src/HOL/Relation.thy
author nipkow
Tue Sep 17 08:42:51 2013 +0200 (2013-09-17)
changeset 53680 c5096c22892b
parent 52749 ed416f4ac34e
child 54147 97a8ff4e4ac9
permissions -rw-r--r--
added lemmas and made concerse executable
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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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header {* Relations – as sets of pairs, and binary predicates *}
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theory Relation
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imports Datatype Finite_Set
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begin
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text {* A preliminary: classical rules for reasoning on predicates *}
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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 INF2_I [intro!]
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declare INF1_D [elim]
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declare INF2_D [elim]
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declare INF1_E [elim]
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declare INF2_E [elim]
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declare SUP1_I [intro]
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declare SUP2_I [intro]
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declare SUP1_E [elim!]
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declare SUP2_E [elim!]
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subsection {* Fundamental *}
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subsubsection {* Relations as sets of pairs *}
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type_synonym 'a rel = "('a * 'a) set"
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lemma subrelI: -- {* Version of @{thm [source] subsetI} for binary relations *}
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  "(\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s) \<Longrightarrow> r \<subseteq> s"
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  by auto
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lemma lfp_induct2: -- {* Version of @{thm [source] lfp_induct} for binary relations *}
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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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  using lfp_induct_set [of "(a, b)" f "prod_case P"] by auto
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subsubsection {* Conversions between set and predicate relations *}
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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 (prod_case ` 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 (prod_case ` 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 {* Properties of relations *}
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subsubsection {* Reflexivity *}
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definition refl_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
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where
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  "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"
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where -- {* reflexivity over a type *}
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  "refl \<equiv> refl_on UNIV"
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definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
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where
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  "reflp r \<longleftrightarrow> (\<forall>x. r x x)"
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lemma reflp_refl_eq [pred_set_conv]:
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  "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: "r \<subseteq> A \<times> A ==> (!!x. x : A ==> (x, x) : r) ==> refl_on A r"
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  by (unfold refl_on_def) (iprover intro!: ballI)
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lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r"
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  by (unfold refl_on_def) blast
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lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A"
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  by (unfold refl_on_def) blast
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lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A"
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  by (unfold refl_on_def) blast
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lemma reflpI:
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  "(\<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:
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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 ==> refl_on B s ==> refl_on (A \<inter> B) (r \<inter> s)"
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  by (unfold refl_on_def) blast
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lemma reflp_inf:
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  "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 ==> refl_on B s ==> refl_on (A \<union> B) (r \<union> s)"
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  by (unfold refl_on_def) blast
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lemma reflp_sup:
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  "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:
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  "ALL x:S. refl_on (A x) (r x) ==> refl_on (INTER S A) (INTER S r)"
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  by (unfold refl_on_def) fast
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lemma refl_on_UNION:
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  "ALL x:S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)"
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  by (unfold refl_on_def) 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_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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subsubsection {* Irreflexivity *}
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definition irrefl :: "'a rel \<Rightarrow> bool"
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where
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  "irrefl r \<longleftrightarrow> (\<forall>x. (x, x) \<notin> r)"
