src/HOL/Transitive_Closure.thy
author blanchet
Tue Nov 24 10:31:01 2009 +0100 (2009-11-24)
changeset 33878 85102f57b4a8
parent 33656 fc1af6753233
child 34909 a799687944af
permissions -rw-r--r--
removed "nitpick_def" attributes from (r)trancl(p), since "Nitpick.thy" overrides these
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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: expand_fun_eq)
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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"
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proof -
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  from a have "a = a --> P b"
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    by (induct "%x y. x = a --> P y" a b) (iprover intro: cases)+
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  then show ?thesis by iprover
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qed
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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_ext)
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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]:
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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:
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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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   309
lemma rtrancl_Un_separator_converseE:
nipkow@31690
   310
  "(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
   311
apply (induct rule:converse_rtrancl_induct)
nipkow@31690
   312
 apply blast
nipkow@31690
   313
apply (blast intro:rtrancl_trans)
nipkow@31690
   314
done
nipkow@31690
   315
wenzelm@12691
   316
wenzelm@12691
   317
subsection {* Transitive closure *}
wenzelm@10331
   318
berghofe@13704
   319
lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
berghofe@23743
   320
  apply (simp add: split_tupled_all)
berghofe@13704
   321
  apply (erule trancl.induct)
wenzelm@26179
   322
   apply (iprover dest: subsetD)+
wenzelm@12691
   323
  done
wenzelm@12691
   324
berghofe@13704
   325
lemma r_into_trancl': "!!p. p : r ==> p : r^+"
berghofe@13704
   326
  by (simp only: split_tupled_all) (erule r_into_trancl)
berghofe@13704
   327
wenzelm@12691
   328
text {*
wenzelm@12691
   329
  \medskip Conversions between @{text trancl} and @{text rtrancl}.
wenzelm@12691
   330
*}
wenzelm@12691
   331
berghofe@23743
   332
lemma tranclp_into_rtranclp: "r^++ a b ==> r^** a b"
berghofe@23743
   333
  by (erule tranclp.induct) iprover+
wenzelm@12691
   334
berghofe@23743
   335
lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]
berghofe@22262
   336
berghofe@23743
   337
lemma rtranclp_into_tranclp1: assumes r: "r^** a b"
berghofe@22262
   338
  shows "!!c. r b c ==> r^++ a c" using r
nipkow@17589
   339
  by induct iprover+
wenzelm@12691
   340
berghofe@23743
   341
lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]
berghofe@22262
   342
berghofe@23743
   343
lemma rtranclp_into_tranclp2: "[| r a b; r^** b c |] ==> r^++ a c"
wenzelm@12691
   344
  -- {* intro rule from @{text r} and @{text rtrancl} *}
wenzelm@26179
   345
  apply (erule rtranclp.cases)
wenzelm@26179
   346
   apply iprover
berghofe@23743
   347
  apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1])
wenzelm@26179
   348
    apply (simp | rule r_into_rtranclp)+
wenzelm@12691
   349
  done
wenzelm@12691
   350
berghofe@23743
   351
lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]
berghofe@22262
   352
wenzelm@26179
   353
text {* Nice induction rule for @{text trancl} *}
wenzelm@26179
   354
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
wenzelm@26179
   355
  assumes "r^++ a b"
berghofe@22262
   356
  and cases: "!!y. r a y ==> P y"
berghofe@22262
   357
    "!!y z. r^++ a y ==> r y z ==> P y ==> P z"
berghofe@13704
   358
  shows "P b"
wenzelm@12691
   359
proof -
wenzelm@26179
   360
  from `r^++ a b` have "a = a --> P b"
nipkow@17589
   361
