src/HOL/Transitive_Closure.thy
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(*  Title:      HOL/Transitive_Closure.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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*)
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header {* Reflexive and Transitive closure of a relation *}
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theory Transitive_Closure
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imports Relation
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begin
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ML_file "~~/src/Provers/trancl.ML"
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text {*
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  @{text rtrancl} is reflexive/transitive closure,
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  @{text trancl} is transitive closure,
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  @{text reflcl} is reflexive closure.
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  These postfix operators have \emph{maximum priority}, forcing their
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  operands to be atomic.
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*}
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inductive_set
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  rtrancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   ("(_^*)" [1000] 999)
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  for r :: "('a \<times> 'a) set"
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where
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    rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) : r^*"
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  | rtrancl_into_rtrancl [Pure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"
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inductive_set
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  trancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"  ("(_^+)" [1000] 999)
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  for r :: "('a \<times> 'a) set"
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where
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    r_into_trancl [intro, Pure.intro]: "(a, b) : r ==> (a, b) : r^+"
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  | trancl_into_trancl [Pure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a, c) : r^+"
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declare rtrancl_def [nitpick_unfold del]
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        rtranclp_def [nitpick_unfold del]
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        trancl_def [nitpick_unfold del]
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        tranclp_def [nitpick_unfold del]
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notation
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  rtranclp  ("(_^**)" [1000] 1000) and
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  tranclp  ("(_^++)" [1000] 1000)
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abbreviation
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  reflclp :: "('a => 'a => bool) => 'a => 'a => bool"  ("(_^==)" [1000] 1000) where
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  "r^== \<equiv> sup r op ="
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abbreviation
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  reflcl :: "('a \<times> 'a) set => ('a \<times> 'a) set"  ("(_^=)" [1000] 999) where
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  "r^= \<equiv> r \<union> Id"
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notation (xsymbols)
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  rtranclp  ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
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  tranclp  ("(_\<^sup>+\<^sup>+)" [1000] 1000) and
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  reflclp  ("(_\<^sup>=\<^sup>=)" [1000] 1000) and
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  rtrancl  ("(_\<^sup>*)" [1000] 999) and
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  trancl  ("(_\<^sup>+)" [1000] 999) and
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  reflcl  ("(_\<^sup>=)" [1000] 999)
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notation (HTML output)
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  rtranclp  ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
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  tranclp  ("(_\<^sup>+\<^sup>+)" [1000] 1000) and
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  reflclp  ("(_\<^sup>=\<^sup>=)" [1000] 1000) and
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  rtrancl  ("(_\<^sup>*)" [1000] 999) and
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  trancl  ("(_\<^sup>+)" [1000] 999) and
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  reflcl  ("(_\<^sup>=)" [1000] 999)
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subsection {* Reflexive closure *}
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lemma refl_reflcl[simp]: "refl(r^=)"
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by(simp add:refl_on_def)
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lemma antisym_reflcl[simp]: "antisym(r^=) = antisym r"
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by(simp add:antisym_def)
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lemma trans_reflclI[simp]: "trans r \<Longrightarrow> trans(r^=)"
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unfolding trans_def by blast
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subsection {* Reflexive-transitive closure *}
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lemma reflcl_set_eq [pred_set_conv]: "(sup (\<lambda>x y. (x, y) \<in> r) op =) = (\<lambda>x y. (x, y) \<in> r \<union> Id)"
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  by (auto simp add: fun_eq_iff)
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lemma r_into_rtrancl [intro]: "!!p. p \<in> r ==> p \<in> r^*"
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  -- {* @{text rtrancl} of @{text r} contains @{text r} *}
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  apply (simp only: split_tupled_all)
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  apply (erule rtrancl_refl [THEN rtrancl_into_rtrancl])
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  done
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lemma r_into_rtranclp [intro]: "r x y ==> r^** x y"
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  -- {* @{text rtrancl} of @{text r} contains @{text r} *}
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  by (erule rtranclp.rtrancl_refl [THEN rtranclp.rtrancl_into_rtrancl])
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lemma rtranclp_mono: "r \<le> s ==> r^** \<le> s^**"
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  -- {* monotonicity of @{text rtrancl} *}
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  apply (rule predicate2I)
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  apply (erule rtranclp.induct)
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   apply (rule_tac [2] rtranclp.rtrancl_into_rtrancl, blast+)
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  done
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lemmas rtrancl_mono = rtranclp_mono [to_set]
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theorem rtranclp_induct [consumes 1, case_names base step, induct set: rtranclp]:
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  assumes a: "r^** a b"
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    and cases: "P a" "!!y z. [| r^** a y; r y z; P y |] ==> P z"
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  shows "P b" using a
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  by (induct x\<equiv>a b) (rule cases)+
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lemmas rtrancl_induct [induct set: rtrancl] = rtranclp_induct [to_set]
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lemmas rtranclp_induct2 =
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  rtranclp_induct[of _ "(ax,ay)" "(bx,by)", split_rule,
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                 consumes 1, case_names refl step]
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lemmas rtrancl_induct2 =
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  rtrancl_induct[of "(ax,ay)" "(bx,by)", split_format (complete),
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                 consumes 1, case_names refl step]
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lemma refl_rtrancl: "refl (r^*)"
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by (unfold refl_on_def) fast
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text {* Transitivity of transitive closure. *}
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lemma trans_rtrancl: "trans (r^*)"
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proof (rule transI)
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  fix x y z
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  assume "(x, y) \<in> r\<^sup>*"
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  assume "(y, z) \<in> r\<^sup>*"
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  then show "(x, z) \<in> r\<^sup>*"
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  proof induct
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    case base
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    show "(x, y) \<in> r\<^sup>*" by fact
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  next
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    case (step u v)
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    from `(x, u) \<in> r\<^sup>*` and `(u, v) \<in> r`
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    show "(x, v) \<in> r\<^sup>*" ..
