src/HOL/Transitive_Closure.thy
author hoelzl
Thu Jan 31 11:31:27 2013 +0100 (2013-01-31)
changeset 50999 3de230ed0547
parent 50616 5b6cf0fbc329
child 51717 9e7d1c139569
permissions -rw-r--r--
introduce order topology
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(*  Title:      HOL/Transitive_Closure.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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*)
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header {* Reflexive and Transitive closure of a relation *}
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theory Transitive_Closure
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imports Relation
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begin
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ML_file "~~/src/Provers/trancl.ML"
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text {*
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  @{text rtrancl} is reflexive/transitive closure,
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  @{text trancl} is transitive closure,
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  @{text reflcl} is reflexive closure.
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  These postfix operators have \emph{maximum priority}, forcing their
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  operands to be atomic.
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*}
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inductive_set
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  rtrancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   ("(_^*)" [1000] 999)
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  for r :: "('a \<times> 'a) set"
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where
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    rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) : r^*"
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  | rtrancl_into_rtrancl [Pure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"
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inductive_set
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  trancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"  ("(_^+)" [1000] 999)
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  for r :: "('a \<times> 'a) set"
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where
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    r_into_trancl [intro, Pure.intro]: "(a, b) : r ==> (a, b) : r^+"
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  | trancl_into_trancl [Pure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a, c) : r^+"
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declare rtrancl_def [nitpick_unfold del]
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        rtranclp_def [nitpick_unfold del]
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        trancl_def [nitpick_unfold del]
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        tranclp_def [nitpick_unfold del]
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notation
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  rtranclp  ("(_^**)" [1000] 1000) and
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  tranclp  ("(_^++)" [1000] 1000)
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abbreviation
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  reflclp :: "('a => 'a => bool) => 'a => 'a => bool"  ("(_^==)" [1000] 1000) where
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  "r^== \<equiv> sup r op ="
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abbreviation
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  reflcl :: "('a \<times> 'a) set => ('a \<times> 'a) set"  ("(_^=)" [1000] 999) where
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  "r^= \<equiv> r \<union> Id"
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notation (xsymbols)
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  rtranclp  ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
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  tranclp  ("(_\<^sup>+\<^sup>+)" [1000] 1000) and
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  reflclp  ("(_\<^sup>=\<^sup>=)" [1000] 1000) and
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  rtrancl  ("(_\<^sup>*)" [1000] 999) and
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  trancl  ("(_\<^sup>+)" [1000] 999) and
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  reflcl  ("(_\<^sup>=)" [1000] 999)
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notation (HTML output)
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  rtranclp  ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
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  tranclp  ("(_\<^sup>+\<^sup>+)" [1000] 1000) and
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  reflclp  ("(_\<^sup>=\<^sup>=)" [1000] 1000) and
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  rtrancl  ("(_\<^sup>*)" [1000] 999) and
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  trancl  ("(_\<^sup>+)" [1000] 999) and
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  reflcl  ("(_\<^sup>=)" [1000] 999)
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subsection {* Reflexive closure *}
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lemma refl_reflcl[simp]: "refl(r^=)"
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by(simp add:refl_on_def)
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lemma antisym_reflcl[simp]: "antisym(r^=) = antisym r"
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by(simp add:antisym_def)
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lemma trans_reflclI[simp]: "trans r \<Longrightarrow> trans(r^=)"
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unfolding trans_def by blast
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lemma reflclp_idemp [simp]: "(P^==)^==  =  P^=="
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by blast
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subsection {* Reflexive-transitive closure *}
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lemma reflcl_set_eq [pred_set_conv]: "(sup (\<lambda>x y. (x, y) \<in> r) op =) = (\<lambda>x y. (x, y) \<in> r \<union> Id)"
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  by (auto simp add: fun_eq_iff)
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lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*"
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  -- {* @{text rtrancl} of @{text r} contains @{text r} *}
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  apply (simp only: split_tupled_all)
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  apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl])
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  done
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lemma r_into_rtranclp [intro]: "r x y ==> r^** x y"
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  -- {* @{text rtrancl} of @{text r} contains @{text r} *}
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  by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl])
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lemma rtranclp_mono: "r \<le> s ==> r^** \<le> s^**"
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  -- {* monotonicity of @{text rtrancl} *}
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  apply (rule predicate2I)
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  apply (erule rtranclp.induct)
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   apply (rule_tac [2] rtranclp.rtrancl_into_rtrancl, blast+)
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  done
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lemmas rtrancl_mono = rtranclp_mono [to_set]
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theorem rtranclp_induct [consumes 1, case_names base step, induct set: rtranclp]:
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  assumes a: "r^** a b"
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    and cases: "P a" "!!y z. [| r^** a y; r y z; P y |] ==> P z"
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  shows "P b" using a
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  by (induct x\<equiv>a b) (rule cases)+
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lemmas rtrancl_induct [induct set: rtrancl] = rtranclp_induct [to_set]
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lemmas rtranclp_induct2 =
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  rtranclp_induct[of _ "(ax,ay)" "(bx,by)", split_rule,
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                 consumes 1, case_names refl step]
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lemmas rtrancl_induct2 =
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  rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
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                 consumes 1, case_names refl step]
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lemma refl_rtrancl: "refl (r^*)"
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by (unfold refl_on_def) fast
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text {* Transitivity of transitive closure. *}
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lemma trans_rtrancl: "trans (r^*)"
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proof (rule transI)
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  fix x y z
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  assume "(x, y) \<in> r\<^sup>*"
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  assume "(y, z) \<in> r\<^sup>*"
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  then show "(x, z) \<in> r\<^sup>*"
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  proof induct
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    case base
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    show "(x, y) \<in> r\<^sup>*" by fact
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  next
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    case (step u v)
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    from `(x, u) \<in> r\<^sup>*` and `(u, v) \<in> r`
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    show "(x, v) \<in> r\<^sup>*" ..
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  qed
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qed
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lemmas rtrancl_trans = trans_rtrancl [THEN transD]
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lemma rtranclp_trans:
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  assumes xy: "r^** x y"
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  and yz: "r^** y z"
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  shows "r^** x z" using yz xy
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  by induct iprover+
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lemma rtranclE [cases set: rtrancl]:
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  assumes major: "(a::'a, b) : r^*"
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  obtains
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    (base) "a = b"
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  | (step) y where "(a, y) : r^*" and "(y, b) : r"
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  -- {* elimination of @{text rtrancl} -- by induction on a special formula *}
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  apply (subgoal_tac "(a::'a) = b | (EX y. (a,y) : r^* & (y,b) : r)")
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   apply (rule_tac [2] major [THEN rtrancl_induct])
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    prefer 2 apply blast
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   prefer 2 apply blast
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  apply (erule asm_rl exE disjE conjE base step)+
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  done
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lemma rtrancl_Int_subset: "[| Id \<subseteq> s; (r^* \<inter> s) O r \<subseteq> s|] ==> r^* \<subseteq> s"
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  apply (rule subsetI)
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  apply (rule_tac p="x" in PairE, clarify)
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  apply (erule rtrancl_induct, auto) 
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  done
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lemma converse_rtranclp_into_rtranclp:
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  "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>*\<^sup>* a c"
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  by (rule rtranclp_trans) iprover+
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lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set]
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text {*
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  \medskip More @{term "r^*"} equations and inclusions.
