src/HOL/Transitive_Closure.thy
author haftmann
Mon, 20 Apr 2009 09:32:40 +0200
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permissions -rw-r--r--
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
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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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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 Un Id)"
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  by (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 O (r^* \<inter> s) \<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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d21db58bcdc2 converted theory Transitive_Closure;
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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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   223
  done
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   224
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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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   230
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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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   234
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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   238
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lemmas rtrancl_converseD = rtranclp_converseD [to_set]
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   240
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   241
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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   246
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lemmas rtrancl_converseI = rtranclp_converseI [to_set]
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   248
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   249
lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
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  by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
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   251
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   252
lemma sym_rtrancl: "sym r ==> sym (r^*)"
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   253
  by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])
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   254
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   255
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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   259
  using rtranclp_converseI [OF major]
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  by induct (iprover intro: cases dest!: conversepD rtranclp_converseD)+
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   261
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9191942c4ead Removed some case_names and consumes attributes that are now no longer
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   262
lemmas converse_rtrancl_induct = converse_rtranclp_induct [to_set]
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   263
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   264
lemmas converse_rtranclp_induct2 =
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   265
  converse_rtranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
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                 consumes 1, case_names refl step]
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   267
14404
4952c5a92e04 Transitive_Closure: added consumes and case_names attributes
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   268
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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   271
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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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   277
  apply (subgoal_tac "x = z | (EX y. r x y & r^** y z)")
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   278
   apply (rule_tac [2] major [THEN converse_rtranclp_induct])
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   279
    prefer 2 apply iprover
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   280
   prefer 2 apply iprover
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   281
  apply (erule asm_rl exE disjE conjE cases)+
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   282
  done
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   283
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lemmas converse_rtranclE = converse_rtranclpE [to_set]
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   285
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   286
lemmas converse_rtranclpE2 = converse_rtranclpE [of _ "(xa,xb)" "(za,zb)", split_rule]
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   287
96ba62dff413 Adapted to new inductive definition package.
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diff changeset
   288
lemmas converse_rtranclE2 = converse_rtranclE [of "(xa,xb)" "(za,zb)", split_rule]
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   289
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   290
lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   291
  by (blast elim: rtranclE converse_rtranclE
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   292
    intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   293
20716
a6686a8e1b68 Changed precedence of "op O" (relation composition) from 60 to 75.
krauss
parents: 19656
diff changeset
   294
lemma rtrancl_unfold: "r^* = Id Un r O r^*"
15551
af78481b37bf unfold theorems for trancl and rtrancl
paulson
parents: 15531
diff changeset
   295
  by (auto intro: rtrancl_into_rtrancl elim: rtranclE)
af78481b37bf unfold theorems for trancl and rtrancl
paulson
parents: 15531
diff changeset
   296
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   297
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   298
subsection {* Transitive closure *}
10331
7411e4659d4a more "xsymbols" syntax;
wenzelm
parents: 10213
diff changeset
   299
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   300
lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   301
  apply (simp add: split_tupled_all)
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   302
  apply (erule trancl.induct)
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   303
   apply (iprover dest: subsetD)+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   304
  done
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   305
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   306
lemma r_into_trancl': "!!p. p : r ==> p : r^+"
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   307
  by (simp only: split_tupled_all) (erule r_into_trancl)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   308
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   309
text {*
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   310
  \medskip Conversions between @{text trancl} and @{text rtrancl}.
