src/HOL/Relation.thy
author Andreas Lochbihler
Fri Jul 29 09:49:23 2016 +0200 (2016-07-29)
changeset 63561 fba08009ff3e
parent 63404 a95e7432d86c
child 63563 0bcd79da075b
permissions -rw-r--r--
add lemmas contributed by Peter Gammie
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(*  Title:      HOL/Relation.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory; 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 [iff]: "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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   279
  assumes "symp r" and "r b a"
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   280
  obtains "r a b"
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   281
  using assms by (rule symE [to_pred])
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   282
haftmann@63376
   283
lemma symD [dest?]:
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   284
  assumes "sym r" and "(b, a) \<in> r"
haftmann@46752
   285
  shows "(a, b) \<in> r"
haftmann@46752
   286
  using assms by (rule symE)
haftmann@46694
   287
haftmann@63376
   288
lemma sympD [dest?]:
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   289
  assumes "symp r" and "r b a"
haftmann@46752
   290
  shows "r a b"
haftmann@46752
   291
  using assms by (rule symD [to_pred])
haftmann@46752
   292
wenzelm@63404
   293
lemma sym_Int: "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<inter> s)"
haftmann@46752
   294
  by (fast intro: symI elim: symE)
haftmann@46692
   295
wenzelm@63404
   296
lemma symp_inf: "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<sqinter> s)"
haftmann@46752
   297
  by (fact sym_Int [to_pred])
haftmann@46752
   298
wenzelm@63404
   299
lemma sym_Un: "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<union> s)"
haftmann@46752
   300
  by (fast intro: symI elim: symE)
haftmann@46752
   301
wenzelm@63404
   302
lemma symp_sup: "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<squnion> s)"
haftmann@46752
   303
  by (fact sym_Un [to_pred])
haftmann@46692
   304
wenzelm@63404
   305
lemma sym_INTER: "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (INTER S r)"
haftmann@46752
   306
  by (fast intro: symI elim: symE)
haftmann@46752
   307
wenzelm@63404
   308
lemma symp_INF: "\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (INFIMUM S r)"
haftmann@46982
   309
  by (fact sym_INTER [to_pred])
haftmann@46692
   310
wenzelm@63404
   311
lemma sym_UNION: "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (UNION S r)"
haftmann@46752
   312
  by (fast intro: symI elim: symE)
haftmann@46752
   313
wenzelm@63404
   314
lemma symp_SUP: "\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (SUPREMUM S r)"
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   315
  by (fact sym_UNION [to_pred])
haftmann@46692
   316
haftmann@46692
   317
wenzelm@60758
   318
subsubsection \<open>Antisymmetry\<close>
haftmann@46694
   319
haftmann@46752
   320
definition antisym :: "'a rel \<Rightarrow> bool"
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   321
  where "antisym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
haftmann@46752
   322
haftmann@46752
   323
abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
wenzelm@63404
   324
  where "antisymP r \<equiv> antisym {(x, y). r x y}" (* FIXME proper logical operation *)
wenzelm@63404
   325
wenzelm@63404
   326
lemma antisymI [intro?]: "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (y, x) \<in> r \<Longrightarrow> x = y) \<Longrightarrow> antisym r"
wenzelm@63404
   327
  unfolding antisym_def by iprover
haftmann@46694
   328
wenzelm@63404
   329
lemma antisymD [dest?]: "antisym r \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r \<Longrightarrow> a = b"
wenzelm@63404
   330
  unfolding antisym_def by iprover
haftmann@46694
   331
wenzelm@63404
   332
lemma antisym_subset: "r \<subseteq> s \<Longrightarrow> antisym s \<Longrightarrow> antisym r"
wenzelm@63404
   333
  unfolding antisym_def by blast
haftmann@46694
   334
haftmann@46694
   335
lemma antisym_empty [simp]: "antisym {}"
wenzelm@63404
   336
  unfolding antisym_def by blast
haftmann@46694
   337
Andreas@60057
   338
lemma antisymP_equality [simp]: "antisymP op ="
wenzelm@63404
   339
  by (auto intro: antisymI)
haftmann@46694
   340
Andreas@63561
   341
lemma antisym_singleton [iff]: "antisym {x}"
Andreas@63561
   342
by (blast intro: antisymI)
Andreas@63561
   343
haftmann@63376
   344
wenzelm@60758
   345
subsubsection \<open>Transitivity\<close>
haftmann@46692
   346
haftmann@46752
   347
definition trans :: "'a rel \<Rightarrow> bool"
wenzelm@63404
   348
  where "trans r \<longleftrightarrow> (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
haftmann@46752
   349
haftmann@46752
   350
definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
wenzelm@63404
   351
  where "transp r \<longleftrightarrow> (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
haftmann@46752
   352
wenzelm@63404
   353
lemma transp_trans_eq [pred_set_conv]: "transp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> trans r"
haftmann@46752
   354
  by (simp add: trans_def transp_def)
haftmann@46752
   355
wenzelm@63404
   356
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
   357
  by (unfold trans_def) iprover
haftmann@46694
   358
wenzelm@63404
   359
lemma transpI [intro?]: "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"
haftmann@46752
   360
  by (fact transI [to_pred])
haftmann@46752
   361
haftmann@46752
   362
lemma transE:
haftmann@46752
   363
  assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
haftmann@46752
   364
  obtains "(x, z) \<in> r"
haftmann@46752
   365
  using assms by (unfold trans_def) iprover
haftmann@46752
   366
haftmann@46694
   367
lemma transpE:
haftmann@46694
   368
  assumes "transp r" and "r x y" and "r y z"
haftmann@46694
   369
  obtains "r x z"
haftmann@46752
   370
  using assms by (rule transE [to_pred])
haftmann@46752
   371
haftmann@63376
   372
lemma transD [dest?]:
haftmann@46752
   373
  assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
haftmann@46752
   374
  shows "(x, z) \<in> r"
haftmann@46752
   375
  using assms by (rule transE)
haftmann@46752
   376
haftmann@63376
   377
lemma transpD [dest?]:
haftmann@46752
   378
  assumes "transp r" and "r x y" and "r y z"
haftmann@46752
   379
  shows "r x z"
haftmann@46752
   380
  using assms by (rule transD [to_pred])
haftmann@46694
   381
wenzelm@63404
   382
lemma trans_Int: "trans r \<Longrightarrow> trans s \<Longrightarrow> trans (r \<inter> s)"
haftmann@46752
   383
