src/HOL/Relation.thy
author haftmann
Wed Dec 21 21:26:26 2016 +0100 (2016-12-21)
changeset 64633 5ebcf6c525f1
parent 64632 9df24b8b6c0a
child 64634 5bd30359e46e
permissions -rw-r--r--
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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   278
haftmann@46694
   279
lemma sympE:
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   280
  assumes "symp r" and "r b a"
haftmann@46752
   281
  obtains "r a b"
haftmann@46752
   282
  using assms by (rule symE [to_pred])
haftmann@46752
   283
haftmann@63376
   284
lemma symD [dest?]:
haftmann@46752
   285
  assumes "sym r" and "(b, a) \<in> r"
haftmann@46752
   286
  shows "(a, b) \<in> r"
haftmann@46752
   287
  using assms by (rule symE)
haftmann@46694
   288
haftmann@63376
   289
lemma sympD [dest?]:
haftmann@46752
   290
  assumes "symp r" and "r b a"
haftmann@46752
   291
  shows "r a b"
haftmann@46752
   292
  using assms by (rule symD [to_pred])
haftmann@46752
   293
wenzelm@63404
   294
lemma sym_Int: "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<inter> s)"
haftmann@46752
   295
  by (fast intro: symI elim: symE)
haftmann@46692
   296
wenzelm@63404
   297
lemma symp_inf: "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<sqinter> s)"
haftmann@46752
   298
  by (fact sym_Int [to_pred])
haftmann@46752
   299
wenzelm@63404
   300
lemma sym_Un: "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<union> s)"
haftmann@46752
   301
  by (fast intro: symI elim: symE)
haftmann@46752
   302
wenzelm@63404
   303
lemma symp_sup: "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<squnion> s)"
haftmann@46752
   304
  by (fact sym_Un [to_pred])
haftmann@46692
   305
wenzelm@63404
   306
lemma sym_INTER: "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (INTER S r)"
haftmann@46752
   307
  by (fast intro: symI elim: symE)
haftmann@46752
   308
wenzelm@63404
   309
lemma symp_INF: "\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (INFIMUM S r)"
haftmann@46982
   310
  by (fact sym_INTER [to_pred])
haftmann@46692
   311
wenzelm@63404
   312
lemma sym_UNION: "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (UNION S r)"
haftmann@46752
   313
  by (fast intro: symI elim: symE)
haftmann@46752
   314
wenzelm@63404
   315
lemma symp_SUP: "\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (SUPREMUM S r)"
haftmann@46982
   316
  by (fact sym_UNION [to_pred])
haftmann@46692
   317
haftmann@46692
   318
wenzelm@60758
   319
subsubsection \<open>Antisymmetry\<close>
haftmann@46694
   320
haftmann@46752
   321
definition antisym :: "'a rel \<Rightarrow> bool"
wenzelm@63404
   322
  where "antisym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
haftmann@46752
   323
haftmann@46752
   324
abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
wenzelm@63404
   325
  where "antisymP r \<equiv> antisym {(x, y). r x y}" (* FIXME proper logical operation *)
wenzelm@63404
   326
wenzelm@63404
   327
lemma antisymI [intro?]: "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (y, x) \<in> r \<Longrightarrow> x = y) \<Longrightarrow> antisym r"
wenzelm@63404
   328
  unfolding antisym_def by iprover
haftmann@46694
   329
wenzelm@63404
   330
lemma antisymD [dest?]: "antisym r \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r \<Longrightarrow> a = b"
wenzelm@63404
   331
  unfolding antisym_def by iprover
haftmann@46694
   332
wenzelm@63404
   333
lemma antisym_subset: "r \<subseteq> s \<Longrightarrow> antisym s \<Longrightarrow> antisym r"
wenzelm@63404
   334
  unfolding antisym_def by blast
haftmann@46694
   335
haftmann@46694
   336
lemma antisym_empty [simp]: "antisym {}"
wenzelm@63404
   337
  unfolding antisym_def by blast
haftmann@46694
   338
Andreas@60057
   339
lemma antisymP_equality [simp]: "antisymP op ="
wenzelm@63404
   340
  by (auto intro: antisymI)
haftmann@46694
   341
Andreas@63563
   342
lemma antisym_singleton [simp]: "antisym {x}"
Andreas@63561
   343
by (blast intro: antisymI)
Andreas@63561
   344
haftmann@63376
   345
wenzelm@60758
   346
subsubsection \<open>Transitivity\<close>
haftmann@46692
   347
haftmann@46752
   348
definition trans :: "'a rel \<Rightarrow> bool"
wenzelm@63404
   349
  where "trans r \<longleftrightarrow> (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
haftmann@46752
   350
haftmann@46752
   351
definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
wenzelm@63404
   352
  where "transp r \<longleftrightarrow> (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
haftmann@46752
   353
wenzelm@63404
   354
lemma transp_trans_eq [pred_set_conv]: "transp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> trans r"
haftmann@46752
   355
  by (simp add: trans_def transp_def)
haftmann@46752
   356
wenzelm@63404
   357
lemma transI [intro?]: "(\<And>x y z. (x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r) \<Longrightarrow> trans r"
haftmann@46752
   358
  by (unfold trans_def) iprover
haftmann@46694
   359
wenzelm@63404
   360
lemma transpI [intro?]: "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"
haftmann@46752
   361
  by (fact transI [to_pred])
haftmann@46752
   362
haftmann@46752
   363
lemma transE:
haftmann@46752
   364
  assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
haftmann@46752
   365
  obtains "(x, z) \<in> r"
haftmann@46752
   366
  using assms by (unfold trans_def) iprover
haftmann@46752
   367
haftmann@46694
   368
lemma transpE:
haftmann@46694
   369
  assumes "transp r" and "r x y" and "r y z"
haftmann@46694
   370
  obtains "r x z"
haftmann@46752
   371
  using assms by (rule transE [to_pred])
haftmann@46752
   372
haftmann@63376
   373
lemma transD [dest?]:
haftmann@46752
   374
  assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
haftmann@46752
   375
  shows "(x, z) \<in> r"
haftmann@46752
   376
  using assms by (rule transE)
haftmann@46752
   377
haftmann@63376
   378
lemma transpD [dest?]:
haftmann@46752
   379
  assumes "transp r" and "r x y" and "r y z"
haftmann@46752
   380
  shows "r x z"
haftmann@46752
   381
  using assms by (rule transD [to_pred])
haftmann@46694
   382
wenzelm@63404
   383
lemma trans_Int: "trans r \<Longrightarrow> trans s \<Longrightarrow> trans (r \<inter> s)"
haftmann@46752
   384
  by (fast intro: transI elim: transE)
haftmann@46692
   385
wenzelm@63404
   386
lemma transp_inf: "transp r \<Longrightarrow> transp s \<Longrightarrow> transp (r \<sqinter> s)"
haftmann@46752
   387
  by (fact trans_Int [to_pred])
haftmann@46752
   388
wenzelm@63404
   389
lemma trans_INTER: "\<forall>x\<in>S. trans (r x) \<Longrightarrow> trans (INTER S r)"
haftmann@46752
   390
  by (fast intro: transI elim: transD)
haftmann@46752
   391
haftmann@64584
   392
lemma transp_INF: "\<forall>x\<in>S. transp (r x) \<Longrightarrow> transp (INFIMUM S r)"
haftmann@64584
   393
  by (fact trans_INTER [to_pred])
haftmann@64584
   394
    
wenzelm@63404
   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)"
haftmann@46694
   396
  by (auto simp add: trans_def)
haftmann@46692
   397
wenzelm@63404
   398
lemma transp_trans: "transp r \<longleftrightarrow> trans {(x, y). r x y}"
haftmann@46752
   399
  by (simp add: trans_def transp_def)
haftmann@46752
   400
Andreas@59518
   401
lemma transp_equality [simp]: "transp op ="
wenzelm@63404
   402
  by (auto intro: transpI)
haftmann@46692
   403
Andreas@63563
   404
lemma trans_empty [simp]: "trans {}"
wenzelm@63612
   405
  by (blast intro: transI)
Andreas@63561
   406
Andreas@63563
   407
lemma transp_empty [simp]: "transp (\<lambda>x y. False)"
wenzelm@63612
   408
  using trans_empty[to_pred] by (simp add: bot_fun_def)
Andreas@63561
   409
Andreas@63563
   410
lemma trans_singleton [simp]: "trans {(a, a)}"
wenzelm@63612
   411
  by (blast intro: transI)
Andreas@63561
   412
Andreas@63563
   413
lemma transp_singleton [simp]: "transp (\<lambda>x y. x = a \<and> y = a)"
wenzelm@63612
   414
  by (simp add: transp_def)
wenzelm@63612
   415
haftmann@63376
   416
wenzelm@60758
   417
subsubsection \<open>Totality\<close>
haftmann@46692
   418
haftmann@46752
   419
definition total_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
wenzelm@63404
   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)"
nipkow@29859
   421
Andreas@63561
   422
lemma total_onI [intro?]:
Andreas@63561
   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"
wenzelm@63612
   424
  unfolding total_on_def by blast
Andreas@63561
   425
nipkow@29859
   426
abbreviation "total \<equiv> total_on UNIV"
nipkow@29859
   427
haftmann@46752
   428
lemma total_on_empty [simp]: "total_on {} r"
haftmann@46752
   429
  by (simp add: total_on_def)
haftmann@46692
   430
Andreas@63563
   431
lemma total_on_singleton [simp]: "total_on {x} {(x, x)}"
wenzelm@63612
   432
  unfolding total_on_def by blast
wenzelm@63612
   433
haftmann@46692
   434
wenzelm@60758
   435
subsubsection \<open>Single valued relations\<close>
haftmann@46692
   436
haftmann@46752
   437
definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool"
wenzelm@63404
   438
  where "single_valued r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
haftmann@46692
   439
haftmann@63376
   440
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
wenzelm@63404
   441
  where "single_valuedP r \<equiv> single_valued {(x, y). r x y}" (* FIXME proper logical operation *)
haftmann@46694
   442
wenzelm@63404
   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"
wenzelm@63404
   444
  unfolding single_valued_def .
haftmann@46752
   445
wenzelm@63404
   446
lemma single_valuedD: "single_valued r \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (x, z) \<in> r \<Longrightarrow> y = z"
haftmann@46752
   447
  by (simp add: single_valued_def)
haftmann@46752
   448
nipkow@57111
   449
lemma single_valued_empty[simp]: "single_valued {}"
wenzelm@63404
   450
  by (simp add: single_valued_def)
nipkow@52392
   451
wenzelm@63404
   452
lemma single_valued_subset: "r \<subseteq> s \<Longrightarrow> single_valued s \<Longrightarrow> single_valued r"
wenzelm@63404
   453
  unfolding single_valued_def by blast
oheimb@11136
   454
berghofe@12905
   455
wenzelm@60758
   456
subsection \<open>Relation operations\<close>
haftmann@46694
   457
wenzelm@60758
   458
subsubsection \<open>The identity relation\<close>
berghofe@12905
   459
haftmann@46752
   460
definition Id :: "'a rel"
wenzelm@63404
   461
  where [code del]: "Id = {p. \<exists>x. p = (x, x)}"
haftmann@46692
   462
wenzelm@63404
   463
lemma IdI [intro]: "(a, a) \<in> Id"
haftmann@46752
   464
  by (simp add: Id_def)
berghofe@12905
   465
wenzelm@63404
   466
lemma IdE [elim!]: "p \<in> Id \<Longrightarrow> (\<And>x. p = (x, x) \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63404
   467
  unfolding Id_def by (iprover elim: CollectE)
berghofe@12905
   468
wenzelm@63404
   469
lemma pair_in_Id_conv [iff]: "(a, b) \<in> Id \<longleftrightarrow> a = b"
wenzelm@63404
   470
  unfolding Id_def by blast
berghofe@12905
   471
nipkow@30198
   472
lemma refl_Id: "refl Id"
haftmann@46752
   473
  by (simp add: refl_on_def)
berghofe@12905
   474
berghofe@12905
   475
lemma antisym_Id: "antisym Id"
wenzelm@61799
   476
  \<comment> \<open>A strange result, since \<open>Id\<close> is also symmetric.\<close>
haftmann@46752
   477
  by (simp add: antisym_def)
berghofe@12905
   478
huffman@19228
   479
lemma sym_Id: "sym Id"
haftmann@46752
   480
  by (simp add: sym_def)
huffman@19228
   481
berghofe@12905
   482
lemma trans_Id: "trans Id"
haftmann@46752
   483
  by (simp add: trans_def)
berghofe@12905
   484
haftmann@46692
   485
lemma single_valued_Id [simp]: "single_valued Id"
haftmann@46692
   486
  by (unfold single_valued_def) blast
haftmann@46692
   487
haftmann@46692
   488
lemma irrefl_diff_Id [simp]: "irrefl (r - Id)"
wenzelm@63404
   489
  by (simp add: irrefl_def)
