src/HOL/Transitive_Closure.thy
author wenzelm
Mon Aug 31 21:28:08 2015 +0200 (2015-08-31)
changeset 61070 b72a990adfe2
parent 61032 b57df8eecad6
child 61378 3e04c9ca001a
permissions -rw-r--r--
prefer symbols;
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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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  @{text rtrancl} is reflexive/transitive closure,
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  @{text trancl} is transitive closure,
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  @{text reflcl} is reflexive closure.
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  These postfix operators have \emph{maximum priority}, forcing their
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  operands to be atomic.
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\<close>
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inductive_set
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  rtrancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   ("(_^*)" [1000] 999)
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  for r :: "('a \<times> 'a) set"
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where
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    rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) : r^*"
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  | rtrancl_into_rtrancl [Pure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"
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inductive_set
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  trancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"  ("(_^+)" [1000] 999)
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  for r :: "('a \<times> 'a) set"
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where
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    r_into_trancl [intro, Pure.intro]: "(a, b) : r ==> (a, b) : r^+"
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  | trancl_into_trancl [Pure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a, c) : r^+"
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declare rtrancl_def [nitpick_unfold del]
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        rtranclp_def [nitpick_unfold del]
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        trancl_def [nitpick_unfold del]
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        tranclp_def [nitpick_unfold del]
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notation
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  rtranclp  ("(_^**)" [1000] 1000) and
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  tranclp  ("(_^++)" [1000] 1000)
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abbreviation
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  reflclp :: "('a => 'a => bool) => 'a => 'a => bool"  ("(_^==)" [1000] 1000) where
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  "r^== \<equiv> sup r op ="
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abbreviation
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  reflcl :: "('a \<times> 'a) set => ('a \<times> 'a) set"  ("(_^=)" [1000] 999) where
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  "r^= \<equiv> r \<union> Id"
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notation (xsymbols)
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  rtranclp  ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
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  tranclp  ("(_\<^sup>+\<^sup>+)" [1000] 1000) and
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  reflclp  ("(_\<^sup>=\<^sup>=)" [1000] 1000) and
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  rtrancl  ("(_\<^sup>*)" [1000] 999) and
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  trancl  ("(_\<^sup>+)" [1000] 999) and
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  reflcl  ("(_\<^sup>=)" [1000] 999)
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notation (HTML output)
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  rtranclp  ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
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  tranclp  ("(_\<^sup>+\<^sup>+)" [1000] 1000) and
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  reflclp  ("(_\<^sup>=\<^sup>=)" [1000] 1000) and
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  rtrancl  ("(_\<^sup>*)" [1000] 999) and
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  trancl  ("(_\<^sup>+)" [1000] 999) and
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  reflcl  ("(_\<^sup>=)" [1000] 999)
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subsection \<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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  -- \<open>@{text rtrancl} of @{text r} contains @{text r}\<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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  -- \<open>@{text rtrancl} of @{text r} contains @{text r}\<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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  -- \<open>monotonicity of @{text rtrancl}\<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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  -- \<open>elimination of @{text rtrancl} -- 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^*"
nipkow@31690
   308
apply (induct rule:rtrancl.induct)
nipkow@31690
   309
 apply blast
nipkow@31690
   310
apply (blast intro:rtrancl_trans)
nipkow@31690
   311
done
nipkow@31690
   312
nipkow@31690
   313
lemma rtrancl_Un_separator_converseE:
nipkow@31690
   314
  "(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
   315
apply (induct rule:converse_rtrancl_induct)
nipkow@31690
   316
 apply blast
nipkow@31690
   317
apply (blast intro:rtrancl_trans)
nipkow@31690
   318
done
nipkow@31690
   319
haftmann@34970
   320
lemma Image_closed_trancl:
haftmann@34970
   321
  assumes "r `` X \<subseteq> X" shows "r\<^sup>* `` X = X"
haftmann@34970
   322
proof -
haftmann@34970
   323
  from assms have **: "{y. \<exists>x\<in>X. (x, y) \<in> r} \<subseteq> X" by auto
haftmann@34970
   324
  have "\<And>x y. (y, x) \<in> r\<^sup>* \<Longrightarrow> y \<in> X \<Longrightarrow> x \<in> X"
haftmann@34970
   325
  proof -
haftmann@34970
   326
    fix x y
haftmann@34970
   327
    assume *: "y \<in> X"
haftmann@34970
   328
    assume "(y, x) \<in> r\<^sup>*"
haftmann@34970
   329
    then show "x \<in> X"
haftmann@34970
   330
    proof induct
haftmann@34970
   331
      case base show ?case by (fact *)
haftmann@34970
   332
    next
haftmann@34970
   333
      case step with ** show ?case by auto
haftmann@34970
   334
    qed
haftmann@34970
   335
  qed
haftmann@34970
   336
  then show ?thesis by auto
haftmann@34970
   337
qed
haftmann@34970
   338
wenzelm@12691
   339
wenzelm@60758
   340
subsection \<open>Transitive closure\<close>
wenzelm@10331
   341
berghofe@13704
   342
lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
berghofe@23743
   343
  apply (simp add: split_tupled_all)
berghofe@13704
   344
  apply (erule trancl.induct)
wenzelm@26179
   345
   apply (iprover dest: subsetD)+
wenzelm@12691
   346
  done
wenzelm@12691
   347
nipkow@57284
   348
lemma r_into_trancl': "!!p. p : r ==> p : r^+"
berghofe@13704
   349
  by (simp only: split_tupled_all) (erule r_into_trancl)
berghofe@13704
   350
wenzelm@60758
   351
text \<open>
wenzelm@12691
   352
  \medskip Conversions between @{text trancl} and @{text rtrancl}.
wenzelm@60758
   353
\<close>
wenzelm@12691
   354
berghofe@23743
   355
lemma tranclp_into_rtranclp: "r^++ a b ==> r^** a b"
berghofe@23743
   356
  by (erule tranclp.induct) iprover+
wenzelm@12691
   357
berghofe@23743
   358
lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]
berghofe@22262
   359
berghofe@23743
   360
lemma rtranclp_into_tranclp1: assumes r: "r^** a b"
berghofe@22262
   361
  shows "!!c. r b c ==> r^++ a c" using r
nipkow@17589
   362
  by induct iprover+
wenzelm@12691
   363
berghofe@23743
   364
lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]
berghofe@22262
   365
berghofe@23743
   366
lemma rtranclp_into_tranclp2: "[| r a b; r^** b c |] ==> r^++ a c"
wenzelm@60758
   367
  -- \<open>intro rule from @{text r} and @{text rtrancl}\<close>
wenzelm@26179
   368
  apply (erule rtranclp.cases)
wenzelm@26179
   369
   apply iprover
berghofe@23743
   370
  apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1])
wenzelm@26179
   371
    apply (simp | rule r_into_rtranclp)+
wenzelm@12691
   372
  done
wenzelm@12691
   373
berghofe@23743
   374
lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]
berghofe@22262
   375
wenzelm@60758
   376
text \<open>Nice induction rule for @{text trancl}\<close>
wenzelm@26179
   377
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
berghofe@34909
   378
  assumes a: "r^++ a b"
berghofe@22262
   379
  and cases: "!!y. r a y ==> P y"
berghofe@22262
   380
    "!!y z. r^++ a y ==> r y z ==> P y ==> P z"
berghofe@34909
   381
  shows "P b" using a
berghofe@34909
   382
  by (induct x\<equiv>a b) (iprover intro: cases)+
wenzelm@12691
   383
berghofe@25425
   384
lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]
berghofe@22262
   385
berghofe@23743
   386
lemmas tranclp_induct2 =
wenzelm@26179
   387
  tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
wenzelm@26179
   388
    consumes 1, case_names base step]
berghofe@22262
   389
paulson@22172
   390
lemmas trancl_induct2 =
wenzelm@26179
   391
  trancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
wenzelm@26179
   392
    consumes 1, case_names base step]
paulson@22172
   393
berghofe@23743
   394
lemma tranclp_trans_induct:
berghofe@22262
   395
  assumes major: "r^++ x y"
berghofe@22262
   396
    and cases: "!!x y. r x y ==> P x y"
berghofe@22262
   397
      "!!x y z. [| r^++ x y; P x y; r^++ y z; P y z |] ==> P x z"
wenzelm@18372
   398
  shows "P x y"
wenzelm@60758
   399
  -- \<open>Another induction rule for trancl, incorporating transitivity\<close>
berghofe@23743
   400
  by (iprover intro: major [THEN tranclp_induct] cases)
wenzelm@12691
   401
berghofe@23743
   402
lemmas trancl_trans_induct = tranclp_trans_induct [to_set]
berghofe@23743
   403
wenzelm@26174
   404
lemma tranclE [cases set: trancl]:
wenzelm@26174
   405
  assumes "(a, b) : r^+"
wenzelm@26174
   406
  obtains
wenzelm@26174
   407
    (base) "(a, b) : r"
wenzelm@26174
   408
  | (step) c where "(a, c) : r^+" and "(c, b) : r"
wenzelm@26174
   409
  using assms by cases simp_all
wenzelm@10980
   410
krauss@32235
   411
lemma trancl_Int_subset: "[| r \<subseteq> s; (r^+ \<inter> s) O r \<subseteq> s|] ==> r^+ \<subseteq> s"
paulson@22080
   412
  apply (rule subsetI)
haftmann@61032
   413
  apply auto
wenzelm@26179
   414
  apply (erule trancl_induct)
haftmann@61032
   415
  apply auto
paulson@22080
   416
  done
paulson@22080
   417
krauss@32235
   418
lemma trancl_unfold: "r^+ = r Un r^+ O r"
paulson@15551
   419
  by (auto intro: trancl_into_trancl elim: tranclE)
paulson@15551
   420
wenzelm@60758
   421
text \<open>Transitivity of @{term "r^+"}\<close>
wenzelm@26179
   422
lemma trans_trancl [simp]: "trans (r^+)"
berghofe@13704
   423
proof (rule transI)
berghofe@13704
   424
  fix x y z
wenzelm@26179
   425
  assume "(x, y) \<in> r^+"
berghofe@13704
   426
  assume "(y, z) \<in> r^+"
wenzelm@26179
   427
  then show "(x, z) \<in> r^+"
wenzelm@26179
   428
  proof induct
wenzelm@26179
   429
    case (base u)
wenzelm@60758
   430
    from \<open>(x, y) \<in> r^+\<close> and \<open>(y, u) \<in> r\<close>
wenzelm@26179
   431
    show "(x, u) \<in> r^+" ..