  by (fast intro: transI elim: transE)
haftmann@46692
   384
wenzelm@63404
   385
lemma transp_inf: "transp r \<Longrightarrow> transp s \<Longrightarrow> transp (r \<sqinter> s)"
haftmann@46752
   386
  by (fact trans_Int [to_pred])
haftmann@46752
   387
wenzelm@63404
   388
lemma trans_INTER: "\<forall>x\<in>S. trans (r x) \<Longrightarrow> trans (INTER S r)"
haftmann@46752
   389
  by (fast intro: transI elim: transD)
haftmann@46752
   390
haftmann@46752
   391
(* FIXME thm trans_INTER [to_pred] *)
haftmann@46692
   392
wenzelm@63404
   393
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
   394
  by (auto simp add: trans_def)
haftmann@46692
   395
wenzelm@63404
   396
lemma transp_trans: "transp r \<longleftrightarrow> trans {(x, y). r x y}"
haftmann@46752
   397
  by (simp add: trans_def transp_def)
haftmann@46752
   398
Andreas@59518
   399
lemma transp_equality [simp]: "transp op ="
wenzelm@63404
   400
  by (auto intro: transpI)
haftmann@46692
   401
Andreas@63561
   402
lemma trans_empty [iff]: "trans {}"
Andreas@63561
   403
by (blast intro: transI)
Andreas@63561
   404
Andreas@63561
   405
lemma transp_empty [iff]: "transp (\<lambda>x y. False)"
Andreas@63561
   406
using trans_empty[to_pred] by(simp add: bot_fun_def)
Andreas@63561
   407
Andreas@63561
   408
lemma trans_singleton [iff]: "trans {(a, a)}"
Andreas@63561
   409
by (blast intro: transI)
Andreas@63561
   410
Andreas@63561
   411
lemma transp_singleton [iff]: "transp (\<lambda>x y. x = a \<and> y = a)"
Andreas@63561
   412
by(simp add: transp_def)
haftmann@63376
   413
wenzelm@60758
   414
subsubsection \<open>Totality\<close>
haftmann@46692
   415
haftmann@46752
   416
definition total_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
wenzelm@63404
   417
  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
   418
Andreas@63561
   419
lemma total_onI [intro?]:
Andreas@63561
   420
  "(\<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"
Andreas@63561
   421
unfolding total_on_def by blast
Andreas@63561
   422
nipkow@29859
   423
abbreviation "total \<equiv> total_on UNIV"
nipkow@29859
   424
haftmann@46752
   425
lemma total_on_empty [simp]: "total_on {} r"
haftmann@46752
   426
  by (simp add: total_on_def)
haftmann@46692
   427
Andreas@63561
   428
lemma total_on_singleton [iff]: "total_on {x} {(x, x)}"
Andreas@63561
   429
unfolding total_on_def by blast
haftmann@46692
   430
wenzelm@60758
   431
subsubsection \<open>Single valued relations\<close>
haftmann@46692
   432
haftmann@46752
   433
definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool"
wenzelm@63404
   434
  where "single_valued r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
haftmann@46692
   435
haftmann@63376
   436
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool"
wenzelm@63404
   437
  where "single_valuedP r \<equiv> single_valued {(x, y). r x y}" (* FIXME proper logical operation *)
haftmann@46694
   438
wenzelm@63404
   439
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
   440
  unfolding single_valued_def .
haftmann@46752
   441
wenzelm@63404
   442
lemma single_valuedD: "single_valued r \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (x, z) \<in> r \<Longrightarrow> y = z"
haftmann@46752
   443
  by (simp add: single_valued_def)
haftmann@46752
   444
nipkow@57111
   445
lemma single_valued_empty[simp]: "single_valued {}"
wenzelm@63404
   446
  by (simp add: single_valued_def)
nipkow@52392
   447
wenzelm@63404
   448
lemma single_valued_subset: "r \<subseteq> s \<Longrightarrow> single_valued s \<Longrightarrow> single_valued r"
wenzelm@63404
   449
  unfolding single_valued_def by blast
oheimb@11136
   450
berghofe@12905
   451
wenzelm@60758
   452
subsection \<open>Relation operations\<close>
haftmann@46694
   453
wenzelm@60758
   454
subsubsection \<open>The identity relation\<close>
berghofe@12905
   455
haftmann@46752
   456
definition Id :: "'a rel"
wenzelm@63404
   457
  where [code del]: "Id = {p. \<exists>x. p = (x, x)}"
haftmann@46692
   458
wenzelm@63404
   459
lemma IdI [intro]: "(a, a) \<in> Id"
haftmann@46752
   460
  by (simp add: Id_def)
berghofe@12905
   461
wenzelm@63404
   462
lemma IdE [elim!]: "p \<in> Id \<Longrightarrow> (\<And>x. p = (x, x) \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63404
   463
  unfolding Id_def by (iprover elim: CollectE)
berghofe@12905
   464
wenzelm@63404
   465
lemma pair_in_Id_conv [iff]: "(a, b) \<in> Id \<longleftrightarrow> a = b"
wenzelm@63404
   466
  unfolding Id_def by blast
berghofe@12905
   467
nipkow@30198
   468
lemma refl_Id: "refl Id"
haftmann@46752
   469
  by (simp add: refl_on_def)
berghofe@12905
   470
berghofe@12905
   471
lemma antisym_Id: "antisym Id"
wenzelm@61799
   472
  \<comment> \<open>A strange result, since \<open>Id\<close> is also symmetric.\<close>
haftmann@46752
   473
  by (simp add: antisym_def)
berghofe@12905
   474
huffman@19228
   475
lemma sym_Id: "sym Id"
haftmann@46752
   476
  by (simp add: sym_def)
huffman@19228
   477
berghofe@12905
   478
lemma trans_Id: "trans Id"
haftmann@46752
   479
  by (simp add: trans_def)
berghofe@12905
   480
haftmann@46692
   481
lemma single_valued_Id [simp]: "single_valued Id"
haftmann@46692
   482
  by (unfold single_valued_def) blast
haftmann@46692
   483
haftmann@46692
   484
lemma irrefl_diff_Id [simp]: "irrefl (r - Id)"
wenzelm@63404
   485
  by (simp add: irrefl_def)
haftmann@46692
   486
haftmann@46692
   487
lemma trans_diff_Id: "trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r - Id)"
haftmann@46692
   488
  unfolding antisym_def trans_def by blast
haftmann@46692
   489
haftmann@46692
   490
lemma total_on_diff_Id [simp]: "total_on A (r - Id) = total_on A r"
haftmann@46692
   491
  by (simp add: total_on_def)
haftmann@46692
   492
paulson@62087
   493
lemma Id_fstsnd_eq: "Id = {x. fst x = snd x}"
paulson@62087
   494
  by force
berghofe@12905
   495
haftmann@63376
   496
wenzelm@60758
   497
subsubsection \<open>Diagonal: identity over a set\<close>
berghofe@12905
   498
haftmann@46752
   499
definition Id_on  :: "'a set \<Rightarrow> 'a rel"
wenzelm@63404
   500
  where "Id_on A = (\<Union>x\<in>A. {(x, x)})"
haftmann@46692
   501
nipkow@30198
   502
lemma Id_on_empty [simp]: "Id_on {} = {}"
wenzelm@63404
   503
  by (simp add: Id_on_def)
paulson@13812
   504
wenzelm@63404
   505
lemma Id_on_eqI: "a = b \<Longrightarrow> a \<in> A \<Longrightarrow> (a, b) \<in> Id_on A"
haftmann@46752
   506
  by (simp add: Id_on_def)
berghofe@12905
   507
wenzelm@63404
   508
lemma Id_onI [intro!]: "a \<in> A \<Longrightarrow> (a, a) \<in> Id_on A"
haftmann@46752
   509
