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(* Title: HOL/Transitive_Closure.thy 
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ID: $Id$ 

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1992 University of Cambridge 

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*) 

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header {* Reflexive and Transitive closure of a relation *} 
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theory Transitive_Closure = Inductive: 

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text {* 

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@{text rtrancl} is reflexive/transitive closure, 

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@{text trancl} is transitive closure, 

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@{text reflcl} is reflexive closure. 

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These postfix operators have \emph{maximum priority}, forcing their 

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operands to be atomic. 

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*} 

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consts 
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rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^*)" [1000] 999) 
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inductive "r^*" 
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intros 
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rtrancl_refl [intro!, CPure.intro!, simp]: "(a, a) : r^*" 
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rtrancl_into_rtrancl [CPure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*" 

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consts 
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trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^+)" [1000] 999) 
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inductive "r^+" 
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intros 
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r_into_trancl [intro, CPure.intro]: "(a, b) : r ==> (a, b) : r^+" 
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trancl_into_trancl [CPure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a,c) : r^+" 
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syntax 

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"_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^=)" [1000] 999) 
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translations 
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"r^=" == "r \<union> Id" 
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syntax (xsymbols) 
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rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>*)" [1000] 999) 
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trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>+)" [1000] 999) 
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"_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_\<^sup>=)" [1000] 999) 
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subsection {* Reflexivetransitive closure *} 

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lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*" 

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 {* @{text rtrancl} of @{text r} contains @{text r} *} 

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apply (simp only: split_tupled_all) 

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apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl]) 

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done 

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lemma rtrancl_mono: "r \<subseteq> s ==> r^* \<subseteq> s^*" 

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 {* monotonicity of @{text rtrancl} *} 

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apply (rule subsetI) 

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apply (simp only: split_tupled_all) 

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apply (erule rtrancl.induct) 

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apply (rule_tac [2] rtrancl_into_rtrancl, blast+) 
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done 
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theorem rtrancl_induct [consumes 1, induct set: rtrancl]: 
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assumes a: "(a, b) : r^*" 
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and cases: "P a" "!!y z. [ (a, y) : r^*; (y, z) : r; P y ] ==> P z" 
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shows "P b" 
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proof  
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from a have "a = a > P b" 

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by (induct "%x y. x = a > P y" a b) (rules intro: cases)+ 
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thus ?thesis by rules 
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qed 

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lemmas rtrancl_induct2 = 
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rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete), 
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consumes 1, case_names refl step] 
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lemma trans_rtrancl: "trans(r^*)" 
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 {* transitivity of transitive closure!!  by induction *} 

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proof (rule transI) 
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fix x y z 

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assume "(x, y) \<in> r\<^sup>*" 

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assume "(y, z) \<in> r\<^sup>*" 

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thus "(x, z) \<in> r\<^sup>*" by induct (rules!)+ 

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qed 

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lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard] 

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lemma rtranclE: 

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"[ (a::'a,b) : r^*; (a = b) ==> P; 

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!!y.[ (a,y) : r^*; (y,b) : r ] ==> P 

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] ==> P" 

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 {* elimination of @{text rtrancl}  by induction on a special formula *} 

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proof  

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assume major: "(a::'a,b) : r^*" 

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case rule_context 

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show ?thesis 

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apply (subgoal_tac "(a::'a) = b  (EX y. (a,y) : r^* & (y,b) : r)") 

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apply (rule_tac [2] major [THEN rtrancl_induct]) 

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prefer 2 apply (blast!) 

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prefer 2 apply (blast!) 

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apply (erule asm_rl exE disjE conjE prems)+ 

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done 

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qed 

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lemma converse_rtrancl_into_rtrancl: 
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"(a, b) \<in> r \<Longrightarrow> (b, c) \<in> r\<^sup>* \<Longrightarrow> (a, c) \<in> r\<^sup>*" 

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by (rule rtrancl_trans) rules+ 

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text {* 

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\medskip More @{term "r^*"} equations and inclusions. 

