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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 
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imports Inductive 
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uses ("../Provers/trancl.ML") 
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begin 
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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!, Pure.intro!, simp]: "(a, a) : r^*" 
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rtrancl_into_rtrancl [Pure.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, Pure.intro]: "(a, b) : r ==> (a, b) : r^+" 
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trancl_into_trancl [Pure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a,c) : r^+" 

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abbreviation 
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reflcl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^=)" [1000] 999) 
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"r^= == r \<union> Id" 
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const_syntax (xsymbols) 
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rtrancl ("(_\<^sup>*)" [1000] 999) 
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trancl ("(_\<^sup>+)" [1000] 999) 
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reflcl ("(_\<^sup>=)" [1000] 999) 
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const_syntax (HTML output) 
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rtrancl ("(_\<^sup>*)" [1000] 999) 
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trancl ("(_\<^sup>+)" [1000] 999) 
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reflcl ("(_\<^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) (iprover intro: cases)+ 
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thus ?thesis by iprover 

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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 reflexive_rtrancl: "reflexive (r^*)" 
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by (unfold refl_def) fast 

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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 (iprover!)+ 
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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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assumes major: "(a::'a,b) : r^*" 
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and cases: "(a = b) ==> P" 

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

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shows P 

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

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done 

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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) iprover+ 
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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 (iprover 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 (iprover 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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lemma sym_rtrancl: "sym r ==> sym (r^*)" 
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by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric]) 

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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 (iprover 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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assumes major: "(x,z):r^*" 
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and cases: "x=z ==> P" 

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

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shows P 

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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 iprover 

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prefer 2 apply iprover 

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apply (erule asm_rl exE disjE conjE cases)+ 

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done 

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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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lemma rtrancl_unfold: "r^* = Id Un (r O r^*)" 
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by (auto intro: rtrancl_into_rtrancl elim: rtranclE) 

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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 (iprover 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) iprover+ 
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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 iprover+ 
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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, iprover) 
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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) (iprover intro: cases)+ 
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thus ?thesis by iprover 

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qed 
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lemma trancl_trans_induct: 

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assumes major: "(x,y) : r^+" 
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and cases: "!!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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shows "P x y" 

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 {* Another induction rule for trancl, incorporating transitivity *} 
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by (iprover intro: r_into_trancl major [THEN trancl_induct] cases) 
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inductive_cases tranclE: "(a, b) : r^+" 
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lemma trancl_unfold: "r^+ = r Un (r O r^+)" 
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by (auto intro: trancl_into_trancl elim: tranclE) 

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lemma trans_trancl[simp]: "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 xy: "(x, y) \<in> r^+" 
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assume "(y, z) \<in> r^+" 
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thus "(x, z) \<in> r^+" by induct (insert xy, iprover)+ 
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qed 
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lemmas trancl_trans = trans_trancl [THEN transD, standard] 

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lemma trancl_id[simp]: "trans r \<Longrightarrow> r^+ = r" 
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apply(auto) 

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apply(erule trancl_induct) 

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apply assumption 

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apply(unfold trans_def) 

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apply(blast) 

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done 

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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 (iprover 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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309 
apply (drule converseD) 
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310 
apply (erule trancl.induct) 
17589  311 
apply (iprover intro: converseI trancl_trans)+ 
12691  312 
done 
313 

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314 
lemma trancl_converseD: "(x, y) \<in> (r^1)^+ ==> (x, y) \<in> (r^+)^1" 
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315 
apply (rule converseI) 
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apply (erule trancl.induct) 
17589  317 
apply (iprover dest: converseD intro: trancl_trans)+ 
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318 
done 
12691  319 

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320 
lemma trancl_converse: "(r^1)^+ = (r^+)^1" 
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321 
by (fastsimp simp add: split_tupled_all 
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intro!: trancl_converseI trancl_converseD) 
12691  323 

19228  324 
lemma sym_trancl: "sym r ==> sym (r^+)" 
325 
by (simp only: sym_conv_converse_eq trancl_converse [symmetric]) 

326 

12691  327 
lemma converse_trancl_induct: 
18372  328 
assumes major: "(a,b) : r^+" 
329 
and cases: "!!y. (y,b) : r ==> P(y)" 

