src/HOL/Transitive_Closure.thy
author paulson <lp15@cam.ac.uk>
Tue Apr 25 16:39:54 2017 +0100 (2017-04-25)
changeset 65578 e4997c181cce
parent 63612 7195acc2fe93
child 67399 eab6ce8368fa
permissions -rw-r--r--
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
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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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section \<open>Reflexive and Transitive closure of a relation\<close>
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theory Transitive_Closure
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  imports Relation
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begin
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ML_file "~~/src/Provers/trancl.ML"
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text \<open>
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  \<open>rtrancl\<close> is reflexive/transitive closure,
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  \<open>trancl\<close> is transitive closure,
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  \<open>reflcl\<close> is reflexive closure.
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  These postfix operators have \<^emph>\<open>maximum priority\<close>, forcing their
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  operands to be atomic.
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\<close>
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context notes [[inductive_internals]]
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begin
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inductive_set rtrancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"  ("(_\<^sup>*)" [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) \<in> r\<^sup>*"
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  | rtrancl_into_rtrancl [Pure.intro]: "(a, b) \<in> r\<^sup>* \<Longrightarrow> (b, c) \<in> r \<Longrightarrow> (a, c) \<in> r\<^sup>*"
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inductive_set trancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"  ("(_\<^sup>+)" [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) \<in> r \<Longrightarrow> (a, b) \<in> r\<^sup>+"
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  | trancl_into_trancl [Pure.intro]: "(a, b) \<in> r\<^sup>+ \<Longrightarrow> (b, c) \<in> r \<Longrightarrow> (a, c) \<in> r\<^sup>+"
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notation
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  rtranclp  ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
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  tranclp  ("(_\<^sup>+\<^sup>+)" [1000] 1000)
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declare
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  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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end
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abbreviation reflcl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"  ("(_\<^sup>=)" [1000] 999)
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  where "r\<^sup>= \<equiv> r \<union> Id"
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abbreviation reflclp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"  ("(_\<^sup>=\<^sup>=)" [1000] 1000)
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  where "r\<^sup>=\<^sup>= \<equiv> sup r op ="
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notation (ASCII)
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  rtrancl  ("(_^*)" [1000] 999) and
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  trancl  ("(_^+)" [1000] 999) and
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  reflcl  ("(_^=)" [1000] 999) and
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  rtranclp  ("(_^**)" [1000] 1000) and
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  tranclp  ("(_^++)" [1000] 1000) and
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  reflclp  ("(_^==)" [1000] 1000)
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subsection \<open>Reflexive closure\<close>
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lemma refl_reflcl[simp]: "refl (r\<^sup>=)"
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  by (simp add: refl_on_def)
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lemma antisym_reflcl[simp]: "antisym (r\<^sup>=) = antisym r"
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  by (simp add: antisym_def)
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lemma trans_reflclI[simp]: "trans r \<Longrightarrow> trans (r\<^sup>=)"
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  unfolding trans_def by blast
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lemma reflclp_idemp [simp]: "(P\<^sup>=\<^sup>=)\<^sup>=\<^sup>= = P\<^sup>=\<^sup>="
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  by blast
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subsection \<open>Reflexive-transitive closure\<close>
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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]: "\<And>p. p \<in> r \<Longrightarrow> p \<in> r\<^sup>*"
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  \<comment> \<open>\<open>rtrancl\<close> of \<open>r\<close> contains \<open>r\<close>\<close>
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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 \<Longrightarrow> r\<^sup>*\<^sup>* x y"
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  \<comment> \<open>\<open>rtrancl\<close> of \<open>r\<close> contains \<open>r\<close>\<close>
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  by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl])
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lemma rtranclp_mono: "r \<le> s \<Longrightarrow> r\<^sup>*\<^sup>* \<le> s\<^sup>*\<^sup>*"
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  \<comment> \<open>monotonicity of \<open>rtrancl\<close>\<close>
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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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lemma mono_rtranclp[mono]: "(\<And>a b. x a b \<longrightarrow> y a b) \<Longrightarrow> x\<^sup>*\<^sup>* a b \<longrightarrow> y\<^sup>*\<^sup>* a b"
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   using rtranclp_mono[of x y] by auto
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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\<^sup>*\<^sup>* a b"
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    and cases: "P a" "\<And>y z. r\<^sup>*\<^sup>* a y \<Longrightarrow> r y z \<Longrightarrow> P y \<Longrightarrow> P z"
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  shows "P b"
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  using a 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, 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), consumes 1, case_names refl step]
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lemma refl_rtrancl: "refl (r\<^sup>*)"
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  unfolding refl_on_def by fast
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text \<open>Transitivity of transitive closure.\<close>
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lemma trans_rtrancl: "trans (r\<^sup>*)"
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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 \<open>(x, u) \<in> r\<^sup>*\<close> and \<open>(u, v) \<in> r\<close>
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    show "(x, v) \<in> r\<^sup>*" ..
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  qed
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qed
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lemmas rtrancl_trans = trans_rtrancl [THEN transD]
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lemma rtranclp_trans:
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  assumes "r\<^sup>*\<^sup>* x y"
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    and "r\<^sup>*\<^sup>* y z"
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  shows "r\<^sup>*\<^sup>* x z"
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  using assms(2,1) by induct iprover+
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lemma rtranclE [cases set: rtrancl]:
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  fixes a b :: 'a
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  assumes major: "(a, b) \<in> r\<^sup>*"
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  obtains
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    (base) "a = b"
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  | (step) y where "(a, y) \<in> r\<^sup>*" and "(y, b) \<in> r"
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  \<comment> \<open>elimination of \<open>rtrancl\<close> -- by induction on a special formula\<close>
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  apply (subgoal_tac "a = b \<or> (\<exists>y. (a, y) \<in> r\<^sup>* \<and> (y, b) \<in> 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 \<Longrightarrow> (r\<^sup>* \<inter> s) O r \<subseteq> s \<Longrightarrow> r\<^sup>* \<subseteq> s"
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  apply (rule subsetI)
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  apply auto
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  apply (erule rtrancl_induct)
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  apply auto
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  done
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lemma converse_rtranclp_into_rtranclp: "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 \<open>\<^medskip> More @{term "r\<^sup>*"} equations and inclusions.\<close>
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lemma rtranclp_idemp [simp]: "(r\<^sup>*\<^sup>*)\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*"
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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\<^sup>* O R\<^sup>* = R\<^sup>*"
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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\<^sup>* \<Longrightarrow> r\<^sup>* \<subseteq> s\<^sup>*"
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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 \<Longrightarrow> S \<le> R\<^sup>*\<^sup>* \<Longrightarrow> S\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
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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\<^sup>*\<^sup>*) (S\<^sup>*\<^sup>*))\<^sup>*\<^sup>* = (sup R S)\<^sup>*\<^sup>*"