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  qed
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qed
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lemmas rtrancl_trans = trans_rtrancl [THEN transD]
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lemma rtranclp_trans:
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  assumes xy: "r^** x y"
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  and yz: "r^** y z"
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  shows "r^** x z" using yz xy
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  by induct iprover+
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lemma rtranclE [cases set: rtrancl]:
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  assumes major: "(a::'a, b) : r^*"
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  obtains
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    (base) "a = b"
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  | (step) y where "(a, y) : r^*" and "(y, b) : r"
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  -- {* elimination of @{text rtrancl} -- by induction on a special formula *}
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  apply (subgoal_tac "(a::'a) = b | (EX y. (a,y) : r^* & (y,b) : r)")
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   apply (rule_tac [2] major [THEN rtrancl_induct])
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    prefer 2 apply blast
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   prefer 2 apply blast
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  apply (erule asm_rl exE disjE conjE base step)+
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  done
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lemma rtrancl_Int_subset: "[| Id \<subseteq> s; (r^* \<inter> s) O r \<subseteq> s|] ==> r^* \<subseteq> s"
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  apply (rule subsetI)
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  apply (rule_tac p="x" in PairE, clarify)
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  apply (erule rtrancl_induct, auto) 
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  done
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lemma converse_rtranclp_into_rtranclp:
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  "r a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>*\<^sup>* a c"
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  by (rule rtranclp_trans) iprover+
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   173
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lemmas converse_rtrancl_into_rtrancl = converse_rtranclp_into_rtranclp [to_set]
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d21db58bcdc2 converted theory Transitive_Closure;
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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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   186
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lemmas rtrancl_idemp [simp] = rtranclp_idemp [to_set]
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   188
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   189
lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"
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  apply (rule set_eqI)
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  apply (simp only: split_tupled_all)
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  apply (blast intro: rtrancl_trans)
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   193
  done
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   194
d21db58bcdc2 converted theory Transitive_Closure;
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   195
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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   205
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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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d21db58bcdc2 converted theory Transitive_Closure;
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lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*"
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  apply (rule sym)
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   220
  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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   223
   apply blast
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   224
  apply blast
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   225
  done
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   226
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lemma rtranclp_r_diff_Id: "(inf r op ~=)^** = r^**"
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  apply (rule sym)
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   229
  apply (rule rtranclp_subset)
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   apply blast+
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  done
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   232
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   233
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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   236
proof -
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   237
  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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   240
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lemmas rtrancl_converseD = rtranclp_converseD [to_set]
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   242
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   243
theorem rtranclp_converseI:
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  assumes "r^** y x"
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  shows "(r^--1)^** x y"
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   246
  using assms
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   247
  by induct (iprover intro: rtranclp_trans conversepI)+
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   248
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lemmas rtrancl_converseI = rtranclp_converseI [to_set]
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   250
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   251
lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
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   252
  by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
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   253
19228
30fce6da8cbe added many simple lemmas
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   254
lemma sym_rtrancl: "sym r ==> sym (r^*)"
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   255
  by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])
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   256
34909
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   257
theorem converse_rtranclp_induct [consumes 1, case_names base step]:
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  assumes major: "r^** a b"
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    and cases: "P b" "!!y z. [| r y z; r^** z b; P z |] ==> P y"
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  shows "P a"
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   261
  using rtranclp_converseI [OF major]
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   262
  by induct (iprover intro: cases dest!: conversepD rtranclp_converseD)+
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   263
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9191942c4ead Removed some case_names and consumes attributes that are now no longer
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   264
lemmas converse_rtrancl_induct = converse_rtranclp_induct [to_set]
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   265
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   266
lemmas converse_rtranclp_induct2 =
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   267
  converse_rtranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
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                 consumes 1, case_names refl step]
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   269
14404
4952c5a92e04 Transitive_Closure: added consumes and case_names attributes
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   270
lemmas converse_rtrancl_induct2 =
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   271
  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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   273
34909
a799687944af Tuned some proofs; nicer case names for some of the induction / cases rules.
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   274
lemma converse_rtranclpE [consumes 1, case_names base step]:
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  assumes major: "r^** x z"
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   276
    and cases: "x=z ==> P"
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      "!!y. [| r x y; r^** y z |] ==> P"
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   278
  shows P
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   279
  apply (subgoal_tac "x = z | (EX y. r x y & r^** y z)")
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   280
   apply (rule_tac [2] major [THEN converse_rtranclp_induct])
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   281
    prefer 2 apply iprover
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   282
   prefer 2 apply iprover
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   283
  apply (erule asm_rl exE disjE conjE cases)+
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   284
  done
12691
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   285
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52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   286
lemmas converse_rtranclE = converse_rtranclpE [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   287
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   288
lemmas converse_rtranclpE2 = converse_rtranclpE [of _ "(xa,xb)" "(za,zb)", split_rule]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   289
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   290
lemmas converse_rtranclE2 = converse_rtranclE [of "(xa,xb)" "(za,zb)", split_rule]
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   291
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   292
lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   293
  by (blast elim: rtranclE converse_rtranclE
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   294
    intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   295
32235
8f9b8d14fc9f "more standard" argument order of relation composition (op O)
krauss
parents: 32215
diff changeset
   296
lemma rtrancl_unfold: "r^* = Id Un r^* O r"
15551
af78481b37bf unfold theorems for trancl and rtrancl
paulson
parents: 15531
diff changeset
   297
  by (auto intro: rtrancl_into_rtrancl elim: rtranclE)
af78481b37bf unfold theorems for trancl and rtrancl
paulson
parents: 15531
diff changeset
   298
31690
cc37bf07f9bb rtrancl lemmas
nipkow
parents: 31577
diff changeset
   299
lemma rtrancl_Un_separatorE:
cc37bf07f9bb rtrancl lemmas
nipkow
parents: 31577
diff changeset
   300
  "(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (a,x) : P^* \<longrightarrow> (x,y) : Q \<longrightarrow> x=y \<Longrightarrow> (a,b) : P^*"
cc37bf07f9bb rtrancl lemmas
nipkow
parents: 31577
diff changeset
   301
apply (induct rule:rtrancl.induct)
cc37bf07f9bb rtrancl lemmas
nipkow
parents: 31577
diff changeset
   302
 apply blast
cc37bf07f9bb rtrancl lemmas
nipkow
parents: 31577
diff changeset
   303
apply (blast intro:rtrancl_trans)
cc37bf07f9bb rtrancl lemmas
nipkow
parents: 31577
diff changeset
   304
done
cc37bf07f9bb rtrancl lemmas
nipkow
parents: 31577
diff changeset
   305
cc37bf07f9bb rtrancl lemmas
nipkow
parents: 31577
diff changeset
   306
lemma rtrancl_Un_separator_converseE:
cc37bf07f9bb rtrancl lemmas
nipkow
parents: 31577
diff changeset
   307
  "(a,b) : (P \<union> Q)^* \<Longrightarrow> \<forall>x y. (x,b) : P^* \<longrightarrow> (y,x) : Q \<longrightarrow> y=x \<Longrightarrow> (a,b) : P^*"
cc37bf07f9bb rtrancl lemmas
nipkow
parents: 31577
diff changeset
   308
apply (induct rule:converse_rtrancl_induct)
cc37bf07f9bb rtrancl lemmas
nipkow
parents: 31577
diff changeset
   309
 apply blast
cc37bf07f9bb rtrancl lemmas
nipkow
parents: 31577
diff changeset
   310
apply (blast intro:rtrancl_trans)
cc37bf07f9bb rtrancl lemmas
nipkow
parents: 31577
diff changeset
   311
done
cc37bf07f9bb rtrancl lemmas
nipkow
parents: 31577
diff changeset
   312
34970
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   313
lemma Image_closed_trancl:
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   314
  assumes "r `` X \<subseteq> X" shows "r\<^sup>* `` X = X"
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   315
proof -
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   316
  from assms have **: "{y. \<exists>x\<in>X. (x, y) \<in> r} \<subseteq> X" by auto
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   317
  have "\<And>x y. (y, x) \<in> r\<^sup>* \<Longrightarrow> y \<in> X \<Longrightarrow> x \<in> X"
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   318
  proof -
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   319
    fix x y
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   320
    assume *: "y \<in> X"
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   321
    assume "(y, x) \<in> r\<^sup>*"
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   322
    then show "x \<in> X"
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   323
    proof induct
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   324
      case base show ?case by (fact *)
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   325
    next
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   326
      case step with ** show ?case by auto
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   327
    qed
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   328
  qed
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   329
  then show ?thesis by auto
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   330
qed
4c316d777461 lemma Image_closed_trancl
haftmann
parents: 34909
diff changeset
   331
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   332
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   333
subsection {* Transitive closure *}
10331
7411e4659d4a more "xsymbols" syntax;
wenzelm
parents: 10213
diff changeset
   334
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   335
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
   336
  apply (simp add: split_tupled_all)
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   337
  apply (erule trancl.induct)
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   338
   apply (iprover dest: subsetD)+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   339
  done
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   340
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   341
lemma r_into_trancl': "!!p. p : r ==> p : r^+"
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   342
  by (simp only: split_tupled_all) (erule r_into_trancl)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   343
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   344
text {*
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   345
  \medskip Conversions between @{text trancl} and @{text rtrancl}.