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*} 

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lemma rtrancl_idemp [simp]: "(r^*)^* = r^*" 

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apply auto 

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apply (erule rtrancl_induct) 

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apply (rule rtrancl_refl) 

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apply (blast intro: rtrancl_trans) 

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done 

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lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*" 

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apply (rule set_ext) 

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apply (simp only: split_tupled_all) 

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apply (blast intro: rtrancl_trans) 

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done 

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lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*" 

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by (drule rtrancl_mono, simp) 
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lemma rtrancl_subset: "R \<subseteq> S ==> S \<subseteq> R^* ==> S^* = R^*" 

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apply (drule rtrancl_mono) 

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apply (drule rtrancl_mono, simp) 
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done 
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lemma rtrancl_Un_rtrancl: "(R^* \<union> S^*)^* = (R \<union> S)^*" 

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by (blast intro!: rtrancl_subset intro: r_into_rtrancl rtrancl_mono [THEN subsetD]) 

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lemma rtrancl_reflcl [simp]: "(R^=)^* = R^*" 

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by (blast intro!: rtrancl_subset intro: r_into_rtrancl) 

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lemma rtrancl_r_diff_Id: "(r  Id)^* = r^*" 

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apply (rule sym) 

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apply (rule rtrancl_subset, blast, clarify) 
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apply (rename_tac a b) 
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apply (case_tac "a = b", blast) 
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apply (blast intro!: r_into_rtrancl) 
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done 

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theorem rtrancl_converseD: 
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assumes r: "(x, y) \<in> (r^1)^*" 
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shows "(y, x) \<in> r^*" 
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proof  
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from r show ?thesis 

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by induct (rules intro: rtrancl_trans dest!: converseD)+ 

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qed 

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theorem rtrancl_converseI: 
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assumes r: "(y, x) \<in> r^*" 
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shows "(x, y) \<in> (r^1)^*" 
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proof  
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from r show ?thesis 

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by induct (rules intro: rtrancl_trans converseI)+ 

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qed 

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lemma rtrancl_converse: "(r^1)^* = (r^*)^1" 

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by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI) 

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theorem converse_rtrancl_induct[consumes 1]: 
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assumes major: "(a, b) : r^*" 
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and cases: "P b" "!!y z. [ (y, z) : r; (z, b) : r^*; P z ] ==> P y" 
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shows "P a" 
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proof  
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from rtrancl_converseI [OF major] 
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show ?thesis 
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by induct (rules intro: cases dest!: converseD rtrancl_converseD)+ 
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qed 
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lemmas converse_rtrancl_induct2 = 
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converse_rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete), 
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consumes 1, case_names refl step] 
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lemma converse_rtranclE: 

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"[ (x,z):r^*; 

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x=z ==> P; 

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!!y. [ (x,y):r; (y,z):r^* ] ==> P 

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] ==> P" 

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proof  

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assume major: "(x,z):r^*" 

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case rule_context 

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show ?thesis 

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apply (subgoal_tac "x = z  (EX y. (x,y) : r & (y,z) : r^*)") 

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apply (rule_tac [2] major [THEN converse_rtrancl_induct]) 

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prefer 2 apply rules 
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prefer 2 apply rules 

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apply (erule asm_rl exE disjE conjE prems)+ 
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done 

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qed 

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ML_setup {* 

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bind_thm ("converse_rtranclE2", split_rule 

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(read_instantiate [("x","(xa,xb)"), ("z","(za,zb)")] (thm "converse_rtranclE"))); 

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*} 

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lemma r_comp_rtrancl_eq: "r O r^* = r^* O r" 

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by (blast elim: rtranclE converse_rtranclE 

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intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl) 

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subsection {* Transitive closure *} 

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lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+" 
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apply (simp only: split_tupled_all) 
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apply (erule trancl.induct) 
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apply (rules dest: subsetD)+ 
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done 
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lemma r_into_trancl': "!!p. p : r ==> p : r^+" 
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by (simp only: split_tupled_all) (erule r_into_trancl) 
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text {* 
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\medskip Conversions between @{text trancl} and @{text rtrancl}. 

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*} 

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lemma trancl_into_rtrancl: "(a, b) \<in> r^+ ==> (a, b) \<in> r^*" 
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by (erule trancl.induct) rules+ 
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lemma rtrancl_into_trancl1: assumes r: "(a, b) \<in> r^*" 
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shows "!!c. (b, c) \<in> r ==> (a, c) \<in> r^+" using r 
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by induct rules+ 
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lemma rtrancl_into_trancl2: "[ (a,b) : r; (b,c) : r^* ] ==> (a,c) : r^+" 

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 {* intro rule from @{text r} and @{text rtrancl} *} 

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apply (erule rtranclE, rules) 
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apply (rule rtrancl_trans [THEN rtrancl_into_trancl1]) 
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apply (assumption  rule r_into_rtrancl)+ 

