author | wenzelm |
Tue, 02 Jun 2015 09:10:05 +0200 | |
changeset 60357 | bc0827281dc1 |
parent 60057 | 86fa63ce8156 |
child 60758 | d8d85a8172b5 |
permissions | -rw-r--r-- |
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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section {* Relations -- as sets of pairs, and binary predicates *} |
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|
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theory Relation |
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imports Finite_Set |
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begin |
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text {* A preliminary: classical rules for reasoning on predicates *} |
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declare predicate1I [Pure.intro!, intro!] |
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declare predicate1D [Pure.dest, dest] |
|
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declare predicate2I [Pure.intro!, intro!] |
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declare predicate2D [Pure.dest, dest] |
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declare bot1E [elim!] |
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declare bot2E [elim!] |
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declare top1I [intro!] |
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declare top2I [intro!] |
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declare inf1I [intro!] |
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declare inf2I [intro!] |
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declare inf1E [elim!] |
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declare inf2E [elim!] |
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declare sup1I1 [intro?] |
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declare sup2I1 [intro?] |
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declare sup1I2 [intro?] |
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declare sup2I2 [intro?] |
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declare sup1E [elim!] |
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declare sup2E [elim!] |
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declare sup1CI [intro!] |
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declare sup2CI [intro!] |
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declare Inf1_I [intro!] |
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declare INF1_I [intro!] |
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declare Inf2_I [intro!] |
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declare INF2_I [intro!] |
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declare Inf1_D [elim] |
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declare INF1_D [elim] |
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declare Inf2_D [elim] |
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declare INF2_D [elim] |
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declare Inf1_E [elim] |
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declare INF1_E [elim] |
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declare Inf2_E [elim] |
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declare INF2_E [elim] |
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declare Sup1_I [intro] |
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declare SUP1_I [intro] |
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declare Sup2_I [intro] |
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declare SUP2_I [intro] |
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declare Sup1_E [elim!] |
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declare SUP1_E [elim!] |
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declare Sup2_E [elim!] |
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declare SUP2_E [elim!] |
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subsection {* Fundamental *} |
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subsubsection {* Relations as sets of pairs *} |
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||
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type_synonym 'a rel = "('a * 'a) set" |
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lemma subrelI: -- {* Version of @{thm [source] subsetI} for binary relations *} |
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"(\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s) \<Longrightarrow> r \<subseteq> s" |
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by auto |
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lemma lfp_induct2: -- {* Version of @{thm [source] lfp_induct} for binary relations *} |
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"(a, b) \<in> lfp f \<Longrightarrow> mono f \<Longrightarrow> |
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(\<And>a b. (a, b) \<in> f (lfp f \<inter> {(x, y). P x y}) \<Longrightarrow> P a b) \<Longrightarrow> P a b" |
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using lfp_induct_set [of "(a, b)" f "case_prod P"] by auto |
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subsubsection {* Conversions between set and predicate relations *} |
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lemma pred_equals_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) = (\<lambda>x. x \<in> S) \<longleftrightarrow> R = S" |
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by (simp add: set_eq_iff fun_eq_iff) |
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lemma pred_equals_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S) \<longleftrightarrow> R = S" |
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by (simp add: set_eq_iff fun_eq_iff) |
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lemma pred_subset_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S) \<longleftrightarrow> R \<subseteq> S" |
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by (simp add: subset_iff le_fun_def) |
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lemma pred_subset_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S) \<longleftrightarrow> R \<subseteq> S" |
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by (simp add: subset_iff le_fun_def) |
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lemma bot_empty_eq [pred_set_conv]: "\<bottom> = (\<lambda>x. x \<in> {})" |
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by (auto simp add: fun_eq_iff) |
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||
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lemma bot_empty_eq2 [pred_set_conv]: "\<bottom> = (\<lambda>x y. (x, y) \<in> {})" |
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by (auto simp add: fun_eq_iff) |
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lemma top_empty_eq [pred_set_conv]: "\<top> = (\<lambda>x. x \<in> UNIV)" |
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by (auto simp add: fun_eq_iff) |
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lemma top_empty_eq2 [pred_set_conv]: "\<top> = (\<lambda>x y. (x, y) \<in> UNIV)" |
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by (auto simp add: fun_eq_iff) |
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lemma inf_Int_eq [pred_set_conv]: "(\<lambda>x. x \<in> R) \<sqinter> (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)" |
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by (simp add: inf_fun_def) |
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lemma inf_Int_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<sqinter> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<inter> S)" |
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by (simp add: inf_fun_def) |
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|
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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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105 |
lemma sup_Un_eq2 [pred_set_conv]: "(\<lambda>x y. (x, y) \<in> R) \<squnion> (\<lambda>x y. (x, y) \<in> S) = (\<lambda>x y. (x, y) \<in> R \<union> S)" |
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by (simp add: sup_fun_def) |
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lemma INF_INT_eq [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Inter>i\<in>S. r i))" |
109 |
by (simp add: fun_eq_iff) |
|
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||
111 |
lemma INF_INT_eq2 [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Inter>i\<in>S. r i))" |
|
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by (simp add: fun_eq_iff) |
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lemma SUP_UN_eq [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x. x \<in> r i)) = (\<lambda>x. x \<in> (\<Union>i\<in>S. r i))" |
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by (simp add: fun_eq_iff) |
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||
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lemma SUP_UN_eq2 [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x y. (x, y) \<in> r i)) = (\<lambda>x y. (x, y) \<in> (\<Union>i\<in>S. r i))" |
|
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by (simp add: fun_eq_iff) |
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||
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lemma Inf_INT_eq [pred_set_conv]: "\<Sqinter>S = (\<lambda>x. x \<in> INTER S Collect)" |
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by (simp add: fun_eq_iff) |
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|
123 |
lemma INF_Int_eq [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x. x \<in> i)) = (\<lambda>x. x \<in> \<Inter>S)" |
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by (simp add: fun_eq_iff) |
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lemma Inf_INT_eq2 [pred_set_conv]: "\<Sqinter>S = (\<lambda>x y. (x, y) \<in> INTER (case_prod ` S) Collect)" |
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by (simp add: fun_eq_iff) |
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|
129 |
lemma INF_Int_eq2 [pred_set_conv]: "(\<Sqinter>i\<in>S. (\<lambda>x y. (x, y) \<in> i)) = (\<lambda>x y. (x, y) \<in> \<Inter>S)" |
|
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by (simp add: fun_eq_iff) |
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|
132 |
lemma Sup_SUP_eq [pred_set_conv]: "\<Squnion>S = (\<lambda>x. x \<in> UNION S Collect)" |
|
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by (simp add: fun_eq_iff) |
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|
135 |
lemma SUP_Sup_eq [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x. x \<in> i)) = (\<lambda>x. x \<in> \<Union>S)" |
|
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by (simp add: fun_eq_iff) |
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|
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lemma Sup_SUP_eq2 [pred_set_conv]: "\<Squnion>S = (\<lambda>x y. (x, y) \<in> UNION (case_prod ` S) Collect)" |
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by (simp add: fun_eq_iff) |