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lemma irrefl_distinct [code]:
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  "irrefl r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<noteq> y)"
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  by (auto simp add: irrefl_def)
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subsubsection {* Symmetry *}
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definition sym :: "'a rel \<Rightarrow> bool"
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where
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  "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
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  "symp r \<longleftrightarrow> (\<forall>x y. r x y \<longrightarrow> r y x)"
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lemma symp_sym_eq [pred_set_conv]:
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  "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:
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  "(\<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:
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  "(\<And>a b. r a b \<Longrightarrow> r b a) \<Longrightarrow> symp r"
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  by (fact symI [to_pred])
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lemma symE:
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  assumes "sym r" and "(b, a) \<in> r"
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  obtains "(a, b) \<in> r"
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  using assms by (simp add: sym_def)
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lemma sympE:
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  assumes "symp r" and "r b a"
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  obtains "r a b"
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  using assms by (rule symE [to_pred])
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lemma symD:
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  assumes "sym r" and "(b, a) \<in> r"
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  shows "(a, b) \<in> r"
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  using assms by (rule symE)
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lemma sympD:
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  assumes "symp r" and "r b a"
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  shows "r a b"
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  using assms by (rule symD [to_pred])
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lemma sym_Int:
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  "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<inter> s)"
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  by (fast intro: symI elim: symE)
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lemma symp_inf:
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  "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<sqinter> s)"
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  by (fact sym_Int [to_pred])
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lemma sym_Un:
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  "sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<union> s)"
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  by (fast intro: symI elim: symE)
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lemma symp_sup:
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  "symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<squnion> s)"
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  by (fact sym_Un [to_pred])
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lemma sym_INTER:
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  "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (INTER S r)"
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  by (fast intro: symI elim: symE)
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lemma symp_INF:
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  "\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (INFI S r)"
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  by (fact sym_INTER [to_pred])
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lemma sym_UNION:
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  "\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (UNION S r)"
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  by (fast intro: symI elim: symE)
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lemma symp_SUP:
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  "\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (SUPR S r)"
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  by (fact sym_UNION [to_pred])
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   296
subsubsection {* Antisymmetry *}
haftmann@46694
   297
haftmann@46752
   298
definition antisym :: "'a rel \<Rightarrow> bool"
haftmann@46752
   299
where
haftmann@46752
   300
  "antisym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)"
haftmann@46752
   301
haftmann@46752
   302
abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
haftmann@46752
   303
where
haftmann@46752
   304
  "antisymP r \<equiv> antisym {(x, y). r x y}"
haftmann@46694
   305
haftmann@46694
   306
lemma antisymI:
haftmann@46694
   307
  "(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r"
haftmann@46752
   308
  by (unfold antisym_def) iprover
haftmann@46694
   309
haftmann@46694
   310
lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b"
haftmann@46752
   311
  by (unfold antisym_def) iprover
haftmann@46694
   312
haftmann@46694
   313
lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r"
haftmann@46752
   314
  by (unfold antisym_def) blast
haftmann@46694
   315
haftmann@46694
   316
lemma antisym_empty [simp]: "antisym {}"
haftmann@46752
   317
  by (unfold antisym_def) blast
haftmann@46694
   318
haftmann@46694
   319
haftmann@46692
   320
subsubsection {* Transitivity *}
haftmann@46692
   321
haftmann@46752
   322
definition trans :: "'a rel \<Rightarrow> bool"
haftmann@46752
   323
where
haftmann@46752
   324
  "trans r \<longleftrightarrow> (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
haftmann@46752
   325
haftmann@46752
   326
definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
haftmann@46752
   327
where
haftmann@46752
   328
  "transp r \<longleftrightarrow> (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
haftmann@46752
   329
haftmann@46752
   330
lemma transp_trans_eq [pred_set_conv]:
haftmann@46752
   331