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done 

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lemma trancl_induct [consumes 1, induct set: trancl]: 
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assumes a: "(a,b) : r^+" 
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and cases: "!!y. (a, y) : r ==> P y" 
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"!!y z. (a,y) : r^+ ==> (y, z) : r ==> P y ==> P z" 
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shows "P b" 
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 {* Nice induction rule for @{text trancl} *} 
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proof  

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from a have "a = a > P b" 
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by (induct "%x y. x = a > P y" a b) (rules intro: cases)+ 
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thus ?thesis by rules 
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qed 
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lemma trancl_trans_induct: 

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"[ (x,y) : r^+; 

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!!x y. (x,y) : r ==> P x y; 

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!!x y z. [ (x,y) : r^+; P x y; (y,z) : r^+; P y z ] ==> P x z 

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] ==> P x y" 

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 {* Another induction rule for trancl, incorporating transitivity *} 

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proof  

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assume major: "(x,y) : r^+" 

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case rule_context 

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show ?thesis 

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by (rules intro: r_into_trancl major [THEN trancl_induct] prems) 
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qed 
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inductive_cases tranclE: "(a, b) : r^+" 
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lemma trans_trancl: "trans(r^+)" 
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 {* Transitivity of @{term "r^+"} *} 

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proof (rule transI) 
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fix x y z 
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assume "(x, y) \<in> r^+" 
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assume "(y, z) \<in> r^+" 
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thus "(x, z) \<in> r^+" by induct (rules!)+ 
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qed 
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lemmas trancl_trans = trans_trancl [THEN transD, standard] 

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lemma rtrancl_trancl_trancl: assumes r: "(x, y) \<in> r^*" 
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shows "!!z. (y, z) \<in> r^+ ==> (x, z) \<in> r^+" using r 
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by induct (rules intro: trancl_trans)+ 
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lemma trancl_into_trancl2: "(a, b) \<in> r ==> (b, c) \<in> r^+ ==> (a, c) \<in> r^+" 

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by (erule transD [OF trans_trancl r_into_trancl]) 

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lemma trancl_insert: 

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"(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}" 

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 {* primitive recursion for @{text trancl} over finite relations *} 

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apply (rule equalityI) 

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apply (rule subsetI) 

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apply (simp only: split_tupled_all) 

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apply (erule trancl_induct, blast) 
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apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl r_into_trancl trancl_trans) 
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apply (rule subsetI) 

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apply (blast intro: trancl_mono rtrancl_mono 

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[THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2) 

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done 

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lemma trancl_converseI: "(x, y) \<in> (r^+)^1 ==> (x, y) \<in> (r^1)^+" 
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apply (drule converseD) 
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apply (erule trancl.induct) 
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apply (rules intro: converseI trancl_trans)+ 
12691  299 
done 
300 

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lemma trancl_converseD: "(x, y) \<in> (r^1)^+ ==> (x, y) \<in> (r^+)^1" 
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apply (rule converseI) 
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apply (erule trancl.induct) 
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apply (rules dest: converseD intro: trancl_trans)+ 
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done 
12691  306 

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lemma trancl_converse: "(r^1)^+ = (r^+)^1" 
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by (fastsimp simp add: split_tupled_all 
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intro!: trancl_converseI trancl_converseD) 
12691  310 

311 
lemma converse_trancl_induct: 

312 
"[ (a,b) : r^+; !!y. (y,b) : r ==> P(y); 

313 
!!y z.[ (y,z) : r; (z,b) : r^+; P(z) ] ==> P(y) ] 

314 
==> P(a)" 

315 
proof  

316 
assume major: "(a,b) : r^+" 

317 
case rule_context 

318 
show ?thesis 

319 
apply (rule major [THEN converseI, THEN trancl_converseI [THEN trancl_induct]]) 

320 
apply (rule prems) 

321 
apply (erule converseD) 

322 
apply (blast intro: prems dest!: trancl_converseD) 

323 
done 

324 
qed 

325 

326 
lemma tranclD: "(x, y) \<in> R^+ ==> EX z. (x, z) \<in> R \<and> (z, y) \<in> R^*" 

14208  327 
apply (erule converse_trancl_induct, auto) 
12691  328 
apply (blast intro: rtrancl_trans) 
329 
done 

330 

13867  331 
lemma irrefl_tranclI: "r^1 \<inter> r^* = {} ==> (x, x) \<notin> r^+" 
332 
by(blast elim: tranclE dest: trancl_into_rtrancl) 

12691  333 

334 
lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y" 

335 
by (blast dest: r_into_trancl) 

336 

337 
lemma trancl_subset_Sigma_aux: 

338 
"(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A" 

