src/HOL/Transitive_Closure.thy
author wenzelm
Fri Dec 17 17:43:54 2010 +0100 (2010-12-17)
changeset 41229 d797baa3d57c
parent 39302 d7728f65b353
child 41792 ff3cb0c418b7
permissions -rw-r--r--
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
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(*  Title:      HOL/Transitive_Closure.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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*)
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header {* Reflexive and Transitive closure of a relation *}
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theory Transitive_Closure
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imports Predicate
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uses "~~/src/Provers/trancl.ML"
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begin
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text {*
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  @{text rtrancl} is reflexive/transitive closure,
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  @{text trancl} is transitive closure,
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  @{text reflcl} is reflexive closure.
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  These postfix operators have \emph{maximum priority}, forcing their
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  operands to be atomic.
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*}
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inductive_set
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  rtrancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   ("(_^*)" [1000] 999)
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  for r :: "('a \<times> 'a) set"
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where
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    rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) : r^*"
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  | rtrancl_into_rtrancl [Pure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"
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inductive_set
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  trancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"  ("(_^+)" [1000] 999)
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  for r :: "('a \<times> 'a) set"
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where
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    r_into_trancl [intro, Pure.intro]: "(a, b) : r ==> (a, b) : r^+"
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  | trancl_into_trancl [Pure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a, c) : r^+"
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declare rtrancl_def [nitpick_def del]
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        rtranclp_def [nitpick_def del]
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        trancl_def [nitpick_def del]
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        tranclp_def [nitpick_def 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^== == 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^= == r \<union> Id"
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notation (xsymbols)
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  rtranclp  ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
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  tranclp  ("(_\<^sup>+\<^sup>+)" [1000] 1000) and
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  reflclp  ("(_\<^sup>=\<^sup>=)" [1000] 1000) and
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  rtrancl  ("(_\<^sup>*)" [1000] 999) and
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  trancl  ("(_\<^sup>+)" [1000] 999) and
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  reflcl  ("(_\<^sup>=)" [1000] 999)
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notation (HTML output)
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  rtranclp  ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
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  tranclp  ("(_\<^sup>+\<^sup>+)" [1000] 1000) and
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  reflclp  ("(_\<^sup>=\<^sup>=)" [1000] 1000) and
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  rtrancl  ("(_\<^sup>*)" [1000] 999) and
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  trancl  ("(_\<^sup>+)" [1000] 999) and
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  reflcl  ("(_\<^sup>=)" [1000] 999)
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subsection {* Reflexive closure *}
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lemma refl_reflcl[simp]: "refl(r^=)"
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by(simp add:refl_on_def)
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lemma antisym_reflcl[simp]: "antisym(r^=) = antisym r"
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by(simp add:antisym_def)
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lemma trans_reflclI[simp]: "trans r \<Longrightarrow> trans(r^=)"
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unfolding trans_def by blast
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subsection {* Reflexive-transitive closure *}
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lemma reflcl_set_eq [pred_set_conv]: "(sup (\<lambda>x y. (x, y) \<in> r) op =) = (\<lambda>x y. (x, y) \<in> r \<union> Id)"
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  by (auto simp add: fun_eq_iff)
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lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*"
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  -- {* @{text rtrancl} of @{text r} contains @{text r} *}
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  apply (simp only: split_tupled_all)
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  apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl])
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  done
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lemma r_into_rtranclp [intro]: "r x y ==> r^** x y"
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  -- {* @{text rtrancl} of @{text r} contains @{text r} *}
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  by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl])
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lemma rtranclp_mono: "r \<le> s ==> r^** \<le> s^**"
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  -- {* monotonicity of @{text rtrancl} *}
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  apply (rule predicate2I)
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  apply (erule rtranclp.induct)
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   apply (rule_tac [2] rtranclp.rtrancl_into_rtrancl, blast+)
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  done
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lemmas rtrancl_mono = rtranclp_mono [to_set]
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theorem rtranclp_induct [consumes 1, case_names base step, induct set: rtranclp]:
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  assumes a: "r^** a b"
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    and cases: "P a" "!!y z. [| r^** a y; r y z; P y |] ==> P z"
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  shows "P b" using a
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  by (induct x\<equiv>a b) (rule cases)+
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lemmas rtrancl_induct [induct set: rtrancl] = rtranclp_induct [to_set]
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lemmas rtranclp_induct2 =
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  rtranclp_induct[of _ "(ax,ay)" "(bx,by)", split_rule,
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                 consumes 1, case_names refl step]
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lemmas rtrancl_induct2 =
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  rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
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                 consumes 1, case_names refl step]
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lemma refl_rtrancl: "refl (r^*)"
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by (unfold refl_on_def) fast
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text {* Transitivity of transitive closure. *}
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lemma trans_rtrancl: "trans (r^*)"
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proof (rule transI)
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  fix x y z
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  assume "(x, y) \<in> r\<^sup>*"
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  assume "(y, z) \<in> r\<^sup>*"
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  then show "(x, z) \<in> r\<^sup>*"
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  proof induct
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    case base
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    show "(x, y) \<in> r\<^sup>*" by fact
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  next
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    case (step u v)
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    from `(x, u) \<in> r\<^sup>*` and `(u, v) \<in> r`
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    show "(x, v) \<in> r\<^sup>*" ..
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  qed
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qed
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lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard]
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lemma rtranclp_trans:
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  assumes xy: "r^** x y"
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  and yz: "r^** y z"
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  shows "r^** x z" using yz xy
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  by induct iprover+
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lemma rtranclE [cases set: rtrancl]:
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  assumes major: "(a::'a, b) : r^*"
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  obtains
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    (base) "a = b"
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  | (step) y where "(a, y) : r^*" and "(y, b) : r"
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  -- {* elimination of @{text rtrancl} -- by induction on a special formula *}
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  apply (subgoal_tac "(a::'a) = b | (EX y. (a,y) : r^* & (y,b) : r)")
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   apply (rule_tac [2] major [THEN rtrancl_induct])
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    prefer 2 apply blast
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   prefer 2 apply blast
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  apply (erule asm_rl exE disjE conjE base step)+
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  done
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lemma rtrancl_Int_subset: "[| Id \<subseteq> s; (r^* \<inter> s) O r \<subseteq> s|] ==> r^* \<subseteq> s"
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  apply (rule subsetI)
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  apply (rule_tac p="x" in PairE, clarify)
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  apply (erule rtrancl_induct, auto) 
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  done
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lemma converse_rtranclp_into_rtranclp:
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  "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>*\<^sup>* a c"
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  by (rule rtranclp_trans) iprover+
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lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set]
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text {*
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  \medskip More @{term "r^*"} equations and inclusions.
