src/HOL/Product_Type.thy
author haftmann
Tue Sep 22 11:48:22 2015 +0200 (2015-09-22)
changeset 61226 af7bed1360f3
parent 61144 5e94dfead1c2
child 61378 3e04c9ca001a
permissions -rw-r--r--
effective revert of e6b1236f9b3d: spontaneous eta-contraction happens on the print translation level and can only be suppressed on the print translation level
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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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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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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 uncurry 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
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  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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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 uncurry (\<lambda>x (y, zs). b)"
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  "\<lambda>(x, y). b" \<rightleftharpoons> "CONST uncurry (\<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 "uncurry f"} as @{term "\<lambda>(x, y). f x y"} and
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  @{term "uncurry (\<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 uncurry_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match
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      | uncurry_guess_names_tr' T [Abs (x, xT, t)] =
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          (case (head_of t) of
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            Const (@{const_syntax uncurry}, _) => 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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      | uncurry_guess_names_tr' T [t] =
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          (case head_of t of
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            Const (@{const_syntax uncurry}, _) => 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)
haftmann@61226
   345
      | uncurry_guess_names_tr' _ _ = raise Match;
haftmann@61226
   346
  in [(@{const_syntax uncurry}, K uncurry_guess_names_tr')] end
haftmann@61226
   347
\<close>
haftmann@61226
   348
nipkow@10213
   349
wenzelm@60758
   350
subsubsection \<open>Code generator setup\<close>
haftmann@37166
   351
haftmann@52435
   352
code_printing
haftmann@52435
   353
  type_constructor prod \<rightharpoonup>
haftmann@52435
   354
    (SML) infix 2 "*"
haftmann@52435
   355
    and (OCaml) infix 2 "*"
haftmann@52435
   356
    and (Haskell) "!((_),/ (_))"
haftmann@52435
   357
    and (Scala) "((_),/ (_))"
haftmann@52435
   358
| constant Pair \<rightharpoonup>
haftmann@52435
   359
    (SML) "!((_),/ (_))"
haftmann@52435
   360
    and (OCaml) "!((_),/ (_))"
haftmann@52435
   361
    and (Haskell) "!((_),/ (_))"
haftmann@52435
   362
    and (Scala) "!((_),/ (_))"
haftmann@52435
   363
| class_instance  prod :: equal \<rightharpoonup>
haftmann@52435
   364
    (Haskell) -
haftmann@52435
   365
| constant "HOL.equal :: 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" \<rightharpoonup>
haftmann@52435
   366
    (Haskell) infix 4 "=="
haftmann@37166
   367
haftmann@37166
   368
wenzelm@60758
   369
subsubsection \<open>Fundamental operations and properties\<close>
wenzelm@11838
   370
bulwahn@49897
   371
lemma Pair_inject:
bulwahn@49897
   372
  assumes "(a, b) = (a', b')"
bulwahn@49897
   373
    and "a = a' ==> b = b' ==> R"
bulwahn@49897
   374
  shows R
bulwahn@49897
   375
  using assms by simp
bulwahn@49897
   376
haftmann@26358
   377
lemma surj_pair [simp]: "EX x y. p = (x, y)"
haftmann@37166
   378
  by (cases p) simp
nipkow@10213
   379
haftmann@52435
   380
code_printing
haftmann@52435
   381
  constant fst \<rightharpoonup> (Haskell) "fst"
haftmann@52435
   382
| constant snd \<rightharpoonup> (Haskell) "snd"
haftmann@26358
   383
haftmann@61125
   384
lemma case_prod_unfold [nitpick_unfold]: "uncurry = (%c p. c (fst p) (snd p))"
nipkow@39302
   385
  by (simp add: fun_eq_iff split: prod.split)
haftmann@26358
   386
wenzelm@11838
   387
lemma fst_eqD: "fst (x, y) = a ==> x = a"
wenzelm@11838
   388
  by simp
wenzelm@11838
   389
wenzelm@11838
   390
lemma snd_eqD: "snd (x, y) = a ==> y = a"
wenzelm@11838
   391
  by simp
wenzelm@11838
   392
blanchet@55393
   393
lemmas surjective_pairing = prod.collapse [symmetric]
wenzelm@11838
   394
huffman@44066
   395
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
haftmann@37166
   396
  by (cases s, cases t) simp
haftmann@37166
   397
haftmann@37166
   398
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
huffman@44066
   399
  by (simp add: prod_eq_iff)
haftmann@37166
   400
haftmann@61127
   401
lemma split_conv [simp, code]: "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
blanchet@55642
   402
  by (fact prod.case)
haftmann@37166
   403
haftmann@61127
   404
lemma splitI: "f a b \<Longrightarrow> case (a, b) of (c, d) \<Rightarrow> f c d"
haftmann@37166
   405
  by (rule split_conv [THEN iffD2])
haftmann@37166
   406
haftmann@61127
   407
lemma splitD: "(case (a, b) of (c, d) \<Rightarrow> f c d) \<Longrightarrow> f a b"
haftmann@37166
   408
  by (rule split_conv [THEN iffD1])
haftmann@37166
   409
haftmann@61127
   410
lemma split_Pair [simp]: "uncurry Pair = id"
nipkow@39302
   411
  by (simp add: fun_eq_iff split: prod.split)
haftmann@37166
   412
haftmann@37166
   413
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
wenzelm@60758
   414
  -- \<open>Subsumes the old @{text split_Pair} when @{term f} is the identity function.\<close>
nipkow@39302
   415
  by (simp add: fun_eq_iff split: prod.split)
haftmann@37166
   416
haftmann@61127
   417
lemma split_comp: "(case x of (a, b) \<Rightarrow> (f \<circ> g) a b) = f (g (fst x)) (snd x)"
haftmann@37166
   418
  by (cases x) simp
haftmann@37166
   419
haftmann@61125
   420
lemma split_twice: "uncurry f (uncurry g p) = uncurry (\<lambda>x y. uncurry f (g x y)) p"
haftmann@61032
   421
  by (fact prod.case_distrib)
haftmann@37166
   422
haftmann@61125
   423
lemma The_split: "The (uncurry P) = (THE xy. P (fst xy) (snd xy))"
blanchet@55414
   424
  by (simp add: case_prod_unfold)
haftmann@37166
   425
haftmann@58468
   426
lemmas split_weak_cong = prod.case_cong_weak
wenzelm@60758
   427
  -- \<open>Prevents simplification of @{term c}: much faster\<close>
haftmann@37166
   428
haftmann@37166
   429
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
haftmann@37166
   430
  by (simp add: split_eta)
haftmann@37166
   431
blanchet@47740
   432
lemma split_paired_all [no_atp]: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
wenzelm@11820
   433
proof
wenzelm@11820
   434
  fix a b
wenzelm@11820
   435
  assume "!!x. PROP P x"
wenzelm@19535
   436
  then show "PROP P (a, b)" .
wenzelm@11820
   437
next
wenzelm@11820
   438
  fix x
wenzelm@11820
   439
  assume "!!a b. PROP P (a, b)"
wenzelm@60758
   440
  from \<open>PROP P (fst x, snd x)\<close> show "PROP P x" by simp
wenzelm@11820
   441
qed
wenzelm@11820
   442
haftmann@61125
   443
lemma uncurry_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))"
hoelzl@50104
   444
  by (cases x) simp
hoelzl@50104
   445
wenzelm@60758
   446
text \<open>
wenzelm@11838
   447
  The rule @{thm [source] split_paired_all} does not work with the
wenzelm@11838
   448
  Simplifier because it also affects premises in congrence rules,
wenzelm@11838
   449
  where this can lead to premises of the form @{text "!!a b. ... =
wenzelm@11838
   450
  ?P(a, b)"} which cannot be solved by reflexivity.