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   311
*}
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   312
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   313
lemma tranclp_into_rtranclp: "r^++ a b ==> r^** a b"
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   314
  by (erule tranclp.induct) iprover+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   315
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   316
lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   317
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   318
lemma rtranclp_into_tranclp1: assumes r: "r^** a b"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   319
  shows "!!c. r b c ==> r^++ a c" using r
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 16514
diff changeset
   320
  by induct iprover+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   321
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   322
lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   323
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   324
lemma rtranclp_into_tranclp2: "[| r a b; r^** b c |] ==> r^++ a c"
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   325
  -- {* intro rule from @{text r} and @{text rtrancl} *}
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   326
  apply (erule rtranclp.cases)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   327
   apply iprover
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   328
  apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1])
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   329
    apply (simp | rule r_into_rtranclp)+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   330
  done
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   331
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   332
lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   333
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   334
text {* Nice induction rule for @{text trancl} *}
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   335
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   336
  assumes "r^++ a b"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   337
  and cases: "!!y. r a y ==> P y"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   338
    "!!y z. r^++ a y ==> r y z ==> P y ==> P z"
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   339
  shows "P b"
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   340
proof -
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   341
  from `r^++ a b` have "a = a --> P b"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 16514
diff changeset
   342
    by (induct "%x y. x = a --> P y" a b) (iprover intro: cases)+
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   343
  then show ?thesis by iprover
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   344
qed
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   345
25425
9191942c4ead Removed some case_names and consumes attributes that are now no longer
berghofe
parents: 25295
diff changeset
   346
lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   347
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   348
lemmas tranclp_induct2 =
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   349
  tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   350
    consumes 1, case_names base step]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   351
22172
e7d6cb237b5e some new lemmas
paulson
parents: 22080
diff changeset
   352
lemmas trancl_induct2 =
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   353
  trancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   354
    consumes 1, case_names base step]
22172
e7d6cb237b5e some new lemmas
paulson
parents: 22080
diff changeset
   355
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   356
lemma tranclp_trans_induct:
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   357
  assumes major: "r^++ x y"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   358
    and cases: "!!x y. r x y ==> P x y"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   359
      "!!x y z. [| r^++ x y; P x y; r^++ y z; P y z |] ==> P x z"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   360
  shows "P x y"
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   361
  -- {* Another induction rule for trancl, incorporating transitivity *}
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   362
  by (iprover intro: major [THEN tranclp_induct] cases)
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   363
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   364
lemmas trancl_trans_induct = tranclp_trans_induct [to_set]
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   365
26174
9efd4c04eaa4 rtranclE, tranclE: tuned statement, added case_names;
wenzelm
parents: 25425
diff changeset
   366
lemma tranclE [cases set: trancl]:
9efd4c04eaa4 rtranclE, tranclE: tuned statement, added case_names;
wenzelm
parents: 25425
diff changeset
   367
  assumes "(a, b) : r^+"
9efd4c04eaa4 rtranclE, tranclE: tuned statement, added case_names;
wenzelm
parents: 25425
diff changeset
   368
  obtains
9efd4c04eaa4 rtranclE, tranclE: tuned statement, added case_names;
wenzelm
parents: 25425
diff changeset
   369
    (base) "(a, b) : r"
9efd4c04eaa4 rtranclE, tranclE: tuned statement, added case_names;
wenzelm
parents: 25425
diff changeset
   370
  | (step) c where "(a, c) : r^+" and "(c, b) : r"
9efd4c04eaa4 rtranclE, tranclE: tuned statement, added case_names;
wenzelm
parents: 25425
diff changeset
   371
  using assms by cases simp_all
10980
0a45f2efaaec Transitive_Closure turned into new-style theory;
wenzelm
parents: 10827
diff changeset
   372
22080
7bf8868ab3e4 induction rules for trancl/rtrancl expressed using subsets
paulson
parents: 21589
diff changeset
   373
lemma trancl_Int_subset: "[| r \<subseteq> s; r O (r^+ \<inter> s) \<subseteq> s|] ==> r^+ \<subseteq> s"
7bf8868ab3e4 induction rules for trancl/rtrancl expressed using subsets
paulson
parents: 21589
diff changeset
   374
  apply (rule subsetI)
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   375
  apply (rule_tac p = x in PairE)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   376
  apply clarify
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   377
  apply (erule trancl_induct)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   378
   apply auto
22080
7bf8868ab3e4 induction rules for trancl/rtrancl expressed using subsets
paulson
parents: 21589
diff changeset
   379
  done
7bf8868ab3e4 induction rules for trancl/rtrancl expressed using subsets
paulson
parents: 21589
diff changeset
   380
20716
a6686a8e1b68 Changed precedence of "op O" (relation composition) from 60 to 75.