wenzelm@26179
   432
  next
wenzelm@26179
   433
    case (step u v)
wenzelm@60758
   434
    from \<open>(x, u) \<in> r^+\<close> and \<open>(u, v) \<in> r\<close>
wenzelm@26179
   435
    show "(x, v) \<in> r^+" ..
wenzelm@26179
   436
  qed
berghofe@13704
   437
qed
wenzelm@12691
   438
wenzelm@45607
   439
lemmas trancl_trans = trans_trancl [THEN transD]
wenzelm@12691
   440
berghofe@23743
   441
lemma tranclp_trans:
berghofe@22262
   442
  assumes xy: "r^++ x y"
berghofe@22262
   443
  and yz: "r^++ y z"
berghofe@22262
   444
  shows "r^++ x z" using yz xy
berghofe@22262
   445
  by induct iprover+
berghofe@22262
   446
wenzelm@26179
   447
lemma trancl_id [simp]: "trans r \<Longrightarrow> r^+ = r"
wenzelm@26179
   448
  apply auto
wenzelm@26179
   449
  apply (erule trancl_induct)
wenzelm@26179
   450
   apply assumption
wenzelm@26179
   451
  apply (unfold trans_def)
wenzelm@26179
   452
  apply blast
wenzelm@26179
   453
  done
nipkow@19623
   454
wenzelm@26179
   455
lemma rtranclp_tranclp_tranclp:
wenzelm@26179
   456
  assumes "r^** x y"
wenzelm@26179
   457
  shows "!!z. r^++ y z ==> r^++ x z" using assms
berghofe@23743
   458
  by induct (iprover intro: tranclp_trans)+
wenzelm@12691
   459
berghofe@23743
   460
lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]
berghofe@22262
   461
berghofe@23743
   462
lemma tranclp_into_tranclp2: "r a b ==> r^++ b c ==> r^++ a c"
berghofe@23743
   463
  by (erule tranclp_trans [OF tranclp.r_into_trancl])
berghofe@22262
   464
berghofe@23743
   465
lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]
wenzelm@12691
   466
berghofe@23743
   467
lemma tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ x y"
berghofe@22262
   468
  apply (drule conversepD)
berghofe@23743
   469
  apply (erule tranclp_induct)
berghofe@23743
   470
  apply (iprover intro: conversepI tranclp_trans)+
wenzelm@12691
   471
  done
wenzelm@12691
   472
berghofe@23743
   473
lemmas trancl_converseI = tranclp_converseI [to_set]
berghofe@22262
   474
berghofe@23743
   475
lemma tranclp_converseD: "(r^--1)^++ x y ==> (r^++)^--1 x y"
berghofe@22262
   476
  apply (rule conversepI)
berghofe@23743
   477
  apply (erule tranclp_induct)
berghofe@23743
   478
  apply (iprover dest: conversepD intro: tranclp_trans)+
berghofe@13704
   479
  done
wenzelm@12691
   480
berghofe@23743
   481
lemmas trancl_converseD = tranclp_converseD [to_set]
berghofe@22262
   482
berghofe@23743
   483
lemma tranclp_converse: "(r^--1)^++ = (r^++)^--1"
nipkow@44890
   484
  by (fastforce simp add: fun_eq_iff
berghofe@23743
   485
    intro!: tranclp_converseI dest!: tranclp_converseD)
berghofe@22262
   486
berghofe@23743
   487
lemmas trancl_converse = tranclp_converse [to_set]
wenzelm@12691
   488
huffman@19228
   489
lemma sym_trancl: "sym r ==> sym (r^+)"
huffman@19228
   490
  by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
huffman@19228
   491
berghofe@34909
   492
lemma converse_tranclp_induct [consumes 1, case_names base step]:
berghofe@22262
   493
  assumes major: "r^++ a b"
berghofe@22262
   494
    and cases: "!!y. r y b ==> P(y)"
berghofe@22262
   495
      "!!y z.[| r y z;  r^++ z b;  P(z) |] ==> P(y)"
wenzelm@18372
   496
  shows "P a"
berghofe@23743
   497
  apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])
wenzelm@18372
   498
   apply (rule cases)
berghofe@22262
   499
   apply (erule conversepD)
huffman@35216
   500
  apply (blast intro: assms dest!: tranclp_converseD)
wenzelm@18372
   501
  done
wenzelm@12691
   502
berghofe@23743
   503
lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
berghofe@22262
   504
berghofe@23743
   505
lemma tranclpD: "R^++ x y ==> EX z. R x z \<and> R^** z y"
wenzelm@26179
   506
  apply (erule converse_tranclp_induct)
wenzelm@26179
   507
   apply auto
berghofe@23743
   508
  apply (blast intro: rtranclp_trans)
wenzelm@12691
   509
  done
wenzelm@12691
   510
berghofe@23743
   511
lemmas tranclD = tranclpD [to_set]
berghofe@22262
   512
bulwahn@31577
   513
lemma converse_tranclpE:
bulwahn@31577
   514
  assumes major: "tranclp r x z"
bulwahn@31577
   515
  assumes base: "r x z ==> P"
bulwahn@31577
   516
  assumes step: "\<And> y. [| r x y; tranclp r y z |] ==> P"
bulwahn@31577
   517
  shows P
bulwahn@31577
   518
proof -
bulwahn@31577
   519
  from tranclpD[OF major]
bulwahn@31577
   520
  obtain y where "r x y" and "rtranclp r y z" by iprover
bulwahn@31577
   521
  from this(2) show P
bulwahn@31577
   522
  proof (cases rule: rtranclp.cases)
bulwahn@31577
   523
    case rtrancl_refl
wenzelm@60758
   524
    with \<open>r x y\<close> base show P by iprover
bulwahn@31577
   525
  next
bulwahn@31577
   526
    case rtrancl_into_rtrancl
bulwahn@31577
   527
    from this have "tranclp r y z"
bulwahn@31577
   528
      by (iprover intro: rtranclp_into_tranclp1)
wenzelm@60758
   529
    with \<open>r x y\<close> step show P by iprover
bulwahn@31577
   530
  qed
bulwahn@31577
   531
qed
bulwahn@31577
   532
bulwahn@31577
   533
lemmas converse_tranclE = converse_tranclpE [to_set]
bulwahn@31577
   534
kleing@25295
   535
lemma tranclD2:
kleing@25295
   536
  "(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
kleing@25295
   537
  by (blast elim: tranclE intro: trancl_into_rtrancl)
kleing@25295
   538
nipkow@13867
   539
lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
wenzelm@18372
   540
  by (blast elim: tranclE dest: trancl_into_rtrancl)
wenzelm@12691
   541
wenzelm@12691
   542
lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"
wenzelm@12691
   543
  by (blast dest: r_into_trancl)
wenzelm@12691
   544
wenzelm@12691
   545
lemma trancl_subset_Sigma_aux:
wenzelm@12691
   546
    "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
wenzelm@18372
   547
  by (induct rule: rtrancl_induct) auto
wenzelm@12691
   548
wenzelm@12691
   549
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"
berghofe@13704
   550
  apply (rule subsetI)
berghofe@13704
   551
  apply (simp only: split_tupled_all)
berghofe@13704
   552
  apply (erule tranclE)
wenzelm@26179
   553
   apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
wenzelm@12691
   554
  done
nipkow@10996
   555
nipkow@50616
   556
lemma reflclp_tranclp [simp]: "(r^++)^== = r^**"
berghofe@22262
   557
  apply (safe intro!: order_antisym)
berghofe@23743
   558
   apply (erule tranclp_into_rtranclp)
berghofe@23743
   559
  apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)
wenzelm@11084
   560
  done
nipkow@10996
   561
nipkow@50616
   562
lemmas reflcl_trancl [simp] = reflclp_tranclp [to_set]
berghofe@22262
   563
wenzelm@11090
   564
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
wenzelm@11084
   565