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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_reflclp [simp]: "(R\<^sup>=\<^sup>=)\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
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  by (blast intro!: rtranclp_subset)
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lemmas rtrancl_reflcl [simp] = rtranclp_reflclp [to_set]
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lemma rtrancl_r_diff_Id: "(r - Id)\<^sup>* = r\<^sup>*"
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  apply (rule sym)
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  apply (rule rtrancl_subset)
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   apply blast
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  apply 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 \<noteq>)\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*"
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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\<inverse>\<inverse>)\<^sup>*\<^sup>* x y"
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  shows "r\<^sup>*\<^sup>* y x"
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  using assms by induct (iprover intro: rtranclp_trans dest!: conversepD)+
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lemmas rtrancl_converseD = rtranclp_converseD [to_set]
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theorem rtranclp_converseI:
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  assumes "r\<^sup>*\<^sup>* y x"
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  shows "(r\<inverse>\<inverse>)\<^sup>*\<^sup>* x y"
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  using assms 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)\<^sup>* = (r\<^sup>*)^-1"
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  by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
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lemma sym_rtrancl: "sym r \<Longrightarrow> sym (r\<^sup>*)"
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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\<^sup>*\<^sup>* a b"
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    and cases: "P b" "\<And>y z. r y z \<Longrightarrow> r\<^sup>*\<^sup>* z b \<Longrightarrow> P z \<Longrightarrow> 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, 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\<^sup>*\<^sup>* x z"
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    and cases: "x = z \<Longrightarrow> P" "\<And>y. r x y \<Longrightarrow> r\<^sup>*\<^sup>* y z \<Longrightarrow> P"
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  shows P
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  apply (subgoal_tac "x = z \<or> (\<exists>y. r x y \<and> r\<^sup>*\<^sup>* 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\<^sup>* = r\<^sup>* O r"
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   286
  by (blast elim: rtranclE converse_rtranclE
wenzelm@63612
   287
      intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
wenzelm@12691
   288
wenzelm@63404
   289
lemma rtrancl_unfold: "r\<^sup>* = Id \<union> r\<^sup>* O r"
paulson@15551
   290
  by (auto intro: rtrancl_into_rtrancl elim: rtranclE)
paulson@15551
   291
nipkow@31690
   292
lemma rtrancl_Un_separatorE:
wenzelm@63404
   293
  "(a, b) \<in> (P \<union> Q)\<^sup>* \<Longrightarrow> \<forall>x y. (a, x) \<in> P\<^sup>* \<longrightarrow> (x, y) \<in> Q \<longrightarrow> x = y \<Longrightarrow> (a, b) \<in> P\<^sup>*"
wenzelm@63612
   294
proof (induct rule: rtrancl.induct)
wenzelm@63612
   295
  case rtrancl_refl
wenzelm@63612
   296
  then show ?case by blast
wenzelm@63612
   297
next
wenzelm@63612
   298
  case rtrancl_into_rtrancl
wenzelm@63612
   299
  then show ?case by (blast intro: rtrancl_trans)
wenzelm@63612
   300
qed
nipkow@31690
   301
nipkow@31690
   302
lemma rtrancl_Un_separator_converseE:
wenzelm@63404
   303
  "(a, b) \<in> (P \<union> Q)\<^sup>* \<Longrightarrow> \<forall>x y. (x, b) \<in> P\<^sup>* \<longrightarrow> (y, x) \<in> Q \<longrightarrow> y = x \<Longrightarrow> (a, b) \<in> P\<^sup>*"
wenzelm@63612
   304
proof (induct rule: converse_rtrancl_induct)
wenzelm@63612
   305
  case base
wenzelm@63612
   306
  then show ?case by blast
wenzelm@63612
   307
next
wenzelm@63612
   308
  case step
wenzelm@63612
   309
  then show ?case by (blast intro: rtrancl_trans)
wenzelm@63612
   310
qed
nipkow@31690
   311
haftmann@34970
   312
lemma Image_closed_trancl:
wenzelm@63404
   313
  assumes "r `` X \<subseteq> X"
wenzelm@63404
   314
  shows "r\<^sup>* `` X = X"
haftmann@34970
   315
proof -
wenzelm@63404
   316
  from assms have **: "{y. \<exists>x\<in>X. (x, y) \<in> r} \<subseteq> X"
wenzelm@63404
   317
    by auto
wenzelm@63404
   318
  have "x \<in> X" if 1: "(y, x) \<in> r\<^sup>*" and 2: "y \<in> X" for x y
haftmann@34970
   319
  proof -
wenzelm@63404
   320
    from 1 show "x \<in> X"
haftmann@34970
   321
    proof induct
wenzelm@63404
   322
      case base
wenzelm@63404
   323
      show ?case by (fact 2)
haftmann@34970
   324
    next
wenzelm@63404
   325
      case step
wenzelm@63404
   326
      with ** show ?case by auto
haftmann@34970
   327
    qed
haftmann@34970
   328
  qed
haftmann@34970
   329
  then show ?thesis by auto
haftmann@34970
   330
qed
haftmann@34970
   331
wenzelm@12691
   332
wenzelm@60758
   333
subsection \<open>Transitive closure\<close>
wenzelm@10331
   334
wenzelm@63404
   335
lemma trancl_mono: "\<And>p. p \<in> r\<^sup>+ \<Longrightarrow> r \<subseteq> s \<Longrightarrow> p \<in> s\<^sup>+"
berghofe@23743
   336
  apply (simp add: split_tupled_all)
berghofe@13704
   337
  apply (erule trancl.induct)
wenzelm@26179
   338
   apply (iprover dest: subsetD)+
wenzelm@12691
   339
  done
wenzelm@12691
   340
wenzelm@63404
   341
lemma r_into_trancl': "\<And>p. p \<in> r \<Longrightarrow> p \<in> r\<^sup>+"
berghofe@13704
   342
  by (simp only: split_tupled_all) (erule r_into_trancl)
berghofe@13704
   343
wenzelm@63404
   344
text \<open>\<^medskip> Conversions between \<open>trancl\<close> and \<open>rtrancl\<close>.\<close>
wenzelm@12691
   345
wenzelm@63404
   346
lemma tranclp_into_rtranclp: "r\<^sup>+\<^sup>+ a b \<Longrightarrow> r\<^sup>*\<^sup>* a b"
berghofe@23743
   347
  by (erule tranclp.induct) iprover+
wenzelm@12691
   348
berghofe@23743
   349
lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]
berghofe@22262
   350
wenzelm@63404
   351
lemma rtranclp_into_tranclp1:
wenzelm@63404
   352
  assumes "r\<^sup>*\<^sup>* a b"
wenzelm@63404
   353
  shows "r b c \<Longrightarrow> r\<^sup>+\<^sup>+ a c"
wenzelm@63404
   354
  using assms by (induct arbitrary: c) iprover+
wenzelm@12691
   355
berghofe@23743
   356
lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]
berghofe@22262
   357
wenzelm@63404
   358
lemma rtranclp_into_tranclp2: "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>+\<^sup>+ a c"
wenzelm@61799
   359
  \<comment> \<open>intro rule from \<open>r\<close> and \<open>rtrancl\<close>\<close>
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@61799
   368
text \<open>Nice induction rule for \<open>trancl\<close>\<close>
wenzelm@26179
   369
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
wenzelm@63404
   370
  assumes a: "r\<^sup>+\<^sup>+ a b"
wenzelm@63404
   371
    and cases: "\<And>y. r a y \<Longrightarrow> P y" "\<And>y z. r\<^sup>+\<^sup>+ a y \<Longrightarrow> r y z \<Longrightarrow> P y \<Longrightarrow> P z"
wenzelm@63404
   372
  shows "P b"
wenzelm@63404
   373
  using a by (induct x\<equiv>a b) (iprover intro: cases)+
wenzelm@12691
   374
berghofe@25425
   375
lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]
berghofe@22262
   376
berghofe@23743
   377
lemmas tranclp_induct2 =
wenzelm@63612
   378
  tranclp_induct [of _ "(ax, ay)" "(bx, by)", split_rule, consumes 1, case_names base step]
berghofe@22262
   379
paulson@22172
   380
lemmas trancl_induct2 =
wenzelm@63612
   381
  trancl_induct [of "(ax, ay)" "(bx, by)", split_format (complete),
wenzelm@26179
   382
    consumes 1, case_names base step]
paulson@22172
   383
berghofe@23743
   384
lemma tranclp_trans_induct:
wenzelm@63404
   385
  assumes major: "r\<^sup>+\<^sup>+ x y"
wenzelm@63404
   386
    and cases: "\<And>x y. r x y \<Longrightarrow> P x y" "\<And>x y z. r\<^sup>+\<^sup>+ x y \<Longrightarrow> P x y \<Longrightarrow> r\<^sup>+\<^sup>+ y z \<Longrightarrow> P y z \<Longrightarrow> P x z"
wenzelm@18372
   387
  shows "P x y"
wenzelm@61799
   388
  \<comment> \<open>Another induction rule for trancl, incorporating transitivity\<close>
berghofe@23743
   389
  by (iprover intro: major [THEN tranclp_induct] cases)
wenzelm@12691
   390
berghofe@23743
   391
lemmas trancl_trans_induct = tranclp_trans_induct [to_set]
berghofe@23743
   392
wenzelm@26174
   393
lemma tranclE [cases set: trancl]:
wenzelm@63404
   394
  assumes "(a, b) \<in> r\<^sup>+"
wenzelm@26174
   395
  obtains
wenzelm@63404
   396
    (base) "(a, b) \<in> r"
wenzelm@63404
   397
  | (step) c where "(a, c) \<in> r\<^sup>+" and "(c, b) \<in> r"
wenzelm@26174
   398
  using assms by cases simp_all
wenzelm@10980
   399
wenzelm@63404
   400
lemma trancl_Int_subset: "r \<subseteq> s \<Longrightarrow> (r\<^sup>+ \<inter> s) O r \<subseteq> s \<Longrightarrow> r\<^sup>+ \<subseteq> s"
paulson@22080
   401
  apply (rule subsetI)
haftmann@61032
   402
  apply auto
wenzelm@26179
   403
  apply (erule trancl_induct)
wenzelm@63612
   404
   apply auto
paulson@22080
   405
  done
paulson@22080
   406
wenzelm@63404
   407
lemma trancl_unfold: "r\<^sup>+ = r \<union> r\<^sup>+ O r"
paulson@15551
   408
  by (auto intro: trancl_into_trancl elim: tranclE)
paulson@15551
   409
wenzelm@63404
   410
text \<open>Transitivity of @{term "r\<^sup>+"}\<close>
wenzelm@63404
   411
lemma trans_trancl [simp]: "trans (r\<^sup>+)"
berghofe@13704
   412
proof (rule transI)
berghofe@13704
   413
  fix x y z
wenzelm@63404
   414
  assume "(x, y) \<in> r\<^sup>+"
wenzelm@63404
   415
  assume "(y, z) \<in> r\<^sup>+"
wenzelm@63404
   416
  then show "(x, z) \<in> r\<^sup>+"
wenzelm@26179
   417
  proof induct
wenzelm@26179
   418
    case (base u)
wenzelm@63404
   419
    from \<open>(x, y) \<in> r\<^sup>+\<close> and \<open>(y, u) \<in> r\<close>
wenzelm@63404
   420
    show "(x, u) \<in> r\<^sup>+" ..
wenzelm@26179
   421
  next
wenzelm@26179
   422
    case (step u v)
wenzelm@63404
   423
    from \<open>(x, u) \<in> r\<^sup>+\<close> and \<open>(u, v) \<in> r\<close>
wenzelm@63404
   424
    show "(x, v) \<in> r\<^sup>+" ..
