src/HOL/Product_Type.thy
author traytel
Fri Nov 07 11:28:37 2014 +0100 (2014-11-07)
changeset 58916 229765cc3414
parent 58889 5b7a9633cfa8
child 59000 6eb0725503fc
permissions -rw-r--r--
more complete fp_sugars for sum and prod;
tuned;
removed theorem duplicates;
removed obsolete Lifting_{Option,Product,Sum} theories
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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 {* Cartesian products *}
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theory Product_Type
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imports Typedef Inductive Fun
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keywords "inductive_set" "coinductive_set" :: thy_decl
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begin
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subsection {* @{typ bool} is a datatype *}
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free_constructors case_bool for True | False
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  by auto
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text {* Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}. *}
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setup {* Sign.mandatory_path "old" *}
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old_rep_datatype True False by (auto intro: bool_induct)
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setup {* Sign.parent_path *}
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text {* But erase the prefix for properties that are not generated by @{text free_constructors}. *}
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setup {* Sign.mandatory_path "bool" *}
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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 {* Sign.parent_path *}
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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 {* Code.add_case @{thm If_case_cert} *}
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code_printing
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  constant "HOL.equal :: bool \<Rightarrow> bool \<Rightarrow> bool" \<rightharpoonup> (Haskell) infix 4 "=="
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| class_instance "bool" :: "equal" \<rightharpoonup> (Haskell) -
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subsection {* The @{text unit} type *}
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typedef unit = "{True}"
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  by auto
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definition Unity :: unit  ("'(')")
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  where "() = Abs_unit True"
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lemma unit_eq [no_atp]: "u = ()"
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  by (induct u) (simp add: Unity_def)
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text {*
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  Simplification procedure for @{thm [source] unit_eq}.  Cannot use
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  this rule directly --- it loops!
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*}
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simproc_setup unit_eq ("x::unit") = {*
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  fn _ => fn _ => fn ct =>
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    if HOLogic.is_unit (term_of ct) then NONE
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    else SOME (mk_meta_eq @{thm unit_eq})
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*}
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free_constructors case_unit for "()"
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  by auto
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text {* Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}. *}
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setup {* Sign.mandatory_path "old" *}
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old_rep_datatype "()" by simp
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setup {* Sign.parent_path *}
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text {* But erase the prefix for properties that are not generated by @{text free_constructors}. *}
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setup {* Sign.mandatory_path "unit" *}
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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 {* Sign.parent_path *}
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lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
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  by simp
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lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
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  by (rule triv_forall_equality)
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text {*
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  This rewrite counters the effect of simproc @{text unit_eq} on @{term
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  [source] "%u::unit. f u"}, replacing it by @{term [source]
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  f} rather than by @{term [source] "%u. f ()"}.
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*}
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lemma unit_abs_eta_conv [simp]: "(%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\<Colon>unit) v \<longleftrightarrow> True" unfolding equal unit_eq [of u] unit_eq [of v] by rule+
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code_printing
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  type_constructor unit \<rightharpoonup>
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    (SML) "unit"
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    and (OCaml) "unit"
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    and (Haskell) "()"
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    and (Scala) "Unit"
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| constant Unity \<rightharpoonup>
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    (SML) "()"
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    and (OCaml) "()"
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    and (Haskell) "()"
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    and (Scala) "()"
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| class_instance unit :: equal \<rightharpoonup>
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    (Haskell) -
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| constant "HOL.equal :: unit \<Rightarrow> unit \<Rightarrow> bool" \<rightharpoonup>
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    (Haskell) infix 4 "=="
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code_reserved SML
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  unit
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code_reserved OCaml
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  unit
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code_reserved Scala
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  Unit
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subsection {* The product type *}
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subsubsection {* Type definition *}
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definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" where
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  "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
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definition "prod = {f. \<exists>a b. f = Pair_Rep (a\<Colon>'a) (b\<Colon>'b)}"
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typedef ('a, 'b) prod (infixr "*" 20) = "prod :: ('a \<Rightarrow> 'b \<Rightarrow> bool) set"
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  unfolding prod_def by auto
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type_notation (xsymbols)
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  "prod"  ("(_ \<times>/ _)" [21, 20] 20)
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type_notation (HTML output)
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  "prod"  ("(_ \<times>/ _)" [21, 20] 20)
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definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b" where
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  "Pair a b = Abs_prod (Pair_Rep a b)"
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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 {* Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}. *}
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setup {* Sign.mandatory_path "old" *}
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old_rep_datatype Pair
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by (erule prod_cases) (rule prod.inject)
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setup {* Sign.parent_path *}
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text {* But erase the prefix for properties that are not generated by @{text free_constructors}. *}
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setup {* Sign.mandatory_path "prod" *}
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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 {* Sign.parent_path *}
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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 {* Tuple syntax *}
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abbreviation (input) split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
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  "split \<equiv> case_prod"
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text {*
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  Patterns -- extends pre-defined type @{typ pttrn} used in
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  abstractions.
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*}
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nonterminal tuple_args and patterns
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syntax
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  "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
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  "_tuple_arg"  :: "'a => tuple_args"                   ("_")
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  "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
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  "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
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  ""            :: "pttrn => patterns"                  ("_")
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  "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
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translations
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  "(x, y)" == "CONST Pair x y"
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  "_pattern x y" => "CONST Pair x y"
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  "_patterns x y" => "CONST Pair x y"
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  "_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