  by (rule Id_on_eqI) (rule refl)
berghofe@12905
   510
wenzelm@63404
   511
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
   512
  \<comment> \<open>The general elimination rule.\<close>
wenzelm@63404
   513
  unfolding Id_on_def by (iprover elim!: UN_E singletonE)
berghofe@12905
   514
wenzelm@63404
   515
lemma Id_on_iff: "(x, y) \<in> Id_on A \<longleftrightarrow> x = y \<and> x \<in> A"
haftmann@46752
   516
  by blast
berghofe@12905
   517
wenzelm@63404
   518
lemma Id_on_def' [nitpick_unfold]: "Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)"
haftmann@46752
   519
  by auto
bulwahn@40923
   520
nipkow@30198
   521
lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
haftmann@46752
   522
  by blast
berghofe@12905
   523
haftmann@46692
   524
lemma refl_on_Id_on: "refl_on A (Id_on A)"
haftmann@46752
   525
  by (rule refl_onI [OF Id_on_subset_Times Id_onI])
haftmann@46692
   526
haftmann@46692
   527
lemma antisym_Id_on [simp]: "antisym (Id_on A)"
wenzelm@63404
   528
  unfolding antisym_def by blast
haftmann@46692
   529
haftmann@46692
   530
lemma sym_Id_on [simp]: "sym (Id_on A)"
haftmann@46752
   531
  by (rule symI) clarify
haftmann@46692
   532
haftmann@46692
   533
lemma trans_Id_on [simp]: "trans (Id_on A)"
haftmann@46752
   534
  by (fast intro: transI elim: transD)
haftmann@46692
   535
haftmann@46692
   536
lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
wenzelm@63404
   537
  unfolding single_valued_def by blast
haftmann@46692
   538
berghofe@12905
   539
wenzelm@60758
   540
subsubsection \<open>Composition\<close>
berghofe@12905
   541
wenzelm@63404
   542
inductive_set relcomp  :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a \<times> 'c) set"  (infixr "O" 75)
haftmann@46752
   543
  for r :: "('a \<times> 'b) set" and s :: "('b \<times> 'c) set"
wenzelm@63404
   544
  where relcompI [intro]: "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> s \<Longrightarrow> (a, c) \<in> r O s"
haftmann@46692
   545
griff@47434
   546
notation relcompp (infixr "OO" 75)
berghofe@12905
   547
griff@47434
   548
lemmas relcomppI = relcompp.intros
berghofe@12905
   549
wenzelm@60758
   550
text \<open>
haftmann@46752
   551
  For historic reasons, the elimination rules are not wholly corresponding.
haftmann@46752
   552
  Feel free to consolidate this.
wenzelm@60758
   553
\<close>
haftmann@46694
   554
griff@47433
   555
inductive_cases relcompEpair: "(a, c) \<in> r O s"
griff@47434
   556
inductive_cases relcomppE [elim!]: "(r OO s) a c"
haftmann@46694
   557
griff@47433
   558
lemma relcompE [elim!]: "xz \<in> r O s \<Longrightarrow>
haftmann@46752
   559
  (\<And>x y z. xz = (x, z) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (y, z) \<in> s  \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63404
   560
  apply (cases xz)
wenzelm@63404
   561
  apply simp
wenzelm@63404
   562
  apply (erule relcompEpair)
wenzelm@63404
   563
  apply iprover
wenzelm@63404
   564
  done
haftmann@46752
   565
wenzelm@63404
   566
lemma R_O_Id [simp]: "R O Id = R"
haftmann@46752
   567
  by fast
haftmann@46694
   568
wenzelm@63404
   569
lemma Id_O_R [simp]: "Id O R = R"
haftmann@46752
   570
  by fast
haftmann@46752
   571
wenzelm@63404
   572
lemma relcomp_empty1 [simp]: "{} O R = {}"
haftmann@46752
   573
  by blast
berghofe@12905
   574
wenzelm@63404
   575
lemma relcompp_bot1 [simp]: "\<bottom> OO R = \<bottom>"
griff@47433
   576
  by (fact relcomp_empty1 [to_pred])
berghofe@12905
   577
wenzelm@63404
   578
lemma relcomp_empty2 [simp]: "R O {} = {}"
haftmann@46752
   579
  by blast
berghofe@12905
   580
wenzelm@63404
   581
lemma relcompp_bot2 [simp]: "R OO \<bottom> = \<bottom>"
griff@47433
   582
  by (fact relcomp_empty2 [to_pred])
krauss@23185
   583
wenzelm@63404
   584
lemma O_assoc: "(R O S) O T = R O (S O T)"
haftmann@46752
   585
  by blast
haftmann@46752
   586
wenzelm@63404
   587
lemma relcompp_assoc: "(r OO s) OO t = r OO (s OO t)"
haftmann@46752
   588
  by (fact O_assoc [to_pred])
krauss@23185
   589
wenzelm@63404
   590
lemma trans_O_subset: "trans r \<Longrightarrow> r O r \<subseteq> r"
haftmann@46752
   591
  by (unfold trans_def) blast
haftmann@46752
   592
wenzelm@63404
   593
lemma transp_relcompp_less_eq: "transp r \<Longrightarrow> r OO r \<le> r "
haftmann@46752
   594
  by (fact trans_O_subset [to_pred])
berghofe@12905
   595
wenzelm@63404
   596
lemma relcomp_mono: "r' \<subseteq> r \<Longrightarrow> s' \<subseteq> s \<Longrightarrow> r' O s' \<subseteq> r O s"
haftmann@46752
   597
  by blast
berghofe@12905
   598
wenzelm@63404
   599
lemma relcompp_mono: "r' \<le> r \<Longrightarrow> s' \<le> s \<Longrightarrow> r' OO s' \<le> r OO s "
griff@47433
   600
  by (fact relcomp_mono [to_pred])
berghofe@12905
   601
wenzelm@63404
   602
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
   603
  by blast
haftmann@46752
   604
wenzelm@63404
   605
lemma relcomp_distrib [simp]: "R O (S \<union> T) = (R O S) \<union> (R O T)"
haftmann@46752
   606
  by auto
berghofe@12905
   607
wenzelm@63404
   608
lemma relcompp_distrib [simp]: "R OO (S \<squnion> T) = R OO S \<squnion> R OO T"
griff@47433
   609
  by (fact relcomp_distrib [to_pred])
haftmann@46752
   610
wenzelm@63404
   611
lemma relcomp_distrib2 [simp]: "(S \<union> T) O R = (S O R) \<union> (T O R)"
haftmann@46752
   612
  by auto
krauss@28008
   613
wenzelm@63404
   614
lemma relcompp_distrib2 [simp]: "(S \<squnion> T) OO R = S OO R \<squnion> T OO R"
griff@47433
   615
  by (fact relcomp_distrib2 [to_pred])
haftmann@46752
   616
wenzelm@63404
   617
lemma relcomp_UNION_distrib: "s O UNION I r = (\<Union>i\<in>I. s O r i) "
haftmann@46752
   618
  by auto
krauss@28008
   619
griff@47433
   620
(* FIXME thm relcomp_UNION_distrib [to_pred] *)
krauss@36772
   621
wenzelm@63404
   622
lemma relcomp_UNION_distrib2: "UNION I r O s = (\<Union>i\<in>I. r i O s) "
haftmann@46752
   623
  by auto
haftmann@46752
   624
griff@47433
   625
(* FIXME thm relcomp_UNION_distrib2 [to_pred] *)
krauss@36772
   626
wenzelm@63404
   627
lemma single_valued_relcomp: "single_valued r \<Longrightarrow> single_valued s \<Longrightarrow> single_valued (r O s)"
wenzelm@63404
   628
  unfolding single_valued_def by blast
haftmann@46752
   629
wenzelm@63404
   630
lemma relcomp_unfold: "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
haftmann@46752
   631
  by (auto simp add: set_eq_iff)
berghofe@12905
   632
haftmann@58195
   633
lemma relcompp_apply: "(R OO S) a c \<longleftrightarrow> (\<exists>b. R a b \<and> S b c)"
haftmann@58195
   634
  unfolding relcomp_unfold [to_pred] ..