krauss
parents: 19656
diff changeset
   381
lemma trancl_unfold: "r^+ = r Un r O r^+"
15551
af78481b37bf unfold theorems for trancl and rtrancl
paulson
parents: 15531
diff changeset
   382
  by (auto intro: trancl_into_trancl elim: tranclE)
af78481b37bf unfold theorems for trancl and rtrancl
paulson
parents: 15531
diff changeset
   383
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   384
text {* Transitivity of @{term "r^+"} *}
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   385
lemma trans_trancl [simp]: "trans (r^+)"
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   386
proof (rule transI)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   387
  fix x y z
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   388
  assume "(x, y) \<in> r^+"
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   389
  assume "(y, z) \<in> r^+"
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   390
  then show "(x, z) \<in> r^+"
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   391
  proof induct
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   392
    case (base u)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   393
    from `(x, y) \<in> r^+` and `(y, u) \<in> r`
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   394
    show "(x, u) \<in> r^+" ..
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   395
  next
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   396
    case (step u v)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   397
    from `(x, u) \<in> r^+` and `(u, v) \<in> r`
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   398
    show "(x, v) \<in> r^+" ..
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   399
  qed
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   400
qed
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   401
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   402
lemmas trancl_trans = trans_trancl [THEN transD, standard]
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   403
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   404
lemma tranclp_trans:
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   405
  assumes xy: "r^++ x y"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   406
  and yz: "r^++ y z"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   407
  shows "r^++ x z" using yz xy
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   408
  by induct iprover+
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   409
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   410
lemma trancl_id [simp]: "trans r \<Longrightarrow> r^+ = r"
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   411
  apply auto
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   412
  apply (erule trancl_induct)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   413
   apply assumption
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   414
  apply (unfold trans_def)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   415
  apply blast
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   416
  done
19623
12e6cc4382ae added lemma in_measure
nipkow
parents: 19228
diff changeset
   417
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   418
lemma rtranclp_tranclp_tranclp:
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   419
  assumes "r^** x y"
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   420
  shows "!!z. r^++ y z ==> r^++ x z" using assms
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   421
  by induct (iprover intro: tranclp_trans)+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   422
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   423
lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   424
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   425
lemma tranclp_into_tranclp2: "r a b ==> r^++ b c ==> r^++ a c"
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   426
  by (erule tranclp_trans [OF tranclp.r_into_trancl])
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   427
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   428
lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   429
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   430
lemma trancl_insert:
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   431
  "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   432
  -- {* primitive recursion for @{text trancl} over finite relations *}
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   433
  apply (rule equalityI)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   434
   apply (rule subsetI)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   435
   apply (simp only: split_tupled_all)
14208
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   436
   apply (erule trancl_induct, blast)
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   437
   apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl r_into_trancl trancl_trans)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   438
  apply (rule subsetI)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   439
  apply (blast intro: trancl_mono rtrancl_mono
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   440
    [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   441
  done
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   442
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   443
lemma tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ x y"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   444
  apply (drule conversepD)
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   445
  apply (erule tranclp_induct)
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   446
  apply (iprover intro: conversepI tranclp_trans)+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   447
  done
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   448
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   449
lemmas trancl_converseI = tranclp_converseI [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   450
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   451
lemma tranclp_converseD: "(r^--1)^++ x y ==> (r^++)^--1 x y"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   452
  apply (rule conversepI)
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   453
  apply (erule tranclp_induct)
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   454
  apply (iprover dest: conversepD intro: tranclp_trans)+
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   455
  done
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   456
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   457
lemmas trancl_converseD = tranclp_converseD [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   458
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   459
lemma tranclp_converse: "(r^--1)^++ = (r^++)^--1"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   460
  by (fastsimp simp add: expand_fun_eq
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   461
    intro!: tranclp_converseI dest!: tranclp_converseD)
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   462
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   463
lemmas trancl_converse = tranclp_converse [to_set]
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   464
