author  noschinl 
Mon, 12 Mar 2012 15:12:22 +0100  
changeset 46883  eec472dae593 
parent 46882  6242b4bc05bc 
child 46884  154dc6ec0041 
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 
15131  9 
begin 
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text {* A preliminary: classical rules for reasoning on predicates *} 
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declare predicate1I [Pure.intro!, intro!] 
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declare predicate1D [Pure.dest, dest] 

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declare predicate2I [Pure.intro!, intro!] 
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declare predicate2D [Pure.dest, dest] 
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declare bot1E [elim!] 
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declare bot2E [elim!] 
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declare top1I [intro!] 
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declare top2I [intro!] 
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declare inf1I [intro!] 
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declare inf2I [intro!] 
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declare inf1E [elim!] 
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declare inf2E [elim!] 
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declare sup1I1 [intro?] 
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declare sup2I1 [intro?] 
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declare sup1I2 [intro?] 
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declare sup2I2 [intro?] 
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declare sup1E [elim!] 
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declare sup2E [elim!] 
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declare sup1CI [intro!] 
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declare sup2CI [intro!] 
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declare INF1_I [intro!] 
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declare INF2_I [intro!] 
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declare INF1_D [elim] 
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declare INF2_D [elim] 
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declare INF1_E [elim] 
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declare INF2_E [elim] 
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declare SUP1_I [intro] 
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declare SUP2_I [intro] 
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declare SUP1_E [elim!] 
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declare SUP2_E [elim!] 
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46694  44 
subsection {* Fundamental *} 
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46694  46 
subsubsection {* Relations as sets of pairs *} 
47 

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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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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 Inf_apply) 

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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 INF_apply) 

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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 Inf_apply INF_apply) 

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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 INF_apply) 

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lemma Sup_SUP_eq [pred_set_conv]: "\<Squnion>S = (\<lambda>x. x \<in> UNION S Collect)" 

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by (simp add: fun_eq_iff Sup_apply) 

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113 
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 SUP_apply) 

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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 Sup_apply SUP_apply) 

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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 SUP_apply) 

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lemma INF_INT_eq [pred_set_conv]: "(\<Sqinter>i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Inter>i. r i))" 
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by (simp add: INF_apply fun_eq_iff) 
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lemma INF_INT_eq2 [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Inter>i\<in>S. r i))" 

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by (simp add: INF_apply fun_eq_iff) 

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lemma SUP_UN_eq [pred_set_conv]: "(\<Squnion>i. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Union>i. r i))" 
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by (simp add: SUP_apply 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: SUP_apply 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> refl {(x, y). r x y}" 
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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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158 

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lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r" 
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161 

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lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A" 
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164 

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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) 

171 

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lemma reflpE: 

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assumes "reflp r" 

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obtains "r x x" 

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

176 

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lemma refl_on_Int: "refl_on A r ==> refl_on B s ==> refl_on (A \<inter> B) (r \<inter> s)" 
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by (unfold refl_on_def) blast 
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lemma reflp_inf: 
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"reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<sqinter> s)" 
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by (auto intro: reflpI elim: reflpE) 
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lemma refl_on_Un: "refl_on A r ==> refl_on B s ==> refl_on (A \<union> B) (r \<union> s)" 
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by (unfold refl_on_def) blast 
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lemma reflp_sup: 
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"reflp r \<Longrightarrow> reflp s \<Longrightarrow> reflp (r \<squnion> s)" 
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by (auto intro: reflpI elim: reflpE) 
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lemma refl_on_INTER: 
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"ALL x:S. refl_on (A x) (r x) ==> refl_on (INTER S A) (INTER S r)" 
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by (unfold refl_on_def) fast 
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lemma refl_on_UNION: 
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"ALL x:S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)" 
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lemma refl_on_empty [simp]: "refl_on {} {}" 
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by (simp add:refl_on_def) 
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lemma refl_on_def' [nitpick_unfold, code]: 
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"refl_on A r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<in> A \<and> y \<in> A) \<and> (\<forall>x \<in> A. (x, x) \<in> r)" 
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by (auto intro: refl_onI dest: refl_onD refl_onD1 refl_onD2) 
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subsubsection {* Irreflexivity *} 
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definition irrefl :: "'a rel \<Rightarrow> bool" 
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where 
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"irrefl r \<longleftrightarrow> (\<forall>x. (x, x) \<notin> r)" 
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lemma irrefl_distinct [code]: 
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"irrefl r \<longleftrightarrow> (\<forall>(x, y) \<in> r. x \<noteq> y)" 

