src/HOL/Product_Type.thy
author blanchet
Fri Sep 05 00:41:01 2014 +0200 (2014-09-05)
changeset 58189 9d714be4f028
parent 57983 6edc3529bb4e
child 58195 1fee63e0377d
permissions -rw-r--r--
added 'plugins' option to control which hooks are enabled
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(*  Title:      HOL/Product_Type.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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*)
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header {* Cartesian products *}
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theory Product_Type
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imports Typedef Inductive Fun
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keywords "inductive_set" "coinductive_set" :: thy_decl
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begin
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subsection {* @{typ bool} is a datatype *}
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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 rep_datatype} with @{text old}. *}
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setup {* Sign.mandatory_path "old" *}
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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 {*
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  Code.add_case @{thm If_case_cert}
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*}
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code_printing
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  constant "HOL.equal :: bool \<Rightarrow> bool \<Rightarrow> bool" \<rightharpoonup> (Haskell) infix 4 "=="
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| class_instance "bool" :: "equal" \<rightharpoonup> (Haskell) -
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subsection {* The @{text unit} type *}
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typedef unit = "{True}"
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  by auto
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definition Unity :: unit  ("'(')")
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  where "() = Abs_unit True"
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lemma unit_eq [no_atp]: "u = ()"
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  by (induct u) (simp add: Unity_def)
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text {*
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  Simplification procedure for @{thm [source] unit_eq}.  Cannot use
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  this rule directly --- it loops!
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*}
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simproc_setup unit_eq ("x::unit") = {*
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  fn _ => fn _ => fn ct =>
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    if HOLogic.is_unit (term_of ct) then NONE
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    else SOME (mk_meta_eq @{thm unit_eq})
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*}
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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 rep_datatype} with @{text old}. *}
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setup {* Sign.mandatory_path "old" *}
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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 rep_datatype} with @{text old}. *}
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setup {* Sign.mandatory_path "old" *}
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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
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          end
wenzelm@52143
   351
      | split_tr' _ = raise Match;
blanchet@55414
   352
  in [(@{const_syntax case_prod}, K split_tr')] end
schirmer@14359
   353
*}
schirmer@14359
   354
schirmer@15422
   355
(* 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
   356
typed_print_translation {*
wenzelm@52143
   357
  let
wenzelm@52143
   358
    fun split_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match
wenzelm@52143
   359
      | split_guess_names_tr' T [Abs (x, xT, t)] =
wenzelm@52143
   360
          (case (head_of t) of
blanchet@55414
   361
            Const (@{const_syntax case_prod}, _) => raise Match
wenzelm@52143
   362
          | _ =>
wenzelm@52143
   363
            let 
wenzelm@52143
   364
              val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
wenzelm@52143
   365
              val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0);
wenzelm@52143
   366
              val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t');
wenzelm@52143
   367
            in
wenzelm@52143
   368
              Syntax.const @{syntax_const "_abs"} $
wenzelm@52143
   369
                (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
wenzelm@52143
   370
            end)
wenzelm@52143
   371
      | split_guess_names_tr' T [t] =
wenzelm@52143
   372
          (case head_of t of
blanchet@55414
   373
            Const (@{const_syntax case_prod}, _) => raise Match
wenzelm@52143
   374
          | _ =>
wenzelm@52143
   375
            let
wenzelm@52143
   376
              val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
wenzelm@52143
   377
              val (y, t') =
wenzelm@52143
   378
                Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0);
wenzelm@52143
   379
              val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t');
wenzelm@52143
   380
            in
wenzelm@52143
   381
              Syntax.const @{syntax_const "_abs"} $
wenzelm@52143
   382
                (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
wenzelm@52143
   383
            end)
wenzelm@52143
   384
      | split_guess_names_tr' _ _ = raise Match;
blanchet@55414
   385
  in [(@{const_syntax case_prod}, K split_guess_names_tr')] end
schirmer@15422
   386
*}
schirmer@15422
   387
blanchet@55414
   388
(* Force eta-contraction for terms of the form "Q A (%p. case_prod P p)"
nipkow@42059
   389
   where Q is some bounded quantifier or set operator.
nipkow@42059
   390
   Reason: the above prints as "Q p : A. case p of (x,y) \<Rightarrow> P x y"
nipkow@42059
   391
   whereas we want "Q (x,y):A. P x y".
nipkow@42059
   392
   Otherwise prevent eta-contraction.
nipkow@42059
   393
*)
nipkow@42059
   394
print_translation {*
wenzelm@52143
   395
  let
wenzelm@52143
   396
    fun contract Q tr ctxt ts =
wenzelm@52143
   397
      (case ts of
blanchet@55414
   398
        [A, Abs (_, _, (s as Const (@{const_syntax case_prod},_) $ t) $ Bound 0)] =>
wenzelm@52143
   399
          if Term.is_dependent t then tr ctxt ts
wenzelm@52143
   400
          else Syntax.const Q $ A $ s
wenzelm@52143
   401
      | _ => tr ctxt ts);
wenzelm@52143
   402
  in
wenzelm@42284
   403
    [Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Ball} @{syntax_const "_Ball"},
wenzelm@42284
   404
     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax Bex} @{syntax_const "_Bex"},
haftmann@56218
   405
     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax INFIMUM} @{syntax_const "_INF"},
haftmann@56218
   406
     Syntax_Trans.preserve_binder_abs2_tr' @{const_syntax SUPREMUM} @{syntax_const "_SUP"}]
wenzelm@52143
   407
    |> map (fn (Q, tr) => (Q, contract Q tr))
wenzelm@52143
   408
  end
nipkow@42059
   409
*}
nipkow@10213
   410
haftmann@37166
   411
subsubsection {* Code generator setup *}
haftmann@37166
   412
haftmann@52435
   413
code_printing
haftmann@52435
   414
  type_constructor prod \<rightharpoonup>
haftmann@52435
   415
    (SML) infix 2 "*"
haftmann@52435
   416
    and (OCaml) infix 2 "*"
