src/HOL/Transitive_Closure.thy
author paulson <lp15@cam.ac.uk>
Mon Feb 22 14:37:56 2016 +0000 (2016-02-22)
changeset 62379 340738057c8c
parent 62343 24106dc44def
child 62957 a9c40cf517d1
permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
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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{maximum priority}, forcing their
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  operands to be atomic.
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\<close>
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context
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  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^=)"
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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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lemma reflclp_idemp [simp]: "(P^==)^==  =  P^=="
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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]: "!!p. p \<in> r ==> p \<in> r^*"
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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 ==> r^** 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 ==> r^** \<le> s^**"
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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]:
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   "(\<And>a b. x a b \<longrightarrow> y a b) \<Longrightarrow> x^** a b \<longrightarrow> y^** 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^** 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 \<open>Transitivity of transitive closure.\<close>
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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 \<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 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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  \<comment> \<open>elimination of \<open>rtrancl\<close> -- by induction on a special formula\<close>
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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 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:
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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 \<open>
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  \medskip More @{term "r^*"} equations and inclusions.
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\<close>
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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_reflclp [simp]: "(R^==)^** = R^**"
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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)^* = 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)
nipkow@31690
   306
done
nipkow@31690
   307
nipkow@31690
   308
lemma rtrancl_Un_separator_converseE:
nipkow@31690
   309
  "(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
   310
apply (induct rule:converse_rtrancl_induct)
nipkow@31690
   311
 apply blast
nipkow@31690
   312
apply (blast intro:rtrancl_trans)
nipkow@31690
   313
done
nipkow@31690
   314
haftmann@34970
   315
lemma Image_closed_trancl:
haftmann@34970
   316
  assumes "r `` X \<subseteq> X" shows "r\<^sup>* `` X = X"
haftmann@34970
   317
proof -
haftmann@34970
   318
  from assms have **: "{y. \<exists>x\<in>X. (x, y) \<in> r} \<subseteq> X" by auto
haftmann@34970
   319
  have "\<And>x y. (y, x) \<in> r\<^sup>* \<Longrightarrow> y \<in> X \<Longrightarrow> x \<in> X"
haftmann@34970
   320
  proof -
haftmann@34970
   321
    fix x y
haftmann@34970
   322
    assume *: "y \<in> X"
haftmann@34970
   323
    assume "(y, x) \<in> r\<^sup>*"
haftmann@34970
   324
    then show "x \<in> X"
haftmann@34970
   325
    proof induct
haftmann@34970
   326
      case base show ?case by (fact *)
haftmann@34970
   327
    next
haftmann@34970
   328
      case step with ** show ?case by auto
haftmann@34970
   329
    qed
haftmann@34970
   330
  qed
haftmann@34970
   331
  then show ?thesis by auto
haftmann@34970
   332
qed
haftmann@34970
   333
wenzelm@12691
   334
wenzelm@60758
   335
subsection \<open>Transitive closure\<close>
wenzelm@10331
   336
berghofe@13704
   337
lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
berghofe@23743
   338
  apply (simp add: split_tupled_all)
berghofe@13704
   339
  apply (erule trancl.induct)
wenzelm@26179
   340
   apply (iprover dest: subsetD)+
wenzelm@12691
   341
  done
wenzelm@12691
   342
nipkow@57284
   343
lemma r_into_trancl': "!!p. p : r ==> p : r^+"
berghofe@13704
   344
  by (simp only: split_tupled_all) (erule r_into_trancl)
berghofe@13704
   345
wenzelm@60758
   346
text \<open>
wenzelm@61799
   347
  \medskip Conversions between \<open>trancl\<close> and \<open>rtrancl\<close>.
wenzelm@60758
   348
\<close>
wenzelm@12691
   349
berghofe@23743
   350
lemma tranclp_into_rtranclp: "r^++ a b ==> r^** a b"
berghofe@23743
   351
  by (erule tranclp.induct) iprover+
wenzelm@12691
   352
berghofe@23743
   353
lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]
berghofe@22262
   354
berghofe@23743
   355
lemma rtranclp_into_tranclp1: assumes r: "r^** a b"
berghofe@22262
   356
  shows "!!c. r b c ==> r^++ a c" using r
nipkow@17589
   357
  by induct iprover+
wenzelm@12691
   358
berghofe@23743
   359
lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]
berghofe@22262
   360
berghofe@23743
   361
lemma rtranclp_into_tranclp2: "[| r a b; r^** b c |] ==> r^++ a c"
wenzelm@61799
   362
  \<comment> \<open>intro rule from \<open>r\<close> and \<open>rtrancl\<close>\<close>
wenzelm@26179
   363
  apply (erule rtranclp.cases)
wenzelm@26179
   364
   apply iprover
berghofe@23743
   365
  apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1])
wenzelm@26179
   366
    apply (simp | rule r_into_rtranclp)+
wenzelm@12691
   367
  done
wenzelm@12691
   368
berghofe@23743
   369
lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]
berghofe@22262
   370
wenzelm@61799
   371
text \<open>Nice induction rule for \<open>trancl\<close>\<close>
wenzelm@26179
   372
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
berghofe@34909
   373
  assumes a: "r^++ a b"
berghofe@22262
   374
  and cases: "!!y. r a y ==> P y"
berghofe@22262
   375
    "!!y z. r^++ a y ==> r y z ==> P y ==> P z"
berghofe@34909
   376
  shows "P b" using a
berghofe@34909
   377
  by (induct x\<equiv>a b) (iprover intro: cases)+
wenzelm@12691
   378
berghofe@25425
   379
lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]
berghofe@22262
   380
berghofe@23743
   381
lemmas tranclp_induct2 =
wenzelm@26179
   382
  tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
wenzelm@26179
   383
    consumes 1, case_names base step]
berghofe@22262
   384
paulson@22172
   385
lemmas trancl_induct2 =
wenzelm@26179
   386
  trancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
wenzelm@26179
   387
    consumes 1, case_names base step]
paulson@22172
   388
berghofe@23743
   389
lemma tranclp_trans_induct:
berghofe@22262
   390
  assumes major: "r^++ x y"
berghofe@22262
   391
    and cases: "!!x y. r x y ==> P x y"
berghofe@22262
   392
      "!!x y z. [| r^++ x y; P x y; r^++ y z; P y z |] ==> P x z"
wenzelm@18372
   393
  shows "P x y"
wenzelm@61799
   394
  \<comment> \<open>Another induction rule for trancl, incorporating transitivity\<close>
berghofe@23743
   395
  by (iprover intro: major [THEN tranclp_induct] cases)
wenzelm@12691
   396
berghofe@23743
   397
lemmas trancl_trans_induct = tranclp_trans_induct [to_set]
berghofe@23743
   398
wenzelm@26174
   399
lemma tranclE [cases set: trancl]:
wenzelm@26174
   400
  assumes "(a, b) : r^+"
wenzelm@26174
   401
  obtains
wenzelm@26174
   402
    (base) "(a, b) : r"
wenzelm@26174
   403
  | (step) c where "(a, c) : r^+" and "(c, b) : r"
wenzelm@26174
   404
  using assms by cases simp_all
wenzelm@10980
   405
krauss@32235
   406
lemma trancl_Int_subset: "[| r \<subseteq> s; (r^+ \<inter> s) O r \<subseteq> s|] ==> r^+ \<subseteq> s"
paulson@22080
   407
  apply (rule subsetI)
haftmann@61032
   408
  apply auto
wenzelm@26179
   409
  apply (erule trancl_induct)
haftmann@61032
   410
  apply auto
paulson@22080
   411
  done
paulson@22080
   412
krauss@32235
   413
lemma trancl_unfold: "r^+ = r Un r^+ O r"
paulson@15551
   414
  by (auto intro: trancl_into_trancl elim: tranclE)
paulson@15551
   415
wenzelm@60758
   416
text \<open>Transitivity of @{term "r^+"}\<close>
wenzelm@26179
   417
lemma trans_trancl [simp]: "trans (r^+)"
berghofe@13704
   418
proof (rule transI)
berghofe@13704
   419
  fix x y z
wenzelm@26179
   420
  assume "(x, y) \<in> r^+"
berghofe@13704
   421
  assume "(y, z) \<in> r^+"
wenzelm@26179
   422
  then show "(x, z) \<in> r^+"
wenzelm@26179
   423
  proof induct
wenzelm@26179
   424
    case (base u)
wenzelm@60758
   425
    from \<open>(x, y) \<in> r^+\<close> and \<open>(y, u) \<in> r\<close>
wenzelm@26179
   426
    show "(x, u) \<in> r^+" ..
