src/HOL/Transitive_Closure.thy
author haftmann
Sun Oct 16 14:48:00 2011 +0200 (2011-10-16)
changeset 45153 93e290c11b0f
parent 45141 b2eb87bd541b
child 45607 16b4f5774621
permissions -rw-r--r--
tuned type annnotation
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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 {* Reflexive-transitive 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, standard]
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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]
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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]
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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
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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
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  done
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lemmas rtrancl_subset = rtranclp_subset [to_set]
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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])
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lemmas rtrancl_Un_rtrancl = rtranclp_sup_rtranclp [to_set]
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lemma rtranclp_reflcl [simp]: "(R^==)^** = R^**"
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  by (blast intro!: rtranclp_subset)
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lemmas rtrancl_reflcl [simp] = rtranclp_reflcl [to_set]
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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")
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   apply blast
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  apply blast
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  done
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lemma rtranclp_r_diff_Id: "(inf r op ~=)^** = r^**"
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  apply (rule sym)
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  apply (rule rtranclp_subset)
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   apply blast+
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  done
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theorem rtranclp_converseD:
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  assumes r: "(r^--1)^** x y"
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  shows "r^** y x"
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proof -
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  from r show ?thesis
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    by induct (iprover intro: rtranclp_trans dest!: conversepD)+
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qed
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lemmas rtrancl_converseD = rtranclp_converseD [to_set]
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theorem rtranclp_converseI:
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  assumes "r^** y x"
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  shows "(r^--1)^** x y"
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  using assms
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  by induct (iprover intro: rtranclp_trans conversepI)+
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lemmas rtrancl_converseI = rtranclp_converseI [to_set]
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lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
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  by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
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lemma sym_rtrancl: "sym r ==> sym (r^*)"
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  by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])
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theorem converse_rtranclp_induct [consumes 1, case_names base step]:
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  assumes major: "r^** a b"
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    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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  using rtranclp_converseI [OF major]
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  by induct (iprover intro: cases dest!: conversepD rtranclp_converseD)+
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lemmas converse_rtrancl_induct = converse_rtranclp_induct [to_set]
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lemmas converse_rtranclp_induct2 =
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  converse_rtranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
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                 consumes 1, case_names refl step]
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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_rtranclpE [consumes 1, case_names base step]:
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  assumes major: "r^** x z"
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    and cases: "x=z ==> P"
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      "!!y. [| r x y; r^** y z |] ==> P"
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  shows P
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  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])
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    prefer 2 apply iprover
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   prefer 2 apply iprover
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  apply (erule asm_rl exE disjE conjE cases)+
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  done
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lemmas converse_rtranclE = converse_rtranclpE [to_set]
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lemmas converse_rtranclpE2 = converse_rtranclpE [of _ "(xa,xb)" "(za,zb)", split_rule]
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lemmas converse_rtranclE2 = converse_rtranclE [of "(xa,xb)" "(za,zb)", split_rule]
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lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"
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  by (blast elim: rtranclE converse_rtranclE
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    intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
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lemma rtrancl_unfold: "r^* = Id Un r^* O r"
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  by (auto intro: rtrancl_into_rtrancl elim: rtranclE)
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lemma rtrancl_Un_separatorE:
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  "(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (a,x) : P^* \<longrightarrow> (x,y) : Q \<longrightarrow> x=y \<Longrightarrow> (a,b) : P^*"
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apply (induct rule:rtrancl.induct)
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 apply blast
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apply (blast intro:rtrancl_trans)
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done
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lemma rtrancl_Un_separator_converseE:
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  "(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (x,b) : P^* \<longrightarrow> (y,x) : Q \<longrightarrow> y=x \<Longrightarrow> (a,b) : P^*"
nipkow@31690
   307
apply (induct rule:converse_rtrancl_induct)
nipkow@31690
   308
 apply blast
nipkow@31690
   309
apply (blast intro:rtrancl_trans)
nipkow@31690
   310
done
nipkow@31690
   311
haftmann@34970
   312
lemma Image_closed_trancl:
haftmann@34970
   313
  assumes "r `` X \<subseteq> X" shows "r\<^sup>* `` X = X"
haftmann@34970
   314
proof -
haftmann@34970
   315
  from assms have **: "{y. \<exists>x\<in>X. (x, y) \<in> r} \<subseteq> X" by auto
haftmann@34970
   316
  have "\<And>x y. (y, x) \<in> r\<^sup>* \<Longrightarrow> y \<in> X \<Longrightarrow> x \<in> X"
haftmann@34970
   317
  proof -
haftmann@34970
   318
    fix x y
haftmann@34970
   319
    assume *: "y \<in> X"
haftmann@34970
   320
    assume "(y, x) \<in> r\<^sup>*"
haftmann@34970
   321
    then show "x \<in> X"
haftmann@34970
   322
    proof induct
haftmann@34970
   323
      case base show ?case by (fact *)
haftmann@34970
   324
    next
haftmann@34970
   325
      case step with ** show ?case by auto
haftmann@34970
   326
    qed
haftmann@34970
   327
  qed
haftmann@34970
   328
  then show ?thesis by auto
haftmann@34970
   329
qed
haftmann@34970
   330
wenzelm@12691
   331
wenzelm@12691
   332
subsection {* Transitive closure *}
wenzelm@10331
   333
berghofe@13704
   334
lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
berghofe@23743
   335
  apply (simp add: split_tupled_all)
berghofe@13704
   336
  apply (erule trancl.induct)
wenzelm@26179
   337
   apply (iprover dest: subsetD)+
wenzelm@12691
   338
  done
wenzelm@12691
   339
berghofe@13704
   340
lemma r_into_trancl': "!!p. p : r ==> p : r^+"
berghofe@13704
   341
  by (simp only: split_tupled_all) (erule r_into_trancl)
berghofe@13704
   342
wenzelm@12691
   343
text {*
wenzelm@12691
   344
  \medskip Conversions between @{text trancl} and @{text rtrancl}.