wenzelm@26179
   425
  qed
berghofe@13704
   426
qed
wenzelm@12691
   427
wenzelm@45607
   428
lemmas trancl_trans = trans_trancl [THEN transD]
wenzelm@12691
   429
berghofe@23743
   430
lemma tranclp_trans:
wenzelm@63404
   431
  assumes "r\<^sup>+\<^sup>+ x y"
wenzelm@63404
   432
    and "r\<^sup>+\<^sup>+ y z"
wenzelm@63404
   433
  shows "r\<^sup>+\<^sup>+ x z"
wenzelm@63404
   434
  using assms(2,1) by induct iprover+
berghofe@22262
   435
wenzelm@63404
   436
lemma trancl_id [simp]: "trans r \<Longrightarrow> r\<^sup>+ = r"
wenzelm@26179
   437
  apply auto
wenzelm@26179
   438
  apply (erule trancl_induct)
wenzelm@26179
   439
   apply assumption
wenzelm@26179
   440
  apply (unfold trans_def)
wenzelm@26179
   441
  apply blast
wenzelm@26179
   442
  done
nipkow@19623
   443
wenzelm@26179
   444
lemma rtranclp_tranclp_tranclp:
wenzelm@63404
   445
  assumes "r\<^sup>*\<^sup>* x y"
wenzelm@63404
   446
  shows "\<And>z. r\<^sup>+\<^sup>+ y z \<Longrightarrow> r\<^sup>+\<^sup>+ x z"
wenzelm@63404
   447
  using assms by induct (iprover intro: tranclp_trans)+
wenzelm@12691
   448
berghofe@23743
   449
lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]
berghofe@22262
   450
wenzelm@63404
   451
lemma tranclp_into_tranclp2: "r a b \<Longrightarrow> r\<^sup>+\<^sup>+ b c \<Longrightarrow> r\<^sup>+\<^sup>+ a c"
berghofe@23743
   452
  by (erule tranclp_trans [OF tranclp.r_into_trancl])
berghofe@22262
   453
berghofe@23743
   454
lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]
wenzelm@12691
   455
wenzelm@63404
   456
lemma tranclp_converseI: "(r\<^sup>+\<^sup>+)\<inverse>\<inverse> x y \<Longrightarrow> (r\<inverse>\<inverse>)\<^sup>+\<^sup>+ x y"
berghofe@22262
   457
  apply (drule conversepD)
berghofe@23743
   458
  apply (erule tranclp_induct)
wenzelm@63612
   459
   apply (iprover intro: conversepI tranclp_trans)+
wenzelm@12691
   460
  done
wenzelm@12691
   461
berghofe@23743
   462
lemmas trancl_converseI = tranclp_converseI [to_set]
berghofe@22262
   463
wenzelm@63404
   464
lemma tranclp_converseD: "(r\<inverse>\<inverse>)\<^sup>+\<^sup>+ x y \<Longrightarrow> (r\<^sup>+\<^sup>+)\<inverse>\<inverse> x y"
berghofe@22262
   465
  apply (rule conversepI)
berghofe@23743
   466
  apply (erule tranclp_induct)
wenzelm@63612
   467
   apply (iprover dest: conversepD intro: tranclp_trans)+
berghofe@13704
   468
  done
wenzelm@12691
   469
berghofe@23743
   470
lemmas trancl_converseD = tranclp_converseD [to_set]
berghofe@22262
   471
wenzelm@63404
   472
lemma tranclp_converse: "(r\<inverse>\<inverse>)\<^sup>+\<^sup>+ = (r\<^sup>+\<^sup>+)\<inverse>\<inverse>"
wenzelm@63404
   473
  by (fastforce simp add: fun_eq_iff intro!: tranclp_converseI dest!: tranclp_converseD)
berghofe@22262
   474
berghofe@23743
   475
lemmas trancl_converse = tranclp_converse [to_set]
wenzelm@12691
   476
wenzelm@63404
   477
lemma sym_trancl: "sym r \<Longrightarrow> sym (r\<^sup>+)"
huffman@19228
   478
  by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
huffman@19228
   479
berghofe@34909
   480
lemma converse_tranclp_induct [consumes 1, case_names base step]:
wenzelm@63404
   481
  assumes major: "r\<^sup>+\<^sup>+ a b"
wenzelm@63404
   482
    and cases: "\<And>y. r y b \<Longrightarrow> P y" "\<And>y z. r y z \<Longrightarrow> r\<^sup>+\<^sup>+ z b \<Longrightarrow> P z \<Longrightarrow> P y"
wenzelm@18372
   483
  shows "P a"
berghofe@23743
   484
  apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])
wenzelm@18372
   485
   apply (rule cases)
berghofe@22262
   486
   apply (erule conversepD)
huffman@35216
   487
  apply (blast intro: assms dest!: tranclp_converseD)
wenzelm@18372
   488
  done
wenzelm@12691
   489
berghofe@23743
   490
lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
berghofe@22262
   491
wenzelm@63404
   492
lemma tranclpD: "R\<^sup>+\<^sup>+ x y \<Longrightarrow> \<exists>z. R x z \<and> R\<^sup>*\<^sup>* z y"
wenzelm@26179
   493
  apply (erule converse_tranclp_induct)
wenzelm@26179
   494
   apply auto
berghofe@23743
   495
  apply (blast intro: rtranclp_trans)
wenzelm@12691
   496
  done
wenzelm@12691
   497
berghofe@23743
   498
lemmas tranclD = tranclpD [to_set]
berghofe@22262
   499
bulwahn@31577
   500
lemma converse_tranclpE:
bulwahn@31577
   501
  assumes major: "tranclp r x z"
wenzelm@63404
   502
    and base: "r x z \<Longrightarrow> P"
wenzelm@63612
   503
    and step: "\<And>y. r x y \<Longrightarrow> tranclp r y z \<Longrightarrow> P"
bulwahn@31577
   504
  shows P
bulwahn@31577
   505
proof -
wenzelm@63404
   506
  from tranclpD [OF major] obtain y where "r x y" and "rtranclp r y z"
wenzelm@63404
   507
    by iprover
bulwahn@31577
   508
  from this(2) show P
bulwahn@31577
   509
  proof (cases rule: rtranclp.cases)
bulwahn@31577
   510
    case rtrancl_refl
wenzelm@63404
   511
    with \<open>r x y\<close> base show P
wenzelm@63404
   512
      by iprover
bulwahn@31577
   513
  next
bulwahn@31577
   514
    case rtrancl_into_rtrancl
bulwahn@31577
   515
    from this have "tranclp r y z"
bulwahn@31577
   516
      by (iprover intro: rtranclp_into_tranclp1)
wenzelm@63404
   517
    with \<open>r x y\<close> step show P
wenzelm@63404
   518
      by iprover
bulwahn@31577
   519
  qed
bulwahn@31577
   520
qed
bulwahn@31577
   521
bulwahn@31577
   522
lemmas converse_tranclE = converse_tranclpE [to_set]
bulwahn@31577
   523
wenzelm@63404
   524
lemma tranclD2: "(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
kleing@25295
   525
  by (blast elim: tranclE intro: trancl_into_rtrancl)
kleing@25295
   526
wenzelm@63404
   527
lemma irrefl_tranclI: "r\<inverse> \<inter> r\<^sup>* = {} \<Longrightarrow> (x, x) \<notin> r\<^sup>+"
wenzelm@18372
   528
  by (blast elim: tranclE dest: trancl_into_rtrancl)
wenzelm@12691
   529
wenzelm@63404
   530
lemma irrefl_trancl_rD: "\<forall>x. (x, x) \<notin> r\<^sup>+ \<Longrightarrow> (x, y) \<in> r \<Longrightarrow> x \<noteq> y"
wenzelm@12691
   531
  by (blast dest: r_into_trancl)
wenzelm@12691
   532
wenzelm@63404
   533
lemma trancl_subset_Sigma_aux: "(a, b) \<in> r\<^sup>* \<Longrightarrow> r \<subseteq> A \<times> A \<Longrightarrow> a = b \<or> a \<in> A"
wenzelm@18372
   534
  by (induct rule: rtrancl_induct) auto
wenzelm@12691
   535
wenzelm@63404
   536
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A \<Longrightarrow> r\<^sup>+ \<subseteq> A \<times> A"
berghofe@13704
   537
  apply (rule subsetI)
berghofe@13704
   538
  apply (simp only: split_tupled_all)
berghofe@13704
   539
  apply (erule tranclE)
wenzelm@26179
   540
   apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
wenzelm@12691
   541
  done
nipkow@10996
   542
wenzelm@63404
   543
lemma reflclp_tranclp [simp]: "(r\<^sup>+\<^sup>+)\<^sup>=\<^sup>= = r\<^sup>*\<^sup>*"
berghofe@22262
   544
  apply (safe intro!: order_antisym)
berghofe@23743
   545
   apply (erule tranclp_into_rtranclp)
berghofe@23743
   546
  apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)
wenzelm@11084
   547
  done
nipkow@10996
   548
nipkow@50616
   549
lemmas reflcl_trancl [simp] = reflclp_tranclp [to_set]
berghofe@22262
   550
wenzelm@63404
   551
lemma trancl_reflcl [simp]: "(r\<^sup>=)\<^sup>+ = r\<^sup>*"
wenzelm@11084
   552
  apply safe
paulson@14208
   553
   apply (drule trancl_into_rtrancl, simp)
paulson@14208
   554
  apply (erule rtranclE, safe)
paulson@14208
   555
   apply (rule r_into_trancl, simp)
wenzelm@11084
   556
  apply (rule rtrancl_into_trancl1)
paulson@14208
   557
   apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
wenzelm@11084
   558
  done
nipkow@10996
   559
wenzelm@63404
   560
lemma rtrancl_trancl_reflcl [code]: "r\<^sup>* = (r\<^sup>+)\<^sup>="
haftmann@45140
   561
  by simp
haftmann@45140
   562
wenzelm@63404
   563
lemma trancl_empty [simp]: "{}\<^sup>+ = {}"
wenzelm@11084
   564
  by (auto elim: trancl_induct)
nipkow@10996
   565
wenzelm@63404
   566
lemma rtrancl_empty [simp]: "{}\<^sup>* = Id"
wenzelm@11084
   567
  by (rule subst [OF reflcl_trancl]) simp
nipkow@10996
   568
wenzelm@63404
   569
lemma rtranclpD: "R\<^sup>*\<^sup>* a b \<Longrightarrow> a = b \<or> a \<noteq> b \<and> R\<^sup>+\<^sup>+ a b"
wenzelm@63404
   570
  by (force simp add: reflclp_tranclp [symmetric] simp del: reflclp_tranclp)