haftmann@46692
   490
haftmann@46692
   491
lemma trans_diff_Id: "trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r - Id)"
haftmann@46692
   492
  unfolding antisym_def trans_def by blast
haftmann@46692
   493
haftmann@46692
   494
lemma total_on_diff_Id [simp]: "total_on A (r - Id) = total_on A r"
haftmann@46692
   495
  by (simp add: total_on_def)
haftmann@46692
   496
paulson@62087
   497
lemma Id_fstsnd_eq: "Id = {x. fst x = snd x}"
paulson@62087
   498
  by force
berghofe@12905
   499
haftmann@63376
   500
wenzelm@60758
   501
subsubsection \<open>Diagonal: identity over a set\<close>
berghofe@12905
   502
wenzelm@63612
   503
definition Id_on :: "'a set \<Rightarrow> 'a rel"
wenzelm@63404
   504
  where "Id_on A = (\<Union>x\<in>A. {(x, x)})"
haftmann@46692
   505
nipkow@30198
   506
lemma Id_on_empty [simp]: "Id_on {} = {}"
wenzelm@63404
   507
  by (simp add: Id_on_def)
paulson@13812
   508
wenzelm@63404
   509
lemma Id_on_eqI: "a = b \<Longrightarrow> a \<in> A \<Longrightarrow> (a, b) \<in> Id_on A"
haftmann@46752
   510
  by (simp add: Id_on_def)
berghofe@12905
   511
wenzelm@63404
   512
lemma Id_onI [intro!]: "a \<in> A \<Longrightarrow> (a, a) \<in> Id_on A"
haftmann@46752
   513
  by (rule Id_on_eqI) (rule refl)
berghofe@12905
   514
wenzelm@63404
   515
lemma Id_onE [elim!]: "c \<in> Id_on A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> c = (x, x) \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@61799
   516
  \<comment> \<open>The general elimination rule.\<close>
wenzelm@63404
   517
  unfolding Id_on_def by (iprover elim!: UN_E singletonE)
berghofe@12905
   518
wenzelm@63404
   519
lemma Id_on_iff: "(x, y) \<in> Id_on A \<longleftrightarrow> x = y \<and> x \<in> A"
haftmann@46752
   520
  by blast
berghofe@12905
   521
wenzelm@63404
   522
lemma Id_on_def' [nitpick_unfold]: "Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)"
haftmann@46752
   523
  by auto
bulwahn@40923
   524
nipkow@30198
   525
lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
haftmann@46752
   526
  by blast
berghofe@12905
   527
haftmann@46692
   528
lemma refl_on_Id_on: "refl_on A (Id_on A)"
haftmann@46752
   529
  by (rule refl_onI [OF Id_on_subset_Times Id_onI])
haftmann@46692
   530
haftmann@46692
   531
lemma antisym_Id_on [simp]: "antisym (Id_on A)"
wenzelm@63404
   532
  unfolding antisym_def by blast
haftmann@46692
   533
haftmann@46692
   534
lemma sym_Id_on [simp]: "sym (Id_on A)"
haftmann@46752
   535
  by (rule symI) clarify
haftmann@46692
   536
haftmann@46692
   537
lemma trans_Id_on [simp]: "trans (Id_on A)"
haftmann@46752
   538
  by (fast intro: transI elim: transD)
haftmann@46692
   539
haftmann@46692
   540
lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
wenzelm@63404
   541
  unfolding single_valued_def by blast
haftmann@46692
   542
berghofe@12905
   543
wenzelm@60758
   544
subsubsection \<open>Composition\<close>
berghofe@12905
   545
wenzelm@63404
   546
inductive_set relcomp  :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a \<times> 'c) set"  (infixr "O" 75)
haftmann@46752
   547
  for r :: "('a \<times> 'b) set" and s :: "('b \<times> 'c) set"
wenzelm@63404
   548
  where relcompI [intro]: "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> s \<Longrightarrow> (a, c) \<in> r O s"
haftmann@46692
   549
griff@47434
   550
notation relcompp (infixr "OO" 75)
berghofe@12905
   551
griff@47434
   552
lemmas relcomppI = relcompp.intros
berghofe@12905
   553
wenzelm@60758
   554
text \<open>
haftmann@46752
   555
  For historic reasons, the elimination rules are not wholly corresponding.
haftmann@46752
   556
  Feel free to consolidate this.
wenzelm@60758
   557
\<close>
haftmann@46694
   558
griff@47433
   559
inductive_cases relcompEpair: "(a, c) \<in> r O s"
griff@47434
   560
inductive_cases relcomppE [elim!]: "(r OO s) a c"
haftmann@46694
   561
griff@47433
   562
lemma relcompE [elim!]: "xz \<in> r O s \<Longrightarrow>
haftmann@46752
   563
  (\<And>x y z. xz = (x, z) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (y, z) \<in> s  \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63404
   564
  apply (cases xz)
wenzelm@63404
   565
  apply simp
wenzelm@63404
   566
  apply (erule relcompEpair)
wenzelm@63404
   567
  apply iprover
wenzelm@63404
   568
  done
haftmann@46752
   569
wenzelm@63404
   570
lemma R_O_Id [simp]: "R O Id = R"
haftmann@46752
   571
  by fast
haftmann@46694
   572
wenzelm@63404
   573
lemma Id_O_R [simp]: "Id O R = R"
haftmann@46752
   574
  by fast
haftmann@46752
   575
wenzelm@63404
   576
lemma relcomp_empty1 [simp]: "{} O R = {}"
haftmann@46752
   577
  by blast
berghofe@12905
   578
wenzelm@63404
   579
lemma relcompp_bot1 [simp]: "\<bottom> OO R = \<bottom>"
griff@47433
   580
  by (fact relcomp_empty1 [to_pred])
berghofe@12905
   581
wenzelm@63404
   582
lemma relcomp_empty2 [simp]: "R O {} = {}"
haftmann@46752
   583
  by blast
berghofe@12905
   584
wenzelm@63404
   585
lemma relcompp_bot2 [simp]: "R OO \<bottom> = \<bottom>"
griff@47433
   586
  by (fact relcomp_empty2 [to_pred])
krauss@23185
   587
wenzelm@63404
   588
lemma O_assoc: "(R O S) O T = R O (S O T)"
haftmann@46752
   589
  by blast
haftmann@46752
   590
wenzelm@63404
   591
lemma relcompp_assoc: "(r OO s) OO t = r OO (s OO t)"
haftmann@46752
   592
  by (fact O_assoc [to_pred])
krauss@23185
   593
wenzelm@63404
   594
lemma trans_O_subset: "trans r \<Longrightarrow> r O r \<subseteq> r"
haftmann@46752
   595
  by (unfold trans_def) blast
haftmann@46752
   596
wenzelm@63404
   597
lemma transp_relcompp_less_eq: "transp r \<Longrightarrow> r OO r \<le> r "
haftmann@46752
   598
  by (fact trans_O_subset [to_pred])
berghofe@12905
   599
wenzelm@63404
   600
lemma relcomp_mono: "r' \<subseteq> r \<Longrightarrow> s' \<subseteq> s \<Longrightarrow> r' O s' \<subseteq> r O s"
haftmann@46752
   601
  by blast
berghofe@12905
   602
wenzelm@63404
   603
