src/HOL/Transitive_Closure.thy
author kleing
Wed Apr 14 14:13:05 2004 +0200 (2004-04-14)
changeset 14565 c6dc17aab88a
parent 14404 4952c5a92e04
child 15076 4b3d280ef06a
permissions -rw-r--r--
use more symbols in HTML output
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(*  Title:      HOL/Transitive_Closure.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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*)
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header {* Reflexive and Transitive closure of a relation *}
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theory Transitive_Closure = Inductive:
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text {*
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  @{text rtrancl} is reflexive/transitive closure,
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  @{text trancl} is transitive closure,
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  @{text reflcl} is reflexive closure.
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  These postfix operators have \emph{maximum priority}, forcing their
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  operands to be atomic.
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*}
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consts
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  rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^*)" [1000] 999)
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inductive "r^*"
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  intros
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    rtrancl_refl [intro!, CPure.intro!, simp]: "(a, a) : r^*"
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    rtrancl_into_rtrancl [CPure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"
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consts
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  trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^+)" [1000] 999)
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inductive "r^+"
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  intros
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    r_into_trancl [intro, CPure.intro]: "(a, b) : r ==> (a, b) : r^+"
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    trancl_into_trancl [CPure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a,c) : r^+"
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syntax
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  "_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^=)" [1000] 999)
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translations
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  "r^=" == "r \<union> Id"
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syntax (xsymbols)
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  rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>*)" [1000] 999)
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  trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>+)" [1000] 999)
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  "_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>=)" [1000] 999)
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syntax (HTML output)
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  rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>*)" [1000] 999)
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  trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>+)" [1000] 999)
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  "_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>=)" [1000] 999)
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subsection {* Reflexive-transitive closure *}
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lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*"
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  -- {* @{text rtrancl} of @{text r} contains @{text r} *}
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  apply (simp only: split_tupled_all)
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  apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl])
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  done
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lemma rtrancl_mono: "r \<subseteq> s ==> r^* \<subseteq> s^*"
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  -- {* monotonicity of @{text rtrancl} *}
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  apply (rule subsetI)
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  apply (simp only: split_tupled_all)
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  apply (erule rtrancl.induct)
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   apply (rule_tac [2] rtrancl_into_rtrancl, blast+)
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  done
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theorem rtrancl_induct [consumes 1, induct set: rtrancl]:
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  assumes a: "(a, b) : r^*"
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    and cases: "P a" "!!y z. [| (a, y) : r^*; (y, z) : r; P y |] ==> P z"
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  shows "P b"
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proof -
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  from a have "a = a --> P b"
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    by (induct "%x y. x = a --> P y" a b) (rules intro: cases)+
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  thus ?thesis by rules
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qed
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lemmas rtrancl_induct2 =
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  rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
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                 consumes 1, case_names refl step]
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lemma trans_rtrancl: "trans(r^*)"
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  -- {* transitivity of transitive closure!! -- by induction *}
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proof (rule transI)
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  fix x y z
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  assume "(x, y) \<in> r\<^sup>*"
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  assume "(y, z) \<in> r\<^sup>*"
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  thus "(x, z) \<in> r\<^sup>*" by induct (rules!)+
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qed
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lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard]
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lemma rtranclE:
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  "[| (a::'a,b) : r^*;  (a = b) ==> P;
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      !!y.[| (a,y) : r^*; (y,b) : r |] ==> P
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   |] ==> P"
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  -- {* elimination of @{text rtrancl} -- by induction on a special formula *}
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proof -
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  assume major: "(a::'a,b) : r^*"
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  case rule_context
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  show ?thesis
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    apply (subgoal_tac "(a::'a) = b | (EX y. (a,y) : r^* & (y,b) : r)")
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     apply (rule_tac [2] major [THEN rtrancl_induct])
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      prefer 2 apply (blast!)
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      prefer 2 apply (blast!)
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    apply (erule asm_rl exE disjE conjE prems)+
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    done
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qed
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lemma converse_rtrancl_into_rtrancl:
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  "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> r\<^sup>* \<Longrightarrow> (a, c) \<in> r\<^sup>*"
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  by (rule rtrancl_trans) rules+
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text {*
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  \medskip More @{term "r^*"} equations and inclusions.