  apply safe
paulson@14208
   566
   apply (drule trancl_into_rtrancl, simp)
paulson@14208
   567
  apply (erule rtranclE, safe)
paulson@14208
   568
   apply (rule r_into_trancl, simp)
wenzelm@11084
   569
  apply (rule rtrancl_into_trancl1)
paulson@14208
   570
   apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
wenzelm@11084
   571
  done
nipkow@10996
   572
haftmann@45140
   573
lemma rtrancl_trancl_reflcl [code]: "r^* = (r^+)^="
haftmann@45140
   574
  by simp
haftmann@45140
   575
wenzelm@11090
   576
lemma trancl_empty [simp]: "{}^+ = {}"
wenzelm@11084
   577
  by (auto elim: trancl_induct)
nipkow@10996
   578
wenzelm@11090
   579
lemma rtrancl_empty [simp]: "{}^* = Id"
wenzelm@11084
   580
  by (rule subst [OF reflcl_trancl]) simp
nipkow@10996
   581
berghofe@23743
   582
lemma rtranclpD: "R^** a b ==> a = b \<or> a \<noteq> b \<and> R^++ a b"
nipkow@50616
   583
by (force simp add: reflclp_tranclp [symmetric] simp del: reflclp_tranclp)
berghofe@22262
   584
berghofe@23743
   585
lemmas rtranclD = rtranclpD [to_set]
wenzelm@11084
   586
kleing@16514
   587
lemma rtrancl_eq_or_trancl:
kleing@16514
   588
  "(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"
kleing@16514
   589
  by (fast elim: trancl_into_rtrancl dest: rtranclD)
nipkow@10996
   590
krauss@33656
   591
lemma trancl_unfold_right: "r^+ = r^* O r"
krauss@33656
   592
by (auto dest: tranclD2 intro: rtrancl_into_trancl1)
krauss@33656
   593
krauss@33656
   594
lemma trancl_unfold_left: "r^+ = r O r^*"
krauss@33656
   595
by (auto dest: tranclD intro: rtrancl_into_trancl2)
krauss@33656
   596
nipkow@57178
   597
lemma trancl_insert:
nipkow@57178
   598
  "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
wenzelm@60758
   599
  -- \<open>primitive recursion for @{text trancl} over finite relations\<close>
nipkow@57178
   600
  apply (rule equalityI)
nipkow@57178
   601
   apply (rule subsetI)
nipkow@57178
   602
   apply (simp only: split_tupled_all)
nipkow@57178
   603
   apply (erule trancl_induct, blast)
nipkow@57178
   604
   apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)
nipkow@57178
   605
  apply (rule subsetI)
nipkow@57178
   606
  apply (blast intro: trancl_mono rtrancl_mono
nipkow@57178
   607
    [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
nipkow@57178
   608
  done
nipkow@57178
   609
nipkow@57178
   610
lemma trancl_insert2:
nipkow@57178
   611
  "(insert (a,b) r)^+ = r^+ \<union> {(x,y). ((x,a) : r^+ \<or> x=a) \<and> ((b,y) \<in> r^+ \<or> y=b)}"
nipkow@57178
   612
by(auto simp add: trancl_insert rtrancl_eq_or_trancl)
nipkow@57178
   613
nipkow@57178
   614
lemma rtrancl_insert:
nipkow@57178
   615
  "(insert (a,b) r)^* = r^* \<union> {(x,y). (x,a) : r^* \<and> (b,y) \<in> r^*}"
nipkow@57178
   616
using trancl_insert[of a b r]
nipkow@57178
   617
by(simp add: rtrancl_trancl_reflcl del: reflcl_trancl) blast
nipkow@57178
   618
krauss@33656
   619
wenzelm@60758
   620
text \<open>Simplifying nested closures\<close>
krauss@33656
   621
krauss@33656
   622
lemma rtrancl_trancl_absorb[simp]: "(R^*)^+ = R^*"
krauss@33656
   623
by (simp add: trans_rtrancl)
krauss@33656
   624
krauss@33656
   625
lemma trancl_rtrancl_absorb[simp]: "(R^+)^* = R^*"
krauss@33656
   626
by (subst reflcl_trancl[symmetric]) simp
krauss@33656
   627
krauss@33656
   628
lemma rtrancl_reflcl_absorb[simp]: "(R^*)^= = R^*"
krauss@33656
   629
by auto
krauss@33656
   630
krauss@33656
   631
wenzelm@60758
   632
text \<open>@{text Domain} and @{text Range}\<close>
nipkow@10996
   633
wenzelm@11090
   634
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
wenzelm@11084
   635
  by blast
nipkow@10996
   636
wenzelm@11090
   637
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
wenzelm@11084
   638
  by blast
nipkow@10996
   639
wenzelm@11090
   640
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
wenzelm@11084
   641
  by (rule rtrancl_Un_rtrancl [THEN subst]) fast
nipkow@10996
   642
wenzelm@11090
   643
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
wenzelm@11084
   644
  by (blast intro: subsetD [OF rtrancl_Un_subset])
nipkow@10996
   645
wenzelm@11090
   646
lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
haftmann@46752
   647
  by (unfold Domain_unfold) (blast dest: tranclD)
nipkow@10996
   648
wenzelm@11090
   649
lemma trancl_range [simp]: "Range (r^+) = Range r"
haftmann@46752
   650
  unfolding Domain_converse [symmetric] by (simp add: trancl_converse [symmetric])
nipkow@10996
   651
paulson@11115
   652
lemma Not_Domain_rtrancl:
wenzelm@12691
   653
    "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
wenzelm@12691
   654
  apply auto
wenzelm@26179
   655
  apply (erule rev_mp)
wenzelm@26179
   656
  apply (erule rtrancl_induct)
wenzelm@26179
   657
   apply auto
wenzelm@26179
   658
  done
berghofe@11327
   659
haftmann@29609
   660
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
haftmann@29609
   661
  apply clarify
haftmann@29609
   662
  apply (erule trancl_induct)
haftmann@29609
   663
   apply (auto simp add: Field_def)
haftmann@29609
   664
  done
haftmann@29609
   665
nipkow@41987
   666
lemma finite_trancl[simp]: "finite (r^+) = finite r"
haftmann@29609
   667
  apply auto
haftmann@29609
   668
   prefer 2
haftmann@29609
   669
   apply (rule trancl_subset_Field2 [THEN finite_subset])
haftmann@29609
   670
   apply (rule finite_SigmaI)
haftmann@29609
   671
    prefer 3
haftmann@29609
   672
    apply (blast intro: r_into_trancl' finite_subset)
haftmann@29609
   673
   apply (auto simp add: finite_Field)
haftmann@29609
   674
  done
haftmann@29609
   675
wenzelm@60758
   676
text \<open>More about converse @{text rtrancl} and @{text trancl}, should
wenzelm@60758
   677
  be merged with main body.\<close>
kleing@12428
   678
nipkow@14337
   679
lemma single_valued_confluent:
nipkow@14337
   680
  "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>
nipkow@14337
   681
  \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
wenzelm@26179
   682
  apply (erule rtrancl_induct)
wenzelm@26179
   683
  apply simp
wenzelm@26179
   684
  apply (erule disjE)
wenzelm@26179
   685
   apply (blast elim:converse_rtranclE dest:single_valuedD)
wenzelm@26179
   686
  apply(blast intro:rtrancl_trans)
wenzelm@26179
   687
  done
nipkow@14337
   688
wenzelm@12691
   689
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"
kleing@12428
   690
  by (fast intro: trancl_trans)
kleing@12428
   691
kleing@12428
   692
lemma trancl_into_trancl [rule_format]:
wenzelm@12691
   693
    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
wenzelm@12691
   694
  apply (erule trancl_induct)
kleing@12428
   695
   apply (fast intro: r_r_into_trancl)