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*}
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lemma rtranclp_idemp [simp]: "(r^**)^** = r^**"
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  apply (auto intro!: order_antisym)
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  apply (erule rtranclp_induct)
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   apply (rule rtranclp.rtrancl_refl)
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  apply (blast intro: rtranclp_trans)
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  done
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lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set]
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lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"
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  apply (rule set_eqI)
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  apply (simp only: split_tupled_all)
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  apply (blast intro: rtrancl_trans)
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  done
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lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*"
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  apply (drule rtrancl_mono)
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  apply simp
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  done
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lemma rtranclp_subset: "R \<le> S ==> S \<le> R^** ==> S^** = R^**"
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  apply (drule rtranclp_mono)
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  apply (drule rtranclp_mono)
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  apply simp
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  done
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lemmas rtrancl_subset = rtranclp_subset [to_set]
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lemma rtranclp_sup_rtranclp: "(sup (R^**) (S^**))^** = (sup R S)^**"
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  by (blast intro!: rtranclp_subset intro: rtranclp_mono [THEN predicate2D])
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lemmas rtrancl_Un_rtrancl = rtranclp_sup_rtranclp [to_set]
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lemma rtranclp_reflcl [simp]: "(R^==)^** = R^**"
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  by (blast intro!: rtranclp_subset)
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lemmas rtrancl_reflcl [simp] = rtranclp_reflcl [to_set]
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lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*"
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  apply (rule sym)
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  apply (rule rtrancl_subset, blast, clarify)
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  apply (rename_tac a b)
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  apply (case_tac "a = b")
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   apply blast
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  apply (blast intro!: r_into_rtrancl)
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  done
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lemma rtranclp_r_diff_Id: "(inf r op ~=)^** = r^**"
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  apply (rule sym)
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  apply (rule rtranclp_subset)
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   apply blast+
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  done
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theorem rtranclp_converseD:
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  assumes r: "(r^--1)^** x y"
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  shows "r^** y x"
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proof -
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  from r show ?thesis
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    by induct (iprover intro: rtranclp_trans dest!: conversepD)+
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qed
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lemmas rtrancl_converseD = rtranclp_converseD [to_set]
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theorem rtranclp_converseI:
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  assumes "r^** y x"
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  shows "(r^--1)^** x y"
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  using assms
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  by induct (iprover intro: rtranclp_trans conversepI)+
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lemmas rtrancl_converseI = rtranclp_converseI [to_set]
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lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
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  by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
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lemma sym_rtrancl: "sym r ==> sym (r^*)"
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  by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])
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theorem converse_rtranclp_induct [consumes 1, case_names base step]:
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  assumes major: "r^** a b"
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    and cases: "P b" "!!y z. [| r y z; r^** z b; P z |] ==> P y"
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  shows "P a"
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  using rtranclp_converseI [OF major]
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  by induct (iprover intro: cases dest!: conversepD rtranclp_converseD)+
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lemmas converse_rtrancl_induct = converse_rtranclp_induct [to_set]
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lemmas converse_rtranclp_induct2 =
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  converse_rtranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
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                 consumes 1, case_names refl step]
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lemmas converse_rtrancl_induct2 =
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  converse_rtrancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
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                 consumes 1, case_names refl step]
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lemma converse_rtranclpE [consumes 1, case_names base step]:
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  assumes major: "r^** x z"
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    and cases: "x=z ==> P"
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      "!!y. [| r x y; r^** y z |] ==> P"
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  shows P
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  apply (subgoal_tac "x = z | (EX y. r x y & r^** y z)")
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   apply (rule_tac [2] major [THEN converse_rtranclp_induct])
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    prefer 2 apply iprover
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   prefer 2 apply iprover
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  apply (erule asm_rl exE disjE conjE cases)+
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  done
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lemmas converse_rtranclE = converse_rtranclpE [to_set]
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lemmas converse_rtranclpE2 = converse_rtranclpE [of _ "(xa,xb)" "(za,zb)", split_rule]
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lemmas converse_rtranclE2 = converse_rtranclE [of "(xa,xb)" "(za,zb)", split_rule]
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lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"
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  by (blast elim: rtranclE converse_rtranclE
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    intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
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lemma rtrancl_unfold: "r^* = Id Un r^* O r"
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  by (auto intro: rtrancl_into_rtrancl elim: rtranclE)
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lemma rtrancl_Un_separatorE:
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  "(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (a,x) : P^* \<longrightarrow> (x,y) : Q \<longrightarrow> x=y \<Longrightarrow> (a,b) : P^*"
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apply (induct rule:rtrancl.induct)
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 apply blast
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apply (blast intro:rtrancl_trans)
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done
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lemma rtrancl_Un_separator_converseE:
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  "(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (x,b) : P^* \<longrightarrow> (y,x) : Q \<longrightarrow> y=x \<Longrightarrow> (a,b) : P^*"
nipkow@31690
   307
apply (induct rule:converse_rtrancl_induct)
nipkow@31690
   308
 apply blast
nipkow@31690
   309
apply (blast intro:rtrancl_trans)
nipkow@31690
   310
done
nipkow@31690
   311
haftmann@34970
   312
lemma Image_closed_trancl:
haftmann@34970
   313
  assumes "r `` X \<subseteq> X" shows "r\<^sup>* `` X = X"
haftmann@34970
   314
proof -
haftmann@34970
   315
  from assms have **: "{y. \<exists>x\<in>X. (x, y) \<in> r} \<subseteq> X" by auto
haftmann@34970
   316
  have "\<And>x y. (y, x) \<in> r\<^sup>* \<Longrightarrow> y \<in> X \<Longrightarrow> x \<in> X"
haftmann@34970
   317
  proof -
haftmann@34970
   318
    fix x y
haftmann@34970
   319
    assume *: "y \<in> X"
haftmann@34970
   320
    assume "(y, x) \<in> r\<^sup>*"
haftmann@34970
   321
    then show "x \<in> X"
haftmann@34970
   322
    proof induct
haftmann@34970
   323
      case base show ?case by (fact *)
haftmann@34970
   324
    next
haftmann@34970
   325
      case step with ** show ?case by auto
haftmann@34970
   326
    qed
haftmann@34970
   327
  qed
haftmann@34970
   328
  then show ?thesis by auto
haftmann@34970
   329
qed
haftmann@34970
   330
wenzelm@12691
   331
wenzelm@12691
   332
subsection {* Transitive closure *}
wenzelm@10331
   333
berghofe@13704
   334
lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
berghofe@23743
   335
  apply (simp add: split_tupled_all)
berghofe@13704
   336
  apply (erule trancl.induct)
wenzelm@26179
   337
   apply (iprover dest: subsetD)+
wenzelm@12691
   338
  done
wenzelm@12691
   339
berghofe@13704
   340
lemma r_into_trancl': "!!p. p : r ==> p : r^+"
berghofe@13704
   341
  by (simp only: split_tupled_all) (erule r_into_trancl)
berghofe@13704
   342
wenzelm@12691
   343
text {*
wenzelm@12691
   344
  \medskip Conversions between @{text trancl} and @{text rtrancl}.