haftmann@52435
   417
    and (Haskell) "!((_),/ (_))"
haftmann@52435
   418
    and (Scala) "((_),/ (_))"
haftmann@52435
   419
| constant Pair \<rightharpoonup>
haftmann@52435
   420
    (SML) "!((_),/ (_))"
haftmann@52435
   421
    and (OCaml) "!((_),/ (_))"
haftmann@52435
   422
    and (Haskell) "!((_),/ (_))"
haftmann@52435
   423
    and (Scala) "!((_),/ (_))"
haftmann@52435
   424
| class_instance  prod :: equal \<rightharpoonup>
haftmann@52435
   425
    (Haskell) -
haftmann@52435
   426
| constant "HOL.equal :: 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" \<rightharpoonup>
haftmann@52435
   427
    (Haskell) infix 4 "=="
haftmann@37166
   428
haftmann@37166
   429
haftmann@37166
   430
subsubsection {* Fundamental operations and properties *}
wenzelm@11838
   431
bulwahn@49897
   432
lemma Pair_inject:
bulwahn@49897
   433
  assumes "(a, b) = (a', b')"
bulwahn@49897
   434
    and "a = a' ==> b = b' ==> R"
bulwahn@49897
   435
  shows R
bulwahn@49897
   436
  using assms by simp
bulwahn@49897
   437
haftmann@26358
   438
lemma surj_pair [simp]: "EX x y. p = (x, y)"
haftmann@37166
   439
  by (cases p) simp
nipkow@10213
   440
haftmann@52435
   441
code_printing
haftmann@52435
   442
  constant fst \<rightharpoonup> (Haskell) "fst"
haftmann@52435
   443
| constant snd \<rightharpoonup> (Haskell) "snd"
haftmann@26358
   444
blanchet@55414
   445
lemma case_prod_unfold [nitpick_unfold]: "case_prod = (%c p. c (fst p) (snd p))"
nipkow@39302
   446
  by (simp add: fun_eq_iff split: prod.split)
haftmann@26358
   447
wenzelm@11838
   448
lemma fst_eqD: "fst (x, y) = a ==> x = a"
wenzelm@11838
   449
  by simp
wenzelm@11838
   450
wenzelm@11838
   451
lemma snd_eqD: "snd (x, y) = a ==> y = a"
wenzelm@11838
   452
  by simp
wenzelm@11838
   453
blanchet@55393
   454
lemmas surjective_pairing = prod.collapse [symmetric]
wenzelm@11838
   455
huffman@44066
   456
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
haftmann@37166
   457
  by (cases s, cases t) simp
haftmann@37166
   458
haftmann@37166
   459
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
huffman@44066
   460
  by (simp add: prod_eq_iff)
haftmann@37166
   461
haftmann@37166
   462
lemma split_conv [simp, code]: "split f (a, b) = f a b"
blanchet@55642
   463
  by (fact prod.case)
haftmann@37166
   464
haftmann@37166
   465
lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
haftmann@37166
   466
  by (rule split_conv [THEN iffD2])
haftmann@37166
   467
haftmann@37166
   468
lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
haftmann@37166
   469
  by (rule split_conv [THEN iffD1])
haftmann@37166
   470
haftmann@37166
   471
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
nipkow@39302
   472
  by (simp add: fun_eq_iff split: prod.split)
haftmann@37166
   473
haftmann@37166
   474
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
haftmann@37166
   475
  -- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
nipkow@39302
   476
  by (simp add: fun_eq_iff split: prod.split)
haftmann@37166
   477
haftmann@37166
   478
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
haftmann@37166
   479
  by (cases x) simp
haftmann@37166
   480
haftmann@37166
   481
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
haftmann@37166
   482
  by (cases p) simp
haftmann@37166
   483
haftmann@37166
   484
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
blanchet@55414
   485
  by (simp add: case_prod_unfold)
haftmann@37166
   486
haftmann@37166
   487
lemma split_weak_cong: "p = q \<Longrightarrow> split c p = split c q"
haftmann@37166
   488
  -- {* Prevents simplification of @{term c}: much faster *}
blanchet@57983
   489
  by (fact prod.case_cong_weak)
haftmann@37166
   490
haftmann@37166
   491
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
haftmann@37166
   492
  by (simp add: split_eta)
haftmann@37166
   493
blanchet@47740
   494
lemma split_paired_all [no_atp]: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
wenzelm@11820
   495
proof
wenzelm@11820
   496
  fix a b
wenzelm@11820
   497
  assume "!!x. PROP P x"
wenzelm@19535
   498
  then show "PROP P (a, b)" .
wenzelm@11820
   499
next
wenzelm@11820
   500
  fix x
wenzelm@11820
   501
  assume "!!a b. PROP P (a, b)"
wenzelm@19535
   502
  from `PROP P (fst x, snd x)` show "PROP P x" by simp
wenzelm@11820
   503
qed
wenzelm@11820
   504
hoelzl@50104
   505
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
   506
  by (cases x) simp
hoelzl@50104
   507
wenzelm@11838
   508
text {*
wenzelm@11838
   509
  The rule @{thm [source] split_paired_all} does not work with the
wenzelm@11838
   510
  Simplifier because it also affects premises in congrence rules,
wenzelm@11838
   511
  where this can lead to premises of the form @{text "!!a b. ... =
wenzelm@11838
   512
  ?P(a, b)"} which cannot be solved by reflexivity.
wenzelm@11838
   513
*}
wenzelm@11838
   514
haftmann@26358
   515
lemmas split_tupled_all = split_paired_all unit_all_eq2
haftmann@26358
   516
wenzelm@26480
   517
ML {*
wenzelm@11838
   518
  (* replace parameters of product type by individual component parameters *)
wenzelm@11838
   519
  local (* filtering with exists_paired_all is an essential optimization *)
wenzelm@56245
   520
    fun exists_paired_all (Const (@{const_name Pure.all}, _) $ Abs (_, T, t)) =
wenzelm@11838
   521
          can HOLogic.dest_prodT T orelse exists_paired_all t
wenzelm@11838
   522
      | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
wenzelm@11838
   523
      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
wenzelm@11838
   524
      | exists_paired_all _ = false;
wenzelm@51717
   525
    val ss =
wenzelm@51717
   526
      simpset_of
wenzelm@51717
   527
       (put_simpset HOL_basic_ss @{context}
wenzelm@51717
   528
        addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
wenzelm@51717
   529
        addsimprocs [@{simproc unit_eq}]);
wenzelm@11838
   530
  in
wenzelm@51717
   531
    fun split_all_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   532
      if exists_paired_all t then safe_full_simp_tac (put_simpset ss ctxt) i else no_tac);
wenzelm@51717
   533
wenzelm@51717
   534
    fun unsafe_split_all_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   535
      if exists_paired_all t then full_simp_tac (put_simpset ss ctxt) i else no_tac);
wenzelm@51717
   536
wenzelm@51717
   537
    fun split_all ctxt th =
wenzelm@51717
   538
      if exists_paired_all (Thm.prop_of th)
wenzelm@51717
   539
      then full_simplify (put_simpset ss ctxt) th else th;
wenzelm@11838
   540
  end;
wenzelm@26340
   541
*}
wenzelm@11838
   542
wenzelm@51703
   543
setup {* map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac)) *}
wenzelm@11838
   544
blanchet@47740
   545
lemma split_paired_All [simp, no_atp]: "(ALL x. P x) = (ALL a b. P (a, b))"
wenzelm@11838
   546
  -- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
wenzelm@11838
   547
  by fast
wenzelm@11838
   548
blanchet@47740
   549
lemma split_paired_Ex [simp, no_atp]: "(EX x. P x) = (EX a b. P (a, b))"
haftmann@26358
   550
  by fast
haftmann@26358
   551
blanchet@47740
   552
lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))"
wenzelm@11838
   553
  -- {* Can't be added to simpset: loops! *}
haftmann@26358
   554
  by (simp add: split_eta)
wenzelm@11838
   555
wenzelm@11838
   556
text {*
wenzelm@11838
   557
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
wenzelm@11838
   558
  @{thm [source] split_eta} as a rewrite rule is not general enough,
wenzelm@11838
   559
  and using @{thm [source] cond_split_eta} directly would render some
wenzelm@11838
   560
  existing proofs very inefficient; similarly for @{text
haftmann@26358
   561
  split_beta}.