kleing@12428
   696
  apply (fast intro: r_r_into_trancl trancl_trans)
kleing@12428
   697
  done
kleing@12428
   698
berghofe@23743
   699
lemma tranclp_rtranclp_tranclp:
berghofe@22262
   700
    "r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c"
berghofe@23743
   701
  apply (drule tranclpD)
wenzelm@26179
   702
  apply (elim exE conjE)
berghofe@23743
   703
  apply (drule rtranclp_trans, assumption)
berghofe@23743
   704
  apply (drule rtranclp_into_tranclp2, assumption, assumption)
kleing@12428
   705
  done
kleing@12428
   706
berghofe@23743
   707
lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]
berghofe@22262
   708
wenzelm@12691
   709
lemmas transitive_closure_trans [trans] =
wenzelm@12691
   710
  r_r_into_trancl trancl_trans rtrancl_trans
berghofe@23743
   711
  trancl.trancl_into_trancl trancl_into_trancl2
berghofe@23743
   712
  rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
wenzelm@12691
   713
  rtrancl_trancl_trancl trancl_rtrancl_trancl
kleing@12428
   714
berghofe@23743
   715
lemmas transitive_closurep_trans' [trans] =
berghofe@23743
   716
  tranclp_trans rtranclp_trans
berghofe@23743
   717
  tranclp.trancl_into_trancl tranclp_into_tranclp2
berghofe@23743
   718
  rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp
berghofe@23743
   719
  rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp
berghofe@22262
   720
kleing@12428
   721
declare trancl_into_rtrancl [elim]
berghofe@11327
   722
wenzelm@60758
   723
subsection \<open>The power operation on relations\<close>
haftmann@30954
   724
wenzelm@60758
   725
text \<open>@{text "R ^^ n = R O ... O R"}, the n-fold composition of @{text R}\<close>
haftmann@30954
   726
haftmann@30971
   727
overloading
haftmann@30971
   728
  relpow == "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
haftmann@47202
   729
  relpowp == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
haftmann@30971
   730
begin
haftmann@30954
   731
blanchet@55575
   732
primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
haftmann@30971
   733
    "relpow 0 R = Id"
krauss@32235
   734
  | "relpow (Suc n) R = (R ^^ n) O R"
haftmann@30954
   735
blanchet@55575
   736
primrec relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)" where
haftmann@47202
   737
    "relpowp 0 R = HOL.eq"
haftmann@47202
   738
  | "relpowp (Suc n) R = (R ^^ n) OO R"
haftmann@47202
   739
haftmann@30971
   740
end
haftmann@30954
   741
haftmann@47202
   742
lemma relpowp_relpow_eq [pred_set_conv]:
haftmann@47202
   743
  fixes R :: "'a rel"
haftmann@47202
   744
  shows "(\<lambda>x y. (x, y) \<in> R) ^^ n = (\<lambda>x y. (x, y) \<in> R ^^ n)"
griff@47433
   745
  by (induct n) (simp_all add: relcompp_relcomp_eq)
haftmann@47202
   746
wenzelm@60758
   747
text \<open>for code generation\<close>
bulwahn@46360
   748
bulwahn@46360
   749
definition relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
bulwahn@46360
   750
  relpow_code_def [code_abbrev]: "relpow = compow"
bulwahn@46360
   751
Christian@47492
   752
definition relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)" where
Christian@47492
   753
  relpowp_code_def [code_abbrev]: "relpowp = compow"
Christian@47492
   754
bulwahn@46360
   755
lemma [code]:
bulwahn@46360
   756
  "relpow (Suc n) R = (relpow n R) O R"
bulwahn@46360
   757
  "relpow 0 R = Id"
bulwahn@46360
   758
  by (simp_all add: relpow_code_def)
bulwahn@46360
   759
Christian@47492
   760
lemma [code]:
Christian@47492
   761
  "relpowp (Suc n) R = (R ^^ n) OO R"
Christian@47492
   762
  "relpowp 0 R = HOL.eq"
Christian@47492
   763
  by (simp_all add: relpowp_code_def)
Christian@47492
   764
bulwahn@46360
   765
hide_const (open) relpow
Christian@47492
   766
hide_const (open) relpowp
bulwahn@46360
   767
bulwahn@46362
   768
lemma relpow_1 [simp]:
haftmann@30971
   769
  fixes R :: "('a \<times> 'a) set"
haftmann@30971
   770
  shows "R ^^ 1 = R"
haftmann@30954
   771
  by simp
haftmann@30954
   772
Christian@47492
   773
lemma relpowp_1 [simp]:
Christian@47492
   774
  fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
Christian@47492
   775
  shows "P ^^ 1 = P"
Christian@47492
   776
  by (fact relpow_1 [to_pred])
Christian@47492
   777
bulwahn@46362
   778
lemma relpow_0_I: 
haftmann@30954
   779
  "(x, x) \<in> R ^^ 0"
haftmann@30954
   780
  by simp
haftmann@30954
   781
Christian@47492
   782
lemma relpowp_0_I:
Christian@47492
   783
  "(P ^^ 0) x x"
Christian@47492
   784
  by (fact relpow_0_I [to_pred])
Christian@47492
   785
bulwahn@46362
   786
lemma relpow_Suc_I:
haftmann@30954
   787
  "(x, y) \<in>  R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
haftmann@30954
   788
  by auto
haftmann@30954
   789
Christian@47492
   790
lemma relpowp_Suc_I:
Christian@47492
   791
  "(P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> (P ^^ Suc n) x z"
Christian@47492
   792
  by (fact relpow_Suc_I [to_pred])
Christian@47492
   793
bulwahn@46362
   794
lemma relpow_Suc_I2:
haftmann@30954
   795
  "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
nipkow@44890
   796
  by (induct n arbitrary: z) (simp, fastforce)
haftmann@30954
   797
Christian@47492
   798
lemma relpowp_Suc_I2:
Christian@47492
   799
  "P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> (P ^^ Suc n) x z"
Christian@47492
   800
  by (fact relpow_Suc_I2 [to_pred])
Christian@47492
   801
bulwahn@46362
   802
lemma relpow_0_E:
haftmann@30954
   803
  "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30954
   804
  by simp
haftmann@30954
   805
Christian@47492
   806
lemma relpowp_0_E:
Christian@47492
   807
  "(P ^^ 0) x y \<Longrightarrow> (x = y \<Longrightarrow> Q) \<Longrightarrow> Q"
Christian@47492
   808
  by (fact relpow_0_E [to_pred])
Christian@47492
   809
bulwahn@46362
   810
lemma relpow_Suc_E:
haftmann@30954
   811
  "(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
   812
  by auto
haftmann@30954
   813
Christian@47492
   814
lemma relpowp_Suc_E:
Christian@47492
   815
  "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. (P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> Q) \<Longrightarrow> Q"
Christian@47492
   816
  by (fact relpow_Suc_E [to_pred])
Christian@47492
   817
bulwahn@46362
   818
lemma relpow_E:
haftmann@30954
   819
  "(x, z) \<in>  R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
haftmann@30954
   820
   \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in>  R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P)
haftmann@30954
   821
   \<Longrightarrow> P"
haftmann@30954
   822
  by (cases n) auto
haftmann@30954
   823
Christian@47492
   824
lemma relpowp_E:
Christian@47492
   825