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*}
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lemma rtranclp_idemp [simp]: "(r^**)^** = r^**"
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  apply (auto intro!: order_antisym)
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  apply (erule rtranclp_induct)
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   apply (rule rtranclp.rtrancl_refl)
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  apply (blast intro: rtranclp_trans)
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  done
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lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set]
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lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"
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  apply (rule set_eqI)
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  apply (simp only: split_tupled_all)
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  apply (blast intro: rtrancl_trans)
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  done
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lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*"
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  apply (drule rtrancl_mono)
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  apply simp
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  done
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lemma rtranclp_subset: "R \<le> S ==> S \<le> R^** ==> S^** = R^**"
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  apply (drule rtranclp_mono)
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  apply (drule rtranclp_mono)
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  apply simp
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  done
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lemmas rtrancl_subset = rtranclp_subset [to_set]
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lemma rtranclp_sup_rtranclp: "(sup (R^**) (S^**))^** = (sup R S)^**"
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by (blast intro!: rtranclp_subset intro: rtranclp_mono [THEN predicate2D])
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lemmas rtrancl_Un_rtrancl = rtranclp_sup_rtranclp [to_set]
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lemma rtranclp_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)
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done
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lemma rtrancl_Un_separator_converseE:
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   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@12691
   335
subsection {* Transitive closure *}
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
berghofe@13704
   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@12691
   346
text {*
wenzelm@12691
   347
  \medskip Conversions between @{text trancl} and @{text rtrancl}.
wenzelm@12691
   348
*}
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@12691
   362
  -- {* intro rule from @{text r} and @{text rtrancl} *}
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@26179
   371
text {* Nice induction rule for @{text trancl} *}
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@12691
   394
  -- {* Another induction rule for trancl, incorporating transitivity *}
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)
wenzelm@26179
   408
  apply (rule_tac p = x in PairE)
wenzelm@26179
   409
  apply clarify
wenzelm@26179
   410
  apply (erule trancl_induct)
wenzelm@26179
   411
   apply auto
paulson@22080
   412
  done
paulson@22080
   413
krauss@32235
   414
lemma trancl_unfold: "r^+ = r Un r^+ O r"
paulson@15551
   415
  by (auto intro: trancl_into_trancl elim: tranclE)
paulson@15551
   416
wenzelm@26179
   417
text {* Transitivity of @{term "r^+"} *}
wenzelm@26179
   418
lemma trans_trancl [simp]: "trans (r^+)"
berghofe@13704
   419
proof (rule transI)
berghofe@13704
   420
  fix x y z
wenzelm@26179
   421
  assume "(x, y) \<in> r^+"
berghofe@13704
   422
  assume "(y, z) \<in> r^+"
wenzelm@26179
   423
  then show "(x, z) \<in> r^+"
wenzelm@26179
   424
  proof induct
wenzelm@26179
   425
    case (base u)
wenzelm@26179
   426
    from `(x, y) \<in> r^+` and `(y, u) \<in> r`
wenzelm@26179
   427
    show "(x, u) \<in> r^+" ..
wenzelm@26179
   428
  next
wenzelm@26179
   429
    case (step u v)
wenzelm@26179
   430
    from `(x, u) \<in> r^+` and `(u, v) \<in> r`
wenzelm@26179
   431
    show "(x, v) \<in> r^+" ..
wenzelm@26179
   432
  qed
berghofe@13704
   433
qed
wenzelm@12691
   434
wenzelm@45607
   435
lemmas trancl_trans = trans_trancl [THEN transD]
wenzelm@12691
   436
berghofe@23743
   437
lemma tranclp_trans:
berghofe@22262
   438
  assumes xy: "r^++ x y"
berghofe@22262
   439
  and yz: "r^++ y z"
berghofe@22262
   440
  shows "r^++ x z" using yz xy
berghofe@22262
   441
  by induct iprover+
berghofe@22262
   442
wenzelm@26179
   443
lemma trancl_id [simp]: "trans r \<Longrightarrow> r^+ = r"
wenzelm@26179
   444
  apply auto
wenzelm@26179
   445
  apply (erule trancl_induct)
wenzelm@26179
   446
   apply assumption
wenzelm@26179
   447
  apply (unfold trans_def)
wenzelm@26179
   448
  apply blast
wenzelm@26179
   449
  done
nipkow@19623
   450
wenzelm@26179
   451
lemma rtranclp_tranclp_tranclp:
wenzelm@26179
   452
  assumes "r^** x y"
wenzelm@26179
   453
  shows "!!z. r^++ y z ==> r^++ x z" using assms
berghofe@23743
   454
  by induct (iprover intro: tranclp_trans)+
wenzelm@12691
   455
berghofe@23743
   456
lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]
berghofe@22262
   457
berghofe@23743
   458
lemma tranclp_into_tranclp2: "r a b ==> r^++ b c ==> r^++ a c"
berghofe@23743
   459
  by (erule tranclp_trans [OF tranclp.r_into_trancl])
berghofe@22262
   460
berghofe@23743
   461
lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]
wenzelm@12691
   462
wenzelm@12691
   463
lemma trancl_insert:
wenzelm@12691
   464
  "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
wenzelm@12691
   465
  -- {* primitive recursion for @{text trancl} over finite relations *}
wenzelm@12691
   466
  apply (rule equalityI)
wenzelm@12691
   467
   apply (rule subsetI)
wenzelm@12691
   468
   apply (simp only: split_tupled_all)
paulson@14208
   469
   apply (erule trancl_induct, blast)
huffman@35216
   470
   apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)
wenzelm@12691
   471
  apply (rule subsetI)
wenzelm@12691
   472
  apply (blast intro: trancl_mono rtrancl_mono
wenzelm@12691
   473
    [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
wenzelm@12691
   474
  done
wenzelm@12691
   475
berghofe@23743
   476
lemma tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ x y"
berghofe@22262
   477
  apply (drule conversepD)
berghofe@23743
   478
  apply (erule tranclp_induct)
berghofe@23743
   479
  apply (iprover intro: conversepI tranclp_trans)+
wenzelm@12691
   480
  done
wenzelm@12691