wenzelm@12691
   345
*}
wenzelm@12691
   346
berghofe@23743
   347
lemma tranclp_into_rtranclp: "r^++ a b ==> r^** a b"
berghofe@23743
   348
  by (erule tranclp.induct) iprover+
wenzelm@12691
   349
berghofe@23743
   350
lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]
berghofe@22262
   351
berghofe@23743
   352
lemma rtranclp_into_tranclp1: assumes r: "r^** a b"
berghofe@22262
   353
  shows "!!c. r b c ==> r^++ a c" using r
nipkow@17589
   354
  by induct iprover+
wenzelm@12691
   355
berghofe@23743
   356
lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]
berghofe@22262
   357
berghofe@23743
   358
lemma rtranclp_into_tranclp2: "[| r a b; r^** b c |] ==> r^++ a c"
wenzelm@12691
   359
  -- {* intro rule from @{text r} and @{text rtrancl} *}
wenzelm@26179
   360
  apply (erule rtranclp.cases)
wenzelm@26179
   361
   apply iprover
berghofe@23743
   362
  apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1])
wenzelm@26179
   363
    apply (simp | rule r_into_rtranclp)+
wenzelm@12691
   364
  done
wenzelm@12691
   365
berghofe@23743
   366
lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]
berghofe@22262
   367
wenzelm@26179
   368
text {* Nice induction rule for @{text trancl} *}
wenzelm@26179
   369
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
berghofe@34909
   370
  assumes a: "r^++ a b"
berghofe@22262
   371
  and cases: "!!y. r a y ==> P y"
berghofe@22262
   372
    "!!y z. r^++ a y ==> r y z ==> P y ==> P z"
berghofe@34909
   373
  shows "P b" using a
berghofe@34909
   374
  by (induct x\<equiv>a b) (iprover intro: cases)+
wenzelm@12691
   375
berghofe@25425
   376
lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]
berghofe@22262
   377
berghofe@23743
   378
lemmas tranclp_induct2 =
wenzelm@26179
   379
  tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
wenzelm@26179
   380
    consumes 1, case_names base step]
berghofe@22262
   381
paulson@22172
   382
lemmas trancl_induct2 =
wenzelm@26179
   383
  trancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
wenzelm@26179
   384
    consumes 1, case_names base step]
paulson@22172
   385
berghofe@23743
   386
lemma tranclp_trans_induct:
berghofe@22262
   387
  assumes major: "r^++ x y"
berghofe@22262
   388
    and cases: "!!x y. r x y ==> P x y"
berghofe@22262
   389
      "!!x y z. [| r^++ x y; P x y; r^++ y z; P y z |] ==> P x z"
wenzelm@18372
   390
  shows "P x y"
wenzelm@12691
   391
  -- {* Another induction rule for trancl, incorporating transitivity *}
berghofe@23743
   392
  by (iprover intro: major [THEN tranclp_induct] cases)
wenzelm@12691
   393
berghofe@23743
   394
lemmas trancl_trans_induct = tranclp_trans_induct [to_set]
berghofe@23743
   395
wenzelm@26174
   396
lemma tranclE [cases set: trancl]:
wenzelm@26174
   397
  assumes "(a, b) : r^+"
wenzelm@26174
   398
  obtains
wenzelm@26174
   399
    (base) "(a, b) : r"
wenzelm@26174
   400
  | (step) c where "(a, c) : r^+" and "(c, b) : r"
wenzelm@26174
   401
  using assms by cases simp_all
wenzelm@10980
   402
krauss@32235
   403
lemma trancl_Int_subset: "[| r \<subseteq> s; (r^+ \<inter> s) O r \<subseteq> s|] ==> r^+ \<subseteq> s"
paulson@22080
   404
  apply (rule subsetI)
wenzelm@26179
   405
  apply (rule_tac p = x in PairE)
wenzelm@26179
   406
  apply clarify
wenzelm@26179
   407
  apply (erule trancl_induct)
wenzelm@26179
   408
   apply auto
paulson@22080
   409
  done
paulson@22080
   410
krauss@32235
   411
lemma trancl_unfold: "r^+ = r Un r^+ O r"
paulson@15551
   412
  by (auto intro: trancl_into_trancl elim: tranclE)
paulson@15551
   413
wenzelm@26179
   414
text {* Transitivity of @{term "r^+"} *}
wenzelm@26179
   415
lemma trans_trancl [simp]: "trans (r^+)"
berghofe@13704
   416
proof (rule transI)
berghofe@13704
   417
  fix x y z
wenzelm@26179
   418
  assume "(x, y) \<in> r^+"
berghofe@13704
   419
  assume "(y, z) \<in> r^+"
wenzelm@26179
   420
  then show "(x, z) \<in> r^+"
wenzelm@26179
   421
  proof induct
wenzelm@26179
   422
    case (base u)
wenzelm@26179
   423
    from `(x, y) \<in> r^+` and `(y, u) \<in> r`
wenzelm@26179
   424
    show "(x, u) \<in> r^+" ..
wenzelm@26179
   425
  next
wenzelm@26179
   426
    case (step u v)
wenzelm@26179
   427
    from `(x, u) \<in> r^+` and `(u, v) \<in> r`
wenzelm@26179
   428
    show "(x, v) \<in> r^+" ..
wenzelm@26179
   429
  qed
berghofe@13704
   430
qed
wenzelm@12691
   431
wenzelm@12691
   432
lemmas trancl_trans = trans_trancl [THEN transD, standard]
wenzelm@12691
   433
berghofe@23743
   434
lemma tranclp_trans:
berghofe@22262
   435
  assumes xy: "r^++ x y"
berghofe@22262
   436
  and yz: "r^++ y z"
berghofe@22262
   437
  shows "r^++ x z" using yz xy
berghofe@22262
   438
  by induct iprover+
berghofe@22262
   439
wenzelm@26179
   440
lemma trancl_id [simp]: "trans r \<Longrightarrow> r^+ = r"
wenzelm@26179
   441
  apply auto
wenzelm@26179
   442
  apply (erule trancl_induct)
wenzelm@26179
   443
   apply assumption
wenzelm@26179
   444
  apply (unfold trans_def)
wenzelm@26179
   445
  apply blast
wenzelm@26179
   446
  done
nipkow@19623
   447
wenzelm@26179
   448
lemma rtranclp_tranclp_tranclp:
wenzelm@26179
   449
  assumes "r^** x y"
wenzelm@26179
   450
  shows "!!z. r^++ y z ==> r^++ x z" using assms
berghofe@23743
   451
  by induct (iprover intro: tranclp_trans)+
wenzelm@12691
   452
berghofe@23743
   453
lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]
berghofe@22262
   454
berghofe@23743
   455
lemma tranclp_into_tranclp2: "r a b ==> r^++ b c ==> r^++ a c"
berghofe@23743
   456
  by (erule tranclp_trans [OF tranclp.r_into_trancl])
berghofe@22262
   457
berghofe@23743
   458
lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]
wenzelm@12691
   459
wenzelm@12691
   460
lemma trancl_insert:
wenzelm@12691
   461
  "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
wenzelm@12691
   462
  -- {* primitive recursion for @{text trancl} over finite relations *}
wenzelm@12691
   463
  apply (rule equalityI)
wenzelm@12691
   464
   apply (rule subsetI)
wenzelm@12691
   465
   apply (simp only: split_tupled_all)
paulson@14208
   466
