src/HOL/Relation.thy
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prefer existing logical constant over abbreviation
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(*  Title:      HOL/Relation.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Author:     Stefan Berghofer, TU Muenchen
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*)
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section \<open>Relations -- as sets of pairs, and binary predicates\<close>
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theory Relation
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  imports Finite_Set
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begin
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text \<open>A preliminary: classical rules for reasoning on predicates\<close>
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declare predicate1I [Pure.intro!, intro!]
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declare predicate1D [Pure.dest, dest]
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declare predicate2I [Pure.intro!, intro!]
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declare predicate2D [Pure.dest, dest]
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declare bot1E [elim!]
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declare bot2E [elim!]
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declare top1I [intro!]
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declare top2I [intro!]
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declare inf1I [intro!]
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declare inf2I [intro!]
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declare inf1E [elim!]
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declare inf2E [elim!]
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declare sup1I1 [intro?]
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declare sup2I1 [intro?]
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declare sup1I2 [intro?]
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declare sup2I2 [intro?]
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declare sup1E [elim!]
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declare sup2E [elim!]
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declare sup1CI [intro!]
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declare sup2CI [intro!]
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declare Inf1_I [intro!]
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declare INF1_I [intro!]
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declare Inf2_I [intro!]
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declare INF2_I [intro!]
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declare Inf1_D [elim]
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declare INF1_D [elim]
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declare Inf2_D [elim]
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declare INF2_D [elim]
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declare Inf1_E [elim]
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declare INF1_E [elim]
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declare Inf2_E [elim]
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declare INF2_E [elim]
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declare Sup1_I [intro]
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declare SUP1_I [intro]
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declare Sup2_I [intro]
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declare SUP2_I [intro]
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declare Sup1_E [elim!]
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declare SUP1_E [elim!]
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declare Sup2_E [elim!]
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declare SUP2_E [elim!]
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subsection \<open>Fundamental\<close>
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subsubsection \<open>Relations as sets of pairs\<close>
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type_synonym 'a rel = "('a \<times> 'a) set"
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lemma subrelI: "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s) \<Longrightarrow> r \<subseteq> s"
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  \<comment> \<open>Version of @{thm [source] subsetI} for binary relations\<close>
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  by auto
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lemma lfp_induct2:
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  "(a, b) \<in> lfp f \<Longrightarrow> mono f \<Longrightarrow>
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    (\<And>a b. (a, b) \<in> f (lfp f \<inter> {(x, y). P x y}) \<Longrightarrow> P a b) \<Longrightarrow> P a b"
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  \<comment> \<open>Version of @{thm [source] lfp_induct} for binary relations\<close>
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  using lfp_induct_set [of "(a, b)" f "case_prod P"] by auto
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subsubsection \<open>Conversions between set and predicate relations\<close>
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lemma pred_equals_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S) \<longleftrightarrow> R = S"
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  by (simp add: set_eq_iff fun_eq_iff)
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lemma pred_equals_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S) \<longleftrightarrow> R = S"
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  by (simp add: set_eq_iff fun_eq_iff)
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lemma pred_subset_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S) \<longleftrightarrow> R \<subseteq> S"
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  by (simp add: subset_iff le_fun_def)
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lemma pred_subset_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S) \<longleftrightarrow> R \<subseteq> S"
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  by (simp add: subset_iff le_fun_def)
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lemma bot_empty_eq [pred_set_conv]: "\<bottom> = (\<lambda>x. x \<in> {})"
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  by (auto simp add: fun_eq_iff)
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lemma bot_empty_eq2 [pred_set_conv]: "\<bottom> = (\<lambda>x y. (x, y) \<in> {})"
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  by (auto simp add: fun_eq_iff)
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lemma top_empty_eq [pred_set_conv]: "\<top> = (\<lambda>x. x \<in> UNIV)"
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  by (auto simp add: fun_eq_iff)
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lemma top_empty_eq2 [pred_set_conv]: "\<top> = (\<lambda>x y. (x, y) \<in> UNIV)"
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  by (auto simp add: fun_eq_iff)
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lemma inf_Int_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<sqinter> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
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  by (simp add: inf_fun_def)
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lemma inf_Int_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<sqinter> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)"
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  by (simp add: inf_fun_def)
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lemma sup_Un_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<squnion> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
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  by (simp add: sup_fun_def)
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lemma sup_Un_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<squnion> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)"
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  by (simp add: sup_fun_def)
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lemma INF_INT_eq [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Inter>i\<in>S. r i))"
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  by (simp add: fun_eq_iff)
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lemma INF_INT_eq2 [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Inter>i\<in>S. r i))"
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  by (simp add: fun_eq_iff)
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lemma SUP_UN_eq [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Union>i\<in>S. r i))"
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  by (simp add: fun_eq_iff)
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lemma SUP_UN_eq2 [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Union>i\<in>S. r i))"
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  by (simp add: fun_eq_iff)
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lemma Inf_INT_eq [pred_set_conv]: "\<Sqinter>S = (\<lambda>x. x \<in> INTER S Collect)"
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  by (simp add: fun_eq_iff)
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lemma INF_Int_eq [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x. x \<in> i)) = (\<lambda>x. x \<in> \<Inter>S)"
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  by (simp add: fun_eq_iff)
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lemma Inf_INT_eq2 [pred_set_conv]: "\<Sqinter>S = (\<lambda>x y. (x, y) \<in> INTER (case_prod ` S) Collect)"
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  by (simp add: fun_eq_iff)
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lemma INF_Int_eq2 [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x y. (x, y) \<in> i)) = (\<lambda>x y. (x, y) \<in> \<Inter>S)"
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  by (simp add: fun_eq_iff)
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lemma Sup_SUP_eq [pred_set_conv]: "\<Squnion>S = (\<lambda>x. x \<in> UNION S Collect)"
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  by (simp add: fun_eq_iff)
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lemma SUP_Sup_eq [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x. x \<in> i)) = (\<lambda>x. x \<in> \<Union>S)"
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  by (simp add: fun_eq_iff)
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lemma Sup_SUP_eq2 [pred_set_conv]: "\<Squnion>S = (\<lambda>x y. (x, y) \<in> UNION (case_prod ` S) Collect)"
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  by (simp add: fun_eq_iff)
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lemma SUP_Sup_eq2 [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x y. (x, y) \<in> i)) = (\<lambda>x y. (x, y) \<in> \<Union>S)"
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  by (simp add: fun_eq_iff)
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subsection \<open>Properties of relations\<close>
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subsubsection \<open>Reflexivity\<close>
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definition refl_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
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  where "refl_on A r \<longleftrightarrow> r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. (x, x) \<in> r)"
