src/HOL/Product_Type.thy
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(*  Title:      HOL/Product_Type.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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*)
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section \<open>Cartesian products\<close>
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theory Product_Type
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imports Typedef Inductive Fun
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keywords "inductive_set" "coinductive_set" :: thy_decl
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begin
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subsection \<open>@{typ bool} is a datatype\<close>
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free_constructors case_bool for True | False
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  by auto
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text \<open>Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}.\<close>
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setup \<open>Sign.mandatory_path "old"\<close>
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old_rep_datatype True False by (auto intro: bool_induct)
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setup \<open>Sign.parent_path\<close>
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text \<open>But erase the prefix for properties that are not generated by @{text free_constructors}.\<close>
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setup \<open>Sign.mandatory_path "bool"\<close>
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lemmas induct = old.bool.induct
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lemmas inducts = old.bool.inducts
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lemmas rec = old.bool.rec
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lemmas simps = bool.distinct bool.case bool.rec
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setup \<open>Sign.parent_path\<close>
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declare case_split [cases type: bool]
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  -- "prefer plain propositional version"
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lemma
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  shows [code]: "HOL.equal False P \<longleftrightarrow> \<not> P"
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    and [code]: "HOL.equal True P \<longleftrightarrow> P" 
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    and [code]: "HOL.equal P False \<longleftrightarrow> \<not> P"
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    and [code]: "HOL.equal P True \<longleftrightarrow> P"
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    and [code nbe]: "HOL.equal P P \<longleftrightarrow> True"
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  by (simp_all add: equal)
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lemma If_case_cert:
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  assumes "CASE \<equiv> (\<lambda>b. If b f g)"
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  shows "(CASE True \<equiv> f) &&& (CASE False \<equiv> g)"
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  using assms by simp_all
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setup \<open>Code.add_case @{thm If_case_cert}\<close>
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code_printing
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  constant "HOL.equal :: bool \<Rightarrow> bool \<Rightarrow> bool" \<rightharpoonup> (Haskell) infix 4 "=="
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| class_instance "bool" :: "equal" \<rightharpoonup> (Haskell) -
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subsection \<open>The @{text unit} type\<close>
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typedef unit = "{True}"
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  by auto
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definition Unity :: unit  ("'(')")
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  where "() = Abs_unit True"
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lemma unit_eq [no_atp]: "u = ()"
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  by (induct u) (simp add: Unity_def)
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text \<open>
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  Simplification procedure for @{thm [source] unit_eq}.  Cannot use
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  this rule directly --- it loops!
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\<close>
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simproc_setup unit_eq ("x::unit") = \<open>
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  fn _ => fn _ => fn ct =>
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    if HOLogic.is_unit (Thm.term_of ct) then NONE
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    else SOME (mk_meta_eq @{thm unit_eq})
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\<close>
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free_constructors case_unit for "()"
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  by auto
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text \<open>Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}.\<close>
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setup \<open>Sign.mandatory_path "old"\<close>
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old_rep_datatype "()" by simp
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setup \<open>Sign.parent_path\<close>
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text \<open>But erase the prefix for properties that are not generated by @{text free_constructors}.\<close>
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setup \<open>Sign.mandatory_path "unit"\<close>
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lemmas induct = old.unit.induct
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lemmas inducts = old.unit.inducts
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lemmas rec = old.unit.rec
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lemmas simps = unit.case unit.rec
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setup \<open>Sign.parent_path\<close>
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lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
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  by simp
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lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
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  by (rule triv_forall_equality)
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text \<open>
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  This rewrite counters the effect of simproc @{text unit_eq} on @{term
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  [source] "%u::unit. f u"}, replacing it by @{term [source]
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  f} rather than by @{term [source] "%u. f ()"}.
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\<close>
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lemma unit_abs_eta_conv [simp]: "(%u::unit. f ()) = f"
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  by (rule ext) simp
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lemma UNIV_unit:
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  "UNIV = {()}" by auto
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instantiation unit :: default
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begin
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definition "default = ()"
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instance ..
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end
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instantiation unit :: "{complete_boolean_algebra, complete_linorder, wellorder}"
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begin
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definition less_eq_unit :: "unit \<Rightarrow> unit \<Rightarrow> bool"
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where
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  "(_::unit) \<le> _ \<longleftrightarrow> True"
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lemma less_eq_unit [iff]:
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  "(u::unit) \<le> v"
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  by (simp add: less_eq_unit_def)
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definition less_unit :: "unit \<Rightarrow> unit \<Rightarrow> bool"
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where
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  "(_::unit) < _ \<longleftrightarrow> False"
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lemma less_unit [iff]:
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  "\<not> (u::unit) < v"
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  by (simp_all add: less_eq_unit_def less_unit_def)
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definition bot_unit :: unit
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where
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  [code_unfold]: "\<bottom> = ()"
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definition top_unit :: unit
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where
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  [code_unfold]: "\<top> = ()"
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definition inf_unit :: "unit \<Rightarrow> unit \<Rightarrow> unit"
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where
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  [simp]: "_ \<sqinter> _ = ()"
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definition sup_unit :: "unit \<Rightarrow> unit \<Rightarrow> unit"
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where
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  [simp]: "_ \<squnion> _ = ()"
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definition Inf_unit :: "unit set \<Rightarrow> unit"
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where
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  [simp]: "\<Sqinter>_ = ()"
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definition Sup_unit :: "unit set \<Rightarrow> unit"
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where
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  [simp]: "\<Squnion>_ = ()"
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definition uminus_unit :: "unit \<Rightarrow> unit"
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where
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  [simp]: "- _ = ()"
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declare less_eq_unit_def [abs_def, code_unfold]
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  less_unit_def [abs_def, code_unfold]
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  inf_unit_def [abs_def, code_unfold]
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  sup_unit_def [abs_def, code_unfold]
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  Inf_unit_def [abs_def, code_unfold]
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  Sup_unit_def [abs_def, code_unfold]
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  uminus_unit_def [abs_def, code_unfold]
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instance
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  by intro_classes auto
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end
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lemma [code]:
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  "HOL.equal (u\<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 \<open>The product type\<close>
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subsubsection \<open>Type definition\<close>
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definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" where
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  "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
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definition "prod = {f. \<exists>a b. f = Pair_Rep (a\<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 \<open>Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}.\<close>
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setup \<open>Sign.mandatory_path "old"\<close>
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old_rep_datatype Pair
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by (erule prod_cases) (rule prod.inject)
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setup \<open>Sign.parent_path\<close>
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text \<open>But erase the prefix for properties that are not generated by @{text free_constructors}.\<close>
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setup \<open>Sign.mandatory_path "prod"\<close>
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declare
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  old.prod.inject[iff del]
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lemmas induct = old.prod.induct
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lemmas inducts = old.prod.inducts
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lemmas rec = old.prod.rec
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lemmas simps = prod.inject prod.case prod.rec
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setup \<open>Sign.parent_path\<close>
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declare prod.case [nitpick_simp del]
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declare prod.case_cong_weak [cong del]
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subsubsection \<open>Tuple syntax\<close>
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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 \<open>
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  Patterns -- extends pre-defined type @{typ pttrn} used in
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  abstractions.