  "transp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> trans r" 
haftmann@46752
   332
  by (simp add: trans_def transp_def)
haftmann@46752
   333
haftmann@46752
   334
abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
haftmann@46752
   335
where -- {* FIXME drop *}
haftmann@46752
   336
  "transP r \<equiv> trans {(x, y). r x y}"
paulson@5978
   337
haftmann@46692
   338
lemma transI:
haftmann@46752
   339
  "(\<And>x y z. (x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r) \<Longrightarrow> trans r"
haftmann@46752
   340
  by (unfold trans_def) iprover
haftmann@46694
   341
haftmann@46694
   342
lemma transpI:
haftmann@46694
   343
  "(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r"
haftmann@46752
   344
  by (fact transI [to_pred])
haftmann@46752
   345
haftmann@46752
   346
lemma transE:
haftmann@46752
   347
  assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
haftmann@46752
   348
  obtains "(x, z) \<in> r"
haftmann@46752
   349
  using assms by (unfold trans_def) iprover
haftmann@46752
   350
haftmann@46694
   351
lemma transpE:
haftmann@46694
   352
  assumes "transp r" and "r x y" and "r y z"
haftmann@46694
   353
  obtains "r x z"
haftmann@46752
   354
  using assms by (rule transE [to_pred])
haftmann@46752
   355
haftmann@46752
   356
lemma transD:
haftmann@46752
   357
  assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r"
haftmann@46752
   358
  shows "(x, z) \<in> r"
haftmann@46752
   359
  using assms by (rule transE)
haftmann@46752
   360
haftmann@46752
   361
lemma transpD:
haftmann@46752
   362
  assumes "transp r" and "r x y" and "r y z"
haftmann@46752
   363
  shows "r x z"
haftmann@46752
   364
  using assms by (rule transD [to_pred])
haftmann@46694
   365
haftmann@46752
   366
lemma trans_Int:
haftmann@46752
   367
  "trans r \<Longrightarrow> trans s \<Longrightarrow> trans (r \<inter> s)"
haftmann@46752
   368
  by (fast intro: transI elim: transE)
haftmann@46692
   369
haftmann@46752
   370
lemma transp_inf:
haftmann@46752
   371
  "transp r \<Longrightarrow> transp s \<Longrightarrow> transp (r \<sqinter> s)"
haftmann@46752
   372
  by (fact trans_Int [to_pred])
haftmann@46752
   373
haftmann@46752
   374
lemma trans_INTER:
haftmann@46752
   375
  "\<forall>x\<in>S. trans (r x) \<Longrightarrow> trans (INTER S r)"
haftmann@46752
   376
  by (fast intro: transI elim: transD)
haftmann@46752
   377
haftmann@46752
   378
(* FIXME thm trans_INTER [to_pred] *)
haftmann@46692
   379
haftmann@46694
   380
lemma trans_join [code]:
haftmann@46694
   381
  "trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)"
haftmann@46694
   382
  by (auto simp add: trans_def)
haftmann@46692
   383
haftmann@46752
   384
lemma transp_trans:
haftmann@46752
   385
  "transp r \<longleftrightarrow> trans {(x, y). r x y}"
haftmann@46752
   386
  by (simp add: trans_def transp_def)
haftmann@46752
   387
haftmann@46692
   388
haftmann@46692
   389
subsubsection {* Totality *}
haftmann@46692
   390
haftmann@46752
   391
definition total_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool"
haftmann@46752
   392
where
haftmann@46752
   393
  "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
   394
nipkow@29859
   395
abbreviation "total \<equiv> total_on UNIV"
nipkow@29859
   396
haftmann@46752
   397
lemma total_on_empty [simp]: "total_on {} r"
haftmann@46752
   398
  by (simp add: total_on_def)
haftmann@46692
   399
haftmann@46692
   400
haftmann@46692
   401
subsubsection {* Single valued relations *}
haftmann@46692
   402
haftmann@46752
   403
definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool"
haftmann@46752
   404
where
haftmann@46752
   405
  "single_valued r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
haftmann@46692
   406
haftmann@46694
   407
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where
haftmann@46694
   408
  "single_valuedP r \<equiv> single_valued {(x, y). r x y}"
haftmann@46694
   409
haftmann@46752
   410
lemma single_valuedI:
haftmann@46752
   411
  "ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r"
haftmann@46752
   412
  by (unfold single_valued_def)
haftmann@46752
   413
haftmann@46752
   414
lemma single_valuedD:
haftmann@46752
   415
  "single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z"
haftmann@46752
   416
  by (simp add: single_valued_def)
haftmann@46752
   417
nipkow@52392
   418
lemma simgle_valued_empty[simp]: "single_valued {}"
nipkow@52392
   419
by(simp add: single_valued_def)
nipkow@52392
   420
haftmann@46692
   421
lemma single_valued_subset:
haftmann@46692
   422
  "r \<subseteq> s ==> single_valued s ==> single_valued r"
haftmann@46752
   423
  by (unfold single_valued_def) blast
oheimb@11136
   424
berghofe@12905
   425
haftmann@46694
   426
subsection {* Relation operations *}
haftmann@46694
   427
haftmann@46664
   428
subsubsection {* The identity relation *}
berghofe@12905
   429
haftmann@46752
   430
definition Id :: "'a rel"
haftmann@46752
   431
where
bulwahn@48253
   432
  [code del]: "Id = {p. \<exists>x. p = (x, x)}"
haftmann@46692
   433
berghofe@12905
   434
lemma IdI [intro]: "(a, a) : Id"
haftmann@46752
   435
  by (simp add: Id_def)
berghofe@12905
   436
berghofe@12905
   437
lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P"
haftmann@46752
   438
  by (unfold Id_def) (iprover elim: CollectE)
berghofe@12905
   439
berghofe@12905
   440
lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)"
haftmann@46752
   441
  by (unfold Id_def) blast
berghofe@12905
   442
nipkow@30198
   443
lemma refl_Id: "refl Id"
haftmann@46752
   444
  by (simp add: refl_on_def)
berghofe@12905
   445
berghofe@12905
   446
lemma antisym_Id: "antisym Id"
berghofe@12905
   447
  -- {* A strange result, since @{text Id} is also symmetric. *}
haftmann@46752
   448
  by (simp add: antisym_def)
berghofe@12905
   449
huffman@19228
   450
lemma sym_Id: "sym Id"
haftmann@46752
   451
  by (simp add: sym_def)
huffman@19228
   452
berghofe@12905
   453
lemma trans_Id: "trans Id"
haftmann@46752
   454
  by (simp add: trans_def)
berghofe@12905
   455
haftmann@46692
   456
lemma single_valued_Id [simp]: "single_valued Id"
haftmann@46692
   457
  by (unfold single_valued_def) blast
haftmann@46692
   458
haftmann@46692
   459
lemma irrefl_diff_Id [simp]: "irrefl (r - Id)"
haftmann@46692
   460
  by (simp add:irrefl_def)
haftmann@46692
   461
haftmann@46692
   462
lemma trans_diff_Id: "trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r - Id)"
haftmann@46692
   463
  unfolding antisym_def trans_def by blast
haftmann@46692
   464
haftmann@46692