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   346
*}
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   347
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   348
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
   349
  by (erule tranclp.induct) iprover+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   350
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   351
lemmas trancl_into_rtrancl = tranclp_into_rtranclp [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   352
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   353
lemma rtranclp_into_tranclp1: assumes r: "r^** a b"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   354
  shows "!!c. r b c ==> r^++ a c" using r
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 16514
diff changeset
   355
  by induct iprover+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   356
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   357
lemmas rtrancl_into_trancl1 = rtranclp_into_tranclp1 [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   358
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   359
lemma rtranclp_into_tranclp2: "[| r a b; r^** b c |] ==> r^++ a c"
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   360
  -- {* intro rule from @{text r} and @{text rtrancl} *}
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   361
  apply (erule rtranclp.cases)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   362
   apply iprover
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   363
  apply (rule rtranclp_trans [THEN rtranclp_into_tranclp1])
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   364
    apply (simp | rule r_into_rtranclp)+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   365
  done
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   366
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   367
lemmas rtrancl_into_trancl2 = rtranclp_into_tranclp2 [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   368
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   369
text {* Nice induction rule for @{text trancl} *}
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   370
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
34909
a799687944af Tuned some proofs; nicer case names for some of the induction / cases rules.
berghofe
parents: 33878
diff changeset
   371
  assumes a: "r^++ a b"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   372
  and cases: "!!y. r a y ==> P y"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   373
    "!!y z. r^++ a y ==> r y z ==> P y ==> P z"
34909
a799687944af Tuned some proofs; nicer case names for some of the induction / cases rules.
berghofe
parents: 33878
diff changeset
   374
  shows "P b" using a
a799687944af Tuned some proofs; nicer case names for some of the induction / cases rules.
berghofe
parents: 33878
diff changeset
   375
  by (induct x\<equiv>a b) (iprover intro: cases)+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   376
25425
9191942c4ead Removed some case_names and consumes attributes that are now no longer
berghofe
parents: 25295
diff changeset
   377
lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   378
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   379
lemmas tranclp_induct2 =
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   380
  tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule,
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   381
    consumes 1, case_names base step]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   382
22172
e7d6cb237b5e some new lemmas
paulson
parents: 22080
diff changeset
   383
lemmas trancl_induct2 =
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   384
  trancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   385
    consumes 1, case_names base step]
22172
e7d6cb237b5e some new lemmas
paulson
parents: 22080
diff changeset
   386
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   387
lemma tranclp_trans_induct:
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   388
  assumes major: "r^++ x y"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   389
    and cases: "!!x y. r x y ==> P x y"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   390
      "!!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
   391
  shows "P x y"
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   392
  -- {* Another induction rule for trancl, incorporating transitivity *}
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   393
  by (iprover intro: major [THEN tranclp_induct] cases)
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   394
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   395
lemmas trancl_trans_induct = tranclp_trans_induct [to_set]
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   396
26174
9efd4c04eaa4 rtranclE, tranclE: tuned statement, added case_names;
wenzelm
parents: 25425
diff changeset
   397
lemma tranclE [cases set: trancl]:
9efd4c04eaa4 rtranclE, tranclE: tuned statement, added case_names;
wenzelm
parents: 25425
diff changeset
   398
  assumes "(a, b) : r^+"
9efd4c04eaa4 rtranclE, tranclE: tuned statement, added case_names;
wenzelm
parents: 25425
diff changeset
   399
  obtains
9efd4c04eaa4 rtranclE, tranclE: tuned statement, added case_names;
wenzelm
parents: 25425
diff changeset
   400
    (base) "(a, b) : r"
9efd4c04eaa4 rtranclE, tranclE: tuned statement, added case_names;
wenzelm
parents: 25425
diff changeset
   401
  | (step) c where "(a, c) : r^+" and "(c, b) : r"
9efd4c04eaa4 rtranclE, tranclE: tuned statement, added case_names;
wenzelm
parents: 25425
diff changeset
   402
  using assms by cases simp_all
10980
0a45f2efaaec Transitive_Closure turned into new-style theory;
wenzelm
parents: 10827
diff changeset
   403
32235
8f9b8d14fc9f "more standard" argument order of relation composition (op O)
krauss
parents: 32215
diff changeset
   404
lemma trancl_Int_subset: "[| r \<subseteq> s; (r^+ \<inter> s) O r \<subseteq> s|] ==> r^+ \<subseteq> s"
22080
7bf8868ab3e4 induction rules for trancl/rtrancl expressed using subsets
paulson
parents: 21589
diff changeset
   405
  apply (rule subsetI)
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   406
  apply (rule_tac p = x in PairE)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   407
  apply clarify
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   408
  apply (erule trancl_induct)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   409
   apply auto
22080
7bf8868ab3e4 induction rules for trancl/rtrancl expressed using subsets
paulson
parents: 21589
diff changeset
   410
  done
7bf8868ab3e4 induction rules for trancl/rtrancl expressed using subsets
paulson
parents: 21589
diff changeset
   411
32235
8f9b8d14fc9f "more standard" argument order of relation composition (op O)
krauss
parents: 32215
diff changeset
   412
lemma trancl_unfold: "r^+ = r Un r^+ O r"
15551
af78481b37bf unfold theorems for trancl and rtrancl
paulson
parents: 15531
diff changeset
   413
  by (auto intro: trancl_into_trancl elim: tranclE)
af78481b37bf unfold theorems for trancl and rtrancl
paulson
parents: 15531
diff changeset
   414
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   415
text {* Transitivity of @{term "r^+"} *}
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   416
lemma trans_trancl [simp]: "trans (r^+)"
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   417
proof (rule transI)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   418
  fix x y z
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   419
  assume "(x, y) \<in> r^+"
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   420
  assume "(y, z) \<in> r^+"
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   421
  then show "(x, z) \<in> r^+"
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   422
  proof induct
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   423
    case (base u)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   424
    from `(x, y) \<in> r^+` and `(y, u) \<in> r`
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   425
    show "(x, u) \<in> r^+" ..
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   426
  next
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   427
    case (step u v)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   428
    from `(x, u) \<in> r^+` and `(u, v) \<in> r`
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   429
    show "(x, v) \<in> r^+" ..
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   430
  qed
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   431
qed
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   432
45607
16b4f5774621 eliminated obsolete "standard";
wenzelm
parents: 45153
diff changeset
   433
lemmas trancl_trans = trans_trancl [THEN transD]
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   434
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   435
lemma tranclp_trans:
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   436
  assumes xy: "r^++ x y"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   437
  and yz: "r^++ y z"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   438
  shows "r^++ x z" using yz xy
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   439
  by induct iprover+
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   440
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   441
lemma trancl_id [simp]: "trans r \<Longrightarrow> r^+ = r"
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   442
  apply auto
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   443
  apply (erule trancl_induct)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   444
   apply assumption
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   445
  apply (unfold trans_def)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   446
  apply blast
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   447
  done
19623
12e6cc4382ae added lemma in_measure
nipkow
parents: 19228
diff changeset
   448
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   449
lemma rtranclp_tranclp_tranclp:
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   450
  assumes "r^** x y"
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   451
  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
   452
  by induct (iprover intro: tranclp_trans)+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   453
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   454
lemmas rtrancl_trancl_trancl = rtranclp_tranclp_tranclp [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   455
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   456
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
   457
  by (erule tranclp_trans [OF tranclp.r_into_trancl])
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
lemmas trancl_into_trancl2 = tranclp_into_tranclp2 [to_set]
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   460
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   461
lemma trancl_insert:
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   462
  "(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
   463
  -- {* primitive recursion for @{text trancl} over finite relations *}
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   464
  apply (rule equalityI)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   465
   apply (rule subsetI)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   466
   apply (simp only: split_tupled_all)
14208
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   467
   apply (erule trancl_induct, blast)
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 34970
diff changeset
   468
   apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   469
  apply (rule subsetI)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   470
  apply (blast intro: trancl_mono rtrancl_mono
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   471
    [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   472
  done
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   473
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   474
lemma tranclp_converseI: "(r^++)^--1 x y ==> (r^--1)^++ x y"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   475
  apply (drule conversepD)
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   476
  apply (erule tranclp_induct)
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   477
  apply (iprover intro: conversepI tranclp_trans)+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   478
  done
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   479
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   480
lemmas trancl_converseI = tranclp_converseI [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   481
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   482
lemma tranclp_converseD: "(r^--1)^++ x y ==> (r^++)^--1 x y"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   483
  apply (rule conversepI)
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   484
  apply (erule tranclp_induct)
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   485
  apply (iprover dest: conversepD intro: tranclp_trans)+
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   486
  done
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   487
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   488
lemmas trancl_converseD = tranclp_converseD [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   489
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   490
lemma tranclp_converse: "(r^--1)^++ = (r^++)^--1"
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 43596
diff changeset
   491
  by (fastforce simp add: fun_eq_iff
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   492
    intro!: tranclp_converseI dest!: tranclp_converseD)
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   493
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   494
lemmas trancl_converse = tranclp_converse [to_set]
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   495
19228
30fce6da8cbe added many simple lemmas
huffman
parents: 18372
diff changeset
   496
lemma sym_trancl: "sym r ==> sym (r^+)"
30fce6da8cbe added many simple lemmas
huffman
parents: 18372
diff changeset
   497
  by (simp only: sym_conv_converse_eq trancl_converse [symmetric])
30fce6da8cbe added many simple lemmas
huffman
parents: 18372
diff changeset
   498
34909
a799687944af Tuned some proofs; nicer case names for some of the induction / cases rules.