berghofe@22262
   571
berghofe@23743
   572
lemmas rtranclD = rtranclpD [to_set]
wenzelm@11084
   573
wenzelm@63404
   574
lemma rtrancl_eq_or_trancl: "(x,y) \<in> R\<^sup>* \<longleftrightarrow> x = y \<or> x \<noteq> y \<and> (x, y) \<in> R\<^sup>+"
kleing@16514
   575
  by (fast elim: trancl_into_rtrancl dest: rtranclD)
nipkow@10996
   576
wenzelm@63404
   577
lemma trancl_unfold_right: "r\<^sup>+ = r\<^sup>* O r"
wenzelm@63404
   578
  by (auto dest: tranclD2 intro: rtrancl_into_trancl1)
krauss@33656
   579
wenzelm@63404
   580
lemma trancl_unfold_left: "r\<^sup>+ = r O r\<^sup>*"
wenzelm@63404
   581
  by (auto dest: tranclD intro: rtrancl_into_trancl2)
krauss@33656
   582
wenzelm@63404
   583
lemma trancl_insert: "(insert (y, x) r)\<^sup>+ = r\<^sup>+ \<union> {(a, b). (a, y) \<in> r\<^sup>* \<and> (x, b) \<in> r\<^sup>*}"
wenzelm@61799
   584
  \<comment> \<open>primitive recursion for \<open>trancl\<close> over finite relations\<close>
nipkow@57178
   585
  apply (rule equalityI)
nipkow@57178
   586
   apply (rule subsetI)
nipkow@57178
   587
   apply (simp only: split_tupled_all)
nipkow@57178
   588
   apply (erule trancl_induct, blast)
nipkow@57178
   589
   apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)
nipkow@57178
   590
  apply (rule subsetI)
nipkow@57178
   591
  apply (blast intro: trancl_mono rtrancl_mono
wenzelm@63612
   592
      [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
nipkow@57178
   593
  done
nipkow@57178
   594
nipkow@57178
   595
lemma trancl_insert2:
wenzelm@63404
   596
  "(insert (a, b) r)\<^sup>+ = r\<^sup>+ \<union> {(x, y). ((x, a) \<in> r\<^sup>+ \<or> x = a) \<and> ((b, y) \<in> r\<^sup>+ \<or> y = b)}"
wenzelm@63404
   597
  by (auto simp add: trancl_insert rtrancl_eq_or_trancl)
nipkow@57178
   598
wenzelm@63404
   599
lemma rtrancl_insert: "(insert (a,b) r)\<^sup>* = r\<^sup>* \<union> {(x, y). (x, a) \<in> r\<^sup>* \<and> (b, y) \<in> r\<^sup>*}"
wenzelm@63404
   600
  using trancl_insert[of a b r]
wenzelm@63404
   601
  by (simp add: rtrancl_trancl_reflcl del: reflcl_trancl) blast
nipkow@57178
   602
krauss@33656
   603
wenzelm@60758
   604
text \<open>Simplifying nested closures\<close>
krauss@33656
   605
wenzelm@63404
   606
lemma rtrancl_trancl_absorb[simp]: "(R\<^sup>*)\<^sup>+ = R\<^sup>*"
wenzelm@63404
   607
  by (simp add: trans_rtrancl)
krauss@33656
   608
wenzelm@63404
   609
lemma trancl_rtrancl_absorb[simp]: "(R\<^sup>+)\<^sup>* = R\<^sup>*"
wenzelm@63404
   610
  by (subst reflcl_trancl[symmetric]) simp
krauss@33656
   611
wenzelm@63404
   612
lemma rtrancl_reflcl_absorb[simp]: "(R\<^sup>*)\<^sup>= = R\<^sup>*"
wenzelm@63404
   613
  by auto
krauss@33656
   614
krauss@33656
   615
wenzelm@61799
   616
text \<open>\<open>Domain\<close> and \<open>Range\<close>\<close>
nipkow@10996
   617
wenzelm@63404
   618
lemma Domain_rtrancl [simp]: "Domain (R\<^sup>*) = UNIV"
wenzelm@11084
   619
  by blast
nipkow@10996
   620
wenzelm@63404
   621
lemma Range_rtrancl [simp]: "Range (R\<^sup>*) = UNIV"
wenzelm@11084
   622
  by blast
nipkow@10996
   623
wenzelm@63404
   624
lemma rtrancl_Un_subset: "(R\<^sup>* \<union> S\<^sup>*) \<subseteq> (R \<union> S)\<^sup>*"
wenzelm@11084
   625
  by (rule rtrancl_Un_rtrancl [THEN subst]) fast
nipkow@10996
   626
wenzelm@63404
   627
lemma in_rtrancl_UnI: "x \<in> R\<^sup>* \<or> x \<in> S\<^sup>* \<Longrightarrow> x \<in> (R \<union> S)\<^sup>*"
wenzelm@11084
   628
  by (blast intro: subsetD [OF rtrancl_Un_subset])
nipkow@10996
   629
wenzelm@63404
   630
lemma trancl_domain [simp]: "Domain (r\<^sup>+) = Domain r"
haftmann@46752
   631
  by (unfold Domain_unfold) (blast dest: tranclD)
nipkow@10996
   632
wenzelm@63404
   633
lemma trancl_range [simp]: "Range (r\<^sup>+) = Range r"
haftmann@46752
   634
  unfolding Domain_converse [symmetric] by (simp add: trancl_converse [symmetric])
nipkow@10996
   635
wenzelm@63404
   636
lemma Not_Domain_rtrancl: "x \<notin> Domain R \<Longrightarrow> (x, y) \<in> R\<^sup>* \<longleftrightarrow> x = y"
wenzelm@12691
   637
  apply auto
wenzelm@26179
   638
  apply (erule rev_mp)
wenzelm@26179
   639
  apply (erule rtrancl_induct)
wenzelm@26179
   640
   apply auto
wenzelm@26179
   641
  done
berghofe@11327
   642
wenzelm@63404
   643
lemma trancl_subset_Field2: "r\<^sup>+ \<subseteq> Field r \<times> Field r"
haftmann@29609
   644
  apply clarify
haftmann@29609
   645
  apply (erule trancl_induct)
haftmann@29609
   646
   apply (auto simp add: Field_def)
haftmann@29609
   647
  done
haftmann@29609
   648
wenzelm@63404
   649
lemma finite_trancl[simp]: "finite (r\<^sup>+) = finite r"
haftmann@29609
   650
  apply auto
haftmann@29609
   651
   prefer 2
haftmann@29609
   652
   apply (rule trancl_subset_Field2 [THEN finite_subset])
haftmann@29609
   653
   apply (rule finite_SigmaI)
haftmann@29609
   654
    prefer 3
haftmann@29609
   655
    apply (blast intro: r_into_trancl' finite_subset)
haftmann@29609
   656
   apply (auto simp add: finite_Field)
haftmann@29609
   657
  done
haftmann@29609
   658
wenzelm@61799
   659
text \<open>More about converse \<open>rtrancl\<close> and \<open>trancl\<close>, should
wenzelm@60758
   660
  be merged with main body.\<close>
kleing@12428
   661
nipkow@14337
   662
lemma single_valued_confluent:
wenzelm@63404
   663
  "single_valued r \<Longrightarrow> (x, y) \<in> r\<^sup>* \<Longrightarrow> (x, z) \<in> r\<^sup>* \<Longrightarrow> (y, z) \<in> r\<^sup>* \<or> (z, y) \<in> r\<^sup>*"
wenzelm@26179
   664
  apply (erule rtrancl_induct)
wenzelm@63612
   665
   apply simp
wenzelm@26179
   666
  apply (erule disjE)
wenzelm@26179
   667
   apply (blast elim:converse_rtranclE dest:single_valuedD)
wenzelm@63612
   668
  apply (blast intro:rtrancl_trans)
wenzelm@26179
   669
  done
nipkow@14337
   670
wenzelm@63404
   671
lemma r_r_into_trancl: "(a, b) \<in> R \<Longrightarrow> (b, c) \<in> R \<Longrightarrow> (a, c) \<in> R\<^sup>+"
kleing@12428
   672
  by (fast intro: trancl_trans)
kleing@12428
   673
wenzelm@63404
   674
lemma trancl_into_trancl: "(a, b) \<in> r\<^sup>+ \<Longrightarrow> (b, c) \<in> r \<Longrightarrow> (a, c) \<in> r\<^sup>+"
wenzelm@63612
   675
  by (induct rule: trancl_induct) (fast intro: r_r_into_trancl trancl_trans)+
kleing@12428
   676
wenzelm@63404
   677
lemma tranclp_rtranclp_tranclp: "r\<^sup>+\<^sup>+ a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>+\<^sup>+ a c"
berghofe@23743
   678
  apply (drule tranclpD)
wenzelm@26179
   679
  apply (elim exE conjE)
berghofe@23743
   680
  apply (drule rtranclp_trans, assumption)
wenzelm@63612
   681
  apply (drule (2) rtranclp_into_tranclp2)
kleing@12428
   682
  done
kleing@12428
   683
berghofe@23743
   684
lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]
berghofe@22262
   685
wenzelm@12691
   686
lemmas transitive_closure_trans [trans] =
wenzelm@12691
   687
  r_r_into_trancl trancl_trans rtrancl_trans
berghofe@23743
   688
  trancl.trancl_into_trancl trancl_into_trancl2
berghofe@23743
   689
  rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
wenzelm@12691
   690
  rtrancl_trancl_trancl trancl_rtrancl_trancl
kleing@12428
   691
berghofe@23743
   692
lemmas transitive_closurep_trans' [trans] =
berghofe@23743
   693
  tranclp_trans rtranclp_trans
berghofe@23743
   694
  tranclp.trancl_into_trancl tranclp_into_tranclp2
berghofe@23743
   695
  rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp
berghofe@23743
   696
  rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp
berghofe@22262
   697
kleing@12428
   698
declare trancl_into_rtrancl [elim]
berghofe@11327
   699
wenzelm@63404
   700
wenzelm@60758
   701
subsection \<open>The power operation on relations\<close>
haftmann@30954
   702
wenzelm@63404
   703
text \<open>\<open>R ^^ n = R O \<dots> O R\<close>, the n-fold composition of \<open>R\<close>\<close>
haftmann@30954
   704
haftmann@30971
   705
overloading
wenzelm@63404
   706
  relpow \<equiv> "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
wenzelm@63404
   707
  relpowp \<equiv> "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
haftmann@30971
   708
begin
haftmann@30954
   709
wenzelm@63404
   710
primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
wenzelm@63612
   711
  where
wenzelm@63612