   465
lemma total_on_diff_Id [simp]: "total_on A (r - Id) = total_on A r"
haftmann@46692
   466
  by (simp add: total_on_def)
haftmann@46692
   467
berghofe@12905
   468
haftmann@46664
   469
subsubsection {* Diagonal: identity over a set *}
berghofe@12905
   470
haftmann@46752
   471
definition Id_on  :: "'a set \<Rightarrow> 'a rel"
haftmann@46752
   472
where
haftmann@46752
   473
  "Id_on A = (\<Union>x\<in>A. {(x, x)})"
haftmann@46692
   474
nipkow@30198
   475
lemma Id_on_empty [simp]: "Id_on {} = {}"
haftmann@46752
   476
  by (simp add: Id_on_def) 
paulson@13812
   477
nipkow@30198
   478
lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A"
haftmann@46752
   479
  by (simp add: Id_on_def)
berghofe@12905
   480
blanchet@35828
   481
lemma Id_onI [intro!,no_atp]: "a : A ==> (a, a) : Id_on A"
haftmann@46752
   482
  by (rule Id_on_eqI) (rule refl)
berghofe@12905
   483
nipkow@30198
   484
lemma Id_onE [elim!]:
nipkow@30198
   485
  "c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P"
wenzelm@12913
   486
  -- {* The general elimination rule. *}
haftmann@46752
   487
  by (unfold Id_on_def) (iprover elim!: UN_E singletonE)
berghofe@12905
   488
nipkow@30198
   489
lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)"
haftmann@46752
   490
  by blast
berghofe@12905
   491
haftmann@45967
   492
lemma Id_on_def' [nitpick_unfold]:
haftmann@44278
   493
  "Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)"
haftmann@46752
   494
  by auto
bulwahn@40923
   495
nipkow@30198
   496
lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A"
haftmann@46752
   497
  by blast
berghofe@12905
   498
haftmann@46692
   499
lemma refl_on_Id_on: "refl_on A (Id_on A)"
haftmann@46752
   500
  by (rule refl_onI [OF Id_on_subset_Times Id_onI])
haftmann@46692
   501
haftmann@46692
   502
lemma antisym_Id_on [simp]: "antisym (Id_on A)"
haftmann@46752
   503
  by (unfold antisym_def) blast
haftmann@46692
   504
haftmann@46692
   505
lemma sym_Id_on [simp]: "sym (Id_on A)"
haftmann@46752
   506
  by (rule symI) clarify
haftmann@46692
   507
haftmann@46692
   508
lemma trans_Id_on [simp]: "trans (Id_on A)"
haftmann@46752
   509
  by (fast intro: transI elim: transD)
haftmann@46692
   510
haftmann@46692
   511
lemma single_valued_Id_on [simp]: "single_valued (Id_on A)"
haftmann@46692
   512
  by (unfold single_valued_def) blast
haftmann@46692
   513
berghofe@12905
   514
haftmann@46694
   515
subsubsection {* Composition *}
berghofe@12905
   516
griff@47433
   517
inductive_set relcomp  :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a \<times> 'c) set" (infixr "O" 75)
haftmann@46752
   518
  for r :: "('a \<times> 'b) set" and s :: "('b \<times> 'c) set"
haftmann@46694
   519
where
griff@47433
   520
  relcompI [intro]: "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> s \<Longrightarrow> (a, c) \<in> r O s"
haftmann@46692
   521
griff@47434
   522
notation relcompp (infixr "OO" 75)
berghofe@12905
   523
griff@47434
   524
lemmas relcomppI = relcompp.intros
berghofe@12905
   525
haftmann@46752
   526
text {*
haftmann@46752
   527
  For historic reasons, the elimination rules are not wholly corresponding.
haftmann@46752
   528
  Feel free to consolidate this.
haftmann@46752
   529
*}
haftmann@46694
   530
griff@47433
   531
inductive_cases relcompEpair: "(a, c) \<in> r O s"
griff@47434
   532
inductive_cases relcomppE [elim!]: "(r OO s) a c"
haftmann@46694
   533
griff@47433
   534
lemma relcompE [elim!]: "xz \<in> r O s \<Longrightarrow>
haftmann@46752
   535
  (\<And>x y z. xz = (x, z) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (y, z) \<in> s  \<Longrightarrow> P) \<Longrightarrow> P"
griff@47433
   536
  by (cases xz) (simp, erule relcompEpair, iprover)
haftmann@46752
   537
haftmann@46752
   538
lemma R_O_Id [simp]:
haftmann@46752
   539
  "R O Id = R"
haftmann@46752
   540
  by fast
haftmann@46694
   541
haftmann@46752
   542
lemma Id_O_R [simp]:
haftmann@46752
   543
  "Id O R = R"
haftmann@46752
   544
  by fast
haftmann@46752
   545
griff@47433
   546
lemma relcomp_empty1 [simp]:
haftmann@46752
   547
  "{} O R = {}"
haftmann@46752
   548
  by blast
berghofe@12905
   549
griff@47434
   550
lemma relcompp_bot1 [simp]:
noschinl@46883
   551
  "\<bottom> OO R = \<bottom>"
griff@47433
   552
  by (fact relcomp_empty1 [to_pred])
berghofe@12905
   553
griff@47433
   554
lemma relcomp_empty2 [simp]:
haftmann@46752
   555
  "R O {} = {}"
haftmann@46752
   556
  by blast
berghofe@12905
   557
griff@47434
   558
lemma relcompp_bot2 [simp]:
noschinl@46883
   559
  "R OO \<bottom> = \<bottom>"
griff@47433
   560
  by (fact relcomp_empty2 [to_pred])
krauss@23185
   561
haftmann@46752
   562
lemma O_assoc:
haftmann@46752
   563
  "(R O S) O T = R O (S O T)"
haftmann@46752
   564
  by blast
haftmann@46752
   565
noschinl@46883
   566
griff@47434
   567
lemma relcompp_assoc:
haftmann@46752
   568
  "(r OO s) OO t = r OO (s OO t)"
haftmann@46752
   569
  by (fact O_assoc [to_pred])
krauss@23185
   570
haftmann@46752
   571
lemma trans_O_subset:
haftmann@46752
   572
  "trans r \<Longrightarrow> r O r \<subseteq> r"
haftmann@46752
   573
  by (unfold trans_def) blast
haftmann@46752
   574
griff@47434
   575
lemma transp_relcompp_less_eq:
haftmann@46752
   576
  "transp r \<Longrightarrow> r OO r \<le> r "
haftmann@46752
   577
  by (fact trans_O_subset [to_pred])
berghofe@12905
   578
griff@47433
   579
lemma relcomp_mono:
haftmann@46752
   580
  "r' \<subseteq> r \<Longrightarrow> s' \<subseteq> s \<Longrightarrow> r' O s' \<subseteq> r O s"
haftmann@46752
   581
  by blast
berghofe@12905
   582
griff@47434
   583
lemma relcompp_mono:
haftmann@46752
   584
  "r' \<le> r \<Longrightarrow> s' \<le> s \<Longrightarrow> r' OO s' \<le> r OO s "
griff@47433
   585
  by (fact relcomp_mono [to_pred])
berghofe@12905
   586
griff@47433
   587
lemma relcomp_subset_Sigma:
haftmann@46752
   588
  "r \<subseteq> A \<times> B \<Longrightarrow> s \<subseteq> B \<times> C \<Longrightarrow> r O s \<subseteq> A \<times> C"
haftmann@46752
   589
  by blast
haftmann@46752
   590
griff@47433
   591
lemma relcomp_distrib [simp]:
haftmann@46752
   592
  "R O (S \<union> T) = (R O S) \<union> (R O T)" 
haftmann@46752
   593
  by auto
berghofe@12905
   594
griff@47434
   595
lemma relcompp_distrib [simp]:
haftmann@46752
   596
  "R OO (S \<squnion> T) = R OO S \<squnion> R OO T"
griff@47433
   597
  by (fact relcomp_distrib [to_pred])
haftmann@46752
   598
griff@47433
   599
lemma relcomp_distrib2 [simp]:
haftmann@46752
   600
  "(S \<union> T) O R = (S O R) \<union> (T O R)"
haftmann@46752
