src/HOL/Product_Type.thy
author haftmann
Tue Oct 13 09:21:15 2015 +0200 (2015-10-13)
changeset 61424 c3658c18b7bc
parent 61422 0dfcd0fb4172
child 61425 fb95d06fb21f
permissions -rw-r--r--
prod_case as canonical name for product type eliminator
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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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section \<open>Cartesian products\<close>
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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 \<open>@{typ bool} is a datatype\<close>
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free_constructors case_bool for True | False
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  by auto
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text \<open>Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}.\<close>
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setup \<open>Sign.mandatory_path "old"\<close>
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old_rep_datatype True False by (auto intro: bool_induct)
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setup \<open>Sign.parent_path\<close>
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text \<open>But erase the prefix for properties that are not generated by @{text free_constructors}.\<close>
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setup \<open>Sign.mandatory_path "bool"\<close>
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lemmas induct = old.bool.induct
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lemmas inducts = old.bool.inducts
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lemmas rec = old.bool.rec
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lemmas simps = bool.distinct bool.case bool.rec
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setup \<open>Sign.parent_path\<close>
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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 \<open>Code.add_case @{thm If_case_cert}\<close>
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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 \<open>The @{text unit} type\<close>
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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 \<open>
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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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\<close>
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simproc_setup unit_eq ("x::unit") = \<open>
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  fn _ => fn _ => fn ct =>
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    if HOLogic.is_unit (Thm.term_of ct) then NONE
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    else SOME (mk_meta_eq @{thm unit_eq})
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\<close>
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free_constructors case_unit for "()"
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  by auto
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text \<open>Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}.\<close>
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setup \<open>Sign.mandatory_path "old"\<close>
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old_rep_datatype "()" by simp
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setup \<open>Sign.parent_path\<close>
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text \<open>But erase the prefix for properties that are not generated by @{text free_constructors}.\<close>
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setup \<open>Sign.mandatory_path "unit"\<close>
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lemmas induct = old.unit.induct
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lemmas inducts = old.unit.inducts
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lemmas rec = old.unit.rec
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lemmas simps = unit.case unit.rec
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setup \<open>Sign.parent_path\<close>
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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 \<open>
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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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\<close>
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lemma unit_abs_eta_conv [simp]: "(%u::unit. f ()) = f"
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  by (rule ext) simp
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lemma UNIV_unit:
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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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instantiation unit :: "{complete_boolean_algebra, complete_linorder, wellorder}"
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begin
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definition less_eq_unit :: "unit \<Rightarrow> unit \<Rightarrow> bool"
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where
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  "(_::unit) \<le> _ \<longleftrightarrow> True"
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lemma less_eq_unit [iff]:
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  "(u::unit) \<le> v"
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  by (simp add: less_eq_unit_def)
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definition less_unit :: "unit \<Rightarrow> unit \<Rightarrow> bool"
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where
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  "(_::unit) < _ \<longleftrightarrow> False"
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lemma less_unit [iff]:
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  "\<not> (u::unit) < v"
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  by (simp_all add: less_eq_unit_def less_unit_def)
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definition bot_unit :: unit
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where
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  [code_unfold]: "\<bottom> = ()"
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definition top_unit :: unit
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where
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  [code_unfold]: "\<top> = ()"
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definition inf_unit :: "unit \<Rightarrow> unit \<Rightarrow> unit"
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where
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  [simp]: "_ \<sqinter> _ = ()"
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definition sup_unit :: "unit \<Rightarrow> unit \<Rightarrow> unit"
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where
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  [simp]: "_ \<squnion> _ = ()"
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definition Inf_unit :: "unit set \<Rightarrow> unit"
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where
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  [simp]: "\<Sqinter>_ = ()"
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definition Sup_unit :: "unit set \<Rightarrow> unit"
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where
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  [simp]: "\<Squnion>_ = ()"
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definition uminus_unit :: "unit \<Rightarrow> unit"
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where
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  [simp]: "- _ = ()"
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declare less_eq_unit_def [abs_def, code_unfold]
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  less_unit_def [abs_def, code_unfold]
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  inf_unit_def [abs_def, code_unfold]
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  sup_unit_def [abs_def, code_unfold]
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  Inf_unit_def [abs_def, code_unfold]
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  Sup_unit_def [abs_def, code_unfold]
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  uminus_unit_def [abs_def, code_unfold]
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instance
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  by intro_classes auto
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end
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lemma [code]:
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  "HOL.equal (u::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 \<open>The product type\<close>
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subsubsection \<open>Type definition\<close>
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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::'a) (b::'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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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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lemma prod_cases: "(\<And>a b. P (Pair a b)) \<Longrightarrow> P p"
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  by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def)
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free_constructors case_prod for Pair fst snd
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proof -
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  fix P :: bool and p :: "'a \<times> 'b"
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  show "(\<And>x1 x2. p = Pair x1 x2 \<Longrightarrow> P) \<Longrightarrow> P"
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    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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text \<open>Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}.\<close>
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setup \<open>Sign.mandatory_path "old"\<close>
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old_rep_datatype Pair
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by (erule prod_cases) (rule prod.inject)
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setup \<open>Sign.parent_path\<close>
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text \<open>But erase the prefix for properties that are not generated by @{text free_constructors}.\<close>
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setup \<open>Sign.mandatory_path "prod"\<close>
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declare old.prod.inject [iff del]
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lemmas induct = old.prod.induct
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lemmas inducts = old.prod.inducts
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lemmas rec = old.prod.rec
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lemmas simps = prod.inject prod.case prod.rec
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setup \<open>Sign.parent_path\<close>
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declare prod.case [nitpick_simp del]
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declare prod.case_cong_weak [cong del]
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declare prod.case_eq_if [mono]
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declare prod.split [no_atp]
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declare prod.split_asm [no_atp]
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text \<open>
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  @{thm [source] prod.split} could be declared as @{text "[split]"}
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  done after the Splitter has been speeded up significantly;
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  precompute the constants involved and don't do anything unless the
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  current goal contains one of those constants.
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\<close>
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subsubsection \<open>Tuple syntax\<close>
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text \<open>
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  Patterns -- extends pre-defined type @{typ pttrn} used in
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  abstractions.