haftmann@26358
   562
*}
wenzelm@11838
   563
wenzelm@26480
   564
ML {*
wenzelm@11838
   565
local
wenzelm@51717
   566
  val cond_split_eta_ss =
wenzelm@51717
   567
    simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms cond_split_eta});
wenzelm@35364
   568
  fun Pair_pat k 0 (Bound m) = (m = k)
wenzelm@35364
   569
    | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =
wenzelm@35364
   570
        i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t
wenzelm@35364
   571
    | Pair_pat _ _ _ = false;
wenzelm@35364
   572
  fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t
wenzelm@35364
   573
    | no_args k i (t $ u) = no_args k i t andalso no_args k i u
wenzelm@35364
   574
    | no_args k i (Bound m) = m < k orelse m > k + i
wenzelm@35364
   575
    | no_args _ _ _ = true;
wenzelm@35364
   576
  fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE
blanchet@55414
   577
    | split_pat tp i (Const (@{const_name case_prod}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
wenzelm@35364
   578
    | split_pat tp i _ = NONE;
wenzelm@51717
   579
  fun metaeq ctxt lhs rhs = mk_meta_eq (Goal.prove ctxt [] []
wenzelm@35364
   580
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
wenzelm@51717
   581
        (K (simp_tac (put_simpset cond_split_eta_ss ctxt) 1)));
wenzelm@11838
   582
wenzelm@35364
   583
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t
wenzelm@35364
   584
    | beta_term_pat k i (t $ u) =
wenzelm@35364
   585
        Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)
wenzelm@35364
   586
    | beta_term_pat k i t = no_args k i t;
wenzelm@35364
   587
  fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
wenzelm@35364
   588
    | eta_term_pat _ _ _ = false;
wenzelm@11838
   589
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
wenzelm@35364
   590
    | subst arg k i (t $ u) =
wenzelm@35364
   591
        if Pair_pat k i (t $ u) then incr_boundvars k arg
wenzelm@35364
   592
        else (subst arg k i t $ subst arg k i u)
wenzelm@35364
   593
    | subst arg k i t = t;
wenzelm@43595
   594
in
blanchet@55414
   595
  fun beta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t) $ arg) =
wenzelm@11838
   596
        (case split_pat beta_term_pat 1 t of
wenzelm@51717
   597
          SOME (i, f) => SOME (metaeq ctxt s (subst arg 0 i f))
skalberg@15531
   598
        | NONE => NONE)
wenzelm@35364
   599
    | beta_proc _ _ = NONE;
blanchet@55414
   600
  fun eta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t)) =
wenzelm@11838
   601
        (case split_pat eta_term_pat 1 t of
wenzelm@51717
   602
          SOME (_, ft) => SOME (metaeq ctxt s (let val (f $ arg) = ft in f end))
skalberg@15531
   603
        | NONE => NONE)
wenzelm@35364
   604
    | eta_proc _ _ = NONE;
wenzelm@11838
   605
end;
wenzelm@11838
   606
*}
wenzelm@51717
   607
simproc_setup split_beta ("split f z") = {* fn _ => fn ctxt => fn ct => beta_proc ctxt (term_of ct) *}
wenzelm@51717
   608
simproc_setup split_eta ("split f") = {* fn _ => fn ctxt => fn ct => eta_proc ctxt (term_of ct) *}
wenzelm@11838
   609
berghofe@26798
   610
lemma split_beta [mono]: "(%(x, y). P x y) z = P (fst z) (snd z)"
wenzelm@11838
   611
  by (subst surjective_pairing, rule split_conv)
wenzelm@11838
   612
hoelzl@50104
   613
lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
hoelzl@50104
   614
  by (auto simp: fun_eq_iff)
hoelzl@50104
   615
hoelzl@50104
   616
blanchet@35828
   617
lemma split_split [no_atp]: "R(split c p) = (ALL x y. p = (x, y) --> R(c x y))"
wenzelm@11838
   618
  -- {* For use with @{text split} and the Simplifier. *}
paulson@15481
   619
  by (insert surj_pair [of p], clarify, simp)
wenzelm@11838
   620
wenzelm@11838
   621
text {*
wenzelm@11838
   622
  @{thm [source] split_split} could be declared as @{text "[split]"}
wenzelm@11838
   623
  done after the Splitter has been speeded up significantly;
wenzelm@11838
   624
  precompute the constants involved and don't do anything unless the
wenzelm@11838
   625
  current goal contains one of those constants.
wenzelm@11838
   626
*}
wenzelm@11838
   627
blanchet@35828
   628
lemma split_split_asm [no_atp]: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
paulson@14208
   629
by (subst split_split, simp)
wenzelm@11838
   630
wenzelm@11838
   631
text {*
wenzelm@11838
   632
  \medskip @{term split} used as a logical connective or set former.
wenzelm@11838
   633
wenzelm@11838
   634
  \medskip These rules are for use with @{text blast}; could instead
huffman@40929
   635
  call @{text simp} using @{thm [source] prod.split} as rewrite. *}
wenzelm@11838
   636
wenzelm@11838
   637
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
wenzelm@11838
   638
  apply (simp only: split_tupled_all)
wenzelm@11838
   639
  apply (simp (no_asm_simp))
wenzelm@11838
   640
  done
wenzelm@11838
   641
wenzelm@11838
   642
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
wenzelm@11838
   643
  apply (simp only: split_tupled_all)
wenzelm@11838
   644
  apply (simp (no_asm_simp))
wenzelm@11838
   645
  done
wenzelm@11838
   646
wenzelm@11838
   647
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@37591
   648
  by (induct p) auto
wenzelm@11838
   649
wenzelm@11838
   650
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@37591
   651
  by (induct p) auto
wenzelm@11838
   652
wenzelm@11838
   653
lemma splitE2:
wenzelm@11838
   654
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
wenzelm@11838
   655
proof -
wenzelm@11838
   656
  assume q: "Q (split P z)"
wenzelm@11838
   657
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
wenzelm@11838
   658
  show R
wenzelm@11838
   659
    apply (rule r surjective_pairing)+
wenzelm@11838
   660
    apply (rule split_beta [THEN subst], rule q)
wenzelm@11838
   661
    done
wenzelm@11838
   662
qed
wenzelm@11838
   663
wenzelm@11838
   664
lemma splitD': "split R (a,b) c ==> R a b c"
wenzelm@11838
   665
  by simp
wenzelm@11838
   666
wenzelm@11838
   667
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
wenzelm@11838
   668
  by simp
wenzelm@11838
   669
wenzelm@11838
   670
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
paulson@14208
   671
by (simp only: split_tupled_all, simp)
wenzelm@11838
   672
wenzelm@18372
   673
lemma mem_splitE:
haftmann@37166
   674
  assumes major: "z \<in> split c p"
haftmann@37166
   675
    and cases: "\<And>x y. p = (x, y) \<Longrightarrow> z \<in> c x y \<Longrightarrow> Q"
wenzelm@18372
   676
  shows Q
blanchet@55414
   677
  by (rule major [unfolded case_prod_unfold] cases surjective_pairing)+
wenzelm@11838
   678
wenzelm@11838
   679
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
wenzelm@11838
   680
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
wenzelm@11838
   681
wenzelm@26340
   682
ML {*
wenzelm@11838
   683
local (* filtering with exists_p_split is an essential optimization *)
blanchet@55414
   684
  fun exists_p_split (Const (@{const_name case_prod},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true
wenzelm@11838
   685
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
wenzelm@11838
   686
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
wenzelm@11838
   687
    | exists_p_split _ = false;
wenzelm@11838
   688
in
wenzelm@51717
   689
fun split_conv_tac ctxt = SUBGOAL (fn (t, i) =>
wenzelm@51717
   690
  if exists_p_split t
wenzelm@51717
   691
  then safe_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms split_conv}) i
wenzelm@51717
   692
  else no_tac);
wenzelm@11838
   693
end;
wenzelm@26340
   694
*}
wenzelm@26340
   695
wenzelm@11838
   696