   601
  by auto
krauss@28008
   602
griff@47434
   603
lemma relcompp_distrib2 [simp]:
haftmann@46752
   604
  "(S \<squnion> T) OO R = S OO R \<squnion> T OO R"
griff@47433
   605
  by (fact relcomp_distrib2 [to_pred])
haftmann@46752
   606
griff@47433
   607
lemma relcomp_UNION_distrib:
haftmann@46752
   608
  "s O UNION I r = (\<Union>i\<in>I. s O r i) "
haftmann@46752
   609
  by auto
krauss@28008
   610
griff@47433
   611
(* FIXME thm relcomp_UNION_distrib [to_pred] *)
krauss@36772
   612
griff@47433
   613
lemma relcomp_UNION_distrib2:
haftmann@46752
   614
  "UNION I r O s = (\<Union>i\<in>I. r i O s) "
haftmann@46752
   615
  by auto
haftmann@46752
   616
griff@47433
   617
(* FIXME thm relcomp_UNION_distrib2 [to_pred] *)
krauss@36772
   618
griff@47433
   619
lemma single_valued_relcomp:
haftmann@46752
   620
  "single_valued r \<Longrightarrow> single_valued s \<Longrightarrow> single_valued (r O s)"
haftmann@46752
   621
  by (unfold single_valued_def) blast
haftmann@46752
   622
griff@47433
   623
lemma relcomp_unfold:
haftmann@46752
   624
  "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
haftmann@46752
   625
  by (auto simp add: set_eq_iff)
berghofe@12905
   626
haftmann@46664
   627
haftmann@46664
   628
subsubsection {* Converse *}
wenzelm@12913
   629
haftmann@46752
   630
inductive_set converse :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'a) set" ("(_^-1)" [1000] 999)
haftmann@46752
   631
  for r :: "('a \<times> 'b) set"
haftmann@46752
   632
where
haftmann@46752
   633
  "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r^-1"
haftmann@46692
   634
haftmann@46692
   635
notation (xsymbols)
haftmann@46692
   636
  converse  ("(_\<inverse>)" [1000] 999)
haftmann@46692
   637
haftmann@46752
   638
notation
haftmann@46752
   639
  conversep ("(_^--1)" [1000] 1000)
haftmann@46694
   640
haftmann@46694
   641
notation (xsymbols)
haftmann@46694
   642
  conversep  ("(_\<inverse>\<inverse>)" [1000] 1000)
haftmann@46694
   643
haftmann@46752
   644
lemma converseI [sym]:
haftmann@46752
   645
  "(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>"
haftmann@46752
   646
  by (fact converse.intros)
haftmann@46752
   647
haftmann@46752
   648
lemma conversepI (* CANDIDATE [sym] *):
haftmann@46752
   649
  "r a b \<Longrightarrow> r\<inverse>\<inverse> b a"
haftmann@46752
   650
  by (fact conversep.intros)
haftmann@46752
   651
haftmann@46752
   652
lemma converseD [sym]:
haftmann@46752
   653
  "(a, b) \<in> r\<inverse> \<Longrightarrow> (b, a) \<in> r"
haftmann@46752
   654
  by (erule converse.cases) iprover
haftmann@46752
   655
haftmann@46752
   656
lemma conversepD (* CANDIDATE [sym] *):
haftmann@46752
   657
  "r\<inverse>\<inverse> b a \<Longrightarrow> r a b"
haftmann@46752
   658
  by (fact converseD [to_pred])
haftmann@46752
   659
haftmann@46752
   660
lemma converseE [elim!]:
haftmann@46752
   661
  -- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *}
haftmann@46752
   662
  "yx \<in> r\<inverse> \<Longrightarrow> (\<And>x y. yx = (y, x) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@46752
   663
  by (cases yx) (simp, erule converse.cases, iprover)
haftmann@46694
   664
noschinl@46882
   665
lemmas conversepE [elim!] = conversep.cases
haftmann@46752
   666
haftmann@46752
   667
lemma converse_iff [iff]:
haftmann@46752
   668
  "(a, b) \<in> r\<inverse> \<longleftrightarrow> (b, a) \<in> r"
haftmann@46752
   669
  by (auto intro: converseI)
haftmann@46752
   670
haftmann@46752
   671
lemma conversep_iff [iff]:
haftmann@46752
   672
  "r\<inverse>\<inverse> a b = r b a"
haftmann@46752
   673
  by (fact converse_iff [to_pred])
haftmann@46694
   674
haftmann@46752
   675
lemma converse_converse [simp]:
haftmann@46752
   676
  "(r\<inverse>)\<inverse> = r"
haftmann@46752
   677
  by (simp add: set_eq_iff)
haftmann@46694
   678
haftmann@46752
   679
lemma conversep_conversep [simp]:
haftmann@46752
   680
  "(r\<inverse>\<inverse>)\<inverse>\<inverse> = r"
haftmann@46752
   681
  by (fact converse_converse [to_pred])
haftmann@46752
   682
nipkow@53680
   683
lemma converse_empty[simp]: "{}\<inverse> = {}"
nipkow@53680
   684
by auto
nipkow@53680
   685
nipkow@53680
   686
lemma converse_UNIV[simp]: "UNIV\<inverse> = UNIV"
nipkow@53680
   687
by auto
nipkow@53680
   688
griff@47433
   689
lemma converse_relcomp: "(r O s)^-1 = s^-1 O r^-1"
haftmann@46752
   690
  by blast
haftmann@46694
   691
griff@47434
   692
lemma converse_relcompp: "(r OO s)^--1 = s^--1 OO r^--1"
griff@47434
   693
  by (iprover intro: order_antisym conversepI relcomppI
griff@47434
   694
    elim: relcomppE dest: conversepD)
haftmann@46694
   695
haftmann@46752
   696
lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1"
haftmann@46752
   697
  by blast
haftmann@46752
   698
haftmann@46694
   699
lemma converse_meet: "(r \<sqinter> s)^--1 = r^--1 \<sqinter> s^--1"
haftmann@46694
   700
  by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD)
haftmann@46694
   701
haftmann@46752
   702
lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1"
haftmann@46752
   703
  by blast
haftmann@46752
   704
haftmann@46694
   705
lemma converse_join: "(r \<squnion> s)^--1 = r^--1 \<squnion> s^--1"
haftmann@46694
   706
  by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD)
haftmann@46694
   707
huffman@19228
   708
lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)"
haftmann@46752
   709
  by fast
huffman@19228
   710
huffman@19228
   711
lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)"
haftmann@46752
   712
  by blast
huffman@19228
   713
traytel@52749
   714
lemma converse_mono[simp]: "r^-1 \<subseteq> s ^-1 \<longleftrightarrow> r \<subseteq> s"
traytel@52749
   715
  by auto
traytel@52749
   716
traytel@52749
   717
lemma conversep_mono[simp]: "r^--1 \<le> s ^--1 \<longleftrightarrow> r \<le> s"
traytel@52749
   718
  by (fact converse_mono[to_pred])
traytel@52749
   719
traytel@52749
   720
lemma converse_inject[simp]: "r^-1 = s ^-1 \<longleftrightarrow> r = s"
traytel@52730
   721
  by auto
traytel@52730
   722
traytel@52749
   723
lemma conversep_inject[simp]: "r^--1 = s ^--1 \<longleftrightarrow> r = s"
traytel@52749
   724
  by (fact converse_inject[to_pred])
traytel@52749
   725
traytel@52749
   726
lemma converse_subset_swap: "r \<subseteq> s ^-1 = (r ^-1 \<subseteq> s)"
traytel@52749
   727
  by auto
traytel@52749
   728
traytel@52749
   729
lemma conversep_le_swap: "r \<le> s ^--1 = (r ^--1 \<le> s)"
traytel@52749
   730
  by (fact converse_subset_swap[to_pred])
traytel@52730
   731
berghofe@12905
   732