   481
berghofe@23743
   482
lemmas trancl_converseI = tranclp_converseI [to_set]
berghofe@22262
   483
berghofe@23743
   484
lemma tranclp_converseD: "(r^--1)^++ x y ==> (r^++)^--1 x y"
berghofe@22262
   485
  apply (rule conversepI)
berghofe@23743
   486
  apply (erule tranclp_induct)
berghofe@23743
   487
  apply (iprover dest: conversepD intro: tranclp_trans)+
berghofe@13704
   488
  done
wenzelm@12691
   489
berghofe@23743
   490
lemmas trancl_converseD = tranclp_converseD [to_set]
berghofe@22262
   491
berghofe@23743
   492
lemma tranclp_converse: "(r^--1)^++ = (r^++)^--1"
nipkow@44890
   493
  by (fastforce simp add: fun_eq_iff
berghofe@23743
   494
    intro!: tranclp_converseI dest!: tranclp_converseD)
berghofe@22262
   495
berghofe@23743
   496
lemmas trancl_converse = tranclp_converse [to_set]
wenzelm@12691
   497
huffman@19228
   498
lemma sym_trancl: "sym r ==> sym (r^+)"
huffman@19228
   499
  by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
huffman@19228
   500
berghofe@34909
   501
lemma converse_tranclp_induct [consumes 1, case_names base step]:
berghofe@22262
   502
  assumes major: "r^++ a b"
berghofe@22262
   503
    and cases: "!!y. r y b ==> P(y)"
berghofe@22262
   504
      "!!y z.[| r y z;  r^++ z b;  P(z) |] ==> P(y)"
wenzelm@18372
   505
  shows "P a"
berghofe@23743
   506
  apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])
wenzelm@18372
   507
   apply (rule cases)
berghofe@22262
   508
   apply (erule conversepD)
huffman@35216
   509
  apply (blast intro: assms dest!: tranclp_converseD)
wenzelm@18372
   510
  done
wenzelm@12691
   511
berghofe@23743
   512
lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
berghofe@22262
   513
berghofe@23743
   514
lemma tranclpD: "R^++ x y ==> EX z. R x z \<and> R^** z y"
wenzelm@26179
   515
  apply (erule converse_tranclp_induct)
wenzelm@26179
   516
   apply auto
berghofe@23743
   517
  apply (blast intro: rtranclp_trans)
wenzelm@12691
   518
  done
wenzelm@12691
   519
berghofe@23743
   520
lemmas tranclD = tranclpD [to_set]
berghofe@22262
   521
bulwahn@31577
   522
lemma converse_tranclpE:
bulwahn@31577
   523
  assumes major: "tranclp r x z"
bulwahn@31577
   524
  assumes base: "r x z ==> P"
bulwahn@31577
   525
  assumes step: "\<And> y. [| r x y; tranclp r y z |] ==> P"
bulwahn@31577
   526
  shows P
bulwahn@31577
   527
proof -
bulwahn@31577
   528
  from tranclpD[OF major]
bulwahn@31577
   529
  obtain y where "r x y" and "rtranclp r y z" by iprover
bulwahn@31577
   530
  from this(2) show P
bulwahn@31577
   531
  proof (cases rule: rtranclp.cases)
bulwahn@31577
   532
    case rtrancl_refl
bulwahn@31577
   533
    with `r x y` base show P by iprover
bulwahn@31577
   534
  next
bulwahn@31577
   535
    case rtrancl_into_rtrancl
bulwahn@31577
   536
    from this have "tranclp r y z"
bulwahn@31577
   537
      by (iprover intro: rtranclp_into_tranclp1)
bulwahn@31577
   538
    with `r x y` step show P by iprover
bulwahn@31577
   539
  qed
bulwahn@31577
   540
qed
bulwahn@31577
   541
bulwahn@31577
   542
lemmas converse_tranclE = converse_tranclpE [to_set]
bulwahn@31577
   543
kleing@25295
   544
lemma tranclD2:
kleing@25295
   545
  "(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
kleing@25295
   546
  by (blast elim: tranclE intro: trancl_into_rtrancl)
kleing@25295
   547
nipkow@13867
   548
lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
wenzelm@18372
   549
  by (blast elim: tranclE dest: trancl_into_rtrancl)
wenzelm@12691
   550
wenzelm@12691
   551
lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"
wenzelm@12691
   552
  by (blast dest: r_into_trancl)
wenzelm@12691
   553
wenzelm@12691
   554
lemma trancl_subset_Sigma_aux:
wenzelm@12691
   555
    "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
wenzelm@18372
   556
  by (induct rule: rtrancl_induct) auto
wenzelm@12691
   557
wenzelm@12691
   558
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"
berghofe@13704
   559
  apply (rule subsetI)
berghofe@13704
   560
  apply (simp only: split_tupled_all)
berghofe@13704
   561
  apply (erule tranclE)
wenzelm@26179
   562
   apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
wenzelm@12691
   563
  done
nipkow@10996
   564
nipkow@50616
   565
lemma reflclp_tranclp [simp]: "(r^++)^== = r^**"
berghofe@22262
   566
  apply (safe intro!: order_antisym)
berghofe@23743
   567
   apply (erule tranclp_into_rtranclp)
berghofe@23743
   568
  apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)
wenzelm@11084
   569
  done
nipkow@10996
   570
nipkow@50616
   571
lemmas reflcl_trancl [simp] = reflclp_tranclp [to_set]
berghofe@22262
   572
wenzelm@11090
   573
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
wenzelm@11084
   574
  apply safe
paulson@14208
   575
   apply (drule trancl_into_rtrancl, simp)
paulson@14208
   576
  apply (erule rtranclE, safe)
paulson@14208
   577
   apply (rule r_into_trancl, simp)
wenzelm@11084
   578
  apply (rule rtrancl_into_trancl1)
paulson@14208
   579
   apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
wenzelm@11084
   580
  done
nipkow@10996
   581
haftmann@45140
   582
lemma rtrancl_trancl_reflcl [code]: "r^* = (r^+)^="
haftmann@45140
   583
  by simp
haftmann@45140
   584
wenzelm@11090
   585
lemma trancl_empty [simp]: "{}^+ = {}"
wenzelm@11084
   586
  by (auto elim: trancl_induct)
nipkow@10996
   587
wenzelm@11090
   588
lemma rtrancl_empty [simp]: "{}^* = Id"
wenzelm@11084
   589
  by (rule subst [OF reflcl_trancl]) simp
nipkow@10996
   590
berghofe@23743
   591
lemma rtranclpD: "R^** a b ==> a = b \<or> a \<noteq> b \<and> R^++ a b"
nipkow@50616
   592
by (force simp add: reflclp_tranclp [symmetric] simp del: reflclp_tranclp)
berghofe@22262
   593
berghofe@23743
   594
lemmas rtranclD = rtranclpD [to_set]
wenzelm@11084
   595
kleing@16514
   596
lemma rtrancl_eq_or_trancl:
kleing@16514
   597
  "(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"
kleing@16514
   598
  by (fast elim: trancl_into_rtrancl dest: rtranclD)
nipkow@10996
   599
krauss@33656
   600
lemma trancl_unfold_right: "r^+ = r^* O r"
krauss@33656
   601
by (auto dest: tranclD2 intro: rtrancl_into_trancl1)
krauss@33656
   602
krauss@33656
   603
lemma trancl_unfold_left: "r^+ = r O r^*"
krauss@33656
   604
by (auto dest: tranclD intro: rtrancl_into_trancl2)
krauss@33656
   605
krauss@33656
   606
krauss@33656
   607
text {* Simplifying nested closures *}
krauss@33656
   608
krauss@33656
   609
lemma rtrancl_trancl_absorb[simp]: "(R^*)^+ = R^*"
krauss@33656
   610
by (simp add: trans_rtrancl)
krauss@33656
   611
krauss@33656
   612
lemma trancl_rtrancl_absorb[simp]: "(R^+)^* = R^*"
krauss@33656
   613
by (subst reflcl_trancl[symmetric]) simp
krauss@33656
   614
krauss@33656
   615
lemma rtrancl_reflcl_absorb[simp]: "(R^*)^= = R^*"
krauss@33656
   616
by auto
krauss@33656
   617
krauss@33656
   618
wenzelm@12691
   619
text {* @{text Domain} and @{text Range} *}
nipkow@10996
   620
wenzelm@11090
   621
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