wenzelm@60758
   451
\<close>
wenzelm@11838
   452
haftmann@26358
   453
lemmas split_tupled_all = split_paired_all unit_all_eq2
haftmann@26358
   454
wenzelm@60758
   455
ML \<open>
wenzelm@11838
   456
  (* replace parameters of product type by individual component parameters *)
wenzelm@11838
   457
  local (* filtering with exists_paired_all is an essential optimization *)
wenzelm@56245
   458
    fun exists_paired_all (Const (@{const_name Pure.all}, _) $ Abs (_, T, t)) =
wenzelm@11838
   459
          can HOLogic.dest_prodT T orelse exists_paired_all t
wenzelm@11838
   460
      | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
wenzelm@11838
   461
      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
wenzelm@11838
   462
      | exists_paired_all _ = false;
wenzelm@51717
   463
    val ss =
wenzelm@51717
   464
      simpset_of
wenzelm@51717
   465
       (put_simpset HOL_basic_ss @{context}
wenzelm@51717
   466
        addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
wenzelm@51717
   467
        addsimprocs [@{simproc unit_eq}]);
wenzelm@11838
   468
  in
wenzelm@51717
   469
    fun split_all_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   470
      if exists_paired_all t then safe_full_simp_tac (put_simpset ss ctxt) i else no_tac);
wenzelm@51717
   471
wenzelm@51717
   472
    fun unsafe_split_all_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   473
      if exists_paired_all t then full_simp_tac (put_simpset ss ctxt) i else no_tac);
wenzelm@51717
   474
wenzelm@51717
   475
    fun split_all ctxt th =
wenzelm@51717
   476
      if exists_paired_all (Thm.prop_of th)
wenzelm@51717
   477
      then full_simplify (put_simpset ss ctxt) th else th;
wenzelm@11838
   478
  end;
wenzelm@60758
   479
\<close>
wenzelm@11838
   480
wenzelm@60758
   481
setup \<open>map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac))\<close>
wenzelm@11838
   482
blanchet@47740
   483
lemma split_paired_All [simp, no_atp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@60758
   484
  -- \<open>@{text "[iff]"} is not a good idea because it makes @{text blast} loop\<close>
wenzelm@11838
   485
  by fast
wenzelm@11838
   486
blanchet@47740
   487
lemma split_paired_Ex [simp, no_atp]: "(EX x. P x) = (EX a b. P (a, b))"
haftmann@26358
   488
  by fast
haftmann@26358
   489
blanchet@47740
   490
lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@60758
   491
  -- \<open>Can't be added to simpset: loops!\<close>
haftmann@26358
   492
  by (simp add: split_eta)
wenzelm@11838
   493
wenzelm@60758
   494
text \<open>
wenzelm@11838
   495
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
wenzelm@11838
   496
  @{thm [source] split_eta} as a rewrite rule is not general enough,
wenzelm@11838
   497
  and using @{thm [source] cond_split_eta} directly would render some
wenzelm@11838
   498
  existing proofs very inefficient; similarly for @{text
haftmann@26358
   499
  split_beta}.
wenzelm@60758
   500
\<close>
wenzelm@11838
   501
wenzelm@60758
   502
ML \<open>
wenzelm@11838
   503
local
wenzelm@51717
   504
  val cond_split_eta_ss =
wenzelm@51717
   505
    simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms cond_split_eta});
wenzelm@35364
   506
  fun Pair_pat k 0 (Bound m) = (m = k)
wenzelm@35364
   507
    | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =
wenzelm@35364
   508
        i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t
wenzelm@35364
   509
    | Pair_pat _ _ _ = false;
wenzelm@35364
   510
  fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t
wenzelm@35364
   511
    | no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@35364
   512
    | no_args k i (Bound m) = m < k orelse m > k + i
wenzelm@35364
   513
    | no_args _ _ _ = true;
wenzelm@35364
   514
  fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE
haftmann@61125
   515
    | split_pat tp i (Const (@{const_name uncurry}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
wenzelm@35364
   516
    | split_pat tp i _ = NONE;
wenzelm@51717
   517
  fun metaeq ctxt lhs rhs = mk_meta_eq (Goal.prove ctxt [] []
wenzelm@35364
   518
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
wenzelm@51717
   519
        (K (simp_tac (put_simpset cond_split_eta_ss ctxt) 1)));
wenzelm@11838
   520
wenzelm@35364
   521
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t
wenzelm@35364
   522
    | beta_term_pat k i (t $ u) =
wenzelm@35364
   523
        Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@35364
   524
    | beta_term_pat k i t = no_args k i t;
wenzelm@35364
   525
  fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@35364
   526
    | eta_term_pat _ _ _ = false;
wenzelm@11838
   527
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@35364
   528
    | subst arg k i (t $ u) =
wenzelm@35364
   529
        if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@35364
   530
        else (subst arg k i t $ subst arg k i u)
wenzelm@35364
   531
    | subst arg k i t = t;
wenzelm@43595
   532
in
haftmann@61125
   533
  fun beta_proc ctxt (s as Const (@{const_name uncurry}, _) $ Abs (_, _, t) $ arg) =
wenzelm@11838
   534
        (case split_pat beta_term_pat 1 t of
wenzelm@51717
   535
          SOME (i, f) => SOME (metaeq ctxt s (subst arg 0 i f))
skalberg@15531
   536
        | NONE => NONE)
wenzelm@35364
   537
    | beta_proc _ _ = NONE;
haftmann@61125
   538
  fun eta_proc ctxt (s as Const (@{const_name uncurry}, _) $ Abs (_, _, t)) =
wenzelm@11838
   539
        (case split_pat eta_term_pat 1 t of
haftmann@58468
   540
          SOME (_, ft) => SOME (metaeq ctxt s (let val f $ _ = ft in f end))
skalberg@15531
   541
        | NONE => NONE)
wenzelm@35364
   542
    | eta_proc _ _ = NONE;
wenzelm@11838
   543
end;
wenzelm@60758
   544
\<close>
wenzelm@59582
   545
simproc_setup split_beta ("split f z") =
wenzelm@60758
   546
  \<open>fn _ => fn ctxt => fn ct => beta_proc ctxt (Thm.term_of ct)\<close>
wenzelm@59582
   547
simproc_setup split_eta ("split f") =
wenzelm@60758
   548
  \<open>fn _ => fn ctxt => fn ct => eta_proc ctxt (Thm.term_of ct)\<close>
wenzelm@11838
   549
haftmann@58468
   550
lemmas split_beta [mono] = prod.case_eq_if
wenzelm@11838
   551
hoelzl@50104
   552
lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
hoelzl@50104
   553
  by (auto simp: fun_eq_iff)
hoelzl@50104
   554
haftmann@58468
   555
lemmas split_split [no_atp] = prod.split
wenzelm@60758
   556
  -- \<open>For use with @{text split} and the Simplifier.\<close>
wenzelm@11838
   557
wenzelm@60758
   558
text \<open>
wenzelm@11838
   559
  @{thm [source] split_split} could be declared as @{text "[split]"}
wenzelm@11838
   560
  done after the Splitter has been speeded up significantly;
wenzelm@11838
   561
  precompute the constants involved and don't do anything unless the
wenzelm@11838
   562
  current goal contains one of those constants.
wenzelm@60758
   563
\<close>
wenzelm@11838
   564
haftmann@58468
   565
lemmas split_split_asm [no_atp] = prod.split_asm
wenzelm@11838
   566
wenzelm@60758
   567
text \<open>
haftmann@61125
   568
  \medskip @{const uncurry} used as a logical connective or set former.