haftmann@58195
   635
blanchet@55083
   636
lemma eq_OO: "op= OO R = R"
wenzelm@63404
   637
  by blast
blanchet@55083
   638
Andreas@61630
   639
lemma OO_eq: "R OO op = = R"
wenzelm@63404
   640
  by blast
haftmann@46664
   641
haftmann@63376
   642
wenzelm@60758
   643
subsubsection \<open>Converse\<close>
wenzelm@12913
   644
wenzelm@61955
   645
inductive_set converse :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'a) set"  ("(_\<inverse>)" [1000] 999)
haftmann@46752
   646
  for r :: "('a \<times> 'b) set"
wenzelm@63404
   647
  where "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>"
haftmann@46692
   648
wenzelm@63404
   649
notation conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)
haftmann@46694
   650
wenzelm@61955
   651
notation (ASCII)
wenzelm@61955
   652
  converse  ("(_^-1)" [1000] 999) and
wenzelm@61955
   653
  conversep ("(_^--1)" [1000] 1000)
haftmann@46694
   654
wenzelm@63404
   655
lemma converseI [sym]: "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>"
haftmann@46752
   656
  by (fact converse.intros)
haftmann@46752
   657
wenzelm@63404
   658
lemma conversepI (* CANDIDATE [sym] *): "r a b \<Longrightarrow> r\<inverse>\<inverse> b a"
haftmann@46752
   659
  by (fact conversep.intros)
haftmann@46752
   660
wenzelm@63404
   661
lemma converseD [sym]: "(a, b) \<in> r\<inverse> \<Longrightarrow> (b, a) \<in> r"
haftmann@46752
   662
  by (erule converse.cases) iprover
haftmann@46752
   663
wenzelm@63404
   664
lemma conversepD (* CANDIDATE [sym] *): "r\<inverse>\<inverse> b a \<Longrightarrow> r a b"
haftmann@46752
   665
  by (fact converseD [to_pred])
haftmann@46752
   666
wenzelm@63404
   667
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
   668
  \<comment> \<open>More general than \<open>converseD\<close>, as it ``splits'' the member of the relation.\<close>
wenzelm@63404
   669
  apply (cases yx)
wenzelm@63404
   670
  apply simp
wenzelm@63404
   671
  apply (erule converse.cases)
wenzelm@63404
   672
  apply iprover
wenzelm@63404
   673
  done
haftmann@46694
   674
noschinl@46882
   675
lemmas conversepE [elim!] = conversep.cases
haftmann@46752
   676
wenzelm@63404
   677
lemma converse_iff [iff]: "(a, b) \<in> r\<inverse> \<longleftrightarrow> (b, a) \<in> r"
haftmann@46752
   678
  by (auto intro: converseI)
haftmann@46752
   679
wenzelm@63404
   680
lemma conversep_iff [iff]: "r\<inverse>\<inverse> a b = r b a"
haftmann@46752
   681
  by (fact converse_iff [to_pred])
haftmann@46694
   682
wenzelm@63404
   683
lemma converse_converse [simp]: "(r\<inverse>)\<inverse> = r"
haftmann@46752
   684
  by (simp add: set_eq_iff)
haftmann@46694
   685
wenzelm@63404
   686
lemma conversep_conversep [simp]: "(r\<inverse>\<inverse>)\<inverse>\<inverse> = r"
haftmann@46752
   687
  by (fact converse_converse [to_pred])
haftmann@46752
   688
nipkow@53680
   689
lemma converse_empty[simp]: "{}\<inverse> = {}"
wenzelm@63404
   690
  by auto
nipkow@53680
   691
nipkow@53680
   692
lemma converse_UNIV[simp]: "UNIV\<inverse> = UNIV"
wenzelm@63404
   693
  by auto
nipkow@53680
   694
wenzelm@63404
   695
lemma converse_relcomp: "(r O s)\<inverse> = s\<inverse> O r\<inverse>"
haftmann@46752
   696
  by blast
haftmann@46694
   697
wenzelm@63404
   698
lemma converse_relcompp: "(r OO s)\<inverse>\<inverse> = s\<inverse>\<inverse> OO r\<inverse>\<inverse>"
wenzelm@63404
   699
  by (iprover intro: order_antisym conversepI relcomppI elim: relcomppE dest: conversepD)
haftmann@46694
   700
wenzelm@63404
   701
lemma converse_Int: "(r \<inter> s)\<inverse> = r\<inverse> \<inter> s\<inverse>"
haftmann@46752
   702
  by blast
haftmann@46752
   703
wenzelm@63404
   704
lemma converse_meet: "(r \<sqinter> s)\<inverse>\<inverse> = r\<inverse>\<inverse> \<sqinter> s\<inverse>\<inverse>"
haftmann@46694
   705
  by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
haftmann@46694
   706
wenzelm@63404
   707
lemma converse_Un: "(r \<union> s)\<inverse> = r\<inverse> \<union> s\<inverse>"
haftmann@46752
   708
  by blast
haftmann@46752
   709
wenzelm@63404
   710
lemma converse_join: "(r \<squnion> s)\<inverse>\<inverse> = r\<inverse>\<inverse> \<squnion> s\<inverse>\<inverse>"
haftmann@46694
   711
  by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)
haftmann@46694
   712
wenzelm@63404
   713
lemma converse_INTER: "(INTER S r)\<inverse> = (INT x:S. (r x)\<inverse>)"
haftmann@46752
   714
  by fast
huffman@19228
   715
wenzelm@63404
   716
lemma converse_UNION: "(UNION S r)\<inverse> = (UN x:S. (r x)\<inverse>)"
haftmann@46752
   717
  by blast
huffman@19228
   718
wenzelm@63404
   719
lemma converse_mono[simp]: "r\<inverse> \<subseteq> s \<inverse> \<longleftrightarrow> r \<subseteq> s"
traytel@52749
   720
  by auto
traytel@52749
   721
wenzelm@63404
   722
lemma conversep_mono[simp]: "r\<inverse>\<inverse> \<le> s \<inverse>\<inverse> \<longleftrightarrow> r \<le> s"
traytel@52749
   723
  by (fact converse_mono[to_pred])
traytel@52749
   724
wenzelm@63404
   725
lemma converse_inject[simp]: "r\<inverse> = s \<inverse> \<longleftrightarrow> r = s"
traytel@52730
   726
  by auto
traytel@52730
   727
wenzelm@63404
   728
lemma conversep_inject[simp]: "r\<inverse>\<inverse> = s \<inverse>\<inverse> \<longleftrightarrow> r = s"
traytel@52749
   729
  by (fact converse_inject[to_pred])
traytel@52749
   730
wenzelm@63404
   731
lemma converse_subset_swap: "r \<subseteq> s \<inverse> = (r \<inverse> \<subseteq> s)"
traytel@52749
   732
  by auto
traytel@52749