    by (induct "%x y. x = a --> P y" a b) (iprover intro: cases)+
wenzelm@26179
   362
  then show ?thesis by iprover
wenzelm@12691
   363
qed
wenzelm@12691
   364
berghofe@25425
   365
lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]
berghofe@22262
   366
berghofe@23743
   367
lemmas tranclp_induct2 =
wenzelm@26179
   368
  tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
wenzelm@26179
   369
    consumes 1, case_names base step]
berghofe@22262
   370
paulson@22172
   371
lemmas trancl_induct2 =
wenzelm@26179
   372
  trancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
wenzelm@26179
   373
    consumes 1, case_names base step]
paulson@22172
   374
berghofe@23743
   375
lemma tranclp_trans_induct:
berghofe@22262
   376
  assumes major: "r^++ x y"
berghofe@22262
   377
    and cases: "!!x y. r x y ==> P x y"
berghofe@22262
   378
      "!!x y z. [| r^++ x y; P x y; r^++ y z; P y z |] ==> P x z"
wenzelm@18372
   379
  shows "P x y"
wenzelm@12691
   380
  -- {* Another induction rule for trancl, incorporating transitivity *}
berghofe@23743
   381
  by (iprover intro: major [THEN tranclp_induct] cases)
wenzelm@12691
   382
berghofe@23743
   383
lemmas trancl_trans_induct = tranclp_trans_induct [to_set]
berghofe@23743
   384
wenzelm@26174
   385
lemma tranclE [cases set: trancl]:
wenzelm@26174
   386
  assumes "(a, b) : r^+"
wenzelm@26174
   387
  obtains
wenzelm@26174
   388
    (base) "(a, b) : r"
wenzelm@26174
   389
  | (step) c where "(a, c) : r^+" and "(c, b) : r"
wenzelm@26174
   390
  using assms by cases simp_all
wenzelm@10980
   391
krauss@32235
   392
lemma trancl_Int_subset: "[| r \<subseteq> s; (r^+ \<inter> s) O r \<subseteq> s|] ==> r^+ \<subseteq> s"
paulson@22080
   393
  apply (rule subsetI)
wenzelm@26179
   394
  apply (rule_tac p = x in PairE)
wenzelm@26179
   395
  apply clarify
wenzelm@26179
   396
  apply (erule trancl_induct)
wenzelm@26179
   397
   apply auto
paulson@22080
   398
  done
paulson@22080
   399
krauss@32235
   400
lemma trancl_unfold: "r^+ = r Un r^+ O r"
paulson@15551
   401
  by (auto intro: trancl_into_trancl elim: tranclE)
paulson@15551
   402
wenzelm@26179
   403
text {* Transitivity of @{term "r^+"} *}
wenzelm@26179
   404
lemma trans_trancl [simp]: "trans (r^+)"
berghofe@13704
   405
proof (rule transI)
berghofe@13704
   406
  fix x y z
wenzelm@26179
   407
  assume "(x, y) \<in> r^+"
berghofe@13704
   408
  assume "(y, z) \<in> r^+"
wenzelm@26179
   409
  then show "(x, z) \<in> r^+"
wenzelm@26179
   410
  proof induct
wenzelm@26179
   411
    case (base u)
wenzelm@26179
   412
    from `(x, y) \<in> r^+` and `(y, u) \<in> r`
wenzelm@26179
   413
    show "(x, u) \<in> r^+" ..
wenzelm@26179
   414
  next
wenzelm@26179
   415
    case (step u v)
wenzelm@26179
   416
    from `(x, u) \<in> r^+` and `(u, v) \<in> r`
wenzelm@26179
   417
    show "(x, v) \<in> r^+" ..
wenzelm@26179
   418
  qed
berghofe@13704
   419
qed
wenzelm@12691
   420
wenzelm@12691
   421
lemmas trancl_trans = trans_trancl [THEN transD, standard]
wenzelm@12691
   422
berghofe@23743
   423
lemma tranclp_trans:
berghofe@22262
   424
  assumes xy: "r^++ x y"
berghofe@22262
   425
  and yz: "r^++ y z"
berghofe@22262
   426
  shows "r^++ x z" using yz xy
berghofe@22262
   427
  by induct iprover+
berghofe@22262
   428
wenzelm@26179
   429
lemma trancl_id [simp]: "trans r \<Longrightarrow> r^+ = r"
wenzelm@26179
   430
  apply auto
wenzelm@26179
   431
  apply (erule trancl_induct)
wenzelm@26179
   432
   apply assumption
wenzelm@26179
   433
  apply (unfold trans_def)
wenzelm@26179
   434
  apply blast
wenzelm@26179
   435
  done
nipkow@19623
   436
wenzelm@26179
   437
lemma rtranclp_tranclp_tranclp:
wenzelm@26179
   438
  assumes "r^** x y"
wenzelm@26179
   439
  shows "!!z. r^++ y z ==> r^++ x z" using assms
berghofe@23743
   440
  by induct (iprover intro: tranclp_trans)+
wenzelm@12691
   441
berghofe@23743
   442
lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]
berghofe@22262
   443
berghofe@23743
   444