berghofe
parents: 33878
diff changeset
   499
lemma converse_tranclp_induct [consumes 1, case_names base step]:
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   500
  assumes major: "r^++ a b"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   501
    and cases: "!!y. r y b ==> P(y)"
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   502
      "!!y z.[| r y z;  r^++ z b;  P(z) |] ==> P(y)"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   503
  shows "P a"
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   504
  apply (rule tranclp_induct [OF tranclp_converseI, OF conversepI, OF major])
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   505
   apply (rule cases)
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   506
   apply (erule conversepD)
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 34970
diff changeset
   507
  apply (blast intro: assms dest!: tranclp_converseD)
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   508
  done
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   509
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   510
lemmas converse_trancl_induct = converse_tranclp_induct [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   511
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   512
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
   513
  apply (erule converse_tranclp_induct)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   514
   apply auto
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   515
  apply (blast intro: rtranclp_trans)
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   516
  done
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   517
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   518
lemmas tranclD = tranclpD [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   519
31577
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   520
lemma converse_tranclpE:
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   521
  assumes major: "tranclp r x z"
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   522
  assumes base: "r x z ==> P"
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   523
  assumes step: "\<And> y. [| r x y; tranclp r y z |] ==> P"
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   524
  shows P
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   525
proof -
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   526
  from tranclpD[OF major]
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   527
  obtain y where "r x y" and "rtranclp r y z" by iprover
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   528
  from this(2) show P
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   529
  proof (cases rule: rtranclp.cases)
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   530
    case rtrancl_refl
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   531
    with `r x y` base show P by iprover
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   532
  next
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   533
    case rtrancl_into_rtrancl
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   534
    from this have "tranclp r y z"
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   535
      by (iprover intro: rtranclp_into_tranclp1)
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   536
    with `r x y` step show P by iprover
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   537
  qed
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   538
qed
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   539
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   540
lemmas converse_tranclE = converse_tranclpE [to_set]
ce3721fa1e17 added lemma
bulwahn
parents: 31576
diff changeset
   541
25295
12985023be5e tranclD2 (tranclD at the other end) + trancl_power
kleing
parents: 23743
diff changeset
   542
lemma tranclD2:
12985023be5e tranclD2 (tranclD at the other end) + trancl_power
kleing
parents: 23743
diff changeset
   543
  "(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
   544
  by (blast elim: tranclE intro: trancl_into_rtrancl)
12985023be5e tranclD2 (tranclD at the other end) + trancl_power
kleing
parents: 23743
diff changeset
   545
13867
1fdecd15437f just a few mods to a few thms
nipkow
parents: 13726
diff changeset
   546
lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   547
  by (blast elim: tranclE dest: trancl_into_rtrancl)
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   548
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   549
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
   550
  by (blast dest: r_into_trancl)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   551
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   552
lemma trancl_subset_Sigma_aux:
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   553
    "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
18372
2bffdf62fe7f tuned proofs;
wenzelm
parents: 17876
diff changeset
   554
  by (induct rule: rtrancl_induct) auto
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   555
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   556
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
   557
  apply (rule subsetI)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   558
  apply (simp only: split_tupled_all)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   559
  apply (erule tranclE)
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   560
   apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   561
  done
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   562
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   563
lemma reflcl_tranclp [simp]: "(r^++)^== = r^**"
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   564
  apply (safe intro!: order_antisym)
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   565
   apply (erule tranclp_into_rtranclp)
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   566
  apply (blast elim: rtranclp.cases dest: rtranclp_into_tranclp1)
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   567
  done
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   568
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   569
lemmas reflcl_trancl [simp] = reflcl_tranclp [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   570
11090
wenzelm
parents: 11084
diff changeset
   571
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   572
  apply safe
14208
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   573
   apply (drule trancl_into_rtrancl, simp)
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   574
  apply (erule rtranclE, safe)
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   575
   apply (rule r_into_trancl, simp)
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   576
  apply (rule rtrancl_into_trancl1)
14208
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   577
   apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   578
  done
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   579
45140
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
   580
lemma rtrancl_trancl_reflcl [code]: "r^* = (r^+)^="
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
   581
  by simp
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
   582
11090
wenzelm
parents: 11084
diff changeset
   583
lemma trancl_empty [simp]: "{}^+ = {}"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   584
  by (auto elim: trancl_induct)
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   585
11090
wenzelm
parents: 11084
diff changeset
   586
lemma rtrancl_empty [simp]: "{}^* = Id"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   587
  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
   588
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   589
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
   590
  by (force simp add: reflcl_tranclp [symmetric] simp del: reflcl_tranclp)
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   591
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   592
lemmas rtranclD = rtranclpD [to_set]
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   593
16514
090c6a98c704 lemma, equation between rtrancl and trancl
kleing
parents: 16417
diff changeset
   594
lemma rtrancl_eq_or_trancl:
090c6a98c704 lemma, equation between rtrancl and trancl
kleing
parents: 16417
diff changeset
   595
  "(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
   596
  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
   597
33656
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   598
lemma trancl_unfold_right: "r^+ = r^* O r"
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   599
by (auto dest: tranclD2 intro: rtrancl_into_trancl1)
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   600
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   601
lemma trancl_unfold_left: "r^+ = r O r^*"
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   602
by (auto dest: tranclD intro: rtrancl_into_trancl2)
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   603
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   604
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   605
text {* Simplifying nested closures *}
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   606
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   607
lemma rtrancl_trancl_absorb[simp]: "(R^*)^+ = R^*"
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   608
by (simp add: trans_rtrancl)
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   609
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   610
lemma trancl_rtrancl_absorb[simp]: "(R^+)^* = R^*"
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   611
by (subst reflcl_trancl[symmetric]) simp
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   612
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   613
lemma rtrancl_reflcl_absorb[simp]: "(R^*)^= = R^*"
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   614
by auto
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   615
fc1af6753233 a few lemmas for point-free reasoning about transitive closure
krauss
parents: 32901
diff changeset
   616
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   617
text {* @{text Domain} and @{text Range} *}
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   618
11090
wenzelm
parents: 11084
diff changeset
   619
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   620
  by blast
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   621
11090
wenzelm
parents: 11084
diff changeset
   622
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   623
  by blast
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   624
11090
wenzelm
parents: 11084
diff changeset
   625
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   626
  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
   627
11090
wenzelm
parents: 11084
diff changeset
   628
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   629
  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
   630
11090
wenzelm
parents: 11084
diff changeset
   631
lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46664
diff changeset
   632
  by (unfold Domain_unfold) (blast dest: tranclD)
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   633
11090
wenzelm
parents: 11084
diff changeset
   634
lemma trancl_range [simp]: "Range (r^+) = Range r"
46752
e9e7209eb375 more fundamental pred-to-set conversions, particularly by means of inductive_set; associated consolidation of some theorem names (c.f. NEWS)
haftmann
parents: 46664
diff changeset
   635
  unfolding Domain_converse [symmetric] by (simp add: trancl_converse [symmetric])
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   636
11115
285b31e9e026 a new theorem from Bryan Ford
paulson
parents: 11090
diff changeset
   637
lemma Not_Domain_rtrancl:
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   638
    "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   639
  apply auto
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   640
  apply (erule rev_mp)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   641
  apply (erule rtrancl_induct)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   642
   apply auto
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   643
  done
11327
cd2c27a23df1 Transitive closure is now defined via "inductive".