lemma relcompp_mono: "r' \<le> r \<Longrightarrow> s' \<le> s \<Longrightarrow> r' OO s' \<le> r OO s "
griff@47433
   604
  by (fact relcomp_mono [to_pred])
berghofe@12905
   605
wenzelm@63404
   606
lemma relcomp_subset_Sigma: "r \<subseteq> A \<times> B \<Longrightarrow> s \<subseteq> B \<times> C \<Longrightarrow> r O s \<subseteq> A \<times> C"
haftmann@46752
   607
  by blast
haftmann@46752
   608
wenzelm@63404
   609
lemma relcomp_distrib [simp]: "R O (S \<union> T) = (R O S) \<union> (R O T)"
haftmann@46752
   610
  by auto
berghofe@12905
   611
wenzelm@63404
   612
lemma relcompp_distrib [simp]: "R OO (S \<squnion> T) = R OO S \<squnion> R OO T"
griff@47433
   613
  by (fact relcomp_distrib [to_pred])
haftmann@46752
   614
wenzelm@63404
   615
lemma relcomp_distrib2 [simp]: "(S \<union> T) O R = (S O R) \<union> (T O R)"
haftmann@46752
   616
  by auto
krauss@28008
   617
wenzelm@63404
   618
lemma relcompp_distrib2 [simp]: "(S \<squnion> T) OO R = S OO R \<squnion> T OO R"
griff@47433
   619
  by (fact relcomp_distrib2 [to_pred])
haftmann@46752
   620
wenzelm@63404
   621
lemma relcomp_UNION_distrib: "s O UNION I r = (\<Union>i\<in>I. s O r i) "
haftmann@46752
   622
  by auto
krauss@28008
   623
haftmann@64584
   624
lemma relcompp_SUP_distrib: "s OO SUPREMUM I r = (\<Squnion>i\<in>I. s OO r i)"
haftmann@64584
   625
  by (fact relcomp_UNION_distrib [to_pred])
haftmann@64584
   626
    
wenzelm@63404
   627
lemma relcomp_UNION_distrib2: "UNION I r O s = (\<Union>i\<in>I. r i O s) "
haftmann@46752
   628
  by auto
haftmann@46752
   629
haftmann@64584
   630
lemma relcompp_SUP_distrib2: "SUPREMUM I r OO s = (\<Squnion>i\<in>I. r i OO s)"
haftmann@64584
   631
  by (fact relcomp_UNION_distrib2 [to_pred])
haftmann@64584
   632
    
wenzelm@63404
   633
lemma single_valued_relcomp: "single_valued r \<Longrightarrow> single_valued s \<Longrightarrow> single_valued (r O s)"
wenzelm@63404
   634
  unfolding single_valued_def by blast
haftmann@46752
   635
wenzelm@63404
   636
lemma relcomp_unfold: "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
haftmann@46752
   637
  by (auto simp add: set_eq_iff)
berghofe@12905
   638
haftmann@58195
   639
lemma relcompp_apply: "(R OO S) a c \<longleftrightarrow> (\<exists>b. R a b \<and> S b c)"
haftmann@58195
   640
  unfolding relcomp_unfold [to_pred] ..
haftmann@58195
   641
wenzelm@63612
   642
lemma eq_OO: "op = OO R = R"
wenzelm@63404
   643
  by blast
blanchet@55083
   644
Andreas@61630
   645
lemma OO_eq: "R OO op = = R"
wenzelm@63404
   646
  by blast
haftmann@46664
   647
haftmann@63376
   648
wenzelm@60758
   649
subsubsection \<open>Converse\<close>
wenzelm@12913
   650
wenzelm@61955
   651
inductive_set converse :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'a) set"  ("(_\<inverse>)" [1000] 999)
haftmann@46752
   652
  for r :: "('a \<times> 'b) set"
wenzelm@63404
   653
  where "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>"
haftmann@46692
   654
wenzelm@63404
   655
notation conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)
haftmann@46694
   656
wenzelm@61955
   657
notation (ASCII)
wenzelm@61955
   658
  converse  ("(_^-1)" [1000] 999) and
wenzelm@61955
   659
  conversep ("(_^--1)" [1000] 1000)
haftmann@46694
   660
wenzelm@63404
   661
lemma converseI [sym]: "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>"
haftmann@46752
   662
  by (fact converse.intros)
haftmann@46752
   663
wenzelm@63404
   664
lemma conversepI (* CANDIDATE [sym] *): "r a b \<Longrightarrow> r\<inverse>\<inverse> b a"
haftmann@46752
   665
  by (fact conversep.intros)
haftmann@46752
   666
wenzelm@63404
   667
lemma converseD [sym]: "(a, b) \<in> r\<inverse> \<Longrightarrow> (b, a) \<in> r"
haftmann@46752
   668
  by (erule converse.cases) iprover
haftmann@46752
   669
wenzelm@63404
   670
lemma conversepD (* CANDIDATE [sym] *): "r\<inverse>\<inverse> b a \<Longrightarrow> r a b"
haftmann@46752
   671
  by (fact converseD [to_pred])
haftmann@46752
   672
wenzelm@63404
   673
lemma converseE [elim!]: "yx \<in> r\<inverse> \<Longrightarrow> (\<And>x y. yx = (y, x) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@61799
   674
  \<comment> \<open>More general than \<open>converseD\<close>, as it ``splits'' the member of the relation.\<close>
wenzelm@63404
   675
  apply (cases yx)
wenzelm@63404
   676
  apply simp
wenzelm@63404
   677
  apply (erule converse.cases)
wenzelm@63404
   678
  apply iprover
wenzelm@63404
   679
  done
haftmann@46694
   680
noschinl@46882
   681
lemmas conversepE [elim!] = conversep.cases
haftmann@46752
   682
wenzelm@63404
   683
lemma converse_iff [iff]: "(a, b) \<in> r\<inverse> \<longleftrightarrow> (b, a) \<in> r"
haftmann@46752
   684
  by (auto intro: converseI)
haftmann@46752
   685
wenzelm@63404
   686
lemma conversep_iff [iff]: "r\<inverse>\<inverse> a b = r b a"
haftmann@46752
   687
  by (fact converse_iff [to_pred])
haftmann@46694
   688
wenzelm@63404
   689
lemma converse_converse [simp]: "(r\<inverse>)\<inverse> = r"
haftmann@46752
   690
  by (simp add: set_eq_iff)
haftmann@46694
   691
wenzelm@63404
   692
lemma conversep_conversep [simp]: "(r\<inverse>\<inverse>)\<inverse>\<inverse> = r"
haftmann@46752
   693
  by (fact converse_converse [to_pred])
haftmann@46752
   694
nipkow@53680
   695
lemma converse_empty[simp]: "{}\<inverse> = {}"
wenzelm@63404
   696
  by auto
nipkow@53680
   697
nipkow@53680
   698
lemma converse_UNIV[simp]: "UNIV\<inverse> = UNIV"
wenzelm@63404
   699
  by auto
nipkow@53680
   700
wenzelm@63404
   701
lemma converse_relcomp: "(r O s)\<inverse> = s\<inverse> O r\<inverse>"
haftmann@46752
   702
  by blast
haftmann@46694
   703
wenzelm@63404
   704
lemma converse_relcompp: "(r OO s)\<inverse>\<inverse> = s\<inverse>\<inverse> OO r\<inverse>\<inverse>"
wenzelm@63404
   705
  by (iprover intro: order_antisym conversepI relcomppI elim: relcomppE dest: conversepD)
haftmann@46694