(* This prevents applications of splitE for already splitted arguments leading
wenzelm@11838
   697
   to quite time-consuming computations (in particular for nested tuples) *)
wenzelm@51703
   698
setup {* map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac)) *}
wenzelm@11838
   699
blanchet@54147
   700
lemma split_eta_SetCompr [simp, no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
wenzelm@18372
   701
  by (rule ext) fast
wenzelm@11838
   702
blanchet@54147
   703
lemma split_eta_SetCompr2 [simp, no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
wenzelm@18372
   704
  by (rule ext) fast
wenzelm@11838
   705
wenzelm@11838
   706
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
wenzelm@11838
   707
  -- {* Allows simplifications of nested splits in case of independent predicates. *}
wenzelm@18372
   708
  by (rule ext) blast
wenzelm@11838
   709
nipkow@14337
   710
(* Do NOT make this a simp rule as it
nipkow@14337
   711
   a) only helps in special situations
nipkow@14337
   712
   b) can lead to nontermination in the presence of split_def
nipkow@14337
   713
*)
nipkow@14337
   714
lemma split_comp_eq: 
paulson@20415
   715
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
paulson@20415
   716
  shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
wenzelm@18372
   717
  by (rule ext) auto
oheimb@14101
   718
haftmann@26358
   719
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
haftmann@26358
   720
  apply (rule_tac x = "(a, b)" in image_eqI)
haftmann@26358
   721
   apply auto
haftmann@26358
   722
  done
haftmann@26358
   723
wenzelm@11838
   724
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
wenzelm@11838
   725
  by blast
wenzelm@11838
   726
wenzelm@11838
   727
(*
wenzelm@11838
   728
the following  would be slightly more general,
wenzelm@11838
   729
but cannot be used as rewrite rule:
wenzelm@11838
   730
### Cannot add premise as rewrite rule because it contains (type) unknowns:
wenzelm@11838
   731
### ?y = .x
wenzelm@11838
   732
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
paulson@14208
   733
by (rtac some_equality 1)
paulson@14208
   734
by ( Simp_tac 1)
paulson@14208
   735
by (split_all_tac 1)
paulson@14208
   736
by (Asm_full_simp_tac 1)
wenzelm@11838
   737
qed "The_split_eq";
wenzelm@11838
   738
*)
wenzelm@11838
   739
wenzelm@11838
   740
text {*
wenzelm@11838
   741
  Setup of internal @{text split_rule}.
wenzelm@11838
   742
*}
wenzelm@11838
   743
blanchet@55642
   744
lemmas case_prodI = prod.case [THEN iffD2]
haftmann@24699
   745
blanchet@55414
   746
lemma case_prodI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> case_prod c p"
haftmann@37678
   747
  by (fact splitI2)
haftmann@24699
   748
blanchet@55414
   749
lemma case_prodI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> case_prod c p x"
haftmann@37678
   750
  by (fact splitI2')
haftmann@24699
   751
blanchet@55414
   752
lemma case_prodE: "case_prod c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
haftmann@37678
   753
  by (fact splitE)
haftmann@24699
   754
blanchet@55414
   755
lemma case_prodE': "case_prod c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
haftmann@37678
   756
  by (fact splitE')
haftmann@24699
   757
blanchet@55414
   758
declare case_prodI [intro!]
haftmann@24699
   759
blanchet@55414
   760
lemma case_prod_beta:
blanchet@55414
   761
  "case_prod f p = f (fst p) (snd p)"
haftmann@37591
   762
  by (fact split_beta)
bulwahn@26143
   763
blanchet@55417
   764
lemma prod_cases3 [cases type]:
haftmann@24699
   765
  obtains (fields) a b c where "y = (a, b, c)"
haftmann@24699
   766
  by (cases y, case_tac b) blast
haftmann@24699
   767
haftmann@24699
   768
lemma prod_induct3 [case_names fields, induct type]:
haftmann@24699
   769
    "(!!a b c. P (a, b, c)) ==> P x"
haftmann@24699
   770
  by (cases x) blast
haftmann@24699
   771
blanchet@55417
   772
lemma prod_cases4 [cases type]:
haftmann@24699
   773
  obtains (fields) a b c d where "y = (a, b, c, d)"
haftmann@24699
   774
  by (cases y, case_tac c) blast
haftmann@24699
   775
haftmann@24699
   776
lemma prod_induct4 [case_names fields, induct type]:
haftmann@24699
   777
    "(!!a b c d. P (a, b, c, d)) ==> P x"
haftmann@24699
   778
  by (cases x) blast
haftmann@24699
   779
blanchet@55417
   780
lemma prod_cases5 [cases type]:
haftmann@24699
   781
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
haftmann@24699
   782
  by (cases y, case_tac d) blast
haftmann@24699
   783
haftmann@24699
   784
lemma prod_induct5 [case_names fields, induct type]:
haftmann@24699
   785
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
haftmann@24699
   786
  by (cases x) blast
haftmann@24699
   787
blanchet@55417
   788
lemma prod_cases6 [cases type]:
haftmann@24699
   789
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
haftmann@24699
   790
  by (cases y, case_tac e) blast
haftmann@24699
   791
haftmann@24699
   792
lemma prod_induct6 [case_names fields, induct type]:
haftmann@24699
   793
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
haftmann@24699
   794
  by (cases x) blast
haftmann@24699
   795
blanchet@55417
   796
lemma prod_cases7 [cases type]:
haftmann@24699
   797
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
haftmann@24699
   798
  by (cases y, case_tac f) blast
haftmann@24699
   799
haftmann@24699
   800
lemma prod_induct7 [case_names fields, induct type]:
haftmann@24699
   801
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
haftmann@24699
   802
  by (cases x) blast
haftmann@24699
   803
haftmann@37166
   804
lemma split_def:
haftmann@37166
   805
  "split = (\<lambda>c p. c (fst p) (snd p))"
blanchet@55414
   806
  by (fact case_prod_unfold)
haftmann@37166
   807
haftmann@37166
   808
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
haftmann@37166
   809
  "internal_split == split"
haftmann@37166
   810
haftmann@37166
   811
lemma internal_split_conv: "internal_split c (a, b) = c a b"
haftmann@37166
   812
  by (simp only: internal_split_def split_conv)
haftmann@37166
   813
wenzelm@48891
   814
ML_file "Tools/split_rule.ML"
haftmann@37166
   815
setup Split_Rule.setup
haftmann@37166
   816
haftmann@37166
   817
hide_const internal_split
haftmann@37166
   818
haftmann@24699
   819
haftmann@26358
   820
subsubsection {* Derived operations *}
haftmann@26358
   821
haftmann@37387
   822
definition curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where
haftmann@37387
   823
  "curry = (\<lambda>c x y. c (x, y))"
haftmann@37166
   824
haftmann@37166
   825
lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
haftmann@37166
   826
  by (simp add: curry_def)
haftmann@37166
   827
haftmann@37166
   828
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
haftmann@37166
   829
  by (simp add: curry_def)
haftmann@37166
   830
haftmann@37166
   831
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
haftmann@37166
   832
  by (simp add: curry_def)
haftmann@37166
   833
haftmann@37166
   834
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
haftmann@37166
   835
  by (simp add: curry_def)
haftmann@37166
   836
haftmann@37166
   837
lemma curry_split [simp]: "curry (split f) = f"
haftmann@37166
   838
  by (simp add: curry_def split_def)
haftmann@37166
   839
haftmann@37166
   840
lemma split_curry [simp]: "split (curry f) = f"
haftmann@37166
   841
  by (simp add: curry_def split_def)
haftmann@37166
   842
Andreas@54630
   843
lemma curry_K: "curry (\<lambda>x. c) = (\<lambda>x y. c)"
Andreas@54630
   844
by(simp add: fun_eq_iff)
Andreas@54630
   845
haftmann@26358
   846
text {*
haftmann@26358
   847
  The composition-uncurry combinator.
