author  webertj 
Fri, 19 Oct 2012 15:12:52 +0200  
changeset 49962  a8cc904a6820 
parent 48620  fc9be489e2fb 
child 50420  f1a27e82af16 
permissions  rwrr 
10358  1 
(* 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 *} 
12905  6 

15131  7 
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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46882  13 
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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46694  44 
subsection {* Fundamental *} 
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46694  46 
subsubsection {* Relations as sets of pairs *} 
47 

48 
type_synonym 'a rel = "('a * 'a) set" 

49 

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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 

58 

59 

60 
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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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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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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46981  98 
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) 

100 

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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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46833  110 
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)" 

46884  123 
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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157 

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lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r" 
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160 

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lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A" 
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lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A" 
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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) 

170 

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lemma reflpE: 

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assumes "reflp r" 

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obtains "r x x" 

174 
using assms by (auto dest: refl_onD simp add: reflp_def) 

175 

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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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183 

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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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190 

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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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194 

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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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202 

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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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46694  217 
lemma irrefl_distinct [code]: 
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"irrefl r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<noteq> y)" 

219 
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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227 

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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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231 

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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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235 

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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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240 
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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243 

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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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249 
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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253 

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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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263 

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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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267 

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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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46982  284 
lemma symp_INF: 
285 
"\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (INFI S r)" 

286 
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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291 

46982  292 
lemma symp_SUP: 
293 
"\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (SUPR S r)" 

294 
by (fact sym_UNION [to_pred]) 

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46694  297 
subsubsection {* Antisymmetry *} 
298 

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definition antisym :: "'a rel \<Rightarrow> bool" 
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where 
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"antisym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)" 
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302 

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abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" 
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304 
where 
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"antisymP r \<equiv> antisym {(x, y). r x y}" 
46694  306 

307 
lemma antisymI: 

308 
"(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r" 

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by (unfold antisym_def) iprover 
46694  310 

311 
lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b" 

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312 
by (unfold antisym_def) iprover 
46694  313 

314 
lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r" 

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by (unfold antisym_def) blast 
46694  316 

317 
lemma antisym_empty [simp]: "antisym {}" 

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318 
by (unfold antisym_def) blast 
46694  319 

320 

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subsubsection {* Transitivity *} 
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322 

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definition trans :: "'a rel \<Rightarrow> bool" 
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where 
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"trans r \<longleftrightarrow> (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)" 
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326 

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definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" 
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328 
where 
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329 
"transp r \<longleftrightarrow> (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)" 
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330 

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331 
lemma transp_trans_eq [pred_set_conv]: 
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"transp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> trans r" 
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333 
by (simp add: trans_def transp_def) 
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334 

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abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" 
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336 
where  {* FIXME drop *} 
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337 
"transP r \<equiv> trans {(x, y). r x y}" 
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338 

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339 
lemma transI: 
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340 
"(\<And>x y z. (x, y) \<in> r \<Longrightarrow> (y, z) \<in> r \<Longrightarrow> (x, z) \<in> r) \<Longrightarrow> trans r" 
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341 
by (unfold trans_def) iprover 
46694  342 

343 
lemma transpI: 

344 
"(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r" 

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345 
by (fact transI [to_pred]) 
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346 

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347 
lemma transE: 
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348 
assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r" 
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349 
obtains "(x, z) \<in> r" 
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350 
using assms by (unfold trans_def) iprover 
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351 

46694  352 
lemma transpE: 
353 
assumes "transp r" and "r x y" and "r y z" 

354 
obtains "r x z" 

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355 
using assms by (rule transE [to_pred]) 
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356 

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357 
lemma transD: 
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358 
assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r" 
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359 
shows "(x, z) \<in> r" 
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360 
using assms by (rule transE) 
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361 

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362 
lemma transpD: 
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363 
assumes "transp r" and "r x y" and "r y z" 
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364 
shows "r x z" 
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365 
using assms by (rule transD [to_pred]) 
46694  366 

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367 
lemma trans_Int: 
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368 
"trans r \<Longrightarrow> trans s \<Longrightarrow> trans (r \<inter> s)" 
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369 
by (fast intro: transI elim: transE) 
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370 

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371 
lemma transp_inf: 
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372 
"transp r \<Longrightarrow> transp s \<Longrightarrow> transp (r \<sqinter> s)" 
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373 
by (fact trans_Int [to_pred]) 
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374 

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375 
lemma trans_INTER: 
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376 
"\<forall>x\<in>S. trans (r x) \<Longrightarrow> trans (INTER S r)" 
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377 
by (fast intro: transI elim: transD) 
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378 

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379 
(* FIXME thm trans_INTER [to_pred] *) 
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380 

46694  381 
lemma trans_join [code]: 
382 
"trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)" 

383 
by (auto simp add: trans_def) 

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384 

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385 
lemma transp_trans: 
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386 
"transp r \<longleftrightarrow> trans {(x, y). r x y}" 
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387 
by (simp add: trans_def transp_def) 
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388 