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  "%(x, y, zs). b" == "CONST case_prod (%x (y, zs). b)"
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  "%(x, y). b" == "CONST case_prod (%x y. b)"
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  "_abs (CONST Pair x y) t" => "%(x, y). t"
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  -- {* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
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     The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *}
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(*reconstruct pattern from (nested) splits, avoiding eta-contraction of body;
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  works best with enclosing "let", if "let" does not avoid eta-contraction*)
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print_translation {*
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  let
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    fun split_tr' [Abs (x, T, t as (Abs abs))] =
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          (* split (%x y. t) => %(x,y) t *)
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          let
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            val (y, t') = Syntax_Trans.atomic_abs_tr' abs;
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            val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t');
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          in
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            Syntax.const @{syntax_const "_abs"} $
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              (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
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          end
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      | split_tr' [Abs (x, T, (s as Const (@{const_syntax case_prod}, _) $ t))] =
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          (* split (%x. (split (%y z. t))) => %(x,y,z). t *)
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          let
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            val Const (@{syntax_const "_abs"}, _) $
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              (Const (@{syntax_const "_pattern"}, _) $ y $ z) $ t' = split_tr' [t];
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            val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t');
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          in
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            Syntax.const @{syntax_const "_abs"} $
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              (Syntax.const @{syntax_const "_pattern"} $ x' $
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                (Syntax.const @{syntax_const "_patterns"} $ y $ z)) $ t''
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          end
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      | split_tr' [Const (@{const_syntax case_prod}, _) $ t] =
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          (* split (split (%x y z. t)) => %((x, y), z). t *)
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          split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *)
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      | split_tr' [Const (@{syntax_const "_abs"}, _) $ x_y $ Abs abs] =
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          (* split (%pttrn z. t) => %(pttrn,z). t *)
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          let val (z, t) = Syntax_Trans.atomic_abs_tr' abs in
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            Syntax.const @{syntax_const "_abs"} $
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              (Syntax.const @{syntax_const "_pattern"} $ x_y $ z) $ t
wenzelm@52143
   348
          end
wenzelm@52143
   349
      | split_tr' _ = raise Match;
blanchet@55414
   350
  in [(@{const_syntax case_prod}, K split_tr')] end
schirmer@14359
   351
*}
schirmer@14359
   352
schirmer@15422
   353
(* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
schirmer@15422
   354
typed_print_translation {*
wenzelm@52143
   355
  let
wenzelm@52143
   356
    fun split_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match
wenzelm@52143
   357
      | split_guess_names_tr' T [Abs (x, xT, t)] =
wenzelm@52143
   358
          (case (head_of t) of
blanchet@55414
   359
            Const (@{const_syntax case_prod}, _) => raise Match
wenzelm@52143
   360
          | _ =>
wenzelm@52143
   361
            let 
wenzelm@52143
   362
              val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
wenzelm@52143
   363
              val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0);
wenzelm@52143
   364
              val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t');
wenzelm@52143
   365
            in
wenzelm@52143
   366
              Syntax.const @{syntax_const "_abs"} $
wenzelm@52143
   367
                (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
wenzelm@52143
   368
            end)
wenzelm@52143
   369
      | split_guess_names_tr' T [t] =
wenzelm@52143
   370
          (case head_of t of
blanchet@55414
   371
            Const (@{const_syntax case_prod}, _) => raise Match
wenzelm@52143
   372
          | _ =>
wenzelm@52143
   373
            let
wenzelm@52143
   374
              val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
wenzelm@52143
   375
              val (y, t') =
wenzelm@52143
   376
                Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0);
wenzelm@52143
   377
              val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t');
wenzelm@52143
   378
            in
wenzelm@52143
   379
              Syntax.const @{syntax_const "_abs"} $
wenzelm@52143
   380
                (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
wenzelm@52143
   381
            end)
wenzelm@52143
   382
      | split_guess_names_tr' _ _ = raise Match;
blanchet@55414
   383
  in [(@{const_syntax case_prod}, K split_guess_names_tr')] end
schirmer@15422
   384
*}
schirmer@15422
   385
nipkow@10213
   386
haftmann@37166
   387
subsubsection {* Code generator setup *}
haftmann@37166
   388
haftmann@52435
   389
code_printing
haftmann@52435
   390
  type_constructor prod \<rightharpoonup>
haftmann@52435
   391
    (SML) infix 2 "*"
haftmann@52435
   392
    and (OCaml) infix 2 "*"
haftmann@52435
   393
    and (Haskell) "!((_),/ (_))"
haftmann@52435
   394
    and (Scala) "((_),/ (_))"
haftmann@52435
   395
| constant Pair \<rightharpoonup>
haftmann@52435
   396
    (SML) "!((_),/ (_))"
haftmann@52435
   397
    and (OCaml) "!((_),/ (_))"
haftmann@52435
   398
    and (Haskell) "!((_),/ (_))"
haftmann@52435
   399
    and (Scala) "!((_),/ (_))"
haftmann@52435
   400
| class_instance  prod :: equal \<rightharpoonup>
haftmann@52435
   401
    (Haskell) -
haftmann@52435
   402
| constant "HOL.equal :: 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" \<rightharpoonup>
haftmann@52435
   403
    (Haskell) infix 4 "=="
haftmann@37166
   404
haftmann@37166
   405
haftmann@37166
   406
subsubsection {* Fundamental operations and properties *}
wenzelm@11838
   407
bulwahn@49897
   408
lemma Pair_inject:
bulwahn@49897
   409
  assumes "(a, b) = (a', b')"
bulwahn@49897
   410
    and "a = a' ==> b = b' ==> R"
bulwahn@49897
   411
  shows R
bulwahn@49897
   412
  using assms by simp
bulwahn@49897
   413
haftmann@26358
   414
lemma surj_pair [simp]: "EX x y. p = (x, y)"
haftmann@37166
   415
  by (cases p) simp
nipkow@10213
   416
haftmann@52435
   417
code_printing
haftmann@52435
   418
  constant fst \<rightharpoonup> (Haskell) "fst"
haftmann@52435
   419
| constant snd \<rightharpoonup> (Haskell) "snd"
haftmann@26358
   420
blanchet@55414
   421
lemma case_prod_unfold [nitpick_unfold]: "case_prod = (%c p. c (fst p) (snd p))"
nipkow@39302
   422
  by (simp add: fun_eq_iff split: prod.split)
haftmann@26358
   423
wenzelm@11838
   424
lemma fst_eqD: "fst (x, y) = a ==> x = a"
wenzelm@11838
   425
  by simp
wenzelm@11838
   426
wenzelm@11838
   427
lemma snd_eqD: "snd (x, y) = a ==> y = a"
wenzelm@11838
   428
  by simp
wenzelm@11838
   429
blanchet@55393
   430
lemmas surjective_pairing = prod.collapse [symmetric]
wenzelm@11838
   431
huffman@44066
   432
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
haftmann@37166
   433
  by (cases s, cases t) simp
haftmann@37166
   434
haftmann@37166
   435
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
huffman@44066
   436
  by (simp add: prod_eq_iff)
haftmann@37166
   437
haftmann@37166
   438
lemma split_conv [simp, code]: "split f (a, b) = f a b"
blanchet@55642
   439
  by (fact prod.case)
haftmann@37166
   440
haftmann@37166
   441
lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
haftmann@37166
   442
  by (rule split_conv [THEN iffD2])
haftmann@37166
   443
haftmann@37166
   444
lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
haftmann@37166
   445
  by (rule split_conv [THEN iffD1])
haftmann@37166
   446
haftmann@37166
   447
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
nipkow@39302
   448
  by (simp add: fun_eq_iff split: prod.split)
haftmann@37166
   449
haftmann@37166
   450
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
haftmann@37166
   451
  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
nipkow@39302
   452
  by (simp add: fun_eq_iff split: prod.split)
haftmann@37166
   453
haftmann@37166
   454
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
haftmann@37166
   455
  by (cases x) simp
haftmann@37166
   456
haftmann@37166
   457
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
haftmann@37166
   458
  by (cases p) simp
haftmann@37166
   459
haftmann@37166
   460
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
blanchet@55414
   461
  by (simp add: case_prod_unfold)
haftmann@37166
   462
haftmann@58468
   463
lemmas split_weak_cong = prod.case_cong_weak
haftmann@37166
   464
  -- {* Prevents simplification of @{term c}: much faster *}
haftmann@37166
   465
haftmann@37166
   466
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
haftmann@37166
   467
  by (simp add: split_eta)
haftmann@37166
   468
blanchet@47740
   469
lemma split_paired_all [no_atp]: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
wenzelm@11820
   470
proof
wenzelm@11820
   471
  fix a b
wenzelm@11820
   472
  assume "!!x. PROP P x"
wenzelm@19535
   473
  then show "PROP P (a, b)" .
wenzelm@11820
   474
next
wenzelm@11820
   475
  fix x
wenzelm@11820
   476
  assume "!!a b. PROP P (a, b)"
wenzelm@19535
   477
  from `PROP P (fst x, snd x)` show "PROP P x" by simp
wenzelm@11820
   478
qed
wenzelm@11820
   479
hoelzl@50104
   480
lemma case_prod_distrib: "f (case x of (x, y) \<Rightarrow> g x y) = (case x of (x, y) \<Rightarrow> f (g x y))"
hoelzl@50104
   481
  by (cases x) simp
hoelzl@50104
   482
wenzelm@11838
   483
text {*
wenzelm@11838
   484
  The rule @{thm [source] split_paired_all} does not work with the
wenzelm@11838
   485
  Simplifier because it also affects premises in congrence rules,
wenzelm@11838
   486
  where this can lead to premises of the form @{text "!!a b. ... =
wenzelm@11838
   487
  ?P(a, b)"} which cannot be solved by reflexivity.