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 18372
diff changeset
   465
lemma sym_trancl: "sym r ==> sym (r^+)"
30fce6da8cbe added many simple lemmas
huffman
parents: 18372
diff changeset
   466
  by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
30fce6da8cbe added many simple lemmas
huffman
parents: 18372
diff changeset
   467
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   468
lemma converse_tranclp_induct:
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   469
  assumes major: "r^++ a b"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   470
    and cases: "!!y. r y b ==> P(y)"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   471
      "!!y z.[| r y z;  r^++ z b;  P(z) |] ==> P(y)"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   472
  shows "P a"
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   473
  apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   474
   apply (rule cases)
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   475
   apply (erule conversepD)
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   476
  apply (blast intro: prems dest!: tranclp_converseD conversepD)
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   477
  done
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   478
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   479
lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   480
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   481
lemma tranclpD: "R^++ x y ==> EX z. R x z \<and> R^** z y"
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   482
  apply (erule converse_tranclp_induct)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   483
   apply auto
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   484
  apply (blast intro: rtranclp_trans)
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   485
  done
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   486
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   487
lemmas tranclD = tranclpD [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   488
25295
12985023be5e tranclD2 (tranclD at the other end) + trancl_power
kleing
parents: 23743
diff changeset
   489
lemma tranclD2:
12985023be5e tranclD2 (tranclD at the other end) + trancl_power
kleing
parents: 23743
diff changeset
   490
  "(x, y) \<in> R\<^sup>+ \<Longrightarrow> \<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
12985023be5e tranclD2 (tranclD at the other end) + trancl_power
kleing
parents: 23743
diff changeset
   491
  by (blast elim: tranclE intro: trancl_into_rtrancl)
12985023be5e tranclD2 (tranclD at the other end) + trancl_power
kleing
parents: 23743
diff changeset
   492
13867
1fdecd15437f just a few mods to a few thms
nipkow
parents: 13726
diff changeset
   493
lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   494
  by (blast elim: tranclE dest: trancl_into_rtrancl)
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   495
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   496
lemma irrefl_trancl_rD: "!!X. ALL x. (x, x) \<notin> r^+ ==> (x, y) \<in> r ==> x \<noteq> y"
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   497
  by (blast dest: r_into_trancl)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   498
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   499
lemma trancl_subset_Sigma_aux:
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   500
    "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   501
  by (induct rule: rtrancl_induct) auto
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   502
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   503
lemma trancl_subset_Sigma: "r \<subseteq> A \<times> A ==> r^+ \<subseteq> A \<times> A"
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   504
  apply (rule subsetI)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   505
  apply (simp only: split_tupled_all)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   506
  apply (erule tranclE)
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   507
   apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   508
  done
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   509
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   510
lemma reflcl_tranclp [simp]: "(r^++)^== = r^**"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   511
  apply (safe intro!: order_antisym)
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   512
   apply (erule tranclp_into_rtranclp)
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   513
  apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   514
  done
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   515
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   516
lemmas reflcl_trancl [simp] = reflcl_tranclp [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   517
11090
wenzelm
parents: 11084
diff changeset
   518
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   519
  apply safe
14208
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   520
   apply (drule trancl_into_rtrancl, simp)
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   521
  apply (erule rtranclE, safe)
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   522
   apply (rule r_into_trancl, simp)
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   523
  apply (rule rtrancl_into_trancl1)
14208
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   524
   apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   525
  done
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   526
11090
wenzelm
parents: 11084
diff changeset
   527
lemma trancl_empty [simp]: "{}^+ = {}"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   528
  by (auto elim: trancl_induct)
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   529
11090
wenzelm
parents: 11084
diff changeset
   530
lemma rtrancl_empty [simp]: "{}^* = Id"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   531
  by (rule subst [OF reflcl_trancl]) simp
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   532
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   533
lemma rtranclpD: "R^** a b ==> a = b \<or> a \<noteq> b \<and> R^++ a b"
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   534
  by (force simp add: reflcl_tranclp [symmetric] simp del: reflcl_tranclp)
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   535
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   536
lemmas rtranclD = rtranclpD [to_set]
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   537
16514
090c6a98c704 lemma, equation between rtrancl and trancl
kleing
parents: 16417
diff changeset
   538
lemma rtrancl_eq_or_trancl:
090c6a98c704 lemma, equation between rtrancl and trancl
kleing
parents: 16417
diff changeset
   539
  "(x,y) \<in> R\<^sup>* = (x=y \<or> x\<noteq>y \<and> (x,y) \<in> R\<^sup>+)"
090c6a98c704 lemma, equation between rtrancl and trancl
kleing
parents: 16417
diff changeset
   540
  by (fast elim: trancl_into_rtrancl dest: rtranclD)
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   541
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   542
text {* @{text Domain} and @{text Range} *}
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   543
11090
wenzelm
parents: 11084
diff changeset
   544
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   545
  by blast