   712
    "relpow 0 R = Id"
wenzelm@63612
   713
  | "relpow (Suc n) R = (R ^^ n) O R"
haftmann@30954
   714
wenzelm@63404
   715
primrec relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
wenzelm@63612
   716
  where
wenzelm@63612
   717
    "relpowp 0 R = HOL.eq"
wenzelm@63612
   718
  | "relpowp (Suc n) R = (R ^^ n) OO R"
haftmann@47202
   719
haftmann@30971
   720
end
haftmann@30954
   721
haftmann@47202
   722
lemma relpowp_relpow_eq [pred_set_conv]:
wenzelm@63404
   723
  "(\<lambda>x y. (x, y) \<in> R) ^^ n = (\<lambda>x y. (x, y) \<in> R ^^ n)" for R :: "'a rel"
griff@47433
   724
  by (induct n) (simp_all add: relcompp_relcomp_eq)
haftmann@47202
   725
wenzelm@63404
   726
text \<open>For code generation:\<close>
bulwahn@46360
   727
wenzelm@63404
   728
definition relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
wenzelm@63404
   729
  where relpow_code_def [code_abbrev]: "relpow = compow"
bulwahn@46360
   730
wenzelm@63404
   731
definition relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
wenzelm@63404
   732
  where relpowp_code_def [code_abbrev]: "relpowp = compow"
Christian@47492
   733
bulwahn@46360
   734
lemma [code]:
bulwahn@46360
   735
  "relpow (Suc n) R = (relpow n R) O R"
bulwahn@46360
   736
  "relpow 0 R = Id"
bulwahn@46360
   737
  by (simp_all add: relpow_code_def)
bulwahn@46360
   738
Christian@47492
   739
lemma [code]:
Christian@47492
   740
  "relpowp (Suc n) R = (R ^^ n) OO R"
Christian@47492
   741
  "relpowp 0 R = HOL.eq"
Christian@47492
   742
  by (simp_all add: relpowp_code_def)
Christian@47492
   743
bulwahn@46360
   744
hide_const (open) relpow
Christian@47492
   745
hide_const (open) relpowp
bulwahn@46360
   746
wenzelm@63612
   747
lemma relpow_1 [simp]: "R ^^ 1 = R"
wenzelm@63612
   748
  for R :: "('a \<times> 'a) set"
haftmann@30954
   749
  by simp
haftmann@30954
   750
wenzelm@63612
   751
lemma relpowp_1 [simp]: "P ^^ 1 = P"
wenzelm@63612
   752
  for P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
Christian@47492
   753
  by (fact relpow_1 [to_pred])
Christian@47492
   754
wenzelm@63404
   755
lemma relpow_0_I: "(x, x) \<in> R ^^ 0"
haftmann@30954
   756
  by simp
haftmann@30954
   757
wenzelm@63404
   758
lemma relpowp_0_I: "(P ^^ 0) x x"
Christian@47492
   759
  by (fact relpow_0_I [to_pred])
Christian@47492
   760
wenzelm@63404
   761
lemma relpow_Suc_I: "(x, y) \<in>  R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
haftmann@30954
   762
  by auto
haftmann@30954
   763
wenzelm@63404
   764
lemma relpowp_Suc_I: "(P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> (P ^^ Suc n) x z"
Christian@47492
   765
  by (fact relpow_Suc_I [to_pred])
Christian@47492
   766
wenzelm@63404
   767
lemma relpow_Suc_I2: "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
nipkow@44890
   768
  by (induct n arbitrary: z) (simp, fastforce)
haftmann@30954
   769
wenzelm@63404
   770
lemma relpowp_Suc_I2: "P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> (P ^^ Suc n) x z"
Christian@47492
   771
  by (fact relpow_Suc_I2 [to_pred])
Christian@47492
   772
wenzelm@63404
   773
lemma relpow_0_E: "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30954
   774
  by simp
haftmann@30954
   775
wenzelm@63404
   776
lemma relpowp_0_E: "(P ^^ 0) x y \<Longrightarrow> (x = y \<Longrightarrow> Q) \<Longrightarrow> Q"
Christian@47492
   777
  by (fact relpow_0_E [to_pred])
Christian@47492
   778
wenzelm@63404
   779
lemma relpow_Suc_E: "(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
   780
  by auto
haftmann@30954
   781
wenzelm@63404
   782
lemma relpowp_Suc_E: "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. (P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> Q) \<Longrightarrow> Q"
Christian@47492
   783
  by (fact relpow_Suc_E [to_pred])
Christian@47492
   784
bulwahn@46362
   785
lemma relpow_E:
wenzelm@63612
   786
  "(x, z) \<in>  R ^^ n \<Longrightarrow>
wenzelm@63612
   787
    (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P) \<Longrightarrow>
wenzelm@63612
   788
    (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in>  R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30954
   789
  by (cases n) auto
haftmann@30954
   790
Christian@47492
   791
lemma relpowp_E:
wenzelm@63612
   792
  "(P ^^ n) x z \<Longrightarrow>
wenzelm@63612
   793
    (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q) \<Longrightarrow>
wenzelm@63612
   794
    (\<And>y m. n = Suc m \<Longrightarrow> (P ^^ m) x y \<Longrightarrow> P y z \<Longrightarrow> Q) \<Longrightarrow> Q"
Christian@47492
   795
  by (fact relpow_E [to_pred])
Christian@47492
   796
wenzelm@63404
   797
lemma relpow_Suc_D2: "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
wenzelm@63612
   798
  by (induct n arbitrary: x z)
wenzelm@63612
   799
    (blast intro: relpow_0_I relpow_Suc_I elim: relpow_0_E relpow_Suc_E)+
haftmann@30954
   800
wenzelm@63404
   801
lemma relpowp_Suc_D2: "(P ^^ Suc n) x z \<Longrightarrow> \<exists>y. P x y \<and> (P ^^ n) y z"
Christian@47492
   802
  by (fact relpow_Suc_D2 [to_pred])
Christian@47492
   803
wenzelm@63404
   804
lemma relpow_Suc_E2: "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> P) \<Longrightarrow> P"
bulwahn@46362
   805
  by (blast dest: relpow_Suc_D2)
haftmann@30954
   806
wenzelm@63404
   807
lemma relpowp_Suc_E2: "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> Q) \<Longrightarrow> Q"
Christian@47492
   808
  by (fact relpow_Suc_E2 [to_pred])
Christian@47492
   809
wenzelm@63404
   810
lemma relpow_Suc_D2': "\<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
   811
  by (induct n) (simp_all, blast)
haftmann@30954
   812
wenzelm@63404
   813
lemma relpowp_Suc_D2': "\<forall>x y z. (P ^^ n) x y \<and> P y z \<longrightarrow> (\<exists>w. P x w \<and> (P ^^ n) w z)"
Christian@47492
   814
  by (fact relpow_Suc_D2' [to_pred])
Christian@47492
   815
bulwahn@46362
   816
lemma relpow_E2:
wenzelm@63612
   817
  "(x, z) \<in> R ^^ n \<Longrightarrow>
wenzelm@63612
   818
    (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P) \<Longrightarrow>
wenzelm@63612
   819
    (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P) \<Longrightarrow> P"
wenzelm@63612
   820
  apply (cases n)
wenzelm@63612
   821
   apply simp
blanchet@55417
   822
  apply (rename_tac nat)
wenzelm@63612
   823
  apply (cut_tac n=nat and R=R in relpow_Suc_D2')
wenzelm@63612
   824
  apply simp
wenzelm@63612
   825
  apply blast
haftmann@30954
   826
  done
haftmann@30954
   827
Christian@47492
   828
lemma relpowp_E2:
wenzelm@63612
   829
  "(P ^^ n) x z \<Longrightarrow>
wenzelm@63612
   830
    (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q) \<Longrightarrow>
wenzelm@63612
   831
    (\<And>y m. n = Suc m \<Longrightarrow> P x y \<Longrightarrow> (P ^^ m) y z \<Longrightarrow> Q) \<Longrightarrow> Q"
Christian@47492
   832
  by (fact relpow_E2 [to_pred])
Christian@47492
   833
wenzelm@63404
   834
lemma relpow_add: "R ^^ (m + n) = R^^m O R^^n"
haftmann@45976
   835
  by (induct n) auto
nipkow@31351
   836
Christian@47492
   837
lemma relpowp_add: "P ^^ (m + n) = P ^^ m OO P ^^ n"
Christian@47492
   838
  by (fact relpow_add [to_pred])
Christian@47492
   839
bulwahn@46362
   840
lemma relpow_commute: "R O R ^^ n = R ^^ n O R"
wenzelm@63404
   841
  by (induct n) (simp_all add: O_assoc [symmetric])
krauss@31970
   842
Christian@47492
   843
lemma relpowp_commute: "P OO P ^^ n = P ^^ n OO P"
Christian@47492
   844
  by (fact relpow_commute [to_pred])
Christian@47492
   845
wenzelm@63404
   846
lemma relpow_empty: "0 < n \<Longrightarrow> ({} :: ('a \<times> 'a) set) ^^ n = {}"
haftmann@45153
   847
  by (cases n) auto
bulwahn@45116
   848
wenzelm@63404
   849
lemma relpowp_bot: "0 < n \<Longrightarrow> (\<bottom> :: 'a \<Rightarrow> 'a \<Rightarrow> bool) ^^ n = \<bottom>"
Christian@47492
   850
  by (fact relpow_empty [to_pred])
Christian@47492
   851
bulwahn@46362
   852
lemma rtrancl_imp_UN_relpow:
wenzelm@63404
   853
  assumes "p \<in> R\<^sup>*"
haftmann@30954
   854
  shows "p \<in> (\<Union>n. R ^^ n)"
haftmann@30954
   855
proof (cases p)
haftmann@30954
   856
  case (Pair x y)
wenzelm@63404
   857
  with assms have "(x, y) \<in> R\<^sup>*" by simp
wenzelm@63612
   858