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abbreviation refl :: "'a rel \<Rightarrow> bool" \<comment> \<open>reflexivity over a type\<close>
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  where "refl \<equiv> refl_on UNIV"
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definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
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  where "reflp r \<longleftrightarrow> (\<forall>x. r x x)"
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lemma reflp_refl_eq [pred_set_conv]: "reflp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> refl r"
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  by (simp add: refl_on_def reflp_def)
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lemma refl_onI [intro?]: "r \<subseteq> A \<times> A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> (x, x) \<in> r) \<Longrightarrow> refl_on A r"
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  unfolding refl_on_def by (iprover intro!: ballI)
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lemma refl_onD: "refl_on A r \<Longrightarrow> a \<in> A \<Longrightarrow> (a, a) \<in> r"
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  unfolding refl_on_def by blast
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lemma refl_onD1: "refl_on A r \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> x \<in> A"
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  unfolding refl_on_def by blast
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lemma refl_onD2: "refl_on A r \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> y \<in> A"
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  unfolding refl_on_def by blast
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lemma reflpI [intro?]: "(\<And>x. r x x) \<Longrightarrow> reflp r"
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  by (auto intro: refl_onI simp add: reflp_def)
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lemma reflpE:
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  assumes "reflp r"
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  obtains "r x x"
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  using assms by (auto dest: refl_onD simp add: reflp_def)
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lemma reflpD [dest?]:
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  assumes "reflp r"
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  shows "r x x"
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  using assms by (auto elim: reflpE)
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lemma refl_on_Int: "refl_on A r \<Longrightarrow> refl_on B s \<Longrightarrow> refl_on (A \<inter> B) (r \<inter> s)"
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  unfolding refl_on_def by blast
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lemma reflp_inf: "reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<sqinter> s)"
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  by (auto intro: reflpI elim: reflpE)
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lemma refl_on_Un: "refl_on A r \<Longrightarrow> refl_on B s \<Longrightarrow> refl_on (A \<union> B) (r \<union> s)"
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  unfolding refl_on_def by blast
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lemma reflp_sup: "reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<squnion> s)"
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  by (auto intro: reflpI elim: reflpE)
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lemma refl_on_INTER: "\<forall>x\<in>S. refl_on (A x) (r x) \<Longrightarrow> refl_on (INTER S A) (INTER S r)"
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  unfolding refl_on_def by fast
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lemma refl_on_UNION: "\<forall>x\<in>S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)"
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  unfolding refl_on_def by blast
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lemma refl_on_empty [simp]: "refl_on {} {}"
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  by (simp add: refl_on_def)
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lemma refl_on_singleton [simp]: "refl_on {x} {(x, x)}"
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by (blast intro: refl_onI)
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lemma refl_on_def' [nitpick_unfold, code]:
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  "refl_on A r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<in> A \<and> y \<in> A) \<and> (\<forall>x \<in> A. (x, x) \<in> r)"
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  by (auto intro: refl_onI dest: refl_onD refl_onD1 refl_onD2)
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lemma reflp_equality [simp]: "reflp op ="
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  by (simp add: reflp_def)
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lemma reflp_mono: "reflp R \<Longrightarrow> (\<And>x y. R x y \<longrightarrow> Q x y) \<Longrightarrow> reflp Q"
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  by (auto intro: reflpI dest: reflpD)
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subsubsection \<open>Irreflexivity\<close>
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definition irrefl :: "'a rel \<Rightarrow> bool"
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  where "irrefl r \<longleftrightarrow> (\<forall>a. (a, a) \<notin> r)"
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definition irreflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
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  where "irreflp R \<longleftrightarrow> (\<forall>a. \<not> R a a)"
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lemma irreflp_irrefl_eq [pred_set_conv]: "irreflp (\<lambda>a b. (a, b) \<in> R) \<longleftrightarrow> irrefl R"
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  by (simp add: irrefl_def irreflp_def)
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lemma irreflI [intro?]: "(\<And>a. (a, a) \<notin> R) \<Longrightarrow> irrefl R"
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  by (simp add: irrefl_def)
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lemma irreflpI [intro?]: "(\<And>a. \<not> R a a) \<Longrightarrow> irreflp R"
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  by (fact irreflI [to_pred])
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lemma irrefl_distinct [code]: "irrefl r \<longleftrightarrow> (\<forall>(a, b) \<in> r. a \<noteq> b)"
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  by (auto simp add: irrefl_def)
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subsubsection \<open>Asymmetry\<close>
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inductive asym :: "'a rel \<Rightarrow> bool"
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  where asymI: "irrefl R \<Longrightarrow> (\<And>a b. (a, b) \<in> R \<Longrightarrow> (b, a) \<notin> R) \<Longrightarrow> asym R"
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inductive asymp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
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  where asympI: "irreflp R \<Longrightarrow> (\<And>a b. R a b \<Longrightarrow> \<not> R b a) \<Longrightarrow> asymp R"
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lemma asymp_asym_eq [pred_set_conv]: "asymp (\<lambda>a b. (a, b) \<in> R) \<longleftrightarrow> asym R"
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  by (auto intro!: asymI asympI elim: asym.cases asymp.cases simp add: irreflp_irrefl_eq)
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subsubsection \<open>Symmetry\<close>
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definition sym :: "'a rel \<Rightarrow> bool"
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  where "sym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r)"
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definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
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  where "symp r \<longleftrightarrow> (\<forall>x y. r x y \<longrightarrow> r y x)"
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lemma symp_sym_eq [pred_set_conv]: "symp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> sym r"
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  by (simp add: sym_def symp_def)
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lemma symI [intro?]: "(\<And>a b. (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r) \<Longrightarrow> sym r"
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  by (unfold sym_def) iprover
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lemma sympI [intro?]: "(\<And>a b. r a b \<Longrightarrow> r b a) \<Longrightarrow> symp r"
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  by (fact symI [to_pred])
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lemma symE:
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  assumes "sym r" and "(b, a) \<in> r"
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  obtains "(a, b) \<in> r"
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  using assms by (simp add: sym_def)
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lemma sympE:
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  assumes "symp r" and "r b a"
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  obtains "r a b"
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  using assms by (rule symE [to_pred])
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lemma symD [dest?]:
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  assumes "sym r" and "(b, a) \<in> r"
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  shows "(a, b) \<in> r"
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  using assms by (rule symE)
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lemma sympD [dest?]:
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  assumes "symp r" and "r b a"
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  shows "r a b"
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  using assms by (rule symD [to_pred])
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lemma sym_Int: "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<inter> s)"
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  by (fast intro: symI elim: symE)
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lemma symp_inf: "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<sqinter> s)"
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  by (fact sym_Int [to_pred])
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lemma sym_Un: "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<union> s)"
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  by (fast intro: symI elim: symE)
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lemma symp_sup: "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<squnion> s)"
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  by (fact sym_Un [to_pred])
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lemma sym_INTER: "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (INTER S r)"
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  by (fast intro: symI elim: symE)
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lemma symp_INF: "\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (INFIMUM S r)"
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  by (fact sym_INTER [to_pred])
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lemma sym_UNION: "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (UNION S r)"
