author  haftmann 
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parent 45607  16b4f5774621 
child 46127  af3b95160b59 
permissions  rwrr 
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(* Title: HOL/Transitive_Closure.thy 
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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 Predicate 
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uses "~~/src/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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inductive_set 
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rtrancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_^*)" [1000] 999) 
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for r :: "('a \<times> 'a) set" 
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where 
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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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inductive_set 
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trancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_^+)" [1000] 999) 
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for r :: "('a \<times> 'a) set" 
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where 
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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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declare rtrancl_def [nitpick_unfold del] 
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rtranclp_def [nitpick_unfold del] 
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trancl_def [nitpick_unfold del] 
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tranclp_def [nitpick_unfold del] 
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notation 
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rtranclp ("(_^**)" [1000] 1000) and 
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tranclp ("(_^++)" [1000] 1000) 
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abbreviation 
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reflclp :: "('a => 'a => bool) => 'a => 'a => bool" ("(_^==)" [1000] 1000) where 
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"r^== \<equiv> sup r op =" 
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abbreviation 

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reflcl :: "('a \<times> 'a) set => ('a \<times> 'a) set" ("(_^=)" [1000] 999) where 
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"r^= \<equiv> r \<union> Id" 
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notation (xsymbols) 
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rtranclp ("(_\<^sup>*\<^sup>*)" [1000] 1000) and 
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tranclp ("(_\<^sup>+\<^sup>+)" [1000] 1000) and 
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reflclp ("(_\<^sup>=\<^sup>=)" [1000] 1000) and 
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rtrancl ("(_\<^sup>*)" [1000] 999) and 
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trancl ("(_\<^sup>+)" [1000] 999) and 
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reflcl ("(_\<^sup>=)" [1000] 999) 
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notation (HTML output) 
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rtranclp ("(_\<^sup>*\<^sup>*)" [1000] 1000) and 
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tranclp ("(_\<^sup>+\<^sup>+)" [1000] 1000) and 
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reflclp ("(_\<^sup>=\<^sup>=)" [1000] 1000) and 
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rtrancl ("(_\<^sup>*)" [1000] 999) and 
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trancl ("(_\<^sup>+)" [1000] 999) and 
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reflcl ("(_\<^sup>=)" [1000] 999) 
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subsection {* Reflexive closure *} 
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lemma refl_reflcl[simp]: "refl(r^=)" 
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by(simp add:refl_on_def) 

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lemma antisym_reflcl[simp]: "antisym(r^=) = antisym r" 

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by(simp add:antisym_def) 

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lemma trans_reflclI[simp]: "trans r \<Longrightarrow> trans(r^=)" 

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unfolding trans_def by blast 

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subsection {* Reflexivetransitive closure *} 
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lemma reflcl_set_eq [pred_set_conv]: "(sup (\<lambda>x y. (x, y) \<in> r) op =) = (\<lambda>x y. (x, y) \<in> r \<union> Id)" 
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by (auto simp add: fun_eq_iff) 
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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 r_into_rtranclp [intro]: "r x y ==> r^** x y" 
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 {* @{text rtrancl} of @{text r} contains @{text r} *} 
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by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl]) 
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lemma rtranclp_mono: "r \<le> s ==> r^** \<le> s^**" 
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 {* monotonicity of @{text rtrancl} *} 
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apply (rule predicate2I) 
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apply (erule rtranclp.induct) 
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apply (rule_tac [2] rtranclp.rtrancl_into_rtrancl, blast+) 
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done 
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lemmas rtrancl_mono = rtranclp_mono [to_set] 
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theorem rtranclp_induct [consumes 1, case_names base step, induct set: rtranclp]: 
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assumes a: "r^** a b" 
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and cases: "P a" "!!y z. [ r^** a y; r y z; P y ] ==> P z" 

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shows "P b" using a 
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by (induct x\<equiv>a b) (rule cases)+ 
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lemmas rtrancl_induct [induct set: rtrancl] = rtranclp_induct [to_set] 
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lemmas rtranclp_induct2 = 
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rtranclp_induct[of _ "(ax,ay)" "(bx,by)", split_rule, 
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consumes 1, case_names refl step] 
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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 refl_rtrancl: "refl (r^*)" 
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by (unfold refl_on_def) fast 

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text {* Transitivity of transitive closure. *} 
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lemma trans_rtrancl: "trans (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\<^sup>*" 

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

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then show "(x, z) \<in> r\<^sup>*" 
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proof induct 
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case base 
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show "(x, y) \<in> r\<^sup>*" by fact 
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next 
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case (step u v) 
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from `(x, u) \<in> r\<^sup>*` and `(u, v) \<in> r` 
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show "(x, v) \<in> r\<^sup>*" .. 
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qed 
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qed 
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lemmas rtrancl_trans = trans_rtrancl [THEN transD] 
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lemma rtranclp_trans: 
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assumes xy: "r^** x y" 
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and yz: "r^** y z" 

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shows "r^** x z" using yz xy 

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by induct iprover+ 

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lemma rtranclE [cases set: rtrancl]: 
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assumes major: "(a::'a, b) : r^*" 
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obtains 
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(base) "a = b" 
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 (step) y where "(a, y) : r^*" and "(y, b) : r" 
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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 base step)+ 
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done 
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lemma rtrancl_Int_subset: "[ Id \<subseteq> s; (r^* \<inter> s) O r \<subseteq> s] ==> r^* \<subseteq> s" 
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apply (rule subsetI) 
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apply (rule_tac p="x" in PairE, clarify) 
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apply (erule rtrancl_induct, auto) 
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done 
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lemma converse_rtranclp_into_rtranclp: 
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"r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>*\<^sup>* a c" 
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by (rule rtranclp_trans) iprover+ 
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lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set] 
12691  174 

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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 rtranclp_idemp [simp]: "(r^**)^** = r^**" 
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apply (auto intro!: order_antisym) 
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apply (erule rtranclp_induct) 
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apply (rule rtranclp.rtrancl_refl) 
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apply (blast intro: rtranclp_trans) 
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done 
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lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set] 
22262  187 

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lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*" 
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apply (rule set_eqI) 
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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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apply (drule rtrancl_mono) 
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apply simp 
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done 
12691  198 

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lemma rtranclp_subset: "R \<le> S ==> S \<le> R^** ==> S^** = R^**" 
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apply (drule rtranclp_mono) 
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apply (drule rtranclp_mono) 
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apply simp 
12691  203 
done 
204 

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lemmas rtrancl_subset = rtranclp_subset [to_set] 
22262  206 

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lemma rtranclp_sup_rtranclp: "(sup (R^**) (S^**))^** = (sup R S)^**" 
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by (blast intro!: rtranclp_subset intro: rtranclp_mono [THEN predicate2D]) 
12691  209 

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lemmas rtrancl_Un_rtrancl = rtranclp_sup_rtranclp [to_set] 
22262  211 

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lemma rtranclp_reflcl [simp]: "(R^==)^** = R^**" 
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by (blast intro!: rtranclp_subset) 
22262  214 

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lemmas rtrancl_reflcl [simp] = rtranclp_reflcl [to_set] 
12691  216 

217 
lemma rtrancl_r_diff_Id: "(r  Id)^* = r^*" 

218 
apply (rule sym) 