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|
141 |
lemma SUP_Sup_eq2 [pred_set_conv]: "(\<Squnion>i\<in>S. (\<lambda>x y. (x, y) \<in> i)) = (\<lambda>x y. (x, y) \<in> \<Union>S)" |
|
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by (simp add: fun_eq_iff) |
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|
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subsection {* Properties of relations *} |
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145 |
|
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subsubsection {* Reflexivity *} |
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|
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definition refl_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool" |
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149 |
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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151 |
|
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abbreviation refl :: "'a rel \<Rightarrow> bool" |
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where -- {* reflexivity over a type *} |
45137 | 154 |
"refl \<equiv> refl_on UNIV" |
26297 | 155 |
|
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definition reflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" |
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where |
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"reflp r \<longleftrightarrow> (\<forall>x. r x x)" |
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|
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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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|
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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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166 |
|
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lemma refl_onD: "refl_on A r ==> a : A ==> (a, a) : r" |
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by (unfold refl_on_def) blast |
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169 |
|
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lemma refl_onD1: "refl_on A r ==> (x, y) : r ==> x : A" |
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by (unfold refl_on_def) blast |
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|
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lemma refl_onD2: "refl_on A r ==> (x, y) : r ==> y : A" |
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by (unfold refl_on_def) blast |
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175 |
|
46694 | 176 |
lemma reflpI: |
177 |
"(\<And>x. r x x) \<Longrightarrow> reflp r" |
|
178 |
by (auto intro: refl_onI simp add: reflp_def) |
|
179 |
||
180 |
lemma reflpE: |
|
181 |
assumes "reflp r" |
|
182 |
obtains "r x x" |
|
183 |
using assms by (auto dest: refl_onD simp add: reflp_def) |
|
184 |
||
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lemma reflpD: |
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assumes "reflp r" |
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shows "r x x" |
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using assms by (auto elim: reflpE) |
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189 |
|
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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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|
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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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196 |
|
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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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|
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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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203 |
|
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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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207 |
|
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lemma refl_on_UNION: |
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"ALL x:S. refl_on (A x) (r x) \<Longrightarrow> refl_on (UNION S A) (UNION S r)" |
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by (unfold refl_on_def) blast |
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211 |
|
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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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214 |
|
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lemma refl_on_def' [nitpick_unfold, code]: |
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216 |
"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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218 |
|
60057 | 219 |
lemma reflp_equality [simp]: "reflp op =" |
220 |
by(simp add: reflp_def) |
|
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221 |
|
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subsubsection {* Irreflexivity *} |
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223 |
|
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definition irrefl :: "'a rel \<Rightarrow> bool" |
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225 |
where |
56545 | 226 |
"irrefl r \<longleftrightarrow> (\<forall>a. (a, a) \<notin> r)" |
227 |
||
228 |
definition irreflp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" |
|
229 |
where |
|
230 |
"irreflp R \<longleftrightarrow> (\<forall>a. \<not> R a a)" |
|
231 |
||
232 |
lemma irreflp_irrefl_eq [pred_set_conv]: |
|
233 |
"irreflp (\<lambda>a b. (a, b) \<in> R) \<longleftrightarrow> irrefl R" |
|
234 |
by (simp add: irrefl_def irreflp_def) |
|
235 |
||
236 |
lemma irreflI: |
|
237 |
"(\<And>a. (a, a) \<notin> R) \<Longrightarrow> irrefl R" |
|
238 |
by (simp add: irrefl_def) |
|
239 |
||
240 |
lemma irreflpI: |
|
241 |
"(\<And>a. \<not> R a a) \<Longrightarrow> irreflp R" |
|
242 |
by (fact irreflI [to_pred]) |
|
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243 |
|
46694 | 244 |
lemma irrefl_distinct [code]: |
56545 | 245 |
"irrefl r \<longleftrightarrow> (\<forall>(a, b) \<in> r. a \<noteq> b)" |
46694 | 246 |
by (auto simp add: irrefl_def) |
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247 |
|
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248 |
|
56545 | 249 |
subsubsection {* Asymmetry *} |
250 |
||
251 |
inductive asym :: "'a rel \<Rightarrow> bool" |
|
252 |
where |
|
253 |
asymI: "irrefl R \<Longrightarrow> (\<And>a b. (a, b) \<in> R \<Longrightarrow> (b, a) \<notin> R) \<Longrightarrow> asym R" |
|
254 |
||
255 |
inductive asymp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" |
|
256 |
where |
|
257 |
asympI: "irreflp R \<Longrightarrow> (\<And>a b. R a b \<Longrightarrow> \<not> R b a) \<Longrightarrow> asymp R" |
|
258 |
||
259 |
lemma asymp_asym_eq [pred_set_conv]: |
|
260 |
"asymp (\<lambda>a b. (a, b) \<in> R) \<longleftrightarrow> asym R" |
|
261 |
by (auto intro!: asymI asympI elim: asym.cases asymp.cases simp add: irreflp_irrefl_eq) |
|
262 |
||
263 |
||
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264 |
subsubsection {* Symmetry *} |
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265 |
|
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definition sym :: "'a rel \<Rightarrow> bool" |
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267 |
where |
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268 |
"sym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r)" |
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269 |
|
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270 |
definition symp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" |
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271 |
where |
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272 |
"symp r \<longleftrightarrow> (\<forall>x y. r x y \<longrightarrow> r y x)" |
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273 |
|
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274 |
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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277 |
|
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278 |
lemma symI: |
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279 |
"(\<And>a b. (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r) \<Longrightarrow> sym r" |
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280 |
by (unfold sym_def) iprover |
46694 | 281 |
|
282 |
lemma sympI: |
|
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283 |
"(\<And>a b. r a b \<Longrightarrow> r b a) \<Longrightarrow> symp r" |
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284 |
by (fact symI [to_pred]) |
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285 |
|
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286 |
lemma symE: |
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287 |
assumes "sym r" and "(b, a) \<in> r" |
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288 |
obtains "(a, b) \<in> r" |
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289 |
using assms by (simp add: sym_def) |
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|
291 |
lemma sympE: |
|
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292 |
assumes "symp r" and "r b a" |
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obtains "r a b" |
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294 |
using assms by (rule symE [to_pred]) |
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295 |
|
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296 |
lemma symD: |
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297 |
assumes "sym r" and "(b, a) \<in> r" |
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298 |
shows "(a, b) \<in> r" |
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299 |
using assms by (rule symE) |
46694 | 300 |
|
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301 |
lemma sympD: |
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302 |
assumes "symp r" and "r b a" |
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303 |
shows "r a b" |
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304 |
using assms by (rule symD [to_pred]) |
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305 |
|
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306 |
lemma sym_Int: |
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307 |
"sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<inter> s)" |
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308 |
by (fast intro: symI elim: symE) |
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309 |
|
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310 |
lemma symp_inf: |
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311 |
"symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<sqinter> s)" |
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312 |
by (fact sym_Int [to_pred]) |