wenzelm@12691
   345
*}
wenzelm@12691
   346
berghofe@23743
   347
lemma tranclp_into_rtranclp: "r^++ a b ==> r^** a b"
berghofe@23743
   348
  by (erule tranclp.induct) iprover+
wenzelm@12691
   349
berghofe@23743
   350
lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]
berghofe@22262
   351
berghofe@23743
   352
lemma rtranclp_into_tranclp1: assumes r: "r^** a b"
berghofe@22262
   353
  shows "!!c. r b c ==> r^++ a c" using r
nipkow@17589
   354
  by induct iprover+
wenzelm@12691
   355
berghofe@23743
   356
lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]
berghofe@22262
   357
berghofe@23743
   358
lemma rtranclp_into_tranclp2: "[| r a b; r^** b c |] ==> r^++ a c"
wenzelm@12691
   359
  -- {* intro rule from @{text r} and @{text rtrancl} *}
wenzelm@26179
   360
  apply (erule rtranclp.cases)
wenzelm@26179
   361
   apply iprover
berghofe@23743
   362
  apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1])
wenzelm@26179
   363
    apply (simp | rule r_into_rtranclp)+
wenzelm@12691
   364
  done
wenzelm@12691
   365
berghofe@23743
   366
lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]
berghofe@22262
   367
wenzelm@26179
   368
text {* Nice induction rule for @{text trancl} *}
wenzelm@26179
   369
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
berghofe@34909
   370
  assumes a: "r^++ a b"
berghofe@22262
   371
  and cases: "!!y. r a y ==> P y"
berghofe@22262
   372
    "!!y z. r^++ a y ==> r y z ==> P y ==> P z"
berghofe@34909
   373
  shows "P b" using a
berghofe@34909
   374
  by (induct x\<equiv>a b) (iprover intro: cases)+
wenzelm@12691
   375
berghofe@25425
   376
lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]
berghofe@22262
   377
berghofe@23743
   378
lemmas tranclp_induct2 =
wenzelm@26179
   379
  tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
wenzelm@26179
   380
    consumes 1, case_names base step]
berghofe@22262
   381
paulson@22172
   382
lemmas trancl_induct2 =
wenzelm@26179
   383
  trancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
wenzelm@26179
   384
    consumes 1, case_names base step]
paulson@22172
   385
berghofe@23743
   386
lemma tranclp_trans_induct:
berghofe@22262
   387
  assumes major: "r^++ x y"
berghofe@22262
   388
    and cases: "!!x y. r x y ==> P x y"
berghofe@22262
   389
      "!!x y z. [| r^++ x y; P x y; r^++ y z; P y z |] ==> P x z"
wenzelm@18372
   390
  shows "P x y"
wenzelm@12691
   391
  -- {* Another induction rule for trancl, incorporating transitivity *}
berghofe@23743
   392
  by (iprover intro: major [THEN tranclp_induct] cases)
wenzelm@12691
   393
berghofe@23743
   394
lemmas trancl_trans_induct = tranclp_trans_induct [to_set]
berghofe@23743
   395
wenzelm@26174
   396
lemma tranclE [cases set: trancl]:
wenzelm@26174
   397
  assumes "(a, b) : r^+"
wenzelm@26174
   398
  obtains
wenzelm@26174
   399
    (base) "(a, b) : r"
wenzelm@26174
   400
  | (step) c where "(a, c) : r^+" and "(c, b) : r"
wenzelm@26174
   401
  using assms by cases simp_all
wenzelm@10980
   402
krauss@32235
   403
lemma trancl_Int_subset: "[| r \<subseteq> s; (r^+ \<inter> s) O r \<subseteq> s|] ==> r^+ \<subseteq> s"
paulson@22080
   404
  apply (rule subsetI)
wenzelm@26179
   405
  apply (rule_tac p = x in PairE)
wenzelm@26179
   406
  apply clarify
wenzelm@26179
   407
  apply (erule trancl_induct)
wenzelm@26179
   408
   apply auto
paulson@22080
   409
  done
paulson@22080
   410
krauss@32235
   411
lemma trancl_unfold: "r^+ = r Un r^+ O r"
paulson@15551
   412
  by (auto intro: trancl_into_trancl elim: tranclE)
paulson@15551
   413
wenzelm@26179
   414
text {* Transitivity of @{term "r^+"} *}
wenzelm@26179
   415
lemma trans_trancl [simp]: "trans (r^+)"
berghofe@13704
   416
proof (rule transI)
berghofe@13704
   417
  fix x y z
wenzelm@26179
   418
  assume "(x, y) \<in> r^+"
berghofe@13704
   419
  assume "(y, z) \<in> r^+"
wenzelm@26179
   420
  then show "(x, z) \<in> r^+"
wenzelm@26179
   421
  proof induct
wenzelm@26179
   422
    case (base u)
wenzelm@26179
   423
    from `(x, y) \<in> r^+` and `(y, u) \<in> r`
wenzelm@26179
   424
    show "(x, u) \<in> r^+" ..