wenzelm@26179
   427
  next
wenzelm@26179
   428
    case (step u v)
wenzelm@60758
   429
    from \<open>(x, u) \<in> r^+\<close> and \<open>(u, v) \<in> r\<close>
wenzelm@26179
   430
    show "(x, v) \<in> r^+" ..
wenzelm@26179
   431
  qed
berghofe@13704
   432
qed
wenzelm@12691
   433
wenzelm@45607
   434
lemmas trancl_trans = trans_trancl [THEN transD]
wenzelm@12691
   435
berghofe@23743
   436
lemma tranclp_trans:
berghofe@22262
   437
  assumes xy: "r^++ x y"
berghofe@22262
   438
  and yz: "r^++ y z"
berghofe@22262
   439
  shows "r^++ x z" using yz xy
berghofe@22262
   440
  by induct iprover+
berghofe@22262
   441
wenzelm@26179
   442
lemma trancl_id [simp]: "trans r \<Longrightarrow> r^+ = r"
wenzelm@26179
   443
  apply auto
wenzelm@26179
   444
  apply (erule trancl_induct)
wenzelm@26179
   445
   apply assumption
wenzelm@26179
   446
  apply (unfold trans_def)
wenzelm@26179
   447
  apply blast
wenzelm@26179
   448
  done
nipkow@19623
   449
wenzelm@26179
   450
lemma rtranclp_tranclp_tranclp:
wenzelm@26179
   451
  assumes "r^** x y"
wenzelm@26179
   452
  shows "!!z. r^++ y z ==> r^++ x z" using assms
berghofe@23743
   453
  by induct (iprover intro: tranclp_trans)+
wenzelm@12691
   454
berghofe@23743
   455
lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]
berghofe@22262
   456
berghofe@23743
   457
lemma tranclp_into_tranclp2: "r a b ==> r^++ b c ==> r^++ a c"
berghofe@23743
   458
  by (erule tranclp_trans [OF tranclp.r_into_trancl])
berghofe@22262
   459
berghofe@23743
   460
lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]
wenzelm@12691
   461
berghofe@23743
   462
lemma tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ x y"
berghofe@22262
   463
  apply (drule conversepD)
berghofe@23743
   464
  apply (erule tranclp_induct)
berghofe@23743
   465
  apply (iprover intro: conversepI tranclp_trans)+
wenzelm@12691
   466
  done
wenzelm@12691
   467
berghofe@23743
   468
lemmas trancl_converseI = tranclp_converseI [to_set]
berghofe@22262
   469
berghofe@23743
   470
lemma tranclp_converseD: "(r^--1)^++ x y ==> (r^++)^--1 x y"
berghofe@22262
   471
  apply (rule conversepI)
berghofe@23743
   472
  apply (erule tranclp_induct)
berghofe@23743
   473
  apply (iprover dest: conversepD intro: tranclp_trans)+
berghofe@13704
   474
  done
wenzelm@12691
   475
berghofe@23743
   476
lemmas trancl_converseD = tranclp_converseD [to_set]
berghofe@22262
   477
berghofe@23743
   478
lemma tranclp_converse: "(r^--1)^++ = (r^++)^--1"
nipkow@44890
   479
  by (fastforce simp add: fun_eq_iff
berghofe@23743
   480
    intro!: tranclp_converseI dest!: tranclp_converseD)
berghofe@22262
   481
berghofe@23743
   482
lemmas trancl_converse = tranclp_converse [to_set]
wenzelm@12691
   483
huffman@19228
   484
lemma sym_trancl: "sym r ==> sym (r^+)"
huffman@19228
   485
  by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
huffman@19228
   486
berghofe@34909
   487
lemma converse_tranclp_induct [consumes 1, case_names base step]:
berghofe@22262
   488
  assumes major: "r^++ a b"
berghofe@22262
   489
    and cases: "!!y. r y b ==> P(y)"
berghofe@22262
   490
      "!!y z.[| r y z;  r^++ z b;  P(z) |] ==> P(y)"
wenzelm@18372
   491
  shows "P a"
berghofe@23743
   492
  apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])
wenzelm@18372
   493
   apply (rule cases)
berghofe@22262
   494
   apply (erule conversepD)
huffman@35216
   495
  apply (blast intro: assms dest!: tranclp_converseD)
wenzelm@18372
   496
  done
wenzelm@12691
   497
berghofe@23743
   498
lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
berghofe@22262
   499
berghofe@23743
   500
lemma tranclpD: "R^++ x y ==> EX z. R x z \<and> R^** z y"
wenzelm@26179
   501
  apply (erule converse_tranclp_induct)
wenzelm@26179
   502
   apply auto
berghofe@23743
   503
  apply (blast intro: rtranclp_trans)
wenzelm@12691
   504
  done
wenzelm@12691
   505
berghofe@23743
   506
lemmas tranclD = tranclpD [to_set]
berghofe@22262
   507
bulwahn@31577
   508
lemma converse_tranclpE:
bulwahn@31577
   509
  assumes major: "tranclp r x z"
bulwahn@31577
   510
  assumes base: "r x z ==> P"
bulwahn@31577
   511
  assumes step: "\<And> y. [| r x y; tranclp r y z |] ==> P"
bulwahn@31577
   512
  shows P
bulwahn@31577
   513
proof -
bulwahn@31577
   514
  from tranclpD[OF major]
bulwahn@31577
   515
  obtain y where "r x y" and "rtranclp r y z" by iprover
bulwahn@31577
   516
  from this(2) show P
bulwahn@31577
   517
  proof (cases rule: rtranclp.cases)
bulwahn@31577
   518
    case rtrancl_refl
wenzelm@60758
   519
    with \<open>r x y\<close> base show P by iprover
bulwahn@31577
   520
  next
bulwahn@31577
   521
    case rtrancl_into_rtrancl
bulwahn@31577
   522
    from this have "tranclp r y z"
bulwahn@31577
   523
      by (iprover intro: rtranclp_into_tranclp1)
wenzelm@60758
   524
    with \<open>r x y\<close> step show P by iprover
bulwahn@31577
   525
  qed
bulwahn@31577
   526
qed
bulwahn@31577
   527
bulwahn@31577
   528
lemmas converse_tranclE = converse_tranclpE [to_set]
bulwahn@31577
   529
kleing@25295
   530
lemma tranclD2:
kleing@25295
   531
  "(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
kleing@25295
   532
  by (blast elim: tranclE intro: trancl_into_rtrancl)
kleing@25295
   533
nipkow@13867
   534
lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
wenzelm@18372
   535
  by (blast elim: tranclE dest: trancl_into_rtrancl)
wenzelm@12691
   536
wenzelm@12691
   537
lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"
wenzelm@12691
   538
  by (blast dest: r_into_trancl)
wenzelm@12691
   539
wenzelm@12691
   540
lemma trancl_subset_Sigma_aux:
wenzelm@12691
   541
    "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
wenzelm@18372
   542
  by (induct rule: rtrancl_induct) auto
wenzelm@12691
   543
wenzelm@12691
   544
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"
berghofe@13704
   545
  apply (rule subsetI)
berghofe@13704
   546
  apply (simp only: split_tupled_all)
berghofe@13704
   547
  apply (erule tranclE)
wenzelm@26179
   548
   apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
wenzelm@12691
   549
  done
nipkow@10996
   550
nipkow@50616
   551
lemma reflclp_tranclp [simp]: "(r^++)^== = r^**"
berghofe@22262
   552
  apply (safe intro!: order_antisym)
berghofe@23743
   553
   apply (erule tranclp_into_rtranclp)
berghofe@23743
   554
  apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)
wenzelm@11084
   555
  done
nipkow@10996
   556
nipkow@50616
   557
lemmas reflcl_trancl [simp] = reflclp_tranclp [to_set]
berghofe@22262
   558
wenzelm@11090
   559
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
wenzelm@11084
   560
  apply safe
paulson@14208
   561