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   546
11090
wenzelm
parents: 11084
diff changeset
   547
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   548
  by blast
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   549
11090
wenzelm
parents: 11084
diff changeset
   550
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   551
  by (rule rtrancl_Un_rtrancl [THEN subst]) fast
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   552
11090
wenzelm
parents: 11084
diff changeset
   553
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   554
  by (blast intro: subsetD [OF rtrancl_Un_subset])
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   555
11090
wenzelm
parents: 11084
diff changeset
   556
lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   557
  by (unfold Domain_def) (blast dest: tranclD)
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   558
11090
wenzelm
parents: 11084
diff changeset
   559
lemma trancl_range [simp]: "Range (r^+) = Range r"
26271
e324f8918c98 Added lemmas
nipkow
parents: 26179
diff changeset
   560
unfolding Range_def by(simp add: trancl_converse [symmetric])
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   561
11115
285b31e9e026 a new theorem from Bryan Ford
paulson
parents: 11090
diff changeset
   562
lemma Not_Domain_rtrancl:
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   563
    "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   564
  apply auto
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   565
  apply (erule rev_mp)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   566
  apply (erule rtrancl_induct)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   567
   apply auto
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   568
  done
11327
cd2c27a23df1 Transitive closure is now defined via "inductive".
berghofe
parents: 11115
diff changeset
   569
29609
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   570
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   571
  apply clarify
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   572
  apply (erule trancl_induct)
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   573
   apply (auto simp add: Field_def)
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   574
  done
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   575
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   576
lemma finite_trancl: "finite (r^+) = finite r"
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   577
  apply auto
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   578
   prefer 2
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   579
   apply (rule trancl_subset_Field2 [THEN finite_subset])
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   580
   apply (rule finite_SigmaI)
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   581
    prefer 3
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   582
    apply (blast intro: r_into_trancl' finite_subset)
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   583
   apply (auto simp add: finite_Field)
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   584
  done
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   585
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   586
text {* More about converse @{text rtrancl} and @{text trancl}, should
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   587
  be merged with main body. *}
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   588
14337
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   589
lemma single_valued_confluent:
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   590
  "\<lbrakk> single_valued r; (x,y) \<in> r^*; (x,z) \<in> r^* \<rbrakk>
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   591
  \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   592
  apply (erule rtrancl_induct)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   593
  apply simp
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   594
  apply (erule disjE)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   595
   apply (blast elim:converse_rtranclE dest:single_valuedD)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   596
  apply(blast intro:rtrancl_trans)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   597
  done
14337
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   598
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   599
lemma r_r_into_trancl: "(a, b) \<in> R ==> (b, c) \<in> R ==> (a, c) \<in> R^+"
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   600
  by (fast intro: trancl_trans)
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   601
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   602
lemma trancl_into_trancl [rule_format]:
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   603
    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   604
  apply (erule trancl_induct)
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   605
   apply (fast intro: r_r_into_trancl)
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   606
  apply (fast intro: r_r_into_trancl trancl_trans)
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   607
  done
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   608
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   609
lemma tranclp_rtranclp_tranclp:
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   610
    "r\<^sup>+\<^sup>+ a b ==> r\<^sup>*\<^sup>* b c ==> r\<^sup>+\<^sup>+ a c"
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   611
  apply (drule tranclpD)
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   612
  apply (elim exE conjE)
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   613
  apply (drule rtranclp_trans, assumption)
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   614
  apply (drule rtranclp_into_tranclp2, assumption, assumption)
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   615
  done
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   616
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   617
lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   618
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   619
lemmas transitive_closure_trans [trans] =
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   620
  r_r_into_trancl trancl_trans rtrancl_trans
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   621
  trancl.trancl_into_trancl trancl_into_trancl2
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   622
  rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   623
  rtrancl_trancl_trancl trancl_rtrancl_trancl
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   624
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   625
lemmas transitive_closurep_trans' [trans] =
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   626
  tranclp_trans rtranclp_trans
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   627
  tranclp.trancl_into_trancl tranclp_into_tranclp2
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   628
  rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   629
  rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   630
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   631
declare trancl_into_rtrancl [elim]
11327
cd2c27a23df1 Transitive closure is now defined via "inductive".