   706
wenzelm@63404
   707
lemma converse_Int: "(r \<inter> s)\<inverse> = r\<inverse> \<inter> s\<inverse>"
haftmann@46752
   708
  by blast
haftmann@46752
   709
wenzelm@63404
   710
lemma converse_meet: "(r \<sqinter> s)\<inverse>\<inverse> = r\<inverse>\<inverse> \<sqinter> s\<inverse>\<inverse>"
haftmann@46694
   711
  by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
haftmann@46694
   712
wenzelm@63404
   713
lemma converse_Un: "(r \<union> s)\<inverse> = r\<inverse> \<union> s\<inverse>"
haftmann@46752
   714
  by blast
haftmann@46752
   715
wenzelm@63404
   716
lemma converse_join: "(r \<squnion> s)\<inverse>\<inverse> = r\<inverse>\<inverse> \<squnion> s\<inverse>\<inverse>"
haftmann@46694
   717
  by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)
haftmann@46694
   718
wenzelm@63404
   719
lemma converse_INTER: "(INTER S r)\<inverse> = (INT x:S. (r x)\<inverse>)"
haftmann@46752
   720
  by fast
huffman@19228
   721
wenzelm@63404
   722
lemma converse_UNION: "(UNION S r)\<inverse> = (UN x:S. (r x)\<inverse>)"
haftmann@46752
   723
  by blast
huffman@19228
   724
wenzelm@63404
   725
lemma converse_mono[simp]: "r\<inverse> \<subseteq> s \<inverse> \<longleftrightarrow> r \<subseteq> s"
traytel@52749
   726
  by auto
traytel@52749
   727
wenzelm@63404
   728
lemma conversep_mono[simp]: "r\<inverse>\<inverse> \<le> s \<inverse>\<inverse> \<longleftrightarrow> r \<le> s"
traytel@52749
   729
  by (fact converse_mono[to_pred])
traytel@52749
   730
wenzelm@63404
   731
lemma converse_inject[simp]: "r\<inverse> = s \<inverse> \<longleftrightarrow> r = s"
traytel@52730
   732
  by auto
traytel@52730
   733
wenzelm@63404
   734
lemma conversep_inject[simp]: "r\<inverse>\<inverse> = s \<inverse>\<inverse> \<longleftrightarrow> r = s"
traytel@52749
   735
  by (fact converse_inject[to_pred])
traytel@52749
   736
wenzelm@63612
   737
lemma converse_subset_swap: "r \<subseteq> s \<inverse> \<longleftrightarrow> r \<inverse> \<subseteq> s"
traytel@52749
   738
  by auto
traytel@52749
   739
wenzelm@63612
   740
lemma conversep_le_swap: "r \<le> s \<inverse>\<inverse> \<longleftrightarrow> r \<inverse>\<inverse> \<le> s"
traytel@52749
   741
  by (fact converse_subset_swap[to_pred])
traytel@52730
   742
wenzelm@63404
   743
lemma converse_Id [simp]: "Id\<inverse> = Id"
haftmann@46752
   744
  by blast
berghofe@12905
   745
wenzelm@63404
   746
lemma converse_Id_on [simp]: "(Id_on A)\<inverse> = Id_on A"
haftmann@46752
   747
  by blast
berghofe@12905
   748
nipkow@30198
   749
lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"
wenzelm@63404
   750
  by (auto simp: refl_on_def)
berghofe@12905
   751
huffman@19228
   752
lemma sym_converse [simp]: "sym (converse r) = sym r"
wenzelm@63404
   753
  unfolding sym_def by blast
huffman@19228
   754
huffman@19228
   755
lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
wenzelm@63404
   756
  unfolding antisym_def by blast
berghofe@12905
   757
huffman@19228
   758
lemma trans_converse [simp]: "trans (converse r) = trans r"
wenzelm@63404
   759
  unfolding trans_def by blast
berghofe@12905
   760
wenzelm@63404
   761
lemma sym_conv_converse_eq: "sym r \<longleftrightarrow> r\<inverse> = r"
wenzelm@63404
   762
  unfolding sym_def by fast
huffman@19228
   763
wenzelm@63404
   764
lemma sym_Un_converse: "sym (r \<union> r\<inverse>)"
wenzelm@63404
   765
  unfolding sym_def by blast
huffman@19228
   766
wenzelm@63404
   767
lemma sym_Int_converse: "sym (r \<inter> r\<inverse>)"
wenzelm@63404
   768
  unfolding sym_def by blast
huffman@19228
   769
wenzelm@63404
   770
lemma total_on_converse [simp]: "total_on A (r\<inverse>) = total_on A r"
haftmann@46752
   771
  by (auto simp: total_on_def)
nipkow@29859
   772
wenzelm@63404
   773
lemma finite_converse [iff]: "finite (r\<inverse>) = finite r"
traytel@54611
   774
  unfolding converse_def conversep_iff using [[simproc add: finite_Collect]]
traytel@54611
   775
  by (auto elim: finite_imageD simp: inj_on_def)
wenzelm@12913
   776
wenzelm@63404
   777
lemma conversep_noteq [simp]: "(op \<noteq>)\<inverse>\<inverse> = op \<noteq>"
haftmann@46752
   778
  by (auto simp add: fun_eq_iff)
haftmann@46752
   779
wenzelm@63404
   780
lemma conversep_eq [simp]: "(op =)\<inverse>\<inverse> = op ="
haftmann@46752
   781
  by (auto simp add: fun_eq_iff)
haftmann@46752
   782
wenzelm@63404
   783
lemma converse_unfold [code]: "r\<inverse> = {(y, x). (x, y) \<in> r}"
haftmann@46752
   784
  by (simp add: set_eq_iff)
haftmann@46752
   785
haftmann@46692
   786
wenzelm@60758
   787
subsubsection \<open>Domain, range and field\<close>
haftmann@46692
   788
wenzelm@63404
   789
inductive_set Domain :: "('a \<times> 'b) set \<Rightarrow> 'a set" for r :: "('a \<times> 'b) set"
wenzelm@63404
   790
  where DomainI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> Domain r"
haftmann@46767
   791
haftmann@46767
   792
lemmas DomainPI = Domainp.DomainI
haftmann@46767
   793
haftmann@46767
   794
inductive_cases DomainE [elim!]: "a \<in> Domain r"
haftmann@46767
   795
inductive_cases DomainpE [elim!]: "Domainp r a"
haftmann@46692
   796
wenzelm@63404
   797
inductive_set Range :: "('a \<times> 'b) set \<Rightarrow> 'b set" for r :: "('a \<times> 'b) set"
wenzelm@63404
   798
  where RangeI [intro]: "(a, b) \<in> r \<Longrightarrow> b \<in> Range r"
haftmann@46767
   799
haftmann@46767
   800
lemmas RangePI = Rangep.RangeI
haftmann@46767
   801
haftmann@46767
   802
inductive_cases RangeE [elim!]: "b \<in> Range r"
haftmann@46767
   803
inductive_cases RangepE [elim!]: "Rangep r b"
haftmann@46692
   804
haftmann@46752
   805
definition Field :: "'a rel \<Rightarrow> 'a set"
wenzelm@63404
   806
  where "Field r = Domain r \<union> Range r"
berghofe@12905
   807
Andreas@63561
   808
lemma FieldI1: "(i, j) \<in> R \<Longrightarrow> i \<in> Field R"
wenzelm@63612
   809
  unfolding Field_def by blast
Andreas@63561
   810