wenzelm@11084
   622
  by blast
nipkow@10996
   623
wenzelm@11090
   624
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
wenzelm@11084
   625
  by blast
nipkow@10996
   626
wenzelm@11090
   627
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
wenzelm@11084
   628
  by (rule rtrancl_Un_rtrancl [THEN subst]) fast
nipkow@10996
   629
wenzelm@11090
   630
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
wenzelm@11084
   631
  by (blast intro: subsetD [OF rtrancl_Un_subset])
nipkow@10996
   632
wenzelm@11090
   633
lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
haftmann@46752
   634
  by (unfold Domain_unfold) (blast dest: tranclD)
nipkow@10996
   635
wenzelm@11090
   636
lemma trancl_range [simp]: "Range (r^+) = Range r"
haftmann@46752
   637
  unfolding Domain_converse [symmetric] by (simp add: trancl_converse [symmetric])
nipkow@10996
   638
paulson@11115
   639
lemma Not_Domain_rtrancl:
wenzelm@12691
   640
    "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
wenzelm@12691
   641
  apply auto
wenzelm@26179
   642
  apply (erule rev_mp)
wenzelm@26179
   643
  apply (erule rtrancl_induct)
wenzelm@26179
   644
   apply auto
wenzelm@26179
   645
  done
berghofe@11327
   646
haftmann@29609
   647
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
haftmann@29609
   648
  apply clarify
haftmann@29609
   649
  apply (erule trancl_induct)
haftmann@29609
   650
   apply (auto simp add: Field_def)
haftmann@29609
   651
  done
haftmann@29609
   652
nipkow@41987
   653
lemma finite_trancl[simp]: "finite (r^+) = finite r"
haftmann@29609
   654
  apply auto
haftmann@29609
   655
   prefer 2
haftmann@29609
   656
   apply (rule trancl_subset_Field2 [THEN finite_subset])
haftmann@29609
   657
   apply (rule finite_SigmaI)
haftmann@29609
   658
    prefer 3
haftmann@29609
   659
    apply (blast intro: r_into_trancl' finite_subset)
haftmann@29609
   660
   apply (auto simp add: finite_Field)
haftmann@29609
   661
  done
haftmann@29609
   662
wenzelm@12691
   663
text {* More about converse @{text rtrancl} and @{text trancl}, should
wenzelm@12691
   664
  be merged with main body. *}
kleing@12428
   665
nipkow@14337
   666
lemma single_valued_confluent:
nipkow@14337
   667
  "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>
nipkow@14337
   668
  \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
wenzelm@26179
   669
  apply (erule rtrancl_induct)
wenzelm@26179
   670
  apply simp
wenzelm@26179
   671
  apply (erule disjE)
wenzelm@26179
   672
   apply (blast elim:converse_rtranclE dest:single_valuedD)
wenzelm@26179
   673
  apply(blast intro:rtrancl_trans)
wenzelm@26179
   674
  done
nipkow@14337
   675
wenzelm@12691
   676
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"
kleing@12428
   677
  by (fast intro: trancl_trans)
kleing@12428
   678
kleing@12428
   679
lemma trancl_into_trancl [rule_format]:
wenzelm@12691
   680
    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
wenzelm@12691
   681
  apply (erule trancl_induct)
kleing@12428
   682
   apply (fast intro: r_r_into_trancl)
kleing@12428
   683
  apply (fast intro: r_r_into_trancl trancl_trans)
kleing@12428
   684
  done
kleing@12428
   685
berghofe@23743
   686
lemma tranclp_rtranclp_tranclp:
berghofe@22262
   687
    "r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c"
berghofe@23743
   688
  apply (drule tranclpD)
wenzelm@26179
   689
  apply (elim exE conjE)
berghofe@23743
   690
  apply (drule rtranclp_trans, assumption)
berghofe@23743
   691
  apply (drule rtranclp_into_tranclp2, assumption, assumption)
kleing@12428
   692
  done
kleing@12428
   693
berghofe@23743
   694
lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]
berghofe@22262
   695
wenzelm@12691
   696
lemmas transitive_closure_trans [trans] =
wenzelm@12691
   697
  r_r_into_trancl trancl_trans rtrancl_trans
berghofe@23743
   698
  trancl.trancl_into_trancl trancl_into_trancl2
berghofe@23743
   699
  rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
wenzelm@12691
   700
  rtrancl_trancl_trancl trancl_rtrancl_trancl
kleing@12428
   701
berghofe@23743
   702
lemmas transitive_closurep_trans' [trans] =
berghofe@23743
   703
  tranclp_trans rtranclp_trans
berghofe@23743
   704
  tranclp.trancl_into_trancl tranclp_into_tranclp2
berghofe@23743
   705
  rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp
berghofe@23743
   706
  rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp
berghofe@22262
   707
kleing@12428
   708
declare trancl_into_rtrancl [elim]
berghofe@11327
   709
haftmann@30954
   710
subsection {* The power operation on relations *}
haftmann@30954
   711
haftmann@30954
   712
text {* @{text "R ^^ n = R O ... O R"}, the n-fold composition of @{text R} *}
haftmann@30954
   713
haftmann@30971
   714
overloading
haftmann@30971
   715
  relpow == "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
haftmann@47202
   716
  relpowp == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
haftmann@30971
   717
begin
haftmann@30954
   718
haftmann@30971
   719
primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
haftmann@30971
   720
    "relpow 0 R = Id"
krauss@32235
   721
  | "relpow (Suc n) R = (R ^^ n) O R"
haftmann@30954
   722
haftmann@47202
   723
primrec relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)" where
haftmann@47202
   724
    "relpowp 0 R = HOL.eq"
haftmann@47202
   725
  | "relpowp (Suc n) R = (R ^^ n) OO R"
haftmann@47202
   726
haftmann@30971
   727
end
haftmann@30954
   728
haftmann@47202
   729
lemma relpowp_relpow_eq [pred_set_conv]:
haftmann@47202
   730
  fixes R :: "'a rel"
haftmann@47202
   731
  shows "(\<lambda>x y. (x, y) \<in> R) ^^ n = (\<lambda>x y. (x, y) \<in> R ^^ n)"
griff@47433
   732
  by (induct n) (simp_all add: relcompp_relcomp_eq)
haftmann@47202
   733
bulwahn@46360
   734
text {* for code generation *}
bulwahn@46360
   735
bulwahn@46360
   736
definition relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
bulwahn@46360
   737
  relpow_code_def [code_abbrev]: "relpow = compow"
bulwahn@46360
   738
Christian@47492
   739
definition relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)" where
Christian@47492
   740
  relpowp_code_def [code_abbrev]: "relpowp = compow"
Christian@47492
   741
bulwahn@46360
   742
lemma [code]:
bulwahn@46360
   743
  "relpow (Suc n) R = (relpow n R) O R"
bulwahn@46360
   744
  "relpow 0 R = Id"
bulwahn@46360
   745
  by (simp_all add: relpow_code_def)
bulwahn@46360
   746
Christian@47492
   747
lemma [code]:
Christian@47492
   748
  "relpowp (Suc n) R = (R ^^ n) OO R"
Christian@47492
   749
  "relpowp 0 R = HOL.eq"
Christian@47492
   750
  by (simp_all add: relpowp_code_def)
Christian@47492
   751
bulwahn@46360
   752
hide_const (open) relpow
Christian@47492
   753
hide_const (open) relpowp
bulwahn@46360
   754
bulwahn@46362
   755
lemma relpow_1 [simp]:
haftmann@30971
   756
  fixes R :: "('a \<times> 'a) set"
haftmann@30971
   757
  shows "R ^^ 1 = R"
haftmann@30954