wenzelm@11838
   569
wenzelm@11838
   570
  \medskip These rules are for use with @{text blast}; could instead
wenzelm@60758
   571
  call @{text simp} using @{thm [source] prod.split} as rewrite.\<close>
wenzelm@11838
   572
haftmann@61127
   573
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> case p of (a, b) \<Rightarrow> c a b"
wenzelm@11838
   574
  apply (simp only: split_tupled_all)
wenzelm@11838
   575
  apply (simp (no_asm_simp))
wenzelm@11838
   576
  done
wenzelm@11838
   577
haftmann@61127
   578
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> (case p of (a, b) \<Rightarrow> c a b) x"
wenzelm@11838
   579
  apply (simp only: split_tupled_all)
wenzelm@11838
   580
  apply (simp (no_asm_simp))
wenzelm@11838
   581
  done
wenzelm@11838
   582
haftmann@61127
   583
lemma splitE: "(case p of (a, b) \<Rightarrow> c a b) ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@37591
   584
  by (induct p) auto
wenzelm@11838
   585
haftmann@61127
   586
lemma splitE': "(case p of (a, b) \<Rightarrow> c a b) z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@37591
   587
  by (induct p) auto
wenzelm@11838
   588
wenzelm@11838
   589
lemma splitE2:
haftmann@61127
   590
  "[| Q (case z of (a, b) \<Rightarrow> P a b);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
wenzelm@11838
   591
proof -
haftmann@61125
   592
  assume q: "Q (uncurry P z)"
wenzelm@11838
   593
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
wenzelm@11838
   594
  show R
wenzelm@11838
   595
    apply (rule r surjective_pairing)+
wenzelm@11838
   596
    apply (rule split_beta [THEN subst], rule q)
wenzelm@11838
   597
    done
wenzelm@11838
   598
qed
wenzelm@11838
   599
haftmann@61127
   600
lemma splitD':
haftmann@61127
   601
  "(case (a, b) of (c, d) \<Rightarrow> R c d) c \<Longrightarrow> R a b c"
wenzelm@11838
   602
  by simp
wenzelm@11838
   603
haftmann@61127
   604
lemma mem_splitI:
haftmann@61127
   605
  "z \<in> c a b \<Longrightarrow> z \<in> (case (a, b) of (d, e) \<Rightarrow> c d e)"
wenzelm@11838
   606
  by simp
wenzelm@11838
   607
haftmann@61127
   608
lemma mem_splitI2:
haftmann@61127
   609
  "\<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
   610
  by (simp only: split_tupled_all) simp
wenzelm@11838
   611
wenzelm@18372
   612
lemma mem_splitE:
haftmann@61125
   613
  assumes "z \<in> uncurry c p"
haftmann@58468
   614
  obtains x y where "p = (x, y)" and "z \<in> c x y"
haftmann@58468
   615
  using assms by (rule splitE2)
wenzelm@11838
   616
wenzelm@11838
   617
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
wenzelm@11838
   618
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
wenzelm@11838
   619
wenzelm@60758
   620
ML \<open>
wenzelm@11838
   621
local (* filtering with exists_p_split is an essential optimization *)
haftmann@61125
   622
  fun exists_p_split (Const (@{const_name uncurry},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true
wenzelm@11838
   623
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838
   624
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838
   625
    | exists_p_split _ = false;
wenzelm@11838
   626
in
wenzelm@51717
   627
fun split_conv_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   628
  if exists_p_split t
wenzelm@51717
   629
  then safe_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms split_conv}) i
wenzelm@51717
   630
  else no_tac);
wenzelm@11838
   631
end;
wenzelm@60758
   632
\<close>
wenzelm@26340
   633
wenzelm@11838
   634
(* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838
   635
   to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@60758
   636
setup \<open>map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac))\<close>
wenzelm@11838
   637
blanchet@54147
   638
lemma split_eta_SetCompr [simp, no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
wenzelm@18372
   639
  by (rule ext) fast
wenzelm@11838
   640
haftmann@61125
   641
lemma split_eta_SetCompr2 [simp, no_atp]: "(%u. EX x y. u = (x, y) & P x y) = uncurry P"
wenzelm@18372
   642
  by (rule ext) fast
wenzelm@11838
   643
haftmann@61125
   644
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & uncurry Q ab)"
wenzelm@60758
   645
  -- \<open>Allows simplifications of nested splits in case of independent predicates.\<close>
wenzelm@18372
   646
  by (rule ext) blast
wenzelm@11838
   647
nipkow@14337
   648
(* Do NOT make this a simp rule as it
nipkow@14337
   649
   a) only helps in special situations
nipkow@14337
   650
   b) can lead to nontermination in the presence of split_def
nipkow@14337
   651
*)
nipkow@14337
   652
lemma split_comp_eq: 
paulson@20415
   653
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
haftmann@61125
   654
  shows "(%u. f (g (fst u)) (snd u)) = (uncurry (%x. f (g x)))"
wenzelm@18372
   655
  by (rule ext) auto
oheimb@14101
   656
haftmann@26358
   657
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
haftmann@26358
   658
  apply (rule_tac x = "(a, b)" in image_eqI)
haftmann@26358
   659
   apply auto
haftmann@26358
   660
  done
haftmann@26358
   661
wenzelm@11838
   662
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838
   663
  by blast
wenzelm@11838
   664
wenzelm@11838
   665
(*
wenzelm@11838
   666
the following  would be slightly more general,
wenzelm@11838
   667
but cannot be used as rewrite rule:
wenzelm@11838
   668
### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838
   669
### ?y = .x
wenzelm@11838
   670
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208
   671
by (rtac some_equality 1)
paulson@14208
   672
by ( Simp_tac 1)
paulson@14208
   673
by (split_all_tac 1)
paulson@14208
   674
by (Asm_full_simp_tac 1)
wenzelm@11838
   675
qed "The_split_eq";
wenzelm@11838
   676
*)
wenzelm@11838
   677
wenzelm@60758
   678
text \<open>
wenzelm@11838
   679
  Setup of internal @{text split_rule}.
wenzelm@60758
   680
\<close>
wenzelm@11838
   681
blanchet@55642
   682
lemmas case_prodI = prod.case [THEN iffD2]
haftmann@24699
   683
haftmann@61125
   684
lemma case_prodI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> uncurry c p"
haftmann@37678
   685
  by (fact splitI2)
haftmann@24699
   686
haftmann@61125
   687
lemma case_prodI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> uncurry c p x"
haftmann@37678
   688
  by (fact splitI2')
haftmann@24699
   689
haftmann@61125
   690
lemma case_prodE: "uncurry c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@37678
   691
  by (fact splitE)
haftmann@24699
   692
haftmann@61125
   693
lemma case_prodE': "uncurry c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@37678
   694
  by (fact splitE')
haftmann@24699
   695
blanchet@55414
   696
declare case_prodI [intro!]