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389 

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390 
subsubsection {* Totality *} 
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391 

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392 
definition total_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool" 
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393 
where 
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394 
"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)" 
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395 

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396 
abbreviation "total \<equiv> total_on UNIV" 
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397 

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398 
lemma total_on_empty [simp]: "total_on {} r" 
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399 
by (simp add: total_on_def) 
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400 

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401 

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402 
subsubsection {* Single valued relations *} 
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403 

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404 
definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool" 
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405 
where 
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406 
"single_valued r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))" 
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407 

46694  408 
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where 
409 
"single_valuedP r \<equiv> single_valued {(x, y). r x y}" 

410 

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411 
lemma single_valuedI: 
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412 
"ALL x y. (x,y):r > (ALL z. (x,z):r > y=z) ==> single_valued r" 
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413 
by (unfold single_valued_def) 
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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414 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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changeset

415 
lemma single_valuedD: 
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haftmann
parents:
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416 
"single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z" 
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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changeset

417 
by (simp add: single_valued_def) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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418 

46692
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haftmann
parents:
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changeset

419 
lemma single_valued_subset: 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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diff
changeset

420 
"r \<subseteq> s ==> single_valued s ==> single_valued r" 
46752
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haftmann
parents:
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diff
changeset

421 
by (unfold single_valued_def) blast 
11136  422 

12905  423 

46694  424 
subsection {* Relation operations *} 
425 

46664
1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

426 
subsubsection {* The identity relation *} 
12905  427 

46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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428 
definition Id :: "'a rel" 
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

429 
where 
48253
4410a709913c
a first guess to avoid the Codegenerator_Test to loop infinitely
bulwahn
parents:
47937
diff
changeset

430 
[code del]: "Id = {p. \<exists>x. p = (x, x)}" 
46692
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haftmann
parents:
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431 

12905  432 
lemma IdI [intro]: "(a, a) : Id" 
46752
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parents:
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433 
by (simp add: Id_def) 
12905  434 

435 
lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P" 

46752
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parents:
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436 
by (unfold Id_def) (iprover elim: CollectE) 
12905  437 

438 
lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)" 

46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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changeset

439 
by (unfold Id_def) blast 
12905  440 

30198  441 
lemma refl_Id: "refl Id" 
46752
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442 
by (simp add: refl_on_def) 
12905  443 

444 
lemma antisym_Id: "antisym Id" 

445 
 {* A strange result, since @{text Id} is also symmetric. *} 

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parents:
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446 
by (simp add: antisym_def) 
12905  447 

19228  448 
lemma sym_Id: "sym Id" 
46752
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parents:
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449 
by (simp add: sym_def) 
19228  450 

12905  451 
lemma trans_Id: "trans Id" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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changeset

452 
by (simp add: trans_def) 
12905  453 

46692
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haftmann
parents:
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diff
changeset

454 
lemma single_valued_Id [simp]: "single_valued Id" 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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diff
changeset

455 
by (unfold single_valued_def) blast 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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diff
changeset

456 

1f8b766224f6
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parents:
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diff
changeset

457 
lemma irrefl_diff_Id [simp]: "irrefl (r  Id)" 
1f8b766224f6
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haftmann
parents:
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diff
changeset

458 
by (simp add:irrefl_def) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

459 

1f8b766224f6
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parents:
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diff
changeset

460 
lemma trans_diff_Id: "trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r  Id)" 
1f8b766224f6
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haftmann
parents:
46691
diff
changeset

461 
unfolding antisym_def trans_def by blast 
1f8b766224f6
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haftmann
parents:
46691
diff
changeset

462 

1f8b766224f6
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haftmann
parents:
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diff
changeset

463 
lemma total_on_diff_Id [simp]: "total_on A (r  Id) = total_on A r" 
1f8b766224f6
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haftmann
parents:
46691
diff
changeset

464 
by (simp add: total_on_def) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

465 

12905  466 

46664
1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

467 
subsubsection {* Diagonal: identity over a set *} 
12905  468 

46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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469 
definition Id_on :: "'a set \<Rightarrow> 'a rel" 
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haftmann
parents:
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470 
where 
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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changeset

471 
"Id_on A = (\<Union>x\<in>A. {(x, x)})" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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changeset

472 

30198  473 
lemma Id_on_empty [simp]: "Id_on {} = {}" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

474 
by (simp add: Id_on_def) 
13812
91713a1915ee
converting HOL/UNITY to use unconditional fairness
paulson
parents:
13639
diff
changeset

475 

30198  476 
lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A" 
46752
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parents:
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diff
changeset

477 
by (simp add: Id_on_def) 
12905  478 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
33218
diff
changeset

479 
lemma Id_onI [intro!,no_atp]: "a : A ==> (a, a) : Id_on A" 
46752
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parents:
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diff
changeset

480 
by (rule Id_on_eqI) (rule refl) 
12905  481 

30198  482 
lemma Id_onE [elim!]: 
483 
"c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P" 

12913  484 
 {* The general elimination rule. *} 
46752
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parents:
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diff
changeset