berghofe
parents: 11115
diff changeset
   644
29609
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   645
lemma trancl_subset_Field2: "r^+ <= Field r \<times> Field r"
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   646
  apply clarify
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   647
  apply (erule trancl_induct)
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   648
   apply (auto simp add: Field_def)
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   649
  done
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   650
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   651
lemma finite_trancl[simp]: "finite (r^+) = finite r"
29609
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   652
  apply auto
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   653
   prefer 2
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   654
   apply (rule trancl_subset_Field2 [THEN finite_subset])
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   655
   apply (rule finite_SigmaI)
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   656
    prefer 3
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   657
    apply (blast intro: r_into_trancl' finite_subset)
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   658
   apply (auto simp add: finite_Field)
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   659
  done
a010aab5bed0 changed import hierarchy
haftmann
parents: 26801
diff changeset
   660
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   661
text {* More about converse @{text rtrancl} and @{text trancl}, should
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   662
  be merged with main body. *}
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   663
14337
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   664
lemma single_valued_confluent:
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   665
  "\<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
   666
  \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   667
  apply (erule rtrancl_induct)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   668
  apply simp
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   669
  apply (erule disjE)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   670
   apply (blast elim:converse_rtranclE dest:single_valuedD)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   671
  apply(blast intro:rtrancl_trans)
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   672
  done
14337
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   673
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   674
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
   675
  by (fast intro: trancl_trans)
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   676
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   677
lemma trancl_into_trancl [rule_format]:
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   678
    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   679
  apply (erule trancl_induct)
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   680
   apply (fast intro: r_r_into_trancl)
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   681
  apply (fast intro: r_r_into_trancl trancl_trans)
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   682
  done
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   683
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   684
lemma tranclp_rtranclp_tranclp:
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   685
    "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
   686
  apply (drule tranclpD)
26179
bc5d582d6cfe rtranclp_induct, tranclp_induct: added case_names;
wenzelm
parents: 26174
diff changeset
   687
  apply (elim exE conjE)
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   688
  apply (drule rtranclp_trans, assumption)
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   689
  apply (drule rtranclp_into_tranclp2, assumption, assumption)
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   690
  done
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   691
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   692
lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   693
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   694
lemmas transitive_closure_trans [trans] =
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   695
  r_r_into_trancl trancl_trans rtrancl_trans
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   696
  trancl.trancl_into_trancl trancl_into_trancl2
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   697
  rtrancl.rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   698
  rtrancl_trancl_trancl trancl_rtrancl_trancl
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   699
23743
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   700
lemmas transitive_closurep_trans' [trans] =
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   701
  tranclp_trans rtranclp_trans
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   702
  tranclp.trancl_into_trancl tranclp_into_tranclp2
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   703
  rtranclp.rtrancl_into_rtrancl converse_rtranclp_into_rtranclp
52fbc991039f rtrancl and trancl are now defined using inductive_set.
berghofe
parents: 22422
diff changeset
   704
  rtranclp_tranclp_tranclp tranclp_rtranclp_tranclp
22262
96ba62dff413 Adapted to new inductive definition package.
berghofe
parents: 22172
diff changeset
   705
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   706
declare trancl_into_rtrancl [elim]
11327
cd2c27a23df1 Transitive closure is now defined via "inductive".
berghofe
parents: 11115
diff changeset
   707
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   708
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
   709
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   710
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
   711
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30954
diff changeset
   712
overloading
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30954
diff changeset
   713
  relpow == "compow :: nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
47202
69cee87927f0 power on predicate relations
haftmann
parents: 46752
diff changeset
   714
  relpowp == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30954
diff changeset
   715
begin
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   716
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30954
diff changeset
   717
primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30954
diff changeset
   718
    "relpow 0 R = Id"
32235
8f9b8d14fc9f "more standard" argument order of relation composition (op O)
krauss
parents: 32215
diff changeset
   719
  | "relpow (Suc n) R = (R ^^ n) O R"
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   720
47202
69cee87927f0 power on predicate relations
haftmann
parents: 46752
diff changeset
   721
primrec relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)" where
69cee87927f0 power on predicate relations
haftmann
parents: 46752
diff changeset
   722
    "relpowp 0 R = HOL.eq"
69cee87927f0 power on predicate relations
haftmann
parents: 46752
diff changeset
   723
  | "relpowp (Suc n) R = (R ^^ n) OO R"
69cee87927f0 power on predicate relations
haftmann
parents: 46752
diff changeset
   724
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30954
diff changeset
   725
end
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   726
47202
69cee87927f0 power on predicate relations
haftmann
parents: 46752
diff changeset
   727
lemma relpowp_relpow_eq [pred_set_conv]:
69cee87927f0 power on predicate relations
haftmann
parents: 46752
diff changeset
   728
  fixes R :: "'a rel"
69cee87927f0 power on predicate relations
haftmann
parents: 46752
diff changeset
   729
  shows "(\<lambda>x y. (x, y) \<in> R) ^^ n = (\<lambda>x y. (x, y) \<in> R ^^ n)"
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47202
diff changeset
   730
  by (induct n) (simp_all add: relcompp_relcomp_eq)
47202
69cee87927f0 power on predicate relations
haftmann
parents: 46752
diff changeset
   731
46360
5cb81e3fa799 adding code generation for relpow by copying the ideas for code generation of funpow
bulwahn
parents: 46347
diff changeset
   732
text {* for code generation *}
5cb81e3fa799 adding code generation for relpow by copying the ideas for code generation of funpow
bulwahn
parents: 46347
diff changeset
   733
5cb81e3fa799 adding code generation for relpow by copying the ideas for code generation of funpow
bulwahn
parents: 46347
diff changeset
   734
definition relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" where
5cb81e3fa799 adding code generation for relpow by copying the ideas for code generation of funpow
bulwahn
parents: 46347
diff changeset
   735
  relpow_code_def [code_abbrev]: "relpow = compow"
5cb81e3fa799 adding code generation for relpow by copying the ideas for code generation of funpow
bulwahn
parents: 46347
diff changeset
   736
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   737
definition relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)" where
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   738
  relpowp_code_def [code_abbrev]: "relpowp = compow"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   739
46360
5cb81e3fa799 adding code generation for relpow by copying the ideas for code generation of funpow
bulwahn
parents: 46347
diff changeset
   740
lemma [code]:
5cb81e3fa799 adding code generation for relpow by copying the ideas for code generation of funpow
bulwahn
parents: 46347
diff changeset
   741
  "relpow (Suc n) R = (relpow n R) O R"
5cb81e3fa799 adding code generation for relpow by copying the ideas for code generation of funpow
bulwahn
parents: 46347
diff changeset
   742
  "relpow 0 R = Id"
5cb81e3fa799 adding code generation for relpow by copying the ideas for code generation of funpow
bulwahn
parents: 46347
diff changeset
   743
  by (simp_all add: relpow_code_def)
5cb81e3fa799 adding code generation for relpow by copying the ideas for code generation of funpow
bulwahn
parents: 46347
diff changeset
   744
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   745
lemma [code]:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   746
  "relpowp (Suc n) R = (R ^^ n) OO R"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   747
  "relpowp 0 R = HOL.eq"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   748
  by (simp_all add: relpowp_code_def)
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   749
46360
5cb81e3fa799 adding code generation for relpow by copying the ideas for code generation of funpow
bulwahn
parents: 46347
diff changeset
   750
hide_const (open) relpow
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   751
hide_const (open) relpowp
46360
5cb81e3fa799 adding code generation for relpow by copying the ideas for code generation of funpow
bulwahn
parents: 46347
diff changeset
   752
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   753
lemma relpow_1 [simp]:
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30954
diff changeset
   754
  fixes R :: "('a \<times> 'a) set"
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30954
diff changeset
   755
  shows "R ^^ 1 = R"
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   756
  by simp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   757
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   758
lemma relpowp_1 [simp]:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   759
  fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   760