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  by (fast intro: symI elim: symE)
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lemma symp_SUP: "\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (SUPREMUM S r)"
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  by (fact sym_UNION [to_pred])
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subsubsection \<open>Antisymmetry\<close>
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definition antisym :: "'a rel \<Rightarrow> bool"
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  where "antisym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
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abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
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  where "antisymP r \<equiv> antisym {(x, y). r x y}" (* FIXME proper logical operation *)
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lemma antisymI [intro?]: "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (y, x) \<in> r \<Longrightarrow> x = y) \<Longrightarrow> antisym r"
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  unfolding antisym_def by iprover
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lemma antisymD [dest?]: "antisym r \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r \<Longrightarrow> a = b"
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  unfolding antisym_def by iprover
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lemma antisym_subset: "r \<subseteq> s \<Longrightarrow> antisym s \<Longrightarrow> antisym r"
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  unfolding antisym_def by blast
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lemma antisym_empty [simp]: "antisym {}"
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  unfolding antisym_def by blast
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lemma antisymP_equality [simp]: "antisymP op ="
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  by (auto intro: antisymI)
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lemma antisym_singleton [simp]: "antisym {x}"
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by (blast intro: antisymI)
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subsubsection \<open>Transitivity\<close>
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definition trans :: "'a rel \<Rightarrow> bool"
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  where "trans r \<longleftrightarrow> (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
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definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
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  where "transp r \<longleftrightarrow> (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
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lemma transp_trans_eq [pred_set_conv]: "transp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> trans r"
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  by (simp add: trans_def transp_def)
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lemma transI [intro?]: "(\<And>x y z. (x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r) \<Longrightarrow> trans r"
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  by (unfold trans_def) iprover
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lemma transpI [intro?]: "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"
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  by (fact transI [to_pred])
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lemma transE:
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  assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
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  obtains "(x, z) \<in> r"
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  using assms by (unfold trans_def) iprover
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lemma transpE:
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  assumes "transp r" and "r x y" and "r y z"
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  obtains "r x z"
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  using assms by (rule transE [to_pred])
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lemma transD [dest?]:
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  assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
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  shows "(x, z) \<in> r"
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  using assms by (rule transE)
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lemma transpD [dest?]:
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  assumes "transp r" and "r x y" and "r y z"
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diff changeset
   380
  shows "r x z"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   381
  using assms by (rule transD [to_pred])
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   382
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   383
lemma trans_Int: "trans r \<Longrightarrow> trans s \<Longrightarrow> trans (r \<inter> s)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   384
  by (fast intro: transI elim: transE)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   385
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   386
lemma transp_inf: "transp r \<Longrightarrow> transp s \<Longrightarrow> transp (r \<sqinter> s)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   387
  by (fact trans_Int [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   388
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   389
lemma trans_INTER: "\<forall>x\<in>S. trans (r x) \<Longrightarrow> trans (INTER S r)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   390
  by (fast intro: transI elim: transD)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   391
64584
142ac30b68fe added lemmas demanded by FIXMEs
haftmann
parents: 63612
diff changeset
   392
lemma transp_INF: "\<forall>x\<in>S. transp (r x) \<Longrightarrow> transp (INFIMUM S r)"
142ac30b68fe added lemmas demanded by FIXMEs
haftmann
parents: 63612
diff changeset
   393
  by (fact trans_INTER [to_pred])
142ac30b68fe added lemmas demanded by FIXMEs
haftmann
parents: 63612
diff changeset
   394
    
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   395
lemma trans_join [code]: "trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   396
  by (auto simp add: trans_def)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   397
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   398
lemma transp_trans: "transp r \<longleftrightarrow> trans {(x, y). r x y}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   399
  by (simp add: trans_def transp_def)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   400
59518
28cfc60dea7a add lemma
Andreas Lochbihler
parents: 58889
diff changeset
   401
lemma transp_equality [simp]: "transp op ="
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   402
  by (auto intro: transpI)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   403
63563
0bcd79da075b prefer [simp] over [iff] as [iff] break HOL-UNITY
Andreas Lochbihler
parents: 63561
diff changeset
   404
lemma trans_empty [simp]: "trans {}"
63612
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   405
  by (blast intro: transI)
63561
fba08009ff3e add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents: 63404
diff changeset
   406
63563
0bcd79da075b prefer [simp] over [iff] as [iff] break HOL-UNITY
Andreas Lochbihler
parents: 63561
diff changeset
   407
lemma transp_empty [simp]: "transp (\<lambda>x y. False)"
63612
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   408
  using trans_empty[to_pred] by (simp add: bot_fun_def)
63561
fba08009ff3e add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents: 63404
diff changeset
   409
63563
0bcd79da075b prefer [simp] over [iff] as [iff] break HOL-UNITY
Andreas Lochbihler
parents: 63561
diff changeset
   410
lemma trans_singleton [simp]: "trans {(a, a)}"
63612
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   411
  by (blast intro: transI)
63561
fba08009ff3e add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents: 63404
diff changeset
   412
63563
0bcd79da075b prefer [simp] over [iff] as [iff] break HOL-UNITY
Andreas Lochbihler
parents: 63561
diff changeset
   413
lemma transp_singleton [simp]: "transp (\<lambda>x y. x = a \<and> y = a)"
63612
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   414
  by (simp add: transp_def)
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   415
63376
4c0cc2b356f0 default one-step rules for predicates on relations;
haftmann
parents: 62343
diff changeset
   416
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   417
subsubsection \<open>Totality\<close>
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   418
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   419
definition total_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   420
  where "total_on A r \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"
29859
33bff35f1335 Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents: 29609
diff changeset
   421
63561
fba08009ff3e add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents: 63404
diff changeset
   422
lemma total_onI [intro?]:
fba08009ff3e add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents: 63404
diff changeset
   423
  "(\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y\<rbrakk> \<Longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r) \<Longrightarrow> total_on A r"
63612
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   424
  unfolding total_on_def by blast
63561
fba08009ff3e add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents: 63404
diff changeset
   425
29859
33bff35f1335 Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents: 29609
diff changeset
   426
abbreviation "total \<equiv> total_on UNIV"
33bff35f1335 Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents: 29609
diff changeset
   427
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   428
lemma total_on_empty [simp]: "total_on {} r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   429
  by (simp add: total_on_def)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   430
63563
0bcd79da075b prefer [simp] over [iff] as [iff] break HOL-UNITY
Andreas Lochbihler
parents: 63561
diff changeset
   431
lemma total_on_singleton [simp]: "total_on {x} {(x, x)}"
63612
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   432
  unfolding total_on_def by blast
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   433
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   434
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   435
subsubsection \<open>Single valued relations\<close>
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   436
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   437
definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   438
  where "single_valued r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   439
63376
4c0cc2b356f0 default one-step rules for predicates on relations;
haftmann
parents: 62343
diff changeset
   440
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   441
  where "single_valuedP r \<equiv> single_valued {(x, y). r x y}" (* FIXME proper logical operation *)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   442
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   443
lemma single_valuedI: "\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z) \<Longrightarrow> single_valued r"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   444
  unfolding single_valued_def .