485 
by (unfold Id_on_def) (iprover elim!: UN_E singletonE) 
12905  486 

30198  487 
lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)" 
46752
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haftmann
parents:
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diff
changeset

488 
by blast 
12905  489 

45967  490 
lemma Id_on_def' [nitpick_unfold]: 
44278
1220ecb81e8f
observe distinction between sets and predicates more properly
haftmann
parents:
41792
diff
changeset

491 
"Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

492 
by auto 
40923
be80c93ac0a2
adding a nice definition of Id_on for quickcheck and nitpick
bulwahn
parents:
36772
diff
changeset

493 

30198  494 
lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

495 
by blast 
12905  496 

46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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diff
changeset

497 
lemma refl_on_Id_on: "refl_on A (Id_on A)" 
46752
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haftmann
parents:
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diff
changeset

498 
by (rule refl_onI [OF Id_on_subset_Times Id_onI]) 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

499 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

500 
lemma antisym_Id_on [simp]: "antisym (Id_on A)" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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changeset

501 
by (unfold antisym_def) blast 
46692
1f8b766224f6
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haftmann
parents:
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diff
changeset

502 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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diff
changeset

503 
lemma sym_Id_on [simp]: "sym (Id_on A)" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

504 
by (rule symI) clarify 
46692
1f8b766224f6
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haftmann
parents:
46691
diff
changeset

505 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
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diff
changeset

506 
lemma trans_Id_on [simp]: "trans (Id_on A)" 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

507 
by (fast intro: transI elim: transD) 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

508 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

509 
lemma single_valued_Id_on [simp]: "single_valued (Id_on A)" 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

510 
by (unfold single_valued_def) blast 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

511 

12905  512 

46694  513 
subsubsection {* Composition *} 
12905  514 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

515 
inductive_set relcomp :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a \<times> 'c) set" (infixr "O" 75) 
46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

516 
for r :: "('a \<times> 'b) set" and s :: "('b \<times> 'c) set" 
46694  517 
where 
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

518 
relcompI [intro]: "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> s \<Longrightarrow> (a, c) \<in> r O s" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

519 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

520 
notation relcompp (infixr "OO" 75) 
12905  521 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

522 
lemmas relcomppI = relcompp.intros 
12905  523 

46752
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more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

524 
text {* 
e9e7209eb375
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parents:
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diff
changeset

525 
For historic reasons, the elimination rules are not wholly corresponding. 
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haftmann
parents:
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diff
changeset

526 
Feel free to consolidate this. 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

527 
*} 
46694  528 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

529 
inductive_cases relcompEpair: "(a, c) \<in> r O s" 
47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

530 
inductive_cases relcomppE [elim!]: "(r OO s) a c" 
46694  531 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

532 
lemma relcompE [elim!]: "xz \<in> r O s \<Longrightarrow> 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

533 
(\<And>x y z. xz = (x, z) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (y, z) \<in> s \<Longrightarrow> P) \<Longrightarrow> P" 
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

534 
by (cases xz) (simp, erule relcompEpair, iprover) 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

535 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

536 
lemma R_O_Id [simp]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

537 
"R O Id = R" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

538 
by fast 
46694  539 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

540 
lemma Id_O_R [simp]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

541 
"Id O R = R" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

542 
by fast 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

543 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

544 
lemma relcomp_empty1 [simp]: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

545 
"{} O R = {}" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

546 
by blast 
12905  547 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

548 
lemma relcompp_bot1 [simp]: 
46883
eec472dae593
tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents:
46882
diff
changeset

549 
"\<bottom> OO R = \<bottom>" 
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

550 
by (fact relcomp_empty1 [to_pred]) 
12905  551 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

552 
lemma relcomp_empty2 [simp]: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

553 
"R O {} = {}" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

554 
by blast 
12905  555 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

556 
lemma relcompp_bot2 [simp]: 
46883
eec472dae593
tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents:
46882
diff
changeset

557 
"R OO \<bottom> = \<bottom>" 
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

558 
by (fact relcomp_empty2 [to_pred]) 
23185  559 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

560 
lemma O_assoc: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

561 
"(R O S) O T = R O (S O T)" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

562 
by blast 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

563 

46883
eec472dae593
tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents:
46882
diff
changeset

564 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

565 
lemma relcompp_assoc: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

566 
"(r OO s) OO t = r OO (s OO t)" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

567 
by (fact O_assoc [to_pred]) 
23185  568 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

569 
lemma trans_O_subset: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

570 
"trans r \<Longrightarrow> r O r \<subseteq> r" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

571 
by (unfold trans_def) blast 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

572 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

573 
lemma transp_relcompp_less_eq: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

574 
"transp r \<Longrightarrow> r OO r \<le> r " 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

575 
by (fact trans_O_subset [to_pred]) 
12905  576 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

577 
lemma relcomp_mono: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

578 
"r' \<subseteq> r \<Longrightarrow> s' \<subseteq> s \<Longrightarrow> r' O s' \<subseteq> r O s" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