215 
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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223 

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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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227 

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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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231 

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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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236 
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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239 

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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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245 
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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249 

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lemma symD: 
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assumes "sym r" and "(b, a) \<in> r" 
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shows "(a, b) \<in> r" 
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using assms by (rule symE) 
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lemma sympD: 
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assumes "symp r" and "r b a" 
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shows "r a b" 
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using assms by (rule symD [to_pred]) 
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lemma sym_Int: 
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"sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<inter> s)" 
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by (fast intro: symI elim: symE) 
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263 

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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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275 

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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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279 

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(* FIXME thm sym_INTER [to_pred] *) 
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281 

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lemma sym_UNION: 
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283 
"\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (UNION S r)" 
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284 
by (fast intro: symI elim: symE) 
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285 

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(* FIXME thm sym_UNION [to_pred] *) 
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287 

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288 

46694  289 
subsubsection {* Antisymmetry *} 
290 

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definition antisym :: "'a rel \<Rightarrow> bool" 
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292 
where 
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293 
"antisym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)" 
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294 

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abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" 
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296 
where 
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297 
"antisymP r \<equiv> antisym {(x, y). r x y}" 
46694  298 

299 
lemma antisymI: 

300 
"(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r" 

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301 
by (unfold antisym_def) iprover 
46694  302 

303 
lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b" 

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304 
by (unfold antisym_def) iprover 
46694  305 

306 
lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r" 

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307 
by (unfold antisym_def) blast 
46694  308 

309 
lemma antisym_empty [simp]: "antisym {}" 

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310 
by (unfold antisym_def) blast 
46694  311 

312 

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313 
subsubsection {* Transitivity *} 
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314 

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definition trans :: "'a rel \<Rightarrow> bool" 
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316 
where 
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317 
"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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318 

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319 
definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" 
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320 
where 
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321 
"transp r \<longleftrightarrow> (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)" 
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322 

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323 
lemma transp_trans_eq [pred_set_conv]: 
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324 
"transp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> trans r" 
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325 
by (simp add: trans_def transp_def) 
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326 

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327 
abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" 
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328 
where  {* FIXME drop *} 
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329 
"transP r \<equiv> trans {(x, y). r x y}" 
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330 

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331 
lemma transI: 
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332 
"(\<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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333 
by (unfold trans_def) iprover 
46694  334 

335 
lemma transpI: 

336 
"(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r" 

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337 
by (fact transI [to_pred]) 
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338 

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339 
lemma transE: 
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340 
assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r" 
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341 
obtains "(x, z) \<in> r" 
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342 
using assms by (unfold trans_def) iprover 
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343 

46694  344 
lemma transpE: 
345 
assumes "transp r" and "r x y" and "r y z" 

346 
obtains "r x z" 

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347 
using assms by (rule transE [to_pred]) 
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348 

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349 
lemma transD: 
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350 
assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r" 
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351 
shows "(x, z) \<in> r" 
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352 
using assms by (rule transE) 
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353 

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354 
lemma transpD: 
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355 
assumes "transp r" and "r x y" and "r y z" 
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356 
shows "r x z" 
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357 
using assms by (rule transD [to_pred]) 
46694  358 

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359 
lemma trans_Int: 
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360 
"trans r \<Longrightarrow> trans s \<Longrightarrow> trans (r \<inter> s)" 
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361 
by (fast intro: transI elim: transE) 
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362 

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363 
lemma transp_inf: 
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364 
"transp r \<Longrightarrow> transp s \<Longrightarrow> transp (r \<sqinter> s)" 
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365 
by (fact trans_Int [to_pred]) 
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366 