   758
  by simp
haftmann@30954
   759
Christian@47492
   760
lemma relpowp_1 [simp]:
Christian@47492
   761
  fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
Christian@47492
   762
  shows "P ^^ 1 = P"
Christian@47492
   763
  by (fact relpow_1 [to_pred])
Christian@47492
   764
bulwahn@46362
   765
lemma relpow_0_I: 
haftmann@30954
   766
  "(x, x) \<in> R ^^ 0"
haftmann@30954
   767
  by simp
haftmann@30954
   768
Christian@47492
   769
lemma relpowp_0_I:
Christian@47492
   770
  "(P ^^ 0) x x"
Christian@47492
   771
  by (fact relpow_0_I [to_pred])
Christian@47492
   772
bulwahn@46362
   773
lemma relpow_Suc_I:
haftmann@30954
   774
  "(x, y) \<in>  R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
haftmann@30954
   775
  by auto
haftmann@30954
   776
Christian@47492
   777
lemma relpowp_Suc_I:
Christian@47492
   778
  "(P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> (P ^^ Suc n) x z"
Christian@47492
   779
  by (fact relpow_Suc_I [to_pred])
Christian@47492
   780
bulwahn@46362
   781
lemma relpow_Suc_I2:
haftmann@30954
   782
  "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
nipkow@44890
   783
  by (induct n arbitrary: z) (simp, fastforce)
haftmann@30954
   784
Christian@47492
   785
lemma relpowp_Suc_I2:
Christian@47492
   786
  "P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> (P ^^ Suc n) x z"
Christian@47492
   787
  by (fact relpow_Suc_I2 [to_pred])
Christian@47492
   788
bulwahn@46362
   789
lemma relpow_0_E:
haftmann@30954
   790
  "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30954
   791
  by simp
haftmann@30954
   792
Christian@47492
   793
lemma relpowp_0_E:
Christian@47492
   794
  "(P ^^ 0) x y \<Longrightarrow> (x = y \<Longrightarrow> Q) \<Longrightarrow> Q"
Christian@47492
   795
  by (fact relpow_0_E [to_pred])
Christian@47492
   796
bulwahn@46362
   797
lemma relpow_Suc_E:
haftmann@30954
   798
  "(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
   799
  by auto
haftmann@30954
   800
Christian@47492
   801
lemma relpowp_Suc_E:
Christian@47492
   802
  "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. (P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> Q) \<Longrightarrow> Q"
Christian@47492
   803
  by (fact relpow_Suc_E [to_pred])
Christian@47492
   804
bulwahn@46362
   805
lemma relpow_E:
haftmann@30954
   806
  "(x, z) \<in>  R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
haftmann@30954
   807
   \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in>  R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P)
haftmann@30954
   808
   \<Longrightarrow> P"
haftmann@30954
   809
  by (cases n) auto
haftmann@30954
   810
Christian@47492
   811
lemma relpowp_E:
Christian@47492
   812
  "(P ^^ n) x z \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q)
Christian@47492
   813
  \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (P ^^ m) x y \<Longrightarrow> P y z \<Longrightarrow> Q)
Christian@47492
   814
  \<Longrightarrow> Q"
Christian@47492
   815
  by (fact relpow_E [to_pred])
Christian@47492
   816
bulwahn@46362
   817
lemma relpow_Suc_D2:
haftmann@30954
   818
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
haftmann@30954
   819
  apply (induct n arbitrary: x z)
bulwahn@46362
   820
   apply (blast intro: relpow_0_I elim: relpow_0_E relpow_Suc_E)
bulwahn@46362
   821
  apply (blast intro: relpow_Suc_I elim: relpow_0_E relpow_Suc_E)
haftmann@30954
   822
  done
haftmann@30954
   823
Christian@47492
   824
lemma relpowp_Suc_D2:
Christian@47492
   825
  "(P ^^ Suc n) x z \<Longrightarrow> \<exists>y. P x y \<and> (P ^^ n) y z"
Christian@47492
   826
  by (fact relpow_Suc_D2 [to_pred])
Christian@47492
   827
bulwahn@46362
   828
lemma relpow_Suc_E2:
haftmann@30954
   829
  "(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
   830
  by (blast dest: relpow_Suc_D2)
haftmann@30954
   831
Christian@47492
   832
lemma relpowp_Suc_E2:
Christian@47492
   833
  "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> Q) \<Longrightarrow> Q"
Christian@47492
   834
  by (fact relpow_Suc_E2 [to_pred])
Christian@47492
   835
bulwahn@46362
   836
lemma relpow_Suc_D2':
haftmann@30954
   837
  "\<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
   838
  by (induct n) (simp_all, blast)
haftmann@30954
   839
Christian@47492
   840
lemma relpowp_Suc_D2':
Christian@47492
   841
  "\<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
   842
  by (fact relpow_Suc_D2' [to_pred])
Christian@47492
   843
bulwahn@46362
   844
lemma relpow_E2:
haftmann@30954
   845
  "(x, z) \<in> R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
haftmann@30954
   846
     \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P)
haftmann@30954
   847
   \<Longrightarrow> P"
haftmann@30954
   848
  apply (cases n, simp)
bulwahn@46362
   849
  apply (cut_tac n=nat and R=R in relpow_Suc_D2', simp, blast)
haftmann@30954
   850
  done
haftmann@30954
   851
Christian@47492
   852
lemma relpowp_E2:
Christian@47492
   853
  "(P ^^ n) x z \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q)
Christian@47492
   854
    \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> P x y \<Longrightarrow> (P ^^ m) y z \<Longrightarrow> Q)
Christian@47492
   855
  \<Longrightarrow> Q"
Christian@47492
   856
  by (fact relpow_E2 [to_pred])
Christian@47492
   857
bulwahn@46362
   858
lemma relpow_add: "R ^^ (m+n) = R^^m O R^^n"
haftmann@45976
   859
  by (induct n) auto
nipkow@31351
   860
Christian@47492
   861
lemma relpowp_add: "P ^^ (m + n) = P ^^ m OO P ^^ n"
Christian@47492
   862
  by (fact relpow_add [to_pred])
Christian@47492
   863
bulwahn@46362
   864
lemma relpow_commute: "R O R ^^ n = R ^^ n O R"
haftmann@45976
   865
  by (induct n) (simp, simp add: O_assoc [symmetric])
krauss@31970
   866
Christian@47492
   867
lemma relpowp_commute: "P OO P ^^ n = P ^^ n OO P"
Christian@47492
   868
  by (fact relpow_commute [to_pred])
Christian@47492
   869
bulwahn@46362
   870
lemma relpow_empty:
haftmann@45153
   871
  "0 < n \<Longrightarrow> ({} :: ('a \<times> 'a) set) ^^ n = {}"
haftmann@45153
   872
  by (cases n) auto
bulwahn@45116
   873
Christian@47492
   874
lemma relpowp_bot:
Christian@47492
   875
  "0 < n \<Longrightarrow> (\<bottom> :: 'a \<Rightarrow> 'a \<Rightarrow> bool) ^^ n = \<bottom>"
Christian@47492
   876
  by (fact relpow_empty [to_pred])
Christian@47492
   877
bulwahn@46362
   878
lemma rtrancl_imp_UN_relpow:
haftmann@30954
   879
  assumes "p \<in> R^*"
haftmann@30954
   880
  shows "p \<in> (\<Union>n. R ^^ n)"
haftmann@30954
   881
proof (cases p)
haftmann@30954
   882
  case (Pair x y)
haftmann@30954
   883
  with assms have "(x, y) \<in> R^*" by simp
haftmann@30954
   884
  then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct
bulwahn@46362
   885
    case base show ?case by (blast intro: relpow_0_I)
haftmann@30954
   886
  next
bulwahn@46362
   887
    case step then show ?case by (blast intro: relpow_Suc_I)
haftmann@30954
   888
  qed
haftmann@30954
   889