berghofe
parents: 11115
diff changeset
   632
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   633
subsection {* The power operation on relations *}
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   634
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   635
text {* @{text "R ^^ n = R O ... O R"}, the n-fold composition of @{text R} *}
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   636
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   637
primrec relpow :: "('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> ('a \<times> 'a) set" (infixr "^^" 80) where
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   638
    "R ^^ 0 = Id"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   639
  | "R ^^ Suc n = R O (R ^^ n)"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   640
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   641
notation (latex output)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   642
  relpow ("(_\<^bsup>_\<^esup>)" [1000] 1000)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   643
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   644
notation (HTML output)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   645
  relpow ("(_\<^bsup>_\<^esup>)" [1000] 1000)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   646
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   647
lemma rel_pow_1 [simp]:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   648
  "R ^^ 1 = R"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   649
  by simp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   650
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   651
lemma rel_pow_0_I: 
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   652
  "(x, x) \<in> R ^^ 0"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   653
  by simp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   654
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   655
lemma rel_pow_Suc_I:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   656
  "(x, y) \<in>  R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   657
  by auto
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   658
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   659
lemma rel_pow_Suc_I2:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   660
  "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   661
  by (induct n arbitrary: z) (simp, fastsimp)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   662
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   663
lemma rel_pow_0_E:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   664
  "(x, y) \<in> R ^^ 0 \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> P"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   665
  by simp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   666
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   667
lemma rel_pow_Suc_E:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   668
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R ^^ n \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) \<Longrightarrow> P"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   669
  by auto
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   670
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   671
lemma rel_pow_E:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   672
  "(x, z) \<in>  R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   673
   \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in>  R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   674
   \<Longrightarrow> P"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   675
  by (cases n) auto
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   676
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   677
lemma rel_pow_Suc_D2:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   678
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   679
  apply (induct n arbitrary: x z)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   680
   apply (blast intro: rel_pow_0_I elim: rel_pow_0_E rel_pow_Suc_E)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   681
  apply (blast intro: rel_pow_Suc_I elim: rel_pow_0_E rel_pow_Suc_E)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   682
  done
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   683
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   684
lemma rel_pow_Suc_E2:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   685
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> P) \<Longrightarrow> P"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   686
  by (blast dest: rel_pow_Suc_D2)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   687
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   688
lemma rel_pow_Suc_D2':
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   689
  "\<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)"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   690
  by (induct n) (simp_all, blast)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   691
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   692
lemma rel_pow_E2:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   693
  "(x, z) \<in> R ^^ n \<Longrightarrow>  (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   694
     \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   695
   \<Longrightarrow> P"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   696
  apply (cases n, simp)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   697
  apply (cut_tac n=nat and R=R in rel_pow_Suc_D2', simp, blast)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   698
  done
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   699
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   700
lemma rtrancl_imp_UN_rel_pow:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   701
  assumes "p \<in> R^*"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   702
  shows "p \<in> (\<Union>n. R ^^ n)"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   703
proof (cases p)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   704
  case (Pair x y)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   705
  with assms have "(x, y) \<in> R^*" by simp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   706
  then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   707
