src/HOL/Lambda/Commutation.thy
author wenzelm
Sat Apr 08 22:51:06 2006 +0200 (2006-04-08)
changeset 19363 667b5ea637dd
parent 19086 1b3780be6cc2
child 21404 eb85850d3eb7
permissions -rw-r--r--
refined 'abbreviation';
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(*  Title:      HOL/Lambda/Commutation.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1995  TU Muenchen
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*)
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header {* Abstract commutation and confluence notions *}
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theory Commutation imports Main begin
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subsection {* Basic definitions *}
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definition
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  square :: "[('a \<times> 'a) set, ('a \<times> 'a) set, ('a \<times> 'a) set, ('a \<times> 'a) set] => bool"
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  "square R S T U =
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    (\<forall>x y. (x, y) \<in> R --> (\<forall>z. (x, z) \<in> S --> (\<exists>u. (y, u) \<in> T \<and> (z, u) \<in> U)))"
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  commute :: "[('a \<times> 'a) set, ('a \<times> 'a) set] => bool"
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  "commute R S = square R S S R"
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  diamond :: "('a \<times> 'a) set => bool"
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  "diamond R = commute R R"
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  Church_Rosser :: "('a \<times> 'a) set => bool"
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  "Church_Rosser R =
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    (\<forall>x y. (x, y) \<in> (R \<union> R^-1)^* --> (\<exists>z. (x, z) \<in> R^* \<and> (y, z) \<in> R^*))"
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abbreviation
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  confluent :: "('a \<times> 'a) set => bool"
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  "confluent R == diamond (R^*)"
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subsection {* Basic lemmas *}
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subsubsection {* square *}
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lemma square_sym: "square R S T U ==> square S R U T"
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  apply (unfold square_def)
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  apply blast
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  done
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lemma square_subset:
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    "[| square R S T U; T \<subseteq> T' |] ==> square R S T' U"
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  apply (unfold square_def)
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  apply blast
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  done
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lemma square_reflcl:
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    "[| square R S T (R^=); S \<subseteq> T |] ==> square (R^=) S T (R^=)"
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  apply (unfold square_def)
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  apply blast
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  done
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lemma square_rtrancl:
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    "square R S S T ==> square (R^*) S S (T^*)"
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  apply (unfold square_def)
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  apply (intro strip)
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  apply (erule rtrancl_induct)
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   apply blast
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  apply (blast intro: rtrancl_into_rtrancl)
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  done
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lemma square_rtrancl_reflcl_commute:
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    "square R S (S^*) (R^=) ==> commute (R^*) (S^*)"
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  apply (unfold commute_def)
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  apply (fastsimp dest: square_reflcl square_sym [THEN square_rtrancl]
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    elim: r_into_rtrancl)
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  done
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subsubsection {* commute *}
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lemma commute_sym: "commute R S ==> commute S R"
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  apply (unfold commute_def)
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  apply (blast intro: square_sym)
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  done
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lemma commute_rtrancl: "commute R S ==> commute (R^*) (S^*)"
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  apply (unfold commute_def)
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  apply (blast intro: square_rtrancl square_sym)
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  done
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lemma commute_Un:
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    "[| commute R T; commute S T |] ==> commute (R \<union> S) T"
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  apply (unfold commute_def square_def)
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  apply blast
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  done
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subsubsection {* diamond, confluence, and union *}
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lemma diamond_Un:
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    "[| diamond R; diamond S; commute R S |] ==> diamond (R \<union> S)"
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  apply (unfold diamond_def)
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  apply (assumption | rule commute_Un commute_sym)+
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  done
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lemma diamond_confluent: "diamond R ==> confluent R"
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  apply (unfold diamond_def)
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  apply (erule commute_rtrancl)
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  done
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lemma square_reflcl_confluent:
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    "square R R (R^=) (R^=) ==> confluent R"
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  apply (unfold diamond_def)
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  apply (fast intro: square_rtrancl_reflcl_commute r_into_rtrancl
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    elim: square_subset)
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  done
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lemma confluent_Un:
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    "[| confluent R; confluent S; commute (R^*) (S^*) |] ==> confluent (R \<union> S)"
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  apply (rule rtrancl_Un_rtrancl [THEN subst])
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  apply (blast dest: diamond_Un intro: diamond_confluent)
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  done
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lemma diamond_to_confluence:
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    "[| diamond R; T \<subseteq> R; R \<subseteq> T^* |] ==> confluent T"
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  apply (force intro: diamond_confluent
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    dest: rtrancl_subset [symmetric])
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  done
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subsection {* Church-Rosser *}
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lemma Church_Rosser_confluent: "Church_Rosser R = confluent R"
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  apply (unfold square_def commute_def diamond_def Church_Rosser_def)
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  apply (tactic {* safe_tac HOL_cs *})
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   apply (tactic {*
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     blast_tac (HOL_cs addIs
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       [Un_upper2 RS rtrancl_mono RS subsetD RS rtrancl_trans,
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       rtrancl_converseI, converseI, Un_upper1 RS rtrancl_mono RS subsetD]) 1 *})
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  apply (erule rtrancl_induct)
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   apply blast
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  apply (blast del: rtrancl_refl intro: rtrancl_trans)
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  done
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subsection {* Newman's lemma *}
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text {* Proof by Stefan Berghofer *}
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theorem newman:
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  assumes wf: "wf (R\<inverse>)"
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  and lc: "\<And>a b c. (a, b) \<in> R \<Longrightarrow> (a, c) \<in> R \<Longrightarrow>
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    \<exists>d. (b, d) \<in> R\<^sup>* \<and> (c, d) \<in> R\<^sup>*"
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  shows "\<And>b c. (a, b) \<in> R\<^sup>* \<Longrightarrow> (a, c) \<in> R\<^sup>* \<Longrightarrow>
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    \<exists>d. (b, d) \<in> R\<^sup>* \<and> (c, d) \<in> R\<^sup>*"
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  using wf
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proof induct
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  case (less x b c)
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  have xc: "(x, c) \<in> R\<^sup>*" .