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367 
lemma trans_INTER: 
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368 
"\<forall>x\<in>S. trans (r x) \<Longrightarrow> trans (INTER S r)" 
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369 
by (fast intro: transI elim: transD) 
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370 

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371 
(* FIXME thm trans_INTER [to_pred] *) 
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372 

46694  373 
lemma trans_join [code]: 
374 
"trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)" 

375 
by (auto simp add: trans_def) 

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376 

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377 
lemma transp_trans: 
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378 
"transp r \<longleftrightarrow> trans {(x, y). r x y}" 
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379 
by (simp add: trans_def transp_def) 
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380 

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381 

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382 
subsubsection {* Totality *} 
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383 

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384 
definition total_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool" 
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385 
where 
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386 
"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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387 

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388 
abbreviation "total \<equiv> total_on UNIV" 
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389 

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390 
lemma total_on_empty [simp]: "total_on {} r" 
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391 
by (simp add: total_on_def) 
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392 

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393 

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394 
subsubsection {* Single valued relations *} 
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395 

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396 
definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool" 
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397 
where 
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398 
"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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399 

46694  400 
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where 
401 
"single_valuedP r \<equiv> single_valued {(x, y). r x y}" 

402 

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403 
lemma single_valuedI: 
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404 
"ALL x y. (x,y):r > (ALL z. (x,z):r > y=z) ==> single_valued r" 
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405 
by (unfold single_valued_def) 
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406 

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407 
lemma single_valuedD: 
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408 
"single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z" 
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409 
by (simp add: single_valued_def) 
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parents:
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410 

46692
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411 
lemma single_valued_subset: 
1f8b766224f6
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412 
"r \<subseteq> s ==> single_valued s ==> single_valued r" 
46752
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413 
by (unfold single_valued_def) blast 
11136  414 

12905  415 

46694  416 
subsection {* Relation operations *} 
417 

46664
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moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
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418 
subsubsection {* The identity relation *} 
12905  419 

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420 
definition Id :: "'a rel" 
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421 
where 
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422 
"Id = {p. \<exists>x. p = (x, x)}" 
46692
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423 

12905  424 
lemma IdI [intro]: "(a, a) : Id" 
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425 
by (simp add: Id_def) 
12905  426 

427 
lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P" 

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428 
by (unfold Id_def) (iprover elim: CollectE) 
12905  429 

430 
lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)" 

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431 
by (unfold Id_def) blast 
12905  432 

30198  433 
lemma refl_Id: "refl Id" 
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434 
by (simp add: refl_on_def) 
12905  435 

436 
lemma antisym_Id: "antisym Id" 

437 
 {* A strange result, since @{text Id} is also symmetric. *} 

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438 
by (simp add: antisym_def) 
12905  439 

19228  440 
lemma sym_Id: "sym Id" 
46752
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441 
by (simp add: sym_def) 
19228  442 

12905  443 
lemma trans_Id: "trans Id" 
46752
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444 
by (simp add: trans_def) 
12905  445 

46692
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446 
lemma single_valued_Id [simp]: "single_valued Id" 
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447 
by (unfold single_valued_def) blast 
1f8b766224f6
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haftmann
parents:
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diff
changeset

448 

1f8b766224f6
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449 
lemma irrefl_diff_Id [simp]: "irrefl (r  Id)" 
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450 
by (simp add:irrefl_def) 
1f8b766224f6
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haftmann
parents:
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diff
changeset

451 

1f8b766224f6
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452 
lemma trans_diff_Id: "trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r  Id)" 
1f8b766224f6
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changeset

453 
unfolding antisym_def trans_def by blast 
1f8b766224f6
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haftmann
parents:
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diff
changeset

454 

1f8b766224f6
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changeset

455 
lemma total_on_diff_Id [simp]: "total_on A (r  Id) = total_on A r" 
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456 
by (simp add: total_on_def) 
1f8b766224f6
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haftmann
parents:
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diff
changeset

457 

12905  458 

46664
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haftmann
parents:
46638
diff
changeset