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*}
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lemma rtrancl_idemp [simp]: "(r^*)^* = r^*"
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  apply auto
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  apply (erule rtrancl_induct)
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   apply (rule rtrancl_refl)
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  apply (blast intro: rtrancl_trans)
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  done
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lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"
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  apply (rule set_ext)
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  apply (simp only: split_tupled_all)
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  apply (blast intro: rtrancl_trans)
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  done
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lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*"
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by (drule rtrancl_mono, simp)
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lemma rtrancl_subset: "R \<subseteq> S ==> S \<subseteq> R^* ==> S^* = R^*"
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  apply (drule rtrancl_mono)
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  apply (drule rtrancl_mono, simp)
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  done
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lemma rtrancl_Un_rtrancl: "(R^* \<union> S^*)^* = (R \<union> S)^*"
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  by (blast intro!: rtrancl_subset intro: r_into_rtrancl rtrancl_mono [THEN subsetD])
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lemma rtrancl_reflcl [simp]: "(R^=)^* = R^*"
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  by (blast intro!: rtrancl_subset intro: r_into_rtrancl)
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lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*"
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  apply (rule sym)
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  apply (rule rtrancl_subset, blast, clarify)
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  apply (rename_tac a b)
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  apply (case_tac "a = b", blast)
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  apply (blast intro!: r_into_rtrancl)
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  done
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theorem rtrancl_converseD:
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  assumes r: "(x, y) \<in> (r^-1)^*"
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  shows "(y, x) \<in> r^*"
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proof -
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  from r show ?thesis
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    by induct (rules intro: rtrancl_trans dest!: converseD)+
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qed
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theorem rtrancl_converseI:
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  assumes r: "(y, x) \<in> r^*"
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  shows "(x, y) \<in> (r^-1)^*"
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proof -
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  from r show ?thesis
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    by induct (rules intro: rtrancl_trans converseI)+
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qed
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lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
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  by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
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theorem converse_rtrancl_induct[consumes 1]:
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  assumes major: "(a, b) : r^*"
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    and cases: "P b" "!!y z. [| (y, z) : r; (z, b) : r^*; P z |] ==> P y"
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  shows "P a"
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proof -
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  from rtrancl_converseI [OF major]
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  show ?thesis
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    by induct (rules intro: cases dest!: converseD rtrancl_converseD)+
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qed
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lemmas converse_rtrancl_induct2 =
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  converse_rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
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                 consumes 1, case_names refl step]
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lemma converse_rtranclE:
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  "[| (x,z):r^*;
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      x=z ==> P;
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      !!y. [| (x,y):r; (y,z):r^* |] ==> P
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   |] ==> P"
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proof -
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  assume major: "(x,z):r^*"
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  case rule_context
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  show ?thesis
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    apply (subgoal_tac "x = z | (EX y. (x,y) : r & (y,z) : r^*)")
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     apply (rule_tac [2] major [THEN converse_rtrancl_induct])
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      prefer 2 apply rules
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     prefer 2 apply rules
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    apply (erule asm_rl exE disjE conjE prems)+
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    done
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qed
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ML_setup {*
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  bind_thm ("converse_rtranclE2", split_rule
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    (read_instantiate [("x","(xa,xb)"), ("z","(za,zb)")] (thm "converse_rtranclE")));
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*}
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lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"
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  by (blast elim: rtranclE converse_rtranclE
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    intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
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subsection {* Transitive closure *}
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lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
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  apply (simp only: split_tupled_all)
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  apply (erule trancl.induct)
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  apply (rules dest: subsetD)+
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  done
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lemma r_into_trancl': "!!p. p : r ==> p : r^+"
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  by (simp only: split_tupled_all) (erule r_into_trancl)
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text {*
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  \medskip Conversions between @{text trancl} and @{text rtrancl}.