    case base show ?case by (blast intro: rel_pow_0_I)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   708
  next
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   709
    case step then show ?case by (blast intro: rel_pow_Suc_I)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   710
  qed
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   711
  with Pair show ?thesis by simp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   712
qed
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   713
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   714
lemma rel_pow_imp_rtrancl:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   715
  assumes "p \<in> R ^^ n"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   716
  shows "p \<in> R^*"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   717
proof (cases p)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   718
  case (Pair x y)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   719
  with assms have "(x, y) \<in> R ^^ n" by simp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   720
  then have "(x, y) \<in> R^*" proof (induct n arbitrary: x y)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   721
    case 0 then show ?case by simp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   722
  next
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   723
    case Suc then show ?case
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   724
      by (blast elim: rel_pow_Suc_E intro: rtrancl_into_rtrancl)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   725
  qed
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   726
  with Pair show ?thesis by simp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   727
qed
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   728
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   729
lemma rtrancl_is_UN_rel_pow:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   730
  "R^* = (\<Union>n. R ^^ n)"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   731
  by (blast intro: rtrancl_imp_UN_rel_pow rel_pow_imp_rtrancl)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   732
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   733
lemma rtrancl_power:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   734
  "p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   735
  by (simp add: rtrancl_is_UN_rel_pow)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   736
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   737
lemma trancl_power:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   738
  "p \<in> R^+ \<longleftrightarrow> (\<exists>n > 0. p \<in> R ^^ n)"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   739
  apply (cases p)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   740
  apply simp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   741
  apply (rule iffI)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   742
   apply (drule tranclD2)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   743
   apply (clarsimp simp: rtrancl_is_UN_rel_pow)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   744
   apply (rule_tac x="Suc x" in exI)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   745
   apply (clarsimp simp: rel_comp_def)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   746
   apply fastsimp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   747
  apply clarsimp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   748
  apply (case_tac n, simp)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   749
  apply clarsimp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   750
  apply (drule rel_pow_imp_rtrancl)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   751
  apply (drule rtrancl_into_trancl1) apply auto
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   752
  done
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   753
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   754
lemma rtrancl_imp_rel_pow:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   755
  "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   756
  by (auto dest: rtrancl_imp_UN_rel_pow)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   757
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   758
lemma single_valued_rel_pow:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   759
  fixes R :: "('a * 'a) set"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   760
  shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   761
  apply (induct n arbitrary: R)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   762
  apply simp_all
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   763
  apply (rule single_valuedI)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   764
  apply (fast dest: single_valuedD elim: rel_pow_Suc_E)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   765
  done
15551
af78481b37bf unfold theorems for trancl and rtrancl
paulson
parents: 15531
diff changeset
   766
15076
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   767
subsection {* Setup of transitivity reasoner *}
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   768
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   769
ML {*
15076
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   770
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   771
structure Trancl_Tac = Trancl_Tac_Fun (
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   772
  struct
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   773
    val r_into_trancl = @{thm trancl.r_into_trancl};
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   774
    val trancl_trans  = @{thm trancl_trans};
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   775
    val rtrancl_refl = @{thm rtrancl.rtrancl_refl};
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   776
    val r_into_rtrancl = @{thm r_into_rtrancl};
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   777
    val trancl_into_rtrancl = @{thm trancl_into_rtrancl};
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   778
    val rtrancl_trancl_trancl = @{thm rtrancl_trancl_trancl};
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   779
    val trancl_rtrancl_trancl = @{thm trancl_rtrancl_trancl};
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   780
    val rtrancl_trans = @{thm rtrancl_trans};
15096
be1d3b8cfbd5 Documentation added; minor improvements.