   apply (erule trancl_induct, blast)
huffman@35216
   467
   apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)
wenzelm@12691
   468
  apply (rule subsetI)
wenzelm@12691
   469
  apply (blast intro: trancl_mono rtrancl_mono
wenzelm@12691
   470
    [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
wenzelm@12691
   471
  done
wenzelm@12691
   472
berghofe@23743
   473
lemma tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ x y"
berghofe@22262
   474
  apply (drule conversepD)
berghofe@23743
   475
  apply (erule tranclp_induct)
berghofe@23743
   476
  apply (iprover intro: conversepI tranclp_trans)+
wenzelm@12691
   477
  done
wenzelm@12691
   478
berghofe@23743
   479
lemmas trancl_converseI = tranclp_converseI [to_set]
berghofe@22262
   480
berghofe@23743
   481
lemma tranclp_converseD: "(r^--1)^++ x y ==> (r^++)^--1 x y"
berghofe@22262
   482
  apply (rule conversepI)
berghofe@23743
   483
  apply (erule tranclp_induct)
berghofe@23743
   484
  apply (iprover dest: conversepD intro: tranclp_trans)+
berghofe@13704
   485
  done
wenzelm@12691
   486
berghofe@23743
   487
lemmas trancl_converseD = tranclp_converseD [to_set]
berghofe@22262
   488
berghofe@23743
   489
lemma tranclp_converse: "(r^--1)^++ = (r^++)^--1"
nipkow@44890
   490
  by (fastforce simp add: fun_eq_iff
berghofe@23743
   491
    intro!: tranclp_converseI dest!: tranclp_converseD)
berghofe@22262
   492
berghofe@23743
   493
lemmas trancl_converse = tranclp_converse [to_set]
wenzelm@12691
   494
huffman@19228
   495
lemma sym_trancl: "sym r ==> sym (r^+)"
huffman@19228
   496
  by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
huffman@19228
   497
berghofe@34909
   498
lemma converse_tranclp_induct [consumes 1, case_names base step]:
berghofe@22262
   499
  assumes major: "r^++ a b"
berghofe@22262
   500
    and cases: "!!y. r y b ==> P(y)"
berghofe@22262
   501
      "!!y z.[| r y z;  r^++ z b;  P(z) |] ==> P(y)"
wenzelm@18372
   502
  shows "P a"
berghofe@23743
   503
  apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])
wenzelm@18372
   504
   apply (rule cases)
berghofe@22262
   505
   apply (erule conversepD)
huffman@35216
   506
  apply (blast intro: assms dest!: tranclp_converseD)
wenzelm@18372
   507
  done
wenzelm@12691
   508
berghofe@23743
   509
lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
berghofe@22262
   510
berghofe@23743
   511
lemma tranclpD: "R^++ x y ==> EX z. R x z \<and> R^** z y"
wenzelm@26179
   512
  apply (erule converse_tranclp_induct)
wenzelm@26179
   513
   apply auto
berghofe@23743
   514
  apply (blast intro: rtranclp_trans)
wenzelm@12691
   515
  done
wenzelm@12691
   516
berghofe@23743
   517
lemmas tranclD = tranclpD [to_set]
berghofe@22262
   518
bulwahn@31577
   519
lemma converse_tranclpE:
bulwahn@31577
   520
  assumes major: "tranclp r x z"
bulwahn@31577
   521
  assumes base: "r x z ==> P"
bulwahn@31577
   522
  assumes step: "\<And> y. [| r x y; tranclp r y z |] ==> P"
bulwahn@31577
   523
  shows P
bulwahn@31577
   524
proof -
bulwahn@31577
   525
  from tranclpD[OF major]
bulwahn@31577
   526
  obtain y where "r x y" and "rtranclp r y z" by iprover
bulwahn@31577
   527
  from this(2) show P
bulwahn@31577
   528
  proof (cases rule: rtranclp.cases)
bulwahn@31577
   529
    case rtrancl_refl
bulwahn@31577
   530
    with `r x y` base show P by iprover
bulwahn@31577
   531
  next
bulwahn@31577
   532
    case rtrancl_into_rtrancl
bulwahn@31577
   533
    from this have "tranclp r y z"
bulwahn@31577
   534
      by (iprover intro: rtranclp_into_tranclp1)
bulwahn@31577
   535
    with `r x y` step show P by iprover
bulwahn@31577
   536
  qed
bulwahn@31577
   537
qed
bulwahn@31577
   538
bulwahn@31577
   539
lemmas converse_tranclE = converse_tranclpE [to_set]
bulwahn@31577
   540
kleing@25295
   541
lemma tranclD2:
kleing@25295
   542
  "(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
kleing@25295
   543
  by (blast elim: tranclE intro: trancl_into_rtrancl)
kleing@25295
   544
nipkow@13867
   545
lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
wenzelm@18372
   546
  by (blast elim: tranclE dest: trancl_into_rtrancl)
wenzelm@12691
   547
wenzelm@12691
   548
lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"
wenzelm@12691
   549
  by (blast dest: r_into_trancl)
wenzelm@12691
   550
wenzelm@12691
   551
lemma trancl_subset_Sigma_aux:
wenzelm@12691
   552
    "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
wenzelm@18372
   553
  by (induct rule: rtrancl_induct) auto
wenzelm@12691
   554
wenzelm@12691
   555
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"
berghofe@13704
   556
  apply (rule subsetI)
berghofe@13704
   557
  apply (simp only: split_tupled_all)
berghofe@13704
   558
  apply (erule tranclE)
wenzelm@26179
   559
   apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
wenzelm@12691
   560
  done
nipkow@10996
   561
berghofe@23743
   562
lemma reflcl_tranclp [simp]: "(r^++)^== = r^**"
berghofe@22262
   563
  apply (safe intro!: order_antisym)
berghofe@23743
   564
   apply (erule tranclp_into_rtranclp)
berghofe@23743
   565
  apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)
wenzelm@11084
   566
  done
nipkow@10996
   567
berghofe@23743
   568
lemmas reflcl_trancl [simp] = reflcl_tranclp [to_set]
berghofe@22262
   569
wenzelm@11090
   570
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
wenzelm@11084
   571
  apply safe
paulson@14208
   572
   apply (drule trancl_into_rtrancl, simp)
paulson@14208
   573
  apply (erule rtranclE, safe)
paulson@14208
   574
   apply (rule r_into_trancl, simp)
wenzelm@11084
   575
  apply (rule rtrancl_into_trancl1)
paulson@14208
   576
   apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
wenzelm@11084
   577
  done
nipkow@10996
   578
haftmann@45140
   579
lemma rtrancl_trancl_reflcl [code]: "r^* = (r^+)^="
haftmann@45140
   580
  by simp
haftmann@45140
   581
wenzelm@11090
   582
lemma trancl_empty [simp]: "{}^+ = {}"
wenzelm@11084
   583
  by (auto elim: trancl_induct)
nipkow@10996
   584
wenzelm@11090
   585
lemma rtrancl_empty [simp]: "{}^* = Id"
wenzelm@11084
   586