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313 |
|
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314 |
lemma sym_Un: |
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315 |
"sym r \<Longrightarrow> sym s \<Longrightarrow> sym (r \<union> s)" |
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316 |
by (fast intro: symI elim: symE) |
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317 |
|
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318 |
lemma symp_sup: |
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319 |
"symp r \<Longrightarrow> symp s \<Longrightarrow> symp (r \<squnion> s)" |
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320 |
by (fact sym_Un [to_pred]) |
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321 |
|
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322 |
lemma sym_INTER: |
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323 |
"\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (INTER S r)" |
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324 |
by (fast intro: symI elim: symE) |
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325 |
|
46982 | 326 |
lemma symp_INF: |
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327 |
"\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (INFIMUM S r)" |
46982 | 328 |
by (fact sym_INTER [to_pred]) |
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329 |
|
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330 |
lemma sym_UNION: |
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331 |
"\<forall>x\<in>S. sym (r x) \<Longrightarrow> sym (UNION S r)" |
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332 |
by (fast intro: symI elim: symE) |
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333 |
|
46982 | 334 |
lemma symp_SUP: |
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335 |
"\<forall>x\<in>S. symp (r x) \<Longrightarrow> symp (SUPREMUM S r)" |
46982 | 336 |
by (fact sym_UNION [to_pred]) |
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337 |
|
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338 |
|
46694 | 339 |
subsubsection {* Antisymmetry *} |
340 |
||
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341 |
definition antisym :: "'a rel \<Rightarrow> bool" |
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342 |
where |
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343 |
"antisym r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r \<longrightarrow> x = y)" |
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344 |
|
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345 |
abbreviation antisymP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" |
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346 |
where |
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347 |
"antisymP r \<equiv> antisym {(x, y). r x y}" |
46694 | 348 |
|
349 |
lemma antisymI: |
|
350 |
"(!!x y. (x, y) : r ==> (y, x) : r ==> x=y) ==> antisym r" |
|
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351 |
by (unfold antisym_def) iprover |
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|
353 |
lemma antisymD: "antisym r ==> (a, b) : r ==> (b, a) : r ==> a = b" |
|
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354 |
by (unfold antisym_def) iprover |
46694 | 355 |
|
356 |
lemma antisym_subset: "r \<subseteq> s ==> antisym s ==> antisym r" |
|
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357 |
by (unfold antisym_def) blast |
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|
359 |
lemma antisym_empty [simp]: "antisym {}" |
|
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360 |
by (unfold antisym_def) blast |
46694 | 361 |
|
60057 | 362 |
lemma antisymP_equality [simp]: "antisymP op =" |
363 |
by(auto intro: antisymI) |
|
46694 | 364 |
|
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365 |
subsubsection {* Transitivity *} |
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366 |
|
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367 |
definition trans :: "'a rel \<Rightarrow> bool" |
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368 |
where |
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369 |
"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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370 |
|
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371 |
definition transp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" |
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372 |
where |
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373 |
"transp r \<longleftrightarrow> (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)" |
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|
374 |
|
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375 |
lemma transp_trans_eq [pred_set_conv]: |
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376 |
"transp (\<lambda>x y. (x, y) \<in> r) \<longleftrightarrow> trans r" |
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377 |
by (simp add: trans_def transp_def) |
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378 |
|
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379 |
abbreviation transP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool" |
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380 |
where -- {* FIXME drop *} |
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381 |
"transP r \<equiv> trans {(x, y). r x y}" |
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|
382 |
|
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383 |
lemma transI: |
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384 |
"(\<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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385 |
by (unfold trans_def) iprover |
46694 | 386 |
|
387 |
lemma transpI: |
|
388 |
"(\<And>x y z. r x y \<Longrightarrow> r y z \<Longrightarrow> r x z) \<Longrightarrow> transp r" |
|
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389 |
by (fact transI [to_pred]) |
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390 |
|
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391 |
lemma transE: |
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392 |
assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r" |
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393 |
obtains "(x, z) \<in> r" |
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394 |
using assms by (unfold trans_def) iprover |
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395 |
|
46694 | 396 |
lemma transpE: |
397 |
assumes "transp r" and "r x y" and "r y z" |
|
398 |
obtains "r x z" |
|
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399 |
using assms by (rule transE [to_pred]) |
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400 |
|
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401 |
lemma transD: |
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more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
402 |
assumes "trans r" and "(x, y) \<in> r" and "(y, z) \<in> r" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
403 |
shows "(x, z) \<in> r" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
404 |
using assms by (rule transE) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
405 |
|
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
406 |
lemma transpD: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
407 |
assumes "transp r" and "r x y" and "r y z" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
408 |
shows "r x z" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
409 |
using assms by (rule transD [to_pred]) |
46694 | 410 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
411 |
lemma trans_Int: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
412 |
"trans r \<Longrightarrow> trans s \<Longrightarrow> trans (r \<inter> s)" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
413 |
by (fast intro: transI elim: transE) |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
414 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
415 |
lemma transp_inf: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
416 |
"transp r \<Longrightarrow> transp s \<Longrightarrow> transp (r \<sqinter> s)" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
417 |
by (fact trans_Int [to_pred]) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
418 |
|
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
419 |
lemma trans_INTER: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
420 |
"\<forall>x\<in>S. trans (r x) \<Longrightarrow> trans (INTER S r)" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
421 |
by (fast intro: transI elim: transD) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
422 |
|
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
423 |
(* FIXME thm trans_INTER [to_pred] *) |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
424 |
|
46694 | 425 |
lemma trans_join [code]: |
426 |
"trans r \<longleftrightarrow> (\<forall>(x, y1) \<in> r. \<forall>(y2, z) \<in> r. y1 = y2 \<longrightarrow> (x, z) \<in> r)" |
|
427 |
by (auto simp add: trans_def) |
|
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
428 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
429 |
lemma transp_trans: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
430 |
"transp r \<longleftrightarrow> trans {(x, y). r x y}" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
431 |
by (simp add: trans_def transp_def) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
432 |
|
59518 | 433 |
lemma transp_equality [simp]: "transp op =" |
434 |
by(auto intro: transpI) |
|
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
435 |
|
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
436 |
subsubsection {* Totality *} |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
437 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
438 |
definition total_on :: "'a set \<Rightarrow> 'a rel \<Rightarrow> bool" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
439 |
where |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
440 |
"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)" |
29859
33bff35f1335
Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents:
29609
diff
changeset
|
441 |
|
33bff35f1335
Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents:
29609
diff
changeset
|
442 |
abbreviation "total \<equiv> total_on UNIV" |
33bff35f1335
Moved Order_Relation into Library and moved some of it into Relation.
nipkow
parents:
29609
diff
changeset
|
443 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
444 |
lemma total_on_empty [simp]: "total_on {} r" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
445 |
by (simp add: total_on_def) |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
446 |
|
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
447 |
|
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
448 |
subsubsection {* Single valued relations *} |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
449 |
|
46752
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more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
450 |