  shows "P ^^ 1 = P"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   761
  by (fact relpow_1 [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   762
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   763
lemma relpow_0_I: 
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   764
  "(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
   765
  by simp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   766
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   767
lemma relpowp_0_I:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   768
  "(P ^^ 0) x x"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   769
  by (fact relpow_0_I [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   770
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   771
lemma relpow_Suc_I:
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   772
  "(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
   773
  by auto
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   774
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   775
lemma relpowp_Suc_I:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   776
  "(P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> (P ^^ Suc n) x z"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   777
  by (fact relpow_Suc_I [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   778
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   779
lemma relpow_Suc_I2:
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   780
  "(x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> (x, z) \<in> R ^^ Suc n"
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 43596
diff changeset
   781
  by (induct n arbitrary: z) (simp, fastforce)
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   782
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   783
lemma relpowp_Suc_I2:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   784
  "P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> (P ^^ Suc n) x z"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   785
  by (fact relpow_Suc_I2 [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   786
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   787
lemma relpow_0_E:
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   788
  "(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
   789
  by simp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   790
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   791
lemma relpowp_0_E:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   792
  "(P ^^ 0) x y \<Longrightarrow> (x = y \<Longrightarrow> Q) \<Longrightarrow> Q"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   793
  by (fact relpow_0_E [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   794
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   795
lemma relpow_Suc_E:
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   796
  "(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
   797
  by auto
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   798
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   799
lemma relpowp_Suc_E:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   800
  "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. (P ^^ n) x y \<Longrightarrow> P y z \<Longrightarrow> Q) \<Longrightarrow> Q"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   801
  by (fact relpow_Suc_E [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   802
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   803
lemma relpow_E:
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   804
  "(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
   805
   \<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
   806
   \<Longrightarrow> P"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   807
  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
   808
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   809
lemma relpowp_E:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   810
  "(P ^^ n) x z \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q)
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   811
  \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (P ^^ m) x y \<Longrightarrow> P y z \<Longrightarrow> Q)
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   812
  \<Longrightarrow> Q"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   813
  by (fact relpow_E [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   814
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   815
lemma relpow_Suc_D2:
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   816
  "(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
   817
  apply (induct n arbitrary: x z)
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   818
   apply (blast intro: relpow_0_I elim: relpow_0_E relpow_Suc_E)
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   819
  apply (blast intro: relpow_Suc_I elim: relpow_0_E relpow_Suc_E)
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   820
  done
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   821
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   822
lemma relpowp_Suc_D2:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   823
  "(P ^^ Suc n) x z \<Longrightarrow> \<exists>y. P x y \<and> (P ^^ n) y z"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   824
  by (fact relpow_Suc_D2 [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   825
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   826
lemma relpow_Suc_E2:
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   827
  "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<And>y. (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ n \<Longrightarrow> P) \<Longrightarrow> P"
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   828
  by (blast dest: relpow_Suc_D2)
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   829
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   830
lemma relpowp_Suc_E2:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   831
  "(P ^^ Suc n) x z \<Longrightarrow> (\<And>y. P x y \<Longrightarrow> (P ^^ n) y z \<Longrightarrow> Q) \<Longrightarrow> Q"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   832
  by (fact relpow_Suc_E2 [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   833
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   834
lemma relpow_Suc_D2':
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   835
  "\<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
   836
  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
   837
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   838
lemma relpowp_Suc_D2':
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   839
  "\<forall>x y z. (P ^^ n) x y \<and> P y z \<longrightarrow> (\<exists>w. P x w \<and> (P ^^ n) w z)"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   840
  by (fact relpow_Suc_D2' [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   841
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   842
lemma relpow_E2:
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   843
  "(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
   844
     \<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
   845
   \<Longrightarrow> P"
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   846
  apply (cases n, simp)
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   847
  apply (cut_tac n=nat and R=R in relpow_Suc_D2', simp, blast)
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   848
  done
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   849
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   850
lemma relpowp_E2:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   851
  "(P ^^ n) x z \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q)
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   852
    \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> P x y \<Longrightarrow> (P ^^ m) y z \<Longrightarrow> Q)
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   853
  \<Longrightarrow> Q"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   854
  by (fact relpow_E2 [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   855
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   856
lemma relpow_add: "R ^^ (m+n) = R^^m O R^^n"
45976
9dc0d950baa9 tuned layout
haftmann
parents: 45607
diff changeset
   857
  by (induct n) auto
31351
b8d856545a02 new lemma
nipkow
parents: 30971
diff changeset
   858
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   859
lemma relpowp_add: "P ^^ (m + n) = P ^^ m OO P ^^ n"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   860
  by (fact relpow_add [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   861
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   862
lemma relpow_commute: "R O R ^^ n = R ^^ n O R"
45976
9dc0d950baa9 tuned layout
haftmann
parents: 45607
diff changeset
   863
  by (induct n) (simp, simp add: O_assoc [symmetric])
31970
ccaadfcf6941 move rel_pow_commute: "R O R ^^ n = R ^^ n O R" to Transitive_Closure
krauss
parents: 31690
diff changeset
   864
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   865
lemma relpowp_commute: "P OO P ^^ n = P ^^ n OO P"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   866
  by (fact relpow_commute [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   867
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   868
lemma relpow_empty:
45153
93e290c11b0f tuned type annnotation
haftmann
parents: 45141
diff changeset
   869
  "0 < n \<Longrightarrow> ({} :: ('a \<times> 'a) set) ^^ n = {}"
93e290c11b0f tuned type annnotation
haftmann
parents: 45141
diff changeset
   870
  by (cases n) auto
45116
f947eeef6b6f adding lemma about rel_pow in Transitive_Closure for executable equation of the (refl) transitive closure
bulwahn
parents: 44921
diff changeset
   871
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   872
lemma relpowp_bot:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   873
  "0 < n \<Longrightarrow> (\<bottom> :: 'a \<Rightarrow> 'a \<Rightarrow> bool) ^^ n = \<bottom>"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   874
  by (fact relpow_empty [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   875
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   876
lemma rtrancl_imp_UN_relpow:
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   877
  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
   878
  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
   879
proof (cases p)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   880
  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
   881
  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
   882
  then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   883
    case base show ?case by (blast intro: relpow_0_I)
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   884
  next
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   885
    case step then show ?case by (blast intro: relpow_Suc_I)
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   886
  qed
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   887
  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
   888
qed
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   889
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   890
lemma rtranclp_imp_Sup_relpowp:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   891
  assumes "(P^**) x y"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   892
  shows "(\<Squnion>n. P ^^ n) x y"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   893