   apply (drule trancl_into_rtrancl, simp)
paulson@14208
   562
  apply (erule rtranclE, safe)
paulson@14208
   563
   apply (rule r_into_trancl, simp)
wenzelm@11084
   564
  apply (rule rtrancl_into_trancl1)
paulson@14208
   565
   apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
wenzelm@11084
   566
  done
nipkow@10996
   567
haftmann@45140
   568
lemma rtrancl_trancl_reflcl [code]: "r^* = (r^+)^="
haftmann@45140
   569
  by simp
haftmann@45140
   570
wenzelm@11090
   571
lemma trancl_empty [simp]: "{}^+ = {}"
wenzelm@11084
   572
  by (auto elim: trancl_induct)
nipkow@10996
   573
wenzelm@11090
   574
lemma rtrancl_empty [simp]: "{}^* = Id"
wenzelm@11084
   575
  by (rule subst [OF reflcl_trancl]) simp
nipkow@10996
   576
berghofe@23743
   577
lemma rtranclpD: "R^** a b ==> a = b \<or> a \<noteq> b \<and> R^++ a b"
nipkow@50616
   578
by (force simp add: reflclp_tranclp [symmetric] simp del: reflclp_tranclp)
berghofe@22262
   579
berghofe@23743
   580
lemmas rtranclD = rtranclpD [to_set]
wenzelm@11084
   581
kleing@16514
   582
lemma rtrancl_eq_or_trancl:
kleing@16514
   583
  "(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"
kleing@16514
   584
  by (fast elim: trancl_into_rtrancl dest: rtranclD)
nipkow@10996
   585
krauss@33656
   586
lemma trancl_unfold_right: "r^+ = r^* O r"
krauss@33656
   587
by (auto dest: tranclD2 intro: rtrancl_into_trancl1)
krauss@33656
   588
krauss@33656
   589
lemma trancl_unfold_left: "r^+ = r O r^*"
krauss@33656
   590
by (auto dest: tranclD intro: rtrancl_into_trancl2)
krauss@33656
   591
nipkow@57178
   592
lemma trancl_insert:
nipkow@57178
   593
  "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
wenzelm@61799
   594
  \<comment> \<open>primitive recursion for \<open>trancl\<close> over finite relations\<close>
nipkow@57178
   595
  apply (rule equalityI)
nipkow@57178
   596
   apply (rule subsetI)
nipkow@57178
   597
   apply (simp only: split_tupled_all)
nipkow@57178
   598
   apply (erule trancl_induct, blast)
nipkow@57178
   599
   apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)
nipkow@57178
   600
  apply (rule subsetI)
nipkow@57178
   601
  apply (blast intro: trancl_mono rtrancl_mono
nipkow@57178
   602
    [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
nipkow@57178
   603
  done
nipkow@57178
   604
nipkow@57178
   605
lemma trancl_insert2:
nipkow@57178
   606
  "(insert (a,b) r)^+ = r^+ \<union> {(x,y). ((x,a) : r^+ \<or> x=a) \<and> ((b,y) \<in> r^+ \<or> y=b)}"
nipkow@57178
   607
by(auto simp add: trancl_insert rtrancl_eq_or_trancl)
nipkow@57178
   608
nipkow@57178
   609
lemma rtrancl_insert:
nipkow@57178
   610
  "(insert (a,b) r)^* = r^* \<union> {(x,y). (x,a) : r^* \<and> (b,y) \<in> r^*}"
nipkow@57178
   611
using trancl_insert[of a b r]
nipkow@57178
   612
by(simp add: rtrancl_trancl_reflcl del: reflcl_trancl) blast
nipkow@57178
   613
krauss@33656
   614
wenzelm@60758
   615
text \<open>Simplifying nested closures\<close>
krauss@33656
   616
krauss@33656
   617
lemma rtrancl_trancl_absorb[simp]: "(R^*)^+ = R^*"
krauss@33656
   618
by (simp add: trans_rtrancl)
krauss@33656
   619
krauss@33656
   620
lemma trancl_rtrancl_absorb[simp]: "(R^+)^* = R^*"
krauss@33656
   621
by (subst reflcl_trancl[symmetric]) simp
krauss@33656
   622
krauss@33656
   623
lemma rtrancl_reflcl_absorb[simp]: "(R^*)^= = R^*"
krauss@33656
   624
by auto
krauss@33656
   625
krauss@33656
   626
wenzelm@61799
   627
text \<open>\<open>Domain\<close> and \<open>Range\<close>\<close>
nipkow@10996
   628
wenzelm@11090
   629
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
wenzelm@11084
   630
  by blast
nipkow@10996
   631
wenzelm@11090
   632
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
wenzelm@11084
   633
  by blast
nipkow@10996
   634
wenzelm@11090
   635
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
wenzelm@11084
   636
  by (rule rtrancl_Un_rtrancl [THEN subst]) fast
nipkow@10996
   637
wenzelm@11090
   638
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
wenzelm@11084
   639
  by (blast intro: subsetD [OF rtrancl_Un_subset])
nipkow@10996
   640
wenzelm@11090
   641
lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
haftmann@46752
   642
  by (unfold Domain_unfold) (blast dest: tranclD)
nipkow@10996
   643
wenzelm@11090
   644
lemma trancl_range [simp]: "Range (r^+) = Range r"
haftmann@46752
   645
  unfolding Domain_converse [symmetric] by (simp add: trancl_converse [symmetric])
nipkow@10996
   646
paulson@11115
   647
lemma Not_Domain_rtrancl:
wenzelm@12691
   648
    "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
wenzelm@12691
   649
  apply auto
wenzelm@26179
   650
  apply (erule rev_mp)
wenzelm@26179
   651
  apply (erule rtrancl_induct)
wenzelm@26179
   652
   apply auto
wenzelm@26179
   653
  done
berghofe@11327
   654
haftmann@29609
   655
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
haftmann@29609
   656
  apply clarify
haftmann@29609
   657
  apply (erule trancl_induct)
haftmann@29609
   658
   apply (auto simp add: Field_def)
haftmann@29609
   659
  done
haftmann@29609
   660
nipkow@41987
   661
lemma finite_trancl[simp]: "finite (r^+) = finite r"
haftmann@29609
   662
  apply auto
haftmann@29609
   663
   prefer 2
haftmann@29609
   664
   apply (rule trancl_subset_Field2 [THEN finite_subset])
haftmann@29609
   665
   apply (rule finite_SigmaI)
haftmann@29609
   666
    prefer 3
haftmann@29609
   667
    apply (blast intro: r_into_trancl' finite_subset)
haftmann@29609
   668
   apply (auto simp add: finite_Field)
haftmann@29609
   669
  done
haftmann@29609
   670
wenzelm@61799
   671
text \<open>More about converse \<open>rtrancl\<close> and \<open>trancl\<close>, should
wenzelm@60758
   672
  be merged with main body.\<close>
kleing@12428
   673
nipkow@14337
   674
lemma single_valued_confluent:
nipkow@14337
   675
  "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>
nipkow@14337
   676
  \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
wenzelm@26179
   677
  apply (erule rtrancl_induct)
wenzelm@26179
   678
  apply simp
wenzelm@26179
   679
  apply (erule disjE)
wenzelm@26179
   680
   apply (blast elim:converse_rtranclE dest:single_valuedD)
wenzelm@26179
   681
  apply(blast intro:rtrancl_trans)
wenzelm@26179
   682
  done
nipkow@14337
   683
wenzelm@12691
   684
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"
kleing@12428
   685
  by (fast intro: trancl_trans)
kleing@12428
   686
kleing@12428
   687
lemma trancl_into_trancl [rule_format]:
wenzelm@12691
   688
    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
wenzelm@12691
   689
  apply (erule trancl_induct)
kleing@12428
   690
   apply (fast intro: r_r_into_trancl)
kleing@12428
   691