haftmann@24699
   697
blanchet@55414
   698
lemma case_prod_beta:
haftmann@61125
   699
  "uncurry f p = f (fst p) (snd p)"
haftmann@37591
   700
  by (fact split_beta)
bulwahn@26143
   701
blanchet@55417
   702
lemma prod_cases3 [cases type]:
haftmann@24699
   703
  obtains (fields) a b c where "y = (a, b, c)"
haftmann@24699
   704
  by (cases y, case_tac b) blast
haftmann@24699
   705
haftmann@24699
   706
lemma prod_induct3 [case_names fields, induct type]:
haftmann@24699
   707
    "(!!a b c. P (a, b, c)) ==> P x"
haftmann@24699
   708
  by (cases x) blast
haftmann@24699
   709
blanchet@55417
   710
lemma prod_cases4 [cases type]:
haftmann@24699
   711
  obtains (fields) a b c d where "y = (a, b, c, d)"
haftmann@24699
   712
  by (cases y, case_tac c) blast
haftmann@24699
   713
haftmann@24699
   714
lemma prod_induct4 [case_names fields, induct type]:
haftmann@24699
   715
    "(!!a b c d. P (a, b, c, d)) ==> P x"
haftmann@24699
   716
  by (cases x) blast
haftmann@24699
   717
blanchet@55417
   718
lemma prod_cases5 [cases type]:
haftmann@24699
   719
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
haftmann@24699
   720
  by (cases y, case_tac d) blast
haftmann@24699
   721
haftmann@24699
   722
lemma prod_induct5 [case_names fields, induct type]:
haftmann@24699
   723
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
haftmann@24699
   724
  by (cases x) blast
haftmann@24699
   725
blanchet@55417
   726
lemma prod_cases6 [cases type]:
haftmann@24699
   727
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
haftmann@24699
   728
  by (cases y, case_tac e) blast
haftmann@24699
   729
haftmann@24699
   730
lemma prod_induct6 [case_names fields, induct type]:
haftmann@24699
   731
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
haftmann@24699
   732
  by (cases x) blast
haftmann@24699
   733
blanchet@55417
   734
lemma prod_cases7 [cases type]:
haftmann@24699
   735
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
haftmann@24699
   736
  by (cases y, case_tac f) blast
haftmann@24699
   737
haftmann@24699
   738
lemma prod_induct7 [case_names fields, induct type]:
haftmann@24699
   739
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
haftmann@24699
   740
  by (cases x) blast
haftmann@24699
   741
haftmann@37166
   742
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
haftmann@61125
   743
  "internal_split == uncurry"
haftmann@37166
   744
haftmann@37166
   745
lemma internal_split_conv: "internal_split c (a, b) = c a b"
haftmann@37166
   746
  by (simp only: internal_split_def split_conv)
haftmann@37166
   747
wenzelm@48891
   748
ML_file "Tools/split_rule.ML"
haftmann@37166
   749
haftmann@37166
   750
hide_const internal_split
haftmann@37166
   751
haftmann@24699
   752
wenzelm@60758
   753
subsubsection \<open>Derived operations\<close>
haftmann@26358
   754
haftmann@37387
   755
definition curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where
haftmann@37387
   756
  "curry = (\<lambda>c x y. c (x, y))"
haftmann@37166
   757
haftmann@37166
   758
lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
haftmann@37166
   759
  by (simp add: curry_def)
haftmann@37166
   760
haftmann@37166
   761
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
haftmann@37166
   762
  by (simp add: curry_def)
haftmann@37166
   763
haftmann@37166
   764
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
haftmann@37166
   765
  by (simp add: curry_def)
haftmann@37166
   766
haftmann@37166
   767
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@37166
   768
  by (simp add: curry_def)
haftmann@37166
   769
haftmann@61125
   770
lemma curry_split [simp]: "curry (uncurry f) = f"
haftmann@61032
   771
  by (simp add: curry_def case_prod_unfold)
haftmann@37166
   772
haftmann@61125
   773
lemma split_curry [simp]: "uncurry (curry f) = f"
haftmann@61032
   774
  by (simp add: curry_def case_prod_unfold)
haftmann@37166
   775
Andreas@54630
   776
lemma curry_K: "curry (\<lambda>x. c) = (\<lambda>x y. c)"
Andreas@54630
   777
by(simp add: fun_eq_iff)
Andreas@54630
   778
wenzelm@60758
   779
text \<open>
haftmann@26358
   780
  The composition-uncurry combinator.
wenzelm@60758
   781
\<close>
haftmann@26358
   782
haftmann@37751
   783
notation fcomp (infixl "\<circ>>" 60)
haftmann@26358
   784
haftmann@37751
   785
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where
haftmann@61125
   786
  "f \<circ>\<rightarrow> g = (\<lambda>x. uncurry g (f x))"
haftmann@26358
   787
haftmann@37678
   788
lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))"
blanchet@55414
   789
  by (simp add: fun_eq_iff scomp_def case_prod_unfold)
haftmann@37678
   790
haftmann@61125
   791
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = uncurry g (f x)"
blanchet@55414
   792
  by (simp add: scomp_unfold case_prod_unfold)
haftmann@26358
   793
haftmann@37751
   794
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x"
huffman@44921
   795
  by (simp add: fun_eq_iff)
haftmann@26358
   796
haftmann@37751
   797
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x"
huffman@44921
   798
  by (simp add: fun_eq_iff)
haftmann@26358
   799
haftmann@37751
   800
lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
nipkow@39302
   801
  by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358
   802
haftmann@37751
   803
lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
nipkow@39302
   804
  by (simp add: fun_eq_iff scomp_unfold fcomp_def)
haftmann@26358
   805
haftmann@37751
   806
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
huffman@44921
   807
  by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358
   808
haftmann@52435
   809
code_printing
haftmann@52435
   810
  constant scomp \<rightharpoonup> (Eval) infixl 3 "#->"
haftmann@31202
   811
haftmann@37751
   812
no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
   813
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@26358
   814
wenzelm@60758
   815
text \<open>
blanchet@55932
   816
  @{term map_prod} --- action of the product functor upon
krauss@36664
   817
  functions.
wenzelm@60758
   818
\<close>
haftmann@21195
   819
blanchet@55932
   820
definition map_prod :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
blanchet@55932
   821
  "map_prod f g = (\<lambda>(x, y). (f x, g y))"
haftmann@26358
   822
blanchet@55932
   823
lemma map_prod_simp [simp, code]:
blanchet@55932
   824
  "map_prod f g (a, b) = (f a, g b)"
blanchet@55932
   825
  by (simp add: map_prod_def)
haftmann@26358
   826
blanchet@55932
   827
functor map_prod: map_prod
huffman@44921
   828
  by (auto simp add: split_paired_all)
nipkow@37278
   829
blanchet@55932
   830
lemma fst_map_prod [simp]:
blanchet@55932
   831
  "fst (map_prod f g x) = f (fst x)"
haftmann@40607
   832
  by (cases x) simp_all
nipkow@37278
   833
traytel@58916
   834
lemma snd_map_prod [simp]:
blanchet@55932
   835
  "snd (map_prod f g x) = g (snd x)"
haftmann@40607
   836
  by (cases x) simp_all
nipkow@37278
   837
blanchet@55932
   838
lemma fst_comp_map_prod [simp]:
blanchet@55932
   839
  "fst \<circ> map_prod f g = f \<circ> fst"
haftmann@40607
   840
  by (rule ext) simp_all