579 
by blast 
12905  580 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

581 
lemma relcompp_mono: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

582 
"r' \<le> r \<Longrightarrow> s' \<le> s \<Longrightarrow> r' OO s' \<le> r OO s " 
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

583 
by (fact relcomp_mono [to_pred]) 
12905  584 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

585 
lemma relcomp_subset_Sigma: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

586 
"r \<subseteq> A \<times> B \<Longrightarrow> s \<subseteq> B \<times> C \<Longrightarrow> r O s \<subseteq> A \<times> C" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

587 
by blast 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

588 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

589 
lemma relcomp_distrib [simp]: 
46752
e9e7209eb375
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parents:
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diff
changeset

590 
"R O (S \<union> T) = (R O S) \<union> (R O T)" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

591 
by auto 
12905  592 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

593 
lemma relcompp_distrib [simp]: 
46752
e9e7209eb375
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parents:
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diff
changeset

594 
"R OO (S \<squnion> T) = R OO S \<squnion> R OO T" 
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

595 
by (fact relcomp_distrib [to_pred]) 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

596 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

597 
lemma relcomp_distrib2 [simp]: 
46752
e9e7209eb375
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parents:
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598 
"(S \<union> T) O R = (S O R) \<union> (T O R)" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

599 
by auto 
28008
f945f8d9ad4d
added distributivity of relation composition over union [simp]
krauss
parents:
26297
diff
changeset

600 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

601 
lemma relcompp_distrib2 [simp]: 
46752
e9e7209eb375
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parents:
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diff
changeset

602 
"(S \<squnion> T) OO R = S OO R \<squnion> T OO R" 
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

603 
by (fact relcomp_distrib2 [to_pred]) 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

604 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

605 
lemma relcomp_UNION_distrib: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

606 
"s O UNION I r = (\<Union>i\<in>I. s O r i) " 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

607 
by auto 
28008
f945f8d9ad4d
added distributivity of relation composition over union [simp]
krauss
parents:
26297
diff
changeset

608 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

609 
(* FIXME thm relcomp_UNION_distrib [to_pred] *) 
36772  610 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

611 
lemma relcomp_UNION_distrib2: 
46752
e9e7209eb375
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parents:
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diff
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612 
"UNION I r O s = (\<Union>i\<in>I. r i O s) " 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

613 
by auto 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

614 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

615 
(* FIXME thm relcomp_UNION_distrib2 [to_pred] *) 
36772  616 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

617 
lemma single_valued_relcomp: 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

618 
"single_valued r \<Longrightarrow> single_valued s \<Longrightarrow> single_valued (r O s)" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

619 
by (unfold single_valued_def) blast 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

620 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

621 
lemma relcomp_unfold: 
46752
e9e7209eb375
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haftmann
parents:
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diff
changeset

622 
"r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

623 
by (auto simp add: set_eq_iff) 
12905  624 

46664
1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

625 

1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

626 
subsubsection {* Converse *} 
12913  627 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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changeset

628 
inductive_set converse :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'a) set" ("(_^1)" [1000] 999) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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629 
for r :: "('a \<times> 'b) set" 
e9e7209eb375
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parents:
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changeset

630 
where 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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changeset

631 
"(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r^1" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

632 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

633 
notation (xsymbols) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

634 
converse ("(_\<inverse>)" [1000] 999) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

635 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

636 
notation 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

637 
conversep ("(_^1)" [1000] 1000) 
46694  638 

639 
notation (xsymbols) 

640 
conversep ("(_\<inverse>\<inverse>)" [1000] 1000) 

641 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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changeset

642 
lemma converseI [sym]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

643 
"(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

644 
by (fact converse.intros) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

645 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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changeset

646 
lemma conversepI (* CANDIDATE [sym] *): 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

647 
"r a b \<Longrightarrow> r\<inverse>\<inverse> b a" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

648 
by (fact conversep.intros) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

649 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

650 
lemma converseD [sym]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

651 
"(a, b) \<in> r\<inverse> \<Longrightarrow> (b, a) \<in> r" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
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parents:
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diff
changeset

652 
by (erule converse.cases) iprover 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

653 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

654 
lemma conversepD (* CANDIDATE [sym] *): 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

655 
"r\<inverse>\<inverse> b a \<Longrightarrow> r a b" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

656 
by (fact converseD [to_pred]) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

657 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

658 
lemma converseE [elim!]: 
e9e7209eb375
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parents:
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diff
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659 
 {* More general than @{text converseD}, as it ``splits'' the member of the relation. *} 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

660 
"yx \<in> r\<inverse> \<Longrightarrow> (\<And>x y. yx = (y, x) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> P) \<Longrightarrow> P" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

661 
by (cases yx) (simp, erule converse.cases, iprover) 
46694  662 

46882  663 
lemmas conversepE [elim!] = conversep.cases 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

664 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

665 
lemma converse_iff [iff]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

666 
"(a, b) \<in> r\<inverse> \<longleftrightarrow> (b, a) \<in> r" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