  "(P ^^ n) x z \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q)
Christian@47492
   826
  \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (P ^^ m) x y \<Longrightarrow> P y z \<Longrightarrow> Q)
Christian@47492
   827
  \<Longrightarrow> Q"
Christian@47492
   828
  by (fact relpow_E [to_pred])
Christian@47492
   829
bulwahn@46362
   830
lemma relpow_Suc_D2:
haftmann@30954
   831
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
haftmann@30954
   832
  apply (induct n arbitrary: x z)
bulwahn@46362
   833
   apply (blast intro: relpow_0_I elim: relpow_0_E relpow_Suc_E)
bulwahn@46362
   834
  apply (blast intro: relpow_Suc_I elim: relpow_0_E relpow_Suc_E)
haftmann@30954
   835
  done
haftmann@30954
   836
Christian@47492
   837
lemma relpowp_Suc_D2:
Christian@47492
   838
  "(P ^^ Suc n) x z \<Longrightarrow> \<exists>y. P x y \<and> (P ^^ n) y z"
Christian@47492
   839
  by (fact relpow_Suc_D2 [to_pred])
Christian@47492
   840
bulwahn@46362
   841
lemma relpow_Suc_E2:
haftmann@30954
   842
  "(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
   843
  by (blast dest: relpow_Suc_D2)
haftmann@30954
   844
Christian@47492
   845
lemma relpowp_Suc_E2:
Christian@47492
   846
  "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> Q) \<Longrightarrow> Q"
Christian@47492
   847
  by (fact relpow_Suc_E2 [to_pred])
Christian@47492
   848
bulwahn@46362
   849
lemma relpow_Suc_D2':
haftmann@30954
   850
  "\<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
   851
  by (induct n) (simp_all, blast)
haftmann@30954
   852
Christian@47492
   853
lemma relpowp_Suc_D2':
Christian@47492
   854
  "\<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
   855
  by (fact relpow_Suc_D2' [to_pred])
Christian@47492
   856
bulwahn@46362
   857
lemma relpow_E2:
haftmann@30954
   858
  "(x, z) \<in> R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
haftmann@30954
   859
     \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P)
haftmann@30954
   860
   \<Longrightarrow> P"
haftmann@30954
   861
  apply (cases n, simp)
blanchet@55417
   862
  apply (rename_tac nat)
bulwahn@46362
   863
  apply (cut_tac n=nat and R=R in relpow_Suc_D2', simp, blast)
haftmann@30954
   864
  done
haftmann@30954
   865
Christian@47492
   866
lemma relpowp_E2:
Christian@47492
   867
  "(P ^^ n) x z \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q)
Christian@47492
   868
    \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> P x y \<Longrightarrow> (P ^^ m) y z \<Longrightarrow> Q)
Christian@47492
   869
  \<Longrightarrow> Q"
Christian@47492
   870
  by (fact relpow_E2 [to_pred])
Christian@47492
   871
bulwahn@46362
   872
lemma relpow_add: "R ^^ (m+n) = R^^m O R^^n"
haftmann@45976
   873
  by (induct n) auto
nipkow@31351
   874
Christian@47492
   875
lemma relpowp_add: "P ^^ (m + n) = P ^^ m OO P ^^ n"
Christian@47492
   876
  by (fact relpow_add [to_pred])
Christian@47492
   877
bulwahn@46362
   878
lemma relpow_commute: "R O R ^^ n = R ^^ n O R"
haftmann@45976
   879
  by (induct n) (simp, simp add: O_assoc [symmetric])
krauss@31970
   880
Christian@47492
   881
lemma relpowp_commute: "P OO P ^^ n = P ^^ n OO P"
Christian@47492
   882
  by (fact relpow_commute [to_pred])
Christian@47492
   883
bulwahn@46362
   884
lemma relpow_empty:
haftmann@45153
   885
  "0 < n \<Longrightarrow> ({} :: ('a \<times> 'a) set) ^^ n = {}"
haftmann@45153
   886
  by (cases n) auto
bulwahn@45116
   887
Christian@47492
   888
lemma relpowp_bot:
Christian@47492
   889
  "0 < n \<Longrightarrow> (\<bottom> :: 'a \<Rightarrow> 'a \<Rightarrow> bool) ^^ n = \<bottom>"
Christian@47492
   890
  by (fact relpow_empty [to_pred])
Christian@47492
   891
bulwahn@46362
   892
lemma rtrancl_imp_UN_relpow:
haftmann@30954
   893
  assumes "p \<in> R^*"
haftmann@30954
   894
  shows "p \<in> (\<Union>n. R ^^ n)"
haftmann@30954
   895
proof (cases p)
haftmann@30954
   896
  case (Pair x y)
haftmann@30954
   897
  with assms have "(x, y) \<in> R^*" by simp
haftmann@30954
   898
  then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct
bulwahn@46362
   899
    case base show ?case by (blast intro: relpow_0_I)
haftmann@30954
   900
  next
bulwahn@46362
   901
    case step then show ?case by (blast intro: relpow_Suc_I)
haftmann@30954
   902
  qed
haftmann@30954
   903
  with Pair show ?thesis by simp
haftmann@30954
   904
qed
haftmann@30954
   905
Christian@47492
   906
lemma rtranclp_imp_Sup_relpowp:
Christian@47492
   907
  assumes "(P^**) x y"
Christian@47492
   908
  shows "(\<Squnion>n. P ^^ n) x y"
Christian@47492
   909
  using assms and rtrancl_imp_UN_relpow [to_pred] by blast
Christian@47492
   910
bulwahn@46362
   911
lemma relpow_imp_rtrancl:
haftmann@30954
   912
  assumes "p \<in> R ^^ n"
haftmann@30954
   913
  shows "p \<in> R^*"
haftmann@30954
   914
proof (cases p)
haftmann@30954
   915
  case (Pair x y)
haftmann@30954
   916
  with assms have "(x, y) \<in> R ^^ n" by simp
haftmann@30954
   917
  then have "(x, y) \<in> R^*" proof (induct n arbitrary: x y)
haftmann@30954
   918
    case 0 then show ?case by simp
haftmann@30954
   919
  next
haftmann@30954
   920
    case Suc then show ?case
bulwahn@46362
   921
      by (blast elim: relpow_Suc_E intro: rtrancl_into_rtrancl)
haftmann@30954
   922
  qed
haftmann@30954
   923
  with Pair show ?thesis by simp
haftmann@30954
   924
qed
haftmann@30954
   925
Christian@47492
   926
lemma relpowp_imp_rtranclp:
Christian@47492
   927
  assumes "(P ^^ n) x y"
Christian@47492
   928
  shows "(P^**) x y"
Christian@47492
   929
  using assms and relpow_imp_rtrancl [to_pred] by blast
Christian@47492
   930
bulwahn@46362
   931
lemma rtrancl_is_UN_relpow:
haftmann@30954
   932
  "R^* = (\<Union>n. R ^^ n)"
bulwahn@46362
   933
  by (blast intro: rtrancl_imp_UN_relpow relpow_imp_rtrancl)
haftmann@30954
   934
Christian@47492
   935
lemma rtranclp_is_Sup_relpowp:
Christian@47492
   936
  "P^** = (\<Squnion>n. P ^^ n)"
Christian@47492
   937
  using rtrancl_is_UN_relpow [to_pred, of P] by auto
Christian@47492
   938
haftmann@30954
   939
lemma rtrancl_power:
haftmann@30954
   940
  "p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
bulwahn@46362
   941
  by (simp add: rtrancl_is_UN_relpow)
haftmann@30954
   942
Christian@47492
   943
lemma rtranclp_power:
Christian@47492
   944
  "(P^**) x y \<longleftrightarrow> (\<exists>n. (P ^^ n) x y)"
Christian@47492
   945
  by (simp add: rtranclp_is_Sup_relpowp)
Christian@47492
   946
haftmann@30954
   947
lemma trancl_power:
haftmann@30954
   948
  "p \<in> R^+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
haftmann@30954
   949
  apply (cases p)
haftmann@30954
   950
  apply simp
haftmann@30954
   951