   733
wenzelm@63404
   734
lemma conversep_le_swap: "r \<le> s \<inverse>\<inverse> = (r \<inverse>\<inverse> \<le> s)"
traytel@52749
   735
  by (fact converse_subset_swap[to_pred])
traytel@52730
   736
wenzelm@63404
   737
lemma converse_Id [simp]: "Id\<inverse> = Id"
haftmann@46752
   738
  by blast
berghofe@12905
   739
wenzelm@63404
   740
lemma converse_Id_on [simp]: "(Id_on A)\<inverse> = Id_on A"
haftmann@46752
   741
  by blast
berghofe@12905
   742
nipkow@30198
   743
lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"
wenzelm@63404
   744
  by (auto simp: refl_on_def)
berghofe@12905
   745
huffman@19228
   746
lemma sym_converse [simp]: "sym (converse r) = sym r"
wenzelm@63404
   747
  unfolding sym_def by blast
huffman@19228
   748
huffman@19228
   749
lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
wenzelm@63404
   750
  unfolding antisym_def by blast
berghofe@12905
   751
huffman@19228
   752
lemma trans_converse [simp]: "trans (converse r) = trans r"
wenzelm@63404
   753
  unfolding trans_def by blast
berghofe@12905
   754
wenzelm@63404
   755
lemma sym_conv_converse_eq: "sym r \<longleftrightarrow> r\<inverse> = r"
wenzelm@63404
   756
  unfolding sym_def by fast
huffman@19228
   757
wenzelm@63404
   758
lemma sym_Un_converse: "sym (r \<union> r\<inverse>)"
wenzelm@63404
   759
  unfolding sym_def by blast
huffman@19228
   760
wenzelm@63404
   761
lemma sym_Int_converse: "sym (r \<inter> r\<inverse>)"
wenzelm@63404
   762
  unfolding sym_def by blast
huffman@19228
   763
wenzelm@63404
   764
lemma total_on_converse [simp]: "total_on A (r\<inverse>) = total_on A r"
haftmann@46752
   765
  by (auto simp: total_on_def)
nipkow@29859
   766
wenzelm@63404
   767
lemma finite_converse [iff]: "finite (r\<inverse>) = finite r"
traytel@54611
   768
  unfolding converse_def conversep_iff using [[simproc add: finite_Collect]]
traytel@54611
   769
  by (auto elim: finite_imageD simp: inj_on_def)
wenzelm@12913
   770
wenzelm@63404
   771
lemma conversep_noteq [simp]: "(op \<noteq>)\<inverse>\<inverse> = op \<noteq>"
haftmann@46752
   772
  by (auto simp add: fun_eq_iff)
haftmann@46752
   773
wenzelm@63404
   774
lemma conversep_eq [simp]: "(op =)\<inverse>\<inverse> = op ="
haftmann@46752
   775
  by (auto simp add: fun_eq_iff)
haftmann@46752
   776
wenzelm@63404
   777
lemma converse_unfold [code]: "r\<inverse> = {(y, x). (x, y) \<in> r}"
haftmann@46752
   778
  by (simp add: set_eq_iff)
haftmann@46752
   779
haftmann@46692
   780
wenzelm@60758
   781
subsubsection \<open>Domain, range and field\<close>
haftmann@46692
   782
wenzelm@63404
   783
inductive_set Domain :: "('a \<times> 'b) set \<Rightarrow> 'a set" for r :: "('a \<times> 'b) set"
wenzelm@63404
   784
  where DomainI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> Domain r"
haftmann@46767
   785
haftmann@46767
   786
lemmas DomainPI = Domainp.DomainI
haftmann@46767
   787
haftmann@46767
   788
inductive_cases DomainE [elim!]: "a \<in> Domain r"
haftmann@46767
   789
inductive_cases DomainpE [elim!]: "Domainp r a"
haftmann@46692
   790
wenzelm@63404
   791
inductive_set Range :: "('a \<times> 'b) set \<Rightarrow> 'b set" for r :: "('a \<times> 'b) set"
wenzelm@63404
   792
  where RangeI [intro]: "(a, b) \<in> r \<Longrightarrow> b \<in> Range r"
haftmann@46767
   793
haftmann@46767
   794
lemmas RangePI = Rangep.RangeI
haftmann@46767
   795
haftmann@46767
   796
inductive_cases RangeE [elim!]: "b \<in> Range r"
haftmann@46767
   797
inductive_cases RangepE [elim!]: "Rangep r b"
haftmann@46692
   798
haftmann@46752
   799
definition Field :: "'a rel \<Rightarrow> 'a set"
wenzelm@63404
   800
  where "Field r = Domain r \<union> Range r"
berghofe@12905
   801
Andreas@63561
   802
lemma FieldI1: "(i, j) \<in> R \<Longrightarrow> i \<in> Field R"
Andreas@63561
   803
unfolding Field_def by blast
Andreas@63561
   804
Andreas@63561
   805
lemma FieldI2: "(i, j) \<in> R \<Longrightarrow> j \<in> Field R"
Andreas@63561
   806
  unfolding Field_def by auto
Andreas@63561
   807
wenzelm@63404
   808
lemma Domain_fst [code]: "Domain r = fst ` r"
haftmann@46767
   809
  by force
haftmann@46767
   810
wenzelm@63404
   811
lemma Range_snd [code]: "Range r = snd ` r"
haftmann@46767
   812
  by force
haftmann@46767
   813
haftmann@46767
   814
lemma fst_eq_Domain: "fst ` R = Domain R"
haftmann@46767
   815
  by force
haftmann@46767
   816
haftmann@46767
   817
lemma snd_eq_Range: "snd ` R = Range R"
haftmann@46767
   818
  by force
haftmann@46694
   819
paulson@62087
   820
lemma range_fst [simp]: "range fst = UNIV"
paulson@62087
   821
  by (auto simp: fst_eq_Domain)
paulson@62087
   822
paulson@62087
   823
lemma range_snd [simp]: "range snd = UNIV"
paulson@62087
   824
  by (auto simp: snd_eq_Range)
paulson@62087
   825
haftmann@46694
   826
lemma Domain_empty [simp]: "Domain {} = {}"
haftmann@46767
   827
  by auto
haftmann@46767
   828
haftmann@46767
   829
lemma Range_empty [simp]: "Range {} = {}"
haftmann@46767
   830
  by auto
haftmann@46767
   831
haftmann@46767
   832
lemma Field_empty [simp]: "Field {} = {}"
haftmann@46767
   833
  by (simp add: Field_def)
haftmann@46694
   834
haftmann@46694
   835
lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}"
haftmann@46694
   836
  by auto
haftmann@46694
   837
haftmann@46767
   838
lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}"
haftmann@46767
   839
  by auto
haftmann@46767
   840
noschinl@46882
   841
lemma Domain_insert [simp]: "Domain (insert (a, b) r) = insert a (Domain r)"