wenzelm@11838
   488
*}
wenzelm@11838
   489
haftmann@26358
   490
lemmas split_tupled_all = split_paired_all unit_all_eq2
haftmann@26358
   491
wenzelm@26480
   492
ML {*
wenzelm@11838
   493
  (* replace parameters of product type by individual component parameters *)
wenzelm@11838
   494
  local (* filtering with exists_paired_all is an essential optimization *)
wenzelm@56245
   495
    fun exists_paired_all (Const (@{const_name Pure.all}, _) $ Abs (_, T, t)) =
wenzelm@11838
   496
          can HOLogic.dest_prodT T orelse exists_paired_all t
wenzelm@11838
   497
      | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
wenzelm@11838
   498
      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
wenzelm@11838
   499
      | exists_paired_all _ = false;
wenzelm@51717
   500
    val ss =
wenzelm@51717
   501
      simpset_of
wenzelm@51717
   502
       (put_simpset HOL_basic_ss @{context}
wenzelm@51717
   503
        addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
wenzelm@51717
   504
        addsimprocs [@{simproc unit_eq}]);
wenzelm@11838
   505
  in
wenzelm@51717
   506
    fun split_all_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   507
      if exists_paired_all t then safe_full_simp_tac (put_simpset ss ctxt) i else no_tac);
wenzelm@51717
   508
wenzelm@51717
   509
    fun unsafe_split_all_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   510
      if exists_paired_all t then full_simp_tac (put_simpset ss ctxt) i else no_tac);
wenzelm@51717
   511
wenzelm@51717
   512
    fun split_all ctxt th =
wenzelm@51717
   513
      if exists_paired_all (Thm.prop_of th)
wenzelm@51717
   514
      then full_simplify (put_simpset ss ctxt) th else th;
wenzelm@11838
   515
  end;
wenzelm@26340
   516
*}
wenzelm@11838
   517
wenzelm@51703
   518
setup {* map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac)) *}
wenzelm@11838
   519
blanchet@47740
   520
lemma split_paired_All [simp, no_atp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@11838
   521
  -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
wenzelm@11838
   522
  by fast
wenzelm@11838
   523
blanchet@47740
   524
lemma split_paired_Ex [simp, no_atp]: "(EX x. P x) = (EX a b. P (a, b))"
haftmann@26358
   525
  by fast
haftmann@26358
   526
blanchet@47740
   527
lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@11838
   528
  -- {* Can't be added to simpset: loops! *}
haftmann@26358
   529
  by (simp add: split_eta)
wenzelm@11838
   530
wenzelm@11838
   531
text {*
wenzelm@11838
   532
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
wenzelm@11838
   533
  @{thm [source] split_eta} as a rewrite rule is not general enough,
wenzelm@11838
   534
  and using @{thm [source] cond_split_eta} directly would render some
wenzelm@11838
   535
  existing proofs very inefficient; similarly for @{text
haftmann@26358
   536
  split_beta}.
haftmann@26358
   537
*}
wenzelm@11838
   538
wenzelm@26480
   539
ML {*
wenzelm@11838
   540
local
wenzelm@51717
   541
  val cond_split_eta_ss =
wenzelm@51717
   542
    simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms cond_split_eta});
wenzelm@35364
   543
  fun Pair_pat k 0 (Bound m) = (m = k)
wenzelm@35364
   544
    | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =
wenzelm@35364
   545
        i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t
wenzelm@35364
   546
    | Pair_pat _ _ _ = false;
wenzelm@35364
   547
  fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t
wenzelm@35364
   548
    | no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@35364
   549
    | no_args k i (Bound m) = m < k orelse m > k + i
wenzelm@35364
   550
    | no_args _ _ _ = true;
wenzelm@35364
   551
  fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE
blanchet@55414
   552
    | split_pat tp i (Const (@{const_name case_prod}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
wenzelm@35364
   553
    | split_pat tp i _ = NONE;
wenzelm@51717
   554
  fun metaeq ctxt lhs rhs = mk_meta_eq (Goal.prove ctxt [] []
wenzelm@35364
   555
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
wenzelm@51717
   556
        (K (simp_tac (put_simpset cond_split_eta_ss ctxt) 1)));
wenzelm@11838
   557
wenzelm@35364
   558
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t
wenzelm@35364
   559
    | beta_term_pat k i (t $ u) =
wenzelm@35364
   560
        Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@35364
   561
    | beta_term_pat k i t = no_args k i t;
wenzelm@35364
   562
  fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@35364
   563
    | eta_term_pat _ _ _ = false;
wenzelm@11838
   564
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@35364
   565
    | subst arg k i (t $ u) =
wenzelm@35364
   566
        if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@35364
   567
        else (subst arg k i t $ subst arg k i u)
wenzelm@35364
   568
    | subst arg k i t = t;
wenzelm@43595
   569
in
blanchet@55414
   570
  fun beta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t) $ arg) =
wenzelm@11838
   571
        (case split_pat beta_term_pat 1 t of
wenzelm@51717
   572
          SOME (i, f) => SOME (metaeq ctxt s (subst arg 0 i f))
skalberg@15531
   573
        | NONE => NONE)
wenzelm@35364
   574
    | beta_proc _ _ = NONE;
blanchet@55414
   575
  fun eta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t)) =
wenzelm@11838
   576
        (case split_pat eta_term_pat 1 t of
haftmann@58468
   577
          SOME (_, ft) => SOME (metaeq ctxt s (let val f $ _ = ft in f end))
skalberg@15531
   578
        | NONE => NONE)
wenzelm@35364
   579
    | eta_proc _ _ = NONE;
wenzelm@11838
   580
end;
wenzelm@11838
   581
*}
wenzelm@51717
   582
simproc_setup split_beta ("split f z") = {* fn _ => fn ctxt => fn ct => beta_proc ctxt (term_of ct) *}
wenzelm@51717
   583
simproc_setup split_eta ("split f") = {* fn _ => fn ctxt => fn ct => eta_proc ctxt (term_of ct) *}
wenzelm@11838
   584
haftmann@58468
   585
lemmas split_beta [mono] = prod.case_eq_if
wenzelm@11838
   586
hoelzl@50104
   587
lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
hoelzl@50104
   588
  by (auto simp: fun_eq_iff)
hoelzl@50104
   589
haftmann@58468
   590
lemmas split_split [no_atp] = prod.split
wenzelm@11838
   591
  -- {* For use with @{text split} and the Simplifier. *}
wenzelm@11838
   592
wenzelm@11838
   593
text {*
wenzelm@11838
   594
  @{thm [source] split_split} could be declared as @{text "[split]"}
wenzelm@11838
   595
  done after the Splitter has been speeded up significantly;
wenzelm@11838
   596
  precompute the constants involved and don't do anything unless the
wenzelm@11838
   597
  current goal contains one of those constants.
wenzelm@11838
   598
*}
wenzelm@11838
   599
haftmann@58468
   600
lemmas split_split_asm [no_atp] = prod.split_asm
wenzelm@11838
   601
wenzelm@11838
   602
text {*
wenzelm@11838
   603
  \medskip @{term split} used as a logical connective or set former.