haftmann@26358
   848
*}
haftmann@26358
   849
haftmann@37751
   850
notation fcomp (infixl "\<circ>>" 60)
haftmann@26358
   851
haftmann@37751
   852
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where
blanchet@55414
   853
  "f \<circ>\<rightarrow> g = (\<lambda>x. case_prod g (f x))"
haftmann@26358
   854
haftmann@37678
   855
lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))"
blanchet@55414
   856
  by (simp add: fun_eq_iff scomp_def case_prod_unfold)
haftmann@37678
   857
blanchet@55414
   858
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = case_prod g (f x)"
blanchet@55414
   859
  by (simp add: scomp_unfold case_prod_unfold)
haftmann@26358
   860
haftmann@37751
   861
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x"
huffman@44921
   862
  by (simp add: fun_eq_iff)
haftmann@26358
   863
haftmann@37751
   864
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x"
huffman@44921
   865
  by (simp add: fun_eq_iff)
haftmann@26358
   866
haftmann@37751
   867
lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
nipkow@39302
   868
  by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358
   869
haftmann@37751
   870
lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
nipkow@39302
   871
  by (simp add: fun_eq_iff scomp_unfold fcomp_def)
haftmann@26358
   872
haftmann@37751
   873
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
huffman@44921
   874
  by (simp add: fun_eq_iff scomp_unfold)
haftmann@26358
   875
haftmann@52435
   876
code_printing
haftmann@52435
   877
  constant scomp \<rightharpoonup> (Eval) infixl 3 "#->"
haftmann@31202
   878
haftmann@37751
   879
no_notation fcomp (infixl "\<circ>>" 60)
haftmann@37751
   880
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
haftmann@26358
   881
haftmann@26358
   882
text {*
blanchet@55932
   883
  @{term map_prod} --- action of the product functor upon
krauss@36664
   884
  functions.
haftmann@26358
   885
*}
haftmann@21195
   886
blanchet@55932
   887
definition map_prod :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
blanchet@55932
   888
  "map_prod f g = (\<lambda>(x, y). (f x, g y))"
haftmann@26358
   889
blanchet@55932
   890
lemma map_prod_simp [simp, code]:
blanchet@55932
   891
  "map_prod f g (a, b) = (f a, g b)"
blanchet@55932
   892
  by (simp add: map_prod_def)
haftmann@26358
   893
blanchet@55932
   894
functor map_prod: map_prod
huffman@44921
   895
  by (auto simp add: split_paired_all)
nipkow@37278
   896
blanchet@55932
   897
lemma fst_map_prod [simp]:
blanchet@55932
   898
  "fst (map_prod f g x) = f (fst x)"
haftmann@40607
   899
  by (cases x) simp_all
nipkow@37278
   900
haftmann@40607
   901
lemma snd_prod_fun [simp]:
blanchet@55932
   902
  "snd (map_prod f g x) = g (snd x)"
haftmann@40607
   903
  by (cases x) simp_all
nipkow@37278
   904
blanchet@55932
   905
lemma fst_comp_map_prod [simp]:
blanchet@55932
   906
  "fst \<circ> map_prod f g = f \<circ> fst"
haftmann@40607
   907
  by (rule ext) simp_all
nipkow@37278
   908
blanchet@55932
   909
lemma snd_comp_map_prod [simp]:
blanchet@55932
   910
  "snd \<circ> map_prod f g = g \<circ> snd"
haftmann@40607
   911
  by (rule ext) simp_all
haftmann@26358
   912
blanchet@55932
   913
lemma map_prod_compose:
blanchet@55932
   914
  "map_prod (f1 o f2) (g1 o g2) = (map_prod f1 g1 o map_prod f2 g2)"
blanchet@55932
   915
  by (rule ext) (simp add: map_prod.compositionality comp_def)
haftmann@26358
   916
blanchet@55932
   917
lemma map_prod_ident [simp]:
blanchet@55932
   918
  "map_prod (%x. x) (%y. y) = (%z. z)"
blanchet@55932
   919
  by (rule ext) (simp add: map_prod.identity)
haftmann@40607
   920
blanchet@55932
   921
lemma map_prod_imageI [intro]:
blanchet@55932
   922
  "(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_prod f g ` R"
haftmann@40607
   923
  by (rule image_eqI) simp_all
haftmann@21195
   924
haftmann@26358
   925
lemma prod_fun_imageE [elim!]:
blanchet@55932
   926
  assumes major: "c \<in> map_prod f g ` R"
haftmann@40607
   927
    and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P"
haftmann@26358
   928
  shows P
haftmann@26358
   929
  apply (rule major [THEN imageE])
haftmann@37166
   930
  apply (case_tac x)
haftmann@26358
   931
  apply (rule cases)
haftmann@40607
   932
  apply simp_all
haftmann@26358
   933
  done
haftmann@26358
   934
haftmann@37166
   935
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where
blanchet@55932
   936
  "apfst f = map_prod f id"
haftmann@26358
   937
haftmann@37166
   938
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where
blanchet@55932
   939
  "apsnd f = map_prod id f"
haftmann@26358
   940
haftmann@26358
   941
lemma apfst_conv [simp, code]:
haftmann@26358
   942
  "apfst f (x, y) = (f x, y)" 
haftmann@26358
   943
  by (simp add: apfst_def)
haftmann@26358
   944
hoelzl@33638
   945
lemma apsnd_conv [simp, code]:
haftmann@26358
   946
  "apsnd f (x, y) = (x, f y)" 
haftmann@26358
   947
  by (simp add: apsnd_def)
haftmann@21195
   948
haftmann@33594
   949
lemma fst_apfst [simp]:
haftmann@33594
   950
  "fst (apfst f x) = f (fst x)"
haftmann@33594
   951
  by (cases x) simp
haftmann@33594
   952
haftmann@51173
   953
lemma fst_comp_apfst [simp]:
haftmann@51173
   954
  "fst \<circ> apfst f = f \<circ> fst"
haftmann@51173
   955
  by (simp add: fun_eq_iff)
haftmann@51173
   956
haftmann@33594
   957
lemma fst_apsnd [simp]:
haftmann@33594
   958
  "fst (apsnd f x) = fst x"
haftmann@33594
   959
  by (cases x) simp
haftmann@33594
   960
haftmann@51173
   961