wenzelm@26179
   425
  next
wenzelm@26179
   426
    case (step u v)
wenzelm@26179
   427
    from `(x, u) \<in> r^+` and `(u, v) \<in> r`
wenzelm@26179
   428
    show "(x, v) \<in> r^+" ..
wenzelm@26179
   429
  qed
berghofe@13704
   430
qed
wenzelm@12691
   431
wenzelm@12691
   432
lemmas trancl_trans = trans_trancl [THEN transD, standard]
wenzelm@12691
   433
berghofe@23743
   434
lemma tranclp_trans:
berghofe@22262
   435
  assumes xy: "r^++ x y"
berghofe@22262
   436
  and yz: "r^++ y z"
berghofe@22262
   437
  shows "r^++ x z" using yz xy
berghofe@22262
   438
  by induct iprover+
berghofe@22262
   439
wenzelm@26179
   440
lemma trancl_id [simp]: "trans r \<Longrightarrow> r^+ = r"
wenzelm@26179
   441
  apply auto
wenzelm@26179
   442
  apply (erule trancl_induct)
wenzelm@26179
   443
   apply assumption
wenzelm@26179
   444
  apply (unfold trans_def)
wenzelm@26179
   445
  apply blast
wenzelm@26179
   446
  done
nipkow@19623
   447
wenzelm@26179
   448
lemma rtranclp_tranclp_tranclp:
wenzelm@26179
   449
  assumes "r^** x y"
wenzelm@26179
   450
  shows "!!z. r^++ y z ==> r^++ x z" using assms
berghofe@23743
   451
  by induct (iprover intro: tranclp_trans)+
wenzelm@12691
   452
berghofe@23743
   453
lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]
berghofe@22262
   454
berghofe@23743
   455
lemma tranclp_into_tranclp2: "r a b ==> r^++ b c ==> r^++ a c"
berghofe@23743
   456
  by (erule tranclp_trans [OF tranclp.r_into_trancl])
berghofe@22262
   457
berghofe@23743
   458
lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]
wenzelm@12691
   459
wenzelm@12691
   460
lemma trancl_insert:
wenzelm@12691
   461
  "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
wenzelm@12691
   462
  -- {* primitive recursion for @{text trancl} over finite relations *}
wenzelm@12691
   463
  apply (rule equalityI)
wenzelm@12691
   464
   apply (rule subsetI)
wenzelm@12691
   465
   apply (simp only: split_tupled_all)
paulson@14208
   466
   apply (erule trancl_induct, blast)
huffman@35216
   467
   apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)
wenzelm@12691
   468
  apply (rule subsetI)
wenzelm@12691
   469
  apply (blast intro: trancl_mono rtrancl_mono
wenzelm@12691
   470
    [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
wenzelm@12691
   471
  done
wenzelm@12691
   472
berghofe@23743
   473
lemma tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ x y"
berghofe@22262
   474
  apply (drule conversepD)
berghofe@23743
   475
  apply (erule tranclp_induct)
berghofe@23743
   476
  apply (iprover intro: conversepI tranclp_trans)+
wenzelm@12691
   477
  done
wenzelm@12691
   478
berghofe@23743
   479
lemmas trancl_converseI = tranclp_converseI [to_set]
berghofe@22262
   480
berghofe@23743
   481
lemma tranclp_converseD: "(r^--1)^++ x y ==> (r^++)^--1 x y"
berghofe@22262
   482
  apply (rule conversepI)
berghofe@23743
   483
  apply (erule tranclp_induct)
berghofe@23743
   484
  apply (iprover dest: conversepD intro: tranclp_trans)+
berghofe@13704
   485
  done
wenzelm@12691
   486
berghofe@23743
   487
lemmas trancl_converseD = tranclp_converseD [to_set]
berghofe@22262
   488
berghofe@23743
   489
lemma tranclp_converse: "(r^--1)^++ = (r^++)^--1"
nipkow@39302
   490
  by (fastsimp simp add: fun_eq_iff
berghofe@23743
   491
    intro!: tranclp_converseI dest!: tranclp_converseD)
berghofe@22262
   492
berghofe@23743
   493
lemmas trancl_converse = tranclp_converse [to_set]
wenzelm@12691
   494
huffman@19228
   495
lemma sym_trancl: "sym r ==> sym (r^+)"
huffman@19228
   496
  by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
huffman@19228
   497
berghofe@34909
   498
lemma converse_tranclp_induct [consumes 1, case_names base step]:
berghofe@22262
   499
  assumes major: "r^++ a b"
berghofe@22262
   500
    and cases: "!!y. r y b ==> P(y)"
berghofe@22262
   501
      "!!y z.[| r y z;  r^++ z b;  P(z) |] ==> P(y)"
wenzelm@18372
   502
  shows "P a"
berghofe@23743
   503
  apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])
wenzelm@18372
   504
   apply (rule cases)
berghofe@22262
   505
   apply (erule conversepD)
huffman@35216
   506
  apply (blast intro: assms dest!: tranclp_converseD)
wenzelm@18372
   507
  done
wenzelm@12691
   508
berghofe@23743
   509
lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
berghofe@22262
   510
berghofe@23743
   511
lemma tranclpD: "R^++ x y ==> EX z. R x z \<and> R^** z y"
wenzelm@26179
   512
  apply (erule converse_tranclp_induct)
wenzelm@26179
   513
   apply auto
berghofe@23743
   514
  apply (blast intro: rtranclp_trans)
wenzelm@12691
   515
  done
wenzelm@12691
   516
berghofe@23743
   517
lemmas tranclD = tranclpD [to_set]
berghofe@22262
   518
bulwahn@31577
   519
lemma converse_tranclpE:
bulwahn@31577
   520
  assumes major: "tranclp r x z"
bulwahn@31577
   521
  assumes base: "r x z ==> P"
bulwahn@31577
   522
  assumes step: "\<And> y. [| r x y; tranclp r y z |] ==> P"
bulwahn@31577
   523
  shows P
bulwahn@31577
   524
proof -
bulwahn@31577
   525
  from tranclpD[OF major]
bulwahn@31577
   526
  obtain y where "r x y" and "rtranclp r y z" by iprover