wenzelm@11838
   604
wenzelm@11838
   605
  \medskip These rules are for use with @{text blast}; could instead
huffman@40929
   606
  call @{text simp} using @{thm [source] prod.split} as rewrite. *}
wenzelm@11838
   607
wenzelm@11838
   608
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
wenzelm@11838
   609
  apply (simp only: split_tupled_all)
wenzelm@11838
   610
  apply (simp (no_asm_simp))
wenzelm@11838
   611
  done
wenzelm@11838
   612
wenzelm@11838
   613
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
wenzelm@11838
   614
  apply (simp only: split_tupled_all)
wenzelm@11838
   615
  apply (simp (no_asm_simp))
wenzelm@11838
   616
  done
wenzelm@11838
   617
wenzelm@11838
   618
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@37591
   619
  by (induct p) auto
wenzelm@11838
   620
wenzelm@11838
   621
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@37591
   622
  by (induct p) auto
wenzelm@11838
   623
wenzelm@11838
   624
lemma splitE2:
wenzelm@11838
   625
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
wenzelm@11838
   626
proof -
wenzelm@11838
   627
  assume q: "Q (split P z)"
wenzelm@11838
   628
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
wenzelm@11838
   629
  show R
wenzelm@11838
   630
    apply (rule r surjective_pairing)+
wenzelm@11838
   631
    apply (rule split_beta [THEN subst], rule q)
wenzelm@11838
   632
    done
wenzelm@11838
   633
qed
wenzelm@11838
   634
wenzelm@11838
   635
lemma splitD': "split R (a,b) c ==> R a b c"
wenzelm@11838
   636
  by simp
wenzelm@11838
   637
wenzelm@11838
   638
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
wenzelm@11838
   639
  by simp
wenzelm@11838
   640
wenzelm@11838
   641
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
paulson@14208
   642
by (simp only: split_tupled_all, simp)
wenzelm@11838
   643
wenzelm@18372
   644
lemma mem_splitE:
haftmann@58468
   645
  assumes "z \<in> split c p"
haftmann@58468
   646
  obtains x y where "p = (x, y)" and "z \<in> c x y"
haftmann@58468
   647
  using assms by (rule splitE2)
wenzelm@11838
   648
wenzelm@11838
   649
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
wenzelm@11838
   650
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
wenzelm@11838
   651
wenzelm@26340
   652
ML {*
wenzelm@11838
   653
local (* filtering with exists_p_split is an essential optimization *)
blanchet@55414
   654
  fun exists_p_split (Const (@{const_name case_prod},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true
wenzelm@11838
   655
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838
   656
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838
   657
    | exists_p_split _ = false;
wenzelm@11838
   658
in
wenzelm@51717
   659
fun split_conv_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   660
  if exists_p_split t
wenzelm@51717
   661
  then safe_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms split_conv}) i
wenzelm@51717
   662
  else no_tac);
wenzelm@11838
   663
end;
wenzelm@26340
   664
*}
wenzelm@26340
   665
wenzelm@11838
   666
(* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838
   667
   to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@51703
   668
setup {* map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac)) *}
wenzelm@11838
   669
blanchet@54147
   670
lemma split_eta_SetCompr [simp, no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
wenzelm@18372
   671
  by (rule ext) fast
wenzelm@11838
   672
blanchet@54147
   673
lemma split_eta_SetCompr2 [simp, no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
wenzelm@18372
   674
  by (rule ext) fast
wenzelm@11838
   675
wenzelm@11838
   676
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
wenzelm@11838
   677
  -- {* Allows simplifications of nested splits in case of independent predicates. *}
wenzelm@18372
   678
  by (rule ext) blast
wenzelm@11838
   679
nipkow@14337
   680
(* Do NOT make this a simp rule as it
nipkow@14337
   681
   a) only helps in special situations
nipkow@14337
   682
   b) can lead to nontermination in the presence of split_def
nipkow@14337
   683
*)
nipkow@14337
   684
lemma split_comp_eq: 
paulson@20415
   685
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
paulson@20415
   686
  shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
wenzelm@18372
   687
  by (rule ext) auto
oheimb@14101
   688
haftmann@26358
   689
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
haftmann@26358
   690
  apply (rule_tac x = "(a, b)" in image_eqI)
haftmann@26358
   691
   apply auto
haftmann@26358
   692
  done
haftmann@26358
   693
wenzelm@11838
   694
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838
   695
  by blast
wenzelm@11838
   696
wenzelm@11838
   697
(*
wenzelm@11838
   698
the following  would be slightly more general,
wenzelm@11838
   699
but cannot be used as rewrite rule:
wenzelm@11838
   700
### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838
   701
### ?y = .x
wenzelm@11838
   702
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208
   703
by (rtac some_equality 1)
paulson@14208
   704
by ( Simp_tac 1)
paulson@14208
   705
by (split_all_tac 1)
paulson@14208
   706
by (Asm_full_simp_tac 1)
wenzelm@11838
   707
qed "The_split_eq";
wenzelm@11838
   708
*)
wenzelm@11838
   709
wenzelm@11838
   710
text {*
wenzelm@11838
   711
  Setup of internal @{text split_rule}.
wenzelm@11838
   712
*}
wenzelm@11838
   713
blanchet@55642
   714
lemmas case_prodI = prod.case [THEN iffD2]
haftmann@24699
   715
blanchet@55414
   716
lemma case_prodI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> case_prod c p"
haftmann@37678
   717
  by (fact splitI2)
haftmann@24699
   718
blanchet@55414
   719
lemma case_prodI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> case_prod c p x"
haftmann@37678
   720
  by (fact splitI2')
haftmann@24699
   721
blanchet@55414
   722
lemma case_prodE: "case_prod c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@37678
   723
  by (fact splitE)
haftmann@24699
   724
blanchet@55414
   725
lemma case_prodE': "case_prod c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@37678
   726
  by (fact splitE')
haftmann@24699
   727
blanchet@55414
   728
declare case_prodI [intro!]