lemma tranclp_into_tranclp2: "r a b ==> r^++ b c ==> r^++ a c"
berghofe@23743
   445
  by (erule tranclp_trans [OF tranclp.r_into_trancl])
berghofe@22262
   446
berghofe@23743
   447
lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]
wenzelm@12691
   448
wenzelm@12691
   449
lemma trancl_insert:
wenzelm@12691
   450
  "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
wenzelm@12691
   451
  -- {* primitive recursion for @{text trancl} over finite relations *}
wenzelm@12691
   452
  apply (rule equalityI)
wenzelm@12691
   453
   apply (rule subsetI)
wenzelm@12691
   454
   apply (simp only: split_tupled_all)
paulson@14208
   455
   apply (erule trancl_induct, blast)
wenzelm@12691
   456
   apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl r_into_trancl trancl_trans)
wenzelm@12691
   457
  apply (rule subsetI)
wenzelm@12691
   458
  apply (blast intro: trancl_mono rtrancl_mono
wenzelm@12691
   459
    [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
wenzelm@12691
   460
  done
wenzelm@12691
   461
berghofe@23743
   462
lemma tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ x y"
berghofe@22262
   463
  apply (drule conversepD)
berghofe@23743
   464
  apply (erule tranclp_induct)
berghofe@23743
   465
  apply (iprover intro: conversepI tranclp_trans)+
wenzelm@12691
   466
  done
wenzelm@12691
   467
berghofe@23743
   468
lemmas trancl_converseI = tranclp_converseI [to_set]
berghofe@22262
   469
berghofe@23743
   470
lemma tranclp_converseD: "(r^--1)^++ x y ==> (r^++)^--1 x y"
berghofe@22262
   471
  apply (rule conversepI)
berghofe@23743
   472
  apply (erule tranclp_induct)
berghofe@23743
   473
  apply (iprover dest: conversepD intro: tranclp_trans)+
berghofe@13704
   474
  done
wenzelm@12691
   475
berghofe@23743
   476
lemmas trancl_converseD = tranclp_converseD [to_set]
berghofe@22262
   477
berghofe@23743
   478
lemma tranclp_converse: "(r^--1)^++ = (r^++)^--1"
berghofe@22262
   479
  by (fastsimp simp add: expand_fun_eq
berghofe@23743
   480
    intro!: tranclp_converseI dest!: tranclp_converseD)
berghofe@22262
   481
berghofe@23743
   482
lemmas trancl_converse = tranclp_converse [to_set]
wenzelm@12691
   483
huffman@19228
   484
lemma sym_trancl: "sym r ==> sym (r^+)"
huffman@19228
   485
  by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
huffman@19228
   486
berghofe@23743
   487
lemma converse_tranclp_induct:
berghofe@22262
   488
  assumes major: "r^++ a b"
berghofe@22262
   489
    and cases: "!!y. r y b ==> P(y)"
berghofe@22262
   490
      "!!y z.[| r y z;  r^++ z b;  P(z) |] ==> P(y)"
wenzelm@18372
   491
  shows "P a"
berghofe@23743
   492
  apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])
wenzelm@18372
   493
   apply (rule cases)
berghofe@22262
   494
   apply (erule conversepD)
berghofe@23743
   495
  apply (blast intro: prems dest!: tranclp_converseD conversepD)
wenzelm@18372
   496
  done
wenzelm@12691
   497
berghofe@23743
   498
lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
berghofe@22262
   499
berghofe@23743
   500
lemma tranclpD: "R^++ x y ==> EX z. R x z \<and> R^** z y"
wenzelm@26179
   501
  apply (erule converse_tranclp_induct)
wenzelm@26179
   502
   apply auto
berghofe@23743
   503
  apply (blast intro: rtranclp_trans)
wenzelm@12691
   504
  done
wenzelm@12691
   505
berghofe@23743
   506
lemmas tranclD = tranclpD [to_set]
berghofe@22262
   507
bulwahn@31577
   508
lemma converse_tranclpE:
bulwahn@31577
   509
  assumes major: "tranclp r x z"
bulwahn@31577
   510
  assumes base: "r x z ==> P"
bulwahn@31577
   511
  assumes step: "\<And> y. [| r x y; tranclp r y z |] ==> P"
bulwahn@31577
   512
  shows P
bulwahn@31577
   513
proof -
bulwahn@31577
   514
  from tranclpD[OF major]
bulwahn@31577
   515
  obtain y where "r x y" and "rtranclp r y z" by iprover
bulwahn@31577
   516
  from this(2) show P
bulwahn@31577
   517
  proof (cases rule: rtranclp.cases)
bulwahn@31577
   518
    case rtrancl_refl
bulwahn@31577
   519
    with `r x y` base show P by iprover
bulwahn@31577
   520
  next
bulwahn@31577