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\<close>
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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)" \<rightleftharpoons> "CONST Pair x y"
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  "_pattern x y" \<rightleftharpoons> "CONST Pair x y"
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  "_patterns x y" \<rightleftharpoons> "CONST Pair x y"
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  "_tuple x (_tuple_args y z)" \<rightleftharpoons> "_tuple x (_tuple_arg (_tuple y z))"
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  "\<lambda>(x, y, zs). b" \<rightleftharpoons> "CONST case_prod (\<lambda>x (y, zs). b)"
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  "\<lambda>(x, y). b" \<rightleftharpoons> "CONST case_prod (\<lambda>x y. b)"
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  "_abs (CONST Pair x y) t" \<rightharpoonup> "\<lambda>(x, y). t"
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  -- \<open>This rule accommodates tuples in @{text "case C \<dots> (x, y) \<dots> \<Rightarrow> \<dots>"}:
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     The @{text "(x, y)"} is parsed as @{text "Pair x y"} because it is @{text logic},
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     not @{text pttrn}.\<close>
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text \<open>print @{term "case_prod f"} as @{term "\<lambda>(x, y). f x y"} and
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  @{term "case_prod (\<lambda>x. f x)"} as @{term "\<lambda>(x, y). f x y"}\<close>
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typed_print_translation \<open>
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  let
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    fun case_prod_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match
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      | case_prod_guess_names_tr' T [Abs (x, xT, t)] =
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          (case (head_of t) of
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            Const (@{const_syntax case_prod}, _) => 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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      | case_prod_guess_names_tr' T [t] =
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          (case head_of t of
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            Const (@{const_syntax case_prod}, _) => 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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   347
              val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t');
haftmann@61226
   348
            in
haftmann@61226
   349
              Syntax.const @{syntax_const "_abs"} $
haftmann@61226
   350
                (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
haftmann@61226
   351
            end)
haftmann@61424
   352
      | case_prod_guess_names_tr' _ _ = raise Match;
haftmann@61424
   353
  in [(@{const_syntax case_prod}, K case_prod_guess_names_tr')] end
haftmann@61226
   354
\<close>
haftmann@61226
   355
nipkow@10213
   356
wenzelm@60758
   357
subsubsection \<open>Code generator setup\<close>
haftmann@37166
   358
haftmann@52435
   359
code_printing
haftmann@52435
   360
  type_constructor prod \<rightharpoonup>
haftmann@52435
   361
    (SML) infix 2 "*"
haftmann@52435
   362
    and (OCaml) infix 2 "*"
haftmann@52435
   363
    and (Haskell) "!((_),/ (_))"
haftmann@52435
   364
    and (Scala) "((_),/ (_))"
haftmann@52435
   365
| constant Pair \<rightharpoonup>
haftmann@52435
   366
    (SML) "!((_),/ (_))"
haftmann@52435
   367
    and (OCaml) "!((_),/ (_))"
haftmann@52435
   368
    and (Haskell) "!((_),/ (_))"
haftmann@52435
   369
    and (Scala) "!((_),/ (_))"
haftmann@52435
   370
| class_instance  prod :: equal \<rightharpoonup>
haftmann@52435
   371
    (Haskell) -
haftmann@52435
   372
| constant "HOL.equal :: 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" \<rightharpoonup>
haftmann@52435
   373
    (Haskell) infix 4 "=="
haftmann@61424
   374
| constant fst \<rightharpoonup> (Haskell) "fst"
haftmann@61424
   375
| constant snd \<rightharpoonup> (Haskell) "snd"
haftmann@37166
   376
haftmann@37166
   377
wenzelm@60758
   378
subsubsection \<open>Fundamental operations and properties\<close>
wenzelm@11838
   379
bulwahn@49897
   380
lemma Pair_inject:
bulwahn@49897
   381
  assumes "(a, b) = (a', b')"
haftmann@61424
   382
    and "a = a' \<Longrightarrow> b = b' \<Longrightarrow> R"
bulwahn@49897
   383
  shows R
bulwahn@49897
   384
  using assms by simp
bulwahn@49897
   385
haftmann@26358
   386
lemma surj_pair [simp]: "EX x y. p = (x, y)"
haftmann@37166
   387
  by (cases p) simp
nipkow@10213
   388
wenzelm@11838
   389
lemma fst_eqD: "fst (x, y) = a ==> x = a"
wenzelm@11838
   390
  by simp
wenzelm@11838
   391
wenzelm@11838
   392
lemma snd_eqD: "snd (x, y) = a ==> y = a"
wenzelm@11838
   393
  by simp
wenzelm@11838
   394
haftmann@61424
   395
lemma case_prod_unfold [nitpick_unfold]: "case_prod = (\<lambda>c p. c (fst p) (snd p))"
haftmann@61424
   396
  by (simp add: fun_eq_iff split: prod.split)
haftmann@61424
   397
haftmann@61424
   398
lemma case_prod_conv [simp, code]: "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
haftmann@61424
   399
  by (fact prod.case)
haftmann@61424
   400
blanchet@55393
   401
lemmas surjective_pairing = prod.collapse [symmetric]
wenzelm@11838
   402
huffman@44066
   403
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
haftmann@37166
   404
  by (cases s, cases t) simp
haftmann@37166
   405
haftmann@37166
   406
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
huffman@44066
   407
  by (simp add: prod_eq_iff)
haftmann@37166
   408
haftmann@61424
   409
lemma case_prodI: "f a b \<Longrightarrow> case (a, b) of (c, d) \<Rightarrow> f c d"
haftmann@61424
   410
  by (rule prod.case [THEN iffD2])
haftmann@37166
   411
haftmann@61424
   412
lemma case_prodD: "(case (a, b) of (c, d) \<Rightarrow> f c d) \<Longrightarrow> f a b"
haftmann@61424
   413
  by (rule prod.case [THEN iffD1])
haftmann@37166
   414
haftmann@61424
   415
lemma case_prod_Pair [simp]: "case_prod Pair = id"
nipkow@39302
   416
  by (simp add: fun_eq_iff split: prod.split)
haftmann@37166
   417
haftmann@61424
   418
lemma case_prod_eta: "(\<lambda>(x, y). f (x, y)) = f"
wenzelm@60758
   419
  -- \<open>Subsumes the old @{text split_Pair} when @{term f} is the identity function.\<close>
nipkow@39302
   420
  by (simp add: fun_eq_iff split: prod.split)
haftmann@37166
   421
haftmann@61424
   422
lemma case_prod_comp: "(case x of (a, b) \<Rightarrow> (f \<circ> g) a b) = f (g (fst x)) (snd x)"
haftmann@37166
   423
  by (cases x) simp
haftmann@37166
   424
haftmann@61424
   425
lemma The_case_prod: "The (case_prod P) = (THE xy. P (fst xy) (snd xy))"
blanchet@55414
   426
  by (simp add: case_prod_unfold)
haftmann@37166
   427
haftmann@61424
   428
lemma cond_case_prod_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
haftmann@61424
   429
  by (simp add: case_prod_eta)
haftmann@37166
   430
blanchet@47740
   431
lemma split_paired_all [no_atp]: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
wenzelm@11820
   432
proof
wenzelm@11820
   433
  fix a b
wenzelm@11820
   434
  assume "!!x. PROP P x"
wenzelm@19535
   435
  then show "PROP P (a, b)" .
wenzelm@11820
   436
next
wenzelm@11820
   437
  fix x
wenzelm@11820
   438
  assume "!!a b. PROP P (a, b)"
wenzelm@60758
   439
  from \<open>PROP P (fst x, snd x)\<close> show "PROP P x" by simp
wenzelm@11820
   440
qed
wenzelm@11820
   441
wenzelm@60758
   442
text \<open>
wenzelm@11838
   443
  The rule @{thm [source] split_paired_all} does not work with the
wenzelm@11838
   444
  Simplifier because it also affects premises in congrence rules,
wenzelm@11838
   445
  where this can lead to premises of the form @{text "!!a b. ... =
wenzelm@11838
   446
  ?P(a, b)"} which cannot be solved by reflexivity.
wenzelm@60758
   447
\<close>
wenzelm@11838
   448
haftmann@26358
   449
lemmas split_tupled_all = split_paired_all unit_all_eq2
haftmann@26358
   450
wenzelm@60758
   451
ML \<open>
wenzelm@11838
   452
  (* replace parameters of product type by individual component parameters *)
wenzelm@11838
   453
  local (* filtering with exists_paired_all is an essential optimization *)
wenzelm@56245
   454
    fun exists_paired_all (Const (@{const_name Pure.all}, _) $ Abs (_, T, t)) =
wenzelm@11838
   455
          can HOLogic.dest_prodT T orelse exists_paired_all t
wenzelm@11838
   456
      | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
wenzelm@11838
   457
      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
wenzelm@11838
   458
      | exists_paired_all _ = false;
wenzelm@51717
   459
    val ss =
wenzelm@51717
   460
      simpset_of
wenzelm@51717
   461
       (put_simpset HOL_basic_ss @{context}
wenzelm@51717
   462
        addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
wenzelm@51717
   463
        addsimprocs [@{simproc unit_eq}]);
wenzelm@11838
   464
  in
wenzelm@51717
   465
    fun split_all_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   466
      if exists_paired_all t then safe_full_simp_tac (put_simpset ss ctxt) i else no_tac);
wenzelm@51717
   467
wenzelm@51717
   468
    fun unsafe_split_all_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   469
      if exists_paired_all t then full_simp_tac (put_simpset ss ctxt) i else no_tac);
wenzelm@51717
   470
wenzelm@51717
   471
    fun split_all ctxt th =
wenzelm@51717
   472
      if exists_paired_all (Thm.prop_of th)
wenzelm@51717
   473
      then full_simplify (put_simpset ss ctxt) th else th;
wenzelm@11838
   474
  end;
wenzelm@60758
   475
\<close>
wenzelm@11838
   476
wenzelm@60758
   477
setup \<open>map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac))\<close>
wenzelm@11838
   478
blanchet@47740
   479
lemma split_paired_All [simp, no_atp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@60758
   480
  -- \<open>@{text "[iff]"} is not a good idea because it makes @{text blast} loop\<close>
wenzelm@11838
   481
  by fast
wenzelm@11838
   482
blanchet@47740
   483
lemma split_paired_Ex [simp, no_atp]: "(EX x. P x) = (EX a b. P (a, b))"
haftmann@26358
   484
  by fast
haftmann@26358
   485
blanchet@47740
   486
lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@60758
   487
  -- \<open>Can't be added to simpset: loops!\<close>
haftmann@61424
   488
  by (simp add: case_prod_eta)
wenzelm@11838
   489
wenzelm@60758
   490
text \<open>
haftmann@61424
   491
  Simplification procedure for @{thm [source] cond_case_prod_eta}.  Using
haftmann@61424
   492
  @{thm [source] case_prod_eta} as a rewrite rule is not general enough,
haftmann@61424
   493
  and using @{thm [source] cond_case_prod_eta} directly would render some
wenzelm@11838
   494
  existing proofs very inefficient; similarly for @{text
haftmann@61424
   495
  prod.case_eq_if}.