  by (rule subst [OF reflcl_trancl]) simp
nipkow@10996
   587
berghofe@23743
   588
lemma rtranclpD: "R^** a b ==> a = b \<or> a \<noteq> b \<and> R^++ a b"
berghofe@23743
   589
  by (force simp add: reflcl_tranclp [symmetric] simp del: reflcl_tranclp)
berghofe@22262
   590
berghofe@23743
   591
lemmas rtranclD = rtranclpD [to_set]
wenzelm@11084
   592
kleing@16514
   593
lemma rtrancl_eq_or_trancl:
kleing@16514
   594
  "(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"
kleing@16514
   595
  by (fast elim: trancl_into_rtrancl dest: rtranclD)
nipkow@10996
   596
krauss@33656
   597
lemma trancl_unfold_right: "r^+ = r^* O r"
krauss@33656
   598
by (auto dest: tranclD2 intro: rtrancl_into_trancl1)
krauss@33656
   599
krauss@33656
   600
lemma trancl_unfold_left: "r^+ = r O r^*"
krauss@33656
   601
by (auto dest: tranclD intro: rtrancl_into_trancl2)
krauss@33656
   602
krauss@33656
   603
krauss@33656
   604
text {* Simplifying nested closures *}
krauss@33656
   605
krauss@33656
   606
lemma rtrancl_trancl_absorb[simp]: "(R^*)^+ = R^*"
krauss@33656
   607
by (simp add: trans_rtrancl)
krauss@33656
   608
krauss@33656
   609
lemma trancl_rtrancl_absorb[simp]: "(R^+)^* = R^*"
krauss@33656
   610
by (subst reflcl_trancl[symmetric]) simp
krauss@33656
   611
krauss@33656
   612
lemma rtrancl_reflcl_absorb[simp]: "(R^*)^= = R^*"
krauss@33656
   613
by auto
krauss@33656
   614
krauss@33656
   615
wenzelm@12691
   616
text {* @{text Domain} and @{text Range} *}
nipkow@10996
   617
wenzelm@11090
   618
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
wenzelm@11084
   619
  by blast
nipkow@10996
   620
wenzelm@11090
   621
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
wenzelm@11084
   622
  by blast
nipkow@10996
   623
wenzelm@11090
   624
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
wenzelm@11084
   625
  by (rule rtrancl_Un_rtrancl [THEN subst]) fast
nipkow@10996
   626
wenzelm@11090
   627
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
wenzelm@11084
   628
  by (blast intro: subsetD [OF rtrancl_Un_subset])
nipkow@10996
   629
wenzelm@11090
   630
lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
wenzelm@11084
   631
  by (unfold Domain_def) (blast dest: tranclD)
nipkow@10996
   632
wenzelm@11090
   633
lemma trancl_range [simp]: "Range (r^+) = Range r"
nipkow@26271
   634
unfolding Range_def by(simp add: trancl_converse [symmetric])
nipkow@10996
   635
paulson@11115
   636
lemma Not_Domain_rtrancl:
wenzelm@12691
   637
    "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
wenzelm@12691
   638
  apply auto
wenzelm@26179
   639
  apply (erule rev_mp)
wenzelm@26179
   640
  apply (erule rtrancl_induct)
wenzelm@26179
   641
   apply auto
wenzelm@26179
   642
  done
berghofe@11327
   643
haftmann@29609
   644
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
haftmann@29609
   645
  apply clarify
haftmann@29609
   646
  apply (erule trancl_induct)
haftmann@29609
   647
   apply (auto simp add: Field_def)
haftmann@29609
   648
  done
haftmann@29609
   649
nipkow@41987
   650
lemma finite_trancl[simp]: "finite (r^+) = finite r"
haftmann@29609
   651
  apply auto
haftmann@29609
   652
   prefer 2
haftmann@29609
   653
   apply (rule trancl_subset_Field2 [THEN finite_subset])
haftmann@29609
   654
   apply (rule finite_SigmaI)
haftmann@29609
   655
    prefer 3
haftmann@29609
   656
    apply (blast intro: r_into_trancl' finite_subset)
haftmann@29609
   657
   apply (auto simp add: finite_Field)
haftmann@29609
   658
  done
haftmann@29609
   659
wenzelm@12691
   660
text {* More about converse @{text rtrancl} and @{text trancl}, should
wenzelm@12691
   661
  be merged with main body. *}
kleing@12428
   662
nipkow@14337
   663
lemma single_valued_confluent:
nipkow@14337
   664
  "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>
nipkow@14337
   665
  \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
wenzelm@26179
   666
  apply (erule rtrancl_induct)
wenzelm@26179
   667
  apply simp
wenzelm@26179
   668
  apply (erule disjE)
wenzelm@26179
   669
   apply (blast elim:converse_rtranclE dest:single_valuedD)
wenzelm@26179
   670
  apply(blast intro:rtrancl_trans)
wenzelm@26179
   671
  done
nipkow@14337
   672
wenzelm@12691
   673
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"
kleing@12428
   674
  by (fast intro: trancl_trans)
kleing@12428
   675
kleing@12428
   676
lemma trancl_into_trancl [rule_format]:
wenzelm@12691
   677
    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
wenzelm@12691
   678
  apply (erule trancl_induct)
kleing@12428
   679
   apply (fast intro: r_r_into_trancl)
kleing@12428
   680
  apply (fast intro: r_r_into_trancl trancl_trans)
kleing@12428
   681
  done
kleing@12428
   682
berghofe@23743
   683
lemma tranclp_rtranclp_tranclp:
berghofe@22262
   684
    "r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c"
berghofe@23743
   685
  apply (drule tranclpD)
wenzelm@26179
   686
  apply (elim exE conjE)
berghofe@23743
   687
  apply (drule rtranclp_trans, assumption)
berghofe@23743
   688
  apply (drule rtranclp_into_tranclp2, assumption, assumption)
kleing@12428
   689
  done
kleing@12428
   690
berghofe@23743
   691
lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]
berghofe@22262
   692
wenzelm@12691
   693
lemmas transitive_closure_trans [trans] =
wenzelm@12691
   694
  r_r_into_trancl trancl_trans rtrancl_trans
berghofe@23743
   695
  trancl.trancl_into_trancl trancl_into_trancl2
berghofe@23743
   696
  rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
wenzelm@12691
   697
  rtrancl_trancl_trancl trancl_rtrancl_trancl
kleing@12428
   698
berghofe@23743
   699
lemmas transitive_closurep_trans' [trans] =
berghofe@23743
   700
  tranclp_trans rtranclp_trans
berghofe@23743
   701
  tranclp.trancl_into_trancl tranclp_into_tranclp2
berghofe@23743
   702
  rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp
berghofe@23743
   703
  rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp
berghofe@22262
   704
kleing@12428
   705
declare trancl_into_rtrancl [elim]
berghofe@11327
   706
haftmann@30954
   707