  using assms and rtrancl_imp_UN_relpow [to_pred] by blast
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   894
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   895
lemma relpow_imp_rtrancl:
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   896
  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
   897
  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
   898
proof (cases p)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   899
  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
   900
  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
   901
  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
   902
    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
   903
  next
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   904
    case Suc then show ?case
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   905
      by (blast elim: relpow_Suc_E intro: rtrancl_into_rtrancl)
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   906
  qed
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   907
  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
   908
qed
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   909
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   910
lemma relpowp_imp_rtranclp:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   911
  assumes "(P ^^ n) x y"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   912
  shows "(P^**) x y"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   913
  using assms and relpow_imp_rtrancl [to_pred] by blast
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   914
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   915
lemma rtrancl_is_UN_relpow:
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   916
  "R^* = (\<Union>n. R ^^ n)"
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   917
  by (blast intro: rtrancl_imp_UN_relpow relpow_imp_rtrancl)
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   918
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   919
lemma rtranclp_is_Sup_relpowp:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   920
  "P^** = (\<Squnion>n. P ^^ n)"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   921
  using rtrancl_is_UN_relpow [to_pred, of P] by auto
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   922
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   923
lemma rtrancl_power:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   924
  "p \<in> R^* \<longleftrightarrow> (\<exists>n. p \<in> R ^^ n)"
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   925
  by (simp add: rtrancl_is_UN_relpow)
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   926
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   927
lemma rtranclp_power:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   928
  "(P^**) x y \<longleftrightarrow> (\<exists>n. (P ^^ n) x y)"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   929
  by (simp add: rtranclp_is_Sup_relpowp)
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   930
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   931
lemma trancl_power:
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   932
  "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
   933
  apply (cases p)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   934
  apply simp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   935
  apply (rule iffI)
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   936
   apply (drule tranclD2)
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   937
   apply (clarsimp simp: rtrancl_is_UN_relpow)
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30954
diff changeset
   938
   apply (rule_tac x="Suc n" in exI)
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47202
diff changeset
   939
   apply (clarsimp simp: relcomp_unfold)
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 43596
diff changeset
   940
   apply fastforce
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   941
  apply clarsimp
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   942
  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
   943
  apply clarsimp
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   944
  apply (drule relpow_imp_rtrancl)
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   945
  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
   946
  done
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   947
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   948
lemma tranclp_power:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   949
  "(P^++) x y \<longleftrightarrow> (\<exists>n > 0. (P ^^ n) x y)"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   950
  using trancl_power [to_pred, of P "(x, y)"] by simp
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   951
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   952
lemma rtrancl_imp_relpow:
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   953
  "p \<in> R^* \<Longrightarrow> \<exists>n. p \<in> R ^^ n"
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   954
  by (auto dest: rtrancl_imp_UN_relpow)
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
   955
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   956
lemma rtranclp_imp_relpowp:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   957
  "(P^**) x y \<Longrightarrow> \<exists>n. (P ^^ n) x y"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   958
  by (auto dest: rtranclp_imp_Sup_relpowp)
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   959
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   960
text{* By Sternagel/Thiemann: *}
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   961
lemma relpow_fun_conv:
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   962
  "((a,b) \<in> R ^^ n) = (\<exists>f. f 0 = a \<and> f n = b \<and> (\<forall>i<n. (f i, f(Suc i)) \<in> R))"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   963
proof (induct n arbitrary: b)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   964
  case 0 show ?case by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   965
next
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   966
  case (Suc n)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   967
  show ?case
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47202
diff changeset
   968
  proof (simp add: relcomp_unfold Suc)
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   969
    show "(\<exists>y. (\<exists>f. f 0 = a \<and> f n = y \<and> (\<forall>i<n. (f i,f(Suc i)) \<in> R)) \<and> (y,b) \<in> R)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   970
     = (\<exists>f. f 0 = a \<and> f(Suc n) = b \<and> (\<forall>i<Suc n. (f i, f (Suc i)) \<in> R))"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   971
    (is "?l = ?r")
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   972
    proof
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   973
      assume ?l
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   974
      then obtain c f where 1: "f 0 = a"  "f n = c"  "\<And>i. i < n \<Longrightarrow> (f i, f (Suc i)) \<in> R"  "(c,b) \<in> R" by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   975
      let ?g = "\<lambda> m. if m = Suc n then b else f m"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   976
      show ?r by (rule exI[of _ ?g], simp add: 1)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   977
    next
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   978
      assume ?r
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   979
      then obtain f where 1: "f 0 = a"  "b = f (Suc n)"  "\<And>i. i < Suc n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   980
      show ?l by (rule exI[of _ "f n"], rule conjI, rule exI[of _ f], insert 1, auto)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   981
    qed
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   982
  qed
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   983
qed
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   984
47492
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   985
lemma relpowp_fun_conv:
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   986
  "(P ^^ n) x y \<longleftrightarrow> (\<exists>f. f 0 = x \<and> f n = y \<and> (\<forall>i<n. P (f i) (f (Suc i))))"
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   987
  by (fact relpow_fun_conv [to_pred])
2631a12fb2d1 duplicate "relpow" facts for "relpowp" (to emphasize that both worlds exist and obtain better search results with "find_theorems")
Christian Sternagel
parents: 47433
diff changeset
   988
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
   989
lemma relpow_finite_bounded1:
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   990
assumes "finite(R :: ('a*'a)set)" and "k>0"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   991
shows "R^^k \<subseteq> (UN n:{n. 0<n & n <= card R}. R^^n)" (is "_ \<subseteq> ?r")
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   992
proof-
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   993
  { fix a b k
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   994
    have "(a,b) : R^^(Suc k) \<Longrightarrow> EX n. 0<n & n <= card R & (a,b) : R^^n"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   995
    proof(induct k arbitrary: b)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   996
      case 0
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   997
      hence "R \<noteq> {}" by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   998
      with card_0_eq[OF `finite R`] have "card R >= Suc 0" by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
   999
      thus ?case using 0 by force
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1000
    next
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1001
      case (Suc k)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1002
      then obtain a' where "(a,a') : R^^(Suc k)" and "(a',b) : R" by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1003
      from Suc(1)[OF `(a,a') : R^^(Suc k)`]
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1004
      obtain n where "n \<le> card R" and "(a,a') \<in> R ^^ n" by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1005
      have "(a,b) : R^^(Suc n)" using `(a,a') \<in> R^^n` and `(a',b)\<in> R` by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1006
      { assume "n < card R"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1007
        hence ?case using `(a,b): R^^(Suc n)` Suc_leI[OF `n < card R`] by blast
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1008
      } moreover
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1009
      { assume "n = card R"
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
  1010
        from `(a,b) \<in> R ^^ (Suc n)`[unfolded relpow_fun_conv]
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1011
        obtain f where "f 0 = a" and "f(Suc n) = b"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1012
          and steps: "\<And>i. i <= n \<Longrightarrow> (f i, f (Suc i)) \<in> R" by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1013
        let ?p = "%i. (f i, f(Suc i))"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1014
        let ?N = "{i. i \<le> n}"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1015
        have "?p ` ?N <= R" using steps by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1016
        from card_mono[OF assms(1) this]
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1017
        have "card(?p ` ?N) <= card R" .