haftmann@46767
   842
  by blast
haftmann@46767
   843
noschinl@46882
   844
lemma Range_insert [simp]: "Range (insert (a, b) r) = insert b (Range r)"
haftmann@46767
   845
  by blast
haftmann@46767
   846
haftmann@46767
   847
lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \<union> Field r"
noschinl@46884
   848
  by (auto simp add: Field_def)
haftmann@46767
   849
haftmann@46767
   850
lemma Domain_iff: "a \<in> Domain r \<longleftrightarrow> (\<exists>y. (a, y) \<in> r)"
haftmann@46767
   851
  by blast
haftmann@46767
   852
haftmann@46767
   853
lemma Range_iff: "a \<in> Range r \<longleftrightarrow> (\<exists>y. (y, a) \<in> r)"
haftmann@46694
   854
  by blast
haftmann@46694
   855
haftmann@46694
   856
lemma Domain_Id [simp]: "Domain Id = UNIV"
haftmann@46694
   857
  by blast
haftmann@46694
   858
haftmann@46767
   859
lemma Range_Id [simp]: "Range Id = UNIV"
haftmann@46767
   860
  by blast
haftmann@46767
   861
haftmann@46694
   862
lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
haftmann@46694
   863
  by blast
haftmann@46694
   864
haftmann@46767
   865
lemma Range_Id_on [simp]: "Range (Id_on A) = A"
haftmann@46767
   866
  by blast
haftmann@46767
   867
haftmann@46767
   868
lemma Domain_Un_eq: "Domain (A \<union> B) = Domain A \<union> Domain B"
haftmann@46694
   869
  by blast
haftmann@46694
   870
haftmann@46767
   871
lemma Range_Un_eq: "Range (A \<union> B) = Range A \<union> Range B"
haftmann@46767
   872
  by blast
haftmann@46767
   873
haftmann@46767
   874
lemma Field_Un [simp]: "Field (r \<union> s) = Field r \<union> Field s"
haftmann@46767
   875
  by (auto simp: Field_def)
haftmann@46767
   876
haftmann@46767
   877
lemma Domain_Int_subset: "Domain (A \<inter> B) \<subseteq> Domain A \<inter> Domain B"
haftmann@46694
   878
  by blast
haftmann@46694
   879
haftmann@46767
   880
lemma Range_Int_subset: "Range (A \<inter> B) \<subseteq> Range A \<inter> Range B"
haftmann@46767
   881
  by blast
haftmann@46767
   882
haftmann@46767
   883
lemma Domain_Diff_subset: "Domain A - Domain B \<subseteq> Domain (A - B)"
haftmann@46767
   884
  by blast
haftmann@46767
   885
haftmann@46767
   886
lemma Range_Diff_subset: "Range A - Range B \<subseteq> Range (A - B)"
haftmann@46694
   887
  by blast
haftmann@46694
   888
haftmann@46767
   889
lemma Domain_Union: "Domain (\<Union>S) = (\<Union>A\<in>S. Domain A)"
haftmann@46694
   890
  by blast
haftmann@46694
   891
haftmann@46767
   892
lemma Range_Union: "Range (\<Union>S) = (\<Union>A\<in>S. Range A)"
haftmann@46767
   893
  by blast
haftmann@46767
   894
haftmann@46767
   895
lemma Field_Union [simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
haftmann@46767
   896
  by (auto simp: Field_def)
haftmann@46767
   897
haftmann@46752
   898
lemma Domain_converse [simp]: "Domain (r\<inverse>) = Range r"
haftmann@46752
   899
  by auto
haftmann@46694
   900
haftmann@46767
   901
lemma Range_converse [simp]: "Range (r\<inverse>) = Domain r"
haftmann@46694
   902
  by blast
haftmann@46694
   903
haftmann@46767
   904
lemma Field_converse [simp]: "Field (r\<inverse>) = Field r"
haftmann@46767
   905
  by (auto simp: Field_def)
haftmann@46767
   906
wenzelm@63404
   907
lemma Domain_Collect_case_prod [simp]: "Domain {(x, y). P x y} = {x. \<exists>y. P x y}"
haftmann@46767
   908
  by auto
haftmann@46767
   909
wenzelm@63404
   910
lemma Range_Collect_case_prod [simp]: "Range {(x, y). P x y} = {y. \<exists>x. P x y}"
haftmann@46767
   911
  by auto
haftmann@46767
   912
haftmann@46767
   913
lemma finite_Domain: "finite r \<Longrightarrow> finite (Domain r)"
noschinl@46884
   914
  by (induct set: finite) auto
haftmann@46767
   915
haftmann@46767
   916
lemma finite_Range: "finite r \<Longrightarrow> finite (Range r)"
noschinl@46884
   917
  by (induct set: finite) auto
haftmann@46767
   918
haftmann@46767
   919
lemma finite_Field: "finite r \<Longrightarrow> finite (Field r)"
haftmann@46767
   920
  by (simp add: Field_def finite_Domain finite_Range)
haftmann@46767
   921
haftmann@46767
   922
lemma Domain_mono: "r \<subseteq> s \<Longrightarrow> Domain r \<subseteq> Domain s"
haftmann@46767
   923
  by blast
haftmann@46767
   924
haftmann@46767
   925
lemma Range_mono: "r \<subseteq> s \<Longrightarrow> Range r \<subseteq> Range s"
haftmann@46767
   926
  by blast
haftmann@46767
   927
haftmann@46767
   928
lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
haftmann@46767
   929
  by (auto simp: Field_def Domain_def Range_def)
haftmann@46767
   930
wenzelm@63404
   931
lemma Domain_unfold: "Domain r = {x. \<exists>y. (x, y) \<in> r}"
haftmann@46767
   932
  by blast
haftmann@46694
   933
Andreas@63561
   934
lemma Field_square [iff]: "Field (x \<times> x) = x"
Andreas@63561
   935
unfolding Field_def by blast
Andreas@63561
   936
berghofe@12905
   937
wenzelm@60758
   938
subsubsection \<open>Image of a set under a relation\<close>
berghofe@12905
   939
wenzelm@63404
   940
definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set"  (infixr "``" 90)
wenzelm@63404
   941
  where "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
haftmann@46692
   942
wenzelm@63404
   943
lemma Image_iff: "b \<in> r``A \<longleftrightarrow> (\<exists>x\<in>A. (x, b) \<in> r)"
haftmann@46752
   944
  by (simp add: Image_def)
berghofe@12905
   945
wenzelm@63404
   946
lemma Image_singleton: "r``{a} = {b. (a, b) \<in> r}"
haftmann@46752
   947
  by (simp add: Image_def)
berghofe@12905