  apply (rule iffI)
haftmann@30954
   952
   apply (drule tranclD2)
bulwahn@46362
   953
   apply (clarsimp simp: rtrancl_is_UN_relpow)
haftmann@30971
   954
   apply (rule_tac x="Suc n" in exI)
griff@47433
   955
   apply (clarsimp simp: relcomp_unfold)
nipkow@44890
   956
   apply fastforce
haftmann@30954
   957
  apply clarsimp
haftmann@30954
   958
  apply (case_tac n, simp)
haftmann@30954
   959
  apply clarsimp
bulwahn@46362
   960
  apply (drule relpow_imp_rtrancl)
haftmann@30954
   961
  apply (drule rtrancl_into_trancl1) apply auto
haftmann@30954
   962
  done
haftmann@30954
   963
Christian@47492
   964
lemma tranclp_power:
Christian@47492
   965
  "(P^++) x y \<longleftrightarrow> (\<exists>n > 0. (P ^^ n) x y)"
Christian@47492
   966
  using trancl_power [to_pred, of P "(x, y)"] by simp
Christian@47492
   967
bulwahn@46362
   968
lemma rtrancl_imp_relpow:
haftmann@30954
   969
  "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
bulwahn@46362
   970
  by (auto dest: rtrancl_imp_UN_relpow)
haftmann@30954
   971
Christian@47492
   972
lemma rtranclp_imp_relpowp:
Christian@47492
   973
  "(P^**) x y \<Longrightarrow> \<exists>n. (P ^^ n) x y"
Christian@47492
   974
  by (auto dest: rtranclp_imp_Sup_relpowp)
Christian@47492
   975
wenzelm@60758
   976
text\<open>By Sternagel/Thiemann:\<close>
bulwahn@46362
   977
lemma relpow_fun_conv:
nipkow@41987
   978
  "((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
   979
proof (induct n arbitrary: b)
nipkow@41987
   980
  case 0 show ?case by auto
nipkow@41987
   981
next
nipkow@41987
   982
  case (Suc n)
nipkow@41987
   983
  show ?case
griff@47433
   984
  proof (simp add: relcomp_unfold Suc)
nipkow@41987
   985
    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
   986
     = (\<exists>f. f 0 = a \<and> f(Suc n) = b \<and> (\<forall>i<Suc n. (f i, f (Suc i)) \<in> R))"
nipkow@41987
   987
    (is "?l = ?r")
nipkow@41987
   988
    proof
nipkow@41987
   989
      assume ?l
nipkow@41987
   990
      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
   991
      let ?g = "\<lambda> m. if m = Suc n then b else f m"
nipkow@41987
   992
      show ?r by (rule exI[of _ ?g], simp add: 1)
nipkow@41987
   993
    next
nipkow@41987
   994
      assume ?r
nipkow@41987
   995
      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
   996
      show ?l by (rule exI[of _ "f n"], rule conjI, rule exI[of _ f], insert 1, auto)
nipkow@41987
   997
    qed
nipkow@41987
   998
  qed
nipkow@41987
   999
qed
nipkow@41987
  1000
Christian@47492
  1001
lemma relpowp_fun_conv:
Christian@47492
  1002
  "(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
  1003
  by (fact relpow_fun_conv [to_pred])
Christian@47492
  1004
bulwahn@46362
  1005
lemma relpow_finite_bounded1:
nipkow@41987
  1006
assumes "finite(R :: ('a*'a)set)" and "k>0"
nipkow@41987
  1007
shows "R^^k \<subseteq> (UN n:{n. 0<n & n <= card R}. R^^n)" (is "_ \<subseteq> ?r")
nipkow@41987
  1008
proof-
nipkow@41987
  1009
  { fix a b k
nipkow@41987
  1010
    have "(a,b) : R^^(Suc k) \<Longrightarrow> EX n. 0<n & n <= card R & (a,b) : R^^n"
nipkow@41987
  1011
    proof(induct k arbitrary: b)
nipkow@41987
  1012
      case 0
nipkow@41987
  1013
      hence "R \<noteq> {}" by auto
wenzelm@60758
  1014
      with card_0_eq[OF \<open>finite R\<close>] have "card R >= Suc 0" by auto
nipkow@41987
  1015
      thus ?case using 0 by force
nipkow@41987
  1016
    next
nipkow@41987
  1017
      case (Suc k)
nipkow@41987
  1018
      then obtain a' where "(a,a') : R^^(Suc k)" and "(a',b) : R" by auto
wenzelm@60758
  1019
      from Suc(1)[OF \<open>(a,a') : R^^(Suc k)\<close>]
nipkow@41987
  1020
      obtain n where "n \<le> card R" and "(a,a') \<in> R ^^ n" by auto
wenzelm@60758
  1021
      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
  1022
      { assume "n < card R"
wenzelm@60758
  1023
        hence ?case using \<open>(a,b): R^^(Suc n)\<close> Suc_leI[OF \<open>n < card R\<close>] by blast
nipkow@41987
  1024
      } moreover
nipkow@41987
  1025
      { assume "n = card R"
wenzelm@60758
  1026
        from \<open>(a,b) \<in> R ^^ (Suc n)\<close>[unfolded relpow_fun_conv]
nipkow@41987
  1027
        obtain f where "f 0 = a" and "f(Suc n) = b"
nipkow@41987
  1028
          and steps: "\<And>i. i <= n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto
nipkow@41987
  1029
        let ?p = "%i. (f i, f(Suc i))"
nipkow@41987
  1030
        let ?N = "{i. i \<le> n}"
nipkow@41987
  1031
        have "?p ` ?N <= R" using steps by auto
nipkow@41987
  1032
        from card_mono[OF assms(1) this]
nipkow@41987
  1033
        have "card(?p ` ?N) <= card R" .
wenzelm@60758
  1034
        also have "\<dots> < card ?N" using \<open>n = card R\<close> by simp
nipkow@41987
  1035
        finally have "~ inj_on ?p ?N" by(rule pigeonhole)
nipkow@41987
  1036
        then obtain i j where i: "i <= n" and j: "j <= n" and ij: "i \<noteq> j" and
nipkow@41987
  1037
          pij: "?p i = ?p j" by(auto simp: inj_on_def)
nipkow@41987
  1038
        let ?i = "min i j" let ?j = "max i j"
nipkow@41987
  1039
        have i: "?i <= n" and j: "?j <= n" and pij: "?p ?i = ?p ?j" 
nipkow@41987
  1040
          and ij: "?i < ?j"
nipkow@41987
  1041
          using i j ij pij unfolding min_def max_def by auto
nipkow@41987
  1042
        from i j pij ij obtain i j where i: "i<=n" and j: "j<=n" and ij: "i<j"
nipkow@41987
  1043
          and pij: "?p i = ?p j" by blast
nipkow@41987
  1044
        let ?g = "\<lambda> l. if l \<le> i then f l else f (l + (j - i))"
nipkow@41987
  1045
        let ?n = "Suc(n - (j - i))"
bulwahn@46362
  1046
        have abl: "(a,b) \<in> R ^^ ?n" unfolding relpow_fun_conv
nipkow@41987
  1047
        proof (rule exI[of _ ?g], intro conjI impI allI)
wenzelm@60758
  1048
          show "?g ?n = b" using \<open>f(Suc n) = b\<close> j ij by auto
nipkow@41987
  1049
        next
nipkow@41987
  1050
          fix k assume "k < ?n"
nipkow@41987
  1051
          show "(?g k, ?g (Suc k)) \<in> R"
nipkow@41987
  1052
          proof (cases "k < i")
nipkow@41987
  1053
            case True
nipkow@41987
  1054
            with i have "k <= n" by auto
nipkow@41987
  1055
            from steps[OF this] show ?thesis using True by simp
nipkow@41987
  1056
          next
nipkow@41987
  1057
            case False
nipkow@41987
  1058
            hence "i \<le> k" by auto
nipkow@41987
  1059
            show ?thesis
nipkow@41987
  1060
            proof (cases "k = i")
nipkow@41987
  1061
              case True
nipkow@41987
  1062
              thus ?thesis using ij pij steps[OF i] by simp
nipkow@41987
  1063
            next
nipkow@41987
  1064