459 
subsubsection {* Diagonal: identity over a set *} 
12905  460 

46752
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461 
definition Id_on :: "'a set \<Rightarrow> 'a rel" 
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462 
where 
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463 
"Id_on A = (\<Union>x\<in>A. {(x, x)})" 
46692
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464 

30198  465 
lemma Id_on_empty [simp]: "Id_on {} = {}" 
46752
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parents:
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466 
by (simp add: Id_on_def) 
13812
91713a1915ee
converting HOL/UNITY to use unconditional fairness
paulson
parents:
13639
diff
changeset

467 

30198  468 
lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A" 
46752
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parents:
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469 
by (simp add: Id_on_def) 
12905  470 

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

471 
lemma Id_onI [intro!,no_atp]: "a : A ==> (a, a) : Id_on A" 
46752
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472 
by (rule Id_on_eqI) (rule refl) 
12905  473 

30198  474 
lemma Id_onE [elim!]: 
475 
"c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P" 

12913  476 
 {* The general elimination rule. *} 
46752
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changeset

477 
by (unfold Id_on_def) (iprover elim!: UN_E singletonE) 
12905  478 

30198  479 
lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)" 
46752
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480 
by blast 
12905  481 

45967  482 
lemma Id_on_def' [nitpick_unfold]: 
44278
1220ecb81e8f
observe distinction between sets and predicates more properly
haftmann
parents:
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diff
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483 
"Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)" 
46752
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parents:
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484 
by auto 
40923
be80c93ac0a2
adding a nice definition of Id_on for quickcheck and nitpick
bulwahn
parents:
36772
diff
changeset

485 

30198  486 
lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A" 
46752
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487 
by blast 
12905  488 

46692
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changeset

489 
lemma refl_on_Id_on: "refl_on A (Id_on A)" 
46752
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490 
by (rule refl_onI [OF Id_on_subset_Times Id_onI]) 
46692
1f8b766224f6
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haftmann
parents:
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diff
changeset

491 

1f8b766224f6
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parents:
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492 
lemma antisym_Id_on [simp]: "antisym (Id_on A)" 
46752
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493 
by (unfold antisym_def) blast 
46692
1f8b766224f6
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haftmann
parents:
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diff
changeset

494 

1f8b766224f6
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parents:
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495 
lemma sym_Id_on [simp]: "sym (Id_on A)" 
46752
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496 
by (rule symI) clarify 
46692
1f8b766224f6
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haftmann
parents:
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diff
changeset

497 

1f8b766224f6
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changeset

498 
lemma trans_Id_on [simp]: "trans (Id_on A)" 
46752
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parents:
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499 
by (fast intro: transI elim: transD) 
46692
1f8b766224f6
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haftmann
parents:
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diff
changeset

500 

1f8b766224f6
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parents:
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diff
changeset

501 
lemma single_valued_Id_on [simp]: "single_valued (Id_on A)" 
1f8b766224f6
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haftmann
parents:
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changeset

502 
by (unfold single_valued_def) blast 
1f8b766224f6
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503 

12905  504 

46694  505 
subsubsection {* Composition *} 
12905  506 

46752
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507 
inductive_set rel_comp :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a \<times> 'c) set" (infixr "O" 75) 
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508 
for r :: "('a \<times> 'b) set" and s :: "('b \<times> 'c) set" 
46694  509 
where 
46752
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parents:
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changeset

510 
rel_compI [intro]: "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> s \<Longrightarrow> (a, c) \<in> r O s" 
46692
1f8b766224f6
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diff
changeset

511 

46752
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512 
abbreviation pred_comp (infixr "OO" 75) where 
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513 
"pred_comp \<equiv> rel_compp" 
12905  514 

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515 
lemmas pred_compI = rel_compp.intros 
12905  516 

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517 
text {* 
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518 
For historic reasons, the elimination rules are not wholly corresponding. 
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519 
Feel free to consolidate this. 
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520 
*} 
46694  521 

46752
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522 
inductive_cases rel_compEpair: "(a, c) \<in> r O s" 
46694  523 
inductive_cases pred_compE [elim!]: "(r OO s) a c" 
524 