lemma fst_comp_apsnd [simp]:
haftmann@51173
   962
  "fst \<circ> apsnd f = fst"
haftmann@51173
   963
  by (simp add: fun_eq_iff)
haftmann@51173
   964
haftmann@33594
   965
lemma snd_apfst [simp]:
haftmann@33594
   966
  "snd (apfst f x) = snd x"
haftmann@33594
   967
  by (cases x) simp
haftmann@33594
   968
haftmann@51173
   969
lemma snd_comp_apfst [simp]:
haftmann@51173
   970
  "snd \<circ> apfst f = snd"
haftmann@51173
   971
  by (simp add: fun_eq_iff)
haftmann@51173
   972
haftmann@33594
   973
lemma snd_apsnd [simp]:
haftmann@33594
   974
  "snd (apsnd f x) = f (snd x)"
haftmann@33594
   975
  by (cases x) simp
haftmann@33594
   976
haftmann@51173
   977
lemma snd_comp_apsnd [simp]:
haftmann@51173
   978
  "snd \<circ> apsnd f = f \<circ> snd"
haftmann@51173
   979
  by (simp add: fun_eq_iff)
haftmann@51173
   980
haftmann@33594
   981
lemma apfst_compose:
haftmann@33594
   982
  "apfst f (apfst g x) = apfst (f \<circ> g) x"
haftmann@33594
   983
  by (cases x) simp
haftmann@33594
   984
haftmann@33594
   985
lemma apsnd_compose:
haftmann@33594
   986
  "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
haftmann@33594
   987
  by (cases x) simp
haftmann@33594
   988
haftmann@33594
   989
lemma apfst_apsnd [simp]:
haftmann@33594
   990
  "apfst f (apsnd g x) = (f (fst x), g (snd x))"
haftmann@33594
   991
  by (cases x) simp
haftmann@33594
   992
haftmann@33594
   993
lemma apsnd_apfst [simp]:
haftmann@33594
   994
  "apsnd f (apfst g x) = (g (fst x), f (snd x))"
haftmann@33594
   995
  by (cases x) simp
haftmann@33594
   996
haftmann@33594
   997
lemma apfst_id [simp] :
haftmann@33594
   998
  "apfst id = id"
nipkow@39302
   999
  by (simp add: fun_eq_iff)
haftmann@33594
  1000
haftmann@33594
  1001
lemma apsnd_id [simp] :
haftmann@33594
  1002
  "apsnd id = id"
nipkow@39302
  1003
  by (simp add: fun_eq_iff)
haftmann@33594
  1004
haftmann@33594
  1005
lemma apfst_eq_conv [simp]:
haftmann@33594
  1006
  "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
haftmann@33594
  1007
  by (cases x) simp
haftmann@33594
  1008
haftmann@33594
  1009
lemma apsnd_eq_conv [simp]:
haftmann@33594
  1010
  "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"
haftmann@33594
  1011
  by (cases x) simp
haftmann@33594
  1012
hoelzl@33638
  1013
lemma apsnd_apfst_commute:
hoelzl@33638
  1014
  "apsnd f (apfst g p) = apfst g (apsnd f p)"
hoelzl@33638
  1015
  by simp
haftmann@21195
  1016
haftmann@56626
  1017
context
haftmann@56626
  1018
begin
haftmann@56626
  1019
haftmann@56626
  1020
local_setup {* Local_Theory.map_naming (Name_Space.mandatory_path "prod") *}
haftmann@56626
  1021
haftmann@56545
  1022
definition swap :: "'a \<times> 'b \<Rightarrow> 'b \<times> 'a"
haftmann@56545
  1023
where
haftmann@56545
  1024
  "swap p = (snd p, fst p)"
haftmann@56545
  1025
haftmann@56626
  1026
end
haftmann@56626
  1027
haftmann@56545
  1028
lemma swap_simp [simp]:
haftmann@56626
  1029
  "prod.swap (x, y) = (y, x)"
haftmann@56626
  1030
  by (simp add: prod.swap_def)
haftmann@56545
  1031
haftmann@56545
  1032
lemma pair_in_swap_image [simp]:
haftmann@56626
  1033
  "(y, x) \<in> prod.swap ` A \<longleftrightarrow> (x, y) \<in> A"
haftmann@56545
  1034
  by (auto intro!: image_eqI)
haftmann@56545
  1035
haftmann@56545
  1036
lemma inj_swap [simp]:
haftmann@56626
  1037
  "inj_on prod.swap A"
haftmann@56626
  1038
  by (rule inj_onI) auto
haftmann@56626
  1039
haftmann@56626
  1040
lemma swap_inj_on:
haftmann@56626
  1041
  "inj_on (\<lambda>(i, j). (j, i)) A"
haftmann@56626
  1042
  by (rule inj_onI) auto
haftmann@56545
  1043
haftmann@56545
  1044
lemma case_swap [simp]:
haftmann@56626
  1045
  "(case prod.swap p of (y, x) \<Rightarrow> f x y) = (case p of (x, y) \<Rightarrow> f x y)"
haftmann@56545
  1046
  by (cases p) simp
haftmann@56545
  1047
haftmann@26358
  1048
text {*
haftmann@26358
  1049
  Disjoint union of a family of sets -- Sigma.
haftmann@26358
  1050
*}
haftmann@26358
  1051
haftmann@45986
  1052
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where
haftmann@26358
  1053
  Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
haftmann@26358
  1054
haftmann@26358
  1055
abbreviation
haftmann@45986
  1056
  Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set"
haftmann@26358
  1057
    (infixr "<*>" 80) where
haftmann@26358
  1058
  "A <*> B == Sigma A (%_. B)"
haftmann@26358
  1059
haftmann@26358
  1060
notation (xsymbols)
haftmann@26358
  1061
  Times  (infixr "\<times>" 80)
berghofe@15394
  1062
haftmann@26358
  1063
notation (HTML output)
haftmann@26358
  1064
  Times  (infixr "\<times>" 80)
haftmann@26358
  1065
nipkow@45662
  1066
hide_const (open) Times
nipkow@45662
  1067
haftmann@26358
  1068
syntax
wenzelm@35115
  1069
  "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
haftmann@26358
  1070
translations
wenzelm@35115
  1071
  "SIGMA x:A. B" == "CONST Sigma A (%x. B)"
haftmann@26358
  1072
haftmann@26358
  1073
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
haftmann@26358
  1074
  by (unfold Sigma_def) blast
haftmann@26358
  1075
haftmann@26358
  1076
lemma SigmaE [elim!]:
haftmann@26358
  1077
    "[| c: Sigma A B;
haftmann@26358
  1078
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
haftmann@26358
  1079
     |] ==> P"
haftmann@26358
  1080
  -- {* The general elimination rule. *}
haftmann@26358
  1081
  by (unfold Sigma_def) blast
haftmann@20588
  1082
haftmann@26358
  1083
text {*
haftmann@26358
  1084
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
haftmann@26358
  1085
  eigenvariables.