14208  219 
apply (rule rtrancl_subset, blast, clarify) 
12691  220 
apply (rename_tac a b) 
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apply (case_tac "a = b") 
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apply blast 
44921  223 
apply blast 
12691  224 
done 
225 

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lemma rtranclp_r_diff_Id: "(inf r op ~=)^** = r^**" 
22262  227 
apply (rule sym) 
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apply (rule rtranclp_subset) 
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229 
apply blast+ 
22262  230 
done 
231 

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theorem rtranclp_converseD: 
22262  233 
assumes r: "(r^1)^** x y" 
234 
shows "r^** y x" 

12823  235 
proof  
236 
from r show ?thesis 

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by induct (iprover intro: rtranclp_trans dest!: conversepD)+ 
12823  238 
qed 
12691  239 

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lemmas rtrancl_converseD = rtranclp_converseD [to_set] 
22262  241 

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theorem rtranclp_converseI: 
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assumes "r^** y x" 
22262  244 
shows "(r^1)^** x y" 
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using assms 
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by induct (iprover intro: rtranclp_trans conversepI)+ 
12691  247 

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lemmas rtrancl_converseI = rtranclp_converseI [to_set] 
22262  249 

12691  250 
lemma rtrancl_converse: "(r^1)^* = (r^*)^1" 
251 
by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI) 

252 

19228  253 
lemma sym_rtrancl: "sym r ==> sym (r^*)" 
254 
by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric]) 

255 

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theorem converse_rtranclp_induct [consumes 1, case_names base step]: 
22262  257 
assumes major: "r^** a b" 
258 
and cases: "P b" "!!y z. [ r y z; r^** z b; P z ] ==> P y" 

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shows "P a" 
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260 
using rtranclp_converseI [OF major] 
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261 
by induct (iprover intro: cases dest!: conversepD rtranclp_converseD)+ 
12691  262 

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lemmas converse_rtrancl_induct = converse_rtranclp_induct [to_set] 
22262  264 

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lemmas converse_rtranclp_induct2 = 
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converse_rtranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule, 
22262  267 
consumes 1, case_names refl step] 
268 

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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] 
12691  272 

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lemma converse_rtranclpE [consumes 1, case_names base step]: 
22262  274 
assumes major: "r^** x z" 
18372  275 
and cases: "x=z ==> P" 
22262  276 
"!!y. [ r x y; r^** y z ] ==> P" 
18372  277 
shows P 
22262  278 
apply (subgoal_tac "x = z  (EX y. r x y & r^** y z)") 
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apply (rule_tac [2] major [THEN converse_rtranclp_induct]) 
18372  280 
prefer 2 apply iprover 
281 
prefer 2 apply iprover 

282 
apply (erule asm_rl exE disjE conjE cases)+ 

283 
done 

12691  284 

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lemmas converse_rtranclE = converse_rtranclpE [to_set] 
22262  286 

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287 
lemmas converse_rtranclpE2 = converse_rtranclpE [of _ "(xa,xb)" "(za,zb)", split_rule] 
22262  288 

289 
lemmas converse_rtranclE2 = converse_rtranclE [of "(xa,xb)" "(za,zb)", split_rule] 

12691  290 

291 
lemma r_comp_rtrancl_eq: "r O r^* = r^* O r" 

292 
by (blast elim: rtranclE converse_rtranclE 

293 
intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl) 

294 

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lemma rtrancl_unfold: "r^* = Id Un r^* O r" 
15551  296 
by (auto intro: rtrancl_into_rtrancl elim: rtranclE) 
297 

31690  298 
lemma rtrancl_Un_separatorE: 
299 
"(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (a,x) : P^* \<longrightarrow> (x,y) : Q \<longrightarrow> x=y \<Longrightarrow> (a,b) : P^*" 

300 
apply (induct rule:rtrancl.induct) 

301 
apply blast 

302 
apply (blast intro:rtrancl_trans) 

303 
done 

304 

305 
lemma rtrancl_Un_separator_converseE: 

306 
"(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (x,b) : P^* \<longrightarrow> (y,x) : Q \<longrightarrow> y=x \<Longrightarrow> (a,b) : P^*" 

307 
apply (induct rule:converse_rtrancl_induct) 

308 
apply blast 

309 
apply (blast intro:rtrancl_trans) 

310 
done 

311 

34970  312 
lemma Image_closed_trancl: 
313 
assumes "r `` X \<subseteq> X" shows "r\<^sup>* `` X = X" 

314 
proof  

315 
from assms have **: "{y. \<exists>x\<in>X. (x, y) \<in> r} \<subseteq> X" by auto 

316 
have "\<And>x y. (y, x) \<in> r\<^sup>* \<Longrightarrow> y \<in> X \<Longrightarrow> x \<in> X" 

317 
proof  

318 
fix x y 

319 
assume *: "y \<in> X" 

320 
assume "(y, x) \<in> r\<^sup>*" 

321 
then show "x \<in> X" 

322 
proof induct 

323 
case base show ?case by (fact *) 

324 
next 

325 
case step with ** show ?case by auto 

326 
qed 

327 
qed 

328 
then show ?thesis by auto 

329 
qed 

330 

12691  331 

332 
subsection {* Transitive closure *} 

10331  333 

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lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+" 
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335 
apply (simp add: split_tupled_all) 
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336 
apply (erule trancl.induct) 
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337 
apply (iprover dest: subsetD)+ 
12691  338 
done 
339 

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340 
lemma r_into_trancl': "!!p. p : r ==> p : r^+" 
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341 
by (simp only: split_tupled_all) (erule r_into_trancl) 
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342 

12691  343 
text {* 
344 
\medskip Conversions between @{text trancl} and @{text rtrancl}. 

345 
*} 

346 

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347 
lemma tranclp_into_rtranclp: "r^++ a b ==> r^** a b" 
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348 
by (erule tranclp.induct) iprover+ 
12691  349 

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350 
lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set] 
22262  351 

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352 
lemma rtranclp_into_tranclp1: assumes r: "r^** a b" 
22262  353 
shows "!!c. r b c ==> r^++ a c" using r 
17589  354 
by induct iprover+ 
12691  355 

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356 
lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set] 
22262  357 

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358 
lemma rtranclp_into_tranclp2: "[ r a b; r^** b c ] ==> r^++ a c" 
12691  359 
 {* intro rule from @{text r} and @{text rtrancl} *} 
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360 
apply (erule rtranclp.cases) 
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361 
apply iprover 
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362 
apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1]) 
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363 
apply (simp  rule r_into_rtranclp)+ 
12691  364 
done 
365 

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366 
lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set] 
22262  367 

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368 
text {* Nice induction rule for @{text trancl} *} 
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369 
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]: 
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370 
assumes a: "r^++ a b" 
22262  371 
and cases: "!!y. r a y ==> P y" 
372 
"!!y z. r^++ a y ==> r y z ==> P y ==> P z" 

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373 
shows "P b" using a 
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374 
by (induct x\<equiv>a b) (iprover intro: cases)+ 
12691  375 

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376 
lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set] 
22262  377 

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378 
lemmas tranclp_induct2 = 
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379 
tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule, 
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380 
consumes 1, case_names base step] 
22262  381 