  apply (fast intro: r_r_into_trancl trancl_trans)
kleing@12428
   692
  done
kleing@12428
   693
berghofe@23743
   694
lemma tranclp_rtranclp_tranclp:
berghofe@22262
   695
    "r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c"
berghofe@23743
   696
  apply (drule tranclpD)
wenzelm@26179
   697
  apply (elim exE conjE)
berghofe@23743
   698
  apply (drule rtranclp_trans, assumption)
berghofe@23743
   699
  apply (drule rtranclp_into_tranclp2, assumption, assumption)
kleing@12428
   700
  done
kleing@12428
   701
berghofe@23743
   702
lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]
berghofe@22262
   703
wenzelm@12691
   704
lemmas transitive_closure_trans [trans] =
wenzelm@12691
   705
  r_r_into_trancl trancl_trans rtrancl_trans
berghofe@23743
   706
  trancl.trancl_into_trancl trancl_into_trancl2
berghofe@23743
   707
  rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
wenzelm@12691
   708
  rtrancl_trancl_trancl trancl_rtrancl_trancl
kleing@12428
   709
berghofe@23743
   710
lemmas transitive_closurep_trans' [trans] =
berghofe@23743
   711
  tranclp_trans rtranclp_trans
berghofe@23743
   712
  tranclp.trancl_into_trancl tranclp_into_tranclp2
berghofe@23743
   713
  rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp
berghofe@23743
   714
  rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp
berghofe@22262
   715
kleing@12428
   716
declare trancl_into_rtrancl [elim]
berghofe@11327
   717
wenzelm@60758
   718
subsection \<open>The power operation on relations\<close>
haftmann@30954
   719
wenzelm@61799
   720
text \<open>\<open>R ^^ n = R O ... O R\<close>, the n-fold composition of \<open>R\<close>\<close>
haftmann@30954
   721
haftmann@30971
   722
overloading
haftmann@30971
   723
  relpow == "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
haftmann@47202
   724
  relpowp == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
haftmann@30971
   725
begin
haftmann@30954
   726
blanchet@55575
   727
primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
haftmann@30971
   728
    "relpow 0 R = Id"
krauss@32235
   729
  | "relpow (Suc n) R = (R ^^ n) O R"
haftmann@30954
   730
blanchet@55575
   731
primrec relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)" where
haftmann@47202
   732
    "relpowp 0 R = HOL.eq"
haftmann@47202
   733
  | "relpowp (Suc n) R = (R ^^ n) OO R"
haftmann@47202
   734
haftmann@30971
   735
end
haftmann@30954
   736
haftmann@47202
   737
lemma relpowp_relpow_eq [pred_set_conv]:
haftmann@47202
   738
  fixes R :: "'a rel"
haftmann@47202
   739
  shows "(\<lambda>x y. (x, y) \<in> R) ^^ n = (\<lambda>x y. (x, y) \<in> R ^^ n)"
griff@47433
   740
  by (induct n) (simp_all add: relcompp_relcomp_eq)
haftmann@47202
   741
wenzelm@60758
   742
text \<open>for code generation\<close>
bulwahn@46360
   743
bulwahn@46360
   744
definition relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
bulwahn@46360
   745
  relpow_code_def [code_abbrev]: "relpow = compow"
bulwahn@46360
   746
Christian@47492
   747
definition relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)" where
Christian@47492
   748
  relpowp_code_def [code_abbrev]: "relpowp = compow"
Christian@47492
   749
bulwahn@46360
   750
lemma [code]:
bulwahn@46360
   751
  "relpow (Suc n) R = (relpow n R) O R"
bulwahn@46360
   752
  "relpow 0 R = Id"
bulwahn@46360
   753
  by (simp_all add: relpow_code_def)
bulwahn@46360
   754
Christian@47492
   755
lemma [code]:
Christian@47492
   756
  "relpowp (Suc n) R = (R ^^ n) OO R"
Christian@47492
   757
  "relpowp 0 R = HOL.eq"
Christian@47492
   758
  by (simp_all add: relpowp_code_def)
Christian@47492
   759
bulwahn@46360
   760
hide_const (open) relpow
Christian@47492
   761
hide_const (open) relpowp
bulwahn@46360
   762
bulwahn@46362
   763
lemma relpow_1 [simp]:
haftmann@30971
   764
  fixes R :: "('a \<times> 'a) set"
haftmann@30971
   765
  shows "R ^^ 1 = R"
haftmann@30954
   766
  by simp
haftmann@30954
   767
Christian@47492
   768
lemma relpowp_1 [simp]:
Christian@47492
   769
  fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
Christian@47492
   770
  shows "P ^^ 1 = P"
Christian@47492
   771
  by (fact relpow_1 [to_pred])
Christian@47492
   772
bulwahn@46362
   773
lemma relpow_0_I: 
haftmann@30954
   774
  "(x, x) \<in> R ^^ 0"
haftmann@30954
   775
  by simp
haftmann@30954
   776
Christian@47492
   777
lemma relpowp_0_I:
Christian@47492
   778
  "(P ^^ 0) x x"
Christian@47492
   779
  by (fact relpow_0_I [to_pred])
Christian@47492
   780
bulwahn@46362
   781
lemma relpow_Suc_I:
haftmann@30954
   782
  "(x, y) \<in>  R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
haftmann@30954
   783
  by auto
haftmann@30954
   784
Christian@47492
   785
lemma relpowp_Suc_I:
Christian@47492
   786
  "(P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> (P ^^ Suc n) x z"
Christian@47492
   787
  by (fact relpow_Suc_I [to_pred])
Christian@47492
   788
bulwahn@46362
   789
lemma relpow_Suc_I2:
haftmann@30954
   790
  "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
nipkow@44890
   791
  by (induct n arbitrary: z) (simp, fastforce)
haftmann@30954
   792
Christian@47492
   793
lemma relpowp_Suc_I2:
Christian@47492
   794
  "P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> (P ^^ Suc n) x z"
Christian@47492
   795
  by (fact relpow_Suc_I2 [to_pred])
Christian@47492
   796
bulwahn@46362
   797
lemma relpow_0_E:
haftmann@30954
   798
  "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30954
   799
  by simp
haftmann@30954
   800
Christian@47492
   801
lemma relpowp_0_E:
Christian@47492
   802
  "(P ^^ 0) x y \<Longrightarrow> (x = y \<Longrightarrow> Q) \<Longrightarrow> Q"
Christian@47492
   803
  by (fact relpow_0_E [to_pred])
Christian@47492
   804
bulwahn@46362
   805
lemma relpow_Suc_E:
haftmann@30954
   806
  "(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
   807
  by auto
haftmann@30954
   808
Christian@47492
   809
lemma relpowp_Suc_E:
Christian@47492
   810
  "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. (P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> Q) \<Longrightarrow> Q"
Christian@47492
   811
  by (fact relpow_Suc_E [to_pred])
Christian@47492
   812
bulwahn@46362
   813
lemma relpow_E:
haftmann@30954
   814
  "(x, z) \<in>  R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
haftmann@30954
   815
   \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in>  R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P)
haftmann@30954
   816
   \<Longrightarrow> P"
haftmann@30954
   817
  by (cases n) auto
haftmann@30954
   818
Christian@47492
   819
lemma relpowp_E:
Christian@47492
   820
  "(P ^^ n) x z \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q)
Christian@47492