46752
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525 
lemma rel_compE [elim!]: "xz \<in> r O s \<Longrightarrow> 
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changeset

526 
(\<And>x y z. xz = (x, z) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (y, z) \<in> s \<Longrightarrow> P) \<Longrightarrow> P" 
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527 
by (cases xz) (simp, erule rel_compEpair, iprover) 
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

528 

e9e7209eb375
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haftmann
parents:
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changeset

529 
lemmas pred_comp_rel_comp_eq = rel_compp_rel_comp_eq 
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

530 

e9e7209eb375
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changeset

531 
lemma R_O_Id [simp]: 
e9e7209eb375
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haftmann
parents:
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532 
"R O Id = R" 
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parents:
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diff
changeset

533 
by fast 
46694  534 

46752
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535 
lemma Id_O_R [simp]: 
e9e7209eb375
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536 
"Id O R = R" 
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parents:
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diff
changeset

537 
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

538 

e9e7209eb375
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haftmann
parents:
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changeset

539 
lemma rel_comp_empty1 [simp]: 
e9e7209eb375
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haftmann
parents:
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changeset

540 
"{} O R = {}" 
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haftmann
parents:
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diff
changeset

541 
by blast 
12905  542 

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

543 
lemma prod_comp_bot1 [simp]: 
eec472dae593
tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents:
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diff
changeset

544 
"\<bottom> OO R = \<bottom>" 
eec472dae593
tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents:
46882
diff
changeset

545 
by (fact rel_comp_empty1 [to_pred]) 
12905  546 

46752
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haftmann
parents:
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changeset

547 
lemma rel_comp_empty2 [simp]: 
e9e7209eb375
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haftmann
parents:
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diff
changeset

548 
"R O {} = {}" 
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

549 
by blast 
12905  550 

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

551 
lemma pred_comp_bot2 [simp]: 
eec472dae593
tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents:
46882
diff
changeset

552 
"R OO \<bottom> = \<bottom>" 
eec472dae593
tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents:
46882
diff
changeset

553 
by (fact rel_comp_empty2 [to_pred]) 
23185  554 

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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changeset

555 
lemma O_assoc: 
e9e7209eb375
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haftmann
parents:
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diff
changeset

556 
"(R O S) O T = R O (S O T)" 
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haftmann
parents:
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diff
changeset

557 
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

558 

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

559 

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

560 
lemma pred_comp_assoc: 
e9e7209eb375
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haftmann
parents:
46696
diff
changeset

561 
"(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

562 
by (fact O_assoc [to_pred]) 
23185  563 

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

564 
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

565 
"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:
46696
diff
changeset

566 
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

567 

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

568 
lemma transp_pred_comp_less_eq: 
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 
"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

570 
by (fact trans_O_subset [to_pred]) 
12905  571 

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

572 
lemma rel_comp_mono: 
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

573 
"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:
46696
diff
changeset

574 
by blast 
12905  575 

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

576 
lemma pred_comp_mono: 
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

577 
"r' \<le> r \<Longrightarrow> s' \<le> s \<Longrightarrow> r' OO s' \<le> r OO s " 
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 
by (fact rel_comp_mono [to_pred]) 
12905  579 

580 
lemma rel_comp_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:
46696
diff
changeset

581 
"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:
46696
diff
changeset

582 
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

583 

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

584 
lemma rel_comp_distrib [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

585 
"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:
46696
diff
changeset

586 
by auto 
12905  587 

46882  588 
lemma pred_comp_distrib [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

589 
"R OO (S \<squnion> T) = R OO S \<squnion> R 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

590 
by (fact rel_comp_distrib [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

591 

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

592 
lemma rel_comp_distrib2 [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

593 
"(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:
46696
diff
changeset

594 
by auto 
28008
f945f8d9ad4d
added distributivity of relation composition over union [simp]
krauss
parents:
26297
diff
changeset

595 

46882  596 
lemma pred_comp_distrib2 [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

597 
"(S \<squnion> T) OO R = S OO R \<squnion> T OO 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

598 
by (fact rel_comp_distrib2 [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

599 

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

600 
lemma rel_comp_UNION_distrib: 
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