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*}
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lemma trancl_into_rtrancl: "(a, b) \<in> r^+ ==> (a, b) \<in> r^*"
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  by (erule trancl.induct) rules+
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lemma rtrancl_into_trancl1: assumes r: "(a, b) \<in> r^*"
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  shows "!!c. (b, c) \<in> r ==> (a, c) \<in> r^+" using r
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  by induct rules+
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lemma rtrancl_into_trancl2: "[| (a,b) : r;  (b,c) : r^* |]   ==>  (a,c) : r^+"
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  -- {* intro rule from @{text r} and @{text rtrancl} *}
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  apply (erule rtranclE, rules)
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  apply (rule rtrancl_trans [THEN rtrancl_into_trancl1])
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   apply (assumption | rule r_into_rtrancl)+
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  done
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lemma trancl_induct [consumes 1, induct set: trancl]:
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  assumes a: "(a,b) : r^+"
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  and cases: "!!y. (a, y) : r ==> P y"
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    "!!y z. (a,y) : r^+ ==> (y, z) : r ==> P y ==> P z"
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  shows "P b"
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  -- {* Nice induction rule for @{text trancl} *}
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proof -
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  from a have "a = a --> P b"
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    by (induct "%x y. x = a --> P y" a b) (rules intro: cases)+
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  thus ?thesis by rules
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qed
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lemma trancl_trans_induct:
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  "[| (x,y) : r^+;
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      !!x y. (x,y) : r ==> P x y;
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      !!x y z. [| (x,y) : r^+; P x y; (y,z) : r^+; P y z |] ==> P x z
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   |] ==> P x y"
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  -- {* Another induction rule for trancl, incorporating transitivity *}
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proof -
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  assume major: "(x,y) : r^+"
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  case rule_context
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  show ?thesis
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    by (rules intro: r_into_trancl major [THEN trancl_induct] prems)
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qed
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inductive_cases tranclE: "(a, b) : r^+"
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lemma trans_trancl: "trans(r^+)"
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  -- {* Transitivity of @{term "r^+"} *}
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proof (rule transI)
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  fix x y z
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  assume "(x, y) \<in> r^+"
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  assume "(y, z) \<in> r^+"
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  thus "(x, z) \<in> r^+" by induct (rules!)+
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qed
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lemmas trancl_trans = trans_trancl [THEN transD, standard]
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lemma rtrancl_trancl_trancl: assumes r: "(x, y) \<in> r^*"
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  shows "!!z. (y, z) \<in> r^+ ==> (x, z) \<in> r^+" using r
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  by induct (rules intro: trancl_trans)+
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lemma trancl_into_trancl2: "(a, b) \<in> r ==> (b, c) \<in> r^+ ==> (a, c) \<in> r^+"
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  by (erule transD [OF trans_trancl r_into_trancl])
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lemma trancl_insert:
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  "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
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  -- {* primitive recursion for @{text trancl} over finite relations *}
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  apply (rule equalityI)
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   apply (rule subsetI)
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   apply (simp only: split_tupled_all)
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   apply (erule trancl_induct, blast)
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   apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl r_into_trancl trancl_trans)
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  apply (rule subsetI)
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  apply (blast intro: trancl_mono rtrancl_mono
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    [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
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  done
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lemma trancl_converseI: "(x, y) \<in> (r^+)^-1 ==> (x, y) \<in> (r^-1)^+"
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  apply (drule converseD)
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  apply (erule trancl.induct)
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  apply (rules intro: converseI trancl_trans)+
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  done
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lemma trancl_converseD: "(x, y) \<in> (r^-1)^+ ==> (x, y) \<in> (r^+)^-1"
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  apply (rule converseI)
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  apply (erule trancl.induct)
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  apply (rules dest: converseD intro: trancl_trans)+
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  done
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lemma trancl_converse: "(r^-1)^+ = (r^+)^-1"
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  by (fastsimp simp add: split_tupled_all
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    intro!: trancl_converseI trancl_converseD)
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lemma converse_trancl_induct:
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  "[| (a,b) : r^+; !!y. (y,b) : r ==> P(y);
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      !!y z.[| (y,z) : r;  (z,b) : r^+;  P(z) |] ==> P(y) |]
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    ==> P(a)"
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proof -
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  assume major: "(a,b) : r^+"
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  case rule_context
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  show ?thesis
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    apply (rule major [THEN converseI, THEN trancl_converseI [THEN trancl_induct]])
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     apply (rule prems)
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     apply (erule converseD)
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    apply (blast intro: prems dest!: trancl_converseD)
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    done
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qed
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lemma tranclD: "(x, y) \<in> R^+ ==> EX z. (x, z) \<in> R \<and> (z, y) \<in> R^*"
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  apply (erule converse_trancl_induct, auto)
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  apply (blast intro: rtrancl_trans)
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  done