              case False
wenzelm@60758
  1065
              with \<open>i \<le> k\<close> have "i < k" by auto
wenzelm@60758
  1066
              hence small: "k + (j - i) <= n" using \<open>k<?n\<close> by arith
wenzelm@60758
  1067
              show ?thesis using steps[OF small] \<open>i<k\<close> by auto
nipkow@41987
  1068
            qed
nipkow@41987
  1069
          qed
wenzelm@60758
  1070
        qed (simp add: \<open>f 0 = a\<close>)
nipkow@41987
  1071
        moreover have "?n <= n" using i j ij by arith
wenzelm@60758
  1072
        ultimately have ?case using \<open>n = card R\<close> by blast
nipkow@41987
  1073
      }
wenzelm@60758
  1074
      ultimately show ?case using \<open>n \<le> card R\<close> by force
nipkow@41987
  1075
    qed
nipkow@41987
  1076
  }
wenzelm@60758
  1077
  thus ?thesis using gr0_implies_Suc[OF \<open>k>0\<close>] by auto
nipkow@41987
  1078
qed
nipkow@41987
  1079
bulwahn@46362
  1080
lemma relpow_finite_bounded:
nipkow@41987
  1081
assumes "finite(R :: ('a*'a)set)"
nipkow@41987
  1082
shows "R^^k \<subseteq> (UN n:{n. n <= card R}. R^^n)"
nipkow@41987
  1083
apply(cases k)
nipkow@41987
  1084
 apply force
bulwahn@46362
  1085
using relpow_finite_bounded1[OF assms, of k] by auto
nipkow@41987
  1086
bulwahn@46362
  1087
lemma rtrancl_finite_eq_relpow:
nipkow@41987
  1088
  "finite R \<Longrightarrow> R^* = (UN n : {n. n <= card R}. R^^n)"
bulwahn@46362
  1089
by(fastforce simp: rtrancl_power dest: relpow_finite_bounded)
nipkow@41987
  1090
bulwahn@46362
  1091
lemma trancl_finite_eq_relpow:
nipkow@41987
  1092
  "finite R \<Longrightarrow> R^+ = (UN n : {n. 0 < n & n <= card R}. R^^n)"
nipkow@41987
  1093
apply(auto simp add: trancl_power)
bulwahn@46362
  1094
apply(auto dest: relpow_finite_bounded1)
nipkow@41987
  1095
done
nipkow@41987
  1096
griff@47433
  1097
lemma finite_relcomp[simp,intro]:
nipkow@41987
  1098
assumes "finite R" and "finite S"
nipkow@41987
  1099
shows "finite(R O S)"
nipkow@41987
  1100
proof-
nipkow@41987
  1101
  have "R O S = (UN (x,y) : R. \<Union>((%(u,v). if u=y then {(x,v)} else {}) ` S))"
nipkow@41987
  1102
    by(force simp add: split_def)
nipkow@41987
  1103
  thus ?thesis using assms by(clarsimp)
nipkow@41987
  1104
qed
nipkow@41987
  1105
nipkow@41987
  1106
lemma finite_relpow[simp,intro]:
nipkow@41987
  1107
  assumes "finite(R :: ('a*'a)set)" shows "n>0 \<Longrightarrow> finite(R^^n)"
nipkow@41987
  1108
apply(induct n)
nipkow@41987
  1109
 apply simp
nipkow@41987
  1110
apply(case_tac n)
nipkow@41987
  1111
 apply(simp_all add: assms)
nipkow@41987
  1112
done
nipkow@41987
  1113
bulwahn@46362
  1114
lemma single_valued_relpow:
haftmann@30954
  1115
  fixes R :: "('a * 'a) set"
haftmann@30954
  1116
  shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
nipkow@41987
  1117
apply (induct n arbitrary: R)
nipkow@41987
  1118
apply simp_all
nipkow@41987
  1119
apply (rule single_valuedI)
bulwahn@46362
  1120
apply (fast dest: single_valuedD elim: relpow_Suc_E)
nipkow@41987
  1121
done
paulson@15551
  1122
haftmann@45140
  1123
wenzelm@60758
  1124
subsection \<open>Bounded transitive closure\<close>
haftmann@45140
  1125
haftmann@45140
  1126
definition ntrancl :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
haftmann@45140
  1127
where
haftmann@45140
  1128
  "ntrancl n R = (\<Union>i\<in>{i. 0 < i \<and> i \<le> Suc n}. R ^^ i)"
haftmann@45140
  1129
haftmann@45140
  1130
lemma ntrancl_Zero [simp, code]:
haftmann@45140
  1131
  "ntrancl 0 R = R"
haftmann@45140
  1132
proof
haftmann@45140
  1133
  show "R \<subseteq> ntrancl 0 R"
haftmann@45140
  1134
    unfolding ntrancl_def by fastforce
haftmann@45140
  1135
next
haftmann@45140
  1136
  { 
haftmann@45140
  1137
    fix i have "0 < i \<and> i \<le> Suc 0 \<longleftrightarrow> i = 1" by auto
haftmann@45140
  1138
  }
haftmann@45140
  1139
  from this show "ntrancl 0 R \<le> R"
haftmann@45140
  1140
    unfolding ntrancl_def by auto
haftmann@45140
  1141
qed
haftmann@45140
  1142
bulwahn@46347
  1143
lemma ntrancl_Suc [simp]:
haftmann@45140
  1144
  "ntrancl (Suc n) R = ntrancl n R O (Id \<union> R)"
haftmann@45140
  1145
proof
haftmann@45140
  1146
  {
haftmann@45140
  1147
    fix a b
haftmann@45140
  1148
    assume "(a, b) \<in> ntrancl (Suc n) R"
haftmann@45140
  1149
    from this obtain i where "0 < i" "i \<le> Suc (Suc n)" "(a, b) \<in> R ^^ i"
haftmann@45140
  1150
      unfolding ntrancl_def by auto
haftmann@45140
  1151
    have "(a, b) \<in> ntrancl n R O (Id \<union> R)"
haftmann@45140
  1152
    proof (cases "i = 1")
haftmann@45140
  1153
      case True
wenzelm@60758
  1154
      from this \<open>(a, b) \<in> R ^^ i\<close> show ?thesis
haftmann@45140
  1155
        unfolding ntrancl_def by auto
haftmann@45140
  1156
    next
haftmann@45140
  1157
      case False
wenzelm@60758
  1158
      from this \<open>0 < i\<close> obtain j where j: "i = Suc j" "0 < j"
haftmann@45140
  1159
        by (cases i) auto
wenzelm@60758
  1160
      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
  1161
        by auto
wenzelm@60758
  1162
      from c1 j \<open>i \<le> Suc (Suc n)\<close> have "(a, c) \<in> ntrancl n R"
haftmann@45140
  1163
        unfolding ntrancl_def by fastforce
haftmann@45140
  1164
      from this c2 show ?thesis by fastforce
haftmann@45140
  1165
    qed
haftmann@45140
  1166
  }
haftmann@45140
  1167
  from this show "ntrancl (Suc n) R \<subseteq> ntrancl n R O (Id \<union> R)"
haftmann@45140
  1168
    by auto
haftmann@45140
  1169
next
haftmann@45140
  1170
  show "ntrancl n R O (Id \<union> R) \<subseteq> ntrancl (Suc n) R"
haftmann@45140
  1171
    unfolding ntrancl_def by fastforce
haftmann@45140
  1172
qed
haftmann@45140
  1173
bulwahn@46347
  1174
lemma [code]:
bulwahn@46347
  1175
  "ntrancl (Suc n) r = (let r' = ntrancl n r in r' Un r' O r)"
bulwahn@46347
  1176
unfolding Let_def by auto
bulwahn@46347
  1177
haftmann@45141
  1178
lemma finite_trancl_ntranl:
haftmann@45140
  1179
  "finite R \<Longrightarrow> trancl R = ntrancl (card R - 1) R"
bulwahn@46362
  1180
  by (cases "card R") (auto simp add: trancl_finite_eq_relpow relpow_empty ntrancl_def)
haftmann@45140
  1181
haftmann@45140
  1182
wenzelm@60758
  1183
subsection \<open>Acyclic relations\<close>
haftmann@45139
  1184
haftmann@45139
  1185
definition acyclic :: "('a * 'a) set => bool" where
haftmann@45139
  1186
  "acyclic r \<longleftrightarrow> (!x. (x,x) ~: r^+)"
haftmann@45139
  1187
haftmann@45139
  1188
abbreviation acyclicP :: "('a => 'a => bool) => bool" where
haftmann@45139
  1189
  "acyclicP r \<equiv> acyclic {(x, y). r x y}"
haftmann@45139
  1190
haftmann@46127
  1191
lemma acyclic_irrefl [code]:
haftmann@45139