haftmann@24699
   729
blanchet@55414
   730
lemma case_prod_beta:
blanchet@55414
   731
  "case_prod f p = f (fst p) (snd p)"
haftmann@37591
   732
  by (fact split_beta)
bulwahn@26143
   733
blanchet@55417
   734
lemma prod_cases3 [cases type]:
haftmann@24699
   735
  obtains (fields) a b c where "y = (a, b, c)"
haftmann@24699
   736
  by (cases y, case_tac b) blast
haftmann@24699
   737
haftmann@24699
   738
lemma prod_induct3 [case_names fields, induct type]:
haftmann@24699
   739
    "(!!a b c. P (a, b, c)) ==> P x"
haftmann@24699
   740
  by (cases x) blast
haftmann@24699
   741
blanchet@55417
   742
lemma prod_cases4 [cases type]:
haftmann@24699
   743
  obtains (fields) a b c d where "y = (a, b, c, d)"
haftmann@24699
   744
  by (cases y, case_tac c) blast
haftmann@24699
   745
haftmann@24699
   746
lemma prod_induct4 [case_names fields, induct type]:
haftmann@24699
   747
    "(!!a b c d. P (a, b, c, d)) ==> P x"
haftmann@24699
   748
  by (cases x) blast
haftmann@24699
   749
blanchet@55417
   750
lemma prod_cases5 [cases type]:
haftmann@24699
   751
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
haftmann@24699
   752
  by (cases y, case_tac d) blast
haftmann@24699
   753
haftmann@24699
   754
lemma prod_induct5 [case_names fields, induct type]:
haftmann@24699
   755
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
haftmann@24699
   756
  by (cases x) blast
haftmann@24699
   757
blanchet@55417
   758
lemma prod_cases6 [cases type]:
haftmann@24699
   759
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
haftmann@24699
   760
  by (cases y, case_tac e) blast
haftmann@24699
   761
haftmann@24699
   762
lemma prod_induct6 [case_names fields, induct type]:
haftmann@24699
   763
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
haftmann@24699
   764
  by (cases x) blast
haftmann@24699
   765
blanchet@55417
   766
lemma prod_cases7 [cases type]:
haftmann@24699
   767
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
haftmann@24699
   768
  by (cases y, case_tac f) blast
haftmann@24699
   769
haftmann@24699
   770
lemma prod_induct7 [case_names fields, induct type]:
haftmann@24699
   771
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
haftmann@24699
   772
  by (cases x) blast
haftmann@24699
   773
haftmann@37166
   774
lemma split_def:
haftmann@37166
   775
  "split = (\<lambda>c p. c (fst p) (snd p))"
blanchet@55414
   776
  by (fact case_prod_unfold)
haftmann@37166
   777
haftmann@37166
   778
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
haftmann@37166
   779
  "internal_split == split"
haftmann@37166
   780
haftmann@37166
   781
lemma internal_split_conv: "internal_split c (a, b) = c a b"
haftmann@37166
   782
  by (simp only: internal_split_def split_conv)
haftmann@37166
   783
wenzelm@48891
   784
ML_file "Tools/split_rule.ML"
haftmann@37166
   785
haftmann@37166
   786
hide_const internal_split
haftmann@37166
   787
haftmann@24699
   788
haftmann@26358
   789
subsubsection {* Derived operations *}
haftmann@26358
   790
haftmann@37387
   791
definition curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where
haftmann@37387
   792
  "curry = (\<lambda>c x y. c (x, y))"
haftmann@37166
   793
haftmann@37166
   794
lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
haftmann@37166
   795
  by (simp add: curry_def)
haftmann@37166
   796
haftmann@37166
   797
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
haftmann@37166
   798
  by (simp add: curry_def)
haftmann@37166
   799
haftmann@37166
   800
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
haftmann@37166
   801
  by (simp add: curry_def)
haftmann@37166
   802
haftmann@37166
   803
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@37166
   804
  by (simp add: curry_def)
haftmann@37166
   805
haftmann@37166
   806
lemma curry_split [simp]: "curry (split f) = f"
haftmann@37166
   807
  by (simp add: curry_def split_def)
haftmann@37166
   808
haftmann@37166
   809
lemma split_curry [simp]: "split (curry f) = f"
haftmann@37166
   810
  by (simp add: curry_def split_def)
haftmann@37166
   811
Andreas@54630
   812
lemma curry_K: "curry (\<lambda>x. c) = (\<lambda>x y. c)"
Andreas@54630
   813
by(simp add: fun_eq_iff)
Andreas@54630
   814
haftmann@26358
   815
text {*
haftmann@26358
   816
  The composition-uncurry combinator.
haftmann@26358
   817
*}
haftmann@26358
   818
haftmann@37751
   819
notation fcomp (infixl "\<circ>>" 60)
haftmann@26358
   820
haftmann@37751
   821
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where
blanchet@55414
   822
  "f \<circ>\<rightarrow> g = (\<lambda>x. case_prod g (f x))"
haftmann@26358
   823
haftmann@37678
   824
lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))"
blanchet@55414
   825
  by (simp add: fun_eq_iff scomp_def case_prod_unfold)
haftmann@37678
   826
blanchet@55414
   827
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = case_prod g (f x)"
blanchet@55414
   828
  by (simp add: scomp_unfold case_prod_unfold)
haftmann@26358
   829
haftmann@37751
   830
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x"
huffman@44921
   831
  by (simp add: fun_eq_iff)
haftmann@26358
   832
haftmann@37751
   833
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x"
huffman@44921
   834
  by (simp add: fun_eq_iff)
haftmann@26358
   835
haftmann@37751
   836
lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
nipkow@39302
   837
  by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358
   838
haftmann@37751
   839
lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
nipkow@39302
   840
  by (simp add: fun_eq_iff scomp_unfold fcomp_def)
haftmann@26358
   841
haftmann@37751
   842
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
huffman@44921
   843
  by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358
   844
haftmann@52435
   845
code_printing
haftmann@52435
   846
  constant scomp \<rightharpoonup> (Eval) infixl 3 "#->"
haftmann@31202
   847
haftmann@37751
   848
no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
   849
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@26358
   850
haftmann@26358
   851
text {*
blanchet@55932
   852
  @{term map_prod} --- action of the product functor upon
krauss@36664
   853
  functions.
haftmann@26358
   854
*}
haftmann@21195
   855
blanchet@55932
   856
definition map_prod :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
blanchet@55932
   857
  "map_prod f g = (\<lambda>(x, y). (f x, g y))"
haftmann@26358
   858
blanchet@55932
   859
lemma map_prod_simp [simp, code]:
blanchet@55932
   860
  "map_prod f g (a, b) = (f a, g b)"
blanchet@55932
   861
  by (simp add: map_prod_def)
haftmann@26358
   862
blanchet@55932
   863
functor map_prod: map_prod
huffman@44921
   864
  by (auto simp add: split_paired_all)
nipkow@37278
   865
blanchet@55932
   866
lemma fst_map_prod [simp]:
blanchet@55932
   867
  "fst (map_prod f g x) = f (fst x)"
haftmann@40607
   868
  by (cases x) simp_all
nipkow@37278
   869
traytel@58916
   870
lemma snd_map_prod [simp]:
blanchet@55932
   871
  "snd (map_prod f g x) = g (snd x)"
haftmann@40607
   872
  by (cases x) simp_all
nipkow@37278
   873
blanchet@55932
   874
lemma fst_comp_map_prod [simp]:
blanchet@55932
   875