   521
    case rtrancl_into_rtrancl
bulwahn@31577
   522
    from this have "tranclp r y z"
bulwahn@31577
   523
      by (iprover intro: rtranclp_into_tranclp1)
bulwahn@31577
   524
    with `r x y` step show P by iprover
bulwahn@31577
   525
  qed
bulwahn@31577
   526
qed
bulwahn@31577
   527
bulwahn@31577
   528
lemmas converse_tranclE = converse_tranclpE [to_set]
bulwahn@31577
   529
kleing@25295
   530
lemma tranclD2:
kleing@25295
   531
  "(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
kleing@25295
   532
  by (blast elim: tranclE intro: trancl_into_rtrancl)
kleing@25295
   533
nipkow@13867
   534
lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
wenzelm@18372
   535
  by (blast elim: tranclE dest: trancl_into_rtrancl)
wenzelm@12691
   536
wenzelm@12691
   537
lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"
wenzelm@12691
   538
  by (blast dest: r_into_trancl)
wenzelm@12691
   539
wenzelm@12691
   540
lemma trancl_subset_Sigma_aux:
wenzelm@12691
   541
    "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
wenzelm@18372
   542
  by (induct rule: rtrancl_induct) auto
wenzelm@12691
   543
wenzelm@12691
   544
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"
berghofe@13704
   545
  apply (rule subsetI)
berghofe@13704
   546
  apply (simp only: split_tupled_all)
berghofe@13704
   547
  apply (erule tranclE)
wenzelm@26179
   548
   apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
wenzelm@12691
   549
  done
nipkow@10996
   550
berghofe@23743
   551
lemma reflcl_tranclp [simp]: "(r^++)^== = r^**"
berghofe@22262
   552
  apply (safe intro!: order_antisym)
berghofe@23743
   553
   apply (erule tranclp_into_rtranclp)
berghofe@23743
   554
  apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)
wenzelm@11084
   555
  done
nipkow@10996
   556
berghofe@23743
   557
lemmas reflcl_trancl [simp] = reflcl_tranclp [to_set]
berghofe@22262
   558
wenzelm@11090
   559
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
wenzelm@11084
   560
  apply safe
paulson@14208
   561
   apply (drule trancl_into_rtrancl, simp)
paulson@14208
   562
  apply (erule rtranclE, safe)
paulson@14208
   563
   apply (rule r_into_trancl, simp)
wenzelm@11084
   564
  apply (rule rtrancl_into_trancl1)
paulson@14208
   565
   apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
wenzelm@11084
   566
  done
nipkow@10996
   567
wenzelm@11090
   568
lemma trancl_empty [simp]: "{}^+ = {}"
wenzelm@11084
   569
  by (auto elim: trancl_induct)
nipkow@10996
   570
wenzelm@11090
   571
lemma rtrancl_empty [simp]: "{}^* = Id"
wenzelm@11084
   572
  by (rule subst [OF reflcl_trancl]) simp
nipkow@10996
   573
berghofe@23743
   574
lemma rtranclpD: "R^** a b ==> a = b \<or> a \<noteq> b \<and> R^++ a b"
berghofe@23743
   575
  by (force simp add: reflcl_tranclp [symmetric] simp del: reflcl_tranclp)
berghofe@22262
   576
berghofe@23743
   577
lemmas rtranclD = rtranclpD [to_set]
wenzelm@11084
   578
kleing@16514
   579
lemma rtrancl_eq_or_trancl:
kleing@16514
   580
  "(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"
kleing@16514
   581
  by (fast elim: trancl_into_rtrancl dest: rtranclD)
nipkow@10996
   582
krauss@33656
   583
lemma trancl_unfold_right: "r^+ = r^* O r"
krauss@33656
   584
by (auto dest: tranclD2 intro: rtrancl_into_trancl1)
krauss@33656
   585
krauss@33656
   586
lemma trancl_unfold_left: "r^+ = r O r^*"
krauss@33656
   587
by (auto dest: tranclD intro: rtrancl_into_trancl2)
krauss@33656
   588
krauss@33656
   589
krauss@33656
   590
text {* Simplifying nested closures *}
krauss@33656
   591
krauss@33656
   592
lemma rtrancl_trancl_absorb[simp]: "(R^*)^+ = R^*"
krauss@33656
   593
by (simp add: trans_rtrancl)
krauss@33656
   594
krauss@33656
   595
lemma trancl_rtrancl_absorb[simp]: "(R^+)^* = R^*"
krauss@33656
   596
by (subst reflcl_trancl[symmetric]) simp
krauss@33656
   597
krauss@33656
   598
lemma rtrancl_reflcl_absorb[simp]: "(R^*)^= = R^*"
krauss@33656
   599
by auto
krauss@33656
   600
krauss@33656
   601