  1192
  "acyclic r \<longleftrightarrow> irrefl (r^+)"
haftmann@45139
  1193
  by (simp add: acyclic_def irrefl_def)
haftmann@45139
  1194
haftmann@45139
  1195
lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
haftmann@45139
  1196
  by (simp add: acyclic_def)
haftmann@45139
  1197
hoelzl@54412
  1198
lemma (in order) acyclicI_order:
hoelzl@54412
  1199
  assumes *: "\<And>a b. (a, b) \<in> r \<Longrightarrow> f b < f a"
hoelzl@54412
  1200
  shows "acyclic r"
hoelzl@54412
  1201
proof -
hoelzl@54412
  1202
  { fix a b assume "(a, b) \<in> r\<^sup>+"
hoelzl@54412
  1203
    then have "f b < f a"
hoelzl@54412
  1204
      by induct (auto intro: * less_trans) }
hoelzl@54412
  1205
  then show ?thesis
hoelzl@54412
  1206
    by (auto intro!: acyclicI)
hoelzl@54412
  1207
qed
hoelzl@54412
  1208
haftmann@45139
  1209
lemma acyclic_insert [iff]:
haftmann@45139
  1210
     "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
haftmann@45139
  1211
apply (simp add: acyclic_def trancl_insert)
haftmann@45139
  1212
apply (blast intro: rtrancl_trans)
haftmann@45139
  1213
done
haftmann@45139
  1214
haftmann@45139
  1215
lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
haftmann@45139
  1216
by (simp add: acyclic_def trancl_converse)
haftmann@45139
  1217
haftmann@45139
  1218
lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
haftmann@45139
  1219
haftmann@45139
  1220
lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
haftmann@45139
  1221
apply (simp add: acyclic_def antisym_def)
haftmann@45139
  1222
apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
haftmann@45139
  1223
done
haftmann@45139
  1224
haftmann@45139
  1225
(* Other direction:
haftmann@45139
  1226
acyclic = no loops
haftmann@45139
  1227
antisym = only self loops
haftmann@45139
  1228
Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
haftmann@45139
  1229
==> antisym( r^* ) = acyclic(r - Id)";
haftmann@45139
  1230
*)
haftmann@45139
  1231
haftmann@45139
  1232
lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
haftmann@45139
  1233
apply (simp add: acyclic_def)
haftmann@45139
  1234
apply (blast intro: trancl_mono)
haftmann@45139
  1235
done
haftmann@45139
  1236
haftmann@45139
  1237
wenzelm@60758
  1238
subsection \<open>Setup of transitivity reasoner\<close>
ballarin@15076
  1239
wenzelm@60758
  1240
ML \<open>
ballarin@15076
  1241
wenzelm@32215
  1242
structure Trancl_Tac = Trancl_Tac
wenzelm@32215
  1243
(
wenzelm@32215
  1244
  val r_into_trancl = @{thm trancl.r_into_trancl};
wenzelm@32215
  1245
  val trancl_trans  = @{thm trancl_trans};
wenzelm@32215
  1246
  val rtrancl_refl = @{thm rtrancl.rtrancl_refl};
wenzelm@32215
  1247
  val r_into_rtrancl = @{thm r_into_rtrancl};
wenzelm@32215
  1248
  val trancl_into_rtrancl = @{thm trancl_into_rtrancl};
wenzelm@32215
  1249
  val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl};
wenzelm@32215
  1250
  val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl};
wenzelm@32215
  1251
  val rtrancl_trans = @{thm rtrancl_trans};
ballarin@15096
  1252
berghofe@30107
  1253
  fun decomp (@{const Trueprop} $ t) =
haftmann@37677
  1254
    let fun dec (Const (@{const_name Set.member}, _) $ (Const (@{const_name Pair}, _) $ a $ b) $ rel ) =
wenzelm@56257
  1255
        let fun decr (Const (@{const_name rtrancl}, _ ) $ r) = (r,"r*")
wenzelm@56257
  1256
              | decr (Const (@{const_name trancl}, _ ) $ r)  = (r,"r+")
wenzelm@18372
  1257
              | decr r = (r,"r");
berghofe@26801
  1258
            val (rel,r) = decr (Envir.beta_eta_contract rel);
wenzelm@18372
  1259
        in SOME (a,b,rel,r) end
wenzelm@18372
  1260
      | dec _ =  NONE
berghofe@30107
  1261
    in dec t end
berghofe@30107
  1262
    | decomp _ = NONE;
wenzelm@32215
  1263
);
ballarin@15076
  1264
wenzelm@32215
  1265
structure Tranclp_Tac = Trancl_Tac
wenzelm@32215
  1266
(
wenzelm@32215
  1267
  val r_into_trancl = @{thm tranclp.r_into_trancl};
wenzelm@32215
  1268
  val trancl_trans  = @{thm tranclp_trans};
wenzelm@32215
  1269
  val rtrancl_refl = @{thm rtranclp.rtrancl_refl};
wenzelm@32215
  1270
  val r_into_rtrancl = @{thm r_into_rtranclp};
wenzelm@32215
  1271
  val trancl_into_rtrancl = @{thm tranclp_into_rtranclp};
wenzelm@32215
  1272
  val rtrancl_trancl_trancl = @{thm rtranclp_tranclp_tranclp};
wenzelm@32215
  1273
  val trancl_rtrancl_trancl = @{thm tranclp_rtranclp_tranclp};
wenzelm@32215
  1274
  val rtrancl_trans = @{thm rtranclp_trans};
berghofe@22262
  1275
berghofe@30107
  1276
  fun decomp (@{const Trueprop} $ t) =
berghofe@22262
  1277
    let fun dec (rel $ a $ b) =
wenzelm@56257
  1278
        let fun decr (Const (@{const_name rtranclp}, _ ) $ r) = (r,"r*")
wenzelm@56257
  1279
              | decr (Const (@{const_name tranclp}, _ ) $ r)  = (r,"r+")
berghofe@22262
  1280
              | decr r = (r,"r");
berghofe@22262
  1281
            val (rel,r) = decr rel;
berghofe@26801
  1282
        in SOME (a, b, rel, r) end
berghofe@22262
  1283
      | dec _ =  NONE
berghofe@30107
  1284
    in dec t end
berghofe@30107
  1285
    | decomp _ = NONE;
wenzelm@32215
  1286
);
wenzelm@60758
  1287
\<close>
berghofe@22262
  1288
wenzelm@60758
  1289
setup \<open>
wenzelm@51717
  1290
  map_theory_simpset (fn ctxt => ctxt
wenzelm@51717
  1291
    addSolver (mk_solver "Trancl" Trancl_Tac.trancl_tac)
wenzelm@51717
  1292
    addSolver (mk_solver "Rtrancl" Trancl_Tac.rtrancl_tac)
wenzelm@51717
  1293
    addSolver (mk_solver "Tranclp" Tranclp_Tac.trancl_tac)
wenzelm@51717
  1294
    addSolver (mk_solver "Rtranclp" Tranclp_Tac.rtrancl_tac))
wenzelm@60758
  1295
\<close>
ballarin@15076
  1296
wenzelm@32215
  1297
wenzelm@60758
  1298
text \<open>Optional methods.\<close>
ballarin@15076
  1299
ballarin@15076
  1300
method_setup trancl =
wenzelm@60758
  1301
  \<open>Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.trancl_tac)\<close>
wenzelm@60758
  1302
  \<open>simple transitivity reasoner\<close>
ballarin@15076
  1303
method_setup rtrancl =
wenzelm@60758
  1304
  \<open>Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.rtrancl_tac)\<close>
wenzelm@60758
  1305
  \<open>simple transitivity reasoner\<close>
berghofe@22262
  1306
method_setup tranclp =
wenzelm@60758
  1307
  \<open>Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.trancl_tac)\<close>
wenzelm@60758
  1308
  \<open>simple transitivity reasoner (predicate version)\<close>
berghofe@22262
  1309
method_setup rtranclp =
wenzelm@60758
  1310
  \<open>Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.rtrancl_tac)\<close>
wenzelm@60758
  1311
  \<open>simple transitivity reasoner (predicate version)\<close>
ballarin@15076
  1312
nipkow@10213
  1313
end