  "fst \<circ> map_prod f g = f \<circ> fst"
haftmann@40607
   876
  by (rule ext) simp_all
nipkow@37278
   877
blanchet@55932
   878
lemma snd_comp_map_prod [simp]:
blanchet@55932
   879
  "snd \<circ> map_prod f g = g \<circ> snd"
haftmann@40607
   880
  by (rule ext) simp_all
haftmann@26358
   881
blanchet@55932
   882
lemma map_prod_compose:
blanchet@55932
   883
  "map_prod (f1 o f2) (g1 o g2) = (map_prod f1 g1 o map_prod f2 g2)"
blanchet@55932
   884
  by (rule ext) (simp add: map_prod.compositionality comp_def)
haftmann@26358
   885
blanchet@55932
   886
lemma map_prod_ident [simp]:
blanchet@55932
   887
  "map_prod (%x. x) (%y. y) = (%z. z)"
blanchet@55932
   888
  by (rule ext) (simp add: map_prod.identity)
haftmann@40607
   889
blanchet@55932
   890
lemma map_prod_imageI [intro]:
blanchet@55932
   891
  "(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_prod f g ` R"
haftmann@40607
   892
  by (rule image_eqI) simp_all
haftmann@21195
   893
haftmann@26358
   894
lemma prod_fun_imageE [elim!]:
blanchet@55932
   895
  assumes major: "c \<in> map_prod f g ` R"
haftmann@40607
   896
    and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P"
haftmann@26358
   897
  shows P
haftmann@26358
   898
  apply (rule major [THEN imageE])
haftmann@37166
   899
  apply (case_tac x)
haftmann@26358
   900
  apply (rule cases)
haftmann@40607
   901
  apply simp_all
haftmann@26358
   902
  done
haftmann@26358
   903
haftmann@37166
   904
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where
blanchet@55932
   905
  "apfst f = map_prod f id"
haftmann@26358
   906
haftmann@37166
   907
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where
blanchet@55932
   908
  "apsnd f = map_prod id f"
haftmann@26358
   909
haftmann@26358
   910
lemma apfst_conv [simp, code]:
haftmann@26358
   911
  "apfst f (x, y) = (f x, y)" 
haftmann@26358
   912
  by (simp add: apfst_def)
haftmann@26358
   913
hoelzl@33638
   914
lemma apsnd_conv [simp, code]:
haftmann@26358
   915
  "apsnd f (x, y) = (x, f y)" 
haftmann@26358
   916
  by (simp add: apsnd_def)
haftmann@21195
   917
haftmann@33594
   918
lemma fst_apfst [simp]:
haftmann@33594
   919
  "fst (apfst f x) = f (fst x)"
haftmann@33594
   920
  by (cases x) simp
haftmann@33594
   921
haftmann@51173
   922
lemma fst_comp_apfst [simp]:
haftmann@51173
   923
  "fst \<circ> apfst f = f \<circ> fst"
haftmann@51173
   924
  by (simp add: fun_eq_iff)
haftmann@51173
   925
haftmann@33594
   926
lemma fst_apsnd [simp]:
haftmann@33594
   927
  "fst (apsnd f x) = fst x"
haftmann@33594
   928
  by (cases x) simp
haftmann@33594
   929
haftmann@51173
   930
lemma fst_comp_apsnd [simp]:
haftmann@51173
   931
  "fst \<circ> apsnd f = fst"
haftmann@51173
   932
  by (simp add: fun_eq_iff)
haftmann@51173
   933
haftmann@33594
   934
lemma snd_apfst [simp]:
haftmann@33594
   935
  "snd (apfst f x) = snd x"
haftmann@33594
   936
  by (cases x) simp
haftmann@33594
   937
haftmann@51173
   938
lemma snd_comp_apfst [simp]:
haftmann@51173
   939
  "snd \<circ> apfst f = snd"
haftmann@51173
   940
  by (simp add: fun_eq_iff)
haftmann@51173
   941
haftmann@33594
   942
lemma snd_apsnd [simp]:
haftmann@33594
   943
  "snd (apsnd f x) = f (snd x)"
haftmann@33594
   944
  by (cases x) simp
haftmann@33594
   945
haftmann@51173
   946
lemma snd_comp_apsnd [simp]:
haftmann@51173
   947
  "snd \<circ> apsnd f = f \<circ> snd"
haftmann@51173
   948
  by (simp add: fun_eq_iff)
haftmann@51173
   949
haftmann@33594
   950
lemma apfst_compose:
haftmann@33594
   951
  "apfst f (apfst g x) = apfst (f \<circ> g) x"
haftmann@33594
   952
  by (cases x) simp
haftmann@33594
   953
haftmann@33594
   954
lemma apsnd_compose:
haftmann@33594
   955
  "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
haftmann@33594
   956
  by (cases x) simp
haftmann@33594
   957
haftmann@33594
   958
lemma apfst_apsnd [simp]:
haftmann@33594
   959
  "apfst f (apsnd g x) = (f (fst x), g (snd x))"
haftmann@33594
   960
  by (cases x) simp
haftmann@33594
   961
haftmann@33594
   962
lemma apsnd_apfst [simp]:
haftmann@33594
   963
  "apsnd f (apfst g x) = (g (fst x), f (snd x))"
haftmann@33594
   964
  by (cases x) simp
haftmann@33594
   965
haftmann@33594
   966
lemma apfst_id [simp] :
haftmann@33594
   967
  "apfst id = id"
nipkow@39302
   968
  by (simp add: fun_eq_iff)
haftmann@33594
   969
haftmann@33594
   970
lemma apsnd_id [simp] :
haftmann@33594
   971
  "apsnd id = id"
nipkow@39302
   972
  by (simp add: fun_eq_iff)
haftmann@33594
   973
haftmann@33594
   974
lemma apfst_eq_conv [simp]:
haftmann@33594
   975
  "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
haftmann@33594
   976
  by (cases x) simp
haftmann@33594
   977
haftmann@33594
   978
lemma apsnd_eq_conv [simp]:
haftmann@33594
   979
  "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"
haftmann@33594
   980
  by (cases x) simp
haftmann@33594
   981
hoelzl@33638
   982
lemma apsnd_apfst_commute:
hoelzl@33638
   983
  "apsnd f (apfst g p) = apfst g (apsnd f p)"
hoelzl@33638
   984
  by simp
haftmann@21195
   985
haftmann@56626
   986
context
haftmann@56626
   987
begin
haftmann@56626
   988
haftmann@56626
   989
local_setup {* Local_Theory.map_naming (Name_Space.mandatory_path "prod") *}
haftmann@56626
   990
haftmann@56545
   991
definition swap :: "'a \<times> 'b \<Rightarrow> 'b \<times> 'a"
haftmann@56545
   992
where
haftmann@56545
   993
  "swap p = (snd p, fst p)"
haftmann@56545
   994
haftmann@56626
   995
end
haftmann@56626
   996
haftmann@56545
   997
lemma swap_simp [simp]:
haftmann@56626
   998
  "prod.swap (x, y) = (y, x)"
haftmann@56626
   999
  by (simp add: prod.swap_def)
haftmann@56545
  1000
haftmann@58195
  1001
lemma swap_swap [simp]:
haftmann@58195
  1002
  "prod.swap (prod.swap p) = p"
haftmann@58195
  1003
  by (cases p) simp
haftmann@58195
  1004
haftmann@58195
  1005
lemma swap_comp_swap [simp]:
haftmann@58195
  1006
  "prod.swap \<circ> prod.swap = id"
haftmann@58195
  1007
  by (simp add: fun_eq_iff)
haftmann@58195
  1008
haftmann@56545
  1009
lemma pair_in_swap_image [simp]:
haftmann@56626
  1010
  "(y, x) \<in> prod.swap ` A \<longleftrightarrow> (x, y) \<in> A"
haftmann@56545
  1011
  by (auto intro!: image_eqI)
haftmann@56545
  1012
haftmann@56545
  1013
lemma inj_swap [simp]:
haftmann@56626
  1014
  "inj_on prod.swap A"
haftmann@56626
  1015
  by (rule inj_onI) auto
haftmann@56626
  1016
haftmann@56626
  1017
lemma swap_inj_on:
haftmann@56626
  1018
  "inj_on (\<lambda>(i, j). (j, i)) A"
haftmann@56626
  1019
  by (rule inj_onI) auto
haftmann@56545
  1020
haftmann@58195
  1021
lemma surj_swap [simp]:
haftmann@58195
  1022
  "surj prod.swap"
haftmann@58195
  1023
  by (rule surjI [of _ prod.swap]) simp
haftmann@58195
  1024
haftmann@58195
  1025
lemma bij_swap [simp]:
haftmann@58195
  1026
  "bij prod.swap"
haftmann@58195
  1027
  by (simp add: bij_def)
haftmann@58195
  1028
haftmann@56545
  1029
lemma case_swap [simp]:
haftmann@56626
  1030
  "(case prod.swap p of (y, x) \<Rightarrow> f x y) = (case p of (x, y) \<Rightarrow> f x y)"
haftmann@56545
  1031
  by (cases p) simp
haftmann@56545
  1032
haftmann@26358
  1033
text {*
haftmann@26358
  1034
  Disjoint union of a family of sets -- Sigma.