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   445
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   446
lemma single_valuedD: "single_valued r \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (x, z) \<in> r \<Longrightarrow> y = z"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   447
  by (simp add: single_valued_def)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   448
57111
nipkow
parents: 56790
diff changeset
   449
lemma single_valued_empty[simp]: "single_valued {}"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   450
  by (simp add: single_valued_def)
52392
ee996ca08de3 added lemma
nipkow
parents: 50420
diff changeset
   451
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   452
lemma single_valued_subset: "r \<subseteq> s \<Longrightarrow> single_valued s \<Longrightarrow> single_valued r"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   453
  unfolding single_valued_def by blast
11136
e34e7f6d9b57 moved inv_image to Relation
oheimb
parents: 10832
diff changeset
   454
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   455
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   456
subsection \<open>Relation operations\<close>
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   457
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   458
subsubsection \<open>The identity relation\<close>
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   459
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   460
definition Id :: "'a rel"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   461
  where [code del]: "Id = {p. \<exists>x. p = (x, x)}"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   462
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   463
lemma IdI [intro]: "(a, a) \<in> Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   464
  by (simp add: Id_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   465
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   466
lemma IdE [elim!]: "p \<in> Id \<Longrightarrow> (\<And>x. p = (x, x) \<Longrightarrow> P) \<Longrightarrow> P"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   467
  unfolding Id_def by (iprover elim: CollectE)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   468
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   469
lemma pair_in_Id_conv [iff]: "(a, b) \<in> Id \<longleftrightarrow> a = b"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   470
  unfolding Id_def by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   471
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   472
lemma refl_Id: "refl Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   473
  by (simp add: refl_on_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   474
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   475
lemma antisym_Id: "antisym Id"
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61630
diff changeset
   476
  \<comment> \<open>A strange result, since \<open>Id\<close> is also symmetric.\<close>
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   477
  by (simp add: antisym_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   478
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   479
lemma sym_Id: "sym Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   480
  by (simp add: sym_def)
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   481
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   482
lemma trans_Id: "trans Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   483
  by (simp add: trans_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   484
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   485
lemma single_valued_Id [simp]: "single_valued Id"
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   486
  by (unfold single_valued_def) blast
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   487
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   488
lemma irrefl_diff_Id [simp]: "irrefl (r - Id)"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   489
  by (simp add: irrefl_def)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   490
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   491
lemma trans_diff_Id: "trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r - Id)"
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   492
  unfolding antisym_def trans_def by blast
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   493
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   494
lemma total_on_diff_Id [simp]: "total_on A (r - Id) = total_on A r"
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   495
  by (simp add: total_on_def)
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   496
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61955
diff changeset
   497
lemma Id_fstsnd_eq: "Id = {x. fst x = snd x}"
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61955
diff changeset
   498
  by force
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   499
63376
4c0cc2b356f0 default one-step rules for predicates on relations;
haftmann
parents: 62343
diff changeset
   500
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   501
subsubsection \<open>Diagonal: identity over a set\<close>
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   502
63612
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   503
definition Id_on :: "'a set \<Rightarrow> 'a rel"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   504
  where "Id_on A = (\<Union>x\<in>A. {(x, x)})"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   505
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   506
lemma Id_on_empty [simp]: "Id_on {} = {}"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   507
  by (simp add: Id_on_def)
13812
91713a1915ee converting HOL/UNITY to use unconditional fairness
paulson
parents: 13639
diff changeset
   508
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   509
lemma Id_on_eqI: "a = b \<Longrightarrow> a \<in> A \<Longrightarrow> (a, b) \<in> Id_on A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   510
  by (simp add: Id_on_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   511
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   512
lemma Id_onI [intro!]: "a \<in> A \<Longrightarrow> (a, a) \<in> Id_on A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   513
  by (rule Id_on_eqI) (rule refl)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   514
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   515
lemma Id_onE [elim!]: "c \<in> Id_on A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> c = (x, x) \<Longrightarrow> P) \<Longrightarrow> P"
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61630
diff changeset
   516
  \<comment> \<open>The general elimination rule.\<close>
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   517
  unfolding Id_on_def by (iprover elim!: UN_E singletonE)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   518
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   519
lemma Id_on_iff: "(x, y) \<in> Id_on A \<longleftrightarrow> x = y \<and> x \<in> A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   520
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   521
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   522
lemma Id_on_def' [nitpick_unfold]: "Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   523
  by auto
40923
be80c93ac0a2 adding a nice definition of Id_on for quickcheck and nitpick
bulwahn
parents: 36772
diff changeset
   524
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   525
lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   526
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   527
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   528
lemma refl_on_Id_on: "refl_on A (Id_on A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   529
  by (rule refl_onI [OF Id_on_subset_Times Id_onI])
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   530
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   531
lemma antisym_Id_on [simp]: "antisym (Id_on A)"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   532
  unfolding antisym_def by blast
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   533
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   534
lemma sym_Id_on [simp]: "sym (Id_on A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   535
  by (rule symI) clarify
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   536
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   537
lemma trans_Id_on [simp]: "trans (Id_on A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   538
  by (fast intro: transI elim: transD)
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   539
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   540
lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   541
  unfolding single_valued_def by blast
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   542
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   543
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   544
subsubsection \<open>Composition\<close>
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   545
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   546
inductive_set relcomp  :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a \<times> 'c) set"  (infixr "O" 75)
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   547
  for r :: "('a \<times> 'b) set" and s :: "('b \<times> 'c) set"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   548
  where relcompI [intro]: "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> s \<Longrightarrow> (a, c) \<in> r O s"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   549
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   550
notation relcompp (infixr "OO" 75)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   551
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   552
lemmas relcomppI = relcompp.intros
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   553
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   554
text \<open>
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   555
  For historic reasons, the elimination rules are not wholly corresponding.
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   556
  Feel free to consolidate this.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   557
\<close>
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   558
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   559
inductive_cases relcompEpair: "(a, c) \<in> r O s"
47434
b75ce48a93ee dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents: 47433
diff changeset
   560
inductive_cases relcomppE [elim!]: "(r OO s) a c"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   561
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   562
lemma relcompE [elim!]: "xz \<in> r O s \<Longrightarrow>
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   563
  (\<And>x y z. xz = (x, z) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (y, z) \<in> s  \<Longrightarrow> P) \<Longrightarrow> P"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   564
  apply (cases xz)
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   565
  apply simp
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   566
  apply (erule relcompEpair)
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   567
  apply iprover
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   568
  done
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   569
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   570
lemma R_O_Id [simp]: "R O Id = R"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   571
  by fast
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   572
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   573
lemma Id_O_R [simp]: "Id O R = R"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   574
  by fast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   575
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   576
lemma relcomp_empty1 [simp]: "{} O R = {}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   577
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   578
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   579
lemma relcompp_bot1 [simp]: "\<bottom> OO R = \<bottom>"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   580
  by (fact relcomp_empty1 [to_pred])
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   581
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   582
lemma relcomp_empty2 [simp]: "R O {} = {}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   583
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   584
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   585
lemma relcompp_bot2 [simp]: "R OO \<bottom> = \<bottom>"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   586
  by (fact relcomp_empty2 [to_pred])
23185
1fa87978cf27 Added simp-rules: "R O {} = {}" and "{} O R = {}"
krauss
parents: 22172
diff changeset
   587
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   588
lemma O_assoc: "(R O S) O T = R O (S O T)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   589
  by blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   590
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   591
lemma relcompp_assoc: "(r OO s) OO t = r OO (s OO t)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   592
  by (fact O_assoc [to_pred])
23185
1fa87978cf27 Added simp-rules: "R O {} = {}" and "{} O R = {}"
krauss
parents: 22172
diff changeset
   593
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   594
lemma trans_O_subset: "trans r \<Longrightarrow> r O r \<subseteq> r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   595
  by (unfold trans_def) blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   596
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   597
lemma transp_relcompp_less_eq: "transp r \<Longrightarrow> r OO r \<le> r "
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   598