nipkow@37278
   841
blanchet@55932
   842
lemma snd_comp_map_prod [simp]:
blanchet@55932
   843
  "snd \<circ> map_prod f g = g \<circ> snd"
haftmann@40607
   844
  by (rule ext) simp_all
haftmann@26358
   845
blanchet@55932
   846
lemma map_prod_compose:
blanchet@55932
   847
  "map_prod (f1 o f2) (g1 o g2) = (map_prod f1 g1 o map_prod f2 g2)"
blanchet@55932
   848
  by (rule ext) (simp add: map_prod.compositionality comp_def)
haftmann@26358
   849
blanchet@55932
   850
lemma map_prod_ident [simp]:
blanchet@55932
   851
  "map_prod (%x. x) (%y. y) = (%z. z)"
blanchet@55932
   852
  by (rule ext) (simp add: map_prod.identity)
haftmann@40607
   853
blanchet@55932
   854
lemma map_prod_imageI [intro]:
blanchet@55932
   855
  "(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_prod f g ` R"
haftmann@40607
   856
  by (rule image_eqI) simp_all
haftmann@21195
   857
haftmann@26358
   858
lemma prod_fun_imageE [elim!]:
blanchet@55932
   859
  assumes major: "c \<in> map_prod f g ` R"
haftmann@40607
   860
    and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P"
haftmann@26358
   861
  shows P
haftmann@26358
   862
  apply (rule major [THEN imageE])
haftmann@37166
   863
  apply (case_tac x)
haftmann@26358
   864
  apply (rule cases)
haftmann@40607
   865
  apply simp_all
haftmann@26358
   866
  done
haftmann@26358
   867
haftmann@37166
   868
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where
blanchet@55932
   869
  "apfst f = map_prod f id"
haftmann@26358
   870
haftmann@37166
   871
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where
blanchet@55932
   872
  "apsnd f = map_prod id f"
haftmann@26358
   873
haftmann@26358
   874
lemma apfst_conv [simp, code]:
haftmann@26358
   875
  "apfst f (x, y) = (f x, y)" 
haftmann@26358
   876
  by (simp add: apfst_def)
haftmann@26358
   877
hoelzl@33638
   878
lemma apsnd_conv [simp, code]:
haftmann@26358
   879
  "apsnd f (x, y) = (x, f y)" 
haftmann@26358
   880
  by (simp add: apsnd_def)
haftmann@21195
   881
haftmann@33594
   882
lemma fst_apfst [simp]:
haftmann@33594
   883
  "fst (apfst f x) = f (fst x)"
haftmann@33594
   884
  by (cases x) simp
haftmann@33594
   885
haftmann@51173
   886
lemma fst_comp_apfst [simp]:
haftmann@51173
   887
  "fst \<circ> apfst f = f \<circ> fst"
haftmann@51173
   888
  by (simp add: fun_eq_iff)
haftmann@51173
   889
haftmann@33594
   890
lemma fst_apsnd [simp]:
haftmann@33594
   891
  "fst (apsnd f x) = fst x"
haftmann@33594
   892
  by (cases x) simp
haftmann@33594
   893
haftmann@51173
   894
lemma fst_comp_apsnd [simp]:
haftmann@51173
   895
  "fst \<circ> apsnd f = fst"
haftmann@51173
   896
  by (simp add: fun_eq_iff)
haftmann@51173
   897
haftmann@33594
   898
lemma snd_apfst [simp]:
haftmann@33594
   899
  "snd (apfst f x) = snd x"
haftmann@33594
   900
  by (cases x) simp
haftmann@33594
   901
haftmann@51173
   902
lemma snd_comp_apfst [simp]:
haftmann@51173
   903
  "snd \<circ> apfst f = snd"
haftmann@51173
   904
  by (simp add: fun_eq_iff)
haftmann@51173
   905
haftmann@33594
   906
lemma snd_apsnd [simp]:
haftmann@33594
   907
  "snd (apsnd f x) = f (snd x)"
haftmann@33594
   908
  by (cases x) simp
haftmann@33594
   909
haftmann@51173
   910
lemma snd_comp_apsnd [simp]:
haftmann@51173
   911
  "snd \<circ> apsnd f = f \<circ> snd"
haftmann@51173
   912
  by (simp add: fun_eq_iff)
haftmann@51173
   913
haftmann@33594
   914
lemma apfst_compose:
haftmann@33594
   915
  "apfst f (apfst g x) = apfst (f \<circ> g) x"
haftmann@33594
   916
  by (cases x) simp
haftmann@33594
   917
haftmann@33594
   918
lemma apsnd_compose:
haftmann@33594
   919
  "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
haftmann@33594
   920
  by (cases x) simp
haftmann@33594
   921
haftmann@33594
   922
lemma apfst_apsnd [simp]:
haftmann@33594
   923
  "apfst f (apsnd g x) = (f (fst x), g (snd x))"
haftmann@33594
   924
  by (cases x) simp
haftmann@33594
   925
haftmann@33594
   926
lemma apsnd_apfst [simp]:
haftmann@33594
   927
  "apsnd f (apfst g x) = (g (fst x), f (snd x))"
haftmann@33594
   928
  by (cases x) simp
haftmann@33594
   929
haftmann@33594
   930
lemma apfst_id [simp] :
haftmann@33594
   931
  "apfst id = id"
nipkow@39302
   932
  by (simp add: fun_eq_iff)
haftmann@33594
   933
haftmann@33594
   934
lemma apsnd_id [simp] :
haftmann@33594
   935
  "apsnd id = id"
nipkow@39302
   936
  by (simp add: fun_eq_iff)
haftmann@33594
   937
haftmann@33594
   938
lemma apfst_eq_conv [simp]:
haftmann@33594
   939
  "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
haftmann@33594
   940
  by (cases x) simp
haftmann@33594
   941
haftmann@33594
   942
lemma apsnd_eq_conv [simp]:
haftmann@33594
   943
  "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"
haftmann@33594
   944
  by (cases x) simp
haftmann@33594
   945
hoelzl@33638
   946
lemma apsnd_apfst_commute:
hoelzl@33638
   947
  "apsnd f (apfst g p) = apfst g (apsnd f p)"
hoelzl@33638
   948
  by simp
haftmann@21195
   949
haftmann@56626
   950
context
haftmann@56626
   951
begin
haftmann@56626
   952
wenzelm@60758
   953
local_setup \<open>Local_Theory.map_background_naming (Name_Space.mandatory_path "prod")\<close>
haftmann@56626
   954
haftmann@56545
   955
definition swap :: "'a \<times> 'b \<Rightarrow> 'b \<times> 'a"
haftmann@56545
   956
where
haftmann@56545
   957
  "swap p = (snd p, fst p)"
haftmann@56545
   958
haftmann@56626
   959
end
haftmann@56626
   960
haftmann@56545
   961
lemma swap_simp [simp]:
haftmann@56626
   962
  "prod.swap (x, y) = (y, x)"
haftmann@56626
   963
  by (simp add: prod.swap_def)
haftmann@56545
   964
haftmann@58195
   965
lemma swap_swap [simp]:
haftmann@58195
   966
  "prod.swap (prod.swap p) = p"
haftmann@58195
   967
  by (cases p) simp
haftmann@58195
   968
haftmann@58195
   969
lemma swap_comp_swap [simp]:
haftmann@58195
   970
  "prod.swap \<circ> prod.swap = id"
haftmann@58195
   971
  by (simp add: fun_eq_iff)
haftmann@58195
   972
haftmann@56545
   973
lemma pair_in_swap_image [simp]:
haftmann@56626
   974
  "(y, x) \<in> prod.swap ` A \<longleftrightarrow> (x, y) \<in> A"
haftmann@56545
   975
  by (auto intro!: image_eqI)
haftmann@56545
   976
haftmann@56545
   977
lemma inj_swap [simp]:
haftmann@56626
   978
  "inj_on prod.swap A"
haftmann@56626
   979
  by (rule inj_onI) auto
haftmann@56626
   980
haftmann@56626
   981
lemma swap_inj_on:
haftmann@56626
   982
  "inj_on (\<lambda>(i, j). (j, i)) A"
haftmann@56626
   983
  by (rule inj_onI) auto
haftmann@56545
   984
haftmann@58195
   985
lemma surj_swap [simp]:
haftmann@58195
   986
  "surj prod.swap"
haftmann@58195
   987
  by (rule surjI [of _ prod.swap]) simp
haftmann@58195
   988
haftmann@58195
   989
lemma bij_swap [simp]:
haftmann@58195
   990
  "bij prod.swap"
haftmann@58195
   991
  by (simp add: bij_def)
haftmann@58195
   992
haftmann@56545
   993
lemma case_swap [simp]:
haftmann@56626
   994
  "(case prod.swap p of (y, x) \<Rightarrow> f x y) = (case p of (x, y) \<Rightarrow> f x y)"
haftmann@56545
   995
  by (cases p) simp
haftmann@56545
   996
wenzelm@60758
   997
text \<open>
haftmann@26358
   998
  Disjoint union of a family of sets -- Sigma.