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\<close>
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nonterminal tuple_args and patterns
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syntax
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  "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
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  "_tuple_arg"  :: "'a => tuple_args"                   ("_")
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  "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
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  "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
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  ""            :: "pttrn => patterns"                  ("_")
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  "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
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translations
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  "(x, y)" == "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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  -- \<open>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\<close>
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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 \<open>
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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
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      | split_tr' _ = raise Match;
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  in [(@{const_syntax case_prod}, K split_tr')] end
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\<close>
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   352
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(* print "split f" as "\<lambda>(x,y). f x y" and "split (\<lambda>x. f x)" as "\<lambda>(x,y). f x y" *) 
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typed_print_translation \<open>
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  let
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    fun split_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match
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      | split_guess_names_tr' T [Abs (x, xT, t)] =
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          (case (head_of t) of
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            Const (@{const_syntax case_prod}, _) => raise Match
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          | _ =>
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            let 
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              val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
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              val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t $ Bound 0);
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              val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t');
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            in
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              Syntax.const @{syntax_const "_abs"} $
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                (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
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            end)
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      | split_guess_names_tr' T [t] =
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          (case head_of t of
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            Const (@{const_syntax case_prod}, _) => raise Match
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          | _ =>
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            let
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              val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
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              val (y, t') =
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                Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t $ Bound 1 $ Bound 0);
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              val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t');
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            in
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              Syntax.const @{syntax_const "_abs"} $
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   380
                (Syntax.const @{syntax_const "_pattern"} $ x' $ y) $ t''
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   381
            end)
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      | split_guess_names_tr' _ _ = raise Match;
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   383
  in [(@{const_syntax case_prod}, K split_guess_names_tr')] end
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   384
\<close>
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   385
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   387
subsubsection \<open>Code generator setup\<close>
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   388
52435
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   389
code_printing
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   390
  type_constructor prod \<rightharpoonup>
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   391
    (SML) infix 2 "*"
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haftmann
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   392
    and (OCaml) infix 2 "*"
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   393
    and (Haskell) "!((_),/ (_))"
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   394
    and (Scala) "((_),/ (_))"
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   395
| constant Pair \<rightharpoonup>
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   396
    (SML) "!((_),/ (_))"
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   397
    and (OCaml) "!((_),/ (_))"
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   398
    and (Haskell) "!((_),/ (_))"
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   399
    and (Scala) "!((_),/ (_))"
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   400
| class_instance  prod :: equal \<rightharpoonup>
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haftmann
parents: 52143
diff changeset
   401
    (Haskell) -
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haftmann
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diff changeset
   402
| constant "HOL.equal :: 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" \<rightharpoonup>
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diff changeset
   403
    (Haskell) infix 4 "=="
37166
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diff changeset
   404
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   405
60758
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wenzelm
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diff changeset
   406
subsubsection \<open>Fundamental operations and properties\<close>
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diff changeset
   407
49897
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bulwahn
parents: 49834
diff changeset
   408
lemma Pair_inject:
cc69be3c8f87 moving Pair_inject from legacy and duplicate section to general section, as Pair_inject was considered a duplicate in e8400e31528a by mistake (cf. communication on dev mailing list)
bulwahn
parents: 49834
diff changeset
   409
  assumes "(a, b) = (a', b')"
cc69be3c8f87 moving Pair_inject from legacy and duplicate section to general section, as Pair_inject was considered a duplicate in e8400e31528a by mistake (cf. communication on dev mailing list)
bulwahn
parents: 49834
diff changeset
   410
    and "a = a' ==> b = b' ==> R"
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bulwahn
parents: 49834
diff changeset
   411
  shows R
cc69be3c8f87 moving Pair_inject from legacy and duplicate section to general section, as Pair_inject was considered a duplicate in e8400e31528a by mistake (cf. communication on dev mailing list)
bulwahn
parents: 49834
diff changeset
   412
  using assms by simp
cc69be3c8f87 moving Pair_inject from legacy and duplicate section to general section, as Pair_inject was considered a duplicate in e8400e31528a by mistake (cf. communication on dev mailing list)
bulwahn
parents: 49834
diff changeset
   413
26358
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haftmann
parents: 26340
diff changeset
   414
lemma surj_pair [simp]: "EX x y. p = (x, y)"
37166
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haftmann
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diff changeset
   415
  by (cases p) simp
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parents:
diff changeset
   416
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
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diff changeset
   417
code_printing
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haftmann
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diff changeset
   418
  constant fst \<rightharpoonup> (Haskell) "fst"
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haftmann
parents: 52143
diff changeset
   419
| constant snd \<rightharpoonup> (Haskell) "snd"
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haftmann
parents: 26340
diff changeset
   420
55414
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blanchet
parents: 55403
diff changeset
   421
lemma case_prod_unfold [nitpick_unfold]: "case_prod = (%c p. c (fst p) (snd p))"
39302
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nipkow
parents: 39272
diff changeset
   422
  by (simp add: fun_eq_iff split: prod.split)
26358
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haftmann
parents: 26340
diff changeset
   423
11838
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wenzelm
parents: 11820
diff changeset
   424
lemma fst_eqD: "fst (x, y) = a ==> x = a"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   425
  by simp
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   426
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   427
lemma snd_eqD: "snd (x, y) = a ==> y = a"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   428
  by simp
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   429
55393
ce5cebfaedda se 'wrap_free_constructors' to register 'sum' , 'prod', 'unit', 'bool' with their discriminators/selectors
blanchet
parents: 54630
diff changeset
   430
lemmas surjective_pairing = prod.collapse [symmetric]
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   431
44066
d74182c93f04 rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents: 43866
diff changeset
   432
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   433
  by (cases s, cases t) simp
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   434
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   435
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
44066
d74182c93f04 rename Pair_fst_snd_eq to prod_eq_iff (keeping old name too)
huffman
parents: 43866
diff changeset
   436
  by (simp add: prod_eq_iff)
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   437
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   438
lemma split_conv [simp, code]: "split f (a, b) = f a b"
55642
63beb38e9258 adapted to renaming of datatype 'cases' and 'recs' to 'case' and 'rec'
blanchet
parents: 55469
diff changeset
   439
  by (fact prod.case)
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   440
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   441
lemma splitI: "f a b \<Longrightarrow> split f (a, b)"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   442
  by (rule split_conv [THEN iffD2])
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   443
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   444
lemma splitD: "split f (a, b) \<Longrightarrow> f a b"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   445
  by (rule split_conv [THEN iffD1])
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   446
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   447
lemma split_Pair [simp]: "(\<lambda>(x, y). (x, y)) = id"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   448
  by (simp add: fun_eq_iff split: prod.split)
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   449
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   450
lemma split_eta: "(\<lambda>(x, y). f (x, y)) = f"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   451
  -- \<open>Subsumes the old @{text split_Pair} when @{term f} is the identity function.\<close>
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   452
  by (simp add: fun_eq_iff split: prod.split)
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   453
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   454
lemma split_comp: "split (f \<circ> g) x = f (g (fst x)) (snd x)"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   455
  by (cases x) simp
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   456
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   457
lemma split_twice: "split f (split g p) = split (\<lambda>x y. split f (g x y)) p"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   458
  by (cases p) simp
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   459
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   460
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   461
  by (simp add: case_prod_unfold)
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   462
58468
haftmann
parents: 58389
diff changeset
   463
lemmas split_weak_cong = prod.case_cong_weak
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   464
  -- \<open>Prevents simplification of @{term c}: much faster\<close>
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   465
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   466
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   467
  by (simp add: split_eta)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   468
47740
a8989fe9a3a5 added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents: 46950
diff changeset
   469
lemma split_paired_all [no_atp]: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
11820
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
parents: 11777
diff changeset
   470
proof
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
parents: 11777
diff changeset
   471
  fix a b
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
parents: 11777
diff changeset
   472
  assume "!!x. PROP P x"
19535
e4fdeb32eadf replaced syntax/translations by abbreviation;
wenzelm
parents: 19179
diff changeset
   473
  then show "PROP P (a, b)" .