   821
  \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (P ^^ m) x y \<Longrightarrow> P y z \<Longrightarrow> Q)
Christian@47492
   822
  \<Longrightarrow> Q"
Christian@47492
   823
  by (fact relpow_E [to_pred])
Christian@47492
   824
bulwahn@46362
   825
lemma relpow_Suc_D2:
haftmann@30954
   826
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
haftmann@30954
   827
  apply (induct n arbitrary: x z)
bulwahn@46362
   828
   apply (blast intro: relpow_0_I elim: relpow_0_E relpow_Suc_E)
bulwahn@46362
   829
  apply (blast intro: relpow_Suc_I elim: relpow_0_E relpow_Suc_E)
haftmann@30954
   830
  done
haftmann@30954
   831
Christian@47492
   832
lemma relpowp_Suc_D2:
Christian@47492
   833
  "(P ^^ Suc n) x z \<Longrightarrow> \<exists>y. P x y \<and> (P ^^ n) y z"
Christian@47492
   834
  by (fact relpow_Suc_D2 [to_pred])
Christian@47492
   835
bulwahn@46362
   836
lemma relpow_Suc_E2:
haftmann@30954
   837
  "(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
   838
  by (blast dest: relpow_Suc_D2)
haftmann@30954
   839
Christian@47492
   840
lemma relpowp_Suc_E2:
Christian@47492
   841
  "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> Q) \<Longrightarrow> Q"
Christian@47492
   842
  by (fact relpow_Suc_E2 [to_pred])
Christian@47492
   843
bulwahn@46362
   844
lemma relpow_Suc_D2':
haftmann@30954
   845
  "\<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
   846
  by (induct n) (simp_all, blast)
haftmann@30954
   847
Christian@47492
   848
lemma relpowp_Suc_D2':
Christian@47492
   849
  "\<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
   850
  by (fact relpow_Suc_D2' [to_pred])
Christian@47492
   851
bulwahn@46362
   852
lemma relpow_E2:
haftmann@30954
   853
  "(x, z) \<in> R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
haftmann@30954
   854
     \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P)
haftmann@30954
   855
   \<Longrightarrow> P"
haftmann@30954
   856
  apply (cases n, simp)
blanchet@55417
   857
  apply (rename_tac nat)
bulwahn@46362
   858
  apply (cut_tac n=nat and R=R in relpow_Suc_D2', simp, blast)
haftmann@30954
   859
  done
haftmann@30954
   860
Christian@47492
   861
lemma relpowp_E2:
Christian@47492
   862
  "(P ^^ n) x z \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q)
Christian@47492
   863
    \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> P x y \<Longrightarrow> (P ^^ m) y z \<Longrightarrow> Q)
Christian@47492
   864
  \<Longrightarrow> Q"
Christian@47492
   865
  by (fact relpow_E2 [to_pred])
Christian@47492
   866
bulwahn@46362
   867
lemma relpow_add: "R ^^ (m+n) = R^^m O R^^n"
haftmann@45976
   868
  by (induct n) auto
nipkow@31351
   869
Christian@47492
   870
lemma relpowp_add: "P ^^ (m + n) = P ^^ m OO P ^^ n"
Christian@47492
   871
  by (fact relpow_add [to_pred])
Christian@47492
   872
bulwahn@46362
   873
lemma relpow_commute: "R O R ^^ n = R ^^ n O R"
haftmann@45976
   874
  by (induct n) (simp, simp add: O_assoc [symmetric])
krauss@31970
   875
Christian@47492
   876
lemma relpowp_commute: "P OO P ^^ n = P ^^ n OO P"
Christian@47492
   877
  by (fact relpow_commute [to_pred])
Christian@47492
   878
bulwahn@46362
   879
lemma relpow_empty:
haftmann@45153
   880
  "0 < n \<Longrightarrow> ({} :: ('a \<times> 'a) set) ^^ n = {}"
haftmann@45153
   881
  by (cases n) auto
bulwahn@45116
   882
Christian@47492
   883
lemma relpowp_bot:
Christian@47492
   884
  "0 < n \<Longrightarrow> (\<bottom> :: 'a \<Rightarrow> 'a \<Rightarrow> bool) ^^ n = \<bottom>"
Christian@47492
   885
  by (fact relpow_empty [to_pred])
Christian@47492
   886
bulwahn@46362
   887
lemma rtrancl_imp_UN_relpow:
haftmann@30954
   888
  assumes "p \<in> R^*"
haftmann@30954
   889
  shows "p \<in> (\<Union>n. R ^^ n)"
haftmann@30954
   890
proof (cases p)
haftmann@30954
   891
  case (Pair x y)
haftmann@30954
   892
  with assms have "(x, y) \<in> R^*" by simp
haftmann@30954
   893
  then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct
bulwahn@46362
   894
    case base show ?case by (blast intro: relpow_0_I)
haftmann@30954
   895
  next
bulwahn@46362
   896
    case step then show ?case by (blast intro: relpow_Suc_I)
haftmann@30954
   897
  qed
haftmann@30954
   898
  with Pair show ?thesis by simp
haftmann@30954
   899
qed
haftmann@30954
   900
Christian@47492
   901
lemma rtranclp_imp_Sup_relpowp:
Christian@47492
   902
  assumes "(P^**) x y"
Christian@47492
   903
  shows "(\<Squnion>n. P ^^ n) x y"
haftmann@61424
   904
  using assms and rtrancl_imp_UN_relpow [of "(x, y)", to_pred] by simp
Christian@47492
   905
bulwahn@46362
   906
lemma relpow_imp_rtrancl:
haftmann@30954
   907
  assumes "p \<in> R ^^ n"
haftmann@30954
   908
  shows "p \<in> R^*"
haftmann@30954
   909
proof (cases p)
haftmann@30954
   910
  case (Pair x y)
haftmann@30954
   911
  with assms have "(x, y) \<in> R ^^ n" by simp
haftmann@30954
   912
  then have "(x, y) \<in> R^*" proof (induct n arbitrary: x y)
haftmann@30954
   913
    case 0 then show ?case by simp
haftmann@30954
   914
  next
haftmann@30954
   915
    case Suc then show ?case
bulwahn@46362
   916
      by (blast elim: relpow_Suc_E intro: rtrancl_into_rtrancl)
haftmann@30954
   917
  qed
haftmann@30954
   918
  with Pair show ?thesis by simp
haftmann@30954
   919
qed
haftmann@30954
   920
Christian@47492
   921
lemma relpowp_imp_rtranclp:
Christian@47492
   922
  assumes "(P ^^ n) x y"
Christian@47492
   923
  shows "(P^**) x y"
haftmann@61424
   924
  using assms and relpow_imp_rtrancl [of "(x, y)", to_pred] by simp
Christian@47492
   925
bulwahn@46362
   926
lemma rtrancl_is_UN_relpow:
haftmann@30954
   927
  "R^* = (\<Union>n. R ^^ n)"
bulwahn@46362
   928
  by (blast intro: rtrancl_imp_UN_relpow relpow_imp_rtrancl)
haftmann@30954
   929
Christian@47492
   930
lemma rtranclp_is_Sup_relpowp:
Christian@47492
   931
  "P^** = (\<Squnion>n. P ^^ n)"
Christian@47492
   932
  using rtrancl_is_UN_relpow [to_pred, of P] by auto
Christian@47492
   933
haftmann@30954
   934
lemma rtrancl_power:
haftmann@30954
   935
  "p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
bulwahn@46362
   936
  by (simp add: rtrancl_is_UN_relpow)
haftmann@30954
   937
Christian@47492
   938
lemma rtranclp_power:
Christian@47492
   939
  "(P^**) x y \<longleftrightarrow> (\<exists>n. (P ^^ n) x y)"
Christian@47492
   940
  by (simp add: rtranclp_is_Sup_relpowp)
Christian@47492
   941
haftmann@30954
   942
lemma trancl_power:
haftmann@30954
   943
  "p \<in> R^+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
haftmann@30954
   944
  apply (cases p)
haftmann@30954
   945
  apply simp
haftmann@30954
   946
  apply (rule iffI)
haftmann@30954
   947
   apply (drule tranclD2)
bulwahn@46362
   948