wenzelm@60758
   999
\<close>
haftmann@26358
  1000
haftmann@45986
  1001
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where
haftmann@26358
  1002
  Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
haftmann@26358
  1003
haftmann@26358
  1004
abbreviation
haftmann@45986
  1005
  Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set"
haftmann@26358
  1006
    (infixr "<*>" 80) where
haftmann@26358
  1007
  "A <*> B == Sigma A (%_. B)"
haftmann@26358
  1008
haftmann@26358
  1009
notation (xsymbols)
haftmann@26358
  1010
  Times  (infixr "\<times>" 80)
berghofe@15394
  1011
haftmann@26358
  1012
notation (HTML output)
haftmann@26358
  1013
  Times  (infixr "\<times>" 80)
haftmann@26358
  1014
nipkow@45662
  1015
hide_const (open) Times
nipkow@45662
  1016
haftmann@26358
  1017
syntax
wenzelm@35115
  1018
  "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
haftmann@26358
  1019
translations
wenzelm@35115
  1020
  "SIGMA x:A. B" == "CONST Sigma A (%x. B)"
haftmann@26358
  1021
haftmann@26358
  1022
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
haftmann@26358
  1023
  by (unfold Sigma_def) blast
haftmann@26358
  1024
haftmann@26358
  1025
lemma SigmaE [elim!]:
haftmann@26358
  1026
    "[| c: Sigma A B;
haftmann@26358
  1027
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
haftmann@26358
  1028
     |] ==> P"
wenzelm@60758
  1029
  -- \<open>The general elimination rule.\<close>
haftmann@26358
  1030
  by (unfold Sigma_def) blast
haftmann@20588
  1031
wenzelm@60758
  1032
text \<open>
haftmann@26358
  1033
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
haftmann@26358
  1034
  eigenvariables.
wenzelm@60758
  1035
\<close>
haftmann@26358
  1036
haftmann@26358
  1037
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
haftmann@26358
  1038
  by blast
haftmann@26358
  1039
haftmann@26358
  1040
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
haftmann@26358
  1041
  by blast
haftmann@26358
  1042
haftmann@26358
  1043
lemma SigmaE2:
haftmann@26358
  1044
    "[| (a, b) : Sigma A B;
haftmann@26358
  1045
        [| a:A;  b:B(a) |] ==> P
haftmann@26358
  1046
     |] ==> P"
haftmann@26358
  1047
  by blast
haftmann@20588
  1048
haftmann@26358
  1049
lemma Sigma_cong:
haftmann@26358
  1050
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
haftmann@26358
  1051
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
haftmann@26358
  1052
  by auto
haftmann@26358
  1053
haftmann@26358
  1054
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
haftmann@26358
  1055
  by blast
haftmann@26358
  1056
haftmann@26358
  1057
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
haftmann@26358
  1058
  by blast
haftmann@26358
  1059
haftmann@26358
  1060
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
haftmann@26358
  1061
  by blast
haftmann@26358
  1062
haftmann@26358
  1063
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
haftmann@26358
  1064
  by auto
haftmann@21908
  1065
haftmann@26358
  1066
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
haftmann@26358
  1067
  by auto
haftmann@26358
  1068
haftmann@26358
  1069
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
haftmann@26358
  1070
  by auto
haftmann@26358
  1071
haftmann@26358
  1072
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
haftmann@26358
  1073
  by blast
haftmann@26358
  1074
hoelzl@59000
  1075
lemma Sigma_empty_iff: "(SIGMA i:I. X i) = {} \<longleftrightarrow> (\<forall>i\<in>I. X i = {})"
hoelzl@59000
  1076
  by auto
hoelzl@59000
  1077
haftmann@26358
  1078
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
haftmann@26358
  1079
  by blast
haftmann@26358
  1080
haftmann@26358
  1081
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
haftmann@26358
  1082
  by (blast elim: equalityE)
haftmann@20588
  1083
haftmann@26358
  1084
lemma SetCompr_Sigma_eq:
haftmann@61127
  1085
  "{(x, y). P x \<and> Q x y} = (SIGMA x:Collect P. Collect (Q x))"
haftmann@26358
  1086
  by blast
haftmann@26358
  1087
haftmann@61127
  1088
lemma Collect_split [simp]:
haftmann@61127
  1089
  "{(a, b). P a \<and> Q b} = Collect P \<times> Collect Q "
haftmann@61032
  1090
  by (fact SetCompr_Sigma_eq)
haftmann@26358
  1091
haftmann@26358
  1092
lemma UN_Times_distrib:
haftmann@61127
  1093
  "(\<Union>(a, b)\<in>A \<times> B. E a \<times> F b) = UNION A E \<times> UNION B F"
wenzelm@60758
  1094
  -- \<open>Suggested by Pierre Chartier\<close>
haftmann@26358
  1095
  by blast
haftmann@26358
  1096
blanchet@47740
  1097
lemma split_paired_Ball_Sigma [simp, no_atp]:
haftmann@61127
  1098
  "(\<forall>z\<in>Sigma A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>B x. P (x, y))"
haftmann@26358
  1099
  by blast
haftmann@26358
  1100
blanchet@47740
  1101
lemma split_paired_Bex_Sigma [simp, no_atp]:
haftmann@61127
  1102
  "(\<exists>z\<in>Sigma A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>y\<in>B x. P (x, y))"
haftmann@61127
  1103
  by blast
haftmann@61127
  1104
haftmann@61127
  1105
lemma Sigma_Un_distrib1:
haftmann@61127
  1106
  "Sigma (I \<union> J) C = Sigma I C \<union> Sigma J C"
haftmann@26358
  1107
  by blast
haftmann@21908
  1108
haftmann@61127
  1109
lemma Sigma_Un_distrib2:
haftmann@61127
  1110
  "(SIGMA i:I. A i \<union> B i) = Sigma I A \<union> Sigma I B"
haftmann@26358
  1111
  by blast
haftmann@26358
  1112
haftmann@61127
  1113
lemma Sigma_Int_distrib1:
haftmann@61127
  1114
  "Sigma (I \<inter> J) C = Sigma I C \<inter> Sigma J C"
haftmann@26358
  1115
  by blast
haftmann@26358
  1116
haftmann@61127
  1117
lemma Sigma_Int_distrib2:
haftmann@61127
  1118
  "(SIGMA i:I. A i \<inter> B i) = Sigma I A \<inter> Sigma I B"
haftmann@26358
  1119
  by blast
haftmann@26358
  1120
haftmann@61127
  1121
lemma Sigma_Diff_distrib1:
haftmann@61127
  1122
  "Sigma (I - J) C = Sigma I C - Sigma J C"
haftmann@26358
  1123
  by blast
haftmann@26358
  1124
haftmann@61127
  1125
lemma Sigma_Diff_distrib2:
haftmann@61127
  1126
  "(SIGMA i:I. A i - B i) = Sigma I A - Sigma I B"
haftmann@26358
  1127
  by blast
haftmann@21908
  1128
haftmann@61127
  1129
lemma Sigma_Union:
haftmann@61127
  1130
  "Sigma (\<Union>X) B = (\<Union>A\<in>X. Sigma A B)"
haftmann@26358
  1131
  by blast
haftmann@26358
  1132
wenzelm@60758
  1133
text \<open>
haftmann@26358
  1134
  Non-dependent versions are needed to avoid the need for higher-order
haftmann@26358
  1135
  matching, especially when the rules are re-oriented.