22172  382 
lemmas trancl_induct2 = 
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383 
trancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete), 
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384 
consumes 1, case_names base step] 
22172  385 

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386 
lemma tranclp_trans_induct: 
22262  387 
assumes major: "r^++ x y" 
388 
and cases: "!!x y. r x y ==> P x y" 

389 
"!!x y z. [ r^++ x y; P x y; r^++ y z; P y z ] ==> P x z" 

18372  390 
shows "P x y" 
12691  391 
 {* Another induction rule for trancl, incorporating transitivity *} 
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392 
by (iprover intro: major [THEN tranclp_induct] cases) 
12691  393 

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394 
lemmas trancl_trans_induct = tranclp_trans_induct [to_set] 
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395 

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396 
lemma tranclE [cases set: trancl]: 
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397 
assumes "(a, b) : r^+" 
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398 
obtains 
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399 
(base) "(a, b) : r" 
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400 
 (step) c where "(a, c) : r^+" and "(c, b) : r" 
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401 
using assms by cases simp_all 
10980  402 

32235
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403 
lemma trancl_Int_subset: "[ r \<subseteq> s; (r^+ \<inter> s) O r \<subseteq> s] ==> r^+ \<subseteq> s" 
22080
7bf8868ab3e4
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changeset

404 
apply (rule subsetI) 
26179
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changeset

405 
apply (rule_tac p = x in PairE) 
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rtranclp_induct, tranclp_induct: added case_names;
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changeset

406 
apply clarify 
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407 
apply (erule trancl_induct) 
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408 
apply auto 
22080
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409 
done 
7bf8868ab3e4
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changeset

410 

32235
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411 
lemma trancl_unfold: "r^+ = r Un r^+ O r" 
15551  412 
by (auto intro: trancl_into_trancl elim: tranclE) 
413 

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414 
text {* Transitivity of @{term "r^+"} *} 
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415 
lemma trans_trancl [simp]: "trans (r^+)" 
13704
854501b1e957
Transitive closure is now defined inductively as well.
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parents:
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diff
changeset

416 
proof (rule transI) 
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset

417 
fix x y z 
26179
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

418 
assume "(x, y) \<in> r^+" 
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset

419 
assume "(y, z) \<in> r^+" 
26179
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

420 
then show "(x, z) \<in> r^+" 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

421 
proof induct 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

422 
case (base u) 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

423 
from `(x, y) \<in> r^+` and `(y, u) \<in> r` 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
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diff
changeset

424 
show "(x, u) \<in> r^+" .. 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

425 
next 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

426 
case (step u v) 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

427 
from `(x, u) \<in> r^+` and `(u, v) \<in> r` 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

428 
show "(x, v) \<in> r^+" .. 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

429 
qed 
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset

430 
qed 
12691  431 

45607  432 
lemmas trancl_trans = trans_trancl [THEN transD] 
12691  433 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

434 
lemma tranclp_trans: 
22262  435 
assumes xy: "r^++ x y" 
436 
and yz: "r^++ y z" 

437 
shows "r^++ x z" using yz xy 

438 
by induct iprover+ 

439 

26179
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rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
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diff
changeset

440 
lemma trancl_id [simp]: "trans r \<Longrightarrow> r^+ = r" 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

441 
apply auto 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

442 
apply (erule trancl_induct) 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

443 
apply assumption 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

444 
apply (unfold trans_def) 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

445 
apply blast 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

446 
done 
19623  447 

26179
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

448 
lemma rtranclp_tranclp_tranclp: 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

449 
assumes "r^** x y" 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

450 
shows "!!z. r^++ y z ==> r^++ x z" using assms 
23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

451 
by induct (iprover intro: tranclp_trans)+ 
12691  452 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
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diff
changeset

453 
lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set] 
22262  454 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

455 
lemma tranclp_into_tranclp2: "r a b ==> r^++ b c ==> r^++ a c" 
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

456 
by (erule tranclp_trans [OF tranclp.r_into_trancl]) 
22262  457 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

458 
lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set] 
12691  459 

460 
lemma trancl_insert: 

461 
"(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}" 

462 
 {* primitive recursion for @{text trancl} over finite relations *} 

463 
apply (rule equalityI) 

464 
apply (rule subsetI) 

465 
apply (simp only: split_tupled_all) 

14208  466 
apply (erule trancl_induct, blast) 
35216  467 
apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans) 
12691  468 
apply (rule subsetI) 
469 
apply (blast intro: trancl_mono rtrancl_mono 

470 
[THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2) 

471 
done 

472 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

473 
lemma tranclp_converseI: "(r^++)^1 x y ==> (r^1)^++ x y" 
22262  474 
apply (drule conversepD) 
23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

475 
apply (erule tranclp_induct) 
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

476 
apply (iprover intro: conversepI tranclp_trans)+ 
12691  477 
done 
478 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

479 
lemmas trancl_converseI = tranclp_converseI [to_set] 
22262  480 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

481 
lemma tranclp_converseD: "(r^1)^++ x y ==> (r^++)^1 x y" 
22262  482 
apply (rule conversepI) 
23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

483 
apply (erule tranclp_induct) 
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

484 
apply (iprover dest: conversepD intro: tranclp_trans)+ 
13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset

485 
done 
12691  486 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

487 
lemmas trancl_converseD = tranclp_converseD [to_set] 
22262  488 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

489 
lemma tranclp_converse: "(r^1)^++ = (r^++)^1" 
44890
22f665a2e91c
new fastforce replacing fastsimp  less confusing name
nipkow
parents:
43596
diff
changeset

490 
by (fastforce simp add: fun_eq_iff 
23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

491 
intro!: tranclp_converseI dest!: tranclp_converseD) 
22262  492 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

493 
lemmas trancl_converse = tranclp_converse [to_set] 
12691  494 

19228  495 
lemma sym_trancl: "sym r ==> sym (r^+)" 
496 
by (simp only: sym_conv_converse_eq trancl_converse [symmetric]) 

497 

34909
a799687944af
Tuned some proofs; nicer case names for some of the induction / cases rules.
berghofe
parents:
33878
diff
changeset

498 
lemma converse_tranclp_induct [consumes 1, case_names base step]: 
22262  499 
assumes major: "r^++ a b" 
500 
and cases: "!!y. r y b ==> P(y)" 

501 
"!!y z.[ r y z; r^++ z b; P(z) ] ==> P(y)" 

18372  502 
shows "P a" 
23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

503 
apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major]) 
18372  504 
apply (rule cases) 
22262  505 
apply (erule conversepD) 
35216  506 
apply (blast intro: assms dest!: tranclp_converseD) 
18372  507 
done 
12691  508 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

509 
lemmas converse_trancl_induct = converse_tranclp_induct [to_set] 
22262  510 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

511 
lemma tranclpD: "R^++ x y ==> EX z. R x z \<and> R^** z y" 
26179
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

512 
apply (erule converse_tranclp_induct) 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

513 
apply auto 
23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

514 
apply (blast intro: rtranclp_trans) 
12691  515 
done 
516 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

517 
lemmas tranclD = tranclpD [to_set] 
22262  518 

31577  519 
lemma converse_tranclpE: 
520 
assumes major: "tranclp r x z" 

521 
assumes base: "r x z ==> P" 