11820
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
parents: 11777
diff changeset
   474
next
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
parents: 11777
diff changeset
   475
  fix x
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
parents: 11777
diff changeset
   476
  assume "!!a b. PROP P (a, b)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   477
  from \<open>PROP P (fst x, snd x)\<close> show "PROP P x" by simp
11820
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
parents: 11777
diff changeset
   478
qed
015a82d4ee96 proper proof of split_paired_all (presently unused);
wenzelm
parents: 11777
diff changeset
   479
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
   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))"
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
   481
  by (cases x) simp
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
   482
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   483
text \<open>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   484
  The rule @{thm [source] split_paired_all} does not work with the
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   485
  Simplifier because it also affects premises in congrence rules,
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   486
  where this can lead to premises of the form @{text "!!a b. ... =
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   487
  ?P(a, b)"} which cannot be solved by reflexivity.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   488
\<close>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   489
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   490
lemmas split_tupled_all = split_paired_all unit_all_eq2
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   491
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   492
ML \<open>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   493
  (* replace parameters of product type by individual component parameters *)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   494
  local (* filtering with exists_paired_all is an essential optimization *)
56245
84fc7dfa3cd4 more qualified names;
wenzelm
parents: 56218
diff changeset
   495
    fun exists_paired_all (Const (@{const_name Pure.all}, _) $ Abs (_, T, t)) =
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   496
          can HOLogic.dest_prodT T orelse exists_paired_all t
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   497
      | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   498
      | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   499
      | exists_paired_all _ = false;
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   500
    val ss =
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   501
      simpset_of
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   502
       (put_simpset HOL_basic_ss @{context}
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   503
        addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   504
        addsimprocs [@{simproc unit_eq}]);
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   505
  in
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   506
    fun split_all_tac ctxt = SUBGOAL (fn (t, i) =>
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   507
      if exists_paired_all t then safe_full_simp_tac (put_simpset ss ctxt) i else no_tac);
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   508
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   509
    fun unsafe_split_all_tac ctxt = SUBGOAL (fn (t, i) =>
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   510
      if exists_paired_all t then full_simp_tac (put_simpset ss ctxt) i else no_tac);
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   511
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   512
    fun split_all ctxt th =
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   513
      if exists_paired_all (Thm.prop_of th)
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   514
      then full_simplify (put_simpset ss ctxt) th else th;
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   515
  end;
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   516
\<close>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   517
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   518
setup \<open>map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac))\<close>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   519
47740
a8989fe9a3a5 added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents: 46950
diff changeset
   520
lemma split_paired_All [simp, no_atp]: "(ALL x. P x) = (ALL a b. P (a, b))"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   521
  -- \<open>@{text "[iff]"} is not a good idea because it makes @{text blast} loop\<close>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   522
  by fast
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   523
47740
a8989fe9a3a5 added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents: 46950
diff changeset
   524
lemma split_paired_Ex [simp, no_atp]: "(EX x. P x) = (EX a b. P (a, b))"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   525
  by fast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   526
47740
a8989fe9a3a5 added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents: 46950
diff changeset
   527
lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   528
  -- \<open>Can't be added to simpset: loops!\<close>
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   529
  by (simp add: split_eta)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   530
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   531
text \<open>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   532
  Simplification procedure for @{thm [source] cond_split_eta}.  Using
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   533
  @{thm [source] split_eta} as a rewrite rule is not general enough,
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   534
  and using @{thm [source] cond_split_eta} directly would render some
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   535
  existing proofs very inefficient; similarly for @{text
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   536
  split_beta}.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   537
\<close>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   538
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   539
ML \<open>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   540
local
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   541
  val cond_split_eta_ss =
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   542
    simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms cond_split_eta});
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   543
  fun Pair_pat k 0 (Bound m) = (m = k)
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   544
    | Pair_pat k i (Const (@{const_name Pair},  _) $ Bound m $ t) =
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   545
        i > 0 andalso m = k + i andalso Pair_pat k (i - 1) t
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   546
    | Pair_pat _ _ _ = false;
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   547
  fun no_args k i (Abs (_, _, t)) = no_args (k + 1) i t
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   548
    | no_args k i (t $ u) = no_args k i t andalso no_args k i u
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   549
    | no_args k i (Bound m) = m < k orelse m > k + i
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   550
    | no_args _ _ _ = true;
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   551
  fun split_pat tp i (Abs  (_, _, t)) = if tp 0 i t then SOME (i, t) else NONE
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   552
    | split_pat tp i (Const (@{const_name case_prod}, _) $ Abs (_, _, t)) = split_pat tp (i + 1) t
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   553
    | split_pat tp i _ = NONE;
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   554
  fun metaeq ctxt lhs rhs = mk_meta_eq (Goal.prove ctxt [] []
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   555
        (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   556
        (K (simp_tac (put_simpset cond_split_eta_ss ctxt) 1)));
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   557
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   558
  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k + 1) i t
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   559
    | beta_term_pat k i (t $ u) =
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   560
        Pair_pat k i (t $ u) orelse (beta_term_pat k i t andalso beta_term_pat k i u)
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   561
    | beta_term_pat k i t = no_args k i t;
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   562
  fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   563
    | eta_term_pat _ _ _ = false;
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   564
  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   565
    | subst arg k i (t $ u) =
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   566
        if Pair_pat k i (t $ u) then incr_boundvars k arg
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   567
        else (subst arg k i t $ subst arg k i u)
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   568
    | subst arg k i t = t;
43595
7ae4a23b5be6 modernized some simproc setup;
wenzelm
parents: 43594
diff changeset
   569
in
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   570
  fun beta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t) $ arg) =
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   571
        (case split_pat beta_term_pat 1 t of
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   572
          SOME (i, f) => SOME (metaeq ctxt s (subst arg 0 i f))
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15481
diff changeset
   573
        | NONE => NONE)
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   574
    | beta_proc _ _ = NONE;
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   575
  fun eta_proc ctxt (s as Const (@{const_name case_prod}, _) $ Abs (_, _, t)) =
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   576
        (case split_pat eta_term_pat 1 t of
58468
haftmann
parents: 58389
diff changeset
   577
          SOME (_, ft) => SOME (metaeq ctxt s (let val f $ _ = ft in f end))
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15481
diff changeset
   578
        | NONE => NONE)
35364
b8c62d60195c more antiquotations;
wenzelm
parents: 35115
diff changeset
   579
    | eta_proc _ _ = NONE;
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   580
end;
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   581
\<close>
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59498
diff changeset
   582
simproc_setup split_beta ("split f z") =
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   583
  \<open>fn _ => fn ctxt => fn ct => beta_proc ctxt (Thm.term_of ct)\<close>
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59498
diff changeset
   584
simproc_setup split_eta ("split f") =
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   585
  \<open>fn _ => fn ctxt => fn ct => eta_proc ctxt (Thm.term_of ct)\<close>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   586
58468
haftmann
parents: 58389
diff changeset
   587
lemmas split_beta [mono] = prod.case_eq_if
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   588
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
   589
lemma split_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
   590
  by (auto simp: fun_eq_iff)
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
   591
58468
haftmann
parents: 58389
diff changeset
   592