wenzelm@60758
  1136
\<close>
haftmann@21908
  1137
haftmann@61127
  1138
lemma Times_Un_distrib1:
haftmann@61127
  1139
  "(A \<union> B) \<times> C = A \<times> C \<union> B \<times> C "
haftmann@56545
  1140
  by (fact Sigma_Un_distrib1)
haftmann@26358
  1141
haftmann@61127
  1142
lemma Times_Int_distrib1:
haftmann@61127
  1143
  "(A \<inter> B) \<times> C = A \<times> C \<inter> B \<times> C "
haftmann@56545
  1144
  by (fact Sigma_Int_distrib1)
haftmann@26358
  1145
haftmann@61127
  1146
lemma Times_Diff_distrib1:
haftmann@61127
  1147
  "(A - B) \<times> C = A \<times> C - B \<times> C "
haftmann@56545
  1148
  by (fact Sigma_Diff_distrib1)
haftmann@26358
  1149
haftmann@61127
  1150
lemma Times_empty [simp]:
haftmann@61127
  1151
  "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
hoelzl@36622
  1152
  by auto
hoelzl@36622
  1153
haftmann@61127
  1154
lemma times_eq_iff:
haftmann@61127
  1155
  "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> (A = {} \<or> B = {}) \<and> (C = {} \<or> D = {})"
hoelzl@50104
  1156
  by auto
hoelzl@50104
  1157
haftmann@61127
  1158
lemma fst_image_times [simp]:
haftmann@61127
  1159
  "fst ` (A \<times> B) = (if B = {} then {} else A)"
huffman@44921
  1160
  by force
hoelzl@36622
  1161
haftmann@61127
  1162
lemma snd_image_times [simp]:
haftmann@61127
  1163
  "snd ` (A \<times> B) = (if A = {} then {} else B)"
huffman@44921
  1164
  by force
hoelzl@36622
  1165
haftmann@56545
  1166
lemma vimage_fst:
haftmann@56545
  1167
  "fst -` A = A \<times> UNIV"
haftmann@56545
  1168
  by auto
haftmann@56545
  1169
haftmann@56545
  1170
lemma vimage_snd:
haftmann@56545
  1171
  "snd -` A = UNIV \<times> A"
haftmann@56545
  1172
  by auto
haftmann@56545
  1173
nipkow@28719
  1174
lemma insert_times_insert[simp]:
nipkow@28719
  1175
  "insert a A \<times> insert b B =
nipkow@28719
  1176
   insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
haftmann@61127
  1177
  by blast
haftmann@26358
  1178
haftmann@61127
  1179
lemma vimage_Times:
haftmann@61127
  1180
  "f -` (A \<times> B) = (fst \<circ> f) -` A \<inter> (snd \<circ> f) -` B"
haftmann@61127
  1181
proof (rule set_eqI)
haftmann@61127
  1182
  fix x
haftmann@61127
  1183
  show "x \<in> f -` (A \<times> B) \<longleftrightarrow> x \<in> (fst \<circ> f) -` A \<inter> (snd \<circ> f) -` B"
haftmann@61127
  1184
    by (cases "f x") (auto split: prod.split)
haftmann@61127
  1185
qed
paulson@33271
  1186
haftmann@61127
  1187
lemma times_Int_times:
haftmann@61127
  1188
  "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
hoelzl@50104
  1189
  by auto
hoelzl@50104
  1190
haftmann@56626
  1191
lemma product_swap:
haftmann@56626
  1192
  "prod.swap ` (A \<times> B) = B \<times> A"
haftmann@56626
  1193
  by (auto simp add: set_eq_iff)
haftmann@35822
  1194
haftmann@35822
  1195
lemma swap_product:
haftmann@56626
  1196
  "(\<lambda>(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
haftmann@56626
  1197
  by (auto simp add: set_eq_iff)
haftmann@35822
  1198
hoelzl@36622
  1199
lemma image_split_eq_Sigma:
hoelzl@36622
  1200
  "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
haftmann@46128
  1201
proof (safe intro!: imageI)
hoelzl@36622
  1202
  fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
hoelzl@36622
  1203
  show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
hoelzl@36622
  1204
    using * eq[symmetric] by auto
hoelzl@36622
  1205
qed simp_all
haftmann@35822
  1206
Andreas@60057
  1207
lemma inj_on_apfst [simp]: "inj_on (apfst f) (A \<times> UNIV) \<longleftrightarrow> inj_on f A"
Andreas@60057
  1208
by(auto simp add: inj_on_def)
Andreas@60057
  1209
Andreas@60057
  1210
lemma inj_apfst [simp]: "inj (apfst f) \<longleftrightarrow> inj f"
Andreas@60057
  1211
using inj_on_apfst[of f UNIV] by simp
Andreas@60057
  1212
Andreas@60057
  1213
lemma inj_on_apsnd [simp]: "inj_on (apsnd f) (UNIV \<times> A) \<longleftrightarrow> inj_on f A"
Andreas@60057
  1214
by(auto simp add: inj_on_def)
Andreas@60057
  1215
Andreas@60057
  1216
lemma inj_apsnd [simp]: "inj (apsnd f) \<longleftrightarrow> inj f"
Andreas@60057
  1217
using inj_on_apsnd[of f UNIV] by simp
Andreas@60057
  1218
haftmann@61127
  1219
context
haftmann@61127
  1220
begin
haftmann@61127
  1221
haftmann@61127
  1222
qualified definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
haftmann@46128
  1223
  [code_abbrev]: "product A B = A \<times> B"
haftmann@46128
  1224
haftmann@46128
  1225
lemma member_product:
haftmann@46128
  1226
  "x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B"
haftmann@61127
  1227
  by (simp add: Product_Type.product_def)
haftmann@46128
  1228
haftmann@61127
  1229
end
haftmann@61127
  1230
  
wenzelm@60758
  1231
text \<open>The following @{const map_prod} lemmas are due to Joachim Breitner:\<close>
haftmann@40607
  1232
blanchet@55932
  1233
lemma map_prod_inj_on:
haftmann@40607
  1234
  assumes "inj_on f A" and "inj_on g B"
blanchet@55932
  1235
  shows "inj_on (map_prod f g) (A \<times> B)"
haftmann@40607
  1236
proof (rule inj_onI)
haftmann@40607
  1237
  fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c"
haftmann@40607
  1238
  assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto
haftmann@40607
  1239
  assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto
blanchet@55932
  1240
  assume "map_prod f g x = map_prod f g y"
blanchet@55932
  1241
  hence "fst (map_prod f g x) = fst (map_prod f g y)" by (auto)
haftmann@40607
  1242
  hence "f (fst x) = f (fst y)" by (cases x,cases y,auto)
wenzelm@60758
  1243
  with \<open>inj_on f A\<close> and \<open>fst x \<in> A\<close> and \<open>fst y \<in> A\<close>
haftmann@40607
  1244
  have "fst x = fst y" by (auto dest:dest:inj_onD)
wenzelm@60758
  1245
  moreover from \<open>map_prod f g x = map_prod f g y\<close>
blanchet@55932
  1246
  have "snd (map_prod f g x) = snd (map_prod f g y)" by (auto)
haftmann@40607
  1247
  hence "g (snd x) = g (snd y)" by (cases x,cases y,auto)
wenzelm@60758
  1248
  with \<open>inj_on g B\<close> and \<open>snd x \<in> B\<close> and \<open>snd y \<in> B\<close>
haftmann@40607
  1249
  have "snd x = snd y" by (auto dest:dest:inj_onD)
haftmann@40607
  1250
  ultimately show "x = y" by(rule prod_eqI)
haftmann@40607
  1251
qed
haftmann@40607
  1252
blanchet@55932
  1253
lemma map_prod_surj:
hoelzl@40702
  1254
  fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd"
haftmann@40607
  1255
  assumes "surj f" and "surj g"