subsection {* The power operation on relations *}
haftmann@30954
   708
haftmann@30954
   709
text {* @{text "R ^^ n = R O ... O R"}, the n-fold composition of @{text R} *}
haftmann@30954
   710
haftmann@30971
   711
overloading
haftmann@30971
   712
  relpow == "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
haftmann@30971
   713
begin
haftmann@30954
   714
haftmann@30971
   715
primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
haftmann@30971
   716
    "relpow 0 R = Id"
krauss@32235
   717
  | "relpow (Suc n) R = (R ^^ n) O R"
haftmann@30954
   718
haftmann@30971
   719
end
haftmann@30954
   720
haftmann@30954
   721
lemma rel_pow_1 [simp]:
haftmann@30971
   722
  fixes R :: "('a \<times> 'a) set"
haftmann@30971
   723
  shows "R ^^ 1 = R"
haftmann@30954
   724
  by simp
haftmann@30954
   725
haftmann@30954
   726
lemma rel_pow_0_I: 
haftmann@30954
   727
  "(x, x) \<in> R ^^ 0"
haftmann@30954
   728
  by simp
haftmann@30954
   729
haftmann@30954
   730
lemma rel_pow_Suc_I:
haftmann@30954
   731
  "(x, y) \<in>  R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
haftmann@30954
   732
  by auto
haftmann@30954
   733
haftmann@30954
   734
lemma rel_pow_Suc_I2:
haftmann@30954
   735
  "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
nipkow@44890
   736
  by (induct n arbitrary: z) (simp, fastforce)
haftmann@30954
   737
haftmann@30954
   738
lemma rel_pow_0_E:
haftmann@30954
   739
  "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30954
   740
  by simp
haftmann@30954
   741
haftmann@30954
   742
lemma rel_pow_Suc_E:
haftmann@30954
   743
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30954
   744
  by auto
haftmann@30954
   745
haftmann@30954
   746
lemma rel_pow_E:
haftmann@30954
   747
  "(x, z) \<in>  R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
haftmann@30954
   748
   \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in>  R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P)
haftmann@30954
   749
   \<Longrightarrow> P"
haftmann@30954
   750
  by (cases n) auto
haftmann@30954
   751
haftmann@30954
   752
lemma rel_pow_Suc_D2:
haftmann@30954
   753
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
haftmann@30954
   754
  apply (induct n arbitrary: x z)
haftmann@30954
   755
   apply (blast intro: rel_pow_0_I elim: rel_pow_0_E rel_pow_Suc_E)
haftmann@30954
   756
  apply (blast intro: rel_pow_Suc_I elim: rel_pow_0_E rel_pow_Suc_E)
haftmann@30954
   757
  done
haftmann@30954
   758
haftmann@30954
   759
lemma rel_pow_Suc_E2:
haftmann@30954
   760
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30954
   761
  by (blast dest: rel_pow_Suc_D2)
haftmann@30954
   762
haftmann@30954
   763
lemma rel_pow_Suc_D2':
haftmann@30954
   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)"
haftmann@30954
   765
  by (induct n) (simp_all, blast)
haftmann@30954
   766
haftmann@30954
   767
lemma rel_pow_E2:
haftmann@30954
   768
  "(x, z) \<in> R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
haftmann@30954
   769
     \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P)
haftmann@30954
   770
   \<Longrightarrow> P"
haftmann@30954
   771
  apply (cases n, simp)
haftmann@30954
   772
  apply (cut_tac n=nat and R=R in rel_pow_Suc_D2', simp, blast)
haftmann@30954
   773
  done
haftmann@30954
   774
krauss@32235
   775
lemma rel_pow_add: "R ^^ (m+n) = R^^m O R^^n"
nipkow@31351
   776
by(induct n) auto
nipkow@31351
   777
krauss@31970
   778
lemma rel_pow_commute: "R O R ^^ n = R ^^ n O R"
krauss@32235
   779
by (induct n) (simp, simp add: O_assoc [symmetric])
krauss@31970
   780
bulwahn@45116
   781
lemma rel_pow_empty:
haftmann@45153
   782
  "0 < n \<Longrightarrow> ({} :: ('a \<times> 'a) set) ^^ n = {}"
haftmann@45153
   783
  by (cases n) auto
bulwahn@45116
   784
haftmann@30954
   785
lemma rtrancl_imp_UN_rel_pow:
haftmann@30954
   786
  assumes "p \<in> R^*"
haftmann@30954
   787
  shows "p \<in> (\<Union>n. R ^^ n)"
haftmann@30954
   788
proof (cases p)
haftmann@30954
   789
  case (Pair x y)
haftmann@30954
   790
  with assms have "(x, y) \<in> R^*" by simp
haftmann@30954
   791
  then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct
haftmann@30954
   792
    case base show ?case by (blast intro: rel_pow_0_I)
haftmann@30954
   793
  next
haftmann@30954
   794
    case step then show ?case by (blast intro: rel_pow_Suc_I)
haftmann@30954
   795
  qed
haftmann@30954
   796
  with Pair show ?thesis by simp
haftmann@30954
   797
qed
haftmann@30954
   798
haftmann@30954
   799
lemma rel_pow_imp_rtrancl:
haftmann@30954
   800
  assumes "p \<in> R ^^ n"
haftmann@30954
   801
  shows "p \<in> R^*"
haftmann@30954
   802
proof (cases p)
haftmann@30954
   803
  case (Pair x y)
haftmann@30954
   804
  with assms have "(x, y) \<in> R ^^ n" by simp
haftmann@30954
   805
  then have "(x, y) \<in> R^*" proof (induct n arbitrary: x y)
haftmann@30954
   806
    case 0 then show ?case by simp
haftmann@30954
   807
  next
haftmann@30954
   808
    case Suc then show ?case
haftmann@30954
   809
      by (blast elim: rel_pow_Suc_E intro: rtrancl_into_rtrancl)
haftmann@30954
   810
  qed
haftmann@30954
   811
  with Pair show ?thesis by simp
haftmann@30954
   812
qed
haftmann@30954
   813
haftmann@30954
   814
lemma rtrancl_is_UN_rel_pow:
haftmann@30954
   815
  "R^* = (\<Union>n. R ^^ n)"
haftmann@30954
   816
  by (blast intro: rtrancl_imp_UN_rel_pow rel_pow_imp_rtrancl)
haftmann@30954
   817
haftmann@30954
   818
lemma rtrancl_power:
haftmann@30954
   819
  "p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
haftmann@30954
   820
  by (simp add: rtrancl_is_UN_rel_pow)
haftmann@30954
   821
haftmann@30954
   822
lemma trancl_power:
haftmann@30954
   823
  "p \<in> R^+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
haftmann@30954
   824
  apply (cases p)
haftmann@30954
   825
  apply simp
haftmann@30954
   826
  apply (rule iffI)
haftmann@30954
   827
   apply (drule tranclD2)
haftmann@30954
   828
   apply (clarsimp simp: rtrancl_is_UN_rel_pow)
haftmann@30971
   829
   apply (rule_tac x="Suc n" in exI)
haftmann@30954
   830
   apply (clarsimp simp: rel_comp_def)