522 
assumes step: "\<And> y. [ r x y; tranclp r y z ] ==> P" 

523 
shows P 

524 
proof  

525 
from tranclpD[OF major] 

526 
obtain y where "r x y" and "rtranclp r y z" by iprover 

527 
from this(2) show P 

528 
proof (cases rule: rtranclp.cases) 

529 
case rtrancl_refl 

530 
with `r x y` base show P by iprover 

531 
next 

532 
case rtrancl_into_rtrancl 

533 
from this have "tranclp r y z" 

534 
by (iprover intro: rtranclp_into_tranclp1) 

535 
with `r x y` step show P by iprover 

536 
qed 

537 
qed 

538 

539 
lemmas converse_tranclE = converse_tranclpE [to_set] 

540 

25295
12985023be5e
tranclD2 (tranclD at the other end) + trancl_power
kleing
parents:
23743
diff
changeset

541 
lemma tranclD2: 
12985023be5e
tranclD2 (tranclD at the other end) + trancl_power
kleing
parents:
23743
diff
changeset

542 
"(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R" 
12985023be5e
tranclD2 (tranclD at the other end) + trancl_power
kleing
parents:
23743
diff
changeset

543 
by (blast elim: tranclE intro: trancl_into_rtrancl) 
12985023be5e
tranclD2 (tranclD at the other end) + trancl_power
kleing
parents:
23743
diff
changeset

544 

13867  545 
lemma irrefl_tranclI: "r^1 \<inter> r^* = {} ==> (x, x) \<notin> r^+" 
18372  546 
by (blast elim: tranclE dest: trancl_into_rtrancl) 
12691  547 

548 
lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y" 

549 
by (blast dest: r_into_trancl) 

550 

551 
lemma trancl_subset_Sigma_aux: 

552 
"(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A" 

18372  553 
by (induct rule: rtrancl_induct) auto 
12691  554 

555 
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A" 

13704
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset

556 
apply (rule subsetI) 
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset

557 
apply (simp only: split_tupled_all) 
854501b1e957
Transitive closure is now defined inductively as well.
berghofe
parents:
12937
diff
changeset

558 
apply (erule tranclE) 
26179
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

559 
apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+ 
12691  560 
done 
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset

561 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

562 
lemma reflcl_tranclp [simp]: "(r^++)^== = r^**" 
22262  563 
apply (safe intro!: order_antisym) 
23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

564 
apply (erule tranclp_into_rtranclp) 
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

565 
apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1) 
11084  566 
done 
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset

567 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

568 
lemmas reflcl_trancl [simp] = reflcl_tranclp [to_set] 
22262  569 

11090  570 
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*" 
11084  571 
apply safe 
14208  572 
apply (drule trancl_into_rtrancl, simp) 
573 
apply (erule rtranclE, safe) 

574 
apply (rule r_into_trancl, simp) 

11084  575 
apply (rule rtrancl_into_trancl1) 
14208  576 
apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast) 
11084  577 
done 
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset

578 

45140  579 
lemma rtrancl_trancl_reflcl [code]: "r^* = (r^+)^=" 
580 
by simp 

581 

11090  582 
lemma trancl_empty [simp]: "{}^+ = {}" 
11084  583 
by (auto elim: trancl_induct) 
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset

584 

11090  585 
lemma rtrancl_empty [simp]: "{}^* = Id" 
11084  586 
by (rule subst [OF reflcl_trancl]) simp 
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset

587 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

588 
lemma rtranclpD: "R^** a b ==> a = b \<or> a \<noteq> b \<and> R^++ a b" 
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

589 
by (force simp add: reflcl_tranclp [symmetric] simp del: reflcl_tranclp) 
22262  590 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

591 
lemmas rtranclD = rtranclpD [to_set] 
11084  592 

16514  593 
lemma rtrancl_eq_or_trancl: 
594 
"(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)" 

595 
by (fast elim: trancl_into_rtrancl dest: rtranclD) 

10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset

596 

33656
fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

597 
lemma trancl_unfold_right: "r^+ = r^* O r" 
fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

598 
by (auto dest: tranclD2 intro: rtrancl_into_trancl1) 
fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

599 

fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

600 
lemma trancl_unfold_left: "r^+ = r O r^*" 
fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

601 
by (auto dest: tranclD intro: rtrancl_into_trancl2) 
fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

602 

fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

603 

fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

604 
text {* Simplifying nested closures *} 
fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

605 

fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

606 
lemma rtrancl_trancl_absorb[simp]: "(R^*)^+ = R^*" 
fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

607 
by (simp add: trans_rtrancl) 
fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

608 

fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

609 
lemma trancl_rtrancl_absorb[simp]: "(R^+)^* = R^*" 
fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

610 
by (subst reflcl_trancl[symmetric]) simp 
fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

611 

fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

612 
lemma rtrancl_reflcl_absorb[simp]: "(R^*)^= = R^*" 
fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

613 
by auto 
fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

614 

fc1af6753233
a few lemmas for pointfree reasoning about transitive closure
krauss
parents:
32901
diff
changeset

615 

12691  616 
text {* @{text Domain} and @{text Range} *} 
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset

617 

11090  618 
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV" 
11084  619 
by blast 
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset

620 

11090  621 
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV" 
11084  622 
by blast 
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset

623 

11090  624 
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*" 
11084  625 
by (rule rtrancl_Un_rtrancl [THEN subst]) fast 
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset

626 

11090  627 
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*" 
11084  628 
by (blast intro: subsetD [OF rtrancl_Un_subset]) 
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset

629 

11090  630 
lemma trancl_domain [simp]: "Domain (r^+) = Domain r" 
11084  631 
by (unfold Domain_def) (blast dest: tranclD) 
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset

632 

11090  633 
lemma trancl_range [simp]: "Range (r^+) = Range r" 
26271  634 
unfolding Range_def by(simp add: trancl_converse [symmetric]) 
10996
74e970389def
Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents:
10980
diff
changeset

635 

11115  636 
lemma Not_Domain_rtrancl: 
12691  637 
"x ~: Domain R ==> ((x, y) : R^*) = (x = y)" 
638 
apply auto 

26179
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

639 
apply (erule rev_mp) 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

640 
apply (erule rtrancl_induct) 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

641 
apply auto 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

642 
done 
11327
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset

643 

29609  644 
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r" 
645 
apply clarify 

646 
apply (erule trancl_induct) 

647 
apply (auto simp add: Field_def) 

648 
done 

649 

41987  650 
lemma finite_trancl[simp]: "finite (r^+) = finite r" 
29609  651 
apply auto 
652 
prefer 2 

653 
apply (rule trancl_subset_Field2 [THEN finite_subset]) 

654 
apply (rule finite_SigmaI) 

655 
prefer 3 

656 
apply (blast intro: r_into_trancl' finite_subset) 

657 
apply (auto simp add: finite_Field) 

658 
done 

659 

12691  660 
text {* More about converse @{text rtrancl} and @{text trancl}, should 
661 
be merged with main body. *} 

12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset

662 

14337
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

663 
lemma single_valued_confluent: 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

664 
"\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk> 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

665 
\<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*" 
26179
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

666 
apply (erule rtrancl_induct) 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