667 
by (auto intro: converseI) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

668 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

669 
lemma conversep_iff [iff]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

670 
"r\<inverse>\<inverse> a b = r b a" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

671 
by (fact converse_iff [to_pred]) 
46694  672 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

673 
lemma converse_converse [simp]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

674 
"(r\<inverse>)\<inverse> = r" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

675 
by (simp add: set_eq_iff) 
46694  676 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
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diff
changeset

677 
lemma conversep_conversep [simp]: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

678 
"(r\<inverse>\<inverse>)\<inverse>\<inverse> = r" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

679 
by (fact converse_converse [to_pred]) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

680 

47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset

681 
lemma converse_relcomp: "(r O s)^1 = s^1 O r^1" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

682 
by blast 
46694  683 

47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

684 
lemma converse_relcompp: "(r OO s)^1 = s^1 OO r^1" 
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

685 
by (iprover intro: order_antisym conversepI relcomppI 
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset

686 
elim: relcomppE dest: conversepD) 
46694  687 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

688 
lemma converse_Int: "(r \<inter> s)^1 = r^1 \<inter> s^1" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

689 
by blast 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

690 

46694  691 
lemma converse_meet: "(r \<sqinter> s)^1 = r^1 \<sqinter> s^1" 
692 
by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD) 

693 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

694 
lemma converse_Un: "(r \<union> s)^1 = r^1 \<union> s^1" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

695 
by blast 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

696 

46694  697 
lemma converse_join: "(r \<squnion> s)^1 = r^1 \<squnion> s^1" 
698 
by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD) 

699 

19228  700 
lemma converse_INTER: "(INTER S r)^1 = (INT x:S. (r x)^1)" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

701 
by fast 
19228  702 

703 
lemma converse_UNION: "(UNION S r)^1 = (UN x:S. (r x)^1)" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

704 
by blast 
19228  705 

12905  706 
lemma converse_Id [simp]: "Id^1 = Id" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

707 
by blast 
12905  708 

30198  709 
lemma converse_Id_on [simp]: "(Id_on A)^1 = Id_on A" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

710 
by blast 
12905  711 

30198  712 
lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

713 
by (unfold refl_on_def) auto 
12905  714 

19228  715 
lemma sym_converse [simp]: "sym (converse r) = sym r" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

716 
by (unfold sym_def) blast 
19228  717 

718 
lemma antisym_converse [simp]: "antisym (converse r) = antisym r" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

719 
by (unfold antisym_def) blast 
12905  720 

19228  721 
lemma trans_converse [simp]: "trans (converse r) = trans r" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

722 
by (unfold trans_def) blast 
12905  723 

19228  724 
lemma sym_conv_converse_eq: "sym r = (r^1 = r)" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

725 
by (unfold sym_def) fast 
19228  726 

727 
lemma sym_Un_converse: "sym (r \<union> r^1)" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

728 
by (unfold sym_def) blast 
19228  729 

730 
lemma sym_Int_converse: "sym (r \<inter> r^1)" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

731 
by (unfold sym_def) blast 
19228  732 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

733 
lemma total_on_converse [simp]: "total_on A (r^1) = total_on A r" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

734 
by (auto simp: total_on_def) 
29859
33bff35f1335
Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents:
29609
diff
changeset

735 

46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

736 
lemma finite_converse [iff]: "finite (r^1) = finite r" 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

737 
apply (subgoal_tac "r^1 = (%(x,y). (y,x))`r") 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

738 
apply simp 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

739 
apply (rule iffI) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

740 
apply (erule finite_imageD [unfolded inj_on_def]) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

741 
apply (simp split add: split_split) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

742 
apply (erule finite_imageI) 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

743 
apply (simp add: set_eq_iff image_def, auto) 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

744 
apply (rule bexI) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

745 
prefer 2 apply assumption 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

746 
apply simp 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

747 
done 
12913  748 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

749 
lemma conversep_noteq [simp]: "(op \<noteq>)^1 = op \<noteq>" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

750 
by (auto simp add: fun_eq_iff) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

751 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

752 
lemma conversep_eq [simp]: "(op =)^1 = op =" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

753 
by (auto simp add: fun_eq_iff) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

754 

e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

755 
lemma converse_unfold: 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

756 
"r\<inverse> = {(y, x). (x, y) \<in> r}" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

757 
by (simp add: set_eq_iff) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

758 

46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

759 

1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

760 
subsubsection {* Domain, range and field *} 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

761 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

762 
inductive_set Domain :: "('a \<times> 'b) set \<Rightarrow> 'a set" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

763 
for r :: "('a \<times> 'b) set" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

764 
where 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

765 
DomainI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> Domain r" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

766 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

767 
abbreviation (input) "DomainP \<equiv> Domainp" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

768 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

769 
lemmas DomainPI = Domainp.DomainI 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

770 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

771 
inductive_cases DomainE [elim!]: "a \<in> Domain r" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

772 
inductive_cases DomainpE [elim!]: "Domainp r a" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