  with Pair show ?thesis by simp
haftmann@30954
   890
qed
haftmann@30954
   891
Christian@47492
   892
lemma rtranclp_imp_Sup_relpowp:
Christian@47492
   893
  assumes "(P^**) x y"
Christian@47492
   894
  shows "(\<Squnion>n. P ^^ n) x y"
Christian@47492
   895
  using assms and rtrancl_imp_UN_relpow [to_pred] by blast
Christian@47492
   896
bulwahn@46362
   897
lemma relpow_imp_rtrancl:
haftmann@30954
   898
  assumes "p \<in> R ^^ n"
haftmann@30954
   899
  shows "p \<in> R^*"
haftmann@30954
   900
proof (cases p)
haftmann@30954
   901
  case (Pair x y)
haftmann@30954
   902
  with assms have "(x, y) \<in> R ^^ n" by simp
haftmann@30954
   903
  then have "(x, y) \<in> R^*" proof (induct n arbitrary: x y)
haftmann@30954
   904
    case 0 then show ?case by simp
haftmann@30954
   905
  next
haftmann@30954
   906
    case Suc then show ?case
bulwahn@46362
   907
      by (blast elim: relpow_Suc_E intro: rtrancl_into_rtrancl)
haftmann@30954
   908
  qed
haftmann@30954
   909
  with Pair show ?thesis by simp
haftmann@30954
   910
qed
haftmann@30954
   911
Christian@47492
   912
lemma relpowp_imp_rtranclp:
Christian@47492
   913
  assumes "(P ^^ n) x y"
Christian@47492
   914
  shows "(P^**) x y"
Christian@47492
   915
  using assms and relpow_imp_rtrancl [to_pred] by blast
Christian@47492
   916
bulwahn@46362
   917
lemma rtrancl_is_UN_relpow:
haftmann@30954
   918
  "R^* = (\<Union>n. R ^^ n)"
bulwahn@46362
   919
  by (blast intro: rtrancl_imp_UN_relpow relpow_imp_rtrancl)
haftmann@30954
   920
Christian@47492
   921
lemma rtranclp_is_Sup_relpowp:
Christian@47492
   922
  "P^** = (\<Squnion>n. P ^^ n)"
Christian@47492
   923
  using rtrancl_is_UN_relpow [to_pred, of P] by auto
Christian@47492
   924
haftmann@30954
   925
lemma rtrancl_power:
haftmann@30954
   926
  "p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
bulwahn@46362
   927
  by (simp add: rtrancl_is_UN_relpow)
haftmann@30954
   928
Christian@47492
   929
lemma rtranclp_power:
Christian@47492
   930
  "(P^**) x y \<longleftrightarrow> (\<exists>n. (P ^^ n) x y)"
Christian@47492
   931
  by (simp add: rtranclp_is_Sup_relpowp)
Christian@47492
   932
haftmann@30954
   933
lemma trancl_power:
haftmann@30954
   934
  "p \<in> R^+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
haftmann@30954
   935
  apply (cases p)
haftmann@30954
   936
  apply simp
haftmann@30954
   937
  apply (rule iffI)
haftmann@30954
   938
   apply (drule tranclD2)
bulwahn@46362
   939
   apply (clarsimp simp: rtrancl_is_UN_relpow)
haftmann@30971
   940
   apply (rule_tac x="Suc n" in exI)
griff@47433
   941
   apply (clarsimp simp: relcomp_unfold)
nipkow@44890
   942
   apply fastforce
haftmann@30954
   943
  apply clarsimp
haftmann@30954
   944
  apply (case_tac n, simp)
haftmann@30954
   945
  apply clarsimp
bulwahn@46362
   946
  apply (drule relpow_imp_rtrancl)
haftmann@30954
   947
  apply (drule rtrancl_into_trancl1) apply auto
haftmann@30954
   948
  done
haftmann@30954
   949
Christian@47492
   950
lemma tranclp_power:
Christian@47492
   951
  "(P^++) x y \<longleftrightarrow> (\<exists>n > 0. (P ^^ n) x y)"
Christian@47492
   952
  using trancl_power [to_pred, of P "(x, y)"] by simp
Christian@47492
   953
bulwahn@46362
   954
lemma rtrancl_imp_relpow:
haftmann@30954
   955
  "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
bulwahn@46362
   956
  by (auto dest: rtrancl_imp_UN_relpow)
haftmann@30954
   957
Christian@47492
   958
lemma rtranclp_imp_relpowp:
Christian@47492
   959
  "(P^**) x y \<Longrightarrow> \<exists>n. (P ^^ n) x y"
Christian@47492
   960
  by (auto dest: rtranclp_imp_Sup_relpowp)
Christian@47492
   961
nipkow@41987
   962
text{* By Sternagel/Thiemann: *}
bulwahn@46362
   963
lemma relpow_fun_conv:
nipkow@41987
   964
  "((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
   965
proof (induct n arbitrary: b)
nipkow@41987
   966
  case 0 show ?case by auto
nipkow@41987
   967
next
nipkow@41987
   968
  case (Suc n)
nipkow@41987
   969
  show ?case
griff@47433
   970
  proof (simp add: relcomp_unfold Suc)
nipkow@41987
   971
    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
   972
     = (\<exists>f. f 0 = a \<and> f(Suc n) = b \<and> (\<forall>i<Suc n. (f i, f (Suc i)) \<in> R))"
nipkow@41987
   973
    (is "?l = ?r")
nipkow@41987
   974
    proof
nipkow@41987
   975
      assume ?l
nipkow@41987
   976
      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
   977
      let ?g = "\<lambda> m. if m = Suc n then b else f m"
nipkow@41987
   978
      show ?r by (rule exI[of _ ?g], simp add: 1)
nipkow@41987
   979
    next
nipkow@41987
   980
      assume ?r
nipkow@41987
   981
      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
   982
      show ?l by (rule exI[of _ "f n"], rule conjI, rule exI[of _ f], insert 1, auto)
nipkow@41987
   983
    qed
nipkow@41987
   984
  qed
nipkow@41987
   985
qed
nipkow@41987
   986
Christian@47492
   987
lemma relpowp_fun_conv:
Christian@47492
   988
  "(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
   989
  by (fact relpow_fun_conv [to_pred])
Christian@47492
   990
bulwahn@46362
   991
lemma relpow_finite_bounded1:
nipkow@41987
   992
assumes "finite(R :: ('a*'a)set)" and "k>0"
nipkow@41987
   993
shows "R^^k \<subseteq> (UN n:{n. 0<n & n <= card R}. R^^n)" (is "_ \<subseteq> ?r")
nipkow@41987
   994
proof-
nipkow@41987
   995
  { fix a b k
nipkow@41987
   996
    have "(a,b) : R^^(Suc k) \<Longrightarrow> EX n. 0<n & n <= card R & (a,b) : R^^n"
nipkow@41987
   997
    proof(induct k arbitrary: b)
nipkow@41987
   998
      case 0
nipkow@41987
   999
      hence "R \<noteq> {}" by auto
nipkow@41987
  1000
      with card_0_eq[OF `finite R`] have "card R >= Suc 0" by auto
nipkow@41987
  1001
      thus ?case using 0 by force
nipkow@41987
  1002
    next
nipkow@41987
  1003
      case (Suc k)
nipkow@41987
  1004
      then obtain a' where "(a,a') : R^^(Suc k)" and "(a',b) : R" by auto
nipkow@41987
  1005
      from Suc(1)[OF `(a,a') : R^^(Suc k)`]
nipkow@41987
  1006
      obtain n where "n \<le> card R" and "(a,a') \<in> R ^^ n" by auto
nipkow@41987
  1007
      have "(a,b) : R^^(Suc n)" using `(a,a') \<in> R^^n` and `(a',b)\<in> R` by auto
nipkow@41987
  1008
      { assume "n < card R"
nipkow@41987
  1009
        hence ?case using `(a,b): R^^(Suc n)` Suc_leI[OF `n < card R`] by blast
nipkow@41987
  1010
      } moreover
nipkow@41987
  1011
      { assume "n = card R"
bulwahn@46362
  1012
        from `(a,b) \<in> R ^^ (Suc n)`[unfolded relpow_fun_conv]
nipkow@41987
  1013
        obtain f where "f 0 = a" and "f(Suc n) = b"
nipkow@41987
  1014
          and steps: "\<And>i. i <= n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto
nipkow@41987
  1015
        let ?p = "%i. (f i, f(Suc i))"
nipkow@41987
  1016
        let ?N = "{i. i \<le> n}"
nipkow@41987
  1017
        have "?p ` ?N <= R" using steps by auto
nipkow@41987
  1018
        from card_mono[OF assms(1) this]
nipkow@41987
  1019
        have "card(?p ` ?N) <= card R" .