bulwahn@31577
   527
  from this(2) show P
bulwahn@31577
   528
  proof (cases rule: rtranclp.cases)
bulwahn@31577
   529
    case rtrancl_refl
bulwahn@31577
   530
    with `r x y` base show P by iprover
bulwahn@31577
   531
  next
bulwahn@31577
   532
    case rtrancl_into_rtrancl
bulwahn@31577
   533
    from this have "tranclp r y z"
bulwahn@31577
   534
      by (iprover intro: rtranclp_into_tranclp1)
bulwahn@31577
   535
    with `r x y` step show P by iprover
bulwahn@31577
   536
  qed
bulwahn@31577
   537
qed
bulwahn@31577
   538
bulwahn@31577
   539
lemmas converse_tranclE = converse_tranclpE [to_set]
bulwahn@31577
   540
kleing@25295
   541
lemma tranclD2:
kleing@25295
   542
  "(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
kleing@25295
   543
  by (blast elim: tranclE intro: trancl_into_rtrancl)
kleing@25295
   544
nipkow@13867
   545
lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
wenzelm@18372
   546
  by (blast elim: tranclE dest: trancl_into_rtrancl)
wenzelm@12691
   547
wenzelm@12691
   548
lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"
wenzelm@12691
   549
  by (blast dest: r_into_trancl)
wenzelm@12691
   550
wenzelm@12691
   551
lemma trancl_subset_Sigma_aux:
wenzelm@12691
   552
    "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
wenzelm@18372
   553
  by (induct rule: rtrancl_induct) auto
wenzelm@12691
   554
wenzelm@12691
   555
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"
berghofe@13704
   556
  apply (rule subsetI)
berghofe@13704
   557
  apply (simp only: split_tupled_all)
berghofe@13704
   558
  apply (erule tranclE)
wenzelm@26179
   559
   apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
wenzelm@12691
   560
  done
nipkow@10996
   561
berghofe@23743
   562
lemma reflcl_tranclp [simp]: "(r^++)^== = r^**"
berghofe@22262
   563
  apply (safe intro!: order_antisym)
berghofe@23743
   564
   apply (erule tranclp_into_rtranclp)
berghofe@23743
   565
  apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)
wenzelm@11084
   566
  done
nipkow@10996
   567
berghofe@23743
   568
lemmas reflcl_trancl [simp] = reflcl_tranclp [to_set]
berghofe@22262
   569
wenzelm@11090
   570
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
wenzelm@11084
   571
  apply safe
paulson@14208
   572
   apply (drule trancl_into_rtrancl, simp)
paulson@14208
   573
  apply (erule rtranclE, safe)
paulson@14208
   574
   apply (rule r_into_trancl, simp)
wenzelm@11084
   575
  apply (rule rtrancl_into_trancl1)
paulson@14208
   576
   apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
wenzelm@11084
   577
  done
nipkow@10996
   578
wenzelm@11090
   579
lemma trancl_empty [simp]: "{}^+ = {}"
wenzelm@11084
   580
  by (auto elim: trancl_induct)
nipkow@10996
   581
wenzelm@11090
   582
lemma rtrancl_empty [simp]: "{}^* = Id"
wenzelm@11084
   583
  by (rule subst [OF reflcl_trancl]) simp
nipkow@10996
   584
berghofe@23743
   585
lemma rtranclpD: "R^** a b ==> a = b \<or> a \<noteq> b \<and> R^++ a b"
berghofe@23743
   586
  by (force simp add: reflcl_tranclp [symmetric] simp del: reflcl_tranclp)
berghofe@22262
   587
berghofe@23743
   588
lemmas rtranclD = rtranclpD [to_set]
wenzelm@11084
   589
kleing@16514
   590
lemma rtrancl_eq_or_trancl:
kleing@16514
   591
  "(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"
kleing@16514
   592
  by (fast elim: trancl_into_rtrancl dest: rtranclD)
nipkow@10996
   593
krauss@33656
   594
lemma trancl_unfold_right: "r^+ = r^* O r"
krauss@33656
   595
by (auto dest: tranclD2 intro: rtrancl_into_trancl1)
krauss@33656
   596
krauss@33656
   597
lemma trancl_unfold_left: "r^+ = r O r^*"
krauss@33656
   598
by (auto dest: tranclD intro: rtrancl_into_trancl2)
krauss@33656
   599
krauss@33656
   600
krauss@33656
   601
text {* Simplifying nested closures *}
krauss@33656
   602
krauss@33656
   603
lemma rtrancl_trancl_absorb[simp]: "(R^*)^+ = R^*"
krauss@33656
   604
by (simp add: trans_rtrancl)
krauss@33656
   605
krauss@33656
   606
lemma trancl_rtrancl_absorb[simp]: "(R^+)^* = R^*"
krauss@33656
   607
by (subst reflcl_trancl[symmetric]) simp
krauss@33656
   608
krauss@33656
   609
lemma rtrancl_reflcl_absorb[simp]: "(R^*)^= = R^*"
krauss@33656
   610
by auto
krauss@33656
   611
krauss@33656
   612
wenzelm@12691
   613
text {* @{text Domain} and @{text Range} *}
nipkow@10996
   614
wenzelm@11090
   615
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
wenzelm@11084
   616
  by blast
nipkow@10996
   617
wenzelm@11090
   618
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
wenzelm@11084
   619
  by blast
nipkow@10996
   620
wenzelm@11090
   621
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
wenzelm@11084
   622
  by (rule rtrancl_Un_rtrancl [THEN subst]) fast
nipkow@10996
   623
wenzelm@11090
   624
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
wenzelm@11084
   625
  by (blast intro: subsetD [OF rtrancl_Un_subset])
nipkow@10996
   626
wenzelm@11090
   627
lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
wenzelm@11084
   628
  by (unfold Domain_def) (blast dest: tranclD)
nipkow@10996