lemma converse_Id [simp]: "Id^-1 = Id"
haftmann@46752
   733
  by blast
berghofe@12905
   734
nipkow@30198
   735
lemma converse_Id_on [simp]: "(Id_on A)^-1 = Id_on A"
haftmann@46752
   736
  by blast
berghofe@12905
   737
nipkow@30198
   738
lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r"
haftmann@46752
   739
  by (unfold refl_on_def) auto
berghofe@12905
   740
huffman@19228
   741
lemma sym_converse [simp]: "sym (converse r) = sym r"
haftmann@46752
   742
  by (unfold sym_def) blast
huffman@19228
   743
huffman@19228
   744
lemma antisym_converse [simp]: "antisym (converse r) = antisym r"
haftmann@46752
   745
  by (unfold antisym_def) blast
berghofe@12905
   746
huffman@19228
   747
lemma trans_converse [simp]: "trans (converse r) = trans r"
haftmann@46752
   748
  by (unfold trans_def) blast
berghofe@12905
   749
huffman@19228
   750
lemma sym_conv_converse_eq: "sym r = (r^-1 = r)"
haftmann@46752
   751
  by (unfold sym_def) fast
huffman@19228
   752
huffman@19228
   753
lemma sym_Un_converse: "sym (r \<union> r^-1)"
haftmann@46752
   754
  by (unfold sym_def) blast
huffman@19228
   755
huffman@19228
   756
lemma sym_Int_converse: "sym (r \<inter> r^-1)"
haftmann@46752
   757
  by (unfold sym_def) blast
huffman@19228
   758
haftmann@46752
   759
lemma total_on_converse [simp]: "total_on A (r^-1) = total_on A r"
haftmann@46752
   760
  by (auto simp: total_on_def)
nipkow@29859
   761
traytel@52749
   762
lemma finite_converse [iff]: "finite (r^-1) = finite r"  
traytel@52749
   763
  unfolding converse_def conversep_iff by (auto elim: finite_imageD simp: inj_on_def)
wenzelm@12913
   764
haftmann@46752
   765
lemma conversep_noteq [simp]: "(op \<noteq>)^--1 = op \<noteq>"
haftmann@46752
   766
  by (auto simp add: fun_eq_iff)
haftmann@46752
   767
haftmann@46752
   768
lemma conversep_eq [simp]: "(op =)^--1 = op ="
haftmann@46752
   769
  by (auto simp add: fun_eq_iff)
haftmann@46752
   770
nipkow@53680
   771
lemma converse_unfold [code]:
haftmann@46752
   772
  "r\<inverse> = {(y, x). (x, y) \<in> r}"
haftmann@46752
   773
  by (simp add: set_eq_iff)
haftmann@46752
   774
haftmann@46692
   775
haftmann@46692
   776
subsubsection {* Domain, range and field *}
haftmann@46692
   777
haftmann@46767
   778
inductive_set Domain :: "('a \<times> 'b) set \<Rightarrow> 'a set"
haftmann@46767
   779
  for r :: "('a \<times> 'b) set"
haftmann@46752
   780
where
haftmann@46767
   781
  DomainI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> Domain r"
haftmann@46767
   782
haftmann@46767
   783
abbreviation (input) "DomainP \<equiv> Domainp"
haftmann@46767
   784
haftmann@46767
   785
lemmas DomainPI = Domainp.DomainI
haftmann@46767
   786
haftmann@46767
   787
inductive_cases DomainE [elim!]: "a \<in> Domain r"
haftmann@46767
   788
inductive_cases DomainpE [elim!]: "Domainp r a"
haftmann@46692
   789
haftmann@46767
   790
inductive_set Range :: "('a \<times> 'b) set \<Rightarrow> 'b set"
haftmann@46767
   791
  for r :: "('a \<times> 'b) set"
haftmann@46752
   792
where
haftmann@46767
   793
  RangeI [intro]: "(a, b) \<in> r \<Longrightarrow> b \<in> Range r"
haftmann@46767
   794
haftmann@46767
   795
abbreviation (input) "RangeP \<equiv> Rangep"
haftmann@46767
   796
haftmann@46767
   797
lemmas RangePI = Rangep.RangeI
haftmann@46767
   798
haftmann@46767
   799
inductive_cases RangeE [elim!]: "b \<in> Range r"
haftmann@46767
   800
inductive_cases RangepE [elim!]: "Rangep r b"
haftmann@46692
   801
haftmann@46752
   802
definition Field :: "'a rel \<Rightarrow> 'a set"
haftmann@46752
   803
where
haftmann@46692
   804
  "Field r = Domain r \<union> Range r"
berghofe@12905
   805
haftmann@46694
   806
lemma Domain_fst [code]:
haftmann@46694
   807
  "Domain r = fst ` r"
haftmann@46767
   808
  by force
haftmann@46767
   809
haftmann@46767
   810
lemma Range_snd [code]:
haftmann@46767
   811
  "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
haftmann@46694
   820
lemma Domain_empty [simp]: "Domain {} = {}"
haftmann@46767
   821
  by auto
haftmann@46767
   822
haftmann@46767
   823
lemma Range_empty [simp]: "Range {} = {}"
haftmann@46767
   824
  by auto
haftmann@46767
   825
haftmann@46767
   826
lemma Field_empty [simp]: "Field {} = {}"
haftmann@46767
   827
  by (simp add: Field_def)
haftmann@46694
   828
haftmann@46694
   829
lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}"
haftmann@46694
   830
  by auto
haftmann@46694
   831
haftmann@46767
   832
lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}"
haftmann@46767
   833
  by auto
haftmann@46767
   834
noschinl@46882
   835
lemma Domain_insert [simp]: "Domain (insert (a, b) r) = insert a (Domain r)"
haftmann@46767
   836
  by blast
haftmann@46767
   837
noschinl@46882
   838
lemma Range_insert [simp]: "Range (insert (a, b) r) = insert b (Range r)"
haftmann@46767
   839
  by blast
haftmann@46767
   840
haftmann@46767
   841
lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \<union> Field r"
noschinl@46884
   842
  by (auto simp add: Field_def)
haftmann@46767
   843
haftmann@46767
   844
lemma Domain_iff: "a \<in> Domain r \<longleftrightarrow> (\<exists>y. (a, y) \<in> r)"
haftmann@46767
   845
  by blast
haftmann@46767
   846
haftmann@46767
   847
lemma Range_iff: "a \<in> Range r \<longleftrightarrow> (\<exists>y. (y, a) \<in> r)"
haftmann@46694
   848
  by blast
haftmann@46694
   849
haftmann@46694
   850
lemma Domain_Id [simp]: "Domain Id = UNIV"
haftmann@46694
   851
  by blast
haftmann@46694
   852
haftmann@46767
   853
lemma Range_Id [simp]: "Range Id = UNIV"
haftmann@46767
   854
  by blast
haftmann@46767
   855
haftmann@46694
   856
lemma Domain_Id_on [simp]: "Domain (Id_on A) = A"
haftmann@46694
   857
  by blast
haftmann@46694
   858
haftmann@46767
   859
lemma Range_Id_on [simp]: "Range (Id_on A) = A"
haftmann@46767
   860
  by blast
haftmann@46767
   861
haftmann@46767
   862
lemma Domain_Un_eq: "Domain (A \<union> B) = Domain A \<union> Domain B"
haftmann@46694
   863
  by blast
haftmann@46694
   864
haftmann@46767
   865
lemma Range_Un_eq: "Range (A \<union> B) = Range A \<union> Range B"
haftmann@46767
   866
  by blast
haftmann@46767
   867
haftmann@46767
   868
lemma Field_Un [simp]: "Field (r \<union> s) = Field r \<union> Field s"
haftmann@46767
   869
  by (auto simp: Field_def)
haftmann@46767
   870
haftmann@46767
   871
lemma Domain_Int_subset: "Domain (A \<inter> B) \<subseteq> Domain A \<inter> Domain B"
haftmann@46694
   872
  by blast
haftmann@46694
   873
haftmann@46767
   874
lemma Range_Int_subset: "Range (A \<inter> B) \<subseteq> Range A \<inter> Range B"