wenzelm@60758
   496
\<close>
wenzelm@11838
   497
wenzelm@60758
   498
ML \<open>
wenzelm@11838
   499
local
haftmann@61424
   500
  val cond_case_prod_eta_ss =
haftmann@61424
   501
    simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms cond_case_prod_eta});
wenzelm@35364
   502
  fun Pair_pat k 0 (Bound m) = (m = k)
wenzelm@35364
   503
    | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =
wenzelm@35364
   504
        i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t
wenzelm@35364
   505
    | Pair_pat _ _ _ = false;
wenzelm@35364
   506
  fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t
wenzelm@35364
   507
    | no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@35364
   508
    | no_args k i (Bound m) = m < k orelse m > k + i
wenzelm@35364
   509
    | no_args _ _ _ = true;
wenzelm@35364
   510
  fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE
haftmann@61424
   511
    | split_pat tp i (Const (@{const_name case_prod}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
wenzelm@35364
   512
    | split_pat tp i _ = NONE;
wenzelm@51717
   513
  fun metaeq ctxt lhs rhs = mk_meta_eq (Goal.prove ctxt [] []
wenzelm@35364
   514
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
haftmann@61424
   515
        (K (simp_tac (put_simpset cond_case_prod_eta_ss ctxt) 1)));
wenzelm@11838
   516
wenzelm@35364
   517
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t
wenzelm@35364
   518
    | beta_term_pat k i (t $ u) =
wenzelm@35364
   519
        Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@35364
   520
    | beta_term_pat k i t = no_args k i t;
wenzelm@35364
   521
  fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@35364
   522
    | eta_term_pat _ _ _ = false;
wenzelm@11838
   523
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@35364
   524
    | subst arg k i (t $ u) =
wenzelm@35364
   525
        if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@35364
   526
        else (subst arg k i t $ subst arg k i u)
wenzelm@35364
   527
    | subst arg k i t = t;
wenzelm@43595
   528
in
haftmann@61424
   529
  fun beta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t) $ arg) =
wenzelm@11838
   530
        (case split_pat beta_term_pat 1 t of
wenzelm@51717
   531
          SOME (i, f) => SOME (metaeq ctxt s (subst arg 0 i f))
skalberg@15531
   532
        | NONE => NONE)
wenzelm@35364
   533
    | beta_proc _ _ = NONE;
haftmann@61424
   534
  fun eta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t)) =
wenzelm@11838
   535
        (case split_pat eta_term_pat 1 t of
haftmann@58468
   536
          SOME (_, ft) => SOME (metaeq ctxt s (let val f $ _ = ft in f end))
skalberg@15531
   537
        | NONE => NONE)
wenzelm@35364
   538
    | eta_proc _ _ = NONE;
wenzelm@11838
   539
end;
wenzelm@60758
   540
\<close>
haftmann@61424
   541
simproc_setup case_prod_beta ("case_prod f z") =
wenzelm@60758
   542
  \<open>fn _ => fn ctxt => fn ct => beta_proc ctxt (Thm.term_of ct)\<close>
haftmann@61424
   543
simproc_setup case_prod_eta ("case_prod f") =
wenzelm@60758
   544
  \<open>fn _ => fn ctxt => fn ct => eta_proc ctxt (Thm.term_of ct)\<close>
wenzelm@11838
   545
haftmann@61424
   546
lemma case_prod_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
hoelzl@50104
   547
  by (auto simp: fun_eq_iff)
hoelzl@50104
   548
wenzelm@60758
   549
text \<open>
haftmann@61424
   550
  \medskip @{const case_prod} used as a logical connective or set former.
wenzelm@11838
   551
wenzelm@11838
   552
  \medskip These rules are for use with @{text blast}; could instead
wenzelm@60758
   553
  call @{text simp} using @{thm [source] prod.split} as rewrite.\<close>
wenzelm@11838
   554
haftmann@61424
   555
lemma case_prodI2:
haftmann@61424
   556
  "\<And>p. (\<And>a b. p = (a, b) \<Longrightarrow> c a b) \<Longrightarrow> case p of (a, b) \<Rightarrow> c a b"
haftmann@61424
   557
  by (simp add: split_tupled_all)
wenzelm@11838
   558
haftmann@61424
   559
lemma case_prodI2':
haftmann@61424
   560
  "\<And>p. (\<And>a b. (a, b) = p \<Longrightarrow> c a b x) \<Longrightarrow> (case p of (a, b) \<Rightarrow> c a b) x"
haftmann@61424
   561
  by (simp add: split_tupled_all)
wenzelm@11838
   562
haftmann@61424
   563
lemma case_prodE [elim!]:
haftmann@61424
   564
  "(case p of (a, b) \<Rightarrow> c a b) \<Longrightarrow> (\<And>x y. p = (x, y) \<Longrightarrow> c x y \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@61424
   565
  by (induct p) simp
wenzelm@11838
   566
haftmann@61424
   567
lemma case_prodE' [elim!]:
haftmann@61424
   568
  "(case p of (a, b) \<Rightarrow> c a b) z \<Longrightarrow> (\<And>x y. p = (x, y) \<Longrightarrow> c x y z \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@61424
   569
  by (induct p) simp
wenzelm@11838
   570
haftmann@61424
   571
lemma case_prodE2:
haftmann@61424
   572
  assumes q: "Q (case z of (a, b) \<Rightarrow> P a b)"
haftmann@61424
   573
    and r: "\<And>x y. z = (x, y) \<Longrightarrow> Q (P x y) \<Longrightarrow> R"
haftmann@61424
   574
  shows R
haftmann@61424
   575
proof (rule r)
haftmann@61424
   576
  show "z = (fst z, snd z)" by simp
haftmann@61424
   577
  then show "Q (P (fst z) (snd z))"
haftmann@61424
   578
    using q by (simp add: case_prod_unfold)
wenzelm@11838
   579
qed
wenzelm@11838
   580
haftmann@61424
   581
lemma case_prodD':
haftmann@61127
   582
  "(case (a, b) of (c, d) \<Rightarrow> R c d) c \<Longrightarrow> R a b c"
wenzelm@11838
   583
  by simp
wenzelm@11838
   584
haftmann@61424
   585
lemma mem_case_prodI:
haftmann@61127
   586
  "z \<in> c a b \<Longrightarrow> z \<in> (case (a, b) of (d, e) \<Rightarrow> c d e)"
wenzelm@11838
   587
  by simp
wenzelm@11838
   588
haftmann@61424
   589
lemma mem_case_prodI2 [intro!]:
haftmann@61127
   590
  "\<And>p. (\<And>a b. p = (a, b) \<Longrightarrow> z \<in> c a b) \<Longrightarrow> z \<in> (case p of (a, b) \<Rightarrow> c a b)"
haftmann@61127
   591
  by (simp only: split_tupled_all) simp
wenzelm@11838
   592
haftmann@61424
   593
declare mem_case_prodI [intro!] -- \<open>postponed to maintain traditional declaration order!\<close>
haftmann@61424
   594
declare case_prodI2' [intro!] -- \<open>postponed to maintain traditional declaration order!\<close>
haftmann@61424
   595
declare case_prodI2 [intro!] -- \<open>postponed to maintain traditional declaration order!\<close>
haftmann@61424
   596
declare case_prodI [intro!] -- \<open>postponed to maintain traditional declaration order!\<close>
haftmann@61424
   597
  
haftmann@61424
   598
lemma mem_case_prodE [elim!]:
haftmann@61424
   599
  assumes "z \<in> case_prod c p"
haftmann@58468
   600
  obtains x y where "p = (x, y)" and "z \<in> c x y"
haftmann@61424
   601
  using assms by (rule case_prodE2)
wenzelm@11838
   602
wenzelm@60758
   603
ML \<open>
wenzelm@11838
   604
local (* filtering with exists_p_split is an essential optimization *)
haftmann@61424
   605
  fun exists_p_split (Const (@{const_name case_prod},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true
wenzelm@11838
   606
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838
   607
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838
   608
    | exists_p_split _ = false;
wenzelm@11838
   609
in
wenzelm@51717
   610
fun split_conv_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   611
  if exists_p_split t
haftmann@61424
   612
  then safe_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms case_prod_conv}) i
wenzelm@51717
   613
  else no_tac);
wenzelm@11838
   614
end;
wenzelm@60758
   615
\<close>
wenzelm@26340
   616
wenzelm@11838
   617
(* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838
   618
   to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@60758
   619
setup \<open>map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac))\<close>
wenzelm@11838
   620
blanchet@54147
   621
lemma split_eta_SetCompr [simp, no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
wenzelm@18372
   622
  by (rule ext) fast
wenzelm@11838