definition single_valued :: "('a \<times> 'b) set \<Rightarrow> bool" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
451 |
where |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
452 |
"single_valued r \<longleftrightarrow> (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))" |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
453 |
|
46694 | 454 |
abbreviation single_valuedP :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where |
455 |
"single_valuedP r \<equiv> single_valued {(x, y). r x y}" |
|
456 |
||
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
457 |
lemma single_valuedI: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
458 |
"ALL x y. (x,y):r --> (ALL z. (x,z):r --> y=z) ==> single_valued r" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
459 |
by (unfold single_valued_def) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
460 |
|
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
461 |
lemma single_valuedD: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
462 |
"single_valued r ==> (x, y) : r ==> (x, z) : r ==> y = z" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
463 |
by (simp add: single_valued_def) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
464 |
|
57111 | 465 |
lemma single_valued_empty[simp]: "single_valued {}" |
52392 | 466 |
by(simp add: single_valued_def) |
467 |
||
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
468 |
lemma single_valued_subset: |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
469 |
"r \<subseteq> s ==> single_valued s ==> single_valued r" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
470 |
by (unfold single_valued_def) blast |
11136 | 471 |
|
12905 | 472 |
|
46694 | 473 |
subsection {* Relation operations *} |
474 |
||
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
|
475 |
subsubsection {* The identity relation *} |
12905 | 476 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
477 |
definition Id :: "'a rel" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
478 |
where |
48253
4410a709913c
a first guess to avoid the Codegenerator_Test to loop infinitely
bulwahn
parents:
47937
diff
changeset
|
479 |
[code del]: "Id = {p. \<exists>x. p = (x, x)}" |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
480 |
|
12905 | 481 |
lemma IdI [intro]: "(a, a) : Id" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
482 |
by (simp add: Id_def) |
12905 | 483 |
|
484 |
lemma IdE [elim!]: "p : Id ==> (!!x. p = (x, x) ==> P) ==> P" |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
485 |
by (unfold Id_def) (iprover elim: CollectE) |
12905 | 486 |
|
487 |
lemma pair_in_Id_conv [iff]: "((a, b) : Id) = (a = b)" |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
488 |
by (unfold Id_def) blast |
12905 | 489 |
|
30198 | 490 |
lemma refl_Id: "refl Id" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
491 |
by (simp add: refl_on_def) |
12905 | 492 |
|
493 |
lemma antisym_Id: "antisym Id" |
|
494 |
-- {* A strange result, since @{text Id} is also symmetric. *} |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
495 |
by (simp add: antisym_def) |
12905 | 496 |
|
19228 | 497 |
lemma sym_Id: "sym Id" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
498 |
by (simp add: sym_def) |
19228 | 499 |
|
12905 | 500 |
lemma trans_Id: "trans Id" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
501 |
by (simp add: trans_def) |
12905 | 502 |
|
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
503 |
lemma single_valued_Id [simp]: "single_valued Id" |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
504 |
by (unfold single_valued_def) blast |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
505 |
|
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
506 |
lemma irrefl_diff_Id [simp]: "irrefl (r - Id)" |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
507 |
by (simp add:irrefl_def) |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
508 |
|
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
509 |
lemma trans_diff_Id: "trans r \<Longrightarrow> antisym r \<Longrightarrow> trans (r - Id)" |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
510 |
unfolding antisym_def trans_def by blast |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
511 |
|
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
512 |
lemma total_on_diff_Id [simp]: "total_on A (r - Id) = total_on A r" |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
513 |
by (simp add: total_on_def) |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
514 |
|
12905 | 515 |
|
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
|
516 |
subsubsection {* Diagonal: identity over a set *} |
12905 | 517 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
518 |
definition Id_on :: "'a set \<Rightarrow> 'a rel" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
519 |
where |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
520 |
"Id_on A = (\<Union>x\<in>A. {(x, x)})" |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
521 |
|
30198 | 522 |
lemma Id_on_empty [simp]: "Id_on {} = {}" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
523 |
by (simp add: Id_on_def) |
13812
91713a1915ee
converting HOL/UNITY to use unconditional fairness
paulson
parents:
13639
diff
changeset
|
524 |
|
30198 | 525 |
lemma Id_on_eqI: "a = b ==> a : A ==> (a, b) : Id_on A" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
526 |
by (simp add: Id_on_def) |
12905 | 527 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53680
diff
changeset
|
528 |
lemma Id_onI [intro!]: "a : A ==> (a, a) : Id_on A" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
529 |
by (rule Id_on_eqI) (rule refl) |
12905 | 530 |
|
30198 | 531 |
lemma Id_onE [elim!]: |
532 |
"c : Id_on A ==> (!!x. x : A ==> c = (x, x) ==> P) ==> P" |
|
12913 | 533 |
-- {* The general elimination rule. *} |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
534 |
by (unfold Id_on_def) (iprover elim!: UN_E singletonE) |
12905 | 535 |
|
30198 | 536 |
lemma Id_on_iff: "((x, y) : Id_on A) = (x = y & x : A)" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
537 |
by blast |
12905 | 538 |
|
45967 | 539 |
lemma Id_on_def' [nitpick_unfold]: |
44278
1220ecb81e8f
observe distinction between sets and predicates more properly
haftmann
parents:
41792
diff
changeset
|
540 |
"Id_on {x. A x} = Collect (\<lambda>(x, y). x = y \<and> A x)" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
541 |
by auto |
40923
be80c93ac0a2
adding a nice definition of Id_on for quickcheck and nitpick
bulwahn
parents:
36772
diff
changeset
|
542 |
|
30198 | 543 |
lemma Id_on_subset_Times: "Id_on A \<subseteq> A \<times> A" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
544 |
by blast |
12905 | 545 |
|
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
546 |
lemma refl_on_Id_on: "refl_on A (Id_on A)" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
547 |
by (rule refl_onI [OF Id_on_subset_Times Id_onI]) |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
548 |
|
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
549 |
lemma antisym_Id_on [simp]: "antisym (Id_on A)" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
550 |
by (unfold antisym_def) blast |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
551 |
|
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
552 |
lemma sym_Id_on [simp]: "sym (Id_on A)" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
553 |
by (rule symI) clarify |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
554 |
|
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
555 |
lemma trans_Id_on [simp]: "trans (Id_on A)" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
556 |
by (fast intro: transI elim: transD) |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
557 |
|
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
558 |
lemma single_valued_Id_on [simp]: "single_valued (Id_on A)" |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
559 |
by (unfold single_valued_def) blast |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
560 |
|
12905 | 561 |
|
46694 | 562 |
subsubsection {* Composition *} |
12905 | 563 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
564 |
inductive_set relcomp :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'c) set \<Rightarrow> ('a \<times> 'c) set" (infixr "O" 75) |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
565 |
for r :: "('a \<times> 'b) set" and s :: "('b \<times> 'c) set" |
46694 | 566 |
where |
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
567 |
relcompI [intro]: "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> s \<Longrightarrow> (a, c) \<in> r O s" |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
568 |
|
47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset
|
569 |
notation relcompp (infixr "OO" 75) |
12905 | 570 |
|
47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset
|
571 |
lemmas relcomppI = relcompp.intros |
12905 | 572 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
573 |
text {* |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
574 |
For historic reasons, the elimination rules are not wholly corresponding. |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
575 |
Feel free to consolidate this. |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
576 |
*} |
46694 | 577 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
578 |
inductive_cases relcompEpair: "(a, c) \<in> r O s" |
47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset
|
579 |
inductive_cases relcomppE [elim!]: "(r OO s) a c" |
46694 | 580 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
581 |
lemma relcompE [elim!]: "xz \<in> r O s \<Longrightarrow> |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
582 |
(\<And>x y z. xz = (x, z) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> (y, z) \<in> s \<Longrightarrow> P) \<Longrightarrow> P" |
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
583 |
by (cases xz) (simp, erule relcompEpair, iprover) |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
584 |
|
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
585 |
lemma R_O_Id [simp]: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
586 |
"R O Id = R" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
587 |
by fast |
46694 | 588 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
589 |
lemma Id_O_R [simp]: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
590 |
"Id O R = R" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
591 |
by fast |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
592 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
593 |
lemma relcomp_empty1 [simp]: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
594 |
"{} O R = {}" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
595 |
by blast |
12905 | 596 |
|
47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset
|
597 |
lemma relcompp_bot1 [simp]: |
46883
eec472dae593
tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents:
46882
diff
changeset
|
598 |
"\<bottom> OO R = \<bottom>" |
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
599 |