lemmas split_split [no_atp] = prod.split
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   593
  -- \<open>For use with @{text split} and the Simplifier.\<close>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   594
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   595
text \<open>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   596
  @{thm [source] split_split} could be declared as @{text "[split]"}
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   597
  done after the Splitter has been speeded up significantly;
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   598
  precompute the constants involved and don't do anything unless the
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   599
  current goal contains one of those constants.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   600
\<close>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   601
58468
haftmann
parents: 58389
diff changeset
   602
lemmas split_split_asm [no_atp] = prod.split_asm
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   603
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   604
text \<open>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   605
  \medskip @{term split} used as a logical connective or set former.
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   606
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   607
  \medskip These rules are for use with @{text blast}; could instead
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   608
  call @{text simp} using @{thm [source] prod.split} as rewrite.\<close>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   609
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   610
lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   611
  apply (simp only: split_tupled_all)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   612
  apply (simp (no_asm_simp))
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   613
  done
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   614
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   615
lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   616
  apply (simp only: split_tupled_all)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   617
  apply (simp (no_asm_simp))
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   618
  done
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   619
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   620
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   621
  by (induct p) auto
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   622
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   623
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   624
  by (induct p) auto
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   625
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   626
lemma splitE2:
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   627
  "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   628
proof -
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   629
  assume q: "Q (split P z)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   630
  assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   631
  show R
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   632
    apply (rule r surjective_pairing)+
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   633
    apply (rule split_beta [THEN subst], rule q)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   634
    done
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   635
qed
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   636
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   637
lemma splitD': "split R (a,b) c ==> R a b c"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   638
  by simp
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   639
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   640
lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   641
  by simp
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   642
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   643
lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
14208
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   644
by (simp only: split_tupled_all, simp)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   645
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   646
lemma mem_splitE:
58468
haftmann
parents: 58389
diff changeset
   647
  assumes "z \<in> split c p"
haftmann
parents: 58389
diff changeset
   648
  obtains x y where "p = (x, y)" and "z \<in> c x y"
haftmann
parents: 58389
diff changeset
   649
  using assms by (rule splitE2)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   650
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   651
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   652
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   653
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   654
ML \<open>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   655
local (* filtering with exists_p_split is an essential optimization *)
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   656
  fun exists_p_split (Const (@{const_name case_prod},_) $ _ $ (Const (@{const_name Pair},_)$_$_)) = true
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   657
    | exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   658
    | exists_p_split (Abs (_, _, t)) = exists_p_split t
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   659
    | exists_p_split _ = false;
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   660
in
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   661
fun split_conv_tac ctxt = SUBGOAL (fn (t, i) =>
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   662
  if exists_p_split t
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   663
  then safe_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms split_conv}) i
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51703
diff changeset
   664
  else no_tac);
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   665
end;
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   666
\<close>
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26143
diff changeset
   667
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   668
(* This prevents applications of splitE for already splitted arguments leading
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   669
   to quite time-consuming computations (in particular for nested tuples) *)
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   670
setup \<open>map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac))\<close>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   671
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   672
lemma split_eta_SetCompr [simp, no_atp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   673
  by (rule ext) fast
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   674
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 52435
diff changeset
   675
lemma split_eta_SetCompr2 [simp, no_atp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   676
  by (rule ext) fast
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   677
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   678
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   679
  -- \<open>Allows simplifications of nested splits in case of independent predicates.\<close>
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   680
  by (rule ext) blast
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   681
14337
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   682
(* Do NOT make this a simp rule as it
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   683
   a) only helps in special situations
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   684
   b) can lead to nontermination in the presence of split_def
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   685
*)
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   686
lemma split_comp_eq: 
20415
e3d2d7b01279 explicit type variables prevent empty sorts
paulson
parents: 20380
diff changeset
   687
  fixes f :: "'a => 'b => 'c" and g :: "'d => 'a"
e3d2d7b01279 explicit type variables prevent empty sorts
paulson
parents: 20380
diff changeset
   688
  shows "(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 18334
diff changeset
   689
  by (rule ext) auto
14101
d25c23e46173 added upd_fst, upd_snd, some thms
oheimb
parents: 13480
diff changeset
   690
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   691
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   692
  apply (rule_tac x = "(a, b)" in image_eqI)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   693
   apply auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   694
  done
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   695
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   696
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   697
  by blast
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   698
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   699
(*
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   700
the following  would be slightly more general,
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   701
but cannot be used as rewrite rule:
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   702
### Cannot add premise as rewrite rule because it contains (type) unknowns:
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   703
### ?y = .x
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   704
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
14208
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   705
by (rtac some_equality 1)
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   706
by ( Simp_tac 1)
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   707
by (split_all_tac 1)
144f45277d5a misc tidying
paulson
parents: 14190
diff changeset
   708
by (Asm_full_simp_tac 1)
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   709
qed "The_split_eq";
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   710
*)
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   711
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   712
text \<open>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   713
  Setup of internal @{text split_rule}.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   714
\<close>
11838
02d75712061d got rid of ML proof scripts for Product_Type;
wenzelm
parents: 11820
diff changeset
   715
55642
63beb38e9258 adapted to renaming of datatype 'cases' and 'recs' to 'case' and 'rec'
blanchet
parents: 55469
diff changeset
   716
lemmas case_prodI = prod.case [THEN iffD2]
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   717
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   718
lemma case_prodI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> case_prod c p"
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   719
  by (fact splitI2)
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   720
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   721
lemma case_prodI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> case_prod c p x"
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   722
  by (fact splitI2')
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   723
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   724
lemma case_prodE: "case_prod c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   725
  by (fact splitE)
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   726
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   727
lemma case_prodE': "case_prod c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   728
  by (fact splitE')
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   729
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   730
declare case_prodI [intro!]