   apply (clarsimp simp: rtrancl_is_UN_relpow)
haftmann@62343
   949
   apply (rule_tac x="Suc x" in exI)
griff@47433
   950
   apply (clarsimp simp: relcomp_unfold)
nipkow@44890
   951
   apply fastforce
haftmann@30954
   952
  apply clarsimp
haftmann@30954
   953
  apply (case_tac n, simp)
haftmann@30954
   954
  apply clarsimp
bulwahn@46362
   955
  apply (drule relpow_imp_rtrancl)
haftmann@30954
   956
  apply (drule rtrancl_into_trancl1) apply auto
haftmann@30954
   957
  done
haftmann@30954
   958
Christian@47492
   959
lemma tranclp_power:
Christian@47492
   960
  "(P^++) x y \<longleftrightarrow> (\<exists>n > 0. (P ^^ n) x y)"
Christian@47492
   961
  using trancl_power [to_pred, of P "(x, y)"] by simp
Christian@47492
   962
bulwahn@46362
   963
lemma rtrancl_imp_relpow:
haftmann@30954
   964
  "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
bulwahn@46362
   965
  by (auto dest: rtrancl_imp_UN_relpow)
haftmann@30954
   966
Christian@47492
   967
lemma rtranclp_imp_relpowp:
Christian@47492
   968
  "(P^**) x y \<Longrightarrow> \<exists>n. (P ^^ n) x y"
Christian@47492
   969
  by (auto dest: rtranclp_imp_Sup_relpowp)
Christian@47492
   970
wenzelm@60758
   971
text\<open>By Sternagel/Thiemann:\<close>
bulwahn@46362
   972
lemma relpow_fun_conv:
nipkow@41987
   973
  "((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
   974
proof (induct n arbitrary: b)
nipkow@41987
   975
  case 0 show ?case by auto
nipkow@41987
   976
next
nipkow@41987
   977
  case (Suc n)
nipkow@41987
   978
  show ?case
griff@47433
   979
  proof (simp add: relcomp_unfold Suc)
nipkow@41987
   980
    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
   981
     = (\<exists>f. f 0 = a \<and> f(Suc n) = b \<and> (\<forall>i<Suc n. (f i, f (Suc i)) \<in> R))"
nipkow@41987
   982
    (is "?l = ?r")
nipkow@41987
   983
    proof
nipkow@41987
   984
      assume ?l
nipkow@41987
   985
      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
   986
      let ?g = "\<lambda> m. if m = Suc n then b else f m"
nipkow@41987
   987
      show ?r by (rule exI[of _ ?g], simp add: 1)
nipkow@41987
   988
    next
nipkow@41987
   989
      assume ?r
nipkow@41987
   990
      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
   991
      show ?l by (rule exI[of _ "f n"], rule conjI, rule exI[of _ f], insert 1, auto)
nipkow@41987
   992
    qed
nipkow@41987
   993
  qed
nipkow@41987
   994
qed
nipkow@41987
   995
Christian@47492
   996
lemma relpowp_fun_conv:
Christian@47492
   997
  "(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
   998
  by (fact relpow_fun_conv [to_pred])
Christian@47492
   999
bulwahn@46362
  1000
lemma relpow_finite_bounded1:
nipkow@41987
  1001
assumes "finite(R :: ('a*'a)set)" and "k>0"
nipkow@41987
  1002
shows "R^^k \<subseteq> (UN n:{n. 0<n & n <= card R}. R^^n)" (is "_ \<subseteq> ?r")
nipkow@41987
  1003
proof-
nipkow@41987
  1004
  { fix a b k
nipkow@41987
  1005
    have "(a,b) : R^^(Suc k) \<Longrightarrow> EX n. 0<n & n <= card R & (a,b) : R^^n"
nipkow@41987
  1006
    proof(induct k arbitrary: b)
nipkow@41987
  1007
      case 0
nipkow@41987
  1008
      hence "R \<noteq> {}" by auto
wenzelm@60758
  1009
      with card_0_eq[OF \<open>finite R\<close>] have "card R >= Suc 0" by auto
nipkow@41987
  1010
      thus ?case using 0 by force
nipkow@41987
  1011
    next
nipkow@41987
  1012
      case (Suc k)
nipkow@41987
  1013
      then obtain a' where "(a,a') : R^^(Suc k)" and "(a',b) : R" by auto
wenzelm@60758
  1014
      from Suc(1)[OF \<open>(a,a') : R^^(Suc k)\<close>]
nipkow@41987
  1015
      obtain n where "n \<le> card R" and "(a,a') \<in> R ^^ n" by auto
wenzelm@60758
  1016
      have "(a,b) : R^^(Suc n)" using \<open>(a,a') \<in> R^^n\<close> and \<open>(a',b)\<in> R\<close> by auto
nipkow@41987
  1017
      { assume "n < card R"
wenzelm@60758
  1018
        hence ?case using \<open>(a,b): R^^(Suc n)\<close> Suc_leI[OF \<open>n < card R\<close>] by blast
nipkow@41987
  1019
      } moreover
nipkow@41987
  1020
      { assume "n = card R"
wenzelm@60758
  1021
        from \<open>(a,b) \<in> R ^^ (Suc n)\<close>[unfolded relpow_fun_conv]
nipkow@41987
  1022
        obtain f where "f 0 = a" and "f(Suc n) = b"
nipkow@41987
  1023
          and steps: "\<And>i. i <= n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto
nipkow@41987
  1024
        let ?p = "%i. (f i, f(Suc i))"
nipkow@41987
  1025
        let ?N = "{i. i \<le> n}"
nipkow@41987
  1026
        have "?p ` ?N <= R" using steps by auto
nipkow@41987
  1027
        from card_mono[OF assms(1) this]
nipkow@41987
  1028
        have "card(?p ` ?N) <= card R" .
wenzelm@60758
  1029
        also have "\<dots> < card ?N" using \<open>n = card R\<close> by simp
nipkow@41987
  1030
        finally have "~ inj_on ?p ?N" by(rule pigeonhole)
nipkow@41987
  1031
        then obtain i j where i: "i <= n" and j: "j <= n" and ij: "i \<noteq> j" and
nipkow@41987
  1032
          pij: "?p i = ?p j" by(auto simp: inj_on_def)
nipkow@41987
  1033
        let ?i = "min i j" let ?j = "max i j"
nipkow@41987
  1034
        have i: "?i <= n" and j: "?j <= n" and pij: "?p ?i = ?p ?j" 
nipkow@41987
  1035
          and ij: "?i < ?j"
nipkow@41987
  1036
          using i j ij pij unfolding min_def max_def by auto
nipkow@41987
  1037
        from i j pij ij obtain i j where i: "i<=n" and j: "j<=n" and ij: "i<j"
nipkow@41987
  1038
          and pij: "?p i = ?p j" by blast
nipkow@41987
  1039
        let ?g = "\<lambda> l. if l \<le> i then f l else f (l + (j - i))"
nipkow@41987
  1040
        let ?n = "Suc(n - (j - i))"
bulwahn@46362
  1041
        have abl: "(a,b) \<in> R ^^ ?n" unfolding relpow_fun_conv
nipkow@41987
  1042
        proof (rule exI[of _ ?g], intro conjI impI allI)
wenzelm@60758
  1043
          show "?g ?n = b" using \<open>f(Suc n) = b\<close> j ij by auto
nipkow@41987
  1044
        next
nipkow@41987
  1045
          fix k assume "k < ?n"
nipkow@41987
  1046
          show "(?g k, ?g (Suc k)) \<in> R"
nipkow@41987
  1047
          proof (cases "k < i")
nipkow@41987
  1048
            case True
nipkow@41987
  1049
            with i have "k <= n" by auto
nipkow@41987
  1050
            from steps[OF this] show ?thesis using True by simp
nipkow@41987
  1051
          next
nipkow@41987
  1052
            case False
nipkow@41987
  1053
            hence "i \<le> k" by auto
nipkow@41987
  1054
            show ?thesis
nipkow@41987
  1055
            proof (cases "k = i")
nipkow@41987
  1056
              case True
nipkow@41987
  1057
              thus ?thesis using ij pij steps[OF i] by simp
nipkow@41987
  1058
            next
nipkow@41987
  1059
              case False
wenzelm@60758
  1060