nipkow@44890
   831
   apply fastforce
haftmann@30954
   832
  apply clarsimp
haftmann@30954
   833
  apply (case_tac n, simp)
haftmann@30954
   834
  apply clarsimp
haftmann@30954
   835
  apply (drule rel_pow_imp_rtrancl)
haftmann@30954
   836
  apply (drule rtrancl_into_trancl1) apply auto
haftmann@30954
   837
  done
haftmann@30954
   838
haftmann@30954
   839
lemma rtrancl_imp_rel_pow:
haftmann@30954
   840
  "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
haftmann@30954
   841
  by (auto dest: rtrancl_imp_UN_rel_pow)
haftmann@30954
   842
nipkow@41987
   843
text{* By Sternagel/Thiemann: *}
nipkow@41987
   844
lemma rel_pow_fun_conv:
nipkow@41987
   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))"
nipkow@41987
   846
proof (induct n arbitrary: b)
nipkow@41987
   847
  case 0 show ?case by auto
nipkow@41987
   848
next
nipkow@41987
   849
  case (Suc n)
nipkow@41987
   850
  show ?case
nipkow@41987
   851
  proof (simp add: rel_comp_def Suc)
nipkow@41987
   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)
nipkow@41987
   853
     = (\<exists>f. f 0 = a \<and> f(Suc n) = b \<and> (\<forall>i<Suc n. (f i, f (Suc i)) \<in> R))"
nipkow@41987
   854
    (is "?l = ?r")
nipkow@41987
   855
    proof
nipkow@41987
   856
      assume ?l
nipkow@41987
   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
nipkow@41987
   858
      let ?g = "\<lambda> m. if m = Suc n then b else f m"
nipkow@41987
   859
      show ?r by (rule exI[of _ ?g], simp add: 1)
nipkow@41987
   860
    next
nipkow@41987
   861
      assume ?r
nipkow@41987
   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
nipkow@41987
   863
      show ?l by (rule exI[of _ "f n"], rule conjI, rule exI[of _ f], insert 1, auto)
nipkow@41987
   864
    qed
nipkow@41987
   865
  qed
nipkow@41987
   866
qed
nipkow@41987
   867
nipkow@41987
   868
lemma rel_pow_finite_bounded1:
nipkow@41987
   869
assumes "finite(R :: ('a*'a)set)" and "k>0"
nipkow@41987
   870
shows "R^^k \<subseteq> (UN n:{n. 0<n & n <= card R}. R^^n)" (is "_ \<subseteq> ?r")
nipkow@41987
   871
proof-
nipkow@41987
   872
  { fix a b k
nipkow@41987
   873
    have "(a,b) : R^^(Suc k) \<Longrightarrow> EX n. 0<n & n <= card R & (a,b) : R^^n"
nipkow@41987
   874
    proof(induct k arbitrary: b)
nipkow@41987
   875
      case 0
nipkow@41987
   876
      hence "R \<noteq> {}" by auto
nipkow@41987
   877
      with card_0_eq[OF `finite R`] have "card R >= Suc 0" by auto
nipkow@41987
   878
      thus ?case using 0 by force
nipkow@41987
   879
    next
nipkow@41987
   880
      case (Suc k)
nipkow@41987
   881
      then obtain a' where "(a,a') : R^^(Suc k)" and "(a',b) : R" by auto
nipkow@41987
   882
      from Suc(1)[OF `(a,a') : R^^(Suc k)`]
nipkow@41987
   883
      obtain n where "n \<le> card R" and "(a,a') \<in> R ^^ n" by auto
nipkow@41987
   884
      have "(a,b) : R^^(Suc n)" using `(a,a') \<in> R^^n` and `(a',b)\<in> R` by auto
nipkow@41987
   885
      { assume "n < card R"
nipkow@41987
   886
        hence ?case using `(a,b): R^^(Suc n)` Suc_leI[OF `n < card R`] by blast
nipkow@41987
   887
      } moreover
nipkow@41987
   888
      { assume "n = card R"
nipkow@41987
   889
        from `(a,b) \<in> R ^^ (Suc n)`[unfolded rel_pow_fun_conv]
nipkow@41987
   890
        obtain f where "f 0 = a" and "f(Suc n) = b"
nipkow@41987
   891
          and steps: "\<And>i. i <= n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto
nipkow@41987
   892
        let ?p = "%i. (f i, f(Suc i))"
nipkow@41987
   893
        let ?N = "{i. i \<le> n}"
nipkow@41987
   894
        have "?p ` ?N <= R" using steps by auto
nipkow@41987
   895
        from card_mono[OF assms(1) this]
nipkow@41987
   896
        have "card(?p ` ?N) <= card R" .
nipkow@41987
   897
        also have "\<dots> < card ?N" using `n = card R` by simp
nipkow@41987
   898
        finally have "~ inj_on ?p ?N" by(rule pigeonhole)
nipkow@41987
   899
        then obtain i j where i: "i <= n" and j: "j <= n" and ij: "i \<noteq> j" and
nipkow@41987
   900
          pij: "?p i = ?p j" by(auto simp: inj_on_def)
nipkow@41987
   901
        let ?i = "min i j" let ?j = "max i j"
nipkow@41987
   902
        have i: "?i <= n" and j: "?j <= n" and pij: "?p ?i = ?p ?j" 
nipkow@41987
   903
          and ij: "?i < ?j"
nipkow@41987
   904
          using i j ij pij unfolding min_def max_def by auto
nipkow@41987
   905
        from i j pij ij obtain i j where i: "i<=n" and j: "j<=n" and ij: "i<j"
nipkow@41987
   906
          and pij: "?p i = ?p j" by blast
nipkow@41987
   907
        let ?g = "\<lambda> l. if l \<le> i then f l else f (l + (j - i))"
nipkow@41987
   908
        let ?n = "Suc(n - (j - i))"
nipkow@41987
   909
        have abl: "(a,b) \<in> R ^^ ?n" unfolding rel_pow_fun_conv
nipkow@41987
   910
        proof (rule exI[of _ ?g], intro conjI impI allI)
nipkow@41987
   911
          show "?g ?n = b" using `f(Suc n) = b` j ij by auto
nipkow@41987
   912
        next
nipkow@41987
   913
          fix k assume "k < ?n"
nipkow@41987
   914
          show "(?g k, ?g (Suc k)) \<in> R"
nipkow@41987
   915
          proof (cases "k < i")
nipkow@41987
   916
            case True
nipkow@41987
   917
            with i have "k <= n" by auto
nipkow@41987
   918
            from steps[OF this] show ?thesis using True by simp
nipkow@41987
   919
          next
nipkow@41987
   920
            case False
nipkow@41987
   921
            hence "i \<le> k" by auto
nipkow@41987
   922
            show ?thesis
nipkow@41987
   923
            proof (cases "k = i")
nipkow@41987
   924
              case True
nipkow@41987
   925
              thus ?thesis using ij pij steps[OF i] by simp
nipkow@41987
   926
            next
nipkow@41987
   927
              case False
nipkow@41987
   928
              with `i \<le> k` have "i < k" by auto
nipkow@41987
   929
              hence small: "k + (j - i) <= n" using `k<?n` by arith
nipkow@41987
   930
              show ?thesis using steps[OF small] `i<k` by auto
nipkow@41987
   931
            qed
nipkow@41987
   932
          qed
nipkow@41987
   933
        qed (simp add: `f 0 = a`)