330 
"!!y z.[ (y,z) : r; (z,b) : r^+; P(z) ] ==> P(y)" 

331 
shows "P a" 

332 
apply (rule major [THEN converseI, THEN trancl_converseI [THEN trancl_induct]]) 

333 
apply (rule cases) 

334 
apply (erule converseD) 

335 
apply (blast intro: prems dest!: trancl_converseD) 

336 
done 

12691  337 

338 
lemma tranclD: "(x, y) \<in> R^+ ==> EX z. (x, z) \<in> R \<and> (z, y) \<in> R^*" 

14208  339 
apply (erule converse_trancl_induct, auto) 
12691  340 
apply (blast intro: rtrancl_trans) 
341 
done 

342 

13867  343 
lemma irrefl_tranclI: "r^1 \<inter> r^* = {} ==> (x, x) \<notin> r^+" 
18372  344 
by (blast elim: tranclE dest: trancl_into_rtrancl) 
12691  345 

346 
lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y" 

347 
by (blast dest: r_into_trancl) 

348 

349 
lemma trancl_subset_Sigma_aux: 

350 
"(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A" 

18372  351 
by (induct rule: rtrancl_induct) auto 
12691  352 

353 
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A" 

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354 
apply (rule subsetI) 
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355 
apply (simp only: split_tupled_all) 
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356 
apply (erule tranclE) 
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357 
apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+ 
12691  358 
done 
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359 

11090  360 
lemma reflcl_trancl [simp]: "(r^+)^= = r^*" 
11084  361 
apply safe 
12691  362 
apply (erule trancl_into_rtrancl) 
11084  363 
apply (blast elim: rtranclE dest: rtrancl_into_trancl1) 
364 
done 

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365 

11090  366 
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*" 
11084  367 
apply safe 
14208  368 
apply (drule trancl_into_rtrancl, simp) 
369 
apply (erule rtranclE, safe) 

370 
apply (rule r_into_trancl, simp) 

11084  371 
apply (rule rtrancl_into_trancl1) 
14208  372 
apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast) 
11084  373 
done 
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374 

11090  375 
lemma trancl_empty [simp]: "{}^+ = {}" 
11084  376 
by (auto elim: trancl_induct) 
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377 

11090  378 
lemma rtrancl_empty [simp]: "{}^* = Id" 
11084  379 
by (rule subst [OF reflcl_trancl]) simp 
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380 

11090  381 
lemma rtranclD: "(a, b) \<in> R^* ==> a = b \<or> a \<noteq> b \<and> (a, b) \<in> R^+" 
11084  382 
by (force simp add: reflcl_trancl [symmetric] simp del: reflcl_trancl) 
383 

16514  384 
lemma rtrancl_eq_or_trancl: 
385 
"(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)" 

386 
by (fast elim: trancl_into_rtrancl dest: rtranclD) 

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387 

12691  388 
text {* @{text Domain} and @{text Range} *} 
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389 

11090  390 
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV" 
11084  391 
by blast 
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392 

11090  393 
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV" 
11084  394 
by blast 
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395 

11090  396 
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*" 
11084  397 
by (rule rtrancl_Un_rtrancl [THEN subst]) fast 
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398 

11090  399 
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*" 
11084  400 
by (blast intro: subsetD [OF rtrancl_Un_subset]) 
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401 

11090  402 
lemma trancl_domain [simp]: "Domain (r^+) = Domain r" 
11084  403 
by (unfold Domain_def) (blast dest: tranclD) 
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404 

11090  405 
lemma trancl_range [simp]: "Range (r^+) = Range r" 
11084  406 
by (simp add: Range_def trancl_converse [symmetric]) 
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407 

11115  408 
lemma Not_Domain_rtrancl: 
12691  409 
"x ~: Domain R ==> ((x, y) : R^*) = (x = y)" 
410 
apply auto 

411 
by (erule rev_mp, erule rtrancl_induct, auto) 

412 

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413 

12691  414 
text {* More about converse @{text rtrancl} and @{text trancl}, should 
415 
be merged with main body. *} 