Andreas@63561
   811
lemma FieldI2: "(i, j) \<in> R \<Longrightarrow> j \<in> Field R"
Andreas@63561
   812
  unfolding Field_def by auto
Andreas@63561
   813
wenzelm@63404
   814
lemma Domain_fst [code]: "Domain r = fst ` r"
haftmann@46767
   815
  by force
haftmann@46767
   816
wenzelm@63404
   817
lemma Range_snd [code]: "Range r = snd ` r"
haftmann@46767
   818
  by force
haftmann@46767
   819
haftmann@46767
   820
lemma fst_eq_Domain: "fst ` R = Domain R"
haftmann@46767
   821
  by force
haftmann@46767
   822
haftmann@46767
   823
lemma snd_eq_Range: "snd ` R = Range R"
haftmann@46767
   824
  by force
haftmann@46694
   825
paulson@62087
   826
lemma range_fst [simp]: "range fst = UNIV"
paulson@62087
   827
  by (auto simp: fst_eq_Domain)
paulson@62087
   828
paulson@62087
   829
lemma range_snd [simp]: "range snd = UNIV"
paulson@62087
   830
  by (auto simp: snd_eq_Range)
paulson@62087
   831
haftmann@46694
   832
lemma Domain_empty [simp]: "Domain {} = {}"
haftmann@46767
   833
  by auto
haftmann@46767
   834
haftmann@46767
   835
lemma Range_empty [simp]: "Range {} = {}"
haftmann@46767
   836
  by auto
haftmann@46767
   837
haftmann@46767
   838
lemma Field_empty [simp]: "Field {} = {}"
haftmann@46767
   839
  by (simp add: Field_def)
haftmann@46694
   840
haftmann@46694
   841
lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}"
haftmann@46694
   842
  by auto
haftmann@46694
   843
haftmann@46767
   844
lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}"
haftmann@46767
   845
  by auto
haftmann@46767
   846
noschinl@46882
   847
lemma Domain_insert [simp]: "Domain (insert (a, b) r) = insert a (Domain r)"
haftmann@46767
   848
  by blast
haftmann@46767
   849
noschinl@46882
   850
lemma Range_insert [simp]: "Range (insert (a, b) r) = insert b (Range r)"
haftmann@46767
   851
  by blast
haftmann@46767
   852
haftmann@46767
   853
lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \<union> Field r"
noschinl@46884
   854
  by (auto simp add: Field_def)
haftmann@46767
   855
haftmann@46767
   856
lemma Domain_iff: "a \<in> Domain r \<longleftrightarrow> (\<exists>y. (a, y) \<in> r)"
haftmann@46767
   857
  by blast
haftmann@46767
   858
haftmann@46767
   859
lemma Range_iff: "a \<in> Range r \<longleftrightarrow> (\<exists>y. (y, a) \<in> r)"
haftmann@46694
   860
  by blast
haftmann@46694
   861
haftmann@46694
   862
lemma Domain_Id [simp]: "Domain Id = UNIV"
haftmann@46694
   863
  by blast
haftmann@46694
   864
haftmann@46767
   865
lemma Range_Id [simp]: "Range Id = UNIV"
haftmann@46767
   866
  by blast
haftmann@46767
   867
haftmann@46694
   868
lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
haftmann@46694
   869
  by blast
haftmann@46694
   870
haftmann@46767
   871
lemma Range_Id_on [simp]: "Range (Id_on A) = A"
haftmann@46767
   872
  by blast
haftmann@46767
   873
haftmann@46767
   874
lemma Domain_Un_eq: "Domain (A \<union> B) = Domain A \<union> Domain B"
haftmann@46694
   875
  by blast
haftmann@46694
   876
haftmann@46767
   877
lemma Range_Un_eq: "Range (A \<union> B) = Range A \<union> Range B"
haftmann@46767
   878
  by blast
haftmann@46767
   879
haftmann@46767
   880
lemma Field_Un [simp]: "Field (r \<union> s) = Field r \<union> Field s"
haftmann@46767
   881
  by (auto simp: Field_def)
haftmann@46767
   882
haftmann@46767
   883
lemma Domain_Int_subset: "Domain (A \<inter> B) \<subseteq> Domain A \<inter> Domain B"
haftmann@46694
   884
  by blast
haftmann@46694
   885
haftmann@46767
   886
lemma Range_Int_subset: "Range (A \<inter> B) \<subseteq> Range A \<inter> Range B"
haftmann@46767
   887
  by blast
haftmann@46767
   888
haftmann@46767
   889
lemma Domain_Diff_subset: "Domain A - Domain B \<subseteq> Domain (A - B)"
haftmann@46767
   890
  by blast
haftmann@46767
   891
haftmann@46767
   892
lemma Range_Diff_subset: "Range A - Range B \<subseteq> Range (A - B)"
haftmann@46694
   893
  by blast
haftmann@46694
   894
haftmann@46767
   895
lemma Domain_Union: "Domain (\<Union>S) = (\<Union>A\<in>S. Domain A)"
haftmann@46694
   896
  by blast
haftmann@46694
   897
haftmann@46767
   898
lemma Range_Union: "Range (\<Union>S) = (\<Union>A\<in>S. Range A)"
haftmann@46767
   899
  by blast
haftmann@46767
   900
haftmann@46767
   901
lemma Field_Union [simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
haftmann@46767
   902
  by (auto simp: Field_def)
haftmann@46767
   903
haftmann@46752
   904
lemma Domain_converse [simp]: "Domain (r\<inverse>) = Range r"
haftmann@46752
   905
  by auto
haftmann@46694
   906
haftmann@46767
   907
lemma Range_converse [simp]: "Range (r\<inverse>) = Domain r"
haftmann@46694
   908
  by blast
haftmann@46694
   909
haftmann@46767
   910
lemma Field_converse [simp]: "Field (r\<inverse>) = Field r"
haftmann@46767
   911
  by (auto simp: Field_def)
haftmann@46767
   912
wenzelm@63404
   913
lemma Domain_Collect_case_prod [simp]: "Domain {(x, y). P x y} = {x. \<exists>y. P x y}"
haftmann@46767
   914
  by auto
haftmann@46767
   915
wenzelm@63404
   916
lemma Range_Collect_case_prod [simp]: "Range {(x, y). P x y} = {y. \<exists>x. P x y}"
haftmann@46767
   917
  by auto
haftmann@46767
   918
haftmann@46767
   919
lemma finite_Domain: "finite r \<Longrightarrow> finite (Domain r)"
noschinl@46884
   920
  by (induct set: finite) auto
haftmann@46767
   921
haftmann@46767
   922
lemma finite_Range: "finite r \<Longrightarrow> finite (Range r)"
noschinl@46884
   923
  by (induct set: finite) auto
haftmann@46767
   924
haftmann@46767
   925
lemma finite_Field: "finite r \<Longrightarrow> finite (Field r)"
haftmann@46767
   926
  by (simp add: Field_def finite_Domain finite_Range)
haftmann@46767
   927
haftmann@46767
   928
lemma Domain_mono: "r \<subseteq> s \<Longrightarrow> Domain r \<subseteq> Domain s"
haftmann@46767
   929
  by blast
haftmann@46767
   930
haftmann@46767
   931
lemma Range_mono: "r \<subseteq> s \<Longrightarrow> Range r \<subseteq> Range s"