  then have "(x, y) \<in> (\<Union>n. R ^^ n)"
wenzelm@63612
   859
  proof induct
wenzelm@63404
   860
    case base
wenzelm@63404
   861
    show ?case by (blast intro: relpow_0_I)
haftmann@30954
   862
  next
wenzelm@63404
   863
    case step
wenzelm@63404
   864
    then show ?case by (blast intro: relpow_Suc_I)
haftmann@30954
   865
  qed
haftmann@30954
   866
  with Pair show ?thesis by simp
haftmann@30954
   867
qed
haftmann@30954
   868
Christian@47492
   869
lemma rtranclp_imp_Sup_relpowp:
wenzelm@63404
   870
  assumes "(P\<^sup>*\<^sup>*) x y"
Christian@47492
   871
  shows "(\<Squnion>n. P ^^ n) x y"
haftmann@61424
   872
  using assms and rtrancl_imp_UN_relpow [of "(x, y)", to_pred] by simp
Christian@47492
   873
bulwahn@46362
   874
lemma relpow_imp_rtrancl:
haftmann@30954
   875
  assumes "p \<in> R ^^ n"
wenzelm@63404
   876
  shows "p \<in> R\<^sup>*"
haftmann@30954
   877
proof (cases p)
haftmann@30954
   878
  case (Pair x y)
haftmann@30954
   879
  with assms have "(x, y) \<in> R ^^ n" by simp
wenzelm@63612
   880
  then have "(x, y) \<in> R\<^sup>*"
wenzelm@63612
   881
  proof (induct n arbitrary: x y)
wenzelm@63404
   882
    case 0
wenzelm@63404
   883
    then show ?case by simp
haftmann@30954
   884
  next
wenzelm@63404
   885
    case Suc
wenzelm@63404
   886
    then show ?case
bulwahn@46362
   887
      by (blast elim: relpow_Suc_E intro: rtrancl_into_rtrancl)
haftmann@30954
   888
  qed
haftmann@30954
   889
  with Pair show ?thesis by simp
haftmann@30954
   890
qed
haftmann@30954
   891
wenzelm@63404
   892
lemma relpowp_imp_rtranclp: "(P ^^ n) x y \<Longrightarrow> (P\<^sup>*\<^sup>*) x y"
wenzelm@63404
   893
  using relpow_imp_rtrancl [of "(x, y)", to_pred] by simp
Christian@47492
   894
wenzelm@63404
   895
lemma rtrancl_is_UN_relpow: "R\<^sup>* = (\<Union>n. R ^^ n)"
bulwahn@46362
   896
  by (blast intro: rtrancl_imp_UN_relpow relpow_imp_rtrancl)
haftmann@30954
   897
wenzelm@63404
   898
lemma rtranclp_is_Sup_relpowp: "P\<^sup>*\<^sup>* = (\<Squnion>n. P ^^ n)"
Christian@47492
   899
  using rtrancl_is_UN_relpow [to_pred, of P] by auto
Christian@47492
   900
wenzelm@63404
   901
lemma rtrancl_power: "p \<in> R\<^sup>* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
bulwahn@46362
   902
  by (simp add: rtrancl_is_UN_relpow)
haftmann@30954
   903
wenzelm@63404
   904
lemma rtranclp_power: "(P\<^sup>*\<^sup>*) x y \<longleftrightarrow> (\<exists>n. (P ^^ n) x y)"
Christian@47492
   905
  by (simp add: rtranclp_is_Sup_relpowp)
Christian@47492
   906
wenzelm@63404
   907
lemma trancl_power: "p \<in> R\<^sup>+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
haftmann@30954
   908
  apply (cases p)
haftmann@30954
   909
  apply simp
haftmann@30954
   910
  apply (rule iffI)
haftmann@30954
   911
   apply (drule tranclD2)
bulwahn@46362
   912
   apply (clarsimp simp: rtrancl_is_UN_relpow)
haftmann@62343
   913
   apply (rule_tac x="Suc x" in exI)
griff@47433
   914
   apply (clarsimp simp: relcomp_unfold)
nipkow@44890
   915
   apply fastforce
haftmann@30954
   916
  apply clarsimp
wenzelm@63612
   917
  apply (case_tac n)
wenzelm@63612
   918
   apply simp
haftmann@30954
   919
  apply clarsimp
bulwahn@46362
   920
  apply (drule relpow_imp_rtrancl)
wenzelm@63612
   921
  apply (drule rtrancl_into_trancl1)
wenzelm@63612
   922
   apply auto
haftmann@30954
   923
  done
haftmann@30954
   924
wenzelm@63404
   925
lemma tranclp_power: "(P\<^sup>+\<^sup>+) x y \<longleftrightarrow> (\<exists>n > 0. (P ^^ n) x y)"
Christian@47492
   926
  using trancl_power [to_pred, of P "(x, y)"] by simp
Christian@47492
   927
wenzelm@63404
   928
lemma rtrancl_imp_relpow: "p \<in> R\<^sup>* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
bulwahn@46362
   929
  by (auto dest: rtrancl_imp_UN_relpow)
haftmann@30954
   930
wenzelm@63404
   931
lemma rtranclp_imp_relpowp: "(P\<^sup>*\<^sup>*) x y \<Longrightarrow> \<exists>n. (P ^^ n) x y"
Christian@47492
   932
  by (auto dest: rtranclp_imp_Sup_relpowp)
Christian@47492
   933
wenzelm@63404
   934
text \<open>By Sternagel/Thiemann:\<close>
wenzelm@63404
   935
lemma relpow_fun_conv: "(a, b) \<in> R ^^ n \<longleftrightarrow> (\<exists>f. f 0 = a \<and> f n = b \<and> (\<forall>i<n. (f i, f (Suc i)) \<in> R))"
nipkow@41987
   936
proof (induct n arbitrary: b)
wenzelm@63404
   937
  case 0
wenzelm@63404
   938
  show ?case by auto
nipkow@41987
   939
next
nipkow@41987
   940
  case (Suc n)
nipkow@41987
   941
  show ?case
griff@47433
   942
  proof (simp add: relcomp_unfold Suc)
wenzelm@63404
   943
    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) \<longleftrightarrow>
wenzelm@63404
   944
      (\<exists>f. f 0 = a \<and> f(Suc n) = b \<and> (\<forall>i<Suc n. (f i, f (Suc i)) \<in> R))"
nipkow@41987
   945
    (is "?l = ?r")
nipkow@41987
   946
    proof
nipkow@41987
   947
      assume ?l
wenzelm@63404
   948
      then obtain c f
wenzelm@63404
   949
        where 1: "f 0 = a"  "f n = c"  "\<And>i. i < n \<Longrightarrow> (f i, f (Suc i)) \<in> R"  "(c,b) \<in> R"
wenzelm@63404
   950
        by auto
nipkow@41987
   951
      let ?g = "\<lambda> m. if m = Suc n then b else f m"
wenzelm@63404
   952
      show ?r by (rule exI[of _ ?g]) (simp add: 1)
nipkow@41987
   953
    next
nipkow@41987
   954
      assume ?r
wenzelm@63404
   955
      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"
wenzelm@63404
   956
        by auto
nipkow@41987
   957
      show ?l by (rule exI[of _ "f n"], rule conjI, rule exI[of _ f], insert 1, auto)
nipkow@41987
   958
    qed
nipkow@41987
   959
  qed
nipkow@41987
   960
qed
nipkow@41987
   961
wenzelm@63404
   962
lemma relpowp_fun_conv: "(P ^^ n) x y \<longleftrightarrow> (\<exists>f. f 0 = x \<and> f n = y \<and> (\<forall>i<n. P (f i) (f (Suc i))))"
Christian@47492
   963
  by (fact relpow_fun_conv [to_pred])
Christian@47492
   964
bulwahn@46362
   965
lemma relpow_finite_bounded1:
wenzelm@63404
   966
  fixes R :: "('a \<times> 'a) set"
wenzelm@63404
   967
  assumes "finite R" and "k > 0"
wenzelm@63612
   968
  shows "R^^k \<subseteq> (\<Union>n\<in>{n. 0 < n \<and> n \<le> card R}. R^^n)"
wenzelm@63612
   969
    (is "_ \<subseteq> ?r")
wenzelm@63404
   970
proof -
wenzelm@63404
   971
  have "(a, b) \<in> R^^(Suc k) \<Longrightarrow> \<exists>n. 0 < n \<and> n \<le> card R \<and> (a, b) \<in> R^^n" for a b k
wenzelm@63404
   972
  proof (induct k arbitrary: b)
wenzelm@63404
   973
    case 0
wenzelm@63404
   974
    then have "R \<noteq> {}" by auto
wenzelm@63404
   975
    with card_0_eq[OF \<open>finite R\<close>] have "card R \<ge> Suc 0" by auto
wenzelm@63404
   976
    then show ?case using 0 by force
wenzelm@63404
   977
  next
wenzelm@63404
   978
    case (Suc k)
wenzelm@63404
   979
    then obtain a' where "(a, a') \<in> R^^(Suc k)" and "(a', b) \<in> R"
wenzelm@63404
   980
      by auto
wenzelm@63404
   981
    from Suc(1)[OF \<open>(a, a') \<in> R^^(Suc k)\<close>] obtain n where "n \<le> card R" and "(a, a') \<in> R ^^ n"
wenzelm@63404
   982
      by auto
wenzelm@63404
   983
    have "(a, b) \<in> R^^(Suc n)"
wenzelm@63404
   984
      using \<open>(a, a') \<in> R^^n\<close> and \<open>(a', b)\<in> R\<close> by auto
wenzelm@63404
   985
    from \<open>n \<le> card R\<close> consider "n < card R" | "n = card R" by force
wenzelm@63404
   986
    then show ?case
wenzelm@63404
   987
    proof cases
wenzelm@63404
   988
      case 1
wenzelm@63404
   989
      then show ?thesis
wenzelm@63404
   990
        using \<open>(a, b) \<in> R^^(Suc n)\<close> Suc_leI[OF \<open>n < card R\<close>] by blast
nipkow@41987
   991
    next
wenzelm@63404
   992
      case 2
wenzelm@63404
   993
      from \<open>(a, b) \<in> R ^^ (Suc n)\<close> [unfolded relpow_fun_conv]
wenzelm@63404
   994
      obtain f where "f 0 = a" and "f (Suc n) = b"
wenzelm@63404
   995
        and steps: "\<And>i. i \<le> n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto
wenzelm@63404
   996
      let ?p = "\<lambda>i. (f i, f(Suc i))"
wenzelm@63404
   997
      let ?N = "{i. i \<le> n}"
wenzelm@63404
   998
      have "?p ` ?N \<subseteq> R"
wenzelm@63404
   999
        using steps by auto
wenzelm@63404
  1000
      from card_mono[OF assms(1) this] have "card (?p ` ?N) \<le> card R" .