  by (fact trans_O_subset [to_pred])
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   599
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   600
lemma relcomp_mono: "r' \<subseteq> r \<Longrightarrow> s' \<subseteq> s \<Longrightarrow> r' O s' \<subseteq> r O s"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   601
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   602
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   603
lemma relcompp_mono: "r' \<le> r \<Longrightarrow> s' \<le> s \<Longrightarrow> r' OO s' \<le> r OO s "
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   604
  by (fact relcomp_mono [to_pred])
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   605
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   606
lemma relcomp_subset_Sigma: "r \<subseteq> A \<times> B \<Longrightarrow> s \<subseteq> B \<times> C \<Longrightarrow> r O s \<subseteq> A \<times> C"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   607
  by blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   608
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   609
lemma relcomp_distrib [simp]: "R O (S \<union> T) = (R O S) \<union> (R O T)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   610
  by auto
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   611
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   612
lemma relcompp_distrib [simp]: "R OO (S \<squnion> T) = R OO S \<squnion> R OO T"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   613
  by (fact relcomp_distrib [to_pred])
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   614
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   615
lemma relcomp_distrib2 [simp]: "(S \<union> T) O R = (S O R) \<union> (T O R)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   616
  by auto
28008
f945f8d9ad4d added distributivity of relation composition over union [simp]
krauss
parents: 26297
diff changeset
   617
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   618
lemma relcompp_distrib2 [simp]: "(S \<squnion> T) OO R = S OO R \<squnion> T OO R"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47087
diff changeset
   619
  by (fact relcomp_distrib2 [to_pred])
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   620
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   621
lemma relcomp_UNION_distrib: "s O UNION I r = (\<Union>i\<in>I. s O r i) "
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   622
  by auto
28008
f945f8d9ad4d added distributivity of relation composition over union [simp]
krauss
parents: 26297
diff changeset
   623
64584
142ac30b68fe added lemmas demanded by FIXMEs
haftmann
parents: 63612
diff changeset
   624
lemma relcompp_SUP_distrib: "s OO SUPREMUM I r = (\<Squnion>i\<in>I. s OO r i)"
142ac30b68fe added lemmas demanded by FIXMEs
haftmann
parents: 63612
diff changeset
   625
  by (fact relcomp_UNION_distrib [to_pred])
142ac30b68fe added lemmas demanded by FIXMEs
haftmann
parents: 63612
diff changeset
   626
    
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   627
lemma relcomp_UNION_distrib2: "UNION I r O s = (\<Union>i\<in>I. r i O s) "
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   628
  by auto
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   629
64584
142ac30b68fe added lemmas demanded by FIXMEs
haftmann
parents: 63612
diff changeset
   630
lemma relcompp_SUP_distrib2: "SUPREMUM I r OO s = (\<Squnion>i\<in>I. r i OO s)"
142ac30b68fe added lemmas demanded by FIXMEs
haftmann
parents: 63612
diff changeset
   631
  by (fact relcomp_UNION_distrib2 [to_pred])
142ac30b68fe added lemmas demanded by FIXMEs
haftmann
parents: 63612
diff changeset
   632
    
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   633
lemma single_valued_relcomp: "single_valued r \<Longrightarrow> single_valued s \<Longrightarrow> single_valued (r O s)"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   634
  unfolding single_valued_def by blast
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   635
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   636
lemma relcomp_unfold: "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   637
  by (auto simp add: set_eq_iff)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   638
58195
1fee63e0377d added various facts
haftmann
parents: 57111
diff changeset
   639
lemma relcompp_apply: "(R OO S) a c \<longleftrightarrow> (\<exists>b. R a b \<and> S b c)"
1fee63e0377d added various facts
haftmann
parents: 57111
diff changeset
   640
  unfolding relcomp_unfold [to_pred] ..
1fee63e0377d added various facts
haftmann
parents: 57111
diff changeset
   641
63612
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   642
lemma eq_OO: "op = OO R = R"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   643
  by blast
55083
0a689157e3ce move BNF_LFP up the dependency chain
blanchet
parents: 54611
diff changeset
   644
61630
608520e0e8e2 add various lemmas
Andreas Lochbihler
parents: 61424
diff changeset
   645
lemma OO_eq: "R OO op = = R"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   646
  by blast
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
   647
63376
4c0cc2b356f0 default one-step rules for predicates on relations;
haftmann
parents: 62343
diff changeset
   648
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   649
subsubsection \<open>Converse\<close>
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   650
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61799
diff changeset
   651
inductive_set converse :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'a) set"  ("(_\<inverse>)" [1000] 999)
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   652
  for r :: "('a \<times> 'b) set"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   653
  where "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   654
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   655
notation conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   656
61955
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61799
diff changeset
   657
notation (ASCII)
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61799
diff changeset
   658
  converse  ("(_^-1)" [1000] 999) and
e96292f32c3c former "xsymbols" syntax is used by default, and ASCII replacement syntax with print mode "ASCII";
wenzelm
parents: 61799
diff changeset
   659
  conversep ("(_^--1)" [1000] 1000)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   660
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   661
lemma converseI [sym]: "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   662
  by (fact converse.intros)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   663
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   664
lemma conversepI (* CANDIDATE [sym] *): "r a b \<Longrightarrow> r\<inverse>\<inverse> b a"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   665
  by (fact conversep.intros)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   666
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   667
lemma converseD [sym]: "(a, b) \<in> r\<inverse> \<Longrightarrow> (b, a) \<in> r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   668
  by (erule converse.cases) iprover
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   669
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   670
lemma conversepD (* CANDIDATE [sym] *): "r\<inverse>\<inverse> b a \<Longrightarrow> r a b"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   671
  by (fact converseD [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   672
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   673
lemma converseE [elim!]: "yx \<in> r\<inverse> \<Longrightarrow> (\<And>x y. yx = (y, x) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> P) \<Longrightarrow> P"
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61630
diff changeset
   674
  \<comment> \<open>More general than \<open>converseD\<close>, as it ``splits'' the member of the relation.\<close>
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   675
  apply (cases yx)
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   676
  apply simp
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   677
  apply (erule converse.cases)
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   678
  apply iprover
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   679
  done
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   680
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46833
diff changeset
   681
lemmas conversepE [elim!] = conversep.cases
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   682
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   683
lemma converse_iff [iff]: "(a, b) \<in> r\<inverse> \<longleftrightarrow> (b, a) \<in> r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   684
  by (auto intro: converseI)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   685
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   686
lemma conversep_iff [iff]: "r\<inverse>\<inverse> a b = r b a"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   687
  by (fact converse_iff [to_pred])
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   688
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   689
lemma converse_converse [simp]: "(r\<inverse>)\<inverse> = r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   690
  by (simp add: set_eq_iff)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   691
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   692
lemma conversep_conversep [simp]: "(r\<inverse>\<inverse>)\<inverse>\<inverse> = r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   693
  by (fact converse_converse [to_pred])
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   694
53680
c5096c22892b added lemmas and made concerse executable
nipkow
parents: 52749
diff changeset
   695
lemma converse_empty[simp]: "{}\<inverse> = {}"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   696
  by auto
53680
c5096c22892b added lemmas and made concerse executable
nipkow
parents: 52749
diff changeset
   697
c5096c22892b added lemmas and made concerse executable
nipkow
parents: 52749
diff changeset
   698
lemma converse_UNIV[simp]: "UNIV\<inverse> = UNIV"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   699
  by auto
53680
c5096c22892b added lemmas and made concerse executable
nipkow
parents: 52749
diff changeset
   700
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   701
lemma converse_relcomp: "(r O s)\<inverse> = s\<inverse> O r\<inverse>"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   702
  by blast
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   703
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   704
lemma converse_relcompp: "(r OO s)\<inverse>\<inverse> = s\<inverse>\<inverse> OO r\<inverse>\<inverse>"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   705
  by (iprover intro: order_antisym conversepI relcomppI elim: relcomppE dest: conversepD)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   706
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   707
lemma converse_Int: "(r \<inter> s)\<inverse> = r\<inverse> \<inter> s\<inverse>"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   708
  by blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   709
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   710
lemma converse_meet: "(r \<sqinter> s)\<inverse>\<inverse> = r\<inverse>\<inverse> \<sqinter> s\<inverse>\<inverse>"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   711
  by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   712
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   713
lemma converse_Un: "(r \<union> s)\<inverse> = r\<inverse> \<union> s\<inverse>"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   714
  by blast
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   715
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   716
lemma converse_join: "(r \<squnion> s)\<inverse>\<inverse> = r\<inverse>\<inverse> \<squnion> s\<inverse>\<inverse>"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   717
  by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   718
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   719
lemma converse_INTER: "(INTER S r)\<inverse> = (INT x:S. (r x)\<inverse>)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   720
  by fast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   721
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   722
lemma converse_UNION: "(UNION S r)\<inverse> = (UN x:S. (r x)\<inverse>)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   723
  by blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   724
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   725
lemma converse_mono[simp]: "r\<inverse> \<subseteq> s \<inverse> \<longleftrightarrow> r \<subseteq> s"
52749
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   726
  by auto
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   727
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   728
lemma conversep_mono[simp]: "r\<inverse>\<inverse> \<le> s \<inverse>\<inverse> \<longleftrightarrow> r \<le> s"
52749
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   729
  by (fact converse_mono[to_pred])
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   730
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   731
lemma converse_inject[simp]: "r\<inverse> = s \<inverse> \<longleftrightarrow> r = s"
52730
6bf02eb4ddf7 two useful relation theorems
traytel
parents: 52392
diff changeset
   732
  by auto
6bf02eb4ddf7 two useful relation theorems
traytel
parents: 52392
diff changeset
   733
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   734
lemma conversep_inject[simp]: "r\<inverse>\<inverse> = s \<inverse>\<inverse> \<longleftrightarrow> r = s"
52749
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   735
  by (fact converse_inject[to_pred])
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   736
63612