haftmann@46767
   875
  by blast
haftmann@46767
   876
haftmann@46767
   877
lemma Domain_Diff_subset: "Domain A - Domain B \<subseteq> Domain (A - B)"
haftmann@46767
   878
  by blast
haftmann@46767
   879
haftmann@46767
   880
lemma Range_Diff_subset: "Range A - Range B \<subseteq> Range (A - B)"
haftmann@46694
   881
  by blast
haftmann@46694
   882
haftmann@46767
   883
lemma Domain_Union: "Domain (\<Union>S) = (\<Union>A\<in>S. Domain A)"
haftmann@46694
   884
  by blast
haftmann@46694
   885
haftmann@46767
   886
lemma Range_Union: "Range (\<Union>S) = (\<Union>A\<in>S. Range A)"
haftmann@46767
   887
  by blast
haftmann@46767
   888
haftmann@46767
   889
lemma Field_Union [simp]: "Field (\<Union>R) = \<Union>(Field ` R)"
haftmann@46767
   890
  by (auto simp: Field_def)
haftmann@46767
   891
haftmann@46752
   892
lemma Domain_converse [simp]: "Domain (r\<inverse>) = Range r"
haftmann@46752
   893
  by auto
haftmann@46694
   894
haftmann@46767
   895
lemma Range_converse [simp]: "Range (r\<inverse>) = Domain r"
haftmann@46694
   896
  by blast
haftmann@46694
   897
haftmann@46767
   898
lemma Field_converse [simp]: "Field (r\<inverse>) = Field r"
haftmann@46767
   899
  by (auto simp: Field_def)
haftmann@46767
   900
haftmann@46767
   901
lemma Domain_Collect_split [simp]: "Domain {(x, y). P x y} = {x. EX y. P x y}"
haftmann@46767
   902
  by auto
haftmann@46767
   903
haftmann@46767
   904
lemma Range_Collect_split [simp]: "Range {(x, y). P x y} = {y. EX x. P x y}"
haftmann@46767
   905
  by auto
haftmann@46767
   906
haftmann@46767
   907
lemma finite_Domain: "finite r \<Longrightarrow> finite (Domain r)"
noschinl@46884
   908
  by (induct set: finite) auto
haftmann@46767
   909
haftmann@46767
   910
lemma finite_Range: "finite r \<Longrightarrow> finite (Range r)"
noschinl@46884
   911
  by (induct set: finite) auto
haftmann@46767
   912
haftmann@46767
   913
lemma finite_Field: "finite r \<Longrightarrow> finite (Field r)"
haftmann@46767
   914
  by (simp add: Field_def finite_Domain finite_Range)
haftmann@46767
   915
haftmann@46767
   916
lemma Domain_mono: "r \<subseteq> s \<Longrightarrow> Domain r \<subseteq> Domain s"
haftmann@46767
   917
  by blast
haftmann@46767
   918
haftmann@46767
   919
lemma Range_mono: "r \<subseteq> s \<Longrightarrow> Range r \<subseteq> Range s"
haftmann@46767
   920
  by blast
haftmann@46767
   921
haftmann@46767
   922
lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s"
haftmann@46767
   923
  by (auto simp: Field_def Domain_def Range_def)
haftmann@46767
   924
haftmann@46767
   925
lemma Domain_unfold:
haftmann@46767
   926
  "Domain r = {x. \<exists>y. (x, y) \<in> r}"
haftmann@46767
   927
  by blast
haftmann@46694
   928
haftmann@46694
   929
lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)"
haftmann@46694
   930
  by auto
haftmann@46694
   931
haftmann@46694
   932
lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)"
haftmann@46694
   933
  by auto
haftmann@46694
   934
berghofe@12905
   935
haftmann@46664
   936
subsubsection {* Image of a set under a relation *}
berghofe@12905
   937
nipkow@50420
   938
definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "``" 90)
haftmann@46752
   939
where
haftmann@46752
   940
  "r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}"
haftmann@46692
   941
blanchet@35828
   942
declare Image_def [no_atp]
paulson@24286
   943
wenzelm@12913
   944
lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)"
haftmann@46752
   945
  by (simp add: Image_def)
berghofe@12905
   946
wenzelm@12913
   947
lemma Image_singleton: "r``{a} = {b. (a, b) : r}"
haftmann@46752
   948
  by (simp add: Image_def)
berghofe@12905
   949
wenzelm@12913
   950
lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)"
haftmann@46752
   951
  by (rule Image_iff [THEN trans]) simp
berghofe@12905
   952
blanchet@35828
   953
lemma ImageI [intro,no_atp]: "(a, b) : r ==> a : A ==> b : r``A"
haftmann@46752
   954
  by (unfold Image_def) blast
berghofe@12905
   955
berghofe@12905
   956
lemma ImageE [elim!]:
haftmann@46752
   957
  "b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P"
haftmann@46752
   958
  by (unfold Image_def) (iprover elim!: CollectE bexE)
berghofe@12905
   959
berghofe@12905
   960
lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A"
berghofe@12905
   961
  -- {* This version's more effective when we already have the required @{text a} *}
haftmann@46752
   962
  by blast
berghofe@12905
   963
berghofe@12905
   964
lemma Image_empty [simp]: "R``{} = {}"
haftmann@46752
   965
  by blast
berghofe@12905
   966
berghofe@12905
   967
lemma Image_Id [simp]: "Id `` A = A"
haftmann@46752
   968
  by blast
berghofe@12905
   969
nipkow@30198
   970
lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B"
haftmann@46752
   971
  by blast
paulson@13830
   972
paulson@13830
   973
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
haftmann@46752
   974
  by blast
berghofe@12905
   975
paulson@13830
   976
lemma Image_Int_eq:
haftmann@46767
   977
  "single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B"
haftmann@46767
   978
  by (simp add: single_valued_def, blast) 
berghofe@12905
   979
paulson@13830
   980
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
haftmann@46752
   981
  by blast
berghofe@12905
   982
paulson@13812
   983
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A"
haftmann@46752
   984
  by blast
paulson@13812
   985
wenzelm@12913
   986
lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B"
haftmann@46752
   987
  by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2)
berghofe@12905
   988
paulson@13830
   989
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})"
berghofe@12905
   990
  -- {* NOT suitable for rewriting *}
haftmann@46752
   991
  by blast
berghofe@12905
   992
wenzelm@12913
   993
lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)"
haftmann@46752
   994
  by blast
berghofe@12905
   995
paulson@13830
   996
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))"
haftmann@46752
   997
  by blast
paulson@13830
   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
paulson@13830
  1002
text{*Converse inclusion requires some assumptions*}
paulson@13830
  1003
lemma Image_INT_eq:
paulson@13830
  1004
     "[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)"
paulson@13830
  1005
apply (rule equalityI)
paulson@13830
  1006
 apply (rule Image_INT_subset) 
paulson@13830
  1007
apply  (simp add: single_valued_def, blast)
paulson@13830
  1008
done
berghofe@12905
  1009
wenzelm@12913
  1010
lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))"
haftmann@46752
  1011
  by blast
berghofe@12905
  1012
haftmann@46692
  1013
lemma Image_Collect_split [simp]: "{(x, y). P x y} `` A = {y. EX x:A. P x y}"
haftmann@46752
  1014
  by auto
berghofe@12905
  1015
berghofe@12905
  1016
haftmann@46664
  1017
subsubsection {* Inverse image *}
berghofe@12905
  1018
haftmann@46752
  1019
definition inv_image :: "'b rel \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a rel"
haftmann@46752
  1020
where
haftmann@46752
  1021
  "inv_image r f = {(x, y). (f x, f y) \<in> r}"
haftmann@46692
  1022
haftmann@46752
  1023
definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
haftmann@46752
  1024
where
haftmann@46694
  1025
  "inv_imagep r f = (\<lambda>x y. r (f x) (f y))"
haftmann@46694
  1026
haftmann@46694
  1027
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
  1028
  by (simp add: inv_image_def inv_imagep_def)
haftmann@46694
  1029
huffman@19228
  1030
lemma sym_inv_image: "sym r ==> sym (inv_image r f)"
haftmann@46752
  1031
  by (unfold sym_def inv_image_def) blast
huffman@19228
  1032
wenzelm@12913
  1033
lemma trans_inv_image: "trans r ==> trans (inv_image r f)"
berghofe@12905
  1034
  apply (unfold trans_def inv_image_def)
berghofe@12905
  1035
  apply (simp (no_asm))
berghofe@12905
  1036
  apply blast
berghofe@12905
  1037
  done
berghofe@12905
  1038
krauss@32463
  1039
lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)"
krauss@32463
  1040
  by (auto simp:inv_image_def)
krauss@32463
  1041
krauss@33218
  1042
lemma converse_inv_image[simp]: "(inv_image R f)^-1 = inv_image (R^-1) f"
haftmann@46752
  1043
  unfolding inv_image_def converse_unfold by auto
krauss@33218
  1044
haftmann@46664
  1045
lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
haftmann@46664
  1046
  by (simp add: inv_imagep_def)
haftmann@46664
  1047
haftmann@46664
  1048
haftmann@46664
  1049
subsubsection {* Powerset *}
haftmann@46664
  1050
haftmann@46752
  1051
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
haftmann@46752
  1052
where
haftmann@46664
  1053
  "Powp A = (\<lambda>B. \<forall>x \<in> B. A x)"
haftmann@46664
  1054
haftmann@46664
  1055
lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"
haftmann@46664
  1056
  by (auto simp add: Powp_def fun_eq_iff)
haftmann@46664
  1057
haftmann@46664
  1058
lemmas Powp_mono [mono] = Pow_mono [to_pred]
haftmann@46664
  1059
kuncar@48620
  1060
subsubsection {* Expressing relation operations via @{const Finite_Set.fold} *}
kuncar@48620
  1061
kuncar@48620
  1062
lemma Id_on_fold:
kuncar@48620
  1063
  assumes "finite A"
kuncar@48620
  1064
  shows "Id_on A = Finite_Set.fold (\<lambda>x. Set.insert (Pair x x)) {} A"
kuncar@48620
  1065
proof -
kuncar@48620
  1066
  interpret comp_fun_commute "\<lambda>x. Set.insert (Pair x x)" by default auto
kuncar@48620
  1067
  show ?thesis using assms unfolding Id_on_def by (induct A) simp_all
kuncar@48620
  1068
qed
kuncar@48620
  1069
kuncar@48620
  1070
lemma comp_fun_commute_Image_fold:
kuncar@48620
  1071
  "comp_fun_commute (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)"
kuncar@48620
  1072
proof -
kuncar@48620
  1073
  interpret comp_fun_idem Set.insert
kuncar@48620
  1074
      by (fact comp_fun_idem_insert)
kuncar@48620
  1075
  show ?thesis 
kuncar@48620
  1076
  by default (auto simp add: fun_eq_iff comp_fun_commute split:prod.split)
kuncar@48620
  1077
qed
kuncar@48620
  1078
kuncar@48620
  1079
lemma Image_fold:
kuncar@48620
  1080
  assumes "finite R"
kuncar@48620
  1081
  shows "R `` S = Finite_Set.fold (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A) {} R"
kuncar@48620
  1082
proof -
kuncar@48620
  1083
  interpret comp_fun_commute "(\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)" 
kuncar@48620
  1084
    by (rule comp_fun_commute_Image_fold)
kuncar@48620
  1085
  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
  1086
    by (force intro: rev_ImageI)
kuncar@48620
  1087
  show ?thesis using assms by (induct R) (auto simp: *)
kuncar@48620
  1088
qed
kuncar@48620
  1089
kuncar@48620
  1090
lemma insert_relcomp_union_fold:
kuncar@48620
  1091
  assumes "finite S"
kuncar@48620
  1092
  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
  1093
proof -
kuncar@48620
  1094
  interpret comp_fun_commute "\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A'"
kuncar@48620
  1095
  proof - 
kuncar@48620
  1096
    interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert)
kuncar@48620
  1097
    show "comp_fun_commute (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A')"
kuncar@48620
  1098
    by default (auto simp add: fun_eq_iff split:prod.split)
kuncar@48620
  1099
  qed
kuncar@48620
  1100
  have *: "{x} O S = {(x', z). x' = fst x \<and> (snd x,z) \<in> S}" by (auto simp: relcomp_unfold intro!: exI)
kuncar@48620
  1101
  show ?thesis unfolding *
kuncar@48620
  1102
  using `finite S` by (induct S) (auto split: prod.split)
kuncar@48620
  1103
qed
kuncar@48620
  1104
kuncar@48620
  1105
lemma insert_relcomp_fold:
kuncar@48620
  1106
  assumes "finite S"
kuncar@48620
  1107
  shows "Set.insert x R O S = 
kuncar@48620
  1108
    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
  1109
proof -
kuncar@48620
  1110
  have "Set.insert x R O S = ({x} O S) \<union> (R O S)" by auto
kuncar@48620
  1111
  then show ?thesis by (auto simp: insert_relcomp_union_fold[OF assms])
kuncar@48620
  1112
qed
kuncar@48620
  1113
kuncar@48620
  1114
lemma comp_fun_commute_relcomp_fold:
kuncar@48620
  1115
  assumes "finite S"
kuncar@48620
  1116
  shows "comp_fun_commute (\<lambda>(x,y) A. 
kuncar@48620
  1117
    Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S)"
kuncar@48620
  1118
proof -
kuncar@48620
  1119
  have *: "\<And>a b A. 
kuncar@48620
  1120
    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
  1121
    by (auto simp: insert_relcomp_union_fold[OF assms] cong: if_cong)
kuncar@48620
  1122
  show ?thesis by default (auto simp: *)
kuncar@48620
  1123
qed
kuncar@48620
  1124
kuncar@48620
  1125
lemma relcomp_fold:
kuncar@48620
  1126
  assumes "finite R"
kuncar@48620
  1127
  assumes "finite S"
kuncar@48620
  1128
  shows "R O S = Finite_Set.fold 
kuncar@48620
  1129
    (\<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"
traytel@52749
  1130
  using assms by (induct R)
traytel@52749
  1131
    (auto simp: comp_fun_commute.fold_insert comp_fun_commute_relcomp_fold insert_relcomp_fold
kuncar@48620
  1132
      cong: if_cong)
kuncar@48620
  1133
kuncar@48620
  1134
kuncar@48620
  1135
nipkow@1128
  1136
end
haftmann@46689
  1137