wenzelm@63404
  1001
      also have "\<dots> < card ?N"
wenzelm@63404
  1002
        using \<open>n = card R\<close> by simp
wenzelm@63404
  1003
      finally have "\<not> inj_on ?p ?N"
wenzelm@63404
  1004
        by (rule pigeonhole)
wenzelm@63404
  1005
      then obtain i j where i: "i \<le> n" and j: "j \<le> n" and ij: "i \<noteq> j" and pij: "?p i = ?p j"
wenzelm@63404
  1006
        by (auto simp: inj_on_def)
wenzelm@63404
  1007
      let ?i = "min i j"
wenzelm@63404
  1008
      let ?j = "max i j"
wenzelm@63404
  1009
      have i: "?i \<le> n" and j: "?j \<le> n" and pij: "?p ?i = ?p ?j" and ij: "?i < ?j"
wenzelm@63404
  1010
        using i j ij pij unfolding min_def max_def by auto
wenzelm@63404
  1011
      from i j pij ij obtain i j where i: "i \<le> n" and j: "j \<le> n" and ij: "i < j"
wenzelm@63404
  1012
        and pij: "?p i = ?p j"
wenzelm@63404
  1013
        by blast
wenzelm@63404
  1014
      let ?g = "\<lambda>l. if l \<le> i then f l else f (l + (j - i))"
wenzelm@63404
  1015
      let ?n = "Suc (n - (j - i))"
wenzelm@63404
  1016
      have abl: "(a, b) \<in> R ^^ ?n"
wenzelm@63404
  1017
        unfolding relpow_fun_conv
wenzelm@63404
  1018
      proof (rule exI[of _ ?g], intro conjI impI allI)
wenzelm@63404
  1019
        show "?g ?n = b"
wenzelm@63404
  1020
          using \<open>f(Suc n) = b\<close> j ij by auto
wenzelm@63404
  1021
      next
wenzelm@63404
  1022
        fix k
wenzelm@63404
  1023
        assume "k < ?n"
wenzelm@63404
  1024
        show "(?g k, ?g (Suc k)) \<in> R"
wenzelm@63404
  1025
        proof (cases "k < i")
wenzelm@63404
  1026
          case True
wenzelm@63404
  1027
          with i have "k \<le> n"
wenzelm@63404
  1028
            by auto
wenzelm@63404
  1029
          from steps[OF this] show ?thesis
wenzelm@63404
  1030
            using True by simp
nipkow@41987
  1031
        next
wenzelm@63404
  1032
          case False
wenzelm@63404
  1033
          then have "i \<le> k" by auto
wenzelm@63404
  1034
          show ?thesis
wenzelm@63404
  1035
          proof (cases "k = i")
nipkow@41987
  1036
            case True
wenzelm@63404
  1037
            then show ?thesis
wenzelm@63404
  1038
              using ij pij steps[OF i] by simp
nipkow@41987
  1039
          next
nipkow@41987
  1040
            case False
wenzelm@63404
  1041
            with \<open>i \<le> k\<close> have "i < k" by auto
wenzelm@63404
  1042
            then have small: "k + (j - i) \<le> n"
wenzelm@63404
  1043
              using \<open>k<?n\<close> by arith
nipkow@41987
  1044
            show ?thesis
wenzelm@63404
  1045
              using steps[OF small] \<open>i<k\<close> by auto
nipkow@41987
  1046
          qed
wenzelm@63404
  1047
        qed
wenzelm@63404
  1048
      qed (simp add: \<open>f 0 = a\<close>)
wenzelm@63404
  1049
      moreover have "?n \<le> n"
wenzelm@63404
  1050
        using i j ij by arith
wenzelm@63404
  1051
      ultimately show ?thesis
wenzelm@63404
  1052
        using \<open>n = card R\<close> by blast
nipkow@41987
  1053
    qed
wenzelm@63404
  1054
  qed
wenzelm@63404
  1055
  then show ?thesis
wenzelm@63404
  1056
    using gr0_implies_Suc[OF \<open>k > 0\<close>] by auto
nipkow@41987
  1057
qed
nipkow@41987
  1058
bulwahn@46362
  1059
lemma relpow_finite_bounded:
wenzelm@63404
  1060
  fixes R :: "('a \<times> 'a) set"
wenzelm@63404
  1061
  assumes "finite R"
wenzelm@63404
  1062
  shows "R^^k \<subseteq> (UN n:{n. n \<le> card R}. R^^n)"
wenzelm@63404
  1063
  apply (cases k)
wenzelm@63404
  1064
   apply force
wenzelm@63612
  1065
  apply (use relpow_finite_bounded1[OF assms, of k] in auto)
wenzelm@63404
  1066
  done
nipkow@41987
  1067
wenzelm@63404
  1068
lemma rtrancl_finite_eq_relpow: "finite R \<Longrightarrow> R\<^sup>* = (\<Union>n\<in>{n. n \<le> card R}. R^^n)"
wenzelm@63404
  1069
  by (fastforce simp: rtrancl_power dest: relpow_finite_bounded)
nipkow@41987
  1070
wenzelm@63404
  1071
lemma trancl_finite_eq_relpow: "finite R \<Longrightarrow> R\<^sup>+ = (\<Union>n\<in>{n. 0 < n \<and> n \<le> card R}. R^^n)"
wenzelm@63404
  1072
  apply (auto simp: trancl_power)
wenzelm@63404
  1073
  apply (auto dest: relpow_finite_bounded1)
wenzelm@63404
  1074
  done
nipkow@41987
  1075
griff@47433
  1076
lemma finite_relcomp[simp,intro]:
wenzelm@63404
  1077
  assumes "finite R" and "finite S"
wenzelm@63404
  1078
  shows "finite (R O S)"
nipkow@41987
  1079
proof-
haftmann@62343
  1080
  have "R O S = (\<Union>(x, y)\<in>R. \<Union>(u, v)\<in>S. if u = y then {(x, v)} else {})"
haftmann@62343
  1081
    by (force simp add: split_def image_constant_conv split: if_splits)
wenzelm@63404
  1082
  then show ?thesis
wenzelm@63404
  1083
    using assms by clarsimp
nipkow@41987
  1084
qed
nipkow@41987
  1085
wenzelm@63404
  1086
lemma finite_relpow [simp, intro]:
wenzelm@63404
  1087
  fixes R :: "('a \<times> 'a) set"
wenzelm@63404
  1088
  assumes "finite R"
wenzelm@63404
  1089
  shows "n > 0 \<Longrightarrow> finite (R^^n)"
wenzelm@63612
  1090
proof (induct n)
wenzelm@63612
  1091
  case 0
wenzelm@63612
  1092
  then show ?case by simp
wenzelm@63612
  1093
next
wenzelm@63612
  1094
  case (Suc n)
wenzelm@63612
  1095
  then show ?case by (cases n) (use assms in simp_all)
wenzelm@63612
  1096
qed
nipkow@41987
  1097
bulwahn@46362
  1098
lemma single_valued_relpow:
wenzelm@63404
  1099
  fixes R :: "('a \<times> 'a) set"
haftmann@30954
  1100
  shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
wenzelm@63612
  1101
proof (induct n arbitrary: R)
wenzelm@63612
  1102
  case 0
wenzelm@63612
  1103
  then show ?case by simp
wenzelm@63612
  1104
next
wenzelm@63612
  1105
  case (Suc n)
wenzelm@63612
  1106
  show ?case
wenzelm@63612
  1107
    by (rule single_valuedI)
wenzelm@63612
  1108
      (use Suc in \<open>fast dest: single_valuedD elim: relpow_Suc_E\<close>)
wenzelm@63612
  1109
qed
paulson@15551
  1110
haftmann@45140
  1111
wenzelm@60758
  1112
subsection \<open>Bounded transitive closure\<close>
haftmann@45140
  1113
haftmann@45140
  1114
definition ntrancl :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
wenzelm@63404
  1115
  where "ntrancl n R = (\<Union>i\<in>{i. 0 < i \<and> i \<le> Suc n}. R ^^ i)"
haftmann@45140
  1116
wenzelm@63404
  1117
lemma ntrancl_Zero [simp, code]: "ntrancl 0 R = R"
haftmann@45140
  1118
proof
haftmann@45140
  1119
  show "R \<subseteq> ntrancl 0 R"
haftmann@45140
  1120
    unfolding ntrancl_def by fastforce
wenzelm@63404
  1121
  have "0 < i \<and> i \<le> Suc 0 \<longleftrightarrow> i = 1" for i
wenzelm@63404
  1122
    by auto
wenzelm@63404
  1123
  then show "ntrancl 0 R \<le> R"
haftmann@45140
  1124
    unfolding ntrancl_def by auto
haftmann@45140
  1125
qed
haftmann@45140
  1126
wenzelm@63404
  1127
lemma ntrancl_Suc [simp]: "ntrancl (Suc n) R = ntrancl n R O (Id \<union> R)"
haftmann@45140
  1128
proof
wenzelm@63612
  1129
  have "(a, b) \<in> ntrancl n R O (Id \<union> R)" if "(a, b) \<in> ntrancl (Suc n) R" for a b
wenzelm@63612
  1130
  proof -
wenzelm@63612
  1131
    from that obtain i where "0 < i" "i \<le> Suc (Suc n)" "(a, b) \<in> R ^^ i"
haftmann@45140
  1132
      unfolding ntrancl_def by auto
wenzelm@63612
  1133
    show ?thesis
haftmann@45140
  1134
    proof (cases "i = 1")
haftmann@45140
  1135
      case True
wenzelm@60758
  1136
      from this \<open>(a, b) \<in> R ^^ i\<close> show ?thesis
wenzelm@63612
  1137
        by (auto simp: ntrancl_def)
haftmann@45140
  1138
    next
haftmann@45140
  1139
      case False
wenzelm@63612
  1140
      with \<open>0 < i\<close> obtain j where j: "i = Suc j" "0 < j"
haftmann@45140
  1141
        by (cases i) auto
wenzelm@63612
  1142
      with \<open>(a, b) \<in> R ^^ i\<close> obtain c where c1: "(a, c) \<in> R ^^ j" and c2: "(c, b) \<in> R"
haftmann@45140
  1143
        by auto
wenzelm@60758
  1144
      from c1 j \<open>i \<le> Suc (Suc n)\<close> have "(a, c) \<in> ntrancl n R"
wenzelm@63612
  1145
        by (fastforce simp: ntrancl_def)
wenzelm@63612
  1146
      with c2 show ?thesis by fastforce
haftmann@45140
  1147
    qed
wenzelm@63612
  1148
  qed
wenzelm@63404
  1149