by (fact relcomp_empty1 [to_pred]) |
12905 | 600 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
601 |
lemma relcomp_empty2 [simp]: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
602 |
"R O {} = {}" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
603 |
by blast |
12905 | 604 |
|
47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset
|
605 |
lemma relcompp_bot2 [simp]: |
46883
eec472dae593
tuned pred_set_conv lemmas. Skipped lemmas changing the lemmas generated by inductive_set
noschinl
parents:
46882
diff
changeset
|
606 |
"R OO \<bottom> = \<bottom>" |
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
607 |
by (fact relcomp_empty2 [to_pred]) |
23185 | 608 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
609 |
lemma O_assoc: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
610 |
"(R O S) O T = R O (S O T)" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
611 |
by blast |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
612 |
|
47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset
|
613 |
lemma relcompp_assoc: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
614 |
"(r OO s) OO t = r OO (s OO t)" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
615 |
by (fact O_assoc [to_pred]) |
23185 | 616 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
617 |
lemma trans_O_subset: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
618 |
"trans r \<Longrightarrow> r O r \<subseteq> r" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
619 |
by (unfold trans_def) blast |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
620 |
|
47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset
|
621 |
lemma transp_relcompp_less_eq: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
622 |
"transp r \<Longrightarrow> r OO r \<le> r " |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
623 |
by (fact trans_O_subset [to_pred]) |
12905 | 624 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
625 |
lemma relcomp_mono: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
626 |
"r' \<subseteq> r \<Longrightarrow> s' \<subseteq> s \<Longrightarrow> r' O s' \<subseteq> r O s" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
627 |
by blast |
12905 | 628 |
|
47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset
|
629 |
lemma relcompp_mono: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
630 |
"r' \<le> r \<Longrightarrow> s' \<le> s \<Longrightarrow> r' OO s' \<le> r OO s " |
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
631 |
by (fact relcomp_mono [to_pred]) |
12905 | 632 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
633 |
lemma relcomp_subset_Sigma: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
634 |
"r \<subseteq> A \<times> B \<Longrightarrow> s \<subseteq> B \<times> C \<Longrightarrow> r O s \<subseteq> A \<times> C" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
635 |
by blast |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
636 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
637 |
lemma relcomp_distrib [simp]: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
638 |
"R O (S \<union> T) = (R O S) \<union> (R O T)" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
639 |
by auto |
12905 | 640 |
|
47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset
|
641 |
lemma relcompp_distrib [simp]: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
642 |
"R OO (S \<squnion> T) = R OO S \<squnion> R OO T" |
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
643 |
by (fact relcomp_distrib [to_pred]) |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
644 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
645 |
lemma relcomp_distrib2 [simp]: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
646 |
"(S \<union> T) O R = (S O R) \<union> (T O R)" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
647 |
by auto |
28008
f945f8d9ad4d
added distributivity of relation composition over union [simp]
krauss
parents:
26297
diff
changeset
|
648 |
|
47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset
|
649 |
lemma relcompp_distrib2 [simp]: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
650 |
"(S \<squnion> T) OO R = S OO R \<squnion> T OO R" |
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
651 |
by (fact relcomp_distrib2 [to_pred]) |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
652 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
653 |
lemma relcomp_UNION_distrib: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
654 |
"s O UNION I r = (\<Union>i\<in>I. s O r i) " |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
655 |
by auto |
28008
f945f8d9ad4d
added distributivity of relation composition over union [simp]
krauss
parents:
26297
diff
changeset
|
656 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
657 |
(* FIXME thm relcomp_UNION_distrib [to_pred] *) |
36772 | 658 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
659 |
lemma relcomp_UNION_distrib2: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
660 |
"UNION I r O s = (\<Union>i\<in>I. r i O s) " |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
661 |
by auto |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
662 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
663 |
(* FIXME thm relcomp_UNION_distrib2 [to_pred] *) |
36772 | 664 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
665 |
lemma single_valued_relcomp: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
666 |
"single_valued r \<Longrightarrow> single_valued s \<Longrightarrow> single_valued (r O s)" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
667 |
by (unfold single_valued_def) blast |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
668 |
|
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
669 |
lemma relcomp_unfold: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
670 |
"r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
671 |
by (auto simp add: set_eq_iff) |
12905 | 672 |
|
58195 | 673 |
lemma relcompp_apply: "(R OO S) a c \<longleftrightarrow> (\<exists>b. R a b \<and> S b c)" |
674 |
unfolding relcomp_unfold [to_pred] .. |
|
675 |
||
55083 | 676 |
lemma eq_OO: "op= OO R = R" |
677 |
by blast |
|
678 |
||
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
|
679 |
|
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
|
680 |
subsubsection {* Converse *} |
12913 | 681 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
682 |
inductive_set converse :: "('a \<times> 'b) set \<Rightarrow> ('b \<times> 'a) set" ("(_^-1)" [1000] 999) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
683 |
for r :: "('a \<times> 'b) set" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
684 |
where |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
685 |
"(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r^-1" |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
686 |
|
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
687 |
notation (xsymbols) |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
688 |
converse ("(_\<inverse>)" [1000] 999) |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
689 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
690 |
notation |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
691 |
conversep ("(_^--1)" [1000] 1000) |
46694 | 692 |
|
693 |
notation (xsymbols) |
|
694 |
conversep ("(_\<inverse>\<inverse>)" [1000] 1000) |
|
695 |
||
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
696 |
lemma converseI [sym]: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
697 |
"(a, b) \<in> r \<Longrightarrow> (b, a) \<in> r\<inverse>" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
698 |
by (fact converse.intros) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
699 |
|
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
700 |
lemma conversepI (* CANDIDATE [sym] *): |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
701 |
"r a b \<Longrightarrow> r\<inverse>\<inverse> b a" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
702 |
by (fact conversep.intros) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
703 |
|
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
704 |
lemma converseD [sym]: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
705 |
"(a, b) \<in> r\<inverse> \<Longrightarrow> (b, a) \<in> r" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
706 |
by (erule converse.cases) iprover |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
707 |
|
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
708 |
lemma conversepD (* CANDIDATE [sym] *): |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
709 |
"r\<inverse>\<inverse> b a \<Longrightarrow> r a b" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
710 |
by (fact converseD [to_pred]) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
711 |
|
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
712 |
lemma converseE [elim!]: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
713 |
-- {* More general than @{text converseD}, as it ``splits'' the member of the relation. *} |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
714 |
"yx \<in> r\<inverse> \<Longrightarrow> (\<And>x y. yx = (y, x) \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> P) \<Longrightarrow> P" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
715 |
by (cases yx) (simp, erule converse.cases, iprover) |
46694 | 716 |
|
46882 | 717 |
lemmas conversepE [elim!] = conversep.cases |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
718 |
|
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
719 |
lemma converse_iff [iff]: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
720 |
"(a, b) \<in> r\<inverse> \<longleftrightarrow> (b, a) \<in> r" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
721 |
by (auto intro: converseI) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
722 |
|
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
723 |
lemma conversep_iff [iff]: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
724 |
"r\<inverse>\<inverse> a b = r b a" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
725 |
by (fact converse_iff [to_pred]) |
46694 | 726 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
727 |
lemma converse_converse [simp]: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
728 |
"(r\<inverse>)\<inverse> = r" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
729 |
by (simp add: set_eq_iff) |
46694 | 730 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
731 |
lemma conversep_conversep [simp]: |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
732 |
"(r\<inverse>\<inverse>)\<inverse>\<inverse> = r" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
733 |
by (fact converse_converse [to_pred]) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
734 |
|
53680 | 735 |
lemma converse_empty[simp]: "{}\<inverse> = {}" |
736 |
by auto |
|
737 |
||
738 |
lemma converse_UNIV[simp]: "UNIV\<inverse> = UNIV" |
|
739 |
by auto |
|
740 |
||
47433
07f4bf913230
renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents:
47087
diff
changeset
|
741 |