nipkow@41987
   934
        moreover have "?n <= n" using i j ij by arith
nipkow@41987
   935
        ultimately have ?case using `n = card R` by blast
nipkow@41987
   936
      }
nipkow@41987
   937
      ultimately show ?case using `n \<le> card R` by force
nipkow@41987
   938
    qed
nipkow@41987
   939
  }
nipkow@41987
   940
  thus ?thesis using gr0_implies_Suc[OF `k>0`] by auto
nipkow@41987
   941
qed
nipkow@41987
   942
nipkow@41987
   943
lemma rel_pow_finite_bounded:
nipkow@41987
   944
assumes "finite(R :: ('a*'a)set)"
nipkow@41987
   945
shows "R^^k \<subseteq> (UN n:{n. n <= card R}. R^^n)"
nipkow@41987
   946
apply(cases k)
nipkow@41987
   947
 apply force
nipkow@41987
   948
using rel_pow_finite_bounded1[OF assms, of k] by auto
nipkow@41987
   949
nipkow@41987
   950
lemma rtrancl_finite_eq_rel_pow:
nipkow@41987
   951
  "finite R \<Longrightarrow> R^* = (UN n : {n. n <= card R}. R^^n)"
nipkow@44890
   952
by(fastforce simp: rtrancl_power dest: rel_pow_finite_bounded)
nipkow@41987
   953
nipkow@41987
   954
lemma trancl_finite_eq_rel_pow:
nipkow@41987
   955
  "finite R \<Longrightarrow> R^+ = (UN n : {n. 0 < n & n <= card R}. R^^n)"
nipkow@41987
   956
apply(auto simp add: trancl_power)
nipkow@41987
   957
apply(auto dest: rel_pow_finite_bounded1)
nipkow@41987
   958
done
nipkow@41987
   959
nipkow@41987
   960
lemma finite_rel_comp[simp,intro]:
nipkow@41987
   961
assumes "finite R" and "finite S"
nipkow@41987
   962
shows "finite(R O S)"
nipkow@41987
   963
proof-
nipkow@41987
   964
  have "R O S = (UN (x,y) : R. \<Union>((%(u,v). if u=y then {(x,v)} else {}) ` S))"
nipkow@41987
   965
    by(force simp add: split_def)
nipkow@41987
   966
  thus ?thesis using assms by(clarsimp)
nipkow@41987
   967
qed
nipkow@41987
   968
nipkow@41987
   969
lemma finite_relpow[simp,intro]:
nipkow@41987
   970
  assumes "finite(R :: ('a*'a)set)" shows "n>0 \<Longrightarrow> finite(R^^n)"
nipkow@41987
   971
apply(induct n)
nipkow@41987
   972
 apply simp
nipkow@41987
   973
apply(case_tac n)
nipkow@41987
   974
 apply(simp_all add: assms)
nipkow@41987
   975
done
nipkow@41987
   976
haftmann@30954
   977
lemma single_valued_rel_pow:
haftmann@30954
   978
  fixes R :: "('a * 'a) set"
haftmann@30954
   979
  shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
nipkow@41987
   980
apply (induct n arbitrary: R)
nipkow@41987
   981
apply simp_all
nipkow@41987
   982
apply (rule single_valuedI)
nipkow@41987
   983
apply (fast dest: single_valuedD elim: rel_pow_Suc_E)
nipkow@41987
   984
done
paulson@15551
   985
haftmann@45140
   986
haftmann@45140
   987
subsection {* Bounded transitive closure *}
haftmann@45140
   988
haftmann@45140
   989
definition ntrancl :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
haftmann@45140
   990
where
haftmann@45140
   991
  "ntrancl n R = (\<Union>i\<in>{i. 0 < i \<and> i \<le> Suc n}. R ^^ i)"
haftmann@45140
   992
haftmann@45140
   993
lemma ntrancl_Zero [simp, code]:
haftmann@45140
   994
  "ntrancl 0 R = R"
haftmann@45140
   995
proof
haftmann@45140
   996
  show "R \<subseteq> ntrancl 0 R"
haftmann@45140
   997
    unfolding ntrancl_def by fastforce
haftmann@45140
   998
next
haftmann@45140
   999
  { 
haftmann@45140
  1000
    fix i have "0 < i \<and> i \<le> Suc 0 \<longleftrightarrow> i = 1" by auto
haftmann@45140
  1001
  }
haftmann@45140
  1002
  from this show "ntrancl 0 R \<le> R"
haftmann@45140
  1003
    unfolding ntrancl_def by auto
haftmann@45140
  1004
qed
haftmann@45140
  1005
haftmann@45140
  1006
lemma ntrancl_Suc [simp, code]:
haftmann@45140
  1007
  "ntrancl (Suc n) R = ntrancl n R O (Id \<union> R)"
haftmann@45140
  1008
proof
haftmann@45140
  1009
  {
haftmann@45140
  1010
    fix a b
haftmann@45140
  1011
    assume "(a, b) \<in> ntrancl (Suc n) R"
haftmann@45140
  1012
    from this obtain i where "0 < i" "i \<le> Suc (Suc n)" "(a, b) \<in> R ^^ i"
haftmann@45140
  1013
      unfolding ntrancl_def by auto
haftmann@45140
  1014
    have "(a, b) \<in> ntrancl n R O (Id \<union> R)"
haftmann@45140
  1015
    proof (cases "i = 1")
haftmann@45140
  1016
      case True
haftmann@45140
  1017
      from this `(a, b) \<in> R ^^ i` show ?thesis
haftmann@45140
  1018
        unfolding ntrancl_def by auto
haftmann@45140
  1019
    next
haftmann@45140
  1020
      case False
haftmann@45140
  1021
      from this `0 < i` obtain j where j: "i = Suc j" "0 < j"
haftmann@45140
  1022
        by (cases i) auto
haftmann@45140
  1023
      from this `(a, b) \<in> R ^^ i` obtain c where c1: "(a, c) \<in> R ^^ j" and c2:"(c, b) \<in> R"
haftmann@45140
  1024
        by auto
haftmann@45140
  1025
      from c1 j `i \<le> Suc (Suc n)` have "(a, c) \<in> ntrancl n R"
haftmann@45140
  1026
        unfolding ntrancl_def by fastforce
haftmann@45140
  1027
      from this c2 show ?thesis by fastforce
haftmann@45140
  1028
    qed
haftmann@45140
  1029
  }
haftmann@45140
  1030
  from this show "ntrancl (Suc n) R \<subseteq> ntrancl n R O (Id \<union> R)"
haftmann@45140
  1031
    by auto
haftmann@45140
  1032
next
haftmann@45140
  1033
  show "ntrancl n R O (Id \<union> R) \<subseteq> ntrancl (Suc n) R"
haftmann@45140
  1034
    unfolding ntrancl_def by fastforce
haftmann@45140
  1035
qed
haftmann@45140
  1036
haftmann@45141
  1037
lemma finite_trancl_ntranl:
haftmann@45140
  1038
  "finite R \<Longrightarrow> trancl R = ntrancl (card R - 1) R"
haftmann@45140
  1039
  by (cases "card R") (auto simp add: trancl_finite_eq_rel_pow rel_pow_empty ntrancl_def)
haftmann@45140
  1040
haftmann@45140
  1041
haftmann@45139
  1042
subsection {* Acyclic relations *}
haftmann@45139
  1043
haftmann@45139
  1044
definition acyclic :: "('a * 'a) set => bool" where
haftmann@45139
  1045
  "acyclic r \<longleftrightarrow> (!x. (x,x) ~: r^+)"
haftmann@45139
  1046
haftmann@45139
  1047
abbreviation acyclicP :: "('a => 'a => bool) => bool" where
haftmann@45139
  1048
  "acyclicP r \<equiv> acyclic {(x, y). r x y}"
haftmann@45139
  1049
haftmann@45139
  1050
lemma acyclic_irrefl:
haftmann@45139
  1051
  "acyclic r \<longleftrightarrow> irrefl (r^+)"
haftmann@45139