601 
"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:
46696
diff
changeset

602 
by auto 
28008
f945f8d9ad4d
added distributivity of relation composition over union [simp]
krauss
parents:
26297
diff
changeset

603 

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

604 
(* FIXME thm rel_comp_UNION_distrib [to_pred] *) 
36772  605 

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

606 
lemma rel_comp_UNION_distrib2: 
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

607 
"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)
haftmann
parents:
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diff
changeset

608 
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

609 

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

610 
(* FIXME thm rel_comp_UNION_distrib2 [to_pred] *) 
36772  611 

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

612 
lemma single_valued_rel_comp: 
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

613 
"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:
46696
diff
changeset

614 
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

615 

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

616 
lemma rel_comp_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

617 
"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:
46696
diff
changeset

618 
by (auto simp add: set_eq_iff) 
12905  619 

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

620 

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

621 
subsubsection {* Converse *} 
12913  622 

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

623 
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)
haftmann
parents:
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diff
changeset

624 
for r :: "('a \<times> 'b) set" 
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

625 
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

626 
"(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r^1" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

627 

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

628 
notation (xsymbols) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

629 
converse ("(_\<inverse>)" [1000] 999) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

630 

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

631 
notation 
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

632 
conversep ("(_^1)" [1000] 1000) 
46694  633 

634 
notation (xsymbols) 

635 
conversep ("(_\<inverse>\<inverse>)" [1000] 1000) 

636 

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

637 
lemma converseI [sym]: 
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

638 
"(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)
haftmann
parents:
46696
diff
changeset

639 
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:
46696
diff
changeset

640 

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

641 
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:
46696
diff
changeset

642 
"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:
46696
diff
changeset

643 
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

644 

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

645 
lemma converseD [sym]: 
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

646 
"(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)
haftmann
parents:
46696
diff
changeset

647 
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

648 

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 
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:
46696
diff
changeset

650 
"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:
46696
diff
changeset

651 
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

652 

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 
lemma converseE [elim!]: 
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 
 {* 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

655 
"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:
46696
diff
changeset

656 
by (cases yx) (simp, erule converse.cases, iprover) 
46694  657 

46882  658 
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:
46696
diff
changeset

659 

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

660 
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

661 
"(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

662 
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

663 

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

664 
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

665 
"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:
46696
diff
changeset

666 
by (fact converse_iff [to_pred]) 
46694  667 

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

668 
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

669 
"(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

670 
by (simp add: set_eq_iff) 
46694  671 

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

672 
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

673 
"(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

674 
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

675 

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

676 
lemma converse_rel_comp: "(r O s)^1 = s^1 O r^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

677 
by blast 
46694  678 

679 
lemma converse_pred_comp: "(r OO s)^1 = s^1 OO r^1" 

680 
by (iprover intro: order_antisym conversepI pred_compI 

681 
elim: pred_compE dest: conversepD) 

682 

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

683 
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

684 
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

685 

46694  686 
lemma converse_meet: "(r \<sqinter> s)^1 = r^1 \<sqinter> s^1" 
687 
by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD) 

688 

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

689 
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

690 
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

691 

46694  692 
lemma converse_join: "(r \<squnion> s)^1 = r^1 \<squnion> s^1" 
693 
by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD) 

694 

19228  695 
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

696 
by fast 
19228  697 

698 
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

699 
by blast 
19228  700 

12905  701 
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

702 
by blast 
12905  703 

30198  704 
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

705 
by blast 
12905  706 

30198  707 
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

708 
by (unfold refl_on_def) auto 
12905  709 

19228  710 
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

711 
by (unfold sym_def) blast 
19228  712 

713 
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

714 
by (unfold antisym_def) blast 
12905  715 

19228  716 
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

717 
by (unfold trans_def) blast 
12905  718 

19228  719 
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

720 
by (unfold sym_def) fast 
19228  721 

722 
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

723 
by (unfold sym_def) blast 
19228  724 

725 
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

726 
by (unfold sym_def) blast 
19228  727 

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 
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

729 
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

730 

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

731 
lemma finite_converse [iff]: "finite (r^1) = finite r" 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