   629
wenzelm@11090
   630
lemma trancl_range [simp]: "Range (r^+) = Range r"
nipkow@26271
   631
unfolding Range_def by(simp add: trancl_converse [symmetric])
nipkow@10996
   632
paulson@11115
   633
lemma Not_Domain_rtrancl:
wenzelm@12691
   634
    "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
wenzelm@12691
   635
  apply auto
wenzelm@26179
   636
  apply (erule rev_mp)
wenzelm@26179
   637
  apply (erule rtrancl_induct)
wenzelm@26179
   638
   apply auto
wenzelm@26179
   639
  done
berghofe@11327
   640
haftmann@29609
   641
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
haftmann@29609
   642
  apply clarify
haftmann@29609
   643
  apply (erule trancl_induct)
haftmann@29609
   644
   apply (auto simp add: Field_def)
haftmann@29609
   645
  done
haftmann@29609
   646
haftmann@29609
   647
lemma finite_trancl: "finite (r^+) = finite r"
haftmann@29609
   648
  apply auto
haftmann@29609
   649
   prefer 2
haftmann@29609
   650
   apply (rule trancl_subset_Field2 [THEN finite_subset])
haftmann@29609
   651
   apply (rule finite_SigmaI)
haftmann@29609
   652
    prefer 3
haftmann@29609
   653
    apply (blast intro: r_into_trancl' finite_subset)
haftmann@29609
   654
   apply (auto simp add: finite_Field)
haftmann@29609
   655
  done
haftmann@29609
   656
wenzelm@12691
   657
text {* More about converse @{text rtrancl} and @{text trancl}, should
wenzelm@12691
   658
  be merged with main body. *}
kleing@12428
   659
nipkow@14337
   660
lemma single_valued_confluent:
nipkow@14337
   661
  "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>
nipkow@14337
   662
  \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
wenzelm@26179
   663
  apply (erule rtrancl_induct)
wenzelm@26179
   664
  apply simp
wenzelm@26179
   665
  apply (erule disjE)
wenzelm@26179
   666
   apply (blast elim:converse_rtranclE dest:single_valuedD)
wenzelm@26179
   667
  apply(blast intro:rtrancl_trans)
wenzelm@26179
   668
  done
nipkow@14337
   669
wenzelm@12691
   670
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"
kleing@12428
   671
  by (fast intro: trancl_trans)
kleing@12428
   672
kleing@12428
   673
lemma trancl_into_trancl [rule_format]:
wenzelm@12691
   674
    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
wenzelm@12691
   675
  apply (erule trancl_induct)
kleing@12428
   676
   apply (fast intro: r_r_into_trancl)
kleing@12428
   677
  apply (fast intro: r_r_into_trancl trancl_trans)
kleing@12428
   678
  done
kleing@12428
   679
berghofe@23743
   680
lemma tranclp_rtranclp_tranclp:
berghofe@22262
   681
    "r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c"
berghofe@23743
   682
  apply (drule tranclpD)
wenzelm@26179
   683
  apply (elim exE conjE)
berghofe@23743
   684
  apply (drule rtranclp_trans, assumption)
berghofe@23743
   685
  apply (drule rtranclp_into_tranclp2, assumption, assumption)
kleing@12428
   686
  done
kleing@12428
   687
berghofe@23743
   688
lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]
berghofe@22262
   689
wenzelm@12691
   690
lemmas transitive_closure_trans [trans] =
wenzelm@12691
   691
  r_r_into_trancl trancl_trans rtrancl_trans
berghofe@23743
   692
  trancl.trancl_into_trancl trancl_into_trancl2
berghofe@23743
   693
  rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
wenzelm@12691
   694
  rtrancl_trancl_trancl trancl_rtrancl_trancl
kleing@12428
   695
berghofe@23743
   696
lemmas transitive_closurep_trans' [trans] =
berghofe@23743
   697
  tranclp_trans rtranclp_trans
berghofe@23743
   698
  tranclp.trancl_into_trancl tranclp_into_tranclp2
berghofe@23743
   699
  rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp
berghofe@23743
   700
  rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp
berghofe@22262
   701
kleing@12428
   702
declare trancl_into_rtrancl [elim]
berghofe@11327
   703
haftmann@30954
   704
subsection {* The power operation on relations *}
haftmann@30954
   705
haftmann@30954
   706
text {* @{text "R ^^ n = R O ... O R"}, the n-fold composition of @{text R} *}
haftmann@30954
   707
haftmann@30971
   708
overloading
haftmann@30971
   709
  relpow == "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
haftmann@30971
   710
begin
haftmann@30954
   711
haftmann@30971
   712
primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
haftmann@30971
   713
    "relpow 0 R = Id"
krauss@32235
   714
  | "relpow (Suc n) R = (R ^^ n) O R"
haftmann@30954
   715
haftmann@30971
   716
end
haftmann@30954
   717
haftmann@30954
   718
lemma rel_pow_1 [simp]:
haftmann@30971
   719
  fixes R :: "('a \<times> 'a) set"
haftmann@30971
   720
  shows "R ^^ 1 = R"
haftmann@30954
   721
  by simp
haftmann@30954
   722
haftmann@30954
   723
lemma rel_pow_0_I: 
haftmann@30954
   724
  "(x, x) \<in> R ^^ 0"
haftmann@30954
   725
  by simp
haftmann@30954
   726
haftmann@30954
   727
lemma rel_pow_Suc_I:
haftmann@30954
   728
  "(x, y) \<in>  R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
haftmann@30954
   729
  by auto
haftmann@30954
   730
haftmann@30954
   731
lemma rel_pow_Suc_I2:
haftmann@30954
   732