14208  339 
apply (erule rtrancl_induct, auto) 
12691  340 
done 
341 

342 
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A" 

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apply (rule subsetI) 
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apply (simp only: split_tupled_all) 
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apply (erule tranclE) 
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apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+ 
12691  347 
done 
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348 

11090  349 
lemma reflcl_trancl [simp]: "(r^+)^= = r^*" 
11084  350 
apply safe 
12691  351 
apply (erule trancl_into_rtrancl) 
11084  352 
apply (blast elim: rtranclE dest: rtrancl_into_trancl1) 
353 
done 

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354 

11090  355 
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*" 
11084  356 
apply safe 
14208  357 
apply (drule trancl_into_rtrancl, simp) 
358 
apply (erule rtranclE, safe) 

359 
apply (rule r_into_trancl, simp) 

11084  360 
apply (rule rtrancl_into_trancl1) 
14208  361 
apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast) 
11084  362 
done 
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363 

11090  364 
lemma trancl_empty [simp]: "{}^+ = {}" 
11084  365 
by (auto elim: trancl_induct) 
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366 

11090  367 
lemma rtrancl_empty [simp]: "{}^* = Id" 
11084  368 
by (rule subst [OF reflcl_trancl]) simp 
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369 

11090  370 
lemma rtranclD: "(a, b) \<in> R^* ==> a = b \<or> a \<noteq> b \<and> (a, b) \<in> R^+" 
11084  371 
by (force simp add: reflcl_trancl [symmetric] simp del: reflcl_trancl) 
372 

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373 

12691  374 
text {* @{text Domain} and @{text Range} *} 
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375 

11090  376 
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV" 
11084  377 
by blast 
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378 

11090  379 
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV" 
11084  380 
by blast 
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381 

11090  382 
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*" 
11084  383 
by (rule rtrancl_Un_rtrancl [THEN subst]) fast 
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384 

11090  385 
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*" 
11084  386 
by (blast intro: subsetD [OF rtrancl_Un_subset]) 
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387 

11090  388 
lemma trancl_domain [simp]: "Domain (r^+) = Domain r" 
11084  389 
by (unfold Domain_def) (blast dest: tranclD) 
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390 

11090  391 
lemma trancl_range [simp]: "Range (r^+) = Range r" 
11084  392 
by (simp add: Range_def trancl_converse [symmetric]) 
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393 

11115  394 
lemma Not_Domain_rtrancl: 
12691  395 
"x ~: Domain R ==> ((x, y) : R^*) = (x = y)" 
396 
apply auto 

397 
by (erule rev_mp, erule rtrancl_induct, auto) 

398 

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399 

12691  400 
text {* More about converse @{text rtrancl} and @{text trancl}, should 
401 
be merged with main body. *} 

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402 

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403 
lemma single_valued_confluent: 
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404 
"\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk> 
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405 
\<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*" 
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apply(erule rtrancl_induct) 
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407 
apply simp 
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408 
apply(erule disjE) 
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409 
apply(blast elim:converse_rtranclE dest:single_valuedD) 
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apply(blast intro:rtrancl_trans) 
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411 
done 
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412 

12691  413 
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+" 
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by (fast intro: trancl_trans) 
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415 

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lemma trancl_into_trancl [rule_format]: 
12691  417 
"(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r > (a,c) \<in> r\<^sup>+" 
418 
apply (erule trancl_induct) 

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419 
apply (fast intro: r_r_into_trancl) 
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420 
apply (fast intro: r_r_into_trancl trancl_trans) 
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421 
done 
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422 

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lemma trancl_rtrancl_trancl: 
12691  424 
"(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r\<^sup>* ==> (a, c) \<in> r\<^sup>+" 
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425 
apply (drule tranclD) 
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426 
apply (erule exE, erule conjE) 
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427 
apply (drule rtrancl_trans, assumption) 
14208  428 
apply (drule rtrancl_into_trancl2, assumption, assumption) 
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429 
done 
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430 

12691  431 
lemmas transitive_closure_trans [trans] = 
432 
r_r_into_trancl trancl_trans rtrancl_trans 

433 
trancl_into_trancl trancl_into_trancl2 

434 
rtrancl_into_rtrancl converse_rtrancl_into_rtrancl 

435 
rtrancl_trancl_trancl trancl_rtrancl_trancl 

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436 

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437 
declare trancl_into_rtrancl [elim] 
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438 

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439 
declare rtranclE [cases set: rtrancl] 
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440 
declare tranclE [cases set: trancl] 
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441 

10213  442 
end 