   623
haftmann@61424
   624
lemma split_eta_SetCompr2 [simp, no_atp]: "(%u. EX x y. u = (x, y) & P x y) = case_prod P"
wenzelm@18372
   625
  by (rule ext) fast
wenzelm@11838
   626
haftmann@61424
   627
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & case_prod Q ab)"
wenzelm@60758
   628
  -- \<open>Allows simplifications of nested splits in case of independent predicates.\<close>
wenzelm@18372
   629
  by (rule ext) blast
wenzelm@11838
   630
nipkow@14337
   631
(* Do NOT make this a simp rule as it
nipkow@14337
   632
   a) only helps in special situations
nipkow@14337
   633
   b) can lead to nontermination in the presence of split_def
nipkow@14337
   634
*)
nipkow@14337
   635
lemma split_comp_eq: 
paulson@20415
   636
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
haftmann@61424
   637
  shows "(%u. f (g (fst u)) (snd u)) = (case_prod (%x. f (g x)))"
wenzelm@18372
   638
  by (rule ext) auto
oheimb@14101
   639
haftmann@26358
   640
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
haftmann@26358
   641
  apply (rule_tac x = "(a, b)" in image_eqI)
haftmann@26358
   642
   apply auto
haftmann@26358
   643
  done
haftmann@26358
   644
wenzelm@11838
   645
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838
   646
  by blast
wenzelm@11838
   647
wenzelm@11838
   648
(*
wenzelm@11838
   649
the following  would be slightly more general,
wenzelm@11838
   650
but cannot be used as rewrite rule:
wenzelm@11838
   651
### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838
   652
### ?y = .x
wenzelm@11838
   653
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208
   654
by (rtac some_equality 1)
paulson@14208
   655
by ( Simp_tac 1)
paulson@14208
   656
by (split_all_tac 1)
paulson@14208
   657
by (Asm_full_simp_tac 1)
wenzelm@11838
   658
qed "The_split_eq";
wenzelm@11838
   659
*)
wenzelm@11838
   660
blanchet@55414
   661
lemma case_prod_beta:
haftmann@61424
   662
  "case_prod f p = f (fst p) (snd p)"
haftmann@61424
   663
  by (fact prod.case_eq_if)
bulwahn@26143
   664
blanchet@55417
   665
lemma prod_cases3 [cases type]:
haftmann@24699
   666
  obtains (fields) a b c where "y = (a, b, c)"
haftmann@24699
   667
  by (cases y, case_tac b) blast
haftmann@24699
   668
haftmann@24699
   669
lemma prod_induct3 [case_names fields, induct type]:
haftmann@24699
   670
    "(!!a b c. P (a, b, c)) ==> P x"
haftmann@24699
   671
  by (cases x) blast
haftmann@24699
   672
blanchet@55417
   673
lemma prod_cases4 [cases type]:
haftmann@24699
   674
  obtains (fields) a b c d where "y = (a, b, c, d)"
haftmann@24699
   675
  by (cases y, case_tac c) blast
haftmann@24699
   676
haftmann@24699
   677
lemma prod_induct4 [case_names fields, induct type]:
haftmann@24699
   678
    "(!!a b c d. P (a, b, c, d)) ==> P x"
haftmann@24699
   679
  by (cases x) blast
haftmann@24699
   680
blanchet@55417
   681
lemma prod_cases5 [cases type]:
haftmann@24699
   682
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
haftmann@24699
   683
  by (cases y, case_tac d) blast
haftmann@24699
   684
haftmann@24699
   685
lemma prod_induct5 [case_names fields, induct type]:
haftmann@24699
   686
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
haftmann@24699
   687
  by (cases x) blast
haftmann@24699
   688
blanchet@55417
   689
lemma prod_cases6 [cases type]:
haftmann@24699
   690
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
haftmann@24699
   691
  by (cases y, case_tac e) blast
haftmann@24699
   692
haftmann@24699
   693
lemma prod_induct6 [case_names fields, induct type]:
haftmann@24699
   694
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
haftmann@24699
   695
  by (cases x) blast
haftmann@24699
   696
blanchet@55417
   697
lemma prod_cases7 [cases type]:
haftmann@24699
   698
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
haftmann@24699
   699
  by (cases y, case_tac f) blast
haftmann@24699
   700
haftmann@24699
   701
lemma prod_induct7 [case_names fields, induct type]:
haftmann@24699
   702
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
haftmann@24699
   703
  by (cases x) blast
haftmann@24699
   704
haftmann@61424
   705
definition internal_case_prod :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
haftmann@61424
   706
  "internal_case_prod == case_prod"
haftmann@37166
   707
haftmann@61424
   708
lemma internal_case_prod_conv: "internal_case_prod c (a, b) = c a b"
haftmann@61424
   709
  by (simp only: internal_case_prod_def case_prod_conv)
haftmann@37166
   710
wenzelm@48891
   711
ML_file "Tools/split_rule.ML"
haftmann@37166
   712
haftmann@61424
   713
hide_const internal_case_prod
haftmann@37166
   714
haftmann@24699
   715
wenzelm@60758
   716
subsubsection \<open>Derived operations\<close>
haftmann@26358
   717
haftmann@61424
   718
definition curry :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where
haftmann@37387
   719
  "curry = (\<lambda>c x y. c (x, y))"
haftmann@37166
   720
haftmann@37166
   721
lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
haftmann@37166
   722
  by (simp add: curry_def)
haftmann@37166
   723
haftmann@37166
   724
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
haftmann@37166
   725
  by (simp add: curry_def)
haftmann@37166
   726
haftmann@37166
   727
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
haftmann@37166
   728
  by (simp add: curry_def)
haftmann@37166
   729
haftmann@37166
   730
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@37166
   731
  by (simp add: curry_def)
haftmann@37166
   732
haftmann@61424
   733
lemma curry_case_prod [simp]: "curry (case_prod f) = f"
haftmann@61032
   734
  by (simp add: curry_def case_prod_unfold)
haftmann@37166
   735
haftmann@61424
   736
lemma case_prod_curry [simp]: "case_prod (curry f) = f"
haftmann@61032
   737
  by (simp add: curry_def case_prod_unfold)
haftmann@37166
   738
Andreas@54630
   739
lemma curry_K: "curry (\<lambda>x. c) = (\<lambda>x y. c)"
Andreas@54630
   740
by(simp add: fun_eq_iff)
Andreas@54630
   741
wenzelm@60758
   742
text \<open>
haftmann@26358
   743
  The composition-uncurry combinator.
wenzelm@60758
   744
\<close>
haftmann@26358
   745
haftmann@37751
   746
notation fcomp (infixl "\<circ>>" 60)
haftmann@26358
   747
haftmann@37751
   748
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where
haftmann@61424
   749
  "f \<circ>\<rightarrow> g = (\<lambda>x. case_prod g (f x))"
haftmann@26358
   750
haftmann@37678
   751
lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))"
blanchet@55414
   752
  by (simp add: fun_eq_iff scomp_def case_prod_unfold)
haftmann@37678
   753
haftmann@61424
   754
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = case_prod g (f x)"
blanchet@55414
   755
  by (simp add: scomp_unfold case_prod_unfold)
haftmann@26358
   756
haftmann@37751
   757
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x"
huffman@44921
   758
  by (simp add: fun_eq_iff)
haftmann@26358
   759
haftmann@37751
   760
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x"
huffman@44921
   761
  by (simp add: fun_eq_iff)
haftmann@26358
   762
haftmann@37751
   763
lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
nipkow@39302
   764
  by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358
   765
haftmann@37751
   766
lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
nipkow@39302
   767
  by (simp add: fun_eq_iff scomp_unfold fcomp_def)
haftmann@26358
   768
haftmann@37751
   769
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
huffman@44921
   770
  by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358
   771
haftmann@52435
   772
code_printing
haftmann@52435
   773
  constant scomp \<rightharpoonup> (Eval) infixl 3 "#->"
haftmann@31202
   774
haftmann@37751
   775
no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
   776
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@26358
   777
wenzelm@60758
   778
text \<open>
blanchet@55932
   779
  @{term map_prod} --- action of the product functor upon
krauss@36664
   780
  functions.