haftmann@46767
   932
  by blast
haftmann@46767
   933
haftmann@46767
   934
lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
haftmann@46767
   935
  by (auto simp: Field_def Domain_def Range_def)
haftmann@46767
   936
wenzelm@63404
   937
lemma Domain_unfold: "Domain r = {x. \<exists>y. (x, y) \<in> r}"
haftmann@46767
   938
  by blast
haftmann@46694
   939
Andreas@63563
   940
lemma Field_square [simp]: "Field (x \<times> x) = x"
wenzelm@63612
   941
  unfolding Field_def by blast
Andreas@63561
   942
berghofe@12905
   943
wenzelm@60758
   944
subsubsection \<open>Image of a set under a relation\<close>
berghofe@12905
   945
wenzelm@63404
   946
definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set"  (infixr "``" 90)
wenzelm@63404
   947
  where "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
haftmann@46692
   948
wenzelm@63404
   949
lemma Image_iff: "b \<in> r``A \<longleftrightarrow> (\<exists>x\<in>A. (x, b) \<in> r)"
haftmann@46752
   950
  by (simp add: Image_def)
berghofe@12905
   951
wenzelm@63404
   952
lemma Image_singleton: "r``{a} = {b. (a, b) \<in> r}"
haftmann@46752
   953
  by (simp add: Image_def)
berghofe@12905
   954
wenzelm@63404
   955
lemma Image_singleton_iff [iff]: "b \<in> r``{a} \<longleftrightarrow> (a, b) \<in> r"
haftmann@46752
   956
  by (rule Image_iff [THEN trans]) simp
berghofe@12905
   957
wenzelm@63404
   958
lemma ImageI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> r``A"
wenzelm@63404
   959
  unfolding Image_def by blast
berghofe@12905
   960
wenzelm@63404
   961
lemma ImageE [elim!]: "b \<in> r `` A \<Longrightarrow> (\<And>x. (x, b) \<in> r \<Longrightarrow> x \<in> A \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63404
   962
  unfolding Image_def by (iprover elim!: CollectE bexE)
berghofe@12905
   963
wenzelm@63404
   964
lemma rev_ImageI: "a \<in> A \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> b \<in> r `` A"
wenzelm@61799
   965
  \<comment> \<open>This version's more effective when we already have the required \<open>a\<close>\<close>
haftmann@46752
   966
  by blast
berghofe@12905
   967
berghofe@12905
   968
lemma Image_empty [simp]: "R``{} = {}"
haftmann@46752
   969
  by blast
berghofe@12905
   970
berghofe@12905
   971
lemma Image_Id [simp]: "Id `` A = A"
haftmann@46752
   972
  by blast
berghofe@12905
   973
nipkow@30198
   974
lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"
haftmann@46752
   975
  by blast
paulson@13830
   976
paulson@13830
   977
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
haftmann@46752
   978
  by blast
berghofe@12905
   979
wenzelm@63404
   980
lemma Image_Int_eq: "single_valued (converse R) \<Longrightarrow> R `` (A \<inter> B) = R `` A \<inter> R `` B"
wenzelm@63612
   981
  by (auto simp: single_valued_def)
berghofe@12905
   982
paulson@13830
   983
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
haftmann@46752
   984
  by blast
berghofe@12905
   985
paulson@13812
   986
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
haftmann@46752
   987
  by blast
paulson@13812
   988
wenzelm@63404
   989
lemma Image_subset: "r \<subseteq> A \<times> B \<Longrightarrow> r``C \<subseteq> B"
haftmann@46752
   990
  by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
berghofe@12905
   991
paulson@13830
   992
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
wenzelm@61799
   993
  \<comment> \<open>NOT suitable for rewriting\<close>
haftmann@46752
   994
  by blast
berghofe@12905
   995
wenzelm@63404
   996
lemma Image_mono: "r' \<subseteq> r \<Longrightarrow> A' \<subseteq> A \<Longrightarrow> (r' `` A') \<subseteq> (r `` A)"
haftmann@46752
   997
  by blast
berghofe@12905
   998
paulson@13830
   999
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
haftmann@46752
  1000
  by blast
paulson@13830
  1001
hoelzl@54410
  1002
lemma UN_Image: "(\<Union>i\<in>I. X i) `` S = (\<Union>i\<in>I. X i `` S)"
hoelzl@54410
  1003
  by auto
hoelzl@54410
  1004
paulson@13830
  1005
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
haftmann@46752
  1006
  by blast
berghofe@12905
  1007
wenzelm@63404
  1008
text \<open>Converse inclusion requires some assumptions\<close>
wenzelm@63404
  1009
lemma Image_INT_eq: "single_valued (r\<inverse>) \<Longrightarrow> A \<noteq> {} \<Longrightarrow> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
wenzelm@63404
  1010
  apply (rule equalityI)
wenzelm@63404
  1011
   apply (rule Image_INT_subset)
wenzelm@63404
  1012
  apply (auto simp add: single_valued_def)
wenzelm@63404
  1013
  apply blast
wenzelm@63404
  1014
  done
berghofe@12905
  1015
wenzelm@63404
  1016
lemma Image_subset_eq: "r``A \<subseteq> B \<longleftrightarrow> A \<subseteq> - ((r\<inverse>) `` (- B))"
haftmann@46752
  1017
  by blast
berghofe@12905
  1018
wenzelm@63404
  1019
lemma Image_Collect_case_prod [simp]: "{(x, y). P x y} `` A = {y. \<exists>x\<in>A. P x y}"
haftmann@46752
  1020
  by auto
berghofe@12905
  1021
hoelzl@54410
  1022
lemma Sigma_Image: "(SIGMA x:A. B x) `` X = (\<Union>x\<in>X \<inter> A. B x)"
hoelzl@54410
  1023
  by auto
hoelzl@54410
  1024
hoelzl@54410
  1025
lemma relcomp_Image: "(X O Y) `` Z = Y `` (X `` Z)"
hoelzl@54410
  1026
  by auto
berghofe@12905
  1027
haftmann@63376
  1028
wenzelm@60758
  1029
subsubsection \<open>Inverse image\<close>
berghofe@12905
  1030
haftmann@46752
  1031
definition inv_image :: "'b rel \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a rel"
wenzelm@63404
  1032
  where "inv_image r f = {(x, y). (f x, f y) \<in> r}"
haftmann@46692
  1033
haftmann@46752
  1034
definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
wenzelm@63404
  1035
  where "inv_imagep r f = (\<lambda>x y. r (f x) (f y))"
haftmann@46694
  1036
haftmann@46694
  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)"
haftmann@46694
  1038
  by (simp add: inv_image_def inv_imagep_def)
haftmann@46694
  1039
wenzelm@63404