blanchet@55932
  1256
  shows "surj (map_prod f g)"
haftmann@40607
  1257
unfolding surj_def
haftmann@40607
  1258
proof
haftmann@40607
  1259
  fix y :: "'b \<times> 'd"
wenzelm@60758
  1260
  from \<open>surj f\<close> obtain a where "fst y = f a" by (auto elim:surjE)
haftmann@40607
  1261
  moreover
wenzelm@60758
  1262
  from \<open>surj g\<close> obtain b where "snd y = g b" by (auto elim:surjE)
blanchet@55932
  1263
  ultimately have "(fst y, snd y) = map_prod f g (a,b)" by auto
blanchet@55932
  1264
  thus "\<exists>x. y = map_prod f g x" by auto
haftmann@40607
  1265
qed
haftmann@40607
  1266
blanchet@55932
  1267
lemma map_prod_surj_on:
haftmann@40607
  1268
  assumes "f ` A = A'" and "g ` B = B'"
blanchet@55932
  1269
  shows "map_prod f g ` (A \<times> B) = A' \<times> B'"
haftmann@40607
  1270
unfolding image_def
haftmann@40607
  1271
proof(rule set_eqI,rule iffI)
haftmann@40607
  1272
  fix x :: "'a \<times> 'c"
wenzelm@61076
  1273
  assume "x \<in> {y::'a \<times> 'c. \<exists>x::'b \<times> 'd\<in>A \<times> B. y = map_prod f g x}"
blanchet@55932
  1274
  then obtain y where "y \<in> A \<times> B" and "x = map_prod f g y" by blast
wenzelm@60758
  1275
  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
  1276
  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
  1277
  ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto
wenzelm@60758
  1278
  with \<open>x = map_prod f g y\<close> show "x \<in> A' \<times> B'" by (cases y, auto)
haftmann@40607
  1279
next
haftmann@40607
  1280
  fix x :: "'a \<times> 'c"
haftmann@40607
  1281
  assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto
wenzelm@60758
  1282
  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
  1283
  then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE)
wenzelm@60758
  1284
  moreover from \<open>image g B = B'\<close> and \<open>snd x \<in> B'\<close>
haftmann@40607
  1285
  obtain b where "b \<in> B" and "snd x = g b" by auto
blanchet@55932
  1286
  ultimately have "(fst x, snd x) = map_prod f g (a,b)" by auto
wenzelm@60758
  1287
  moreover from \<open>a \<in> A\<close> and  \<open>b \<in> B\<close> have "(a , b) \<in> A \<times> B" by auto
blanchet@55932
  1288
  ultimately have "\<exists>y \<in> A \<times> B. x = map_prod f g y" by auto
blanchet@55932
  1289
  thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_prod f g y}" by auto
haftmann@40607
  1290
qed
haftmann@40607
  1291
haftmann@21908
  1292
wenzelm@60758
  1293
subsection \<open>Simproc for rewriting a set comprehension into a pointfree expression\<close>
bulwahn@49764
  1294
bulwahn@49764
  1295
ML_file "Tools/set_comprehension_pointfree.ML"
bulwahn@49764
  1296
wenzelm@60758
  1297
setup \<open>
wenzelm@51717
  1298
  Code_Preproc.map_pre (fn ctxt => ctxt addsimprocs
wenzelm@61144
  1299
    [Simplifier.make_simproc @{context} "set comprehension"
wenzelm@61144
  1300
      {lhss = [@{term "Collect P"}],
wenzelm@61144
  1301
       proc = K Set_Comprehension_Pointfree.code_simproc,
wenzelm@61144
  1302
       identifier = []}])
wenzelm@60758
  1303
\<close>
bulwahn@49764
  1304
bulwahn@49764
  1305
wenzelm@60758
  1306
subsection \<open>Inductively defined sets\<close>
berghofe@15394
  1307
wenzelm@56512
  1308
(* simplify {(x1, ..., xn). (x1, ..., xn) : S} to S *)
wenzelm@60758
  1309
simproc_setup Collect_mem ("Collect t") = \<open>
wenzelm@56512
  1310
  fn _ => fn ctxt => fn ct =>
wenzelm@59582
  1311
    (case Thm.term_of ct of
wenzelm@56512
  1312
      S as Const (@{const_name Collect}, Type (@{type_name fun}, [_, T])) $ t =>
wenzelm@56512
  1313
        let val (u, _, ps) = HOLogic.strip_psplits t in
wenzelm@56512
  1314
          (case u of
wenzelm@56512
  1315
            (c as Const (@{const_name Set.member}, _)) $ q $ S' =>
wenzelm@56512
  1316
              (case try (HOLogic.strip_ptuple ps) q of
wenzelm@56512
  1317
                NONE => NONE
wenzelm@56512
  1318
              | SOME ts =>
wenzelm@56512
  1319
                  if not (Term.is_open S') andalso
wenzelm@56512
  1320
                    ts = map Bound (length ps downto 0)
wenzelm@56512
  1321
                  then
wenzelm@56512
  1322
                    let val simp =
wenzelm@56512
  1323
                      full_simp_tac (put_simpset HOL_basic_ss ctxt
wenzelm@56512
  1324
                        addsimps [@{thm split_paired_all}, @{thm split_conv}]) 1
wenzelm@56512
  1325
                    in
wenzelm@56512
  1326
                      SOME (Goal.prove ctxt [] []
wenzelm@56512
  1327
                        (Const (@{const_name Pure.eq}, T --> T --> propT) $ S $ S')
wenzelm@56512
  1328
                        (K (EVERY
wenzelm@59498
  1329
                          [resolve_tac ctxt [eq_reflection] 1,
wenzelm@59498
  1330
                           resolve_tac ctxt @{thms subset_antisym} 1,
wenzelm@59498
  1331
                           resolve_tac ctxt [subsetI] 1, dresolve_tac ctxt [CollectD] 1, simp,
wenzelm@59498
  1332
                           resolve_tac ctxt [subsetI] 1, resolve_tac ctxt [CollectI] 1, simp])))
wenzelm@56512
  1333
                    end
wenzelm@56512
  1334
                  else NONE)
wenzelm@56512
  1335
          | _ => NONE)
wenzelm@56512
  1336
        end
wenzelm@56512
  1337
    | _ => NONE)
wenzelm@60758
  1338
\<close>
blanchet@58389
  1339
wenzelm@48891
  1340
ML_file "Tools/inductive_set.ML"
haftmann@24699
  1341
haftmann@37166
  1342
wenzelm@60758
  1343
subsection \<open>Legacy theorem bindings and duplicates\<close>
haftmann@37166
  1344
haftmann@61125
  1345
abbreviation (input) case_prod :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
haftmann@61125
  1346
  "case_prod \<equiv> uncurry"
haftmann@61125
  1347
haftmann@61032
  1348
abbreviation (input) split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
haftmann@61125
  1349
  "split \<equiv> uncurry"
haftmann@37166
  1350
haftmann@61032
  1351
lemmas PairE = prod.exhaust
haftmann@37166
  1352
lemmas Pair_eq = prod.inject
blanchet@55393
  1353
lemmas fst_conv = prod.sel(1)
blanchet@55393
  1354
lemmas snd_conv = prod.sel(2)
blanchet@55393
  1355
lemmas pair_collapse = prod.collapse
blanchet@55393
  1356
lemmas split = split_conv
huffman@44066
  1357
lemmas Pair_fst_snd_eq = prod_eq_iff
haftmann@61032
  1358
lemmas split_def = case_prod_unfold
huffman@44066
  1359
huffman@45204
  1360
hide_const (open) prod
huffman@45204
  1361
nipkow@10213
  1362
end