wenzelm@60758
   781
\<close>
haftmann@21195
   782
blanchet@55932
   783
definition map_prod :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
blanchet@55932
   784
  "map_prod f g = (\<lambda>(x, y). (f x, g y))"
haftmann@26358
   785
blanchet@55932
   786
lemma map_prod_simp [simp, code]:
blanchet@55932
   787
  "map_prod f g (a, b) = (f a, g b)"
blanchet@55932
   788
  by (simp add: map_prod_def)
haftmann@26358
   789
blanchet@55932
   790
functor map_prod: map_prod
huffman@44921
   791
  by (auto simp add: split_paired_all)
nipkow@37278
   792
blanchet@55932
   793
lemma fst_map_prod [simp]:
blanchet@55932
   794
  "fst (map_prod f g x) = f (fst x)"
haftmann@40607
   795
  by (cases x) simp_all
nipkow@37278
   796
traytel@58916
   797
lemma snd_map_prod [simp]:
blanchet@55932
   798
  "snd (map_prod f g x) = g (snd x)"
haftmann@40607
   799
  by (cases x) simp_all
nipkow@37278
   800
blanchet@55932
   801
lemma fst_comp_map_prod [simp]:
blanchet@55932
   802
  "fst \<circ> map_prod f g = f \<circ> fst"
haftmann@40607
   803
  by (rule ext) simp_all
nipkow@37278
   804
blanchet@55932
   805
lemma snd_comp_map_prod [simp]:
blanchet@55932
   806
  "snd \<circ> map_prod f g = g \<circ> snd"
haftmann@40607
   807
  by (rule ext) simp_all
haftmann@26358
   808
blanchet@55932
   809
lemma map_prod_compose:
blanchet@55932
   810
  "map_prod (f1 o f2) (g1 o g2) = (map_prod f1 g1 o map_prod f2 g2)"
blanchet@55932
   811
  by (rule ext) (simp add: map_prod.compositionality comp_def)
haftmann@26358
   812
blanchet@55932
   813
lemma map_prod_ident [simp]:
blanchet@55932
   814
  "map_prod (%x. x) (%y. y) = (%z. z)"
blanchet@55932
   815
  by (rule ext) (simp add: map_prod.identity)
haftmann@40607
   816
blanchet@55932
   817
lemma map_prod_imageI [intro]:
blanchet@55932
   818
  "(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_prod f g ` R"
haftmann@40607
   819
  by (rule image_eqI) simp_all
haftmann@21195
   820
haftmann@26358
   821
lemma prod_fun_imageE [elim!]:
blanchet@55932
   822
  assumes major: "c \<in> map_prod f g ` R"
haftmann@40607
   823
    and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P"
haftmann@26358
   824
  shows P
haftmann@26358
   825
  apply (rule major [THEN imageE])
haftmann@37166
   826
  apply (case_tac x)
haftmann@26358
   827
  apply (rule cases)
haftmann@40607
   828
  apply simp_all
haftmann@26358
   829
  done
haftmann@26358
   830
haftmann@37166
   831
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where
blanchet@55932
   832
  "apfst f = map_prod f id"
haftmann@26358
   833
haftmann@37166
   834
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where
blanchet@55932
   835
  "apsnd f = map_prod id f"
haftmann@26358
   836
haftmann@26358
   837
lemma apfst_conv [simp, code]:
haftmann@26358
   838
  "apfst f (x, y) = (f x, y)" 
haftmann@26358
   839
  by (simp add: apfst_def)
haftmann@26358
   840
hoelzl@33638
   841
lemma apsnd_conv [simp, code]:
haftmann@26358
   842
  "apsnd f (x, y) = (x, f y)" 
haftmann@26358
   843
  by (simp add: apsnd_def)
haftmann@21195
   844
haftmann@33594
   845
lemma fst_apfst [simp]:
haftmann@33594
   846
  "fst (apfst f x) = f (fst x)"
haftmann@33594
   847
  by (cases x) simp
haftmann@33594
   848
haftmann@51173
   849
lemma fst_comp_apfst [simp]:
haftmann@51173
   850
  "fst \<circ> apfst f = f \<circ> fst"
haftmann@51173
   851
  by (simp add: fun_eq_iff)
haftmann@51173
   852
haftmann@33594
   853
lemma fst_apsnd [simp]:
haftmann@33594
   854
  "fst (apsnd f x) = fst x"
haftmann@33594
   855
  by (cases x) simp
haftmann@33594
   856
haftmann@51173
   857
lemma fst_comp_apsnd [simp]:
haftmann@51173
   858
  "fst \<circ> apsnd f = fst"
haftmann@51173
   859
  by (simp add: fun_eq_iff)
haftmann@51173
   860
haftmann@33594
   861
lemma snd_apfst [simp]:
haftmann@33594
   862
  "snd (apfst f x) = snd x"
haftmann@33594
   863
  by (cases x) simp
haftmann@33594
   864
haftmann@51173
   865
lemma snd_comp_apfst [simp]:
haftmann@51173
   866
  "snd \<circ> apfst f = snd"
haftmann@51173
   867
  by (simp add: fun_eq_iff)
haftmann@51173
   868
haftmann@33594
   869
lemma snd_apsnd [simp]:
haftmann@33594
   870
  "snd (apsnd f x) = f (snd x)"
haftmann@33594
   871
  by (cases x) simp
haftmann@33594
   872
haftmann@51173
   873
lemma snd_comp_apsnd [simp]:
haftmann@51173
   874
  "snd \<circ> apsnd f = f \<circ> snd"
haftmann@51173
   875
  by (simp add: fun_eq_iff)
haftmann@51173
   876
haftmann@33594
   877
lemma apfst_compose:
haftmann@33594
   878
  "apfst f (apfst g x) = apfst (f \<circ> g) x"
haftmann@33594
   879
  by (cases x) simp
haftmann@33594
   880
haftmann@33594
   881
lemma apsnd_compose:
haftmann@33594
   882
  "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
haftmann@33594
   883
  by (cases x) simp
haftmann@33594
   884
haftmann@33594
   885
lemma apfst_apsnd [simp]:
haftmann@33594
   886
  "apfst f (apsnd g x) = (f (fst x), g (snd x))"
haftmann@33594
   887
  by (cases x) simp
haftmann@33594
   888
haftmann@33594
   889
lemma apsnd_apfst [simp]:
haftmann@33594
   890
  "apsnd f (apfst g x) = (g (fst x), f (snd x))"
haftmann@33594
   891
  by (cases x) simp
haftmann@33594
   892
haftmann@33594
   893
lemma apfst_id [simp] :
haftmann@33594
   894
  "apfst id = id"
nipkow@39302
   895
  by (simp add: fun_eq_iff)
haftmann@33594
   896
haftmann@33594
   897
lemma apsnd_id [simp] :
haftmann@33594
   898
  "apsnd id = id"
nipkow@39302
   899
  by (simp add: fun_eq_iff)
haftmann@33594
   900
haftmann@33594
   901
lemma apfst_eq_conv [simp]:
haftmann@33594
   902
  "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
haftmann@33594
   903
  by (cases x) simp
haftmann@33594
   904
haftmann@33594
   905
lemma apsnd_eq_conv [simp]:
haftmann@33594
   906
  "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"
haftmann@33594
   907
  by (cases x) simp
haftmann@33594
   908
hoelzl@33638
   909
lemma apsnd_apfst_commute:
hoelzl@33638
   910
  "apsnd f (apfst g p) = apfst g (apsnd f p)"
hoelzl@33638
   911
  by simp
haftmann@21195
   912
haftmann@56626
   913
context
haftmann@56626
   914
begin
haftmann@56626
   915
wenzelm@60758
   916
local_setup \<open>Local_Theory.map_background_naming (Name_Space.mandatory_path "prod")\<close>
haftmann@56626
   917
haftmann@56545
   918
definition swap :: "'a \<times> 'b \<Rightarrow> 'b \<times> 'a"
haftmann@56545
   919
where
haftmann@56545
   920
  "swap p = (snd p, fst p)"
haftmann@56545
   921
haftmann@56626
   922
end
haftmann@56626
   923
haftmann@56545
   924
lemma swap_simp [simp]:
haftmann@56626
   925
  "prod.swap (x, y) = (y, x)"
haftmann@56626
   926
  by (simp add: prod.swap_def)
haftmann@56545
   927
haftmann@58195
   928
lemma swap_swap [simp]:
haftmann@58195
   929
  "prod.swap (prod.swap p) = p"
haftmann@58195
   930
  by (cases p) simp
haftmann@58195
   931
haftmann@58195
   932
lemma swap_comp_swap [simp]:
haftmann@58195
   933
  "prod.swap \<circ> prod.swap = id"
haftmann@58195
   934
  by (simp add: fun_eq_iff)
haftmann@58195
   935
haftmann@56545
   936
lemma pair_in_swap_image [simp]:
haftmann@56626
   937
  "(y, x) \<in> prod.swap ` A \<longleftrightarrow> (x, y) \<in> A"
haftmann@56545
   938
  by (auto intro!: image_eqI)
haftmann@56545
   939
haftmann@56545
   940
lemma inj_swap [simp]:
haftmann@56626
   941
  "inj_on prod.swap A"
haftmann@56626
   942
  by (rule inj_onI) auto
haftmann@56626
   943
haftmann@56626
   944
lemma swap_inj_on:
haftmann@56626
   945
  "inj_on (\<lambda>(i, j). (j, i)) A"
haftmann@56626
   946
  by (rule inj_onI) auto
haftmann@56545
   947
haftmann@58195
   948
lemma surj_swap [simp]:
haftmann@58195
   949
  "surj prod.swap"
haftmann@58195
   950
  by (rule surjI [of _ prod.swap]) simp
haftmann@58195
   951
haftmann@58195
   952
lemma bij_swap [simp]:
haftmann@58195
   953
  "bij prod.swap"
haftmann@58195
   954
  by (simp add: bij_def)
haftmann@58195
   955
haftmann@56545
   956
lemma case_swap [simp]:
haftmann@56626
   957
  "(case prod.swap p of (y, x) \<Rightarrow> f x y) = (case p of (x, y) \<Rightarrow> f x y)"
haftmann@56545
   958
  by (cases p) simp
haftmann@56545
   959
wenzelm@60758
   960
text \<open>
haftmann@26358
   961
  Disjoint union of a family of sets -- Sigma.