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lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
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by(blast elim: tranclE dest: trancl_into_rtrancl)
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lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"
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  by (blast dest: r_into_trancl)
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lemma trancl_subset_Sigma_aux:
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    "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
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  apply (erule rtrancl_induct, auto)
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  done
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lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"
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  apply (rule subsetI)
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  apply (simp only: split_tupled_all)
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  apply (erule tranclE)
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  apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
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  done
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lemma reflcl_trancl [simp]: "(r^+)^= = r^*"
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  apply safe
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   apply (erule trancl_into_rtrancl)
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  apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
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  done
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lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
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  apply safe
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   apply (drule trancl_into_rtrancl, simp)
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  apply (erule rtranclE, safe)
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   apply (rule r_into_trancl, simp)
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  apply (rule rtrancl_into_trancl1)
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   apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
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  done
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lemma trancl_empty [simp]: "{}^+ = {}"
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  by (auto elim: trancl_induct)
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lemma rtrancl_empty [simp]: "{}^* = Id"
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  by (rule subst [OF reflcl_trancl]) simp
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lemma rtranclD: "(a, b) \<in> R^* ==> a = b \<or> a \<noteq> b \<and> (a, b) \<in> R^+"
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  by (force simp add: reflcl_trancl [symmetric] simp del: reflcl_trancl)
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text {* @{text Domain} and @{text Range} *}
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lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
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  by blast
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lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
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  by blast
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lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
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  by (rule rtrancl_Un_rtrancl [THEN subst]) fast
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lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
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  by (blast intro: subsetD [OF rtrancl_Un_subset])
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   392
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lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
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  by (unfold Domain_def) (blast dest: tranclD)
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   395
wenzelm@11090
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lemma trancl_range [simp]: "Range (r^+) = Range r"
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  by (simp add: Range_def trancl_converse [symmetric])
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   398
paulson@11115
   399
lemma Not_Domain_rtrancl:
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    "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
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  apply auto
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   402
  by (erule rev_mp, erule rtrancl_induct, auto)
wenzelm@12691
   403
berghofe@11327
   404
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   405
text {* More about converse @{text rtrancl} and @{text trancl}, should
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   406
  be merged with main body. *}
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   407
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   408
lemma single_valued_confluent:
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  "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>
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   410
  \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
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apply(erule rtrancl_induct)
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   412
 apply simp
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   413
apply(erule disjE)
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   414
 apply(blast elim:converse_rtranclE dest:single_valuedD)
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   415
apply(blast intro:rtrancl_trans)
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   416
done
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   417
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   418
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"
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   419
  by (fast intro: trancl_trans)
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   420
kleing@12428
   421
lemma trancl_into_trancl [rule_format]:
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   422
    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
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   423
  apply (erule trancl_induct)
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   424
   apply (fast intro: r_r_into_trancl)
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   425
  apply (fast intro: r_r_into_trancl trancl_trans)
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   426
  done
kleing@12428
   427
kleing@12428
   428
lemma trancl_rtrancl_trancl:
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   429
    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r\<^sup>* ==> (a, c) \<in> r\<^sup>+"
kleing@12428
   430
  apply (drule tranclD)
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   431
  apply (erule exE, erule conjE)
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   432
  apply (drule rtrancl_trans, assumption)
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   433
  apply (drule rtrancl_into_trancl2, assumption, assumption)
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   434
  done
kleing@12428
   435
wenzelm@12691
   436
lemmas transitive_closure_trans [trans] =
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   437
  r_r_into_trancl trancl_trans rtrancl_trans
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   438
  trancl_into_trancl trancl_into_trancl2
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   439
  rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
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   440
  rtrancl_trancl_trancl trancl_rtrancl_trancl
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   441
kleing@12428
   442
declare trancl_into_rtrancl [elim]
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   443
berghofe@11327
   444
declare rtranclE [cases set: rtrancl]
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   445
declare tranclE [cases set: trancl]
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   446
nipkow@10213
   447
end