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   737
lemma converse_subset_swap: "r \<subseteq> s \<inverse> \<longleftrightarrow> r \<inverse> \<subseteq> s"
52749
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   738
  by auto
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   739
63612
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   740
lemma conversep_le_swap: "r \<le> s \<inverse>\<inverse> \<longleftrightarrow> r \<inverse>\<inverse> \<le> s"
52749
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
   741
  by (fact converse_subset_swap[to_pred])
52730
6bf02eb4ddf7 two useful relation theorems
traytel
parents: 52392
diff changeset
   742
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   743
lemma converse_Id [simp]: "Id\<inverse> = Id"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   744
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   745
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   746
lemma converse_Id_on [simp]: "(Id_on A)\<inverse> = Id_on A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   747
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   748
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   749
lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   750
  by (auto simp: refl_on_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   751
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   752
lemma sym_converse [simp]: "sym (converse r) = sym r"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   753
  unfolding sym_def by blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   754
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   755
lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   756
  unfolding antisym_def by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   757
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   758
lemma trans_converse [simp]: "trans (converse r) = trans r"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   759
  unfolding trans_def by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   760
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   761
lemma sym_conv_converse_eq: "sym r \<longleftrightarrow> r\<inverse> = r"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   762
  unfolding sym_def by fast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   763
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   764
lemma sym_Un_converse: "sym (r \<union> r\<inverse>)"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   765
  unfolding sym_def by blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   766
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   767
lemma sym_Int_converse: "sym (r \<inter> r\<inverse>)"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   768
  unfolding sym_def by blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
   769
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   770
lemma total_on_converse [simp]: "total_on A (r\<inverse>) = total_on A r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   771
  by (auto simp: total_on_def)
29859
33bff35f1335 Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents: 29609
diff changeset
   772
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   773
lemma finite_converse [iff]: "finite (r\<inverse>) = finite r"
54611
31afce809794 set_comprehension_pointfree simproc causes to many surprises if enabled by default
traytel
parents: 54555
diff changeset
   774
  unfolding converse_def conversep_iff using [[simproc add: finite_Collect]]
31afce809794 set_comprehension_pointfree simproc causes to many surprises if enabled by default
traytel
parents: 54555
diff changeset
   775
  by (auto elim: finite_imageD simp: inj_on_def)
12913
5ac498bffb6b fixed document;
wenzelm
parents: 12905
diff changeset
   776
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   777
lemma conversep_noteq [simp]: "(op \<noteq>)\<inverse>\<inverse> = op \<noteq>"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   778
  by (auto simp add: fun_eq_iff)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   779
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   780
lemma conversep_eq [simp]: "(op =)\<inverse>\<inverse> = op ="
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   781
  by (auto simp add: fun_eq_iff)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   782
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   783
lemma converse_unfold [code]: "r\<inverse> = {(y, x). (x, y) \<in> r}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   784
  by (simp add: set_eq_iff)
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   785
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   786
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   787
subsubsection \<open>Domain, range and field\<close>
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   788
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   789
inductive_set Domain :: "('a \<times> 'b) set \<Rightarrow> 'a set" for r :: "('a \<times> 'b) set"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   790
  where DomainI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> Domain r"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   791
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   792
lemmas DomainPI = Domainp.DomainI
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   793
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   794
inductive_cases DomainE [elim!]: "a \<in> Domain r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   795
inductive_cases DomainpE [elim!]: "Domainp r a"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   796
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   797
inductive_set Range :: "('a \<times> 'b) set \<Rightarrow> 'b set" for r :: "('a \<times> 'b) set"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   798
  where RangeI [intro]: "(a, b) \<in> r \<Longrightarrow> b \<in> Range r"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   799
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   800
lemmas RangePI = Rangep.RangeI
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   801
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   802
inductive_cases RangeE [elim!]: "b \<in> Range r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   803
inductive_cases RangepE [elim!]: "Rangep r b"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   804
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   805
definition Field :: "'a rel \<Rightarrow> 'a set"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   806
  where "Field r = Domain r \<union> Range r"
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   807
63561
fba08009ff3e add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents: 63404
diff changeset
   808
lemma FieldI1: "(i, j) \<in> R \<Longrightarrow> i \<in> Field R"
63612
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   809
  unfolding Field_def by blast
63561
fba08009ff3e add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents: 63404
diff changeset
   810
fba08009ff3e add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents: 63404
diff changeset
   811
lemma FieldI2: "(i, j) \<in> R \<Longrightarrow> j \<in> Field R"
fba08009ff3e add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents: 63404
diff changeset
   812
  unfolding Field_def by auto
fba08009ff3e add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents: 63404
diff changeset
   813
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   814
lemma Domain_fst [code]: "Domain r = fst ` r"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   815
  by force
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   816
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   817
lemma Range_snd [code]: "Range r = snd ` r"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   818
  by force
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   819
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   820
lemma fst_eq_Domain: "fst ` R = Domain R"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   821
  by force
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   822
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   823
lemma snd_eq_Range: "snd ` R = Range R"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   824
  by force
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   825
62087
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61955
diff changeset
   826
lemma range_fst [simp]: "range fst = UNIV"
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61955
diff changeset
   827
  by (auto simp: fst_eq_Domain)
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61955
diff changeset
   828
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61955
diff changeset
   829
lemma range_snd [simp]: "range snd = UNIV"
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61955
diff changeset
   830
  by (auto simp: snd_eq_Range)
44841d07ef1d revisions to limits and derivatives, plus new lemmas
paulson
parents: 61955
diff changeset
   831
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   832
lemma Domain_empty [simp]: "Domain {} = {}"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   833
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   834
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   835
lemma Range_empty [simp]: "Range {} = {}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   836
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   837
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   838
lemma Field_empty [simp]: "Field {} = {}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   839
  by (simp add: Field_def)
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   840
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   841
lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   842
  by auto
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   843
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   844
lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   845
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   846
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46833
diff changeset
   847
lemma Domain_insert [simp]: "Domain (insert (a, b) r) = insert a (Domain r)"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   848
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   849
46882
6242b4bc05bc tuned simpset
noschinl
parents: 46833
diff changeset
   850
lemma Range_insert [simp]: "Range (insert (a, b) r) = insert b (Range r)"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   851
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   852
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   853
lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \<union> Field r"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   854
  by (auto simp add: Field_def)
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   855
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   856
lemma Domain_iff: "a \<in> Domain r \<longleftrightarrow> (\<exists>y. (a, y) \<in> r)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   857
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   858
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   859
lemma Range_iff: "a \<in> Range r \<longleftrightarrow> (\<exists>y. (y, a) \<in> r)"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   860
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   861
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   862
lemma Domain_Id [simp]: "Domain Id = UNIV"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   863
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   864
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   865
lemma Range_Id [simp]: "Range Id = UNIV"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   866
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   867
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   868
lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   869
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   870
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   871
lemma Range_Id_on [simp]: "Range (Id_on A) = A"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   872
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   873
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   874
lemma Domain_Un_eq: "Domain (A \<union> B) = Domain A \<union> Domain B"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   875
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   876
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   877
lemma Range_Un_eq: "Range (A \<union> B) = Range A \<union> Range B"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   878
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   879
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   880
lemma Field_Un [simp]: "Field (r \<union> s) = Field r \<union> Field s"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   881
  by (auto simp: Field_def)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   882
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   883
lemma Domain_Int_subset: "Domain (A \<inter> B) \<subseteq> Domain A \<inter> Domain B"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   884
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   885
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   886
lemma Range_Int_subset: "Range (A \<inter> B) \<subseteq> Range A \<inter> Range B"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   887
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   888
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   889
lemma Domain_Diff_subset: "Domain A - Domain B \<subseteq> Domain (A - B)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   890
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   891
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   892