lemma converse_relcomp: "(r O s)^-1 = s^-1 O r^-1" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
742 |
by blast |
46694 | 743 |
|
47434
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset
|
744 |
lemma converse_relcompp: "(r OO s)^--1 = s^--1 OO r^--1" |
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset
|
745 |
by (iprover intro: order_antisym conversepI relcomppI |
b75ce48a93ee
dropped abbreviation "pred_comp"; introduced infix notation "P OO Q" for "relcompp P Q"
griff
parents:
47433
diff
changeset
|
746 |
elim: relcomppE dest: conversepD) |
46694 | 747 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
748 |
lemma converse_Int: "(r \<inter> s)^-1 = r^-1 \<inter> s^-1" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
749 |
by blast |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
750 |
|
46694 | 751 |
lemma converse_meet: "(r \<sqinter> s)^--1 = r^--1 \<sqinter> s^--1" |
752 |
by (simp add: inf_fun_def) (iprover intro: conversepI ext dest: conversepD) |
|
753 |
||
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
754 |
lemma converse_Un: "(r \<union> s)^-1 = r^-1 \<union> s^-1" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
755 |
by blast |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
756 |
|
46694 | 757 |
lemma converse_join: "(r \<squnion> s)^--1 = r^--1 \<squnion> s^--1" |
758 |
by (simp add: sup_fun_def) (iprover intro: conversepI ext dest: conversepD) |
|
759 |
||
19228 | 760 |
lemma converse_INTER: "(INTER S r)^-1 = (INT x:S. (r x)^-1)" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
761 |
by fast |
19228 | 762 |
|
763 |
lemma converse_UNION: "(UNION S r)^-1 = (UN x:S. (r x)^-1)" |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
764 |
by blast |
19228 | 765 |
|
52749 | 766 |
lemma converse_mono[simp]: "r^-1 \<subseteq> s ^-1 \<longleftrightarrow> r \<subseteq> s" |
767 |
by auto |
|
768 |
||
769 |
lemma conversep_mono[simp]: "r^--1 \<le> s ^--1 \<longleftrightarrow> r \<le> s" |
|
770 |
by (fact converse_mono[to_pred]) |
|
771 |
||
772 |
lemma converse_inject[simp]: "r^-1 = s ^-1 \<longleftrightarrow> r = s" |
|
52730 | 773 |
by auto |
774 |
||
52749 | 775 |
lemma conversep_inject[simp]: "r^--1 = s ^--1 \<longleftrightarrow> r = s" |
776 |
by (fact converse_inject[to_pred]) |
|
777 |
||
778 |
lemma converse_subset_swap: "r \<subseteq> s ^-1 = (r ^-1 \<subseteq> s)" |
|
779 |
by auto |
|
780 |
||
781 |
lemma conversep_le_swap: "r \<le> s ^--1 = (r ^--1 \<le> s)" |
|
782 |
by (fact converse_subset_swap[to_pred]) |
|
52730 | 783 |
|
12905 | 784 |
lemma converse_Id [simp]: "Id^-1 = Id" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
785 |
by blast |
12905 | 786 |
|
30198 | 787 |
lemma converse_Id_on [simp]: "(Id_on A)^-1 = Id_on A" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
788 |
by blast |
12905 | 789 |
|
30198 | 790 |
lemma refl_on_converse [simp]: "refl_on A (converse r) = refl_on A r" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
791 |
by (unfold refl_on_def) auto |
12905 | 792 |
|
19228 | 793 |
lemma sym_converse [simp]: "sym (converse r) = sym r" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
794 |
by (unfold sym_def) blast |
19228 | 795 |
|
796 |
lemma antisym_converse [simp]: "antisym (converse r) = antisym r" |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
797 |
by (unfold antisym_def) blast |
12905 | 798 |
|
19228 | 799 |
lemma trans_converse [simp]: "trans (converse r) = trans r" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
800 |
by (unfold trans_def) blast |
12905 | 801 |
|
19228 | 802 |
lemma sym_conv_converse_eq: "sym r = (r^-1 = r)" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
803 |
by (unfold sym_def) fast |
19228 | 804 |
|
805 |
lemma sym_Un_converse: "sym (r \<union> r^-1)" |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
806 |
by (unfold sym_def) blast |
19228 | 807 |
|
808 |
lemma sym_Int_converse: "sym (r \<inter> r^-1)" |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
809 |
by (unfold sym_def) blast |
19228 | 810 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
811 |
lemma total_on_converse [simp]: "total_on A (r^-1) = total_on A r" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
812 |
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
|
813 |
|
52749 | 814 |
lemma finite_converse [iff]: "finite (r^-1) = finite r" |
54611
31afce809794
set_comprehension_pointfree simproc causes to many surprises if enabled by default
traytel
parents:
54555
diff
changeset
|
815 |
unfolding converse_def conversep_iff using [[simproc add: finite_Collect]] |
31afce809794
set_comprehension_pointfree simproc causes to many surprises if enabled by default
traytel
parents:
54555
diff
changeset
|
816 |
by (auto elim: finite_imageD simp: inj_on_def) |
12913 | 817 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
818 |
lemma conversep_noteq [simp]: "(op \<noteq>)^--1 = op \<noteq>" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
819 |
by (auto simp add: fun_eq_iff) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
820 |
|
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
821 |
lemma conversep_eq [simp]: "(op =)^--1 = op =" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
822 |
by (auto simp add: fun_eq_iff) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
823 |
|
53680 | 824 |
lemma converse_unfold [code]: |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
825 |
"r\<inverse> = {(y, x). (x, y) \<in> r}" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
826 |
by (simp add: set_eq_iff) |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
827 |
|
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
828 |
|
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
829 |
subsubsection {* Domain, range and field *} |
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
830 |
|
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
831 |
inductive_set Domain :: "('a \<times> 'b) set \<Rightarrow> 'a set" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
832 |
for r :: "('a \<times> 'b) set" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
833 |
where |
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
834 |
DomainI [intro]: "(a, b) \<in> r \<Longrightarrow> a \<in> Domain r" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
835 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
836 |
abbreviation (input) "DomainP \<equiv> Domainp" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
837 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
838 |
lemmas DomainPI = Domainp.DomainI |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
839 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
840 |
inductive_cases DomainE [elim!]: "a \<in> Domain r" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
841 |
inductive_cases DomainpE [elim!]: "Domainp r a" |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
842 |
|
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
843 |
inductive_set Range :: "('a \<times> 'b) set \<Rightarrow> 'b set" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
844 |
for r :: "('a \<times> 'b) set" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
845 |
where |
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
846 |
RangeI [intro]: "(a, b) \<in> r \<Longrightarrow> b \<in> Range r" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
847 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
848 |
abbreviation (input) "RangeP \<equiv> Rangep" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
849 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
850 |
lemmas RangePI = Rangep.RangeI |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
851 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
852 |
inductive_cases RangeE [elim!]: "b \<in> Range r" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
853 |
inductive_cases RangepE [elim!]: "Rangep r b" |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
854 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
855 |
definition Field :: "'a rel \<Rightarrow> 'a set" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
856 |
where |
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
857 |
"Field r = Domain r \<union> Range r" |
12905 | 858 |
|
46694 | 859 |
lemma Domain_fst [code]: |
860 |
"Domain r = fst ` r" |
|
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
861 |
by force |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
862 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
863 |
lemma Range_snd [code]: |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
864 |
"Range r = snd ` r" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
865 |
by force |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
866 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
867 |
lemma fst_eq_Domain: "fst ` R = Domain R" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
868 |
by force |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
869 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
870 |
lemma snd_eq_Range: "snd ` R = Range R" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
871 |
by force |
46694 | 872 |
|
873 |
lemma Domain_empty [simp]: "Domain {} = {}" |
|
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
874 |
by auto |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
875 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
876 |
lemma Range_empty [simp]: "Range {} = {}" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
877 |
by auto |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
878 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
879 |
lemma Field_empty [simp]: "Field {} = {}" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
880 |
by (simp add: Field_def) |
46694 | 881 |
|
882 |
lemma Domain_empty_iff: "Domain r = {} \<longleftrightarrow> r = {}" |
|
883 |
by auto |
|
884 |
||
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
885 |
lemma Range_empty_iff: "Range r = {} \<longleftrightarrow> r = {}" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
886 |
by auto |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
887 |
|
46882 | 888 |
lemma Domain_insert [simp]: "Domain (insert (a, b) r) = insert a (Domain r)" |
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
889 |
by blast |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
890 |
|
46882 | 891 |
lemma Range_insert [simp]: "Range (insert (a, b) r) = insert b (Range r)" |
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
892 |
by blast |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
893 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
894 |
lemma Field_insert [simp]: "Field (insert (a, b) r) = {a, b} \<union> Field r" |
46884 | 895 |
by (auto simp add: Field_def) |
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
896 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
897 |
lemma Domain_iff: "a \<in> Domain r \<longleftrightarrow> (\<exists>y. (a, y) \<in> r)" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
898 |
by blast |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
899 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