667 
apply simp 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

668 
apply (erule disjE) 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

669 
apply (blast elim:converse_rtranclE dest:single_valuedD) 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

670 
apply(blast intro:rtrancl_trans) 
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

671 
done 
14337
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

672 

12691  673 
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+" 
12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset

674 
by (fast intro: trancl_trans) 
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset

675 

f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset

676 
lemma trancl_into_trancl [rule_format]: 
12691  677 
"(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r > (a,c) \<in> r\<^sup>+" 
678 
apply (erule trancl_induct) 

12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset

679 
apply (fast intro: r_r_into_trancl) 
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset

680 
apply (fast intro: r_r_into_trancl trancl_trans) 
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset

681 
done 
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset

682 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

683 
lemma tranclp_rtranclp_tranclp: 
22262  684 
"r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c" 
23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

685 
apply (drule tranclpD) 
26179
bc5d582d6cfe
rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents:
26174
diff
changeset

686 
apply (elim exE conjE) 
23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

687 
apply (drule rtranclp_trans, assumption) 
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

688 
apply (drule rtranclp_into_tranclp2, assumption, assumption) 
12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset

689 
done 
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset

690 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

691 
lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set] 
22262  692 

12691  693 
lemmas transitive_closure_trans [trans] = 
694 
r_r_into_trancl trancl_trans rtrancl_trans 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

695 
trancl.trancl_into_trancl trancl_into_trancl2 
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

696 
rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl 
12691  697 
rtrancl_trancl_trancl trancl_rtrancl_trancl 
12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset

698 

23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

699 
lemmas transitive_closurep_trans' [trans] = 
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

700 
tranclp_trans rtranclp_trans 
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

701 
tranclp.trancl_into_trancl tranclp_into_tranclp2 
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

702 
rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp 
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

703 
rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp 
22262  704 

12428
f3033eed309a
setup [trans] rules for calculational Isar reasoning
kleing
parents:
11327
diff
changeset

705 
declare trancl_into_rtrancl [elim] 
11327
cd2c27a23df1
Transitive closure is now defined via "inductive".
berghofe
parents:
11115
diff
changeset

706 

30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

707 
subsection {* The power operation on relations *} 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

708 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

709 
text {* @{text "R ^^ n = R O ... O R"}, the nfold composition of @{text R} *} 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

710 

30971  711 
overloading 
712 
relpow == "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" 

713 
begin 

30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

714 

30971  715 
primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where 
716 
"relpow 0 R = Id" 

32235
8f9b8d14fc9f
"more standard" argument order of relation composition (op O)
krauss
parents:
32215
diff
changeset

717 
 "relpow (Suc n) R = (R ^^ n) O R" 
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

718 

30971  719 
end 
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

720 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

721 
lemma rel_pow_1 [simp]: 
30971  722 
fixes R :: "('a \<times> 'a) set" 
723 
shows "R ^^ 1 = R" 

30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

724 
by simp 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

725 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

726 
lemma rel_pow_0_I: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

727 
"(x, x) \<in> R ^^ 0" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

728 
by simp 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

729 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

730 
lemma rel_pow_Suc_I: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

731 
"(x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

732 
by auto 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

733 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

734 
lemma rel_pow_Suc_I2: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

735 
"(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n" 
44890
22f665a2e91c
new fastforce replacing fastsimp  less confusing name
nipkow
parents:
43596
diff
changeset

736 
by (induct n arbitrary: z) (simp, fastforce) 
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

737 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

738 
lemma rel_pow_0_E: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

739 
"(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

740 
by simp 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

741 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

742 
lemma rel_pow_Suc_E: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

743 
"(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) \<Longrightarrow> P" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

744 
by auto 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

745 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

746 
lemma rel_pow_E: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

747 
"(x, z) \<in> R ^^ n \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

748 
\<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

749 
\<Longrightarrow> P" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

750 
by (cases n) auto 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

751 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

752 
lemma rel_pow_Suc_D2: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

753 
"(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

754 
apply (induct n arbitrary: x z) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

755 
apply (blast intro: rel_pow_0_I elim: rel_pow_0_E rel_pow_Suc_E) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

756 
apply (blast intro: rel_pow_Suc_I elim: rel_pow_0_E rel_pow_Suc_E) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

757 
done 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

758 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

759 
lemma rel_pow_Suc_E2: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

760 
"(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> P) \<Longrightarrow> P" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

761 
by (blast dest: rel_pow_Suc_D2) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

762 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

763 
lemma rel_pow_Suc_D2': 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

764 
"\<forall>x y z. (x, y) \<in> R ^^ n \<and> (y, z) \<in> R \<longrightarrow> (\<exists>w. (x, w) \<in> R \<and> (w, z) \<in> R ^^ n)" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

765 
by (induct n) (simp_all, blast) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

766 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

767 
lemma rel_pow_E2: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

768 
"(x, z) \<in> R ^^ n \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

769 
\<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

770 
\<Longrightarrow> P" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

771 
apply (cases n, simp) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

772 
apply (cut_tac n=nat and R=R in rel_pow_Suc_D2', simp, blast) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

773 
done 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

774 

32235
8f9b8d14fc9f
"more standard" argument order of relation composition (op O)
krauss
parents:
32215
diff
changeset

775 
lemma rel_pow_add: "R ^^ (m+n) = R^^m O R^^n" 
45976  776 
by (induct n) auto 
31351  777 

31970
ccaadfcf6941
move rel_pow_commute: "R O R ^^ n = R ^^ n O R" to Transitive_Closure
krauss
parents:
31690
diff
changeset

778 
lemma rel_pow_commute: "R O R ^^ n = R ^^ n O R" 
45976  779 
by (induct n) (simp, simp add: O_assoc [symmetric]) 
31970
ccaadfcf6941
move rel_pow_commute: "R O R ^^ n = R ^^ n O R" to Transitive_Closure
krauss
parents:
31690
diff
changeset

780 

45116
f947eeef6b6f
adding lemma about rel_pow in Transitive_Closure for executable equation of the (refl) transitive closure
bulwahn
parents:
44921
diff
changeset

781 
lemma rel_pow_empty: 
45153  782 
"0 < n \<Longrightarrow> ({} :: ('a \<times> 'a) set) ^^ n = {}" 
783 
by (cases n) auto 

45116
f947eeef6b6f
adding lemma about rel_pow in Transitive_Closure for executable equation of the (refl) transitive closure
bulwahn
parents:
44921
diff
changeset

784 

30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

785 
lemma rtrancl_imp_UN_rel_pow: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

786 
assumes "p \<in> R^*" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

787 
shows "p \<in> (\<Union>n. R ^^ n)" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

788 
proof (cases p) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

789 
case (Pair x y) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

790 
with assms have "(x, y) \<in> R^*" by simp 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

791 
then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

792 
case base show ?case by (blast intro: rel_pow_0_I) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

793 
next 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

794 
case step then show ?case by (blast intro: rel_pow_Suc_I) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

795 
qed 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

796 
with Pair show ?thesis by simp 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

797 
qed 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

798 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

799 
lemma rel_pow_imp_rtrancl: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

800 
assumes "p \<in> R ^^ n" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

801 
shows "p \<in> R^*" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

802 
proof (cases p) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

803 
case (Pair x y) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

804 
with assms have "(x, y) \<in> R ^^ n" by simp 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