732 
apply (subgoal_tac "r^1 = (%(x,y). (y,x))`r") 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

733 
apply simp 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

734 
apply (rule iffI) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

735 
apply (erule finite_imageD [unfolded inj_on_def]) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

736 
apply (simp split add: split_split) 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

737 
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

738 
apply (simp add: set_eq_iff image_def, auto) 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

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

740 
prefer 2 apply assumption 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

741 
apply simp 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

742 
done 
12913  743 

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

744 
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

745 
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

746 

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

747 
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

748 
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

749 

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 
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

751 
"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

752 
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

753 

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

754 

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

755 
subsubsection {* Domain, range and field *} 
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

756 

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

757 
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

758 
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

759 
where 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

760 
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

761 

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

762 
abbreviation (input) "DomainP \<equiv> Domainp" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

763 

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

764 
lemmas DomainPI = Domainp.DomainI 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

765 

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

766 
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

767 
inductive_cases DomainpE [elim!]: "Domainp r a" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

768 

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

769 
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

770 
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

771 
where 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

772 
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

773 

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

774 
abbreviation (input) "RangeP \<equiv> Rangep" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

775 

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

776 
lemmas RangePI = Rangep.RangeI 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

777 

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

778 
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

779 
inductive_cases RangepE [elim!]: "Rangep r b" 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

780 

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

781 
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

782 
where 
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset

783 
"Field r = Domain r \<union> Range r" 
12905  784 

46694  785 
lemma Domain_fst [code]: 
786 
"Domain r = fst ` r" 

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

787 
by force 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

788 

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

789 
lemma Range_snd [code]: 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

790 
"Range r = snd ` r" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

791 
by force 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

792 

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

793 
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

794 
by force 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

795 

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

796 
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

797 
by force 
46694  798 

799 
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

800 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

801 

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

802 
lemma Range_empty [simp]: "Range {} = {}" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

803 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

804 

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

805 
lemma Field_empty [simp]: "Field {} = {}" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

806 
by (simp add: Field_def) 
46694  807 

808 
lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}" 

809 
by auto 

810 

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

811 
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

812 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

813 

46882  814 
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

815 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

816 

46882  817 
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

818 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

819 

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

820 
lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \<union> Field r" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

821 
by (auto simp add: Field_def Domain_insert Range_insert) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

822 

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

823 
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

824 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

825 

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

826 
lemma Range_iff: "a \<in> Range r \<longleftrightarrow> (\<exists>y. (y, a) \<in> r)" 
46694  827 
by blast 
828 

829 
lemma Domain_Id [simp]: "Domain Id = UNIV" 

830 
by blast 

831 

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

832 
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

833 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

834 

46694  835 
lemma Domain_Id_on [simp]: "Domain (Id_on A) = A" 
836 
by blast 

837 

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

838 
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

839 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

840 

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

841 
lemma Domain_Un_eq: "Domain (A \<union> B) = Domain A \<union> Domain B" 
46694  842 
by blast 
843 

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

844 
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

845 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

846 

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

847 
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

848 
by (auto simp: Field_def) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

849 

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

850 
lemma Domain_Int_subset: "Domain (A \<inter> B) \<subseteq> Domain A \<inter> Domain B" 
46694  851 
by blast 
852 

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

853 
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

854 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

855 

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

856 
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

857 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

858 

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

859 
lemma Range_Diff_subset: "Range A  Range B \<subseteq> Range (A  B)" 
46694  860 
by blast 
861 

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

862 
lemma Domain_Union: "Domain (\<Union>S) = (\<Union>A\<in>S. Domain A)" 
46694  863 
by blast 
864 

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

865 
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

866 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

867 

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

868 
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

869 
by (auto simp: Field_def) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

870 

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

871 
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

872 
by auto 
46694  873 

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

874 
lemma Range_converse [simp]: "Range (r\<inverse>) = Domain r" 
46694  875 
by blast 
876 

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

877 
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

878 
by (auto simp: Field_def) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