900 |
lemma Range_iff: "a \<in> Range r \<longleftrightarrow> (\<exists>y. (y, a) \<in> r)" |
46694 | 901 |
by blast |
902 |
||
903 |
lemma Domain_Id [simp]: "Domain Id = UNIV" |
|
904 |
by blast |
|
905 |
||
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
906 |
lemma Range_Id [simp]: "Range Id = UNIV" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
907 |
by blast |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
908 |
|
46694 | 909 |
lemma Domain_Id_on [simp]: "Domain (Id_on A) = A" |
910 |
by blast |
|
911 |
||
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
912 |
lemma Range_Id_on [simp]: "Range (Id_on A) = A" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
913 |
by blast |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
914 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
915 |
lemma Domain_Un_eq: "Domain (A \<union> B) = Domain A \<union> Domain B" |
46694 | 916 |
by blast |
917 |
||
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
918 |
lemma Range_Un_eq: "Range (A \<union> B) = Range A \<union> Range B" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
919 |
by blast |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
920 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
921 |
lemma Field_Un [simp]: "Field (r \<union> s) = Field r \<union> Field s" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
922 |
by (auto simp: Field_def) |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
923 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
924 |
lemma Domain_Int_subset: "Domain (A \<inter> B) \<subseteq> Domain A \<inter> Domain B" |
46694 | 925 |
by blast |
926 |
||
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
927 |
lemma Range_Int_subset: "Range (A \<inter> B) \<subseteq> Range A \<inter> Range B" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
928 |
by blast |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
929 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
930 |
lemma Domain_Diff_subset: "Domain A - Domain B \<subseteq> Domain (A - B)" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
931 |
by blast |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
932 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
933 |
lemma Range_Diff_subset: "Range A - Range B \<subseteq> Range (A - B)" |
46694 | 934 |
by blast |
935 |
||
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
936 |
lemma Domain_Union: "Domain (\<Union>S) = (\<Union>A\<in>S. Domain A)" |
46694 | 937 |
by blast |
938 |
||
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
939 |
lemma Range_Union: "Range (\<Union>S) = (\<Union>A\<in>S. Range A)" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
940 |
by blast |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
941 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
942 |
lemma Field_Union [simp]: "Field (\<Union>R) = \<Union>(Field ` R)" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
943 |
by (auto simp: Field_def) |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
944 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
945 |
lemma Domain_converse [simp]: "Domain (r\<inverse>) = Range r" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
946 |
by auto |
46694 | 947 |
|
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
948 |
lemma Range_converse [simp]: "Range (r\<inverse>) = Domain r" |
46694 | 949 |
by blast |
950 |
||
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
951 |
lemma Field_converse [simp]: "Field (r\<inverse>) = Field r" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
952 |
by (auto simp: Field_def) |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
953 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
954 |
lemma Domain_Collect_split [simp]: "Domain {(x, y). P x y} = {x. EX y. P x y}" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
955 |
by auto |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
956 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
957 |
lemma Range_Collect_split [simp]: "Range {(x, y). P x y} = {y. EX x. P x y}" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
958 |
by auto |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
959 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
960 |
lemma finite_Domain: "finite r \<Longrightarrow> finite (Domain r)" |
46884 | 961 |
by (induct set: finite) auto |
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
962 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
963 |
lemma finite_Range: "finite r \<Longrightarrow> finite (Range r)" |
46884 | 964 |
by (induct set: finite) auto |
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
965 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
966 |
lemma finite_Field: "finite r \<Longrightarrow> finite (Field r)" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
967 |
by (simp add: Field_def finite_Domain finite_Range) |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
968 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
969 |
lemma Domain_mono: "r \<subseteq> s \<Longrightarrow> Domain r \<subseteq> Domain s" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
970 |
by blast |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
971 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
972 |
lemma Range_mono: "r \<subseteq> s \<Longrightarrow> Range r \<subseteq> Range s" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
973 |
by blast |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
974 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
975 |
lemma mono_Field: "r \<subseteq> s \<Longrightarrow> Field r \<subseteq> Field s" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
976 |
by (auto simp: Field_def Domain_def Range_def) |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
977 |
|
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
978 |
lemma Domain_unfold: |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
979 |
"Domain r = {x. \<exists>y. (x, y) \<in> r}" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
980 |
by blast |
46694 | 981 |
|
12905 | 982 |
|
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
|
983 |
subsubsection {* Image of a set under a relation *} |
12905 | 984 |
|
50420 | 985 |
definition Image :: "('a \<times> 'b) set \<Rightarrow> 'a set \<Rightarrow> 'b set" (infixr "``" 90) |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
986 |
where |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
987 |
"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
|
988 |
|
12913 | 989 |
lemma Image_iff: "(b : r``A) = (EX x:A. (x, b) : r)" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
990 |
by (simp add: Image_def) |
12905 | 991 |
|
12913 | 992 |
lemma Image_singleton: "r``{a} = {b. (a, b) : r}" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
993 |
by (simp add: Image_def) |
12905 | 994 |
|
12913 | 995 |
lemma Image_singleton_iff [iff]: "(b : r``{a}) = ((a, b) : r)" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
996 |
by (rule Image_iff [THEN trans]) simp |
12905 | 997 |
|
54147
97a8ff4e4ac9
killed most "no_atp", to make Sledgehammer more complete
blanchet
parents:
53680
diff
changeset
|
998 |
lemma ImageI [intro]: "(a, b) : r ==> a : A ==> b : r``A" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
999 |
by (unfold Image_def) blast |
12905 | 1000 |
|
1001 |
lemma ImageE [elim!]: |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1002 |
"b : r `` A ==> (!!x. (x, b) : r ==> x : A ==> P) ==> P" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1003 |
by (unfold Image_def) (iprover elim!: CollectE bexE) |
12905 | 1004 |
|
1005 |
lemma rev_ImageI: "a : A ==> (a, b) : r ==> b : r `` A" |
|
1006 |
-- {* This version's more effective when we already have the required @{text a} *} |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1007 |
by blast |
12905 | 1008 |
|
1009 |
lemma Image_empty [simp]: "R``{} = {}" |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1010 |
by blast |
12905 | 1011 |
|
1012 |
lemma Image_Id [simp]: "Id `` A = A" |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1013 |
by blast |
12905 | 1014 |
|
30198 | 1015 |
lemma Image_Id_on [simp]: "Id_on A `` B = A \<inter> B" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1016 |
by blast |
13830 | 1017 |
|
1018 |
lemma Image_Int_subset: "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B" |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1019 |
by blast |
12905 | 1020 |
|
13830 | 1021 |
lemma Image_Int_eq: |
46767
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
1022 |
"single_valued (converse R) ==> R `` (A \<inter> B) = R `` A \<inter> R `` B" |
807a5d219c23
more fundamental pred-to-set conversions for range and domain by means of inductive_set
haftmann
parents:
46752
diff
changeset
|
1023 |
by (simp add: single_valued_def, blast) |
12905 | 1024 |
|
13830 | 1025 |
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1026 |
by blast |
12905 | 1027 |
|
13812
91713a1915ee
converting HOL/UNITY to use unconditional fairness
paulson
parents:
13639
diff
changeset
|
1028 |
lemma Un_Image: "(R \<union> S) `` A = R `` A \<union> S `` A" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1029 |
by blast |
13812
91713a1915ee
converting HOL/UNITY to use unconditional fairness
paulson
parents:
13639
diff
changeset
|
1030 |
|
12913 | 1031 |
lemma Image_subset: "r \<subseteq> A \<times> B ==> r``C \<subseteq> B" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1032 |
by (iprover intro!: subsetI elim!: ImageE dest!: subsetD SigmaD2) |
12905 | 1033 |
|
13830 | 1034 |
lemma Image_eq_UN: "r``B = (\<Union>y\<in> B. r``{y})" |
12905 | 1035 |
-- {* NOT suitable for rewriting *} |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1036 |
by blast |
12905 | 1037 |
|
12913 | 1038 |
lemma Image_mono: "r' \<subseteq> r ==> A' \<subseteq> A ==> (r' `` A') \<subseteq> (r `` A)" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1039 |
by blast |
12905 | 1040 |
|
13830 | 1041 |
lemma Image_UN: "(r `` (UNION A B)) = (\<Union>x\<in>A. r `` (B x))" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1042 |
by blast |
13830 | 1043 |
|
54410
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset
|
1044 |
lemma UN_Image: "(\<Union>i\<in>I. X i) `` S = (\<Union>i\<in>I. X i `` S)" |
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset
|
1045 |
by auto |
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset
|
1046 |
|
13830 | 1047 |
lemma Image_INT_subset: "(r `` INTER A B) \<subseteq> (\<Inter>x\<in>A. r `` (B x))" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1048 |
by blast |
12905 | 1049 |
|
13830 | 1050 |
text{*Converse inclusion requires some assumptions*} |
1051 |
lemma Image_INT_eq: |
|
1052 |
"[|single_valued (r\<inverse>); A\<noteq>{}|] ==> r `` INTER A B = (\<Inter>x\<in>A. r `` B x)" |
|
1053 |
apply (rule equalityI) |
|
1054 |
apply (rule Image_INT_subset) |
|
1055 |
apply (simp add: single_valued_def, blast) |
|
1056 |
done |
|
12905 | 1057 |
|
12913 | 1058 |
lemma Image_subset_eq: "(r``A \<subseteq> B) = (A \<subseteq> - ((r^-1) `` (-B)))" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1059 |