   948
wenzelm@63404
   949
lemma Image_singleton_iff [iff]: "b \<in> r``{a} \<longleftrightarrow> (a, b) \<in> r"
haftmann@46752
   950
  by (rule Image_iff [THEN trans]) simp
berghofe@12905
   951
wenzelm@63404
   952
lemma ImageI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> A \<Longrightarrow> b \<in> r``A"
wenzelm@63404
   953
  unfolding Image_def by blast
berghofe@12905
   954
wenzelm@63404
   955
lemma ImageE [elim!]: "b \<in> r `` A \<Longrightarrow> (\<And>x. (x, b) \<in> r \<Longrightarrow> x \<in> A \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63404
   956
  unfolding Image_def by (iprover elim!: CollectE bexE)
berghofe@12905
   957
wenzelm@63404
   958
lemma rev_ImageI: "a \<in> A \<Longrightarrow> (a, b) \<in> r \<Longrightarrow> b \<in> r `` A"
wenzelm@61799
   959
  \<comment> \<open>This version's more effective when we already have the required \<open>a\<close>\<close>
haftmann@46752
   960
  by blast
berghofe@12905
   961
berghofe@12905
   962
lemma Image_empty [simp]: "R``{} = {}"
haftmann@46752
   963
  by blast
berghofe@12905
   964
berghofe@12905
   965
lemma Image_Id [simp]: "Id `` A = A"
haftmann@46752
   966
  by blast
berghofe@12905
   967
nipkow@30198
   968
lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"
haftmann@46752
   969
  by blast
paulson@13830
   970
paulson@13830
   971
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
haftmann@46752
   972
  by blast
berghofe@12905
   973
wenzelm@63404
   974
lemma Image_Int_eq: "single_valued (converse R) \<Longrightarrow> R `` (A \<inter> B) = R `` A \<inter> R `` B"
wenzelm@63404
   975
  by (simp add: single_valued_def, blast)
berghofe@12905
   976
paulson@13830
   977
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
haftmann@46752
   978
  by blast
berghofe@12905
   979
paulson@13812
   980
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
haftmann@46752
   981
  by blast
paulson@13812
   982
wenzelm@63404
   983
lemma Image_subset: "r \<subseteq> A \<times> B \<Longrightarrow> r``C \<subseteq> B"
haftmann@46752
   984
  by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
berghofe@12905
   985
paulson@13830
   986
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
wenzelm@61799
   987
  \<comment> \<open>NOT suitable for rewriting\<close>
haftmann@46752
   988
  by blast
berghofe@12905
   989
wenzelm@63404
   990
lemma Image_mono: "r' \<subseteq> r \<Longrightarrow> A' \<subseteq> A \<Longrightarrow> (r' `` A') \<subseteq> (r `` A)"
haftmann@46752
   991
  by blast
berghofe@12905
   992
paulson@13830
   993
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
haftmann@46752
   994
  by blast
paulson@13830
   995
hoelzl@54410
   996
lemma UN_Image: "(\<Union>i\<in>I. X i) `` S = (\<Union>i\<in>I. X i `` S)"
hoelzl@54410
   997
  by auto
hoelzl@54410
   998
paulson@13830
   999
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))"
haftmann@46752
  1000
  by blast
berghofe@12905
  1001
wenzelm@63404
  1002
text \<open>Converse inclusion requires some assumptions\<close>
wenzelm@63404
  1003
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
  1004
  apply (rule equalityI)
wenzelm@63404
  1005
   apply (rule Image_INT_subset)
wenzelm@63404
  1006
  apply (auto simp add: single_valued_def)
wenzelm@63404
  1007
  apply blast
wenzelm@63404
  1008
  done
berghofe@12905
  1009
wenzelm@63404
  1010
lemma Image_subset_eq: "r``A \<subseteq> B \<longleftrightarrow> A \<subseteq> - ((r\<inverse>) `` (- B))"
haftmann@46752
  1011
  by blast
berghofe@12905
  1012
wenzelm@63404
  1013
lemma Image_Collect_case_prod [simp]: "{(x, y). P x y} `` A = {y. \<exists>x\<in>A. P x y}"
haftmann@46752
  1014
  by auto
berghofe@12905
  1015
hoelzl@54410
  1016
lemma Sigma_Image: "(SIGMA x:A. B x) `` X = (\<Union>x\<in>X \<inter> A. B x)"
hoelzl@54410
  1017
  by auto
hoelzl@54410
  1018
hoelzl@54410
  1019
lemma relcomp_Image: "(X O Y) `` Z = Y `` (X `` Z)"
hoelzl@54410
  1020
  by auto
berghofe@12905
  1021
haftmann@63376
  1022
wenzelm@60758
  1023
subsubsection \<open>Inverse image\<close>
berghofe@12905
  1024
haftmann@46752
  1025
definition inv_image :: "'b rel \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a rel"
wenzelm@63404
  1026
  where "inv_image r f = {(x, y). (f x, f y) \<in> r}"
haftmann@46692
  1027
haftmann@46752
  1028
definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
wenzelm@63404
  1029
  where "inv_imagep r f = (\<lambda>x y. r (f x) (f y))"
haftmann@46694
  1030
haftmann@46694
  1031
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
  1032
  by (simp add: inv_image_def inv_imagep_def)
haftmann@46694
  1033
wenzelm@63404
  1034
lemma sym_inv_image: "sym r \<Longrightarrow> sym (inv_image r f)"
wenzelm@63404
  1035
  unfolding sym_def inv_image_def by blast
huffman@19228
  1036
wenzelm@63404
  1037
lemma trans_inv_image: "trans r \<Longrightarrow> trans (inv_image r f)"
wenzelm@63404
  1038
  unfolding trans_def inv_image_def
berghofe@12905
  1039
  apply (simp (no_asm))
berghofe@12905
  1040
  apply blast
berghofe@12905
  1041
  done
berghofe@12905
  1042
wenzelm@63404
  1043
lemma in_inv_image[simp]: "(x, y) \<in> inv_image r f \<longleftrightarrow> (f x, f y) \<in> r"
krauss@32463
  1044
  by (auto simp:inv_image_def)
krauss@32463
  1045
wenzelm@63404
  1046
lemma converse_inv_image[simp]: "(inv_image R f)\<inverse> = inv_image (R\<inverse>) f"
haftmann@46752
  1047
  unfolding inv_image_def converse_unfold by auto
krauss@33218
  1048
haftmann@46664