wenzelm@60758
   962
\<close>
haftmann@26358
   963
haftmann@45986
   964
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where
haftmann@26358
   965
  Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
haftmann@26358
   966
haftmann@26358
   967
abbreviation
haftmann@45986
   968
  Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set"
haftmann@26358
   969
    (infixr "<*>" 80) where
haftmann@26358
   970
  "A <*> B == Sigma A (%_. B)"
haftmann@26358
   971
haftmann@26358
   972
notation (xsymbols)
haftmann@26358
   973
  Times  (infixr "\<times>" 80)
berghofe@15394
   974
nipkow@45662
   975
hide_const (open) Times
nipkow@45662
   976
haftmann@26358
   977
syntax
wenzelm@35115
   978
  "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
haftmann@26358
   979
translations
wenzelm@35115
   980
  "SIGMA x:A. B" == "CONST Sigma A (%x. B)"
haftmann@26358
   981
haftmann@26358
   982
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
haftmann@26358
   983
  by (unfold Sigma_def) blast
haftmann@26358
   984
haftmann@26358
   985
lemma SigmaE [elim!]:
haftmann@26358
   986
    "[| c: Sigma A B;
haftmann@26358
   987
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
haftmann@26358
   988
     |] ==> P"
wenzelm@60758
   989
  -- \<open>The general elimination rule.\<close>
haftmann@26358
   990
  by (unfold Sigma_def) blast
haftmann@20588
   991
wenzelm@60758
   992
text \<open>
haftmann@26358
   993
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
haftmann@26358
   994
  eigenvariables.
wenzelm@60758
   995
\<close>
haftmann@26358
   996
haftmann@26358
   997
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
haftmann@26358
   998
  by blast
haftmann@26358
   999
haftmann@26358
  1000
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
haftmann@26358
  1001
  by blast
haftmann@26358
  1002
haftmann@26358
  1003
lemma SigmaE2:
haftmann@26358
  1004
    "[| (a, b) : Sigma A B;
haftmann@26358
  1005
        [| a:A;  b:B(a) |] ==> P
haftmann@26358
  1006
     |] ==> P"
haftmann@26358
  1007
  by blast
haftmann@20588
  1008
haftmann@26358
  1009
lemma Sigma_cong:
haftmann@26358
  1010
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
haftmann@26358
  1011
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
haftmann@26358
  1012
  by auto
haftmann@26358
  1013
haftmann@26358
  1014
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
haftmann@26358
  1015
  by blast
haftmann@26358
  1016
haftmann@26358
  1017
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
haftmann@26358
  1018
  by blast
haftmann@26358
  1019
haftmann@26358
  1020
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
haftmann@26358
  1021
  by blast
haftmann@26358
  1022
haftmann@26358
  1023
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
haftmann@26358
  1024
  by auto
haftmann@21908
  1025
haftmann@26358
  1026
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
haftmann@26358
  1027
  by auto
haftmann@26358
  1028
haftmann@26358
  1029
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
haftmann@26358
  1030
  by auto
haftmann@26358
  1031
haftmann@26358
  1032
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
haftmann@26358
  1033
  by blast
haftmann@26358
  1034
hoelzl@59000
  1035
lemma Sigma_empty_iff: "(SIGMA i:I. X i) = {} \<longleftrightarrow> (\<forall>i\<in>I. X i = {})"
hoelzl@59000
  1036
  by auto
hoelzl@59000
  1037
haftmann@26358
  1038
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
haftmann@26358
  1039
  by blast
haftmann@26358
  1040
haftmann@26358
  1041
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
haftmann@26358
  1042
  by (blast elim: equalityE)
haftmann@20588
  1043
haftmann@61424
  1044
lemma Collect_case_prod_Sigma:
haftmann@61127
  1045
  "{(x, y). P x \<and> Q x y} = (SIGMA x:Collect P. Collect (Q x))"
haftmann@26358
  1046
  by blast
haftmann@26358
  1047
haftmann@61424
  1048
lemma Collect_case_prod [simp]:
haftmann@61127
  1049
  "{(a, b). P a \<and> Q b} = Collect P \<times> Collect Q "
haftmann@61424
  1050
  by (fact Collect_case_prod_Sigma)
haftmann@26358
  1051
haftmann@61424
  1052
lemma Collect_case_prodD:
haftmann@61424
  1053
  "x \<in> Collect (case_prod A) \<Longrightarrow> A (fst x) (snd x)"
haftmann@61422
  1054
  by auto
haftmann@61422
  1055
haftmann@61424
  1056
lemma Collect_case_prod_mono:
haftmann@61424
  1057
  "A \<le> B \<Longrightarrow> Collect (case_prod A) \<subseteq> Collect (case_prod B)"
haftmann@61422
  1058
  by auto (auto elim!: le_funE)
haftmann@61422
  1059
haftmann@61422
  1060
lemma Collect_split_mono_strong: 
haftmann@61422
  1061
  "X = fst ` A \<Longrightarrow> Y = snd ` A \<Longrightarrow> \<forall>a\<in>X. \<forall>b \<in> Y. P a b \<longrightarrow> Q a b
haftmann@61424
  1062
    \<Longrightarrow> A \<subseteq> Collect (case_prod P) \<Longrightarrow> A \<subseteq> Collect (case_prod Q)"
haftmann@61422
  1063
  by fastforce
haftmann@61422
  1064
  
haftmann@26358
  1065
lemma UN_Times_distrib:
haftmann@61127
  1066
  "(\<Union>(a, b)\<in>A \<times> B. E a \<times> F b) = UNION A E \<times> UNION B F"
wenzelm@60758
  1067
  -- \<open>Suggested by Pierre Chartier\<close>
haftmann@26358
  1068
  by blast
haftmann@26358
  1069
blanchet@47740
  1070
lemma split_paired_Ball_Sigma [simp, no_atp]:
haftmann@61127
  1071
  "(\<forall>z\<in>Sigma A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>B x. P (x, y))"
haftmann@26358
  1072
  by blast
haftmann@26358
  1073
blanchet@47740
  1074
lemma split_paired_Bex_Sigma [simp, no_atp]:
haftmann@61127
  1075
  "(\<exists>z\<in>Sigma A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>y\<in>B x. P (x, y))"
haftmann@61127
  1076
  by blast
haftmann@61127
  1077
haftmann@61127
  1078
lemma Sigma_Un_distrib1:
haftmann@61127
  1079
  "Sigma (I \<union> J) C = Sigma I C \<union> Sigma J C"
haftmann@26358
  1080
  by blast
haftmann@21908
  1081
haftmann@61127
  1082
lemma Sigma_Un_distrib2:
haftmann@61127
  1083
  "(SIGMA i:I. A i \<union> B i) = Sigma I A \<union> Sigma I B"
haftmann@26358
  1084
  by blast
haftmann@26358
  1085
haftmann@61127
  1086
lemma Sigma_Int_distrib1:
haftmann@61127
  1087
  "Sigma (I \<inter> J) C = Sigma I C \<inter> Sigma J C"
haftmann@26358
  1088
  by blast
haftmann@26358
  1089
haftmann@61127
  1090
lemma Sigma_Int_distrib2:
haftmann@61127
  1091
  "(SIGMA i:I. A i \<inter> B i) = Sigma I A \<inter> Sigma I B"
haftmann@26358
  1092
  by blast
haftmann@26358
  1093
haftmann@61127
  1094
lemma Sigma_Diff_distrib1:
haftmann@61127
  1095
  "Sigma (I - J) C = Sigma I C - Sigma J C"
haftmann@26358
  1096
  by blast
haftmann@26358
  1097
haftmann@61127
  1098
lemma Sigma_Diff_distrib2:
haftmann@61127
  1099
  "(SIGMA i:I. A i - B i) = Sigma I A - Sigma I B"
haftmann@26358
  1100
  by blast
haftmann@21908
  1101
haftmann@61127
  1102
lemma Sigma_Union:
haftmann@61127
  1103
  "Sigma (\<Union>X) B = (\<Union>A\<in>X. Sigma A B)"
haftmann@26358
  1104
  by blast
haftmann@26358
  1105
wenzelm@60758
  1106
text \<open>
haftmann@26358
  1107
  Non-dependent versions are needed to avoid the need for higher-order
haftmann@26358
  1108
  matching, especially when the rules are re-oriented.