805 
then have "(x, y) \<in> R^*" proof (induct n arbitrary: x y) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

806 
case 0 then show ?case by simp 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

807 
next 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

808 
case Suc then show ?case 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

809 
by (blast elim: rel_pow_Suc_E intro: rtrancl_into_rtrancl) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

810 
qed 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

811 
with Pair show ?thesis by simp 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

812 
qed 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

813 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

814 
lemma rtrancl_is_UN_rel_pow: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

815 
"R^* = (\<Union>n. R ^^ n)" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

816 
by (blast intro: rtrancl_imp_UN_rel_pow rel_pow_imp_rtrancl) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

817 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

818 
lemma rtrancl_power: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

819 
"p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

820 
by (simp add: rtrancl_is_UN_rel_pow) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

821 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

822 
lemma trancl_power: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

823 
"p \<in> R^+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

824 
apply (cases p) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

825 
apply simp 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

826 
apply (rule iffI) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

827 
apply (drule tranclD2) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

828 
apply (clarsimp simp: rtrancl_is_UN_rel_pow) 
30971  829 
apply (rule_tac x="Suc n" in exI) 
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

830 
apply (clarsimp simp: rel_comp_def) 
44890
22f665a2e91c
new fastforce replacing fastsimp  less confusing name
nipkow
parents:
43596
diff
changeset

831 
apply fastforce 
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

832 
apply clarsimp 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

833 
apply (case_tac n, simp) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

834 
apply clarsimp 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

835 
apply (drule rel_pow_imp_rtrancl) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

836 
apply (drule rtrancl_into_trancl1) apply auto 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

837 
done 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

838 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

839 
lemma rtrancl_imp_rel_pow: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

840 
"p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

841 
by (auto dest: rtrancl_imp_UN_rel_pow) 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

842 

41987  843 
text{* By Sternagel/Thiemann: *} 
844 
lemma rel_pow_fun_conv: 

845 
"((a,b) \<in> R ^^ n) = (\<exists>f. f 0 = a \<and> f n = b \<and> (\<forall>i<n. (f i, f(Suc i)) \<in> R))" 

846 
proof (induct n arbitrary: b) 

847 
case 0 show ?case by auto 

848 
next 

849 
case (Suc n) 

850 
show ?case 

851 
proof (simp add: rel_comp_def Suc) 

852 
show "(\<exists>y. (\<exists>f. f 0 = a \<and> f n = y \<and> (\<forall>i<n. (f i,f(Suc i)) \<in> R)) \<and> (y,b) \<in> R) 

853 
= (\<exists>f. f 0 = a \<and> f(Suc n) = b \<and> (\<forall>i<Suc n. (f i, f (Suc i)) \<in> R))" 

854 
(is "?l = ?r") 

855 
proof 

856 
assume ?l 

857 
then obtain c f where 1: "f 0 = a" "f n = c" "\<And>i. i < n \<Longrightarrow> (f i, f (Suc i)) \<in> R" "(c,b) \<in> R" by auto 

858 
let ?g = "\<lambda> m. if m = Suc n then b else f m" 

859 
show ?r by (rule exI[of _ ?g], simp add: 1) 

860 
next 

861 
assume ?r 

862 
then obtain f where 1: "f 0 = a" "b = f (Suc n)" "\<And>i. i < Suc n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto 

863 
show ?l by (rule exI[of _ "f n"], rule conjI, rule exI[of _ f], insert 1, auto) 

864 
qed 

865 
qed 

866 
qed 

867 

868 
lemma rel_pow_finite_bounded1: 

869 
assumes "finite(R :: ('a*'a)set)" and "k>0" 

870 
shows "R^^k \<subseteq> (UN n:{n. 0<n & n <= card R}. R^^n)" (is "_ \<subseteq> ?r") 

871 
proof 

872 
{ fix a b k 

873 
have "(a,b) : R^^(Suc k) \<Longrightarrow> EX n. 0<n & n <= card R & (a,b) : R^^n" 

874 
proof(induct k arbitrary: b) 

875 
case 0 

876 
hence "R \<noteq> {}" by auto 

877 
with card_0_eq[OF `finite R`] have "card R >= Suc 0" by auto 

878 
thus ?case using 0 by force 

879 
next 

880 
case (Suc k) 

881 
then obtain a' where "(a,a') : R^^(Suc k)" and "(a',b) : R" by auto 

882 
from Suc(1)[OF `(a,a') : R^^(Suc k)`] 

883 
obtain n where "n \<le> card R" and "(a,a') \<in> R ^^ n" by auto 

884 
have "(a,b) : R^^(Suc n)" using `(a,a') \<in> R^^n` and `(a',b)\<in> R` by auto 

885 
{ assume "n < card R" 

886 
hence ?case using `(a,b): R^^(Suc n)` Suc_leI[OF `n < card R`] by blast 

887 
} moreover 

888 
{ assume "n = card R" 

889 
from `(a,b) \<in> R ^^ (Suc n)`[unfolded rel_pow_fun_conv] 

890 
obtain f where "f 0 = a" and "f(Suc n) = b" 

891 
and steps: "\<And>i. i <= n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto 

892 
let ?p = "%i. (f i, f(Suc i))" 

893 
let ?N = "{i. i \<le> n}" 

894 
have "?p ` ?N <= R" using steps by auto 

895 
from card_mono[OF assms(1) this] 

896 
have "card(?p ` ?N) <= card R" . 

897 
also have "\<dots> < card ?N" using `n = card R` by simp 

898 
finally have "~ inj_on ?p ?N" by(rule pigeonhole) 

899 
then obtain i j where i: "i <= n" and j: "j <= n" and ij: "i \<noteq> j" and 

900 
pij: "?p i = ?p j" by(auto simp: inj_on_def) 

901 
let ?i = "min i j" let ?j = "max i j" 

902 
have i: "?i <= n" and j: "?j <= n" and pij: "?p ?i = ?p ?j" 

903 
and ij: "?i < ?j" 

904 
using i j ij pij unfolding min_def max_def by auto 

905 
from i j pij ij obtain i j where i: "i<=n" and j: "j<=n" and ij: "i<j" 

906 
and pij: "?p i = ?p j" by blast 

907 
let ?g = "\<lambda> l. if l \<le> i then f l else f (l + (j  i))" 

908 
let ?n = "Suc(n  (j  i))" 

909 
have abl: "(a,b) \<in> R ^^ ?n" unfolding rel_pow_fun_conv 

910 
proof (rule exI[of _ ?g], intro conjI impI allI) 

911 
show "?g ?n = b" using `f(Suc n) = b` j ij by auto 

912 
next 

913 
fix k assume "k < ?n" 

914 
show "(?g k, ?g (Suc k)) \<in> R" 

915 
proof (cases "k < i") 

916 
case True 

917 
with i have "k <= n" by auto 

918 
from steps[OF this] show ?thesis using True by simp 

919 
next 

920 
case False 

921 
hence "i \<le> k" by auto 

922 
show ?thesis 

923 
proof (cases "k = i") 

924 
case True 

925 
thus ?thesis using ij pij steps[OF i] by simp 

926 
next 

927 
case False 

928 
with `i \<le> k` have "i < k" by auto 

929 
hence small: "k + (j  i) <= n" using `k<?n` by arith 

930 
show ?thesis using steps[OF small] `i<k` by auto 

931 
qed 

932 
qed 

933 
qed (simp add: `f 0 = a`) 