nipkow@41987
  1020
        also have "\<dots> < card ?N" using `n = card R` by simp
nipkow@41987
  1021
        finally have "~ inj_on ?p ?N" by(rule pigeonhole)
nipkow@41987
  1022
        then obtain i j where i: "i <= n" and j: "j <= n" and ij: "i \<noteq> j" and
nipkow@41987
  1023
          pij: "?p i = ?p j" by(auto simp: inj_on_def)
nipkow@41987
  1024
        let ?i = "min i j" let ?j = "max i j"
nipkow@41987
  1025
        have i: "?i <= n" and j: "?j <= n" and pij: "?p ?i = ?p ?j" 
nipkow@41987
  1026
          and ij: "?i < ?j"
nipkow@41987
  1027
          using i j ij pij unfolding min_def max_def by auto
nipkow@41987
  1028
        from i j pij ij obtain i j where i: "i<=n" and j: "j<=n" and ij: "i<j"
nipkow@41987
  1029
          and pij: "?p i = ?p j" by blast
nipkow@41987
  1030
        let ?g = "\<lambda> l. if l \<le> i then f l else f (l + (j - i))"
nipkow@41987
  1031
        let ?n = "Suc(n - (j - i))"
bulwahn@46362
  1032
        have abl: "(a,b) \<in> R ^^ ?n" unfolding relpow_fun_conv
nipkow@41987
  1033
        proof (rule exI[of _ ?g], intro conjI impI allI)
nipkow@41987
  1034
          show "?g ?n = b" using `f(Suc n) = b` j ij by auto
nipkow@41987
  1035
        next
nipkow@41987
  1036
          fix k assume "k < ?n"
nipkow@41987
  1037
          show "(?g k, ?g (Suc k)) \<in> R"
nipkow@41987
  1038
          proof (cases "k < i")
nipkow@41987
  1039
            case True
nipkow@41987
  1040
            with i have "k <= n" by auto
nipkow@41987
  1041
            from steps[OF this] show ?thesis using True by simp
nipkow@41987
  1042
          next
nipkow@41987
  1043
            case False
nipkow@41987
  1044
            hence "i \<le> k" by auto
nipkow@41987
  1045
            show ?thesis
nipkow@41987
  1046
            proof (cases "k = i")
nipkow@41987
  1047
              case True
nipkow@41987
  1048
              thus ?thesis using ij pij steps[OF i] by simp
nipkow@41987
  1049
            next
nipkow@41987
  1050
              case False
nipkow@41987
  1051
              with `i \<le> k` have "i < k" by auto
nipkow@41987
  1052
              hence small: "k + (j - i) <= n" using `k<?n` by arith
nipkow@41987
  1053
              show ?thesis using steps[OF small] `i<k` by auto
nipkow@41987
  1054
            qed
nipkow@41987
  1055
          qed
nipkow@41987
  1056
        qed (simp add: `f 0 = a`)
nipkow@41987
  1057
        moreover have "?n <= n" using i j ij by arith
nipkow@41987
  1058
        ultimately have ?case using `n = card R` by blast
nipkow@41987
  1059
      }
nipkow@41987
  1060
      ultimately show ?case using `n \<le> card R` by force
nipkow@41987
  1061
    qed
nipkow@41987
  1062
  }
nipkow@41987
  1063
  thus ?thesis using gr0_implies_Suc[OF `k>0`] by auto
nipkow@41987
  1064
qed
nipkow@41987
  1065
bulwahn@46362
  1066
lemma relpow_finite_bounded:
nipkow@41987
  1067
assumes "finite(R :: ('a*'a)set)"
nipkow@41987
  1068
shows "R^^k \<subseteq> (UN n:{n. n <= card R}. R^^n)"
nipkow@41987
  1069
apply(cases k)
nipkow@41987
  1070
 apply force
bulwahn@46362
  1071
using relpow_finite_bounded1[OF assms, of k] by auto
nipkow@41987
  1072
bulwahn@46362
  1073
lemma rtrancl_finite_eq_relpow:
nipkow@41987
  1074
  "finite R \<Longrightarrow> R^* = (UN n : {n. n <= card R}. R^^n)"
bulwahn@46362
  1075
by(fastforce simp: rtrancl_power dest: relpow_finite_bounded)
nipkow@41987
  1076
bulwahn@46362
  1077
lemma trancl_finite_eq_relpow:
nipkow@41987
  1078
  "finite R \<Longrightarrow> R^+ = (UN n : {n. 0 < n & n <= card R}. R^^n)"
nipkow@41987
  1079
apply(auto simp add: trancl_power)
bulwahn@46362
  1080
apply(auto dest: relpow_finite_bounded1)
nipkow@41987
  1081
done
nipkow@41987
  1082
griff@47433
  1083
lemma finite_relcomp[simp,intro]:
nipkow@41987
  1084
assumes "finite R" and "finite S"
nipkow@41987
  1085
shows "finite(R O S)"
nipkow@41987
  1086
proof-
nipkow@41987
  1087
  have "R O S = (UN (x,y) : R. \<Union>((%(u,v). if u=y then {(x,v)} else {}) ` S))"
nipkow@41987
  1088
    by(force simp add: split_def)
nipkow@41987
  1089
  thus ?thesis using assms by(clarsimp)
nipkow@41987
  1090
qed
nipkow@41987
  1091
nipkow@41987
  1092
lemma finite_relpow[simp,intro]:
nipkow@41987
  1093
  assumes "finite(R :: ('a*'a)set)" shows "n>0 \<Longrightarrow> finite(R^^n)"
nipkow@41987
  1094
apply(induct n)
nipkow@41987
  1095
 apply simp
nipkow@41987
  1096
apply(case_tac n)
nipkow@41987
  1097
 apply(simp_all add: assms)
nipkow@41987
  1098
done
nipkow@41987
  1099
bulwahn@46362
  1100
lemma single_valued_relpow:
haftmann@30954
  1101
  fixes R :: "('a * 'a) set"
haftmann@30954
  1102
  shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
nipkow@41987
  1103
apply (induct n arbitrary: R)
nipkow@41987
  1104
apply simp_all
nipkow@41987
  1105
apply (rule single_valuedI)
bulwahn@46362
  1106
apply (fast dest: single_valuedD elim: relpow_Suc_E)
nipkow@41987
  1107
done
paulson@15551
  1108
haftmann@45140
  1109
haftmann@45140
  1110
subsection {* Bounded transitive closure *}
haftmann@45140
  1111
haftmann@45140
  1112
definition ntrancl :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
haftmann@45140
  1113
where
haftmann@45140
  1114
  "ntrancl n R = (\<Union>i\<in>{i. 0 < i \<and> i \<le> Suc n}. R ^^ i)"
haftmann@45140
  1115
haftmann@45140
  1116
lemma ntrancl_Zero [simp, code]:
haftmann@45140
  1117
  "ntrancl 0 R = R"
haftmann@45140
  1118
proof
haftmann@45140
  1119
  show "R \<subseteq> ntrancl 0 R"
haftmann@45140
  1120
    unfolding ntrancl_def by fastforce
haftmann@45140
  1121
next
haftmann@45140
  1122
  { 
haftmann@45140
  1123
    fix i have "0 < i \<and> i \<le> Suc 0 \<longleftrightarrow> i = 1" by auto
haftmann@45140
  1124
  }
haftmann@45140
  1125
  from this show "ntrancl 0 R \<le> R"
haftmann@45140
  1126
    unfolding ntrancl_def by auto
haftmann@45140
  1127
qed
haftmann@45140
  1128
bulwahn@46347
  1129
lemma ntrancl_Suc [simp]:
haftmann@45140
  1130
  "ntrancl (Suc n) R = ntrancl n R O (Id \<union> R)"
haftmann@45140
  1131
proof
haftmann@45140
  1132
  {
haftmann@45140
  1133
    fix a b
haftmann@45140
  1134
    assume "(a, b) \<in> ntrancl (Suc n) R"
haftmann@45140
  1135
    from this obtain i where "0 < i" "i \<le> Suc (Suc n)" "(a, b) \<in> R ^^ i"
haftmann@45140
  1136
      unfolding ntrancl_def by auto
haftmann@45140
  1137
    have "(a, b) \<in> ntrancl n R O (Id \<union> R)"
haftmann@45140
  1138
    proof (cases "i = 1")
haftmann@45140
  1139
      case True
haftmann@45140
  1140
      from this `(a, b) \<in> R ^^ i` show ?thesis
haftmann@45140
  1141
        unfolding ntrancl_def by auto
haftmann@45140
  1142
    next
haftmann@45140
  1143
      case False
haftmann@45140
  1144
      from this `0 < i` obtain j where j: "i = Suc j" "0 < j"
haftmann@45140
  1145
        by (cases i) auto
haftmann@45140
  1146
      from this `(a, b) \<in> R ^^ i` obtain c where c1: "(a, c) \<in> R ^^ j" and c2:"(c, b) \<in> R"
haftmann@45140
  1147
        by auto
haftmann@45140
  1148
      from c1 j `i \<le> Suc (Suc n)` have "(a, c) \<in> ntrancl n R"
haftmann@45140
  1149
        unfolding ntrancl_def by fastforce
haftmann@45140
  1150
      from this c2 show ?thesis by fastforce
haftmann@45140
  1151
    qed
haftmann@45140
  1152
  }
haftmann@45140
  1153
  from this show "ntrancl (Suc n) R \<subseteq> ntrancl n R O (Id \<union> R)"
haftmann@45140
  1154
    by auto
haftmann@45140