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  have xb: "(x, b) \<in> R\<^sup>*" . thus ?case
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  proof (rule converse_rtranclE)
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    assume "x = b"
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    with xc have "(b, c) \<in> R\<^sup>*" by simp
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    thus ?thesis by iprover
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  next
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    fix y
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    assume xy: "(x, y) \<in> R"
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    assume yb: "(y, b) \<in> R\<^sup>*"
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    from xc show ?thesis
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    proof (rule converse_rtranclE)
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      assume "x = c"
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      with xb have "(c, b) \<in> R\<^sup>*" by simp
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      thus ?thesis by iprover
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    next
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      fix y'
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      assume y'c: "(y', c) \<in> R\<^sup>*"
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      assume xy': "(x, y') \<in> R"
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      with xy have "\<exists>u. (y, u) \<in> R\<^sup>* \<and> (y', u) \<in> R\<^sup>*" by (rule lc)
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      then obtain u where yu: "(y, u) \<in> R\<^sup>*" and y'u: "(y', u) \<in> R\<^sup>*" by iprover
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      from xy have "(y, x) \<in> R\<inverse>" ..
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      from this and yb yu have "\<exists>d. (b, d) \<in> R\<^sup>* \<and> (u, d) \<in> R\<^sup>*" by (rule less)
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      then obtain v where bv: "(b, v) \<in> R\<^sup>*" and uv: "(u, v) \<in> R\<^sup>*" by iprover
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      from xy' have "(y', x) \<in> R\<inverse>" ..
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      moreover from y'u and uv have "(y', v) \<in> R\<^sup>*" by (rule rtrancl_trans)
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      moreover note y'c
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      ultimately have "\<exists>d. (v, d) \<in> R\<^sup>* \<and> (c, d) \<in> R\<^sup>*" by (rule less)
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      then obtain w where vw: "(v, w) \<in> R\<^sup>*" and cw: "(c, w) \<in> R\<^sup>*" by iprover
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      from bv vw have "(b, w) \<in> R\<^sup>*" by (rule rtrancl_trans)
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      with cw show ?thesis by iprover
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    qed
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  qed
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qed
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text {*
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  \medskip Alternative version.  Partly automated by Tobias
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  Nipkow. Takes 2 minutes (2002).
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  This is the maximal amount of automation possible at the moment.
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*}
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theorem newman':
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  assumes wf: "wf (R\<inverse>)"
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  and lc: "\<And>a b c. (a, b) \<in> R \<Longrightarrow> (a, c) \<in> R \<Longrightarrow>
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    \<exists>d. (b, d) \<in> R\<^sup>* \<and> (c, d) \<in> R\<^sup>*"
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  shows "\<And>b c. (a, b) \<in> R\<^sup>* \<Longrightarrow> (a, c) \<in> R\<^sup>* \<Longrightarrow>
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    \<exists>d. (b, d) \<in> R\<^sup>* \<and> (c, d) \<in> R\<^sup>*"
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  using wf
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proof induct
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  case (less x b c)
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  note IH = `\<And>y b c. \<lbrakk>(y,x) \<in> R\<inverse>; (y,b) \<in> R\<^sup>*; (y,c) \<in> R\<^sup>*\<rbrakk>
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                     \<Longrightarrow> \<exists>d. (b,d) \<in> R\<^sup>* \<and> (c,d) \<in> R\<^sup>*`
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  have xc: "(x, c) \<in> R\<^sup>*" .
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  have xb: "(x, b) \<in> R\<^sup>*" .
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  thus ?case
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  proof (rule converse_rtranclE)
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    assume "x = b"
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    with xc have "(b, c) \<in> R\<^sup>*" by simp
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    thus ?thesis by iprover
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  next
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    fix y
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    assume xy: "(x, y) \<in> R"
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    assume yb: "(y, b) \<in> R\<^sup>*"
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    from xc show ?thesis
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    proof (rule converse_rtranclE)
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      assume "x = c"
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      with xb have "(c, b) \<in> R\<^sup>*" by simp
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      thus ?thesis by iprover
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    next
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      fix y'
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      assume y'c: "(y', c) \<in> R\<^sup>*"
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      assume xy': "(x, y') \<in> R"
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      with xy obtain u where u: "(y, u) \<in> R\<^sup>*" "(y', u) \<in> R\<^sup>*"
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        by (blast dest: lc)
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      from yb u y'c show ?thesis
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        by (blast del: rtrancl_refl
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            intro: rtrancl_trans
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            dest: IH [OF xy [symmetric]] IH [OF xy' [symmetric]])
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    qed
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  qed
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qed
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end