wenzelm@60758
  1109
\<close>
haftmann@21908
  1110
haftmann@61127
  1111
lemma Times_Un_distrib1:
haftmann@61127
  1112
  "(A \<union> B) \<times> C = A \<times> C \<union> B \<times> C "
haftmann@56545
  1113
  by (fact Sigma_Un_distrib1)
haftmann@26358
  1114
haftmann@61127
  1115
lemma Times_Int_distrib1:
haftmann@61127
  1116
  "(A \<inter> B) \<times> C = A \<times> C \<inter> B \<times> C "
haftmann@56545
  1117
  by (fact Sigma_Int_distrib1)
haftmann@26358
  1118
haftmann@61127
  1119
lemma Times_Diff_distrib1:
haftmann@61127
  1120
  "(A - B) \<times> C = A \<times> C - B \<times> C "
haftmann@56545
  1121
  by (fact Sigma_Diff_distrib1)
haftmann@26358
  1122
haftmann@61127
  1123
lemma Times_empty [simp]:
haftmann@61127
  1124
  "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
hoelzl@36622
  1125
  by auto
hoelzl@36622
  1126
haftmann@61127
  1127
lemma times_eq_iff:
haftmann@61127
  1128
  "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> (A = {} \<or> B = {}) \<and> (C = {} \<or> D = {})"
hoelzl@50104
  1129
  by auto
hoelzl@50104
  1130
haftmann@61127
  1131
lemma fst_image_times [simp]:
haftmann@61127
  1132
  "fst ` (A \<times> B) = (if B = {} then {} else A)"
huffman@44921
  1133
  by force
hoelzl@36622
  1134
haftmann@61127
  1135
lemma snd_image_times [simp]:
haftmann@61127
  1136
  "snd ` (A \<times> B) = (if A = {} then {} else B)"
huffman@44921
  1137
  by force
hoelzl@36622
  1138
haftmann@56545
  1139
lemma vimage_fst:
haftmann@56545
  1140
  "fst -` A = A \<times> UNIV"
haftmann@56545
  1141
  by auto
haftmann@56545
  1142
haftmann@56545
  1143
lemma vimage_snd:
haftmann@56545
  1144
  "snd -` A = UNIV \<times> A"
haftmann@56545
  1145
  by auto
haftmann@56545
  1146
nipkow@28719
  1147
lemma insert_times_insert[simp]:
nipkow@28719
  1148
  "insert a A \<times> insert b B =
nipkow@28719
  1149
   insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
haftmann@61127
  1150
  by blast
haftmann@26358
  1151
haftmann@61127
  1152
lemma vimage_Times:
haftmann@61127
  1153
  "f -` (A \<times> B) = (fst \<circ> f) -` A \<inter> (snd \<circ> f) -` B"
haftmann@61127
  1154
proof (rule set_eqI)
haftmann@61127
  1155
  fix x
haftmann@61127
  1156
  show "x \<in> f -` (A \<times> B) \<longleftrightarrow> x \<in> (fst \<circ> f) -` A \<inter> (snd \<circ> f) -` B"
haftmann@61127
  1157
    by (cases "f x") (auto split: prod.split)
haftmann@61127
  1158
qed
paulson@33271
  1159
haftmann@61127
  1160
lemma times_Int_times:
haftmann@61127
  1161
  "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
hoelzl@50104
  1162
  by auto
hoelzl@50104
  1163
haftmann@56626
  1164
lemma product_swap:
haftmann@56626
  1165
  "prod.swap ` (A \<times> B) = B \<times> A"
haftmann@56626
  1166
  by (auto simp add: set_eq_iff)
haftmann@35822
  1167
haftmann@35822
  1168
lemma swap_product:
haftmann@56626
  1169
  "(\<lambda>(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
haftmann@56626
  1170
  by (auto simp add: set_eq_iff)
haftmann@35822
  1171
hoelzl@36622
  1172
lemma image_split_eq_Sigma:
hoelzl@36622
  1173
  "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
haftmann@46128
  1174
proof (safe intro!: imageI)
hoelzl@36622
  1175
  fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
hoelzl@36622
  1176
  show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
hoelzl@36622
  1177
    using * eq[symmetric] by auto
hoelzl@36622
  1178
qed simp_all
haftmann@35822
  1179
Andreas@60057
  1180
lemma inj_on_apfst [simp]: "inj_on (apfst f) (A \<times> UNIV) \<longleftrightarrow> inj_on f A"
Andreas@60057
  1181
by(auto simp add: inj_on_def)
Andreas@60057
  1182
Andreas@60057
  1183
lemma inj_apfst [simp]: "inj (apfst f) \<longleftrightarrow> inj f"
Andreas@60057
  1184
using inj_on_apfst[of f UNIV] by simp
Andreas@60057
  1185
Andreas@60057
  1186
lemma inj_on_apsnd [simp]: "inj_on (apsnd f) (UNIV \<times> A) \<longleftrightarrow> inj_on f A"
Andreas@60057
  1187
by(auto simp add: inj_on_def)
Andreas@60057
  1188
Andreas@60057
  1189
lemma inj_apsnd [simp]: "inj (apsnd f) \<longleftrightarrow> inj f"
Andreas@60057
  1190
using inj_on_apsnd[of f UNIV] by simp
Andreas@60057
  1191
haftmann@61127
  1192
context
haftmann@61127
  1193
begin
haftmann@61127
  1194
haftmann@61127
  1195
qualified definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
haftmann@46128
  1196
  [code_abbrev]: "product A B = A \<times> B"
haftmann@46128
  1197
haftmann@46128
  1198
lemma member_product:
haftmann@46128
  1199
  "x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B"
haftmann@61127
  1200
  by (simp add: Product_Type.product_def)
haftmann@46128
  1201
haftmann@61127
  1202
end
haftmann@61127
  1203
  
wenzelm@60758
  1204
text \<open>The following @{const map_prod} lemmas are due to Joachim Breitner:\<close>
haftmann@40607
  1205
blanchet@55932
  1206
lemma map_prod_inj_on:
haftmann@40607
  1207
  assumes "inj_on f A" and "inj_on g B"
blanchet@55932
  1208
  shows "inj_on (map_prod f g) (A \<times> B)"
haftmann@40607
  1209
proof (rule inj_onI)
haftmann@40607
  1210
  fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c"
haftmann@40607
  1211
  assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto
haftmann@40607
  1212
  assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto
blanchet@55932
  1213
  assume "map_prod f g x = map_prod f g y"
blanchet@55932
  1214
  hence "fst (map_prod f g x) = fst (map_prod f g y)" by (auto)
haftmann@40607
  1215
  hence "f (fst x) = f (fst y)" by (cases x,cases y,auto)
wenzelm@60758
  1216
  with \<open>inj_on f A\<close> and \<open>fst x \<in> A\<close> and \<open>fst y \<in> A\<close>
haftmann@40607
  1217
  have "fst x = fst y" by (auto dest:dest:inj_onD)
wenzelm@60758
  1218
  moreover from \<open>map_prod f g x = map_prod f g y\<close>
blanchet@55932
  1219
  have "snd (map_prod f g x) = snd (map_prod f g y)" by (auto)
haftmann@40607
  1220
  hence "g (snd x) = g (snd y)" by (cases x,cases y,auto)
wenzelm@60758
  1221
  with \<open>inj_on g B\<close> and \<open>snd x \<in> B\<close> and \<open>snd y \<in> B\<close>
haftmann@40607
  1222
  have "snd x = snd y" by (auto dest:dest:inj_onD)
haftmann@40607
  1223
  ultimately show "x = y" by(rule prod_eqI)
haftmann@40607
  1224
qed
haftmann@40607
  1225
blanchet@55932
  1226
lemma map_prod_surj:
hoelzl@40702
  1227
  fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd"
haftmann@40607
  1228
  assumes "surj f" and "surj g"
blanchet@55932
  1229
  shows "surj (map_prod f g)"
haftmann@40607
  1230
unfolding surj_def
haftmann@40607
  1231
proof
haftmann@40607
  1232
  fix y :: "'b \<times> 'd"
wenzelm@60758
  1233
  from \<open>surj f\<close> obtain a where "fst y = f a" by (auto elim:surjE)
haftmann@40607
  1234
  moreover
wenzelm@60758
  1235
  from \<open>surj g\<close> obtain b where "snd y = g b" by (auto elim:surjE)
blanchet@55932
  1236
  ultimately have "(fst y, snd y) = map_prod f g (a,b)" by auto
blanchet@55932
  1237
  thus "\<exists>x. y = map_prod f g x" by auto
haftmann@40607
  1238
qed
haftmann@40607
  1239
blanchet@55932
  1240
lemma map_prod_surj_on:
haftmann@40607
  1241
  assumes "f ` A = A'" and "g ` B = B'"
blanchet@55932
  1242
  shows "map_prod f g ` (A \<times> B) = A' \<times> B'"
haftmann@40607
  1243
unfolding image_def
haftmann@40607
  1244
proof(rule set_eqI,rule iffI)
haftmann@40607
  1245
  fix x :: "'a \<times> 'c"
wenzelm@61076
  1246
  assume "x \<in> {y::'a \<times> 'c. \<exists>x::'b \<times> 'd\<in>A \<times> B. y = map_prod f g x}"
blanchet@55932
  1247
  then obtain y where "y \<in> A \<times> B" and "x = map_prod f g y" by blast
wenzelm@60758
  1248
  from \<open>image f A = A'\<close> and \<open>y \<in> A \<times> B\<close> have "f (fst y) \<in> A'" by auto
wenzelm@60758
  1249
  moreover from \<open>image g B = B'\<close> and \<open>y \<in> A \<times> B\<close> have "g (snd y) \<in> B'" by auto
haftmann@40607
  1250
  ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto
wenzelm@60758
  1251
  with \<open>x = map_prod f g y\<close> show "x \<in> A' \<times> B'" by (cases y, auto)
haftmann@40607
  1252
next
haftmann@40607
  1253
  fix x :: "'a \<times> 'c"
haftmann@40607
  1254
  assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto
wenzelm@60758
  1255
  from \<open>image f A = A'\<close> and \<open>fst x \<in> A'\<close> have "fst x \<in> image f A" by auto
haftmann@40607
  1256
  then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE)
wenzelm@60758
  1257
  moreover from \<open>image g B = B'\<close> and \<open>snd x \<in> B'\<close>
haftmann@40607
  1258
  obtain b where "b \<in> B" and "snd x = g b" by auto
blanchet@55932
  1259
  ultimately have "(fst x, snd x) = map_prod f g (a,b)" by auto
wenzelm@60758
  1260
  moreover from \<open>a \<in> A\<close> and  \<open>b \<in> B\<close> have "(a , b) \<in> A \<times> B" by auto
blanchet@55932
  1261
  ultimately have "\<exists>y \<in> A \<times> B. x = map_prod f g y" by auto
blanchet@55932
  1262
  thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_prod f g y}" by auto
haftmann@40607
  1263
qed
haftmann@40607
  1264
haftmann@21908
  1265
wenzelm@60758
  1266
subsection \<open>Simproc for rewriting a set comprehension into a pointfree expression\<close>
bulwahn@49764
  1267
bulwahn@49764
  1268
ML_file "Tools/set_comprehension_pointfree.ML"
bulwahn@49764
  1269
wenzelm@60758
  1270
setup \<open>
wenzelm@51717
  1271
  Code_Preproc.map_pre (fn ctxt => ctxt addsimprocs
wenzelm@61144
  1272
    [Simplifier.make_simproc @{context} "set comprehension"
wenzelm@61144
  1273
      {lhss = [@{term "Collect P"}],
wenzelm@61144
  1274
       proc = K Set_Comprehension_Pointfree.code_simproc,
wenzelm@61144
  1275
       identifier = []}])
wenzelm@60758
  1276
\<close>
bulwahn@49764
  1277
bulwahn@49764
  1278
wenzelm@60758
  1279
subsection \<open>Inductively defined sets\<close>
berghofe@15394
  1280
wenzelm@56512
  1281
(* simplify {(x1, ..., xn). (x1, ..., xn) : S} to S *)
wenzelm@60758
  1282
simproc_setup Collect_mem ("Collect t") = \<open>
wenzelm@56512
  1283
  fn _ => fn ctxt => fn ct =>
wenzelm@59582
  1284
    (case Thm.term_of ct of
wenzelm@56512
  1285
      S as Const (@{const_name Collect}, Type (@{type_name fun}, [_, T])) $ t =>
haftmann@61424
  1286
        let val (u, _, ps) = HOLogic.strip_ptupleabs t in
wenzelm@56512
  1287
          (case u of
wenzelm@56512
  1288
            (c as Const (@{const_name Set.member}, _)) $ q $ S' =>
wenzelm@56512
  1289
              (case try (HOLogic.strip_ptuple ps) q of
wenzelm@56512
  1290
                NONE => NONE
wenzelm@56512
  1291
              | SOME ts =>
wenzelm@56512
  1292
                  if not (Term.is_open S') andalso
wenzelm@56512
  1293
                    ts = map Bound (length ps downto 0)
wenzelm@56512
  1294
                  then
wenzelm@56512
  1295
                    let val simp =
wenzelm@56512
  1296
                      full_simp_tac (put_simpset HOL_basic_ss ctxt
haftmann@61424
  1297
                        addsimps [@{thm split_paired_all}, @{thm case_prod_conv}]) 1
wenzelm@56512
  1298
                    in
wenzelm@56512
  1299
                      SOME (Goal.prove ctxt [] []
wenzelm@56512
  1300
                        (Const (@{const_name Pure.eq}, T --> T --> propT) $ S $ S')
wenzelm@56512
  1301
                        (K (EVERY
wenzelm@59498
  1302
                          [resolve_tac ctxt [eq_reflection] 1,
wenzelm@59498
  1303
                           resolve_tac ctxt @{thms subset_antisym} 1,
wenzelm@59498
  1304
                           resolve_tac ctxt [subsetI] 1, dresolve_tac ctxt [CollectD] 1, simp,
wenzelm@59498
  1305
                           resolve_tac ctxt [subsetI] 1, resolve_tac ctxt [CollectI] 1, simp])))
wenzelm@56512
  1306
                    end
wenzelm@56512
  1307
                  else NONE)
wenzelm@56512
  1308
          | _ => NONE)
wenzelm@56512
  1309
        end
wenzelm@56512
  1310
    | _ => NONE)
wenzelm@60758
  1311
\<close>
blanchet@58389
  1312
wenzelm@48891
  1313
ML_file "Tools/inductive_set.ML"
haftmann@24699
  1314
haftmann@37166
  1315
wenzelm@60758
  1316
subsection \<open>Legacy theorem bindings and duplicates\<close>
haftmann@37166
  1317
blanchet@55393
  1318
lemmas fst_conv = prod.sel(1)
blanchet@55393
  1319
lemmas snd_conv = prod.sel(2)
haftmann@61032
  1320
lemmas split_def = case_prod_unfold
haftmann@61424
  1321
lemmas split_beta' = case_prod_beta'
haftmann@61424
  1322
lemmas split_beta = prod.case_eq_if
haftmann@61424
  1323
lemmas split_conv = case_prod_conv
haftmann@61424
  1324
lemmas split = case_prod_conv
huffman@44066
  1325
huffman@45204
  1326
hide_const (open) prod
huffman@45204
  1327
nipkow@10213
  1328
end