  1155
next
haftmann@45140
  1156
  show "ntrancl n R O (Id \<union> R) \<subseteq> ntrancl (Suc n) R"
haftmann@45140
  1157
    unfolding ntrancl_def by fastforce
haftmann@45140
  1158
qed
haftmann@45140
  1159
bulwahn@46347
  1160
lemma [code]:
bulwahn@46347
  1161
  "ntrancl (Suc n) r = (let r' = ntrancl n r in r' Un r' O r)"
bulwahn@46347
  1162
unfolding Let_def by auto
bulwahn@46347
  1163
haftmann@45141
  1164
lemma finite_trancl_ntranl:
haftmann@45140
  1165
  "finite R \<Longrightarrow> trancl R = ntrancl (card R - 1) R"
bulwahn@46362
  1166
  by (cases "card R") (auto simp add: trancl_finite_eq_relpow relpow_empty ntrancl_def)
haftmann@45140
  1167
haftmann@45140
  1168
haftmann@45139
  1169
subsection {* Acyclic relations *}
haftmann@45139
  1170
haftmann@45139
  1171
definition acyclic :: "('a * 'a) set => bool" where
haftmann@45139
  1172
  "acyclic r \<longleftrightarrow> (!x. (x,x) ~: r^+)"
haftmann@45139
  1173
haftmann@45139
  1174
abbreviation acyclicP :: "('a => 'a => bool) => bool" where
haftmann@45139
  1175
  "acyclicP r \<equiv> acyclic {(x, y). r x y}"
haftmann@45139
  1176
haftmann@46127
  1177
lemma acyclic_irrefl [code]:
haftmann@45139
  1178
  "acyclic r \<longleftrightarrow> irrefl (r^+)"
haftmann@45139
  1179
  by (simp add: acyclic_def irrefl_def)
haftmann@45139
  1180
haftmann@45139
  1181
lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
haftmann@45139
  1182
  by (simp add: acyclic_def)
haftmann@45139
  1183
haftmann@45139
  1184
lemma acyclic_insert [iff]:
haftmann@45139
  1185
     "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
haftmann@45139
  1186
apply (simp add: acyclic_def trancl_insert)
haftmann@45139
  1187
apply (blast intro: rtrancl_trans)
haftmann@45139
  1188
done
haftmann@45139
  1189
haftmann@45139
  1190
lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
haftmann@45139
  1191
by (simp add: acyclic_def trancl_converse)
haftmann@45139
  1192
haftmann@45139
  1193
lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
haftmann@45139
  1194
haftmann@45139
  1195
lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
haftmann@45139
  1196
apply (simp add: acyclic_def antisym_def)
haftmann@45139
  1197
apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
haftmann@45139
  1198
done
haftmann@45139
  1199
haftmann@45139
  1200
(* Other direction:
haftmann@45139
  1201
acyclic = no loops
haftmann@45139
  1202
antisym = only self loops
haftmann@45139
  1203
Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
haftmann@45139
  1204
==> antisym( r^* ) = acyclic(r - Id)";
haftmann@45139
  1205
*)
haftmann@45139
  1206
haftmann@45139
  1207
lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
haftmann@45139
  1208
apply (simp add: acyclic_def)
haftmann@45139
  1209
apply (blast intro: trancl_mono)
haftmann@45139
  1210
done
haftmann@45139
  1211
haftmann@45139
  1212
ballarin@15076
  1213
subsection {* Setup of transitivity reasoner *}
ballarin@15076
  1214
wenzelm@26340
  1215
ML {*
ballarin@15076
  1216
wenzelm@32215
  1217
structure Trancl_Tac = Trancl_Tac
wenzelm@32215
  1218
(
wenzelm@32215
  1219
  val r_into_trancl = @{thm trancl.r_into_trancl};
wenzelm@32215
  1220
  val trancl_trans  = @{thm trancl_trans};
wenzelm@32215
  1221
  val rtrancl_refl = @{thm rtrancl.rtrancl_refl};
wenzelm@32215
  1222
  val r_into_rtrancl = @{thm r_into_rtrancl};
wenzelm@32215
  1223
  val trancl_into_rtrancl = @{thm trancl_into_rtrancl};
wenzelm@32215
  1224
  val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl};
wenzelm@32215
  1225
  val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl};
wenzelm@32215
  1226
  val rtrancl_trans = @{thm rtrancl_trans};
ballarin@15096
  1227
berghofe@30107
  1228
  fun decomp (@{const Trueprop} $ t) =
haftmann@37677
  1229
    let fun dec (Const (@{const_name Set.member}, _) $ (Const (@{const_name Pair}, _) $ a $ b) $ rel ) =
berghofe@23743
  1230
        let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*")
berghofe@23743
  1231
              | decr (Const ("Transitive_Closure.trancl", _ ) $ r)  = (r,"r+")
wenzelm@18372
  1232
              | decr r = (r,"r");
berghofe@26801
  1233
            val (rel,r) = decr (Envir.beta_eta_contract rel);
wenzelm@18372
  1234
        in SOME (a,b,rel,r) end
wenzelm@18372
  1235
      | dec _ =  NONE
berghofe@30107
  1236
    in dec t end
berghofe@30107
  1237
    | decomp _ = NONE;
wenzelm@32215
  1238
);
ballarin@15076
  1239
wenzelm@32215
  1240
structure Tranclp_Tac = Trancl_Tac
wenzelm@32215
  1241
(
wenzelm@32215
  1242
  val r_into_trancl = @{thm tranclp.r_into_trancl};
wenzelm@32215
  1243
  val trancl_trans  = @{thm tranclp_trans};
wenzelm@32215
  1244
  val rtrancl_refl = @{thm rtranclp.rtrancl_refl};
wenzelm@32215
  1245
  val r_into_rtrancl = @{thm r_into_rtranclp};
wenzelm@32215
  1246
  val trancl_into_rtrancl = @{thm tranclp_into_rtranclp};
wenzelm@32215
  1247
  val rtrancl_trancl_trancl = @{thm rtranclp_tranclp_tranclp};
wenzelm@32215
  1248
  val trancl_rtrancl_trancl = @{thm tranclp_rtranclp_tranclp};
wenzelm@32215
  1249
  val rtrancl_trans = @{thm rtranclp_trans};
berghofe@22262
  1250
berghofe@30107
  1251
  fun decomp (@{const Trueprop} $ t) =
berghofe@22262
  1252
    let fun dec (rel $ a $ b) =
berghofe@23743
  1253
        let fun decr (Const ("Transitive_Closure.rtranclp", _ ) $ r) = (r,"r*")
berghofe@23743
  1254
              | decr (Const ("Transitive_Closure.tranclp", _ ) $ r)  = (r,"r+")
berghofe@22262
  1255
              | decr r = (r,"r");
berghofe@22262
  1256
            val (rel,r) = decr rel;
berghofe@26801
  1257
        in SOME (a, b, rel, r) end
berghofe@22262
  1258
      | dec _ =  NONE
berghofe@30107
  1259
    in dec t end
berghofe@30107
  1260
    | decomp _ = NONE;
wenzelm@32215
  1261
);
wenzelm@26340
  1262
*}
berghofe@22262
  1263
wenzelm@42795
  1264
setup {*
wenzelm@42795
  1265
  Simplifier.map_simpset_global (fn ss => ss
wenzelm@43596
  1266
    addSolver (mk_solver "Trancl" (Trancl_Tac.trancl_tac o Simplifier.the_context))
wenzelm@43596
  1267
    addSolver (mk_solver "Rtrancl" (Trancl_Tac.rtrancl_tac o Simplifier.the_context))
wenzelm@43596
  1268
    addSolver (mk_solver "Tranclp" (Tranclp_Tac.trancl_tac o Simplifier.the_context))
wenzelm@43596
  1269
    addSolver (mk_solver "Rtranclp" (Tranclp_Tac.rtrancl_tac o Simplifier.the_context)))
ballarin@15076
  1270
*}
ballarin@15076
  1271
wenzelm@32215
  1272
wenzelm@32215
  1273
text {* Optional methods. *}
ballarin@15076
  1274
ballarin@15076
  1275
method_setup trancl =
wenzelm@32215
  1276
  {* Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.trancl_tac) *}
wenzelm@18372
  1277
  {* simple transitivity reasoner *}
ballarin@15076
  1278
method_setup rtrancl =
wenzelm@32215
  1279
  {* Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.rtrancl_tac) *}
ballarin@15076
  1280
  {* simple transitivity reasoner *}
berghofe@22262
  1281
method_setup tranclp =
wenzelm@32215
  1282
  {* Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.trancl_tac) *}
berghofe@22262
  1283
  {* simple transitivity reasoner (predicate version) *}
berghofe@22262
  1284
method_setup rtranclp =
wenzelm@32215
  1285
  {* Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.rtrancl_tac) *}
berghofe@22262
  1286
  {* simple transitivity reasoner (predicate version) *}
ballarin@15076
  1287
nipkow@10213
  1288
end
haftmann@46752
  1289