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   731
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   732
lemma case_prod_beta:
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   733
  "case_prod f p = f (fst p) (snd p)"
37591
d3daea901123 merged constants "split" and "prod_case"
haftmann
parents: 37411
diff changeset
   734
  by (fact split_beta)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 25885
diff changeset
   735
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55414
diff changeset
   736
lemma prod_cases3 [cases type]:
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   737
  obtains (fields) a b c where "y = (a, b, c)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   738
  by (cases y, case_tac b) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   739
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   740
lemma prod_induct3 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   741
    "(!!a b c. P (a, b, c)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   742
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   743
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55414
diff changeset
   744
lemma prod_cases4 [cases type]:
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   745
  obtains (fields) a b c d where "y = (a, b, c, d)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   746
  by (cases y, case_tac c) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   747
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   748
lemma prod_induct4 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   749
    "(!!a b c d. P (a, b, c, d)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   750
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   751
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55414
diff changeset
   752
lemma prod_cases5 [cases type]:
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   753
  obtains (fields) a b c d e where "y = (a, b, c, d, e)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   754
  by (cases y, case_tac d) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   755
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   756
lemma prod_induct5 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   757
    "(!!a b c d e. P (a, b, c, d, e)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   758
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   759
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55414
diff changeset
   760
lemma prod_cases6 [cases type]:
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   761
  obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   762
  by (cases y, case_tac e) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   763
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   764
lemma prod_induct6 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   765
    "(!!a b c d e f. P (a, b, c, d, e, f)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   766
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   767
55417
01fbfb60c33e adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
blanchet
parents: 55414
diff changeset
   768
lemma prod_cases7 [cases type]:
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   769
  obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   770
  by (cases y, case_tac f) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   771
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   772
lemma prod_induct7 [case_names fields, induct type]:
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   773
    "(!!a b c d e f g. P (a, b, c, d, e, f, g)) ==> P x"
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   774
  by (cases x) blast
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   775
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   776
lemma split_def:
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   777
  "split = (\<lambda>c p. c (fst p) (snd p))"
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   778
  by (fact case_prod_unfold)
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   779
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   780
definition internal_split :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c" where
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   781
  "internal_split == split"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   782
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   783
lemma internal_split_conv: "internal_split c (a, b) = c a b"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   784
  by (simp only: internal_split_def split_conv)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   785
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 47988
diff changeset
   786
ML_file "Tools/split_rule.ML"
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   787
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   788
hide_const internal_split
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   789
24699
c6674504103f datatype interpretators for size and datatype_realizer
haftmann
parents: 24286
diff changeset
   790
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   791
subsubsection \<open>Derived operations\<close>
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   792
37387
3581483cca6c qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents: 37278
diff changeset
   793
definition curry    :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c" where
3581483cca6c qualified types "+" and nat; qualified constants Ball, Bex, Suc, curry; modernized some specifications
haftmann
parents: 37278
diff changeset
   794
  "curry = (\<lambda>c x y. c (x, y))"
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   795
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   796
lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   797
  by (simp add: curry_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   798
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   799
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   800
  by (simp add: curry_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   801
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   802
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   803
  by (simp add: curry_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   804
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   805
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   806
  by (simp add: curry_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   807
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   808
lemma curry_split [simp]: "curry (split f) = f"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   809
  by (simp add: curry_def split_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   810
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   811
lemma split_curry [simp]: "split (curry f) = f"
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   812
  by (simp add: curry_def split_def)
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   813
54630
9061af4d5ebc restrict admissibility to non-empty chains to allow more syntax-directed proof rules
Andreas Lochbihler
parents: 54147
diff changeset
   814
lemma curry_K: "curry (\<lambda>x. c) = (\<lambda>x y. c)"
9061af4d5ebc restrict admissibility to non-empty chains to allow more syntax-directed proof rules
Andreas Lochbihler
parents: 54147
diff changeset
   815
by(simp add: fun_eq_iff)
9061af4d5ebc restrict admissibility to non-empty chains to allow more syntax-directed proof rules
Andreas Lochbihler
parents: 54147
diff changeset
   816
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   817
text \<open>
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   818
  The composition-uncurry combinator.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   819
\<close>
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   820
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   821
notation fcomp (infixl "\<circ>>" 60)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   822
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   823
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60) where
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   824
  "f \<circ>\<rightarrow> g = (\<lambda>x. case_prod g (f x))"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   825
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   826
lemma scomp_unfold: "scomp = (\<lambda>f g x. g (fst (f x)) (snd (f x)))"
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   827
  by (simp add: fun_eq_iff scomp_def case_prod_unfold)
37678
0040bafffdef "prod" and "sum" replace "*" and "+" respectively
haftmann
parents: 37591
diff changeset
   828
55414
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   829
lemma scomp_apply [simp]: "(f \<circ>\<rightarrow> g) x = case_prod g (f x)"
eab03e9cee8a renamed '{prod,sum,bool,unit}_case' to 'case_...'