wenzelm@12691
   602
text {* @{text Domain} and @{text Range} *}
nipkow@10996
   603
wenzelm@11090
   604
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
wenzelm@11084
   605
  by blast
nipkow@10996
   606
wenzelm@11090
   607
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
wenzelm@11084
   608
  by blast
nipkow@10996
   609
wenzelm@11090
   610
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
wenzelm@11084
   611
  by (rule rtrancl_Un_rtrancl [THEN subst]) fast
nipkow@10996
   612
wenzelm@11090
   613
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
wenzelm@11084
   614
  by (blast intro: subsetD [OF rtrancl_Un_subset])
nipkow@10996
   615
wenzelm@11090
   616
lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
wenzelm@11084
   617
  by (unfold Domain_def) (blast dest: tranclD)
nipkow@10996
   618
wenzelm@11090
   619
lemma trancl_range [simp]: "Range (r^+) = Range r"
nipkow@26271
   620
unfolding Range_def by(simp add: trancl_converse [symmetric])
nipkow@10996
   621
paulson@11115
   622
lemma Not_Domain_rtrancl:
wenzelm@12691
   623
    "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
wenzelm@12691
   624
  apply auto
wenzelm@26179
   625
  apply (erule rev_mp)
wenzelm@26179
   626
  apply (erule rtrancl_induct)
wenzelm@26179
   627
   apply auto
wenzelm@26179
   628
  done
berghofe@11327
   629
haftmann@29609
   630
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
haftmann@29609
   631
  apply clarify
haftmann@29609
   632
  apply (erule trancl_induct)
haftmann@29609
   633
   apply (auto simp add: Field_def)
haftmann@29609
   634
  done
haftmann@29609
   635
haftmann@29609
   636
lemma finite_trancl: "finite (r^+) = finite r"
haftmann@29609
   637
  apply auto
haftmann@29609
   638
   prefer 2
haftmann@29609
   639
   apply (rule trancl_subset_Field2 [THEN finite_subset])
haftmann@29609
   640
   apply (rule finite_SigmaI)
haftmann@29609
   641
    prefer 3
haftmann@29609
   642
    apply (blast intro: r_into_trancl' finite_subset)
haftmann@29609
   643
   apply (auto simp add: finite_Field)
haftmann@29609
   644
  done
haftmann@29609
   645
wenzelm@12691
   646
text {* More about converse @{text rtrancl} and @{text trancl}, should
wenzelm@12691
   647
  be merged with main body. *}
kleing@12428
   648
nipkow@14337
   649
lemma single_valued_confluent:
nipkow@14337
   650
  "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>
nipkow@14337
   651
  \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
wenzelm@26179
   652
  apply (erule rtrancl_induct)
wenzelm@26179
   653
  apply simp
wenzelm@26179
   654
  apply (erule disjE)
wenzelm@26179
   655
   apply (blast elim:converse_rtranclE dest:single_valuedD)
wenzelm@26179
   656
  apply(blast intro:rtrancl_trans)
wenzelm@26179
   657
  done
nipkow@14337
   658
wenzelm@12691
   659
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"
kleing@12428
   660
  by (fast intro: trancl_trans)
kleing@12428
   661
kleing@12428
   662
lemma trancl_into_trancl [rule_format]:
wenzelm@12691
   663
    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
wenzelm@12691
   664
  apply (erule trancl_induct)
kleing@12428
   665
   apply (fast intro: r_r_into_trancl)
kleing@12428
   666
  apply (fast intro: r_r_into_trancl trancl_trans)
kleing@12428
   667
  done
kleing@12428
   668
berghofe@23743
   669
lemma tranclp_rtranclp_tranclp:
berghofe@22262
   670
    "r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c"
berghofe@23743
   671
  apply (drule tranclpD)
wenzelm@26179
   672
  apply (elim exE conjE)
berghofe@23743
   673
  apply (drule rtranclp_trans, assumption)
berghofe@23743
   674
  apply (drule rtranclp_into_tranclp2, assumption, assumption)
kleing@12428
   675
  done
kleing@12428
   676
berghofe@23743
   677
lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]
berghofe@22262
   678
wenzelm@12691
   679
lemmas transitive_closure_trans [trans] =
wenzelm@12691
   680
  r_r_into_trancl trancl_trans rtrancl_trans
berghofe@23743
   681
  trancl.trancl_into_trancl trancl_into_trancl2