934 
moreover have "?n <= n" using i j ij by arith 

935 
ultimately have ?case using `n = card R` by blast 

936 
} 

937 
ultimately show ?case using `n \<le> card R` by force 

938 
qed 

939 
} 

940 
thus ?thesis using gr0_implies_Suc[OF `k>0`] by auto 

941 
qed 

942 

943 
lemma rel_pow_finite_bounded: 

944 
assumes "finite(R :: ('a*'a)set)" 

945 
shows "R^^k \<subseteq> (UN n:{n. n <= card R}. R^^n)" 

946 
apply(cases k) 

947 
apply force 

948 
using rel_pow_finite_bounded1[OF assms, of k] by auto 

949 

950 
lemma rtrancl_finite_eq_rel_pow: 

951 
"finite R \<Longrightarrow> R^* = (UN n : {n. n <= card R}. R^^n)" 

44890
22f665a2e91c
new fastforce replacing fastsimp  less confusing name
nipkow
parents:
43596
diff
changeset

952 
by(fastforce simp: rtrancl_power dest: rel_pow_finite_bounded) 
41987  953 

954 
lemma trancl_finite_eq_rel_pow: 

955 
"finite R \<Longrightarrow> R^+ = (UN n : {n. 0 < n & n <= card R}. R^^n)" 

956 
apply(auto simp add: trancl_power) 

957 
apply(auto dest: rel_pow_finite_bounded1) 

958 
done 

959 

960 
lemma finite_rel_comp[simp,intro]: 

961 
assumes "finite R" and "finite S" 

962 
shows "finite(R O S)" 

963 
proof 

964 
have "R O S = (UN (x,y) : R. \<Union>((%(u,v). if u=y then {(x,v)} else {}) ` S))" 

965 
by(force simp add: split_def) 

966 
thus ?thesis using assms by(clarsimp) 

967 
qed 

968 

969 
lemma finite_relpow[simp,intro]: 

970 
assumes "finite(R :: ('a*'a)set)" shows "n>0 \<Longrightarrow> finite(R^^n)" 

971 
apply(induct n) 

972 
apply simp 

973 
apply(case_tac n) 

974 
apply(simp_all add: assms) 

975 
done 

976 

30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

977 
lemma single_valued_rel_pow: 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

978 
fixes R :: "('a * 'a) set" 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30549
diff
changeset

979 
shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)" 
41987  980 
apply (induct n arbitrary: R) 
981 
apply simp_all 

982 
apply (rule single_valuedI) 

983 
apply (fast dest: single_valuedD elim: rel_pow_Suc_E) 

984 
done 

15551  985 

45140  986 

987 
subsection {* Bounded transitive closure *} 

988 

989 
definition ntrancl :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" 

990 
where 

991 
"ntrancl n R = (\<Union>i\<in>{i. 0 < i \<and> i \<le> Suc n}. R ^^ i)" 

992 

993 
lemma ntrancl_Zero [simp, code]: 

994 
"ntrancl 0 R = R" 

995 
proof 

996 
show "R \<subseteq> ntrancl 0 R" 

997 
unfolding ntrancl_def by fastforce 

998 
next 

999 
{ 

1000 
fix i have "0 < i \<and> i \<le> Suc 0 \<longleftrightarrow> i = 1" by auto 

1001 
} 

1002 
from this show "ntrancl 0 R \<le> R" 

1003 
unfolding ntrancl_def by auto 

1004 
qed 

1005 

1006 
lemma ntrancl_Suc [simp, code]: 

1007 
"ntrancl (Suc n) R = ntrancl n R O (Id \<union> R)" 

1008 
proof 

1009 
{ 

1010 
fix a b 

1011 
assume "(a, b) \<in> ntrancl (Suc n) R" 

1012 
from this obtain i where "0 < i" "i \<le> Suc (Suc n)" "(a, b) \<in> R ^^ i" 

1013 
unfolding ntrancl_def by auto 

1014 
have "(a, b) \<in> ntrancl n R O (Id \<union> R)" 

1015 
proof (cases "i = 1") 

1016 
case True 

1017 
from this `(a, b) \<in> R ^^ i` show ?thesis 

1018 
unfolding ntrancl_def by auto 

1019 
next 

1020 
case False 

1021 
from this `0 < i` obtain j where j: "i = Suc j" "0 < j" 

1022 
by (cases i) auto 

1023 
from this `(a, b) \<in> R ^^ i` obtain c where c1: "(a, c) \<in> R ^^ j" and c2:"(c, b) \<in> R" 

1024 
by auto 

1025 
from c1 j `i \<le> Suc (Suc n)` have "(a, c) \<in> ntrancl n R" 

1026 
unfolding ntrancl_def by fastforce 

1027 
from this c2 show ?thesis by fastforce 

1028 
qed 

1029 
} 

1030 
from this show "ntrancl (Suc n) R \<subseteq> ntrancl n R O (Id \<union> R)" 

1031 
by auto 

1032 
next 

1033 
show "ntrancl n R O (Id \<union> R) \<subseteq> ntrancl (Suc n) R" 

1034 
unfolding ntrancl_def by fastforce 

1035 
qed 

1036 

45141
b2eb87bd541b
avoid very specific code equation for card; corrected spelling
haftmann
parents:
45140
diff
changeset

1037 
lemma finite_trancl_ntranl: 
45140  1038 
"finite R \<Longrightarrow> trancl R = ntrancl (card R  1) R" 
1039 
by (cases "card R") (auto simp add: trancl_finite_eq_rel_pow rel_pow_empty ntrancl_def) 

1040 

1041 

45139  1042 
subsection {* Acyclic relations *} 
1043 

1044 
definition acyclic :: "('a * 'a) set => bool" where 

1045 
"acyclic r \<longleftrightarrow> (!x. (x,x) ~: r^+)" 

1046 

1047 
abbreviation acyclicP :: "('a => 'a => bool) => bool" where 

1048 
"acyclicP r \<equiv> acyclic {(x, y). r x y}" 

1049 

1050 
lemma acyclic_irrefl: 

1051 
"acyclic r \<longleftrightarrow> irrefl (r^+)" 

1052 
by (simp add: acyclic_def irrefl_def) 

1053 

1054 
lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r" 

1055 
by (simp add: acyclic_def) 

1056 

1057 
lemma acyclic_insert [iff]: 

1058 
"acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)" 

1059 
apply (simp add: acyclic_def trancl_insert) 

1060 
apply (blast intro: rtrancl_trans) 

1061 
done 

1062 

1063 
lemma acyclic_converse [iff]: "acyclic(r^1) = acyclic r" 

1064 
by (simp add: acyclic_def trancl_converse) 

1065 

1066 
lemmas acyclicP_converse [iff] = acyclic_converse [to_pred] 

1067 

1068 
lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)" 

1069 
apply (simp add: acyclic_def antisym_def) 

1070 
apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl) 

1071 
done 

1072 

1073 
(* Other direction: 

1074 
acyclic = no loops 

1075 
antisym = only self loops 

1076 
Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r  Id) 