              with \<open>i \<le> k\<close> have "i < k" by auto
wenzelm@60758
  1061
              hence small: "k + (j - i) <= n" using \<open>k<?n\<close> by arith
wenzelm@60758
  1062
              show ?thesis using steps[OF small] \<open>i<k\<close> by auto
nipkow@41987
  1063
            qed
nipkow@41987
  1064
          qed
wenzelm@60758
  1065
        qed (simp add: \<open>f 0 = a\<close>)
nipkow@41987
  1066
        moreover have "?n <= n" using i j ij by arith
wenzelm@60758
  1067
        ultimately have ?case using \<open>n = card R\<close> by blast
nipkow@41987
  1068
      }
wenzelm@60758
  1069
      ultimately show ?case using \<open>n \<le> card R\<close> by force
nipkow@41987
  1070
    qed
nipkow@41987
  1071
  }
wenzelm@60758
  1072
  thus ?thesis using gr0_implies_Suc[OF \<open>k>0\<close>] by auto
nipkow@41987
  1073
qed
nipkow@41987
  1074
bulwahn@46362
  1075
lemma relpow_finite_bounded:
nipkow@41987
  1076
assumes "finite(R :: ('a*'a)set)"
nipkow@41987
  1077
shows "R^^k \<subseteq> (UN n:{n. n <= card R}. R^^n)"
nipkow@41987
  1078
apply(cases k)
nipkow@41987
  1079
 apply force
bulwahn@46362
  1080
using relpow_finite_bounded1[OF assms, of k] by auto
nipkow@41987
  1081
bulwahn@46362
  1082
lemma rtrancl_finite_eq_relpow:
nipkow@41987
  1083
  "finite R \<Longrightarrow> R^* = (UN n : {n. n <= card R}. R^^n)"
bulwahn@46362
  1084
by(fastforce simp: rtrancl_power dest: relpow_finite_bounded)
nipkow@41987
  1085
bulwahn@46362
  1086
lemma trancl_finite_eq_relpow:
nipkow@41987
  1087
  "finite R \<Longrightarrow> R^+ = (UN n : {n. 0 < n & n <= card R}. R^^n)"
nipkow@41987
  1088
apply(auto simp add: trancl_power)
bulwahn@46362
  1089
apply(auto dest: relpow_finite_bounded1)
nipkow@41987
  1090
done
nipkow@41987
  1091
griff@47433
  1092
lemma finite_relcomp[simp,intro]:
nipkow@41987
  1093
assumes "finite R" and "finite S"
nipkow@41987
  1094
shows "finite(R O S)"
nipkow@41987
  1095
proof-
haftmann@62343
  1096
  have "R O S = (\<Union>(x, y)\<in>R. \<Union>(u, v)\<in>S. if u = y then {(x, v)} else {})"
haftmann@62343
  1097
    by (force simp add: split_def image_constant_conv split: if_splits)
haftmann@62343
  1098
  then show ?thesis using assms by clarsimp
nipkow@41987
  1099
qed
nipkow@41987
  1100
nipkow@41987
  1101
lemma finite_relpow[simp,intro]:
nipkow@41987
  1102
  assumes "finite(R :: ('a*'a)set)" shows "n>0 \<Longrightarrow> finite(R^^n)"
nipkow@41987
  1103
apply(induct n)
nipkow@41987
  1104
 apply simp
nipkow@41987
  1105
apply(case_tac n)
nipkow@41987
  1106
 apply(simp_all add: assms)
nipkow@41987
  1107
done
nipkow@41987
  1108
bulwahn@46362
  1109
lemma single_valued_relpow:
haftmann@30954
  1110
  fixes R :: "('a * 'a) set"
haftmann@30954
  1111
  shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
nipkow@41987
  1112
apply (induct n arbitrary: R)
nipkow@41987
  1113
apply simp_all
nipkow@41987
  1114
apply (rule single_valuedI)
bulwahn@46362
  1115
apply (fast dest: single_valuedD elim: relpow_Suc_E)
nipkow@41987
  1116
done
paulson@15551
  1117
haftmann@45140
  1118
wenzelm@60758
  1119
subsection \<open>Bounded transitive closure\<close>
haftmann@45140
  1120
haftmann@45140
  1121
definition ntrancl :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
haftmann@45140
  1122
where
haftmann@45140
  1123
  "ntrancl n R = (\<Union>i\<in>{i. 0 < i \<and> i \<le> Suc n}. R ^^ i)"
haftmann@45140
  1124
haftmann@45140
  1125
lemma ntrancl_Zero [simp, code]:
haftmann@45140
  1126
  "ntrancl 0 R = R"
haftmann@45140
  1127
proof
haftmann@45140
  1128
  show "R \<subseteq> ntrancl 0 R"
haftmann@45140
  1129
    unfolding ntrancl_def by fastforce
haftmann@45140
  1130
next
haftmann@45140
  1131
  { 
haftmann@45140
  1132
    fix i have "0 < i \<and> i \<le> Suc 0 \<longleftrightarrow> i = 1" by auto
haftmann@45140
  1133
  }
haftmann@45140
  1134
  from this show "ntrancl 0 R \<le> R"
haftmann@45140
  1135
    unfolding ntrancl_def by auto
haftmann@45140
  1136
qed
haftmann@45140
  1137
bulwahn@46347
  1138
lemma ntrancl_Suc [simp]:
haftmann@45140
  1139
  "ntrancl (Suc n) R = ntrancl n R O (Id \<union> R)"
haftmann@45140
  1140
proof
haftmann@45140
  1141
  {
haftmann@45140
  1142
    fix a b
haftmann@45140
  1143
    assume "(a, b) \<in> ntrancl (Suc n) R"
haftmann@45140
  1144
    from this obtain i where "0 < i" "i \<le> Suc (Suc n)" "(a, b) \<in> R ^^ i"
haftmann@45140
  1145
      unfolding ntrancl_def by auto
haftmann@45140
  1146
    have "(a, b) \<in> ntrancl n R O (Id \<union> R)"
haftmann@45140
  1147
    proof (cases "i = 1")
haftmann@45140
  1148
      case True
wenzelm@60758
  1149
      from this \<open>(a, b) \<in> R ^^ i\<close> show ?thesis
haftmann@45140
  1150
        unfolding ntrancl_def by auto
haftmann@45140
  1151
    next
haftmann@45140
  1152
      case False
wenzelm@60758
  1153
      from this \<open>0 < i\<close> obtain j where j: "i = Suc j" "0 < j"
haftmann@45140
  1154
        by (cases i) auto
wenzelm@60758
  1155
      from this \<open>(a, b) \<in> R ^^ i\<close> obtain c where c1: "(a, c) \<in> R ^^ j" and c2:"(c, b) \<in> R"
haftmann@45140
  1156
        by auto
wenzelm@60758
  1157
      from c1 j \<open>i \<le> Suc (Suc n)\<close> have "(a, c) \<in> ntrancl n R"
haftmann@45140
  1158
        unfolding ntrancl_def by fastforce
haftmann@45140
  1159
      from this c2 show ?thesis by fastforce
haftmann@45140
  1160
    qed
haftmann@45140
  1161
  }
haftmann@45140
  1162
  from this show "ntrancl (Suc n) R \<subseteq> ntrancl n R O (Id \<union> R)"
haftmann@45140
  1163
    by auto
haftmann@45140
  1164
next
haftmann@45140
  1165
  show "ntrancl n R O (Id \<union> R) \<subseteq> ntrancl (Suc n) R"
haftmann@45140
  1166
    unfolding ntrancl_def by fastforce
haftmann@45140
  1167
qed
haftmann@45140
  1168
bulwahn@46347
  1169
lemma [code]:
bulwahn@46347
  1170
  "ntrancl (Suc n) r = (let r' = ntrancl n r in r' Un r' O r)"
bulwahn@46347
  1171
unfolding Let_def by auto
bulwahn@46347
  1172
haftmann@45141
  1173
lemma finite_trancl_ntranl:
haftmann@45140
  1174
  "finite R \<Longrightarrow> trancl R = ntrancl (card R - 1) R"
bulwahn@46362
  1175
  by (cases "card R") (auto simp add: trancl_finite_eq_relpow relpow_empty ntrancl_def)
haftmann@45140
  1176
haftmann@45140
  1177
wenzelm@60758
  1178
subsection \<open>Acyclic relations\<close>
haftmann@45139
  1179
haftmann@45139
  1180
definition acyclic :: "('a * 'a) set => bool" where
haftmann@45139
  1181
  "acyclic r \<longleftrightarrow> (!x. (x,x) ~: r^+)"
haftmann@45139
  1182
haftmann@45139
  1183
abbreviation acyclicP :: "('a => 'a => bool) => bool" where
haftmann@45139
  1184
  "acyclicP r \<equiv> acyclic {(x, y). r x y}"
haftmann@45139
  1185
haftmann@46127
  1186