by blast |
12905 | 1060 |
|
46692
1f8b766224f6
tuned structure; dropped already existing syntax declarations
haftmann
parents:
46691
diff
changeset
|
1061 |
lemma Image_Collect_split [simp]: "{(x, y). P x y} `` A = {y. EX x:A. P x y}" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1062 |
by auto |
12905 | 1063 |
|
54410
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset
|
1064 |
lemma Sigma_Image: "(SIGMA x:A. B x) `` X = (\<Union>x\<in>X \<inter> A. B x)" |
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset
|
1065 |
by auto |
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset
|
1066 |
|
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset
|
1067 |
lemma relcomp_Image: "(X O Y) `` Z = Y `` (X `` Z)" |
0a578fb7fb73
countability of the image of a reflexive transitive closure
hoelzl
parents:
54147
diff
changeset
|
1068 |
by auto |
12905 | 1069 |
|
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
|
1070 |
subsubsection {* Inverse image *} |
12905 | 1071 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1072 |
definition inv_image :: "'b rel \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a rel" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1073 |
where |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1074 |
"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
|
1075 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1076 |
definition inv_imagep :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1077 |
where |
46694 | 1078 |
"inv_imagep r f = (\<lambda>x y. r (f x) (f y))" |
1079 |
||
1080 |
lemma [pred_set_conv]: "inv_imagep (\<lambda>x y. (x, y) \<in> r) f = (\<lambda>x y. (x, y) \<in> inv_image r f)" |
|
1081 |
by (simp add: inv_image_def inv_imagep_def) |
|
1082 |
||
19228 | 1083 |
lemma sym_inv_image: "sym r ==> sym (inv_image r f)" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1084 |
by (unfold sym_def inv_image_def) blast |
19228 | 1085 |
|
12913 | 1086 |
lemma trans_inv_image: "trans r ==> trans (inv_image r f)" |
12905 | 1087 |
apply (unfold trans_def inv_image_def) |
1088 |
apply (simp (no_asm)) |
|
1089 |
apply blast |
|
1090 |
done |
|
1091 |
||
32463
3a0a65ca2261
moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents:
32235
diff
changeset
|
1092 |
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
|
1093 |
by (auto simp:inv_image_def) |
3a0a65ca2261
moved lemma Wellfounded.in_inv_image to Relation.thy
krauss
parents:
32235
diff
changeset
|
1094 |
|
33218 | 1095 |
lemma converse_inv_image[simp]: "(inv_image R f)^-1 = inv_image (R^-1) f" |
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1096 |
unfolding inv_image_def converse_unfold by auto |
33218 | 1097 |
|
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
|
1098 |
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
|
1099 |
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
|
1100 |
|
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
|
1101 |
|
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
|
1102 |
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
|
1103 |
|
46752
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1104 |
definition Powp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" |
e9e7209eb375
more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents:
46696
diff
changeset
|
1105 |
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
|
1106 |
"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
|
1107 |
|
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
|
1108 |
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
|
1109 |
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
|
1110 |
|
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
|
1111 |
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
|
1112 |
|
48620
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1113 |
subsubsection {* Expressing relation operations via @{const Finite_Set.fold} *} |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1114 |
|
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1115 |
lemma Id_on_fold: |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1116 |
assumes "finite A" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1117 |
shows "Id_on A = Finite_Set.fold (\<lambda>x. Set.insert (Pair x x)) {} A" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1118 |
proof - |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1119 |
interpret comp_fun_commute "\<lambda>x. Set.insert (Pair x x)" by default auto |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1120 |
show ?thesis using assms unfolding Id_on_def by (induct A) simp_all |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1121 |
qed |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1122 |
|
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1123 |
lemma comp_fun_commute_Image_fold: |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1124 |
"comp_fun_commute (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1125 |
proof - |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1126 |
interpret comp_fun_idem Set.insert |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1127 |
by (fact comp_fun_idem_insert) |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1128 |
show ?thesis |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1129 |
by default (auto simp add: fun_eq_iff comp_fun_commute split:prod.split) |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1130 |
qed |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1131 |
|
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1132 |
lemma Image_fold: |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1133 |
assumes "finite R" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1134 |
shows "R `` S = Finite_Set.fold (\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A) {} R" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1135 |
proof - |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1136 |
interpret comp_fun_commute "(\<lambda>(x,y) A. if x \<in> S then Set.insert y A else A)" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1137 |
by (rule comp_fun_commute_Image_fold) |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1138 |
have *: "\<And>x F. Set.insert x F `` S = (if fst x \<in> S then Set.insert (snd x) (F `` S) else (F `` S))" |
52749 | 1139 |
by (force intro: rev_ImageI) |
48620
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1140 |
show ?thesis using assms by (induct R) (auto simp: *) |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1141 |
qed |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1142 |
|
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1143 |
lemma insert_relcomp_union_fold: |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1144 |
assumes "finite S" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1145 |
shows "{x} O S \<union> X = Finite_Set.fold (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') X S" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1146 |
proof - |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1147 |
interpret comp_fun_commute "\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A'" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1148 |
proof - |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1149 |
interpret comp_fun_idem Set.insert by (fact comp_fun_idem_insert) |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1150 |
show "comp_fun_commute (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A')" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1151 |
by default (auto simp add: fun_eq_iff split:prod.split) |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1152 |
qed |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1153 |
have *: "{x} O S = {(x', z). x' = fst x \<and> (snd x,z) \<in> S}" by (auto simp: relcomp_unfold intro!: exI) |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1154 |
show ?thesis unfolding * |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1155 |
using `finite S` by (induct S) (auto split: prod.split) |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1156 |
qed |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1157 |
|
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1158 |
lemma insert_relcomp_fold: |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1159 |
assumes "finite S" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1160 |
shows "Set.insert x R O S = |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1161 |
Finite_Set.fold (\<lambda>(w,z) A'. if snd x = w then Set.insert (fst x,z) A' else A') (R O S) S" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1162 |
proof - |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1163 |
have "Set.insert x R O S = ({x} O S) \<union> (R O S)" by auto |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1164 |
then show ?thesis by (auto simp: insert_relcomp_union_fold[OF assms]) |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1165 |
qed |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1166 |
|
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1167 |
lemma comp_fun_commute_relcomp_fold: |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1168 |
assumes "finite S" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1169 |
shows "comp_fun_commute (\<lambda>(x,y) A. |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1170 |
Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S)" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1171 |
proof - |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1172 |
have *: "\<And>a b A. |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1173 |
Finite_Set.fold (\<lambda>(w, z) A'. if b = w then Set.insert (a, z) A' else A') A S = {(a,b)} O S \<union> A" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1174 |
by (auto simp: insert_relcomp_union_fold[OF assms] cong: if_cong) |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1175 |
show ?thesis by default (auto simp: *) |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1176 |
qed |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1177 |
|
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1178 |
lemma relcomp_fold: |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1179 |
assumes "finite R" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1180 |
assumes "finite S" |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1181 |
shows "R O S = Finite_Set.fold |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1182 |
(\<lambda>(x,y) A. Finite_Set.fold (\<lambda>(w,z) A'. if y = w then Set.insert (x,z) A' else A') A S) {} R" |
52749 | 1183 |
using assms by (induct R) |
1184 |
(auto simp: comp_fun_commute.fold_insert comp_fun_commute_relcomp_fold insert_relcomp_fold |
|
48620
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1185 |
cong: if_cong) |
fc9be489e2fb
more relation operations expressed by Finite_Set.fold
kuncar
parents:
48253
diff
changeset
|
1186 |
|
1128
64b30e3cc6d4
Trancl is now based on Relation which used to be in Integ.
nipkow
parents:
diff
changeset
|
1187 |
end |