1077 
==> antisym( r^* ) = acyclic(r  Id)"; 

1078 
*) 

1079 

1080 
lemma acyclic_subset: "[ acyclic s; r <= s ] ==> acyclic r" 

1081 
apply (simp add: acyclic_def) 

1082 
apply (blast intro: trancl_mono) 

1083 
done 

1084 

1085 

15076
4b3d280ef06a
New prover for transitive and reflexivetransitive closure of relations.
ballarin
parents:
14565
diff
changeset

1086 
subsection {* Setup of transitivity reasoner *} 
4b3d280ef06a
New prover for transitive and reflexivetransitive closure of relations.
ballarin
parents:
14565
diff
changeset

1087 

26340  1088 
ML {* 
15076
4b3d280ef06a
New prover for transitive and reflexivetransitive closure of relations.
ballarin
parents:
14565
diff
changeset

1089 

32215  1090 
structure Trancl_Tac = Trancl_Tac 
1091 
( 

1092 
val r_into_trancl = @{thm trancl.r_into_trancl}; 

1093 
val trancl_trans = @{thm trancl_trans}; 

1094 
val rtrancl_refl = @{thm rtrancl.rtrancl_refl}; 

1095 
val r_into_rtrancl = @{thm r_into_rtrancl}; 

1096 
val trancl_into_rtrancl = @{thm trancl_into_rtrancl}; 

1097 
val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl}; 

1098 
val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl}; 

1099 
val rtrancl_trans = @{thm rtrancl_trans}; 

15096  1100 

30107
f3b3b0e3d184
Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents:
29609
diff
changeset

1101 
fun decomp (@{const Trueprop} $ t) = 
37677  1102 
let fun dec (Const (@{const_name Set.member}, _) $ (Const (@{const_name Pair}, _) $ a $ b) $ rel ) = 
23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

1103 
let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*") 
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

1104 
 decr (Const ("Transitive_Closure.trancl", _ ) $ r) = (r,"r+") 
18372  1105 
 decr r = (r,"r"); 
26801
244184661a09
 Function dec in Trancl_Tac must etacontract relation before calling
berghofe
parents:
26340
diff
changeset

1106 
val (rel,r) = decr (Envir.beta_eta_contract rel); 
18372  1107 
in SOME (a,b,rel,r) end 
1108 
 dec _ = NONE 

30107
f3b3b0e3d184
Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents:
29609
diff
changeset

1109 
in dec t end 
f3b3b0e3d184
Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents:
29609
diff
changeset

1110 
 decomp _ = NONE; 
32215  1111 
); 
15076
4b3d280ef06a
New prover for transitive and reflexivetransitive closure of relations.
ballarin
parents:
14565
diff
changeset

1112 

32215  1113 
structure Tranclp_Tac = Trancl_Tac 
1114 
( 

1115 
val r_into_trancl = @{thm tranclp.r_into_trancl}; 

1116 
val trancl_trans = @{thm tranclp_trans}; 

1117 
val rtrancl_refl = @{thm rtranclp.rtrancl_refl}; 

1118 
val r_into_rtrancl = @{thm r_into_rtranclp}; 

1119 
val trancl_into_rtrancl = @{thm tranclp_into_rtranclp}; 

1120 
val rtrancl_trancl_trancl = @{thm rtranclp_tranclp_tranclp}; 

1121 
val trancl_rtrancl_trancl = @{thm tranclp_rtranclp_tranclp}; 

1122 
val rtrancl_trans = @{thm rtranclp_trans}; 

22262  1123 

30107
f3b3b0e3d184
Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents:
29609
diff
changeset

1124 
fun decomp (@{const Trueprop} $ t) = 
22262  1125 
let fun dec (rel $ a $ b) = 
23743
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

1126 
let fun decr (Const ("Transitive_Closure.rtranclp", _ ) $ r) = (r,"r*") 
52fbc991039f
rtrancl and trancl are now defined using inductive_set.
berghofe
parents:
22422
diff
changeset

1127 
 decr (Const ("Transitive_Closure.tranclp", _ ) $ r) = (r,"r+") 
22262  1128 
 decr r = (r,"r"); 
1129 
val (rel,r) = decr rel; 

26801
244184661a09
 Function dec in Trancl_Tac must etacontract relation before calling
berghofe
parents:
26340
diff
changeset

1130 
in SOME (a, b, rel, r) end 
22262  1131 
 dec _ = NONE 
30107
f3b3b0e3d184
Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents:
29609
diff
changeset

1132 
in dec t end 
f3b3b0e3d184
Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents:
29609
diff
changeset

1133 
 decomp _ = NONE; 
32215  1134 
); 
26340  1135 
*} 
22262  1136 

42795
66fcc9882784
clarified map_simpset versus Simplifier.map_simpset_global;
wenzelm
parents:
41987
diff
changeset

1137 
setup {* 
66fcc9882784
clarified map_simpset versus Simplifier.map_simpset_global;
wenzelm
parents:
41987
diff
changeset

1138 
Simplifier.map_simpset_global (fn ss => ss 
43596  1139 
addSolver (mk_solver "Trancl" (Trancl_Tac.trancl_tac o Simplifier.the_context)) 
1140 
addSolver (mk_solver "Rtrancl" (Trancl_Tac.rtrancl_tac o Simplifier.the_context)) 

1141 
addSolver (mk_solver "Tranclp" (Tranclp_Tac.trancl_tac o Simplifier.the_context)) 

1142 
addSolver (mk_solver "Rtranclp" (Tranclp_Tac.rtrancl_tac o Simplifier.the_context))) 

15076
4b3d280ef06a
New prover for transitive and reflexivetransitive closure of relations.
ballarin
parents:
14565
diff
changeset

1143 
*} 
4b3d280ef06a
New prover for transitive and reflexivetransitive closure of relations.
ballarin
parents:
14565
diff
changeset

1144 

32215  1145 

1146 
text {* Optional methods. *} 

15076
4b3d280ef06a
New prover for transitive and reflexivetransitive closure of relations.
ballarin
parents:
14565
diff
changeset

1147 

4b3d280ef06a
New prover for transitive and reflexivetransitive closure of relations.
ballarin
parents:
14565
diff
changeset

1148 
method_setup trancl = 
32215  1149 
{* Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.trancl_tac) *} 
18372  1150 
{* simple transitivity reasoner *} 
15076
4b3d280ef06a
New prover for transitive and reflexivetransitive closure of relations.
ballarin
parents:
14565
diff
changeset

1151 
method_setup rtrancl = 
32215  1152 
{* Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.rtrancl_tac) *} 
15076
4b3d280ef06a
New prover for transitive and reflexivetransitive closure of relations.
ballarin
parents:
14565
diff
changeset

1153 
{* simple transitivity reasoner *} 
22262  1154 
method_setup tranclp = 
32215  1155 
{* Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.trancl_tac) *} 
22262  1156 
{* simple transitivity reasoner (predicate version) *} 
1157 
method_setup rtranclp = 

32215  1158 
{* Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.rtrancl_tac) *} 
22262  1159 
{* simple transitivity reasoner (predicate version) *} 
15076
4b3d280ef06a
New prover for transitive and reflexivetransitive closure of relations.
ballarin
parents:
14565
diff
changeset

1160 

10213  1161 
end 