lemma Range_Diff_subset: "Range A - Range B \<subseteq> Range (A - B)"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   893
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   894
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   895
lemma Domain_Union: "Domain (\<Union>S) = (\<Union>A\<in>S. Domain A)"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   896
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   897
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   898
lemma Range_Union: "Range (\<Union>S) = (\<Union>A\<in>S. Range A)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   899
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   900
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   901
lemma Field_Union [simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   902
  by (auto simp: Field_def)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   903
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   904
lemma Domain_converse [simp]: "Domain (r\<inverse>) = Range r"
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   905
  by auto
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   906
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   907
lemma Range_converse [simp]: "Range (r\<inverse>) = Domain r"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   908
  by blast
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   909
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   910
lemma Field_converse [simp]: "Field (r\<inverse>) = Field r"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   911
  by (auto simp: Field_def)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   912
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   913
lemma Domain_Collect_case_prod [simp]: "Domain {(x, y). P x y} = {x. \<exists>y. P x y}"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   914
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   915
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   916
lemma Range_Collect_case_prod [simp]: "Range {(x, y). P x y} = {y. \<exists>x. P x y}"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   917
  by auto
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   918
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   919
lemma finite_Domain: "finite r \<Longrightarrow> finite (Domain r)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   920
  by (induct set: finite) auto
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   921
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   922
lemma finite_Range: "finite r \<Longrightarrow> finite (Range r)"
46884
154dc6ec0041 tuned proofs
noschinl
parents: 46883
diff changeset
   923
  by (induct set: finite) auto
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   924
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   925
lemma finite_Field: "finite r \<Longrightarrow> finite (Field r)"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   926
  by (simp add: Field_def finite_Domain finite_Range)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   927
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   928
lemma Domain_mono: "r \<subseteq> s \<Longrightarrow> Domain r \<subseteq> Domain s"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   929
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   930
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   931
lemma Range_mono: "r \<subseteq> s \<Longrightarrow> Range r \<subseteq> Range s"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   932
  by blast
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   933
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   934
lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   935
  by (auto simp: Field_def Domain_def Range_def)
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   936
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   937
lemma Domain_unfold: "Domain r = {x. \<exists>y. (x, y) \<in> r}"
46767
807a5d219c23 more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents: 46752
diff changeset
   938
  by blast
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
   939
63563
0bcd79da075b prefer [simp] over [iff] as [iff] break HOL-UNITY
Andreas Lochbihler
parents: 63561
diff changeset
   940
lemma Field_square [simp]: "Field (x \<times> x) = x"
63612
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   941
  unfolding Field_def by blast
63561
fba08009ff3e add lemmas contributed by Peter Gammie
Andreas Lochbihler
parents: 63404
diff changeset
   942
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   943
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
   944
subsubsection \<open>Image of a set under a relation\<close>
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   945
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   946
definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set"  (infixr "``" 90)
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   947
  where "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
   948
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   949
lemma Image_iff: "b \<in> r``A \<longleftrightarrow> (\<exists>x\<in>A. (x, b) \<in> r)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   950
  by (simp add: Image_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   951
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   952
lemma Image_singleton: "r``{a} = {b. (a, b) \<in> r}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   953
  by (simp add: Image_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   954
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   955
lemma Image_singleton_iff [iff]: "b \<in> r``{a} \<longleftrightarrow> (a, b) \<in> r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   956
  by (rule Image_iff [THEN trans]) simp
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   957
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   958
lemma ImageI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> r``A"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   959
  unfolding Image_def by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   960
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   961
lemma ImageE [elim!]: "b \<in> r `` A \<Longrightarrow> (\<And>x. (x, b) \<in> r \<Longrightarrow> x \<in> A \<Longrightarrow> P) \<Longrightarrow> P"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   962
  unfolding Image_def by (iprover elim!: CollectE bexE)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   963
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   964
lemma rev_ImageI: "a \<in> A \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> b \<in> r `` A"
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61630
diff changeset
   965
  \<comment> \<open>This version's more effective when we already have the required \<open>a\<close>\<close>
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   966
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   967
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   968
lemma Image_empty [simp]: "R``{} = {}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   969
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   970
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   971
lemma Image_Id [simp]: "Id `` A = A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   972
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   973
30198
922f944f03b2 name changes
nipkow
parents: 29859
diff changeset
   974
lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   975
  by blast
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   976
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   977
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   978
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   979
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   980
lemma Image_Int_eq: "single_valued (converse R) \<Longrightarrow> R `` (A \<inter> B) = R `` A \<inter> R `` B"
63612
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
   981
  by (auto simp: single_valued_def)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   982
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   983
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   984
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   985
13812
91713a1915ee converting HOL/UNITY to use unconditional fairness
paulson
parents: 13639
diff changeset
   986
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   987
  by blast
13812
91713a1915ee converting HOL/UNITY to use unconditional fairness
paulson
parents: 13639
diff changeset
   988
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   989
lemma Image_subset: "r \<subseteq> A \<times> B \<Longrightarrow> r``C \<subseteq> B"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   990
  by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   991
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   992
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61630
diff changeset
   993
  \<comment> \<open>NOT suitable for rewriting\<close>
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   994
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   995
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
   996
lemma Image_mono: "r' \<subseteq> r \<Longrightarrow> A' \<subseteq> A \<Longrightarrow> (r' `` A') \<subseteq> (r `` A)"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
   997
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
   998
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
   999
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1000
  by blast
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
  1001
54410
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1002
lemma UN_Image: "(\<Union>i\<in>I. X i) `` S = (\<Union>i\<in>I. X i `` S)"
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1003
  by auto
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1004
13830
7f8c1b533e8b some x-symbols and some new lemmas
paulson
parents: 13812
diff changeset
  1005
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1006
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1007
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1008
text \<open>Converse inclusion requires some assumptions\<close>
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1009
lemma Image_INT_eq: "single_valued (r\<inverse>) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1010
  apply (rule equalityI)
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1011
   apply (rule Image_INT_subset)
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1012
  apply (auto simp add: single_valued_def)
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1013
  apply blast
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1014
  done
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1015
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1016
lemma Image_subset_eq: "r``A \<subseteq> B \<longleftrightarrow> A \<subseteq> - ((r\<inverse>) `` (- B))"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1017
  by blast
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1018
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1019
lemma Image_Collect_case_prod [simp]: "{(x, y). P x y} `` A = {y. \<exists>x\<in>A. P x y}"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1020
  by auto
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1021
54410
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1022
lemma Sigma_Image: "(SIGMA x:A. B x) `` X = (\<Union>x\<in>X \<inter> A. B x)"
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1023
  by auto
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1024
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1025
lemma relcomp_Image: "(X O Y) `` Z = Y `` (X `` Z)"
0a578fb7fb73 countability of the image of a reflexive transitive closure
hoelzl
parents: 54147
diff changeset
  1026
  by auto
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1027
63376
4c0cc2b356f0 default one-step rules for predicates on relations;
haftmann
parents: 62343
diff changeset
  1028
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
  1029
subsubsection \<open>Inverse image\<close>
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1030
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1031
definition inv_image :: "'b rel \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a rel"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1032
  where "inv_image r f = {(x, y). (f x, f y) \<in> r}"
46692
1f8b766224f6 tuned structure; dropped already existing syntax declarations
haftmann
parents: 46691
diff changeset
  1033
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1034
definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1035
  where "inv_imagep r f = (\<lambda>x y. r (f x) (f y))"
46694
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
  1036
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
  1037
lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)"
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
  1038
  by (simp add: inv_image_def inv_imagep_def)
0988b22e2626 tuned structure
haftmann
parents: 46692
diff changeset
  1039
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1040
lemma sym_inv_image: "sym r \<Longrightarrow> sym (inv_image r f)"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1041
  unfolding sym_def inv_image_def by blast
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 17589
diff changeset
  1042
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1043
lemma trans_inv_image: "trans r \<Longrightarrow> trans (inv_image r f)"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1044
  unfolding trans_def inv_image_def
12905
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1045
  apply (simp (no_asm))
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1046
  apply blast
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1047
  done
bbbae3f359e6 Converted to new theory format.