berghofe@23743
   682
  rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
wenzelm@12691
   683
  rtrancl_trancl_trancl trancl_rtrancl_trancl
kleing@12428
   684
berghofe@23743
   685
lemmas transitive_closurep_trans' [trans] =
berghofe@23743
   686
  tranclp_trans rtranclp_trans
berghofe@23743
   687
  tranclp.trancl_into_trancl tranclp_into_tranclp2
berghofe@23743
   688
  rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp
berghofe@23743
   689
  rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp
berghofe@22262
   690
kleing@12428
   691
declare trancl_into_rtrancl [elim]
berghofe@11327
   692
haftmann@30954
   693
subsection {* The power operation on relations *}
haftmann@30954
   694
haftmann@30954
   695
text {* @{text "R ^^ n = R O ... O R"}, the n-fold composition of @{text R} *}
haftmann@30954
   696
haftmann@30971
   697
overloading
haftmann@30971
   698
  relpow == "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
haftmann@30971
   699
begin
haftmann@30954
   700
haftmann@30971
   701
primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
haftmann@30971
   702
    "relpow 0 R = Id"
krauss@32235
   703
  | "relpow (Suc n) R = (R ^^ n) O R"
haftmann@30954
   704
haftmann@30971
   705
end
haftmann@30954
   706
haftmann@30954
   707
lemma rel_pow_1 [simp]:
haftmann@30971
   708
  fixes R :: "('a \<times> 'a) set"
haftmann@30971
   709
  shows "R ^^ 1 = R"
haftmann@30954
   710
  by simp
haftmann@30954
   711
haftmann@30954
   712
lemma rel_pow_0_I: 
haftmann@30954
   713
  "(x, x) \<in> R ^^ 0"
haftmann@30954
   714
  by simp
haftmann@30954
   715
haftmann@30954
   716
lemma rel_pow_Suc_I:
haftmann@30954
   717
  "(x, y) \<in>  R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
haftmann@30954
   718
  by auto
haftmann@30954
   719
haftmann@30954
   720
lemma rel_pow_Suc_I2:
haftmann@30954
   721
  "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
haftmann@30954
   722
  by (induct n arbitrary: z) (simp, fastsimp)
haftmann@30954
   723
haftmann@30954
   724
lemma rel_pow_0_E:
haftmann@30954
   725
  "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
haftmann@30954
   726
  by simp
haftmann@30954
   727
haftmann@30954
   728
lemma rel_pow_Suc_E:
haftmann@30954
   729
  "(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
   730
  by auto
haftmann@30954
   731
haftmann@30954
   732
lemma rel_pow_E:
haftmann@30954
   733
  "(x, z) \<in>  R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
haftmann@30954
   734
   \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in>  R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P)
haftmann@30954
   735
   \<Longrightarrow> P"
haftmann@30954
   736
  by (cases n) auto
haftmann@30954
   737
haftmann@30954
   738
lemma rel_pow_Suc_D2:
haftmann@30954
   739
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
haftmann@30954
   740
  apply (induct n arbitrary: x z)
haftmann@30954
   741
   apply (blast intro: rel_pow_0_I elim: rel_pow_0_E rel_pow_Suc_E)
haftmann@30954
   742
  apply (blast intro: rel_pow_Suc_I elim: rel_pow_0_E rel_pow_Suc_E)
haftmann@30954
   743
  done
haftmann@30954
   744
haftmann@30954
   745
lemma rel_pow_Suc_E2:
haftmann@30954
   746
  "(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
   747
  by (blast dest: rel_pow_Suc_D2)
haftmann@30954
   748
haftmann@30954
   749
lemma rel_pow_Suc_D2':
haftmann@30954
   750
  "\<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
   751
  by (induct n) (simp_all, blast)
haftmann@30954
   752
haftmann@30954
   753
lemma rel_pow_E2:
haftmann@30954
   754
  "(x, z) \<in> R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
haftmann@30954
   755
     \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P)
haftmann@30954
   756
   \<Longrightarrow> P"
haftmann@30954
   757
  apply (cases n, simp)
haftmann@30954
   758
  apply (cut_tac n=nat and R=R in rel_pow_Suc_D2', simp, blast)
haftmann@30954
   759
  done
haftmann@30954
   760
krauss@32235
   761
lemma rel_pow_add: "R ^^ (m+n) = R^^m O R^^n"