12428
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416 

14337
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417 
lemma single_valued_confluent: 
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418 
"\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk> 
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419 
\<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*" 
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420 
apply(erule rtrancl_induct) 
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421 
apply simp 
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422 
apply(erule disjE) 
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423 
apply(blast elim:converse_rtranclE dest:single_valuedD) 
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424 
apply(blast intro:rtrancl_trans) 
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425 
done 
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426 

12691  427 
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+" 
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428 
by (fast intro: trancl_trans) 
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429 

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430 
lemma trancl_into_trancl [rule_format]: 
12691  431 
"(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r > (a,c) \<in> r\<^sup>+" 
432 
apply (erule trancl_induct) 

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433 
apply (fast intro: r_r_into_trancl) 
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434 
apply (fast intro: r_r_into_trancl trancl_trans) 
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435 
done 
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436 

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437 
lemma trancl_rtrancl_trancl: 
12691  438 
"(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r\<^sup>* ==> (a, c) \<in> r\<^sup>+" 
12428
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439 
apply (drule tranclD) 
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440 
apply (erule exE, erule conjE) 
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441 
apply (drule rtrancl_trans, assumption) 
14208  442 
apply (drule rtrancl_into_trancl2, assumption, assumption) 
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443 
done 
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444 

12691  445 
lemmas transitive_closure_trans [trans] = 
446 
r_r_into_trancl trancl_trans rtrancl_trans 

447 
trancl_into_trancl trancl_into_trancl2 

448 
rtrancl_into_rtrancl converse_rtrancl_into_rtrancl 

449 
rtrancl_trancl_trancl trancl_rtrancl_trancl 

12428
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450 

f3033eed309a
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451 
declare trancl_into_rtrancl [elim] 
11327
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452 

cd2c27a23df1
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453 
declare rtranclE [cases set: rtrancl] 
cd2c27a23df1
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454 
declare tranclE [cases set: trancl] 
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455 

15551  456 

457 

458 

459 

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460 
subsection {* Setup of transitivity reasoner *} 
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461 

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462 
use "../Provers/trancl.ML"; 
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463 

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464 
ML_setup {* 
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465 

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466 
structure Trancl_Tac = Trancl_Tac_Fun ( 
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467 
struct 
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468 
val r_into_trancl = thm "r_into_trancl"; 
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469 
val trancl_trans = thm "trancl_trans"; 
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470 
val rtrancl_refl = thm "rtrancl_refl"; 
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471 
val r_into_rtrancl = thm "r_into_rtrancl"; 
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472 
val trancl_into_rtrancl = thm "trancl_into_rtrancl"; 
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473 
val rtrancl_trancl_trancl = thm "rtrancl_trancl_trancl"; 
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474 
val trancl_rtrancl_trancl = thm "trancl_rtrancl_trancl"; 
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475 
val rtrancl_trans = thm "rtrancl_trans"; 
15096  476 

18372  477 
fun decomp (Trueprop $ t) = 
478 
let fun dec (Const ("op :", _) $ (Const ("Pair", _) $ a $ b) $ rel ) = 

479 
let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*") 

480 
 decr (Const ("Transitive_Closure.trancl", _ ) $ r) = (r,"r+") 

481 
 decr r = (r,"r"); 

482 
val (rel,r) = decr rel; 

483 
in SOME (a,b,rel,r) end 

484 
 dec _ = NONE 

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485 
in dec t end; 
18372  486 

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487 
end); (* struct *) 
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488 

17876  489 
change_simpset (fn ss => ss 
490 
addSolver (mk_solver "Trancl" (fn _ => Trancl_Tac.trancl_tac)) 

491 
addSolver (mk_solver "Rtrancl" (fn _ => Trancl_Tac.rtrancl_tac))); 

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492 

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493 
*} 
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494 

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495 
(* Optional methods 
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496 

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497 
method_setup trancl = 
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498 
{* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (trancl_tac)) *} 
18372  499 
{* simple transitivity reasoner *} 
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500 
method_setup rtrancl = 
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501 
{* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (rtrancl_tac)) *} 
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502 
{* simple transitivity reasoner *} 
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503 

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504 
*) 
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505 

10213  506 
end 