  1052
  by (simp add: acyclic_def irrefl_def)
haftmann@45139
  1053
haftmann@45139
  1054
lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
haftmann@45139
  1055
  by (simp add: acyclic_def)
haftmann@45139
  1056
haftmann@45139
  1057
lemma acyclic_insert [iff]:
haftmann@45139
  1058
     "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
haftmann@45139
  1059
apply (simp add: acyclic_def trancl_insert)
haftmann@45139
  1060
apply (blast intro: rtrancl_trans)
haftmann@45139
  1061
done
haftmann@45139
  1062
haftmann@45139
  1063
lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
haftmann@45139
  1064
by (simp add: acyclic_def trancl_converse)
haftmann@45139
  1065
haftmann@45139
  1066
lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
haftmann@45139
  1067
haftmann@45139
  1068
lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
haftmann@45139
  1069
apply (simp add: acyclic_def antisym_def)
haftmann@45139
  1070
apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
haftmann@45139
  1071
done
haftmann@45139
  1072
haftmann@45139
  1073
(* Other direction:
haftmann@45139
  1074
acyclic = no loops
haftmann@45139
  1075
antisym = only self loops
haftmann@45139
  1076
Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
haftmann@45139
  1077
==> antisym( r^* ) = acyclic(r - Id)";
haftmann@45139
  1078
*)
haftmann@45139
  1079
haftmann@45139
  1080
lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
haftmann@45139
  1081
apply (simp add: acyclic_def)
haftmann@45139
  1082
apply (blast intro: trancl_mono)
haftmann@45139
  1083
done
haftmann@45139
  1084
haftmann@45139
  1085
ballarin@15076
  1086
subsection {* Setup of transitivity reasoner *}
ballarin@15076
  1087
wenzelm@26340
  1088
ML {*
ballarin@15076
  1089
wenzelm@32215
  1090
structure Trancl_Tac = Trancl_Tac
wenzelm@32215
  1091
(
wenzelm@32215
  1092
  val r_into_trancl = @{thm trancl.r_into_trancl};
wenzelm@32215
  1093
  val trancl_trans  = @{thm trancl_trans};
wenzelm@32215
  1094
  val rtrancl_refl = @{thm rtrancl.rtrancl_refl};
wenzelm@32215
  1095
  val r_into_rtrancl = @{thm r_into_rtrancl};
wenzelm@32215
  1096
  val trancl_into_rtrancl = @{thm trancl_into_rtrancl};
wenzelm@32215
  1097
  val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl};
wenzelm@32215
  1098
  val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl};
wenzelm@32215
  1099
  val rtrancl_trans = @{thm rtrancl_trans};
ballarin@15096
  1100
berghofe@30107
  1101
  fun decomp (@{const Trueprop} $ t) =
haftmann@37677
  1102
    let fun dec (Const (@{const_name Set.member}, _) $ (Const (@{const_name Pair}, _) $ a $ b) $ rel ) =
berghofe@23743
  1103
        let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*")
berghofe@23743
  1104
              | decr (Const ("Transitive_Closure.trancl", _ ) $ r)  = (r,"r+")
wenzelm@18372
  1105
              | decr r = (r,"r");
berghofe@26801
  1106
            val (rel,r) = decr (Envir.beta_eta_contract rel);
wenzelm@18372
  1107
        in SOME (a,b,rel,r) end
wenzelm@18372
  1108
      | dec _ =  NONE
berghofe@30107
  1109
    in dec t end
berghofe@30107
  1110
    | decomp _ = NONE;
wenzelm@32215
  1111
);
ballarin@15076
  1112
wenzelm@32215
  1113
structure Tranclp_Tac = Trancl_Tac
wenzelm@32215
  1114
(
wenzelm@32215
  1115
  val r_into_trancl = @{thm tranclp.r_into_trancl};
wenzelm@32215
  1116
  val trancl_trans  = @{thm tranclp_trans};
wenzelm@32215
  1117
  val rtrancl_refl = @{thm rtranclp.rtrancl_refl};
wenzelm@32215
  1118
  val r_into_rtrancl = @{thm r_into_rtranclp};
wenzelm@32215
  1119
  val trancl_into_rtrancl = @{thm tranclp_into_rtranclp};
wenzelm@32215
  1120
  val rtrancl_trancl_trancl = @{thm rtranclp_tranclp_tranclp};
wenzelm@32215
  1121
  val trancl_rtrancl_trancl = @{thm tranclp_rtranclp_tranclp};
wenzelm@32215
  1122
  val rtrancl_trans = @{thm rtranclp_trans};
berghofe@22262
  1123
berghofe@30107
  1124
  fun decomp (@{const Trueprop} $ t) =
berghofe@22262
  1125
    let fun dec (rel $ a $ b) =
berghofe@23743
  1126
        let fun decr (Const ("Transitive_Closure.rtranclp", _ ) $ r) = (r,"r*")
berghofe@23743
  1127
              | decr (Const ("Transitive_Closure.tranclp", _ ) $ r)  = (r,"r+")
berghofe@22262
  1128
              | decr r = (r,"r");
berghofe@22262
  1129
            val (rel,r) = decr rel;
berghofe@26801
  1130
        in SOME (a, b, rel, r) end
berghofe@22262
  1131
      | dec _ =  NONE
berghofe@30107
  1132
    in dec t end
berghofe@30107
  1133
    | decomp _ = NONE;
wenzelm@32215
  1134
);
wenzelm@26340
  1135
*}
berghofe@22262
  1136
wenzelm@42795
  1137
setup {*
wenzelm@42795
  1138
  Simplifier.map_simpset_global (fn ss => ss
wenzelm@43596
  1139
    addSolver (mk_solver "Trancl" (Trancl_Tac.trancl_tac o Simplifier.the_context))
wenzelm@43596
  1140
    addSolver (mk_solver "Rtrancl" (Trancl_Tac.rtrancl_tac o Simplifier.the_context))
wenzelm@43596
  1141
    addSolver (mk_solver "Tranclp" (Tranclp_Tac.trancl_tac o Simplifier.the_context))
wenzelm@43596
  1142
    addSolver (mk_solver "Rtranclp" (Tranclp_Tac.rtrancl_tac o Simplifier.the_context)))
ballarin@15076
  1143
*}
ballarin@15076
  1144
wenzelm@32215
  1145
wenzelm@32215
  1146
text {* Optional methods. *}
ballarin@15076
  1147
ballarin@15076
  1148
method_setup trancl =
wenzelm@32215
  1149
  {* Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.trancl_tac) *}
wenzelm@18372
  1150
  {* simple transitivity reasoner *}
ballarin@15076
  1151
method_setup rtrancl =
wenzelm@32215
  1152
  {* Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.rtrancl_tac) *}
ballarin@15076
  1153
  {* simple transitivity reasoner *}
berghofe@22262
  1154
method_setup tranclp =
wenzelm@32215
  1155
  {* Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.trancl_tac) *}
berghofe@22262
  1156
  {* simple transitivity reasoner (predicate version) *}
berghofe@22262
  1157
method_setup rtranclp =
wenzelm@32215
  1158
  {* Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.rtrancl_tac) *}
berghofe@22262
  1159
  {* simple transitivity reasoner (predicate version) *}
ballarin@15076
  1160
nipkow@10213
  1161
end