nipkow@31351
   762
by(induct n) auto
nipkow@31351
   763
krauss@31970
   764
lemma rel_pow_commute: "R O R ^^ n = R ^^ n O R"
krauss@32235
   765
by (induct n) (simp, simp add: O_assoc [symmetric])
krauss@31970
   766
haftmann@30954
   767
lemma rtrancl_imp_UN_rel_pow:
haftmann@30954
   768
  assumes "p \<in> R^*"
haftmann@30954
   769
  shows "p \<in> (\<Union>n. R ^^ n)"
haftmann@30954
   770
proof (cases p)
haftmann@30954
   771
  case (Pair x y)
haftmann@30954
   772
  with assms have "(x, y) \<in> R^*" by simp
haftmann@30954
   773
  then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct
haftmann@30954
   774
    case base show ?case by (blast intro: rel_pow_0_I)
haftmann@30954
   775
  next
haftmann@30954
   776
    case step then show ?case by (blast intro: rel_pow_Suc_I)
haftmann@30954
   777
  qed
haftmann@30954
   778
  with Pair show ?thesis by simp
haftmann@30954
   779
qed
haftmann@30954
   780
haftmann@30954
   781
lemma rel_pow_imp_rtrancl:
haftmann@30954
   782
  assumes "p \<in> R ^^ n"
haftmann@30954
   783
  shows "p \<in> R^*"
haftmann@30954
   784
proof (cases p)
haftmann@30954
   785
  case (Pair x y)
haftmann@30954
   786
  with assms have "(x, y) \<in> R ^^ n" by simp
haftmann@30954
   787
  then have "(x, y) \<in> R^*" proof (induct n arbitrary: x y)
haftmann@30954
   788
    case 0 then show ?case by simp
haftmann@30954
   789
  next
haftmann@30954
   790
    case Suc then show ?case
haftmann@30954
   791
      by (blast elim: rel_pow_Suc_E intro: rtrancl_into_rtrancl)
haftmann@30954
   792
  qed
haftmann@30954
   793
  with Pair show ?thesis by simp
haftmann@30954
   794
qed
haftmann@30954
   795
haftmann@30954
   796
lemma rtrancl_is_UN_rel_pow:
haftmann@30954
   797
  "R^* = (\<Union>n. R ^^ n)"
haftmann@30954
   798
  by (blast intro: rtrancl_imp_UN_rel_pow rel_pow_imp_rtrancl)
haftmann@30954
   799
haftmann@30954
   800
lemma rtrancl_power:
haftmann@30954
   801
  "p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
haftmann@30954
   802
  by (simp add: rtrancl_is_UN_rel_pow)
haftmann@30954
   803
haftmann@30954
   804
lemma trancl_power:
haftmann@30954
   805
  "p \<in> R^+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
haftmann@30954
   806
  apply (cases p)
haftmann@30954
   807
  apply simp
haftmann@30954
   808
  apply (rule iffI)
haftmann@30954
   809
   apply (drule tranclD2)
haftmann@30954
   810
   apply (clarsimp simp: rtrancl_is_UN_rel_pow)
haftmann@30971
   811
   apply (rule_tac x="Suc n" in exI)
haftmann@30954
   812
   apply (clarsimp simp: rel_comp_def)
haftmann@30954
   813
   apply fastsimp
haftmann@30954
   814
  apply clarsimp
haftmann@30954
   815
  apply (case_tac n, simp)
haftmann@30954
   816
  apply clarsimp
haftmann@30954
   817
  apply (drule rel_pow_imp_rtrancl)
haftmann@30954
   818
  apply (drule rtrancl_into_trancl1) apply auto
haftmann@30954
   819
  done
haftmann@30954
   820
haftmann@30954
   821
lemma rtrancl_imp_rel_pow:
haftmann@30954
   822
  "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
haftmann@30954
   823
  by (auto dest: rtrancl_imp_UN_rel_pow)
haftmann@30954
   824
haftmann@30954
   825
lemma single_valued_rel_pow:
haftmann@30954
   826
  fixes R :: "('a * 'a) set"
haftmann@30954
   827
  shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
haftmann@30954
   828
  apply (induct n arbitrary: R)
haftmann@30954
   829
  apply simp_all
haftmann@30954
   830
  apply (rule single_valuedI)
haftmann@30954
   831
  apply (fast dest: single_valuedD elim: rel_pow_Suc_E)
haftmann@30954
   832
  done
paulson@15551
   833
ballarin@15076
   834
subsection {* Setup of transitivity reasoner *}
ballarin@15076
   835
wenzelm@26340
   836
ML {*
ballarin@15076
   837
wenzelm@32215
   838
structure Trancl_Tac = Trancl_Tac
wenzelm@32215
   839
(
wenzelm@32215
   840
  val r_into_trancl = @{thm trancl.r_into_trancl};
wenzelm@32215
   841
  val trancl_trans  = @{thm trancl_trans};
wenzelm@32215
   842
  val rtrancl_refl = @{thm rtrancl.rtrancl_refl};