berghofe
parents: 12487
diff changeset
  1048
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1049
lemma in_inv_image[simp]: "(x, y) \<in> inv_image r f \<longleftrightarrow> (f x, f y) \<in> r"
32463
3a0a65ca2261 moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents: 32235
diff changeset
  1050
  by (auto simp:inv_image_def)
3a0a65ca2261 moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents: 32235
diff changeset
  1051
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1052
lemma converse_inv_image[simp]: "(inv_image R f)\<inverse> = inv_image (R\<inverse>) f"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1053
  unfolding inv_image_def converse_unfold by auto
33218
ecb5cd453ef2 lemma converse_inv_image
krauss
parents: 32876
diff changeset
  1054
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1055
lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1056
  by (simp add: inv_imagep_def)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1057
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1058
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
  1059
subsubsection \<open>Powerset\<close>
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1060
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46696
diff changeset
  1061
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1062
  where "Powp A = (\<lambda>B. \<forall>x \<in> B. A x)"
46664
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1063
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1064
lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1065
  by (auto simp add: Powp_def fun_eq_iff)
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1066
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1067
lemmas Powp_mono [mono] = Pow_mono [to_pred]
1f6c140f9c72 moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents: 46638
diff changeset
  1068
63376
4c0cc2b356f0 default one-step rules for predicates on relations;
haftmann
parents: 62343
diff changeset
  1069
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60057
diff changeset
  1070
subsubsection \<open>Expressing relation operations via @{const Finite_Set.fold}\<close>
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1071
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1072
lemma Id_on_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1073
  assumes "finite A"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1074
  shows "Id_on A = Finite_Set.fold (\<lambda>x. Set.insert (Pair x x)) {} A"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1075
proof -
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1076
  interpret comp_fun_commute "\<lambda>x. Set.insert (Pair x x)"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1077
    by standard auto
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1078
  from assms show ?thesis
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1079
    unfolding Id_on_def by (induct A) simp_all
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1080
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1081
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1082
lemma comp_fun_commute_Image_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1083
  "comp_fun_commute (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1084
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1085
  interpret comp_fun_idem Set.insert
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1086
    by (fact comp_fun_idem_insert)
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1087
  show ?thesis
63612
7195acc2fe93 misc tuning and modernization;
wenzelm
parents: 63563
diff changeset
  1088
    by standard (auto simp: fun_eq_iff comp_fun_commute split: prod.split)
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1089
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1090
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1091
lemma Image_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1092
  assumes "finite R"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1093
  shows "R `` S = Finite_Set.fold (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A) {} R"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1094
proof -
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1095
  interpret comp_fun_commute "(\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)"
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1096
    by (rule comp_fun_commute_Image_fold)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1097
  have *: "\<And>x F. Set.insert x F `` S = (if fst x \<in> S then Set.insert (snd x) (F `` S) else (F `` S))"
52749
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
  1098
    by (force intro: rev_ImageI)
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1099
  show ?thesis
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1100
    using assms by (induct R) (auto simp: *)
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1101
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1102
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1103
lemma insert_relcomp_union_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1104
  assumes "finite S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1105
  shows "{x} O S \<union> X = Finite_Set.fold (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') X S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1106
proof -
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1107
  interpret comp_fun_commute "\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A'"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1108
  proof -
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1109
    interpret comp_fun_idem Set.insert
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1110
      by (fact comp_fun_idem_insert)
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1111
    show "comp_fun_commute (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A')"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1112
      by standard (auto simp add: fun_eq_iff split: prod.split)
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1113
  qed
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1114
  have *: "{x} O S = {(x', z). x' = fst x \<and> (snd x, z) \<in> S}"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1115
    by (auto simp: relcomp_unfold intro!: exI)
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1116
  show ?thesis
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1117
    unfolding * using \<open>finite S\<close> by (induct S) (auto split: prod.split)
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1118
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1119
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1120
lemma insert_relcomp_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1121
  assumes "finite S"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1122
  shows "Set.insert x R O S =
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1123
    Finite_Set.fold (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') (R O S) S"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1124
proof -
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1125
  have "Set.insert x R O S = ({x} O S) \<union> (R O S)"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1126
    by auto
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1127
  then show ?thesis
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1128
    by (auto simp: insert_relcomp_union_fold [OF assms])
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1129
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1130
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1131
lemma comp_fun_commute_relcomp_fold:
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1132
  assumes "finite S"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1133
  shows "comp_fun_commute (\<lambda>(x,y) A.
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1134
    Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S)"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1135
proof -
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1136
  have *: "\<And>a b A.
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1137
    Finite_Set.fold (\<lambda>(w, z) A'. if b = w then Set.insert (a, z) A' else A') A S = {(a,b)} O S \<union> A"
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1138
    by (auto simp: insert_relcomp_union_fold[OF assms] cong: if_cong)
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1139
  show ?thesis
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1140
    by standard (auto simp: *)
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1141
qed
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1142
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1143
lemma relcomp_fold:
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1144
  assumes "finite R" "finite S"
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1145
  shows "R O S = Finite_Set.fold
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1146
    (\<lambda>(x,y) A. Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S) {} R"
63404
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1147
  using assms
a95e7432d86c misc tuning and modernization;
wenzelm
parents: 63376
diff changeset
  1148
  by (induct R)
52749
ed416f4ac34e more converse(p) theorems; tuned proofs;
traytel
parents: 52730
diff changeset
  1149
    (auto simp: comp_fun_commute.fold_insert comp_fun_commute_relcomp_fold insert_relcomp_fold
48620
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1150
      cong: if_cong)
fc9be489e2fb more relation operations expressed by Finite_Set.fold
kuncar
parents: 48253
diff changeset
  1151
1128
64b30e3cc6d4 Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff changeset
  1152
end