blanchet
parents: 55403
diff changeset
   830
  by (simp add: scomp_unfold case_prod_unfold)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   831
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   832
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
   833
  by (simp add: fun_eq_iff)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   834
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   835
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
   836
  by (simp add: fun_eq_iff)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   837
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   838
lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   839
  by (simp add: fun_eq_iff scomp_unfold)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   840
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   841
lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   842
  by (simp add: fun_eq_iff scomp_unfold fcomp_def)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   843
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   844
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
   845
  by (simp add: fun_eq_iff scomp_unfold)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   846
52435
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 52143
diff changeset
   847
code_printing
6646bb548c6b migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
haftmann
parents: 52143
diff changeset
   848
  constant scomp \<rightharpoonup> (Eval) infixl 3 "#->"
31202
52d332f8f909 pretty printing of functional combinators for evaluation code
haftmann
parents: 30924
diff changeset
   849
37751
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   850
no_notation fcomp (infixl "\<circ>>" 60)
89e16802b6cc nicer xsymbol syntax for fcomp and scomp
haftmann
parents: 37704
diff changeset
   851
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   852
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   853
text \<open>
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   854
  @{term map_prod} --- action of the product functor upon
36664
6302f9ad7047 repaired comments where SOMEthing went utterly wrong (cf. 2b04504fcb69)
krauss
parents: 36622
diff changeset
   855
  functions.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   856
\<close>
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   857
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   858
definition map_prod :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd" where
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   859
  "map_prod f g = (\<lambda>(x, y). (f x, g y))"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   860
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   861
lemma map_prod_simp [simp, code]:
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   862
  "map_prod f g (a, b) = (f a, g b)"
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   863
  by (simp add: map_prod_def)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   864
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   865
functor map_prod: map_prod
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
   866
  by (auto simp add: split_paired_all)
37278
307845cc7f51 added lemmas
nipkow
parents: 37166
diff changeset
   867
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   868
lemma fst_map_prod [simp]:
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   869
  "fst (map_prod f g x) = f (fst x)"
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   870
  by (cases x) simp_all
37278
307845cc7f51 added lemmas
nipkow
parents: 37166
diff changeset
   871
58916
229765cc3414 more complete fp_sugars for sum and prod;
traytel
parents: 58889
diff changeset
   872
lemma snd_map_prod [simp]:
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   873
  "snd (map_prod f g x) = g (snd x)"
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   874
  by (cases x) simp_all
37278
307845cc7f51 added lemmas
nipkow
parents: 37166
diff changeset
   875
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   876
lemma fst_comp_map_prod [simp]:
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   877
  "fst \<circ> map_prod f g = f \<circ> fst"
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   878
  by (rule ext) simp_all
37278
307845cc7f51 added lemmas
nipkow
parents: 37166
diff changeset
   879
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   880
lemma snd_comp_map_prod [simp]:
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   881
  "snd \<circ> map_prod f g = g \<circ> snd"
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   882
  by (rule ext) simp_all
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   883
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   884
lemma map_prod_compose:
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   885
  "map_prod (f1 o f2) (g1 o g2) = (map_prod f1 g1 o map_prod f2 g2)"
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   886
  by (rule ext) (simp add: map_prod.compositionality comp_def)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   887
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   888
lemma map_prod_ident [simp]:
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   889
  "map_prod (%x. x) (%y. y) = (%z. z)"
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   890
  by (rule ext) (simp add: map_prod.identity)
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   891
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   892
lemma map_prod_imageI [intro]:
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   893
  "(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_prod f g ` R"
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   894
  by (rule image_eqI) simp_all
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   895
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   896
lemma prod_fun_imageE [elim!]:
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   897
  assumes major: "c \<in> map_prod f g ` R"
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   898
    and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   899
  shows P
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   900
  apply (rule major [THEN imageE])
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   901
  apply (case_tac x)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   902
  apply (rule cases)
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
   903
  apply simp_all
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   904
  done
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   905
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   906
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b" where
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   907
  "apfst f = map_prod f id"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   908
37166
e8400e31528a more coherent theory structure; tuned headings
haftmann
parents: 37136
diff changeset
   909
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c" where
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
   910
  "apsnd f = map_prod id f"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   911
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   912
lemma apfst_conv [simp, code]:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   913
  "apfst f (x, y) = (f x, y)" 
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   914
  by (simp add: apfst_def)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   915
33638
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   916
lemma apsnd_conv [simp, code]:
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   917
  "apsnd f (x, y) = (x, f y)" 
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
   918
  by (simp add: apsnd_def)
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   919
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   920
lemma fst_apfst [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   921
  "fst (apfst f x) = f (fst x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   922
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   923
51173
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   924
lemma fst_comp_apfst [simp]:
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   925
  "fst \<circ> apfst f = f \<circ> fst"
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   926
  by (simp add: fun_eq_iff)
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   927
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   928
lemma fst_apsnd [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   929
  "fst (apsnd f x) = fst x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   930
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   931
51173
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   932
lemma fst_comp_apsnd [simp]:
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   933
  "fst \<circ> apsnd f = fst"
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   934
  by (simp add: fun_eq_iff)
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   935
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   936
lemma snd_apfst [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   937
  "snd (apfst f x) = snd x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   938
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   939
51173
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   940
lemma snd_comp_apfst [simp]:
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   941
  "snd \<circ> apfst f = snd"
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   942
  by (simp add: fun_eq_iff)
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   943
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   944
lemma snd_apsnd [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   945
  "snd (apsnd f x) = f (snd x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   946
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   947
51173
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   948
lemma snd_comp_apsnd [simp]:
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   949
  "snd \<circ> apsnd f = f \<circ> snd"
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   950
  by (simp add: fun_eq_iff)
3cbb4e95a565 Sieve of Eratosthenes
haftmann
parents: 50107
diff changeset
   951
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   952
lemma apfst_compose:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   953
  "apfst f (apfst g x) = apfst (f \<circ> g) x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   954
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   955
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   956
lemma apsnd_compose:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   957
  "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   958
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   959
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   960
lemma apfst_apsnd [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   961
  "apfst f (apsnd g x) = (f (fst x), g (snd x))"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   962
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   963
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   964
lemma apsnd_apfst [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   965
  "apsnd f (apfst g x) = (g (fst x), f (snd x))"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   966
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   967
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   968
lemma apfst_id [simp] :
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   969
  "apfst id = id"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   970