  1049
lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
haftmann@46664
  1050
  by (simp add: inv_imagep_def)
haftmann@46664
  1051
haftmann@46664
  1052
wenzelm@60758
  1053
subsubsection \<open>Powerset\<close>
haftmann@46664
  1054
haftmann@46752
  1055
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
wenzelm@63404
  1056
  where "Powp A = (\<lambda>B. \<forall>x \<in> B. A x)"
haftmann@46664
  1057
haftmann@46664
  1058
lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"
haftmann@46664
  1059
  by (auto simp add: Powp_def fun_eq_iff)
haftmann@46664
  1060
haftmann@46664
  1061
lemmas Powp_mono [mono] = Pow_mono [to_pred]
haftmann@46664
  1062
haftmann@63376
  1063
wenzelm@60758
  1064
subsubsection \<open>Expressing relation operations via @{const Finite_Set.fold}\<close>
kuncar@48620
  1065
kuncar@48620
  1066
lemma Id_on_fold:
kuncar@48620
  1067
  assumes "finite A"
kuncar@48620
  1068
  shows "Id_on A = Finite_Set.fold (\<lambda>x. Set.insert (Pair x x)) {} A"
kuncar@48620
  1069
proof -
wenzelm@63404
  1070
  interpret comp_fun_commute "\<lambda>x. Set.insert (Pair x x)"
wenzelm@63404
  1071
    by standard auto
wenzelm@63404
  1072
  from assms show ?thesis
wenzelm@63404
  1073
    unfolding Id_on_def by (induct A) simp_all
kuncar@48620
  1074
qed
kuncar@48620
  1075
kuncar@48620
  1076
lemma comp_fun_commute_Image_fold:
kuncar@48620
  1077
  "comp_fun_commute (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)"
kuncar@48620
  1078
proof -
kuncar@48620
  1079
  interpret comp_fun_idem Set.insert
wenzelm@63404
  1080
    by (fact comp_fun_idem_insert)
wenzelm@63404
  1081
  show ?thesis
wenzelm@63404
  1082
    by standard (auto simp add: fun_eq_iff comp_fun_commute split: prod.split)
kuncar@48620
  1083
qed
kuncar@48620
  1084
kuncar@48620
  1085
lemma Image_fold:
kuncar@48620
  1086
  assumes "finite R"
kuncar@48620
  1087
  shows "R `` S = Finite_Set.fold (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A) {} R"
kuncar@48620
  1088
proof -
wenzelm@63404
  1089
  interpret comp_fun_commute "(\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)"
kuncar@48620
  1090
    by (rule comp_fun_commute_Image_fold)
kuncar@48620
  1091
  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
  1092
    by (force intro: rev_ImageI)
wenzelm@63404
  1093
  show ?thesis
wenzelm@63404
  1094
    using assms by (induct R) (auto simp: *)
kuncar@48620
  1095
qed
kuncar@48620
  1096
kuncar@48620
  1097
lemma insert_relcomp_union_fold:
kuncar@48620
  1098
  assumes "finite S"
kuncar@48620
  1099
  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
  1100
proof -
kuncar@48620
  1101
  interpret comp_fun_commute "\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A'"
wenzelm@63404
  1102
  proof -
wenzelm@63404
  1103
    interpret comp_fun_idem Set.insert
wenzelm@63404
  1104
      by (fact comp_fun_idem_insert)
kuncar@48620
  1105
    show "comp_fun_commute (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A')"
wenzelm@63404
  1106
      by standard (auto simp add: fun_eq_iff split: prod.split)
kuncar@48620
  1107
  qed
wenzelm@63404
  1108
  have *: "{x} O S = {(x', z). x' = fst x \<and> (snd x, z) \<in> S}"
wenzelm@63404
  1109
    by (auto simp: relcomp_unfold intro!: exI)
wenzelm@63404
  1110
  show ?thesis
wenzelm@63404
  1111
    unfolding * using \<open>finite S\<close> by (induct S) (auto split: prod.split)
kuncar@48620
  1112
qed
kuncar@48620
  1113
kuncar@48620
  1114
lemma insert_relcomp_fold:
kuncar@48620
  1115
  assumes "finite S"
wenzelm@63404
  1116
  shows "Set.insert x R O S =
kuncar@48620
  1117
    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
  1118
proof -
wenzelm@63404
  1119
  have "Set.insert x R O S = ({x} O S) \<union> (R O S)"
wenzelm@63404
  1120
    by auto
wenzelm@63404
  1121
  then show ?thesis
wenzelm@63404
  1122
    by (auto simp: insert_relcomp_union_fold [OF assms])
kuncar@48620
  1123
qed
kuncar@48620
  1124
kuncar@48620
  1125
lemma comp_fun_commute_relcomp_fold:
kuncar@48620
  1126
  assumes "finite S"
wenzelm@63404
  1127
  shows "comp_fun_commute (\<lambda>(x,y) A.
kuncar@48620
  1128
    Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S)"
kuncar@48620
  1129
proof -
wenzelm@63404
  1130
  have *: "\<And>a b A.
kuncar@48620
  1131
    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
  1132
    by (auto simp: insert_relcomp_union_fold[OF assms] cong: if_cong)
wenzelm@63404
  1133
  show ?thesis
wenzelm@63404
  1134
    by standard (auto simp: *)
kuncar@48620
  1135
qed
kuncar@48620
  1136
kuncar@48620
  1137
lemma relcomp_fold:
wenzelm@63404
  1138
  assumes "finite R" "finite S"
wenzelm@63404
  1139
  shows "R O S = Finite_Set.fold
kuncar@48620
  1140
    (\<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
  1141
  using assms
wenzelm@63404
  1142
  by (induct R)
traytel@52749
  1143
    (auto simp: comp_fun_commute.fold_insert comp_fun_commute_relcomp_fold insert_relcomp_fold
kuncar@48620
  1144
      cong: if_cong)
kuncar@48620
  1145
haftmann@63376
  1146
text \<open>Misc\<close>
haftmann@63376
  1147
haftmann@63376
  1148
abbreviation (input) transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
wenzelm@63404
  1149
  where "transP r \<equiv> trans {(x, y). r x y}"  (* FIXME drop *)
haftmann@63376
  1150
wenzelm@63404
  1151
abbreviation (input) "RangeP \<equiv> Rangep"
wenzelm@63404
  1152
abbreviation (input) "DomainP \<equiv> Domainp"
haftmann@63376
  1153
nipkow@1128
  1154
end