  then show "ntrancl (Suc n) R \<subseteq> ntrancl n R O (Id \<union> R)"
haftmann@45140
  1150
    by auto
haftmann@45140
  1151
  show "ntrancl n R O (Id \<union> R) \<subseteq> ntrancl (Suc n) R"
wenzelm@63612
  1152
    by (fastforce simp: ntrancl_def)
haftmann@45140
  1153
qed
haftmann@45140
  1154
wenzelm@63404
  1155
lemma [code]: "ntrancl (Suc n) r = (let r' = ntrancl n r in r' \<union> r' O r)"
wenzelm@63404
  1156
  by (auto simp: Let_def)
bulwahn@46347
  1157
wenzelm@63404
  1158
lemma finite_trancl_ntranl: "finite R \<Longrightarrow> trancl R = ntrancl (card R - 1) R"
bulwahn@46362
  1159
  by (cases "card R") (auto simp add: trancl_finite_eq_relpow relpow_empty ntrancl_def)
haftmann@45140
  1160
haftmann@45140
  1161
wenzelm@60758
  1162
subsection \<open>Acyclic relations\<close>
haftmann@45139
  1163
wenzelm@63404
  1164
definition acyclic :: "('a \<times> 'a) set \<Rightarrow> bool"
wenzelm@63404
  1165
  where "acyclic r \<longleftrightarrow> (\<forall>x. (x,x) \<notin> r\<^sup>+)"
haftmann@45139
  1166
wenzelm@63404
  1167
abbreviation acyclicP :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> bool"
wenzelm@63404
  1168
  where "acyclicP r \<equiv> acyclic {(x, y). r x y}"
haftmann@45139
  1169
wenzelm@63404
  1170
lemma acyclic_irrefl [code]: "acyclic r \<longleftrightarrow> irrefl (r\<^sup>+)"
haftmann@45139
  1171
  by (simp add: acyclic_def irrefl_def)
haftmann@45139
  1172
wenzelm@63404
  1173
lemma acyclicI: "\<forall>x. (x, x) \<notin> r\<^sup>+ \<Longrightarrow> acyclic r"
haftmann@45139
  1174
  by (simp add: acyclic_def)
haftmann@45139
  1175
hoelzl@54412
  1176
lemma (in order) acyclicI_order:
hoelzl@54412
  1177
  assumes *: "\<And>a b. (a, b) \<in> r \<Longrightarrow> f b < f a"
hoelzl@54412
  1178
  shows "acyclic r"
hoelzl@54412
  1179
proof -
wenzelm@63404
  1180
  have "f b < f a" if "(a, b) \<in> r\<^sup>+" for a b
wenzelm@63404
  1181
    using that by induct (auto intro: * less_trans)
hoelzl@54412
  1182
  then show ?thesis
hoelzl@54412
  1183
    by (auto intro!: acyclicI)
hoelzl@54412
  1184
qed
hoelzl@54412
  1185
wenzelm@63404
  1186
lemma acyclic_insert [iff]: "acyclic (insert (y, x) r) \<longleftrightarrow> acyclic r \<and> (x, y) \<notin> r\<^sup>*"
wenzelm@63612
  1187
  by (simp add: acyclic_def trancl_insert) (blast intro: rtrancl_trans)
haftmann@45139
  1188
wenzelm@63404
  1189
lemma acyclic_converse [iff]: "acyclic (r\<inverse>) \<longleftrightarrow> acyclic r"
wenzelm@63404
  1190
  by (simp add: acyclic_def trancl_converse)
haftmann@45139
  1191
haftmann@45139
  1192
lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
haftmann@45139
  1193
wenzelm@63404
  1194
lemma acyclic_impl_antisym_rtrancl: "acyclic r \<Longrightarrow> antisym (r\<^sup>*)"
wenzelm@63612
  1195
  by (simp add: acyclic_def antisym_def)
wenzelm@63612
  1196
    (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
haftmann@45139
  1197
haftmann@45139
  1198
(* Other direction:
haftmann@45139
  1199
acyclic = no loops
haftmann@45139
  1200
antisym = only self loops
wenzelm@63404
  1201
Goalw [acyclic_def,antisym_def] "antisym( r\<^sup>* ) \<Longrightarrow> acyclic(r - Id)
wenzelm@63404
  1202
\<Longrightarrow> antisym( r\<^sup>* ) = acyclic(r - Id)";
haftmann@45139
  1203
*)
haftmann@45139
  1204
wenzelm@63404
  1205
lemma acyclic_subset: "acyclic s \<Longrightarrow> r \<subseteq> s \<Longrightarrow> acyclic r"
wenzelm@63404
  1206
  unfolding acyclic_def by (blast intro: trancl_mono)
haftmann@45139
  1207
haftmann@45139
  1208
wenzelm@60758
  1209
subsection \<open>Setup of transitivity reasoner\<close>
ballarin@15076
  1210
wenzelm@60758
  1211
ML \<open>
wenzelm@32215
  1212
structure Trancl_Tac = Trancl_Tac
wenzelm@32215
  1213
(
wenzelm@32215
  1214
  val r_into_trancl = @{thm trancl.r_into_trancl};
wenzelm@32215
  1215
  val trancl_trans  = @{thm trancl_trans};
wenzelm@32215
  1216
  val rtrancl_refl = @{thm rtrancl.rtrancl_refl};
wenzelm@32215
  1217
  val r_into_rtrancl = @{thm r_into_rtrancl};
wenzelm@32215
  1218
  val trancl_into_rtrancl = @{thm trancl_into_rtrancl};
wenzelm@32215
  1219
  val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl};
wenzelm@32215
  1220
  val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl};
wenzelm@32215
  1221
  val rtrancl_trans = @{thm rtrancl_trans};
ballarin@15096
  1222
berghofe@30107
  1223
  fun decomp (@{const Trueprop} $ t) =
wenzelm@63404
  1224
        let
wenzelm@63404
  1225
          fun dec (Const (@{const_name Set.member}, _) $ (Const (@{const_name Pair}, _) $ a $ b) $ rel) =
wenzelm@63404
  1226
              let
wenzelm@63404
  1227
                fun decr (Const (@{const_name rtrancl}, _ ) $ r) = (r,"r*")
wenzelm@63404
  1228
                  | decr (Const (@{const_name trancl}, _ ) $ r)  = (r,"r+")
wenzelm@63404
  1229
                  | decr r = (r,"r");
wenzelm@63404
  1230
                val (rel,r) = decr (Envir.beta_eta_contract rel);
wenzelm@63404
  1231
              in SOME (a,b,rel,r) end
wenzelm@63404
  1232
          | dec _ =  NONE
wenzelm@63404
  1233
        in dec t end
berghofe@30107
  1234
    | decomp _ = NONE;
wenzelm@32215
  1235
);
ballarin@15076
  1236
wenzelm@32215
  1237
structure Tranclp_Tac = Trancl_Tac
wenzelm@32215
  1238
(
wenzelm@32215
  1239
  val r_into_trancl = @{thm tranclp.r_into_trancl};
wenzelm@32215
  1240
  val trancl_trans  = @{thm tranclp_trans};
wenzelm@32215
  1241
  val rtrancl_refl = @{thm rtranclp.rtrancl_refl};
wenzelm@32215
  1242
  val r_into_rtrancl = @{thm r_into_rtranclp};
wenzelm@32215
  1243
  val trancl_into_rtrancl = @{thm tranclp_into_rtranclp};
wenzelm@32215
  1244
  val rtrancl_trancl_trancl = @{thm rtranclp_tranclp_tranclp};
wenzelm@32215
  1245
  val trancl_rtrancl_trancl = @{thm tranclp_rtranclp_tranclp};
wenzelm@32215
  1246
  val rtrancl_trans = @{thm rtranclp_trans};
berghofe@22262
  1247
berghofe@30107
  1248
  fun decomp (@{const Trueprop} $ t) =
wenzelm@63404
  1249
        let
wenzelm@63404
  1250
          fun dec (rel $ a $ b) =
wenzelm@63404
  1251
            let
wenzelm@63404
  1252
              fun decr (Const (@{const_name rtranclp}, _ ) $ r) = (r,"r*")
wenzelm@63404
  1253
                | decr (Const (@{const_name tranclp}, _ ) $ r)  = (r,"r+")
wenzelm@63404
  1254
                | decr r = (r,"r");
wenzelm@63404
  1255
              val (rel,r) = decr rel;
wenzelm@63404
  1256
            in SOME (a, b, rel, r) end
wenzelm@63404
  1257
          | dec _ =  NONE
wenzelm@63404
  1258
        in dec t end
berghofe@30107
  1259
    | decomp _ = NONE;
wenzelm@32215
  1260
);
wenzelm@60758
  1261
\<close>
berghofe@22262
  1262
wenzelm@60758
  1263
setup \<open>
wenzelm@51717
  1264
  map_theory_simpset (fn ctxt => ctxt
wenzelm@51717
  1265
    addSolver (mk_solver "Trancl" Trancl_Tac.trancl_tac)
wenzelm@51717
  1266
    addSolver (mk_solver "Rtrancl" Trancl_Tac.rtrancl_tac)
wenzelm@51717
  1267
    addSolver (mk_solver "Tranclp" Tranclp_Tac.trancl_tac)
wenzelm@51717
  1268
    addSolver (mk_solver "Rtranclp" Tranclp_Tac.rtrancl_tac))
wenzelm@60758
  1269
\<close>
ballarin@15076
  1270
wenzelm@32215
  1271
wenzelm@60758
  1272
text \<open>Optional methods.\<close>
ballarin@15076
  1273
ballarin@15076
  1274
method_setup trancl =
wenzelm@60758
  1275
  \<open>Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.trancl_tac)\<close>
wenzelm@60758
  1276
  \<open>simple transitivity reasoner\<close>
ballarin@15076
  1277
method_setup rtrancl =
wenzelm@60758
  1278
  \<open>Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.rtrancl_tac)\<close>
wenzelm@60758
  1279
  \<open>simple transitivity reasoner\<close>
berghofe@22262
  1280
method_setup tranclp =
wenzelm@60758
  1281
  \<open>Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.trancl_tac)\<close>
wenzelm@60758
  1282
  \<open>simple transitivity reasoner (predicate version)\<close>
berghofe@22262
  1283
method_setup rtranclp =
wenzelm@60758
  1284
  \<open>Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.rtrancl_tac)\<close>
wenzelm@60758
  1285
  \<open>simple transitivity reasoner (predicate version)\<close>
ballarin@15076
  1286
nipkow@10213
  1287
end