haftmann@26358
  1086
*}
haftmann@26358
  1087
haftmann@26358
  1088
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
haftmann@26358
  1089
  by blast
haftmann@26358
  1090
haftmann@26358
  1091
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
haftmann@26358
  1092
  by blast
haftmann@26358
  1093
haftmann@26358
  1094
lemma SigmaE2:
haftmann@26358
  1095
    "[| (a, b) : Sigma A B;
haftmann@26358
  1096
        [| a:A;  b:B(a) |] ==> P
haftmann@26358
  1097
     |] ==> P"
haftmann@26358
  1098
  by blast
haftmann@20588
  1099
haftmann@26358
  1100
lemma Sigma_cong:
haftmann@26358
  1101
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
haftmann@26358
  1102
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
haftmann@26358
  1103
  by auto
haftmann@26358
  1104
haftmann@26358
  1105
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
haftmann@26358
  1106
  by blast
haftmann@26358
  1107
haftmann@26358
  1108
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
haftmann@26358
  1109
  by blast
haftmann@26358
  1110
haftmann@26358
  1111
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
haftmann@26358
  1112
  by blast
haftmann@26358
  1113
haftmann@26358
  1114
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
haftmann@26358
  1115
  by auto
haftmann@21908
  1116
haftmann@26358
  1117
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
haftmann@26358
  1118
  by auto
haftmann@26358
  1119
haftmann@26358
  1120
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
haftmann@26358
  1121
  by auto
haftmann@26358
  1122
haftmann@26358
  1123
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
haftmann@26358
  1124
  by blast
haftmann@26358
  1125
haftmann@26358
  1126
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
haftmann@26358
  1127
  by blast
haftmann@26358
  1128
haftmann@26358
  1129
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
haftmann@26358
  1130
  by (blast elim: equalityE)
haftmann@20588
  1131
haftmann@26358
  1132
lemma SetCompr_Sigma_eq:
haftmann@26358
  1133
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
haftmann@26358
  1134
  by blast
haftmann@26358
  1135
haftmann@26358
  1136
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
haftmann@26358
  1137
  by blast
haftmann@26358
  1138
haftmann@26358
  1139
lemma UN_Times_distrib:
haftmann@26358
  1140
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
haftmann@26358
  1141
  -- {* Suggested by Pierre Chartier *}
haftmann@26358
  1142
  by blast
haftmann@26358
  1143
blanchet@47740
  1144
lemma split_paired_Ball_Sigma [simp, no_atp]:
haftmann@26358
  1145
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
haftmann@26358
  1146
  by blast
haftmann@26358
  1147
blanchet@47740
  1148
lemma split_paired_Bex_Sigma [simp, no_atp]:
haftmann@26358
  1149
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
haftmann@26358
  1150
  by blast
haftmann@21908
  1151
haftmann@26358
  1152
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
haftmann@26358
  1153
  by blast
haftmann@26358
  1154
haftmann@26358
  1155
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
  1156
  by blast
haftmann@26358
  1157
haftmann@26358
  1158
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
haftmann@26358
  1159
  by blast
haftmann@26358
  1160
haftmann@26358
  1161
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
  1162
  by blast
haftmann@26358
  1163
haftmann@26358
  1164
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
haftmann@26358
  1165
  by blast
haftmann@26358
  1166
haftmann@26358
  1167
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
haftmann@26358
  1168
  by blast
haftmann@21908
  1169
haftmann@26358
  1170
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
haftmann@26358
  1171
  by blast
haftmann@26358
  1172
haftmann@26358
  1173
text {*
haftmann@26358
  1174
  Non-dependent versions are needed to avoid the need for higher-order
haftmann@26358
  1175
  matching, especially when the rules are re-oriented.
haftmann@26358
  1176
*}
haftmann@21908
  1177
haftmann@26358
  1178
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
haftmann@56545
  1179
  by (fact Sigma_Un_distrib1)
haftmann@26358
  1180
haftmann@26358
  1181
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
haftmann@56545
  1182
  by (fact Sigma_Int_distrib1)
haftmann@26358
  1183
haftmann@26358
  1184
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
haftmann@56545
  1185
  by (fact Sigma_Diff_distrib1)
haftmann@26358
  1186
hoelzl@36622
  1187
lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
hoelzl@36622
  1188
  by auto
hoelzl@36622
  1189
hoelzl@50104
  1190
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
  1191
  by auto
hoelzl@50104
  1192
hoelzl@36622
  1193
lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)"
huffman@44921
  1194
  by force
hoelzl@36622
  1195
hoelzl@36622
  1196
lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)"
huffman@44921
  1197
  by force
hoelzl@36622
  1198
haftmann@56545
  1199
lemma vimage_fst:
haftmann@56545
  1200
  "fst -` A = A \<times> UNIV"
haftmann@56545
  1201
  by auto
haftmann@56545
  1202
haftmann@56545
  1203
lemma vimage_snd:
haftmann@56545
  1204
  "snd -` A = UNIV \<times> A"
haftmann@56545
  1205
  by auto
haftmann@56545
  1206
nipkow@28719
  1207
lemma insert_times_insert[simp]:
nipkow@28719
  1208
  "insert a A \<times> insert b B =
nipkow@28719
  1209
   insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
nipkow@28719
  1210
by blast
haftmann@26358
  1211
paulson@33271
  1212
lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)"
wenzelm@47988
  1213
  apply auto
wenzelm@47988
  1214
  apply (case_tac "f x")
wenzelm@47988
  1215
  apply auto
wenzelm@47988
  1216
  done
paulson@33271
  1217
hoelzl@50104
  1218
lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
hoelzl@50104
  1219
  by auto
hoelzl@50104
  1220
haftmann@56626
  1221
lemma product_swap:
haftmann@56626
  1222
  "prod.swap ` (A \<times> B) = B \<times> A"
haftmann@56626
  1223
  by (auto simp add: set_eq_iff)
haftmann@35822
  1224
haftmann@35822
  1225
lemma swap_product:
haftmann@56626
  1226
  "(\<lambda>(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
haftmann@56626
  1227
  by (auto simp add: set_eq_iff)
haftmann@35822
  1228
hoelzl@36622
  1229
lemma image_split_eq_Sigma:
hoelzl@36622
  1230
  "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
haftmann@46128
  1231
proof (safe intro!: imageI)
hoelzl@36622
  1232
  fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
hoelzl@36622
  1233
  show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
hoelzl@36622
  1234
    using * eq[symmetric] by auto
hoelzl@36622
  1235
qed simp_all
haftmann@35822
  1236
haftmann@46128
  1237
definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
haftmann@46128
  1238
  [code_abbrev]: "product A B = A \<times> B"
haftmann@46128
  1239
haftmann@46128
  1240
hide_const (open) product
haftmann@46128
  1241
haftmann@46128
  1242
lemma member_product:
haftmann@46128
  1243
  "x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B"
haftmann@46128
  1244
  by (simp add: product_def)
haftmann@46128
  1245
blanchet@55932
  1246
text {* The following @{const map_prod} lemmas are due to Joachim Breitner: *}
haftmann@40607
  1247
blanchet@55932
  1248
lemma map_prod_inj_on:
haftmann@40607
  1249