ballarin
parents: 15076
diff changeset
   781
30107
f3b3b0e3d184 Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents: 29609
diff changeset
   782
  fun decomp (@{const Trueprop} $ t) =
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   783
    let fun dec (Const ("op :", _) $ (Const ("Pair", _) $ a $ b) $ rel ) =
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   784
        let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*")
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   785
              | decr (Const ("Transitive_Closure.trancl", _ ) $ r)  = (r,"r+")
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   786
              | decr r = (r,"r");
26801
244184661a09 - Function dec in Trancl_Tac must eta-contract relation before calling
berghofe
parents: 26340
diff changeset
   787
            val (rel,r) = decr (Envir.beta_eta_contract rel);
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   788
        in SOME (a,b,rel,r) end
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   789
      | dec _ =  NONE
30107
f3b3b0e3d184 Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents: 29609
diff changeset
   790
    in dec t end
f3b3b0e3d184 Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents: 29609
diff changeset
   791
    | decomp _ = NONE;
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   792
21589
1b02201d7195 simplified method setup;
wenzelm
parents: 21404
diff changeset
   793
  end);
15076
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   794
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   795
structure Tranclp_Tac = Trancl_Tac_Fun (
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   796
  struct
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   797
    val r_into_trancl = @{thm tranclp.r_into_trancl};
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   798
    val trancl_trans  = @{thm tranclp_trans};
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   799
    val rtrancl_refl = @{thm rtranclp.rtrancl_refl};
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   800
    val r_into_rtrancl = @{thm r_into_rtranclp};
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   801
    val trancl_into_rtrancl = @{thm tranclp_into_rtranclp};
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   802
    val rtrancl_trancl_trancl = @{thm rtranclp_tranclp_tranclp};
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   803
    val trancl_rtrancl_trancl = @{thm tranclp_rtranclp_tranclp};
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   804
    val rtrancl_trans = @{thm rtranclp_trans};
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   805
30107
f3b3b0e3d184 Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents: 29609
diff changeset
   806
  fun decomp (@{const Trueprop} $ t) =
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   807
    let fun dec (rel $ a $ b) =
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   808
        let fun decr (Const ("Transitive_Closure.rtranclp", _ ) $ r) = (r,"r*")
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   809
              | decr (Const ("Transitive_Closure.tranclp", _ ) $ r)  = (r,"r+")
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   810
              | decr r = (r,"r");
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   811
            val (rel,r) = decr rel;
26801
244184661a09 - Function dec in Trancl_Tac must eta-contract relation before calling
berghofe
parents: 26340
diff changeset
   812
        in SOME (a, b, rel, r) end
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   813
      | dec _ =  NONE
30107
f3b3b0e3d184 Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents: 29609
diff changeset
   814
    in dec t end
f3b3b0e3d184 Fixed nonexhaustive match problem in decomp, to make it fail more gracefully
berghofe
parents: 29609
diff changeset
   815
    | decomp _ = NONE;
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   816
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   817
  end);
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   818
*}
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   819
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   820
declaration {* fn _ =>
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   821
  Simplifier.map_ss (fn ss => ss
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   822
    addSolver (mk_solver "Trancl" (fn _ => Trancl_Tac.trancl_tac))
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   823
    addSolver (mk_solver "Rtrancl" (fn _ => Trancl_Tac.rtrancl_tac))
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   824
    addSolver (mk_solver "Tranclp" (fn _ => Tranclp_Tac.trancl_tac))
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
   825
    addSolver (mk_solver "Rtranclp" (fn _ => Tranclp_Tac.rtrancl_tac)))
15076
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   826
*}
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   827
21589
1b02201d7195 simplified method setup;
wenzelm
parents: 21404
diff changeset
   828
(* Optional methods *)
15076
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   829
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   830
method_setup trancl =
30549
d2d7874648bd simplified method setup;
wenzelm
parents: 30510
diff changeset
   831
  {* Scan.succeed (K (SIMPLE_METHOD' Trancl_Tac.trancl_tac)) *}
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   832
  {* simple transitivity reasoner *}
15076
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   833
method_setup rtrancl =
30549
d2d7874648bd simplified method setup;
wenzelm
parents: 30510
diff changeset
   834
  {* Scan.succeed (K (SIMPLE_METHOD' Trancl_Tac.rtrancl_tac)) *}
15076
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   835
  {* simple transitivity reasoner *}
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   836
method_setup tranclp =
30549
d2d7874648bd simplified method setup;
wenzelm
parents: 30510
diff changeset
   837
  {* Scan.succeed (K (SIMPLE_METHOD' Tranclp_Tac.trancl_tac)) *}
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   838
  {* simple transitivity reasoner (predicate version) *}
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   839
method_setup rtranclp =
30549
d2d7874648bd simplified method setup;
wenzelm
parents: 30510
diff changeset
   840
  {* Scan.succeed (K (SIMPLE_METHOD' Tranclp_Tac.rtrancl_tac)) *}
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   841
  {* simple transitivity reasoner (predicate version) *}
15076
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   842
10213
01c2744a3786 *** empty log message ***
nipkow
parents:
diff changeset
   843
end