  by (simp add: fun_eq_iff)
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   971
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   972
lemma apsnd_id [simp] :
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   973
  "apsnd id = id"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39272
diff changeset
   974
  by (simp add: fun_eq_iff)
33594
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   975
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   976
lemma apfst_eq_conv [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   977
  "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   978
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   979
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   980
lemma apsnd_eq_conv [simp]:
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   981
  "apsnd f x = apsnd g x \<longleftrightarrow> f (snd x) = g (snd x)"
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   982
  by (cases x) simp
357f74e0090c lemmas about apfst and apsnd
haftmann
parents: 33275
diff changeset
   983
33638
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   984
lemma apsnd_apfst_commute:
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   985
  "apsnd f (apfst g p) = apfst g (apsnd f p)"
548a34929e98 Renamed upd_snd_conv to apsnd_conv to be consistent with apfst_conv; Added apsnd_apfst_commute
hoelzl
parents: 33594
diff changeset
   986
  by simp
21195
0cca8d19557d two further lemmas on split
haftmann
parents: 21046
diff changeset
   987
56626
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
   988
context
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
   989
begin
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
   990
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   991
local_setup \<open>Local_Theory.map_background_naming (Name_Space.mandatory_path "prod")\<close>
56626
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
   992
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
   993
definition swap :: "'a \<times> 'b \<Rightarrow> 'b \<times> 'a"
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
   994
where
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
   995
  "swap p = (snd p, fst p)"
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
   996
56626
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
   997
end
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
   998
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
   999
lemma swap_simp [simp]:
56626
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1000
  "prod.swap (x, y) = (y, x)"
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1001
  by (simp add: prod.swap_def)
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1002
58195
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1003
lemma swap_swap [simp]:
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1004
  "prod.swap (prod.swap p) = p"
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1005
  by (cases p) simp
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1006
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1007
lemma swap_comp_swap [simp]:
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1008
  "prod.swap \<circ> prod.swap = id"
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1009
  by (simp add: fun_eq_iff)
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1010
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1011
lemma pair_in_swap_image [simp]:
56626
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1012
  "(y, x) \<in> prod.swap ` A \<longleftrightarrow> (x, y) \<in> A"
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1013
  by (auto intro!: image_eqI)
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1014
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1015
lemma inj_swap [simp]:
56626
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1016
  "inj_on prod.swap A"
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1017
  by (rule inj_onI) auto
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1018
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1019
lemma swap_inj_on:
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1020
  "inj_on (\<lambda>(i, j). (j, i)) A"
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1021
  by (rule inj_onI) auto
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1022
58195
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1023
lemma surj_swap [simp]:
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1024
  "surj prod.swap"
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1025
  by (rule surjI [of _ prod.swap]) simp
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1026
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1027
lemma bij_swap [simp]:
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1028
  "bij prod.swap"
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1029
  by (simp add: bij_def)
1fee63e0377d added various facts
haftmann
parents: 58189
diff changeset
  1030
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1031
lemma case_swap [simp]:
56626
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1032
  "(case prod.swap p of (y, x) \<Rightarrow> f x y) = (case p of (x, y) \<Rightarrow> f x y)"
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1033
  by (cases p) simp
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1034
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
  1035
text \<open>
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1036
  Disjoint union of a family of sets -- Sigma.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
  1037
\<close>
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1038
45986
c9e50153e5ae moved various set operations to theory Set (resp. Product_Type)
haftmann
parents: 45696
diff changeset
  1039
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set" where
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1040
  Sigma_def: "Sigma A B == UN x:A. UN y:B x. {Pair x y}"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1041
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1042
abbreviation
45986
c9e50153e5ae moved various set operations to theory Set (resp. Product_Type)
haftmann
parents: 45696
diff changeset
  1043
  Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1044
    (infixr "<*>" 80) where
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1045
  "A <*> B == Sigma A (%_. B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1046
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1047
notation (xsymbols)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1048
  Times  (infixr "\<times>" 80)
15394
a2c34e6ca4f8 New code generator for let and split.
berghofe
parents: 15140
diff changeset
  1049
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1050
notation (HTML output)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1051
  Times  (infixr "\<times>" 80)
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1052
45662
4f7c05990420 Hide Product_Type.Times - too precious an identifier
nipkow
parents: 45607
diff changeset
  1053
hide_const (open) Times
4f7c05990420 Hide Product_Type.Times - too precious an identifier
nipkow
parents: 45607
diff changeset
  1054
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1055
syntax
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34900
diff changeset
  1056
  "_Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1057
translations
35115
446c5063e4fd modernized translations;
wenzelm
parents: 34900
diff changeset
  1058
  "SIGMA x:A. B" == "CONST Sigma A (%x. B)"
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1059
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1060
lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1061
  by (unfold Sigma_def) blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1062
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1063
lemma SigmaE [elim!]:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1064
    "[| c: Sigma A B;
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1065
        !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1066
     |] ==> P"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
  1067
  -- \<open>The general elimination rule.\<close>
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1068
  by (unfold Sigma_def) blast
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
  1069
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
  1070
text \<open>
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1071
  Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1072
  eigenvariables.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
  1073
\<close>
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1074
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1075
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1076
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1077
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1078
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1079
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1080
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1081
lemma SigmaE2:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1082
    "[| (a, b) : Sigma A B;
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1083
        [| a:A;  b:B(a) |] ==> P
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1084
     |] ==> P"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1085
  by blast
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
  1086
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1087
lemma Sigma_cong:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1088
     "\<lbrakk>A = B; !!x. x \<in> B \<Longrightarrow> C x = D x\<rbrakk>
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1089
      \<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1090
  by auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1091
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1092
lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1093
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1094
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1095
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1096
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1097
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1098
lemma Sigma_empty2 [simp]: "A <*> {} = {}"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1099
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1100
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1101
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1102
  by auto
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1103
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1104
lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1105
  by auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1106
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1107
lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1108
  by auto
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1109
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1110
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1111
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1112
59000
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58916
diff changeset
  1113
lemma Sigma_empty_iff: "(SIGMA i:I. X i) = {} \<longleftrightarrow> (\<forall>i\<in>I. X i = {})"
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58916
diff changeset
  1114
  by auto
6eb0725503fc import general theorems from AFP/Markov_Models
hoelzl
parents: 58916
diff changeset
  1115
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1116
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1117
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1118