  1040
lemma sym_inv_image: "sym r \<Longrightarrow> sym (inv_image r f)"
wenzelm@63404
  1041
  unfolding sym_def inv_image_def by blast
huffman@19228
  1042
wenzelm@63404
  1043
lemma trans_inv_image: "trans r \<Longrightarrow> trans (inv_image r f)"
wenzelm@63404
  1044
  unfolding trans_def inv_image_def
berghofe@12905
  1045
  apply (simp (no_asm))
berghofe@12905
  1046
  apply blast
berghofe@12905
  1047
  done
berghofe@12905
  1048
wenzelm@63404
  1049
lemma in_inv_image[simp]: "(x, y) \<in> inv_image r f \<longleftrightarrow> (f x, f y) \<in> r"
krauss@32463
  1050
  by (auto simp:inv_image_def)
krauss@32463
  1051
wenzelm@63404
  1052
lemma converse_inv_image[simp]: "(inv_image R f)\<inverse> = inv_image (R\<inverse>) f"
haftmann@46752
  1053
  unfolding inv_image_def converse_unfold by auto
krauss@33218
  1054
haftmann@46664
  1055
lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
haftmann@46664
  1056
  by (simp add: inv_imagep_def)
haftmann@46664
  1057
haftmann@46664
  1058
wenzelm@60758
  1059
subsubsection \<open>Powerset\<close>
haftmann@46664
  1060
haftmann@46752
  1061
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
wenzelm@63404
  1062
  where "Powp A = (\<lambda>B. \<forall>x \<in> B. A x)"
haftmann@46664
  1063
haftmann@46664
  1064
lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"
haftmann@46664
  1065
  by (auto simp add: Powp_def fun_eq_iff)
haftmann@46664
  1066
haftmann@46664
  1067
lemmas Powp_mono [mono] = Pow_mono [to_pred]
haftmann@46664
  1068
haftmann@63376
  1069
wenzelm@60758
  1070
subsubsection \<open>Expressing relation operations via @{const Finite_Set.fold}\<close>
kuncar@48620
  1071
kuncar@48620
  1072
lemma Id_on_fold:
kuncar@48620
  1073
  assumes "finite A"
kuncar@48620
  1074
  shows "Id_on A = Finite_Set.fold (\<lambda>x. Set.insert (Pair x x)) {} A"
kuncar@48620
  1075
proof -
wenzelm@63404
  1076
  interpret comp_fun_commute "\<lambda>x. Set.insert (Pair x x)"
wenzelm@63404
  1077
    by standard auto
wenzelm@63404
  1078
  from assms show ?thesis
wenzelm@63404
  1079
    unfolding Id_on_def by (induct A) simp_all
kuncar@48620
  1080
qed
kuncar@48620
  1081
kuncar@48620
  1082
lemma comp_fun_commute_Image_fold:
kuncar@48620
  1083
  "comp_fun_commute (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)"
kuncar@48620
  1084
proof -
kuncar@48620
  1085
  interpret comp_fun_idem Set.insert
wenzelm@63404
  1086
    by (fact comp_fun_idem_insert)
wenzelm@63404
  1087
  show ?thesis
wenzelm@63612
  1088
    by standard (auto simp: fun_eq_iff comp_fun_commute split: prod.split)
kuncar@48620
  1089
qed
kuncar@48620
  1090
kuncar@48620
  1091
lemma Image_fold:
kuncar@48620
  1092
  assumes "finite R"
kuncar@48620
  1093
  shows "R `` S = Finite_Set.fold (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A) {} R"
kuncar@48620
  1094
proof -
wenzelm@63404
  1095
  interpret comp_fun_commute "(\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)"
kuncar@48620
  1096
    by (rule comp_fun_commute_Image_fold)
kuncar@48620
  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))"
traytel@52749
  1098
    by (force intro: rev_ImageI)
wenzelm@63404
  1099
  show ?thesis
wenzelm@63404
  1100
    using assms by (induct R) (auto simp: *)
kuncar@48620
  1101
qed
kuncar@48620
  1102
kuncar@48620
  1103
lemma insert_relcomp_union_fold:
kuncar@48620
  1104
  assumes "finite S"
kuncar@48620
  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"
kuncar@48620
  1106
proof -
kuncar@48620
  1107
  interpret comp_fun_commute "\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A'"
wenzelm@63404
  1108
  proof -
wenzelm@63404
  1109
    interpret comp_fun_idem Set.insert
wenzelm@63404
  1110
      by (fact comp_fun_idem_insert)
kuncar@48620
  1111
    show "comp_fun_commute (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A')"
wenzelm@63404
  1112
      by standard (auto simp add: fun_eq_iff split: prod.split)
kuncar@48620
  1113
  qed
wenzelm@63404
  1114
  have *: "{x} O S = {(x', z). x' = fst x \<and> (snd x, z) \<in> S}"
wenzelm@63404
  1115
    by (auto simp: relcomp_unfold intro!: exI)
wenzelm@63404
  1116
  show ?thesis
wenzelm@63404
  1117
    unfolding * using \<open>finite S\<close> by (induct S) (auto split: prod.split)
kuncar@48620
  1118
qed
kuncar@48620
  1119
kuncar@48620
  1120
lemma insert_relcomp_fold:
kuncar@48620
  1121
  assumes "finite S"
wenzelm@63404
  1122
  shows "Set.insert x R O S =
kuncar@48620
  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"
kuncar@48620
  1124
proof -
wenzelm@63404
  1125
  have "Set.insert x R O S = ({x} O S) \<union> (R O S)"
wenzelm@63404
  1126
    by auto
wenzelm@63404
  1127
  then show ?thesis
wenzelm@63404
  1128
    by (auto simp: insert_relcomp_union_fold [OF assms])
kuncar@48620
  1129
qed
kuncar@48620
  1130
kuncar@48620
  1131
lemma comp_fun_commute_relcomp_fold:
kuncar@48620
  1132
  assumes "finite S"
wenzelm@63404
  1133
  shows "comp_fun_commute (\<lambda>(x,y) A.
kuncar@48620
  1134
    Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S)"
kuncar@48620
  1135
proof -
wenzelm@63404
  1136
  have *: "\<And>a b A.
kuncar@48620
  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"
kuncar@48620
  1138
    by (auto simp: insert_relcomp_union_fold[OF assms] cong: if_cong)
wenzelm@63404
  1139
  show ?thesis
wenzelm@63404
  1140
    by standard (auto simp: *)
kuncar@48620
  1141
qed
kuncar@48620
  1142
kuncar@48620
  1143
lemma relcomp_fold:
wenzelm@63404
  1144
  assumes "finite R" "finite S"
wenzelm@63404
  1145
  shows "R O S = Finite_Set.fold
kuncar@48620
  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"
wenzelm@63404
  1147
  using assms
wenzelm@63404
  1148
  by (induct R)
traytel@52749
  1149
    (auto simp: comp_fun_commute.fold_insert comp_fun_commute_relcomp_fold insert_relcomp_fold
kuncar@48620
  1150
      cong: if_cong)
kuncar@48620
  1151
nipkow@1128
  1152
end