haftmann@26358
  1035
*}
haftmann@26358
  1036
haftmann@45986
  1037
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where
haftmann@26358
  1038
  Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
haftmann@26358
  1039
haftmann@26358
  1040
abbreviation
haftmann@45986
  1041
  Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set"
haftmann@26358
  1042
    (infixr "<*>" 80) where
haftmann@26358
  1043
  "A <*> B == Sigma A (%_. B)"
haftmann@26358
  1044
haftmann@26358
  1045
notation (xsymbols)
haftmann@26358
  1046
  Times  (infixr "\<times>" 80)
berghofe@15394
  1047
haftmann@26358
  1048
notation (HTML output)
haftmann@26358
  1049
  Times  (infixr "\<times>" 80)
haftmann@26358
  1050
nipkow@45662
  1051
hide_const (open) Times
nipkow@45662
  1052
haftmann@26358
  1053
syntax
wenzelm@35115
  1054
  "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
haftmann@26358
  1055
translations
wenzelm@35115
  1056
  "SIGMA x:A. B" == "CONST Sigma A (%x. B)"
haftmann@26358
  1057
haftmann@26358
  1058
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
haftmann@26358
  1059
  by (unfold Sigma_def) blast
haftmann@26358
  1060
haftmann@26358
  1061
lemma SigmaE [elim!]:
haftmann@26358
  1062
    "[| c: Sigma A B;
haftmann@26358
  1063
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
haftmann@26358
  1064
     |] ==> P"
haftmann@26358
  1065
  -- {* The general elimination rule. *}
haftmann@26358
  1066
  by (unfold Sigma_def) blast
haftmann@20588
  1067
haftmann@26358
  1068
text {*
haftmann@26358
  1069
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
haftmann@26358
  1070
  eigenvariables.
haftmann@26358
  1071
*}
haftmann@26358
  1072
haftmann@26358
  1073
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
haftmann@26358
  1074
  by blast
haftmann@26358
  1075
haftmann@26358
  1076
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
haftmann@26358
  1077
  by blast
haftmann@26358
  1078
haftmann@26358
  1079
lemma SigmaE2:
haftmann@26358
  1080
    "[| (a, b) : Sigma A B;
haftmann@26358
  1081
        [| a:A;  b:B(a) |] ==> P
haftmann@26358
  1082
     |] ==> P"
haftmann@26358
  1083
  by blast
haftmann@20588
  1084
haftmann@26358
  1085
lemma Sigma_cong:
haftmann@26358
  1086
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
haftmann@26358
  1087
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
haftmann@26358
  1088
  by auto
haftmann@26358
  1089
haftmann@26358
  1090
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
haftmann@26358
  1091
  by blast
haftmann@26358
  1092
haftmann@26358
  1093
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
haftmann@26358
  1094
  by blast
haftmann@26358
  1095
haftmann@26358
  1096
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
haftmann@26358
  1097
  by blast
haftmann@26358
  1098
haftmann@26358
  1099
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
haftmann@26358
  1100
  by auto
haftmann@21908
  1101
haftmann@26358
  1102
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
haftmann@26358
  1103
  by auto
haftmann@26358
  1104
haftmann@26358
  1105
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
haftmann@26358
  1106
  by auto
haftmann@26358
  1107
haftmann@26358
  1108
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
haftmann@26358
  1109
  by blast
haftmann@26358
  1110
haftmann@26358
  1111
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
haftmann@26358
  1112
  by blast
haftmann@26358
  1113
haftmann@26358
  1114
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
haftmann@26358
  1115
  by (blast elim: equalityE)
haftmann@20588
  1116
haftmann@26358
  1117
lemma SetCompr_Sigma_eq:
haftmann@26358
  1118
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
haftmann@26358
  1119
  by blast
haftmann@26358
  1120
haftmann@26358
  1121
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
haftmann@26358
  1122
  by blast
haftmann@26358
  1123
haftmann@26358
  1124
lemma UN_Times_distrib:
haftmann@26358
  1125
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
haftmann@26358
  1126
  -- {* Suggested by Pierre Chartier *}
haftmann@26358
  1127
  by blast
haftmann@26358
  1128
blanchet@47740
  1129
lemma split_paired_Ball_Sigma [simp, no_atp]:
haftmann@26358
  1130
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
haftmann@26358
  1131
  by blast
haftmann@26358
  1132
blanchet@47740
  1133
lemma split_paired_Bex_Sigma [simp, no_atp]:
haftmann@26358
  1134
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
haftmann@26358
  1135
  by blast
haftmann@21908
  1136
haftmann@26358
  1137
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
haftmann@26358
  1138
  by blast
haftmann@26358
  1139
haftmann@26358
  1140
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
haftmann@26358
  1141
  by blast
haftmann@26358
  1142
haftmann@26358
  1143
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
haftmann@26358
  1144
  by blast
haftmann@26358
  1145
haftmann@26358
  1146
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
haftmann@26358
  1147
  by blast
haftmann@26358
  1148
haftmann@26358
  1149
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
haftmann@26358
  1150
  by blast
haftmann@26358
  1151
haftmann@26358
  1152
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
haftmann@26358
  1153
  by blast
haftmann@21908
  1154
haftmann@26358
  1155
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
haftmann@26358
  1156
  by blast
haftmann@26358
  1157
haftmann@26358
  1158
text {*
haftmann@26358
  1159
  Non-dependent versions are needed to avoid the need for higher-order
haftmann@26358
  1160
  matching, especially when the rules are re-oriented.