  assumes "inj_on f A" and "inj_on g B"
blanchet@55932
  1250
  shows "inj_on (map_prod f g) (A \<times> B)"
haftmann@40607
  1251
proof (rule inj_onI)
haftmann@40607
  1252
  fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c"
haftmann@40607
  1253
  assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto
haftmann@40607
  1254
  assume "y \<in> A \<times> B" hence "fst y \<in> A" and "snd y \<in> B" by auto
blanchet@55932
  1255
  assume "map_prod f g x = map_prod f g y"
blanchet@55932
  1256
  hence "fst (map_prod f g x) = fst (map_prod f g y)" by (auto)
haftmann@40607
  1257
  hence "f (fst x) = f (fst y)" by (cases x,cases y,auto)
haftmann@40607
  1258
  with `inj_on f A` and `fst x \<in> A` and `fst y \<in> A`
haftmann@40607
  1259
  have "fst x = fst y" by (auto dest:dest:inj_onD)
blanchet@55932
  1260
  moreover from `map_prod f g x = map_prod f g y`
blanchet@55932
  1261
  have "snd (map_prod f g x) = snd (map_prod f g y)" by (auto)
haftmann@40607
  1262
  hence "g (snd x) = g (snd y)" by (cases x,cases y,auto)
haftmann@40607
  1263
  with `inj_on g B` and `snd x \<in> B` and `snd y \<in> B`
haftmann@40607
  1264
  have "snd x = snd y" by (auto dest:dest:inj_onD)
haftmann@40607
  1265
  ultimately show "x = y" by(rule prod_eqI)
haftmann@40607
  1266
qed
haftmann@40607
  1267
blanchet@55932
  1268
lemma map_prod_surj:
hoelzl@40702
  1269
  fixes f :: "'a \<Rightarrow> 'b" and g :: "'c \<Rightarrow> 'd"
haftmann@40607
  1270
  assumes "surj f" and "surj g"
blanchet@55932
  1271
  shows "surj (map_prod f g)"
haftmann@40607
  1272
unfolding surj_def
haftmann@40607
  1273
proof
haftmann@40607
  1274
  fix y :: "'b \<times> 'd"
haftmann@40607
  1275
  from `surj f` obtain a where "fst y = f a" by (auto elim:surjE)
haftmann@40607
  1276
  moreover
haftmann@40607
  1277
  from `surj g` obtain b where "snd y = g b" by (auto elim:surjE)
blanchet@55932
  1278
  ultimately have "(fst y, snd y) = map_prod f g (a,b)" by auto
blanchet@55932
  1279
  thus "\<exists>x. y = map_prod f g x" by auto
haftmann@40607
  1280
qed
haftmann@40607
  1281
blanchet@55932
  1282
lemma map_prod_surj_on:
haftmann@40607
  1283
  assumes "f ` A = A'" and "g ` B = B'"
blanchet@55932
  1284
  shows "map_prod f g ` (A \<times> B) = A' \<times> B'"
haftmann@40607
  1285
unfolding image_def
haftmann@40607
  1286
proof(rule set_eqI,rule iffI)
haftmann@40607
  1287
  fix x :: "'a \<times> 'c"
blanchet@55932
  1288
  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
  1289
  then obtain y where "y \<in> A \<times> B" and "x = map_prod f g y" by blast
haftmann@40607
  1290
  from `image f A = A'` and `y \<in> A \<times> B` have "f (fst y) \<in> A'" by auto
haftmann@40607
  1291
  moreover from `image g B = B'` and `y \<in> A \<times> B` have "g (snd y) \<in> B'" by auto
haftmann@40607
  1292
  ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')" by auto
blanchet@55932
  1293
  with `x = map_prod f g y` show "x \<in> A' \<times> B'" by (cases y, auto)
haftmann@40607
  1294
next
haftmann@40607
  1295
  fix x :: "'a \<times> 'c"
haftmann@40607
  1296
  assume "x \<in> A' \<times> B'" hence "fst x \<in> A'" and "snd x \<in> B'" by auto
haftmann@40607
  1297
  from `image f A = A'` and `fst x \<in> A'` have "fst x \<in> image f A" by auto
haftmann@40607
  1298
  then obtain a where "a \<in> A" and "fst x = f a" by (rule imageE)
haftmann@40607
  1299
  moreover from `image g B = B'` and `snd x \<in> B'`
haftmann@40607
  1300
  obtain b where "b \<in> B" and "snd x = g b" by auto
blanchet@55932
  1301
  ultimately have "(fst x, snd x) = map_prod f g (a,b)" by auto
haftmann@40607
  1302
  moreover from `a \<in> A` and  `b \<in> B` have "(a , b) \<in> A \<times> B" by auto
blanchet@55932
  1303
  ultimately have "\<exists>y \<in> A \<times> B. x = map_prod f g y" by auto
blanchet@55932
  1304
  thus "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_prod f g y}" by auto
haftmann@40607
  1305
qed
haftmann@40607
  1306
haftmann@21908
  1307
bulwahn@49764
  1308
subsection {* Simproc for rewriting a set comprehension into a pointfree expression *}
bulwahn@49764
  1309
bulwahn@49764
  1310
ML_file "Tools/set_comprehension_pointfree.ML"
bulwahn@49764
  1311
bulwahn@49764
  1312
setup {*
wenzelm@51717
  1313
  Code_Preproc.map_pre (fn ctxt => ctxt addsimprocs
bulwahn@49764
  1314
    [Raw_Simplifier.make_simproc {name = "set comprehension", lhss = [@{cpat "Collect ?P"}],
bulwahn@49764
  1315
    proc = K Set_Comprehension_Pointfree.code_simproc, identifier = []}])
bulwahn@49764
  1316
*}
bulwahn@49764
  1317
bulwahn@49764
  1318
haftmann@37166
  1319
subsection {* Inductively defined sets *}
berghofe@15394
  1320
wenzelm@56512
  1321
(* simplify {(x1, ..., xn). (x1, ..., xn) : S} to S *)
wenzelm@56512
  1322
simproc_setup Collect_mem ("Collect t") = {*
wenzelm@56512
  1323
  fn _ => fn ctxt => fn ct =>
wenzelm@56512
  1324
    (case term_of ct of
wenzelm@56512
  1325
      S as Const (@{const_name Collect}, Type (@{type_name fun}, [_, T])) $ t =>
wenzelm@56512
  1326
        let val (u, _, ps) = HOLogic.strip_psplits t in
wenzelm@56512
  1327
          (case u of
wenzelm@56512
  1328
            (c as Const (@{const_name Set.member}, _)) $ q $ S' =>
wenzelm@56512
  1329
              (case try (HOLogic.strip_ptuple ps) q of
wenzelm@56512
  1330
                NONE => NONE
wenzelm@56512
  1331
              | SOME ts =>
wenzelm@56512
  1332
                  if not (Term.is_open S') andalso
wenzelm@56512
  1333
                    ts = map Bound (length ps downto 0)
wenzelm@56512
  1334
                  then
wenzelm@56512
  1335
                    let val simp =
wenzelm@56512
  1336
                      full_simp_tac (put_simpset HOL_basic_ss ctxt
wenzelm@56512
  1337
                        addsimps [@{thm split_paired_all}, @{thm split_conv}]) 1
wenzelm@56512
  1338
                    in
wenzelm@56512
  1339
                      SOME (Goal.prove ctxt [] []
wenzelm@56512
  1340
                        (Const (@{const_name Pure.eq}, T --> T --> propT) $ S $ S')
wenzelm@56512
  1341
                        (K (EVERY
wenzelm@56512
  1342
                          [rtac eq_reflection 1, rtac @{thm subset_antisym} 1,
wenzelm@56512
  1343
                           rtac subsetI 1, dtac CollectD 1, simp,
wenzelm@56512
  1344
                           rtac subsetI 1, rtac CollectI 1, simp])))
wenzelm@56512
  1345
                    end
wenzelm@56512
  1346
                  else NONE)
wenzelm@56512
  1347
          | _ => NONE)
wenzelm@56512
  1348
        end
wenzelm@56512
  1349
    | _ => NONE)
wenzelm@56512
  1350
*}
wenzelm@48891
  1351
ML_file "Tools/inductive_set.ML"
haftmann@24699
  1352
haftmann@37166
  1353
haftmann@37166
  1354
subsection {* Legacy theorem bindings and duplicates *}
haftmann@37166
  1355
haftmann@37166
  1356
lemma PairE:
haftmann@37166
  1357
  obtains x y where "p = (x, y)"
haftmann@37166
  1358
  by (fact prod.exhaust)
haftmann@37166
  1359
haftmann@37166
  1360
lemmas Pair_eq = prod.inject
blanchet@55393
  1361
lemmas fst_conv = prod.sel(1)
blanchet@55393
  1362
lemmas snd_conv = prod.sel(2)
blanchet@55393
  1363
lemmas pair_collapse = prod.collapse
blanchet@55393
  1364
lemmas split = split_conv
huffman@44066
  1365
lemmas Pair_fst_snd_eq = prod_eq_iff
huffman@44066
  1366
huffman@45204
  1367
hide_const (open) prod
huffman@45204
  1368
nipkow@10213
  1369
end