  "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
haftmann@30954
   733
  by (induct n arbitrary: z) (simp, fastsimp)
haftmann@30954
   734
haftmann@30954
   735
lemma rel_pow_0_E:
haftmann@30954
   736
  "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30954
   737
  by simp
haftmann@30954
   738
haftmann@30954
   739
lemma rel_pow_Suc_E:
haftmann@30954
   740
  "(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
   741
  by auto
haftmann@30954
   742
haftmann@30954
   743
lemma rel_pow_E:
haftmann@30954
   744
  "(x, z) \<in>  R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
haftmann@30954
   745
   \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in>  R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P)
haftmann@30954
   746
   \<Longrightarrow> P"
haftmann@30954
   747
  by (cases n) auto
haftmann@30954
   748
haftmann@30954
   749
lemma rel_pow_Suc_D2:
haftmann@30954
   750
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
haftmann@30954
   751
  apply (induct n arbitrary: x z)
haftmann@30954
   752
   apply (blast intro: rel_pow_0_I elim: rel_pow_0_E rel_pow_Suc_E)
haftmann@30954
   753
  apply (blast intro: rel_pow_Suc_I elim: rel_pow_0_E rel_pow_Suc_E)
haftmann@30954
   754
  done
haftmann@30954
   755
haftmann@30954
   756
lemma rel_pow_Suc_E2:
haftmann@30954
   757
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30954
   758
  by (blast dest: rel_pow_Suc_D2)
haftmann@30954
   759
haftmann@30954
   760
lemma rel_pow_Suc_D2':
haftmann@30954
   761
  "\<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
   762
  by (induct n) (simp_all, blast)
haftmann@30954
   763
haftmann@30954
   764
lemma rel_pow_E2:
haftmann@30954
   765
  "(x, z) \<in> R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
haftmann@30954
   766
     \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P)
haftmann@30954
   767
   \<Longrightarrow> P"
haftmann@30954
   768
  apply (cases n, simp)
haftmann@30954
   769
  apply (cut_tac n=nat and R=R in rel_pow_Suc_D2', simp, blast)
haftmann@30954
   770
  done
haftmann@30954
   771
krauss@32235
   772
lemma rel_pow_add: "R ^^ (m+n) = R^^m O R^^n"
nipkow@31351
   773
by(induct n) auto
nipkow@31351
   774
krauss@31970
   775
lemma rel_pow_commute: "R O R ^^ n = R ^^ n O R"
krauss@32235
   776
by (induct n) (simp, simp add: O_assoc [symmetric])
krauss@31970
   777
haftmann@30954
   778
lemma rtrancl_imp_UN_rel_pow:
haftmann@30954
   779
  assumes "p \<in> R^*"
haftmann@30954
   780
  shows "p \<in> (\<Union>n. R ^^ n)"
haftmann@30954
   781
proof (cases p)
haftmann@30954
   782
  case (Pair x y)
haftmann@30954
   783
  with assms have "(x, y) \<in> R^*" by simp
haftmann@30954
   784
  then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct
haftmann@30954
   785
    case base show ?case by (blast intro: rel_pow_0_I)
haftmann@30954
   786
  next
haftmann@30954
   787
    case step then show ?case by (blast intro: rel_pow_Suc_I)
haftmann@30954
   788
  qed
haftmann@30954
   789
  with Pair show ?thesis by simp
haftmann@30954
   790
qed
haftmann@30954
   791
haftmann@30954
   792
lemma rel_pow_imp_rtrancl:
haftmann@30954
   793
  assumes "p \<in> R ^^ n"
haftmann@30954
   794
  shows "p \<in> R^*"
haftmann@30954
   795
proof (cases p)
haftmann@30954
   796
  case (Pair x y)
haftmann@30954
   797
  with assms have "(x, y) \<in> R ^^ n" by simp
haftmann@30954
   798
  then have "(x, y) \<in> R^*" proof (induct n arbitrary: x y)
haftmann@30954
   799
    case 0 then show ?case by simp
haftmann@30954
   800
  next
haftmann@30954
   801
    case Suc then show ?case
haftmann@30954
   802
      by (blast elim: rel_pow_Suc_E intro: rtrancl_into_rtrancl)
haftmann@30954
   803
  qed
haftmann@30954
   804
  with Pair show ?thesis by simp
haftmann@30954
   805
qed
haftmann@30954
   806
haftmann@30954
   807
lemma rtrancl_is_UN_rel_pow:
haftmann@30954
   808
  "R^* = (\<Union>n. R ^^ n)"
haftmann@30954
   809
  by (blast intro: rtrancl_imp_UN_rel_pow rel_pow_imp_rtrancl)
haftmann@30954
   810
haftmann@30954
   811
lemma rtrancl_power:
haftmann@30954
   812
  "p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
haftmann@30954
   813
  by (simp add: rtrancl_is_UN_rel_pow)
haftmann@30954
   814
haftmann@30954
   815
lemma trancl_power:
haftmann@30954
   816
  "p \<in> R^+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
haftmann@30954
   817
  apply (cases p)
haftmann@30954
   818
  apply simp
haftmann@30954
   819
  apply (rule iffI)
haftmann@30954
   820
   apply (drule tranclD2)
haftmann@30954
   821
   apply (clarsimp simp: rtrancl_is_UN_rel_pow)
haftmann@30971
   822
   apply (rule_tac x="Suc n" in exI)
haftmann@30954
   823
   apply (clarsimp simp: rel_comp_def)
haftmann@30954
   824
   apply fastsimp
haftmann@30954
   825
  apply clarsimp
haftmann@30954
   826
  apply (case_tac n, simp)
haftmann@30954
   827
  apply clarsimp
haftmann@30954
   828
  apply (drule rel_pow_imp_rtrancl)
haftmann@30954
   829
  apply (drule rtrancl_into_trancl1) apply auto
haftmann@30954
   830
  done
haftmann@30954
   831
haftmann@30954
   832
lemma rtrancl_imp_rel_pow:
haftmann@30954