773 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

774 
inductive_set Range :: "('a \<times> 'b) set \<Rightarrow> 'b set" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

775 
for r :: "('a \<times> 'b) set" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

776 
where 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

777 
RangeI [intro]: "(a, b) \<in> r \<Longrightarrow> b \<in> Range r" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

778 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

779 
abbreviation (input) "RangeP \<equiv> Rangep" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

780 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

781 
lemmas RangePI = Rangep.RangeI 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

782 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

783 
inductive_cases RangeE [elim!]: "b \<in> Range r" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

784 
inductive_cases RangepE [elim!]: "Rangep r b" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

785 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

786 
definition Field :: "'a rel \<Rightarrow> 'a set" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

787 
where 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

788 
"Field r = Domain r \<union> Range r" 
12905  789 

46694  790 
lemma Domain_fst [code]: 
791 
"Domain r = fst ` r" 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

792 
by force 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

793 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

794 
lemma Range_snd [code]: 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

795 
"Range r = snd ` r" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

796 
by force 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

797 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

798 
lemma fst_eq_Domain: "fst ` R = Domain R" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

799 
by force 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

800 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

801 
lemma snd_eq_Range: "snd ` R = Range R" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

802 
by force 
46694  803 

804 
lemma Domain_empty [simp]: "Domain {} = {}" 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

805 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

806 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

807 
lemma Range_empty [simp]: "Range {} = {}" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

808 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

809 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

810 
lemma Field_empty [simp]: "Field {} = {}" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

811 
by (simp add: Field_def) 
46694  812 

813 
lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}" 

814 
by auto 

815 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

816 
lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

817 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

818 

46882  819 
lemma Domain_insert [simp]: "Domain (insert (a, b) r) = insert a (Domain r)" 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

820 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

821 

46882  822 
lemma Range_insert [simp]: "Range (insert (a, b) r) = insert b (Range r)" 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

823 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

824 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

825 
lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \<union> Field r" 
46884  826 
by (auto simp add: Field_def) 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

827 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

828 
lemma Domain_iff: "a \<in> Domain r \<longleftrightarrow> (\<exists>y. (a, y) \<in> r)" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

829 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

830 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

831 
lemma Range_iff: "a \<in> Range r \<longleftrightarrow> (\<exists>y. (y, a) \<in> r)" 
46694  832 
by blast 
833 

834 
lemma Domain_Id [simp]: "Domain Id = UNIV" 

835 
by blast 

836 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

837 
lemma Range_Id [simp]: "Range Id = UNIV" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

838 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

839 

46694  840 
lemma Domain_Id_on [simp]: "Domain (Id_on A) = A" 
841 
by blast 

842 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

843 
lemma Range_Id_on [simp]: "Range (Id_on A) = A" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

844 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

845 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

846 
lemma Domain_Un_eq: "Domain (A \<union> B) = Domain A \<union> Domain B" 
46694  847 
by blast 
848 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

849 
lemma Range_Un_eq: "Range (A \<union> B) = Range A \<union> Range B" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

850 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

851 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

852 
lemma Field_Un [simp]: "Field (r \<union> s) = Field r \<union> Field s" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

853 
by (auto simp: Field_def) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

854 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

855 
lemma Domain_Int_subset: "Domain (A \<inter> B) \<subseteq> Domain A \<inter> Domain B" 
46694  856 
by blast 
857 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

858 
lemma Range_Int_subset: "Range (A \<inter> B) \<subseteq> Range A \<inter> Range B" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

859 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

860 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

861 
lemma Domain_Diff_subset: "Domain A  Domain B \<subseteq> Domain (A  B)" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

862 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

863 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

864 
lemma Range_Diff_subset: "Range A  Range B \<subseteq> Range (A  B)" 
46694  865 
by blast 
866 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

867 
lemma Domain_Union: "Domain (\<Union>S) = (\<Union>A\<in>S. Domain A)" 
46694  868 
by blast 
869 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

870 
lemma Range_Union: "Range (\<Union>S) = (\<Union>A\<in>S. Range A)" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

871 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

872 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

873 
lemma Field_Union [simp]: "Field (\<Union>R) = \<Union>(Field ` R)" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

874 
by (auto simp: Field_def) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

875 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

876 
lemma Domain_converse [simp]: "Domain (r\<inverse>) = Range r" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

877 
by auto 
46694  878 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

879 
lemma Range_converse [simp]: "Range (r\<inverse>) = Domain r" 
46694  880 
by blast 
881 

46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

882 
lemma Field_converse [simp]: "Field (r\<inverse>) = Field r" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

883 
by (auto simp: Field_def) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

884 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

885 
lemma Domain_Collect_split [simp]: "Domain {(x, y). P x y} = {x. EX y. P x y}" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

886 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

887 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

888 
lemma Range_Collect_split [simp]: "Range {(x, y). P x y} = {y. EX x. P x y}" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

889 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

890 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

891 
lemma finite_Domain: "finite r \<Longrightarrow> finite (Domain r)" 
46884  892 
by (induct set: finite) auto 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