wenzelm@32215
   843
  val r_into_rtrancl = @{thm r_into_rtrancl};
wenzelm@32215
   844
  val trancl_into_rtrancl = @{thm trancl_into_rtrancl};
wenzelm@32215
   845
  val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl};
wenzelm@32215
   846
  val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl};
wenzelm@32215
   847
  val rtrancl_trans = @{thm rtrancl_trans};
ballarin@15096
   848
berghofe@30107
   849
  fun decomp (@{const Trueprop} $ t) =
wenzelm@18372
   850
    let fun dec (Const ("op :", _) $ (Const ("Pair", _) $ a $ b) $ rel ) =
berghofe@23743
   851
        let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*")
berghofe@23743
   852
              | decr (Const ("Transitive_Closure.trancl", _ ) $ r)  = (r,"r+")
wenzelm@18372
   853
              | decr r = (r,"r");
berghofe@26801
   854
            val (rel,r) = decr (Envir.beta_eta_contract rel);
wenzelm@18372
   855
        in SOME (a,b,rel,r) end
wenzelm@18372
   856
      | dec _ =  NONE
berghofe@30107
   857
    in dec t end
berghofe@30107
   858
    | decomp _ = NONE;
wenzelm@32215
   859
);
ballarin@15076
   860
wenzelm@32215
   861
structure Tranclp_Tac = Trancl_Tac
wenzelm@32215
   862
(
wenzelm@32215
   863
  val r_into_trancl = @{thm tranclp.r_into_trancl};
wenzelm@32215
   864
  val trancl_trans  = @{thm tranclp_trans};
wenzelm@32215
   865
  val rtrancl_refl = @{thm rtranclp.rtrancl_refl};
wenzelm@32215
   866
  val r_into_rtrancl = @{thm r_into_rtranclp};
wenzelm@32215
   867
  val trancl_into_rtrancl = @{thm tranclp_into_rtranclp};
wenzelm@32215
   868
  val rtrancl_trancl_trancl = @{thm rtranclp_tranclp_tranclp};
wenzelm@32215
   869
  val trancl_rtrancl_trancl = @{thm tranclp_rtranclp_tranclp};
wenzelm@32215
   870
  val rtrancl_trans = @{thm rtranclp_trans};
berghofe@22262
   871
berghofe@30107
   872
  fun decomp (@{const Trueprop} $ t) =
berghofe@22262
   873
    let fun dec (rel $ a $ b) =
berghofe@23743
   874
        let fun decr (Const ("Transitive_Closure.rtranclp", _ ) $ r) = (r,"r*")
berghofe@23743
   875
              | decr (Const ("Transitive_Closure.tranclp", _ ) $ r)  = (r,"r+")
berghofe@22262
   876
              | decr r = (r,"r");
berghofe@22262
   877
            val (rel,r) = decr rel;
berghofe@26801
   878
        in SOME (a, b, rel, r) end
berghofe@22262
   879
      | dec _ =  NONE
berghofe@30107
   880
    in dec t end
berghofe@30107
   881
    | decomp _ = NONE;
wenzelm@32215
   882
);
wenzelm@26340
   883
*}
berghofe@22262
   884
wenzelm@26340
   885
declaration {* fn _ =>
wenzelm@26340
   886
  Simplifier.map_ss (fn ss => ss
wenzelm@32215
   887
    addSolver (mk_solver' "Trancl" (Trancl_Tac.trancl_tac o Simplifier.the_context))
wenzelm@32215
   888
    addSolver (mk_solver' "Rtrancl" (Trancl_Tac.rtrancl_tac o Simplifier.the_context))
wenzelm@32215
   889
    addSolver (mk_solver' "Tranclp" (Tranclp_Tac.trancl_tac o Simplifier.the_context))
wenzelm@32215
   890
    addSolver (mk_solver' "Rtranclp" (Tranclp_Tac.rtrancl_tac o Simplifier.the_context)))
ballarin@15076
   891
*}
ballarin@15076
   892
wenzelm@32215
   893
wenzelm@32215
   894
text {* Optional methods. *}
ballarin@15076
   895
ballarin@15076
   896
method_setup trancl =
wenzelm@32215
   897
  {* Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.trancl_tac) *}
wenzelm@18372
   898
  {* simple transitivity reasoner *}
ballarin@15076
   899
method_setup rtrancl =
wenzelm@32215
   900
  {* Scan.succeed (SIMPLE_METHOD' o Trancl_Tac.rtrancl_tac) *}
ballarin@15076
   901
  {* simple transitivity reasoner *}
berghofe@22262
   902
method_setup tranclp =
wenzelm@32215
   903
  {* Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.trancl_tac) *}
berghofe@22262
   904
  {* simple transitivity reasoner (predicate version) *}
berghofe@22262
   905
method_setup rtranclp =
wenzelm@32215
   906
  {* Scan.succeed (SIMPLE_METHOD' o Tranclp_Tac.rtrancl_tac) *}
berghofe@22262
   907
  {* simple transitivity reasoner (predicate version) *}
ballarin@15076
   908
nipkow@10213
   909
end