   833
  "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
haftmann@30954
   834
  by (auto dest: rtrancl_imp_UN_rel_pow)
haftmann@30954
   835
haftmann@30954
   836
lemma single_valued_rel_pow:
haftmann@30954
   837
  fixes R :: "('a * 'a) set"
haftmann@30954
   838
  shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
haftmann@30954
   839
  apply (induct n arbitrary: R)
haftmann@30954
   840
  apply simp_all
haftmann@30954
   841
  apply (rule single_valuedI)
haftmann@30954
   842
  apply (fast dest: single_valuedD elim: rel_pow_Suc_E)
haftmann@30954
   843
  done
paulson@15551
   844
ballarin@15076
   845
subsection {* Setup of transitivity reasoner *}
ballarin@15076
   846
wenzelm@26340
   847
ML {*
ballarin@15076
   848
wenzelm@32215
   849
structure Trancl_Tac = Trancl_Tac
wenzelm@32215
   850
(
wenzelm@32215
   851
  val r_into_trancl = @{thm trancl.r_into_trancl};
wenzelm@32215
   852
  val trancl_trans  = @{thm trancl_trans};
wenzelm@32215
   853
  val rtrancl_refl = @{thm rtrancl.rtrancl_refl};
wenzelm@32215
   854
  val r_into_rtrancl = @{thm r_into_rtrancl};
wenzelm@32215
   855
  val trancl_into_rtrancl = @{thm trancl_into_rtrancl};
wenzelm@32215
   856
  val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl};
wenzelm@32215
   857
  val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl};
wenzelm@32215
   858
  val rtrancl_trans = @{thm rtrancl_trans};
ballarin@15096
   859
berghofe@30107
   860
  fun decomp (@{const Trueprop} $ t) =
haftmann@37677
   861
    let fun dec (Const (@{const_name Set.member}, _) $ (Const (@{const_name Pair}, _) $ a $ b) $ rel ) =
berghofe@23743
   862
        let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*")
berghofe@23743
   863
              | decr (Const ("Transitive_Closure.trancl", _ ) $ r)  = (r,"r+")
wenzelm@18372
   864
              | decr r = (r,"r");
berghofe@26801
   865
            val (rel,r) = decr (Envir.beta_eta_contract rel);
wenzelm@18372
   866
        in SOME (a,b,rel,r) end
wenzelm@18372
   867
      | dec _ =  NONE
berghofe@30107
   868
    in dec t end
berghofe@30107
   869
    | decomp _ = NONE;
wenzelm@32215
   870
);
ballarin@15076
   871
wenzelm@32215
   872
structure Tranclp_Tac = Trancl_Tac
wenzelm@32215
   873
(
wenzelm@32215
   874
  val r_into_trancl = @{thm tranclp.r_into_trancl};
wenzelm@32215
   875
  val trancl_trans  = @{thm tranclp_trans};
wenzelm@32215
   876
  val rtrancl_refl = @{thm rtranclp.rtrancl_refl};
wenzelm@32215
   877
  val r_into_rtrancl = @{thm r_into_rtranclp};
wenzelm@32215
   878
  val trancl_into_rtrancl = @{thm tranclp_into_rtranclp};
wenzelm@32215
   879
  val rtrancl_trancl_trancl = @{thm rtranclp_tranclp_tranclp};
wenzelm@32215
   880
  val trancl_rtrancl_trancl = @{thm tranclp_rtranclp_tranclp};
wenzelm@32215
   881
  val rtrancl_trans = @{thm rtranclp_trans};
berghofe@22262
   882
berghofe@30107
   883
  fun decomp (@{const Trueprop} $ t) =
berghofe@22262
   884
    let fun dec (rel $ a $ b) =
berghofe@23743
   885
        let fun decr (Const ("Transitive_Closure.rtranclp", _ ) $ r) = (r,"r*")
berghofe@23743
   886
              | decr (Const ("Transitive_Closure.tranclp", _ ) $ r)  = (r,"r+")
berghofe@22262
   887
              | decr r = (r,"r");
berghofe@22262
   888
            val (rel,r) = decr rel;
berghofe@26801
   889
        in SOME (a, b, rel, r) end
berghofe@22262
   890
      | dec _ =  NONE
berghofe@30107
   891
    in dec t end
berghofe@30107
   892
    | decomp _ = NONE;
wenzelm@32215
   893
);
wenzelm@26340
   894
*}
berghofe@22262
   895
wenzelm@26340
   896
declaration {* fn _ =>
wenzelm@26340
   897
  Simplifier.map_ss (fn ss => ss
wenzelm@32215
   898
    addSolver (mk_solver' "Trancl" (Trancl_Tac.trancl_tac o Simplifier.the_context))
wenzelm@32215
   899
    addSolver (mk_solver' "Rtrancl" (Trancl_Tac.rtrancl_tac o Simplifier.the_context))
wenzelm@32215
   900
    addSolver (mk_solver' "Tranclp" (Tranclp_Tac.trancl_tac o Simplifier.the_context))
wenzelm@32215
   901
    addSolver (mk_solver' "Rtranclp" (Tranclp_Tac.rtrancl_tac o Simplifier.the_context)))
ballarin@15076
   902
*}
ballarin@15076
   903
wenzelm@32215
   904
wenzelm@32215
   905
text {* Optional methods. *}
ballarin@15076
   906
ballarin@15076
   907
method_setup trancl =
wenzelm@32215
   908
  {* Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.trancl_tac) *}
wenzelm@18372
   909
  {* simple transitivity reasoner *}
ballarin@15076
   910
method_setup rtrancl =
wenzelm@32215
   911
  {* Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.rtrancl_tac) *}
ballarin@15076
   912
  {* simple transitivity reasoner *}
berghofe@22262
   913
method_setup tranclp =
wenzelm@32215
   914
  {* Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.trancl_tac) *}
berghofe@22262
   915
  {* simple transitivity reasoner (predicate version) *}
berghofe@22262
   916
method_setup rtranclp =
wenzelm@32215
   917
  {* Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.rtrancl_tac) *}
berghofe@22262
   918
  {* simple transitivity reasoner (predicate version) *}
ballarin@15076
   919
nipkow@10213
   920
end