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1119
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1120
  by (blast elim: equalityE)
20588
c847c56edf0c added operational equality
haftmann
parents: 20415
diff changeset
  1121
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1122
lemma SetCompr_Sigma_eq:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1123
    "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1124
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1125
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1126
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1127
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1128
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1129
lemma UN_Times_distrib:
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1130
  "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
  1131
  -- \<open>Suggested by Pierre Chartier\<close>
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1132
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1133
47740
a8989fe9a3a5 added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents: 46950
diff changeset
  1134
lemma split_paired_Ball_Sigma [simp, no_atp]:
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1135
    "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1136
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1137
47740
a8989fe9a3a5 added "no_atp"s for extremely prolific, useless facts for ATPs
blanchet
parents: 46950
diff changeset
  1138
lemma split_paired_Bex_Sigma [simp, no_atp]:
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1139
    "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1140
  by blast
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1141
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1142
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1143
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1144
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1145
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1146
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1147
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1148
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1149
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1150
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1151
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1152
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1153
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1154
lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1155
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1156
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1157
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1158
  by blast
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1159
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1160
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1161
  by blast
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1162
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
  1163
text \<open>
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1164
  Non-dependent versions are needed to avoid the need for higher-order
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1165
  matching, especially when the rules are re-oriented.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
  1166
\<close>
21908
d02ba728cd56 moved code generator product setup here
haftmann
parents: 21454
diff changeset
  1167
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1168
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1169
  by (fact Sigma_Un_distrib1)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1170
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1171
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1172
  by (fact Sigma_Int_distrib1)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1173
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1174
lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1175
  by (fact Sigma_Diff_distrib1)
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1176
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1177
lemma Times_empty[simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1178
  by auto
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1179
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
  1180
lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> ((A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
  1181
  by auto
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
  1182
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1183
lemma fst_image_times[simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
  1184
  by force
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1185
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1186
lemma snd_image_times[simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)"
44921
58eef4843641 tuned proofs
huffman
parents: 44066
diff changeset
  1187
  by force
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1188
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1189
lemma vimage_fst:
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1190
  "fst -` A = A \<times> UNIV"
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1191
  by auto
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1192
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1193
lemma vimage_snd:
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1194
  "snd -` A = UNIV \<times> A"
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1195
  by auto
8f1e7596deb7 more operations and lemmas
haftmann
parents: 56512
diff changeset
  1196
28719
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1197
lemma insert_times_insert[simp]:
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1198
  "insert a A \<times> insert b B =
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1199
   insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
01e04e41cc7b added lemma
nipkow
parents: 28562
diff changeset
  1200
by blast
26358
d6a508c16908 Product_Type.apfst and Product_Type.apsnd; mbind combinator; tuned
haftmann
parents: 26340
diff changeset
  1201
33271
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 33089
diff changeset
  1202
lemma vimage_Times: "f -` (A \<times> B) = ((fst \<circ> f) -` A) \<inter> ((snd \<circ> f) -` B)"
47988
e4b69e10b990 tuned proofs;
wenzelm
parents: 47740
diff changeset
  1203
  apply auto
e4b69e10b990 tuned proofs;
wenzelm
parents: 47740
diff changeset
  1204
  apply (case_tac "f x")
e4b69e10b990 tuned proofs;
wenzelm
parents: 47740
diff changeset
  1205
  apply auto
e4b69e10b990 tuned proofs;
wenzelm
parents: 47740
diff changeset
  1206
  done
33271
7be66dee1a5a New theory Probability, which contains a development of measure theory
paulson
parents: 33089
diff changeset
  1207
50104
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
  1208
lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
  1209
  by auto
de19856feb54 move theorems to be more generally useable
hoelzl
parents: 49897
diff changeset
  1210
56626
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1211
lemma product_swap:
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1212
  "prod.swap ` (A \<times> B) = B \<times> A"
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1213
  by (auto simp add: set_eq_iff)
35822
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1214
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1215
lemma swap_product:
56626
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1216
  "(\<lambda>(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
6532efd66a70 swap with qualifier;
haftmann
parents: 56545
diff changeset
  1217
  by (auto simp add: set_eq_iff)
35822
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1218
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1219
lemma image_split_eq_Sigma:
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1220
  "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
46128
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1221
proof (safe intro!: imageI)
36622
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1222
  fix a b assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1223
  show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1224
    using * eq[symmetric] by auto
e393a91f86df Generalize swap_inj_on; add simps for Times; add Ex_list_of_length, log_inj; Added missing locale edges for linordered semiring with 1.
hoelzl
parents: 36176
diff changeset
  1225
qed simp_all
35822
67e4de90d2c2 lemma swap_inj_on, swap_product
haftmann
parents: 35427
diff changeset
  1226
60057
86fa63ce8156 add lemmas
Andreas Lochbihler
parents: 59880
diff changeset
  1227
lemma inj_on_apfst [simp]: "inj_on (apfst f) (A \<times> UNIV) \<longleftrightarrow> inj_on f A"
86fa63ce8156 add lemmas
Andreas Lochbihler
parents: 59880
diff changeset
  1228
by(auto simp add: inj_on_def)
86fa63ce8156 add lemmas
Andreas Lochbihler
parents: 59880
diff changeset
  1229
86fa63ce8156 add lemmas
Andreas Lochbihler
parents: 59880
diff changeset
  1230
lemma inj_apfst [simp]: "inj (apfst f) \<longleftrightarrow> inj f"
86fa63ce8156 add lemmas
Andreas Lochbihler
parents: 59880
diff changeset
  1231
using inj_on_apfst[of f UNIV] by simp
86fa63ce8156 add lemmas
Andreas Lochbihler
parents: 59880
diff changeset
  1232
86fa63ce8156 add lemmas
Andreas Lochbihler
parents: 59880
diff changeset
  1233
lemma inj_on_apsnd [simp]: "inj_on (apsnd f) (UNIV \<times> A) \<longleftrightarrow> inj_on f A"
86fa63ce8156 add lemmas
Andreas Lochbihler
parents: 59880
diff changeset
  1234
by(auto simp add: inj_on_def)
86fa63ce8156 add lemmas
Andreas Lochbihler
parents: 59880
diff changeset
  1235
86fa63ce8156 add lemmas
Andreas Lochbihler
parents: 59880
diff changeset
  1236
lemma inj_apsnd [simp]: "inj (apsnd f) \<longleftrightarrow> inj f"
86fa63ce8156 add lemmas
Andreas Lochbihler
parents: 59880
diff changeset
  1237
using inj_on_apsnd[of f UNIV] by simp
86fa63ce8156 add lemmas
Andreas Lochbihler
parents: 59880
diff changeset
  1238
46128
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1239
definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1240
  [code_abbrev]: "product A B = A \<times> B"
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1241
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1242
hide_const (open) product
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1243
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1244
lemma member_product:
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1245
  "x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B"
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1246
  by (simp add: product_def)
53e7cc599f58 interaction of set operations for execution and membership predicate
haftmann
parents: 46028
diff changeset
  1247
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
  1248
text \<open>The following @{const map_prod} lemmas are due to Joachim Breitner:\<close>
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1249
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
  1250
lemma map_prod_inj_on:
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1251
  assumes "inj_on f A" and "inj_on g B"
55932
68c5104d2204 renamed 'map_pair' to 'map_prod'
blanchet
parents: 55642
diff changeset
  1252
  shows "inj_on (map_prod f g) (A \<times> B)"
40607
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1253
proof (rule inj_onI)
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1254
  fix x :: "'a \<times> 'c" and y :: "'a \<times> 'c"
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590
diff changeset
  1255
  assume "x \<in> A \<times> B" hence "fst x \<in> A" and "snd x \<in> B" by auto
30d512bf47a7 map_pair replaces prod_fun
haftmann
parents: 40590