893 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

894 
lemma finite_Range: "finite r \<Longrightarrow> finite (Range r)" 
46884  895 
by (induct set: finite) auto 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

896 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

897 
lemma finite_Field: "finite r \<Longrightarrow> finite (Field r)" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

898 
by (simp add: Field_def finite_Domain finite_Range) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

899 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

900 
lemma Domain_mono: "r \<subseteq> s \<Longrightarrow> Domain r \<subseteq> Domain s" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

901 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

902 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

903 
lemma Range_mono: "r \<subseteq> s \<Longrightarrow> Range r \<subseteq> Range s" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

904 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

905 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

906 
lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

907 
by (auto simp: Field_def Domain_def Range_def) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

908 

807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

909 
lemma Domain_unfold: 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

910 
"Domain r = {x. \<exists>y. (x, y) \<in> r}" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

911 
by blast 
46694  912 

913 
lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)" 

914 
by auto 

915 

916 
lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)" 

917 
by auto 

918 

12905  919 

46664
1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

920 
subsubsection {* Image of a set under a relation *} 
12905  921 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

922 
definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixl "``" 90) 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

923 
where 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

924 
"r `` s = {y. \<exists>x\<in>s. (x, y) \<in> r}" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

925 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
33218
diff
changeset

926 
declare Image_def [no_atp] 
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23709
diff
changeset

927 

12913  928 
lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

929 
by (simp add: Image_def) 
12905  930 

12913  931 
lemma Image_singleton: "r``{a} = {b. (a, b) : r}" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

932 
by (simp add: Image_def) 
12905  933 

12913  934 
lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

935 
by (rule Image_iff [THEN trans]) simp 
12905  936 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
33218
diff
changeset

937 
lemma ImageI [intro,no_atp]: "(a, b) : r ==> a : A ==> b : r``A" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

938 
by (unfold Image_def) blast 
12905  939 

940 
lemma ImageE [elim!]: 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

941 
"b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

942 
by (unfold Image_def) (iprover elim!: CollectE bexE) 
12905  943 

944 
lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A" 

945 
 {* This version's more effective when we already have the required @{text a} *} 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

946 
by blast 
12905  947 

948 
lemma Image_empty [simp]: "R``{} = {}" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

949 
by blast 
12905  950 

951 
lemma Image_Id [simp]: "Id `` A = A" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

952 
by blast 
12905  953 

30198  954 
lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

955 
by blast 
13830  956 

957 
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

958 
by blast 
12905  959 

13830  960 
lemma Image_Int_eq: 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

961 
"single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

962 
by (simp add: single_valued_def, blast) 
12905  963 

13830  964 
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

965 
by blast 
12905  966 

13812
91713a1915ee
converting HOL/UNITY to use unconditional fairness
paulson
parents:
13639
diff
changeset

967 
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

968 
by blast 
13812
91713a1915ee
converting HOL/UNITY to use unconditional fairness
paulson
parents:
13639
diff
changeset

969 

12913  970 
lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

971 
by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2) 
12905  972 

13830  973 
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})" 
12905  974 
 {* NOT suitable for rewriting *} 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

975 
by blast 
12905  976 

12913  977 
lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

978 
by blast 
12905  979 

13830  980 
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

981 
by blast 
13830  982 

983 
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))" 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

984 
by blast 
12905  985 

13830  986 
text{*Converse inclusion requires some assumptions*} 
987 
lemma Image_INT_eq: 

988 
"[single_valued (r\<inverse>); A\<noteq>{}] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)" 

989 
apply (rule equalityI) 

990 
apply (rule Image_INT_subset) 

991 
apply (simp add: single_valued_def, blast) 

992 
done 

12905  993 

12913  994 
lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq>  ((r^1) `` (B)))" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

995 
by blast 
12905  996 

46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

997 
lemma Image_Collect_split [simp]: "{(x, y). P x y} `` A = {y. EX x:A. P x y}" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

998 
by auto 
12905  999 

1000 

46664
1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

1001 
subsubsection {* Inverse image *} 
12905  1002 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1003 
definition inv_image :: "'b rel \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a rel" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1004 
where 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1005 
"inv_image r f = {(x, y). (f x, f y) \<in> r}" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

1006 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1007 
definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1008 
where 
46694  1009 
"inv_imagep r f = (\<lambda>x y. r (f x) (f y))" 
1010 

1011 
lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)" 

1012 
by (simp add: inv_image_def inv_imagep_def) 

1013 

19228  1014 
lemma sym_inv_image: "sym r ==> sym (inv_image r f)" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1015 
by (unfold sym_def inv_image_def) blast 
19228  1016 

12913  1017 
lemma trans_inv_image: "trans r ==> trans (inv_image r f)" 
12905  1018 
apply (unfold trans_def inv_image_def) 
1019 
apply (simp (no_asm)) 

1020 
apply blast 

1021 
done 

1022 

32463
3a0a65ca2261
moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents:
32235
diff
changeset