879 

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

880 
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

881 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

882 

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

883 
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

884 
by auto 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

885 

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

886 
lemma finite_Domain: "finite r \<Longrightarrow> finite (Domain r)" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

887 
by (induct set: finite) (auto simp add: Domain_insert) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

888 

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

889 
lemma finite_Range: "finite r \<Longrightarrow> finite (Range r)" 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

890 
by (induct set: finite) (auto simp add: Range_insert) 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

891 

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

892 
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

893 
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

894 

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

895 
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

896 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

897 

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

898 
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

899 
by blast 
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

900 

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

901 
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

902 
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

903 

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

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

905 
"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

906 
by blast 
46694  907 

908 
lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)" 

909 
by auto 

910 

911 
lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)" 

912 
by auto 

913 

12905  914 

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

915 
subsubsection {* Image of a set under a relation *} 
12905  916 

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

917 
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

918 
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

919 
"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

920 

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

921 
declare Image_def [no_atp] 
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
23709
diff
changeset

922 

12913  923 
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

924 
by (simp add: Image_def) 
12905  925 

12913  926 
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

927 
by (simp add: Image_def) 
12905  928 

12913  929 
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

930 
by (rule Image_iff [THEN trans]) simp 
12905  931 

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

932 
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

933 
by (unfold Image_def) blast 
12905  934 

935 
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

936 
"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

937 
by (unfold Image_def) (iprover elim!: CollectE bexE) 
12905  938 

939 
lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A" 

940 
 {* 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

941 
by blast 
12905  942 

943 
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

944 
by blast 
12905  945 

946 
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

947 
by blast 
12905  948 

30198  949 
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

950 
by blast 
13830  951 

952 
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

953 
by blast 
12905  954 

13830  955 
lemma Image_Int_eq: 
46767
807a5d219c23
more fundamental predtoset conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset

956 
"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

957 
by (simp add: single_valued_def, blast) 
12905  958 

13830  959 
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

960 
by blast 
12905  961 

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

962 
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

963 
by blast 
13812
91713a1915ee
converting HOL/UNITY to use unconditional fairness
paulson
parents:
13639
diff
changeset

964 

12913  965 
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

966 
by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2) 
12905  967 

13830  968 
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})" 
12905  969 
 {* 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

970 
by blast 
12905  971 

12913  972 
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

973 
by blast 
12905  974 

13830  975 
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

976 
by blast 
13830  977 

978 
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

979 
by blast 
12905  980 

13830  981 
text{*Converse inclusion requires some assumptions*} 
982 
lemma Image_INT_eq: 

983 
"[single_valued (r\<inverse>); A\<noteq>{}] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)" 

984 
apply (rule equalityI) 

985 
apply (rule Image_INT_subset) 

986 
apply (simp add: single_valued_def, blast) 

987 
done 

12905  988 

12913  989 
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

990 
by blast 
12905  991 

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

992 
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

993 
by auto 
12905  994 

995 

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

996 
subsubsection {* Inverse image *} 
12905  997 

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 
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

999 
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

1000 
"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

1001 

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

1002 
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

1003 
where 
46694  1004 
"inv_imagep r f = (\<lambda>x y. r (f x) (f y))" 
1005 

1006 
lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)" 

1007 
by (simp add: inv_image_def inv_imagep_def) 

1008 

19228  1009 
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

1010 
by (unfold sym_def inv_image_def) blast 
19228  1011 

12913  1012 
lemma trans_inv_image: "trans r ==> trans (inv_image r f)" 
12905  1013 
apply (unfold trans_def inv_image_def) 
1014 
apply (simp (no_asm)) 

1015 
apply blast 

1016 
done 

1017 

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

1018 
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

1019 
by (auto simp:inv_image_def) 
3a0a65ca2261
moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents:
32235
diff
changeset

1020 

33218  1021 
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

1022 
unfolding inv_image_def converse_unfold by auto 
33218  1023 

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

1024 
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

1025 
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

1026 

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

1027 

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

1028 
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

1029 

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

1030 
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

1031 
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

1032 
"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

1033 

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 
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

1035 
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

1036 

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 
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

1038 

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

1039 
end 
46689  1040 