = A'` and `fst x \<in> A'` have "fst x \<in> image f A" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1145
  then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1146
  moreover from `image g B = B'` and `snd x \<in> B'`
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1147
  obtain b where "b \<in> B" and "snd x = g b" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1148
  ultimately have "(fst x, snd x) = map_pair f g (a,b)" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1149
  moreover from `a \<in> A` and  `b \<in> B` have "(a , b) \<in> A \<times> B" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1150
  ultimately have "\<exists>y \<in> A \<times> B. x = map_pair f g y" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1151
  thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_pair f g y}" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1152
qed
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1153
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1154
49764
9979d64b8016 moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents: 48891
diff changeset
  1155
subsection {* Simproc for rewriting a set comprehension into a pointfree expression *}
9979d64b8016 moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents: 48891
diff changeset
  1156
9979d64b8016 moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents: 48891
diff changeset
  1157
ML_file "Tools/set_comprehension_pointfree.ML"
9979d64b8016 moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents: 48891
diff changeset
  1158
9979d64b8016 moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents: 48891
diff changeset
  1159
setup {*
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
  1160
  Code_Preproc.map_pre (fn ctxt => ctxt addsimprocs
49764
9979d64b8016 moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents: 48891
diff changeset
  1161
    [Raw_Simplifier.make_simproc {name = "set comprehension", lhss = [@{cpat "Collect ?P"}],
9979d64b8016 moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents: 48891
diff changeset
  1162
    proc = K Set_Comprehension_Pointfree.code_simproc, identifier = []}])
9979d64b8016 moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents: 48891
diff changeset
  1163
*}
9979d64b8016 moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents: 48891
diff changeset
  1164
9979d64b8016 moving simproc from Finite_Set to more appropriate Product_Type theory
bulwahn
parents: 48891
diff changeset
  1165
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1166
subsection {* Inductively defined sets *}
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1167
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47988
diff changeset
  1168
ML_file "Tools/inductive_set.ML"
31723
f5cafe803b55 discontinued ancient tradition to suffix certain ML module names with "_package"
haftmann
parents: 31667
diff changeset
  1169
setup Inductive_Set.setup
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
  1170
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1171
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1172
subsection {* Legacy theorem bindings and duplicates *}
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1173
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1174
lemma PairE:
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1175
  obtains x y where "p = (x, y)"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1176
  by (fact prod.exhaust)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1177
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1178
lemmas Pair_eq = prod.inject
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1179
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1180
lemmas split = split_conv  -- {* for backwards compatibility *}
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
  1181
44066
d74182c93f04 rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents: 43866
diff changeset
  1182
lemmas Pair_fst_snd_eq = prod_eq_iff
d74182c93f04 rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents: 43866
diff changeset
  1183
45204
5e4a1270c000 hide typedef-generated constants Product_Type.prod and Sum_Type.sum
huffman
parents: 44921
diff changeset
  1184
hide_const (open) prod
5e4a1270c000 hide typedef-generated constants Product_Type.prod and Sum_Type.sum
huffman
parents: 44921
diff changeset
  1185
10213
01c2744a3786 *** empty log message ***
nipkow
parents:
diff changeset
  1186
end