haftmann@26358
  1161
*}
haftmann@21908
  1162
haftmann@26358
  1163
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
haftmann@56545
  1164
  by (fact Sigma_Un_distrib1)
haftmann@26358
  1165
haftmann@26358
  1166
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
haftmann@56545
  1167
  by (fact Sigma_Int_distrib1)
haftmann@26358
  1168
haftmann@26358
  1169
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
haftmann@56545
  1170
  by (fact Sigma_Diff_distrib1)
haftmann@26358
  1171
hoelzl@36622
  1172
lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
hoelzl@36622
  1173
  by auto
hoelzl@36622
  1174
hoelzl@50104
  1175
lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
hoelzl@50104
  1176
  by auto
hoelzl@50104
  1177
hoelzl@36622
  1178
lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)"
huffman@44921
  1179
  by force
hoelzl@36622
  1180
hoelzl@36622
  1181
lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)"
huffman@44921
  1182
  by force
hoelzl@36622
  1183
haftmann@56545
  1184
lemma vimage_fst:
haftmann@56545
  1185
  "fst -` A = A \<times> UNIV"
haftmann@56545
  1186
  by auto
haftmann@56545
  1187
haftmann@56545
  1188
lemma vimage_snd:
haftmann@56545
  1189
  "snd -` A = UNIV \<times> A"
haftmann@56545
  1190
  by auto
haftmann@56545
  1191
nipkow@28719
  1192
lemma insert_times_insert[simp]:
nipkow@28719
  1193
  "insert a A \<times> insert b B =
nipkow@28719
  1194
   insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
nipkow@28719
  1195
by blast
haftmann@26358
  1196
paulson@33271
  1197
lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)"
wenzelm@47988
  1198
  apply auto
wenzelm@47988
  1199
  apply (case_tac "f x")
wenzelm@47988
  1200
  apply auto
wenzelm@47988
  1201
  done
paulson@33271
  1202
hoelzl@50104
  1203
lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
hoelzl@50104
  1204
  by auto
hoelzl@50104
  1205
haftmann@56626
  1206
lemma product_swap:
haftmann@56626
  1207
  "prod.swap ` (A \<times> B) = B \<times> A"
haftmann@56626
  1208
  by (auto simp add: set_eq_iff)
haftmann@35822
  1209
haftmann@35822
  1210
lemma swap_product:
haftmann@56626
  1211
  "(\<lambda>(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
haftmann@56626
  1212
  by (auto simp add: set_eq_iff)
haftmann@35822
  1213
hoelzl@36622
  1214
lemma image_split_eq_Sigma:
hoelzl@36622
  1215
  "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
haftmann@46128
  1216
proof (safe intro!: imageI)
hoelzl@36622
  1217
  fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
hoelzl@36622
  1218
  show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
hoelzl@36622
  1219
    using * eq[symmetric] by auto
hoelzl@36622
  1220
qed simp_all
haftmann@35822
  1221
haftmann@46128
  1222
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
hide_const (open) product
haftmann@46128
  1226
haftmann@46128
  1227
lemma member_product:
haftmann@46128
  1228
  "x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B"
haftmann@46128
  1229
  by (simp add: product_def)
haftmann@46128
  1230
blanchet@55932
  1231
text {* The following @{const map_prod} lemmas are due to Joachim Breitner: *}
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)
haftmann@40607
  1243
  with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A`
haftmann@40607
  1244
  have "fst x = fst y" by (auto dest:dest:inj_onD)
blanchet@55932
  1245
  moreover from `map_prod f g x = map_prod f g y`
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)
haftmann@40607
  1248
  with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B`
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"
haftmann@40607
  1260
  from `surj f` obtain a where "fst y = f a" by (auto elim:surjE)
haftmann@40607
  1261
  moreover
haftmann@40607
  1262
  from `surj g` 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"
blanchet@55932
  1273
  assume "x \<in> {y\<Colon>'a \<times> 'c. \<exists>x\<Colon>'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
haftmann@40607
  1275
  from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto
haftmann@40607
  1276
  moreover from `image g B = B'` and `y \<in> A \<times> B` 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
blanchet@55932
  1278
  with `x = map_prod f g y` 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
haftmann@40607
  1282
  from `image f A = A'` and `fst x \<in> A'` 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)
haftmann@40607
  1284
  moreover from `image g B = B'` and `snd x \<in> B'`
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
haftmann@40607
  1287
  moreover from `a \<in> A` and  `b \<in> B` 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
bulwahn@49764
  1293
subsection {* Simproc for rewriting a set comprehension into a pointfree expression *}
bulwahn@49764
  1294
bulwahn@49764
  1295
ML_file "Tools/set_comprehension_pointfree.ML"
bulwahn@49764
  1296
bulwahn@49764
  1297
setup {*
wenzelm@51717
  1298
  Code_Preproc.map_pre (fn ctxt => ctxt addsimprocs
bulwahn@49764
  1299
    [Raw_Simplifier.make_simproc {name = "set comprehension", lhss = [@{cpat "Collect ?P"}],
bulwahn@49764
  1300
    proc = K Set_Comprehension_Pointfree.code_simproc, identifier = []}])
bulwahn@49764
  1301
*}
bulwahn@49764
  1302
bulwahn@49764
  1303
haftmann@37166
  1304
subsection {* Inductively defined sets *}
berghofe@15394
  1305
wenzelm@56512
  1306
(* simplify {(x1, ..., xn). (x1, ..., xn) : S} to S *)
wenzelm@56512
  1307
simproc_setup Collect_mem ("Collect t") = {*
wenzelm@56512
  1308
  fn _ => fn ctxt => fn ct =>
wenzelm@56512
  1309
    (case term_of ct of
wenzelm@56512
  1310
      S as Const (@{const_name Collect}, Type (@{type_name fun}, [_, T])) $ t =>
wenzelm@56512
  1311
        let val (u, _, ps) = HOLogic.strip_psplits t in
wenzelm@56512
  1312
          (case u of
wenzelm@56512
  1313
            (c as Const (@{const_name Set.member}, _)) $ q $ S' =>
wenzelm@56512
  1314
              (case try (HOLogic.strip_ptuple ps) q of
wenzelm@56512
  1315
                NONE => NONE
wenzelm@56512
  1316
              | SOME ts =>
wenzelm@56512
  1317
                  if not (Term.is_open S') andalso
wenzelm@56512
  1318
                    ts = map Bound (length ps downto 0)
wenzelm@56512
  1319
                  then
wenzelm@56512
  1320
                    let val simp =
wenzelm@56512
  1321
                      full_simp_tac (put_simpset HOL_basic_ss ctxt
wenzelm@56512
  1322
                        addsimps [@{thm split_paired_all}, @{thm split_conv}]) 1
wenzelm@56512
  1323
                    in
wenzelm@56512
  1324
                      SOME (Goal.prove ctxt [] []
wenzelm@56512
  1325
                        (Const (@{const_name Pure.eq}, T --> T --> propT) $ S $ S')
wenzelm@56512
  1326
                        (K (EVERY
wenzelm@58839
  1327
                          [resolve_tac [eq_reflection] 1,
wenzelm@58839
  1328
                           resolve_tac @{thms subset_antisym} 1,
wenzelm@58839
  1329
                           resolve_tac [subsetI] 1, dresolve_tac [CollectD] 1, simp,
wenzelm@58839
  1330
                           resolve_tac [subsetI] 1, resolve_tac [CollectI] 1, simp])))
wenzelm@56512
  1331
                    end
wenzelm@56512
  1332
                  else NONE)
wenzelm@56512
  1333
          | _ => NONE)
wenzelm@56512
  1334
        end
wenzelm@56512
  1335
    | _ => NONE)
wenzelm@56512
  1336
*}
blanchet@58389
  1337
wenzelm@48891
  1338
ML_file "Tools/inductive_set.ML"
haftmann@24699
  1339
haftmann@37166
  1340
haftmann@37166
  1341
subsection {* Legacy theorem bindings and duplicates *}
haftmann@37166
  1342
haftmann@37166
  1343
lemma PairE:
haftmann@37166
  1344
  obtains x y where "p = (x, y)"
haftmann@37166
  1345
  by (fact prod.exhaust)
haftmann@37166
  1346
haftmann@37166
  1347
lemmas Pair_eq = prod.inject
blanchet@55393
  1348
lemmas fst_conv = prod.sel(1)
blanchet@55393
  1349
lemmas snd_conv = prod.sel(2)
blanchet@55393
  1350
lemmas pair_collapse = prod.collapse
blanchet@55393
  1351
lemmas split = split_conv
huffman@44066
  1352
lemmas Pair_fst_snd_eq = prod_eq_iff
huffman@44066
  1353
huffman@45204
  1354
hide_const (open) prod
huffman@45204
  1355
nipkow@10213
  1356
end