lemma acyclic_irrefl [code]:
haftmann@45139
  1187
  "acyclic r \<longleftrightarrow> irrefl (r^+)"
haftmann@45139
  1188
  by (simp add: acyclic_def irrefl_def)
haftmann@45139
  1189
haftmann@45139
  1190
lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
haftmann@45139
  1191
  by (simp add: acyclic_def)
haftmann@45139
  1192
hoelzl@54412
  1193
lemma (in order) acyclicI_order:
hoelzl@54412
  1194
  assumes *: "\<And>a b. (a, b) \<in> r \<Longrightarrow> f b < f a"
hoelzl@54412
  1195
  shows "acyclic r"
hoelzl@54412
  1196
proof -
hoelzl@54412
  1197
  { fix a b assume "(a, b) \<in> r\<^sup>+"
hoelzl@54412
  1198
    then have "f b < f a"
hoelzl@54412
  1199
      by induct (auto intro: * less_trans) }
hoelzl@54412
  1200
  then show ?thesis
hoelzl@54412
  1201
    by (auto intro!: acyclicI)
hoelzl@54412
  1202
qed
hoelzl@54412
  1203
haftmann@45139
  1204
lemma acyclic_insert [iff]:
haftmann@45139
  1205
     "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
haftmann@45139
  1206
apply (simp add: acyclic_def trancl_insert)
haftmann@45139
  1207
apply (blast intro: rtrancl_trans)
haftmann@45139
  1208
done
haftmann@45139
  1209
haftmann@45139
  1210
lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
haftmann@45139
  1211
by (simp add: acyclic_def trancl_converse)
haftmann@45139
  1212
haftmann@45139
  1213
lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
haftmann@45139
  1214
haftmann@45139
  1215
lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
haftmann@45139
  1216
apply (simp add: acyclic_def antisym_def)
haftmann@45139
  1217
apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
haftmann@45139
  1218
done
haftmann@45139
  1219
haftmann@45139
  1220
(* Other direction:
haftmann@45139
  1221
acyclic = no loops
haftmann@45139
  1222
antisym = only self loops
haftmann@45139
  1223
Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
haftmann@45139
  1224
==> antisym( r^* ) = acyclic(r - Id)";
haftmann@45139
  1225
*)
haftmann@45139
  1226
haftmann@45139
  1227
lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
haftmann@45139
  1228
apply (simp add: acyclic_def)
haftmann@45139
  1229
apply (blast intro: trancl_mono)
haftmann@45139
  1230
done
haftmann@45139
  1231
haftmann@45139
  1232
wenzelm@60758
  1233
subsection \<open>Setup of transitivity reasoner\<close>
ballarin@15076
  1234
wenzelm@60758
  1235
ML \<open>
ballarin@15076
  1236
wenzelm@32215
  1237
structure Trancl_Tac = Trancl_Tac
wenzelm@32215
  1238
(
wenzelm@32215
  1239
  val r_into_trancl = @{thm trancl.r_into_trancl};
wenzelm@32215
  1240
  val trancl_trans  = @{thm trancl_trans};
wenzelm@32215
  1241
  val rtrancl_refl = @{thm rtrancl.rtrancl_refl};
wenzelm@32215
  1242
  val r_into_rtrancl = @{thm r_into_rtrancl};
wenzelm@32215
  1243
  val trancl_into_rtrancl = @{thm trancl_into_rtrancl};
wenzelm@32215
  1244
  val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl};
wenzelm@32215
  1245
  val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl};
wenzelm@32215
  1246
  val rtrancl_trans = @{thm rtrancl_trans};
ballarin@15096
  1247
berghofe@30107
  1248
  fun decomp (@{const Trueprop} $ t) =
haftmann@37677
  1249
    let fun dec (Const (@{const_name Set.member}, _) $ (Const (@{const_name Pair}, _) $ a $ b) $ rel ) =
wenzelm@56257
  1250
        let fun decr (Const (@{const_name rtrancl}, _ ) $ r) = (r,"r*")
wenzelm@56257
  1251
              | decr (Const (@{const_name trancl}, _ ) $ r)  = (r,"r+")
wenzelm@18372
  1252
              | decr r = (r,"r");
berghofe@26801
  1253
            val (rel,r) = decr (Envir.beta_eta_contract rel);
wenzelm@18372
  1254
        in SOME (a,b,rel,r) end
wenzelm@18372
  1255
      | dec _ =  NONE
berghofe@30107
  1256
    in dec t end
berghofe@30107
  1257
    | decomp _ = NONE;
wenzelm@32215
  1258
);
ballarin@15076
  1259
wenzelm@32215
  1260
structure Tranclp_Tac = Trancl_Tac
wenzelm@32215
  1261
(
wenzelm@32215
  1262
  val r_into_trancl = @{thm tranclp.r_into_trancl};
wenzelm@32215
  1263
  val trancl_trans  = @{thm tranclp_trans};
wenzelm@32215
  1264
  val rtrancl_refl = @{thm rtranclp.rtrancl_refl};
wenzelm@32215
  1265
  val r_into_rtrancl = @{thm r_into_rtranclp};
wenzelm@32215
  1266
  val trancl_into_rtrancl = @{thm tranclp_into_rtranclp};
wenzelm@32215
  1267
  val rtrancl_trancl_trancl = @{thm rtranclp_tranclp_tranclp};
wenzelm@32215
  1268
  val trancl_rtrancl_trancl = @{thm tranclp_rtranclp_tranclp};
wenzelm@32215
  1269
  val rtrancl_trans = @{thm rtranclp_trans};
berghofe@22262
  1270
berghofe@30107
  1271
  fun decomp (@{const Trueprop} $ t) =
berghofe@22262
  1272
    let fun dec (rel $ a $ b) =
wenzelm@56257
  1273
        let fun decr (Const (@{const_name rtranclp}, _ ) $ r) = (r,"r*")
wenzelm@56257
  1274
              | decr (Const (@{const_name tranclp}, _ ) $ r)  = (r,"r+")
berghofe@22262
  1275
              | decr r = (r,"r");
berghofe@22262
  1276
            val (rel,r) = decr rel;
berghofe@26801
  1277
        in SOME (a, b, rel, r) end
berghofe@22262
  1278
      | dec _ =  NONE
berghofe@30107
  1279
    in dec t end
berghofe@30107
  1280
    | decomp _ = NONE;
wenzelm@32215
  1281
);
wenzelm@60758
  1282
\<close>
berghofe@22262
  1283
wenzelm@60758
  1284
setup \<open>
wenzelm@51717
  1285
  map_theory_simpset (fn ctxt => ctxt
wenzelm@51717
  1286
    addSolver (mk_solver "Trancl" Trancl_Tac.trancl_tac)
wenzelm@51717
  1287
    addSolver (mk_solver "Rtrancl" Trancl_Tac.rtrancl_tac)
wenzelm@51717
  1288
    addSolver (mk_solver "Tranclp" Tranclp_Tac.trancl_tac)
wenzelm@51717
  1289
    addSolver (mk_solver "Rtranclp" Tranclp_Tac.rtrancl_tac))
wenzelm@60758
  1290
\<close>
ballarin@15076
  1291
wenzelm@32215
  1292
wenzelm@60758
  1293
text \<open>Optional methods.\<close>
ballarin@15076
  1294
ballarin@15076
  1295
method_setup trancl =
wenzelm@60758
  1296
  \<open>Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.trancl_tac)\<close>
wenzelm@60758
  1297
  \<open>simple transitivity reasoner\<close>
ballarin@15076
  1298
method_setup rtrancl =
wenzelm@60758
  1299
  \<open>Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.rtrancl_tac)\<close>
wenzelm@60758
  1300
  \<open>simple transitivity reasoner\<close>
berghofe@22262
  1301
method_setup tranclp =
wenzelm@60758
  1302
  \<open>Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.trancl_tac)\<close>
wenzelm@60758
  1303
  \<open>simple transitivity reasoner (predicate version)\<close>
berghofe@22262
  1304
method_setup rtranclp =
wenzelm@60758
  1305
  \<open>Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.rtrancl_tac)\<close>
wenzelm@60758
  1306
  \<open>simple transitivity reasoner (predicate version)\<close>
ballarin@15076
  1307
nipkow@10213
  1308
end