1023 
lemma in_inv_image[simp]: "((x,y) : inv_image r f) = ((f x, f y) : r)" 
3a0a65ca2261
moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents:
32235
diff
changeset

1024 
by (auto simp:inv_image_def) 
3a0a65ca2261
moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents:
32235
diff
changeset

1025 

33218  1026 
lemma converse_inv_image[simp]: "(inv_image R f)^1 = inv_image (R^1) f" 
46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1027 
unfolding inv_image_def converse_unfold by auto 
33218  1028 

46664
1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

1029 
lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)" 
1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

1030 
by (simp add: inv_imagep_def) 
1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

1031 

1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

1032 

1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

1033 
subsubsection {* Powerset *} 
1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

1034 

46752
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1035 
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" 
e9e7209eb375
more fundamental predtoset conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset

1036 
where 
46664
1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

1037 
"Powp A = (\<lambda>B. \<forall>x \<in> B. A x)" 
1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

1038 

1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

1039 
lemma Powp_Pow_eq [pred_set_conv]: "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)" 
1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

1040 
by (auto simp add: Powp_def fun_eq_iff) 
1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

1041 

1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

1042 
lemmas Powp_mono [mono] = Pow_mono [to_pred] 
1f6c140f9c72
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
haftmann
parents:
46638
diff
changeset

1043 

48620
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1044 
subsubsection {* Expressing relation operations via @{const Finite_Set.fold} *} 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1045 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1046 
lemma Id_on_fold: 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1047 
assumes "finite A" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1048 
shows "Id_on A = Finite_Set.fold (\<lambda>x. Set.insert (Pair x x)) {} A" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1049 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1050 
interpret comp_fun_commute "\<lambda>x. Set.insert (Pair x x)" by default auto 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1051 
show ?thesis using assms unfolding Id_on_def by (induct A) simp_all 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1052 
qed 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1053 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1054 
lemma comp_fun_commute_Image_fold: 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1055 
"comp_fun_commute (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1056 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1057 
interpret comp_fun_idem Set.insert 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1058 
by (fact comp_fun_idem_insert) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1059 
show ?thesis 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1060 
by default (auto simp add: fun_eq_iff comp_fun_commute split:prod.split) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1061 
qed 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1062 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1063 
lemma Image_fold: 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1064 
assumes "finite R" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1065 
shows "R `` S = Finite_Set.fold (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A) {} R" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1066 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1067 
interpret comp_fun_commute "(\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1068 
by (rule comp_fun_commute_Image_fold) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1069 
have *: "\<And>x F. Set.insert x F `` S = (if fst x \<in> S then Set.insert (snd x) (F `` S) else (F `` S))" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1070 
by (auto intro: rev_ImageI) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1071 
show ?thesis using assms by (induct R) (auto simp: *) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1072 
qed 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1073 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1074 
lemma insert_relcomp_union_fold: 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1075 
assumes "finite S" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1076 
shows "{x} O S \<union> X = Finite_Set.fold (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') X S" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1077 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1078 
interpret comp_fun_commute "\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A'" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1079 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1080 
interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1081 
show "comp_fun_commute (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A')" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1082 
by default (auto simp add: fun_eq_iff split:prod.split) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1083 
qed 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1084 
have *: "{x} O S = {(x', z). x' = fst x \<and> (snd x,z) \<in> S}" by (auto simp: relcomp_unfold intro!: exI) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1085 
show ?thesis unfolding * 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1086 
using `finite S` by (induct S) (auto split: prod.split) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1087 
qed 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1088 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1089 
lemma insert_relcomp_fold: 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1090 
assumes "finite S" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1091 
shows "Set.insert x R O S = 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1092 
Finite_Set.fold (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') (R O S) S" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1093 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1094 
have "Set.insert x R O S = ({x} O S) \<union> (R O S)" by auto 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1095 
then show ?thesis by (auto simp: insert_relcomp_union_fold[OF assms]) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1096 
qed 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1097 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1098 
lemma comp_fun_commute_relcomp_fold: 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1099 
assumes "finite S" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1100 
shows "comp_fun_commute (\<lambda>(x,y) A. 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1101 
Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S)" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1102 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1103 
have *: "\<And>a b A. 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1104 
Finite_Set.fold (\<lambda>(w, z) A'. if b = w then Set.insert (a, z) A' else A') A S = {(a,b)} O S \<union> A" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1105 
by (auto simp: insert_relcomp_union_fold[OF assms] cong: if_cong) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1106 
show ?thesis by default (auto simp: *) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1107 
qed 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1108 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1109 
lemma relcomp_fold: 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1110 
assumes "finite R" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1111 
assumes "finite S" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1112 
shows "R O S = Finite_Set.fold 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1113 
(\<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" 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1114 
proof  
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1115 
show ?thesis using assms by (induct R) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1116 
(auto simp: comp_fun_commute.fold_insert comp_fun_commute_relcomp_fold insert_relcomp_fold 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1117 
cong: if_cong) 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1118 
qed 
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1119 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1120 

fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset

1121 

1128
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset

1122 
end 
46689  1123 