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1018
        also have "\<dots> < card ?N" using `n = card R` by simp
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1019
        finally have "~ inj_on ?p ?N" by(rule pigeonhole)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1020
        then obtain i j where i: "i <= n" and j: "j <= n" and ij: "i \<noteq> j" and
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1021
          pij: "?p i = ?p j" by(auto simp: inj_on_def)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1022
        let ?i = "min i j" let ?j = "max i j"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1023
        have i: "?i <= n" and j: "?j <= n" and pij: "?p ?i = ?p ?j" 
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1024
          and ij: "?i < ?j"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1025
          using i j ij pij unfolding min_def max_def by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1026
        from i j pij ij obtain i j where i: "i<=n" and j: "j<=n" and ij: "i<j"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1027
          and pij: "?p i = ?p j" by blast
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1028
        let ?g = "\<lambda> l. if l \<le> i then f l else f (l + (j - i))"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1029
        let ?n = "Suc(n - (j - i))"
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
  1030
        have abl: "(a,b) \<in> R ^^ ?n" unfolding relpow_fun_conv
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1031
        proof (rule exI[of _ ?g], intro conjI impI allI)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1032
          show "?g ?n = b" using `f(Suc n) = b` j ij by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1033
        next
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1034
          fix k assume "k < ?n"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1035
          show "(?g k, ?g (Suc k)) \<in> R"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1036
          proof (cases "k < i")
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1037
            case True
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1038
            with i have "k <= n" by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1039
            from steps[OF this] show ?thesis using True by simp
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1040
          next
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1041
            case False
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1042
            hence "i \<le> k" by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1043
            show ?thesis
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1044
            proof (cases "k = i")
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1045
              case True
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1046
              thus ?thesis using ij pij steps[OF i] by simp
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1047
            next
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1048
              case False
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1049
              with `i \<le> k` have "i < k" by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1050
              hence small: "k + (j - i) <= n" using `k<?n` by arith
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1051
              show ?thesis using steps[OF small] `i<k` by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1052
            qed
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1053
          qed
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1054
        qed (simp add: `f 0 = a`)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1055
        moreover have "?n <= n" using i j ij by arith
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1056
        ultimately have ?case using `n = card R` by blast
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1057
      }
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1058
      ultimately show ?case using `n \<le> card R` by force
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1059
    qed
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1060
  }
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1061
  thus ?thesis using gr0_implies_Suc[OF `k>0`] by auto
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1062
qed
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1063
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
  1064
lemma relpow_finite_bounded:
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1065
assumes "finite(R :: ('a*'a)set)"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1066
shows "R^^k \<subseteq> (UN n:{n. n <= card R}. R^^n)"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1067
apply(cases k)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1068
 apply force
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
  1069
using relpow_finite_bounded1[OF assms, of k] by auto
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1070
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
  1071
lemma rtrancl_finite_eq_relpow:
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1072
  "finite R \<Longrightarrow> R^* = (UN n : {n. n <= card R}. R^^n)"
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
  1073
by(fastforce simp: rtrancl_power dest: relpow_finite_bounded)
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1074
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
  1075
lemma trancl_finite_eq_relpow:
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1076
  "finite R \<Longrightarrow> R^+ = (UN n : {n. 0 < n & n <= card R}. R^^n)"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1077
apply(auto simp add: trancl_power)
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
  1078
apply(auto dest: relpow_finite_bounded1)
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1079
done
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1080
47433
07f4bf913230 renamed "rel_comp" to "relcomp" (to be consistent with, e.g., "relpow")
griff
parents: 47202
diff changeset
  1081
lemma finite_relcomp[simp,intro]:
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1082
assumes "finite R" and "finite S"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1083
shows "finite(R O S)"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1084
proof-
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1085
  have "R O S = (UN (x,y) : R. \<Union>((%(u,v). if u=y then {(x,v)} else {}) ` S))"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1086
    by(force simp add: split_def)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1087
  thus ?thesis using assms by(clarsimp)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1088
qed
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1089
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1090
lemma finite_relpow[simp,intro]:
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1091
  assumes "finite(R :: ('a*'a)set)" shows "n>0 \<Longrightarrow> finite(R^^n)"
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1092
apply(induct n)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1093
 apply simp
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1094
apply(case_tac n)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1095
 apply(simp_all add: assms)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1096
done
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1097
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
  1098
lemma single_valued_relpow:
30954
cf50e67bc1d1 power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents: 30549
diff changeset
  1099
  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
  1100
  shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1101
apply (induct n arbitrary: R)
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1102
apply simp_all
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1103
apply (rule single_valuedI)
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
  1104
apply (fast dest: single_valuedD elim: relpow_Suc_E)
41987
4ad8f1dc2e0b added lemmas
nipkow
parents: 41792
diff changeset
  1105
done
15551
af78481b37bf unfold theorems for trancl and rtrancl
paulson
parents: 15531
diff changeset
  1106
45140
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1107
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1108
subsection {* Bounded transitive closure *}
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1109
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1110
definition ntrancl :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1111
where
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1112
  "ntrancl n R = (\<Union>i\<in>{i. 0 < i \<and> i \<le> Suc n}. R ^^ i)"
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1113
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1114
lemma ntrancl_Zero [simp, code]:
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1115
  "ntrancl 0 R = R"
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1116
proof
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1117
  show "R \<subseteq> ntrancl 0 R"
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1118
    unfolding ntrancl_def by fastforce
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1119
next
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1120
  { 
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1121
    fix i have "0 < i \<and> i \<le> Suc 0 \<longleftrightarrow> i = 1" by auto
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1122
  }
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1123
  from this show "ntrancl 0 R \<le> R"
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1124
    unfolding ntrancl_def by auto
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1125
qed
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1126
46347
54870ad19af4 new code equation for ntrancl that allows computation of the transitive closure of sets on infinite types as well
bulwahn
parents: 46127
diff changeset
  1127
lemma ntrancl_Suc [simp]:
45140
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1128
  "ntrancl (Suc n) R = ntrancl n R O (Id \<union> R)"
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1129
proof
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1130
  {
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1131
    fix a b
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1132
    assume "(a, b) \<in> ntrancl (Suc n) R"
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1133
    from this obtain i where "0 < i" "i \<le> Suc (Suc n)" "(a, b) \<in> R ^^ i"
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1134
      unfolding ntrancl_def by auto
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1135
    have "(a, b) \<in> ntrancl n R O (Id \<union> R)"
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1136
    proof (cases "i = 1")
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1137
      case True
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1138
      from this `(a, b) \<in> R ^^ i` show ?thesis
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1139
        unfolding ntrancl_def by auto
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1140
    next
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1141
      case False
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1142
      from this `0 < i` obtain j where j: "i = Suc j" "0 < j"
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1143
        by (cases i) auto
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1144
      from this `(a, b) \<in> R ^^ i` obtain c where c1: "(a, c) \<in> R ^^ j" and c2:"(c, b) \<in> R"
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1145
        by auto
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1146
      from c1 j `i \<le> Suc (Suc n)` have "(a, c) \<in> ntrancl n R"
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1147
        unfolding ntrancl_def by fastforce
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1148
      from this c2 show ?thesis by fastforce
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1149
    qed
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1150
  }
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1151
  from this show "ntrancl (Suc n) R \<subseteq> ntrancl n R O (Id \<union> R)"
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1152
    by auto
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1153
next
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1154
  show "ntrancl n R O (Id \<union> R) \<subseteq> ntrancl (Suc n) R"
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1155
    unfolding ntrancl_def by fastforce
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1156
qed
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1157
46347
54870ad19af4 new code equation for ntrancl that allows computation of the transitive closure of sets on infinite types as well
bulwahn
parents: 46127
diff changeset
  1158
lemma [code]:
54870ad19af4 new code equation for ntrancl that allows computation of the transitive closure of sets on infinite types as well
bulwahn
parents: 46127
diff changeset
  1159
  "ntrancl (Suc n) r = (let r' = ntrancl n r in r' Un r' O r)"
54870ad19af4 new code equation for ntrancl that allows computation of the transitive closure of sets on infinite types as well
bulwahn
parents: 46127
diff changeset
  1160
unfolding Let_def by auto
54870ad19af4 new code equation for ntrancl that allows computation of the transitive closure of sets on infinite types as well
bulwahn
parents: 46127
diff changeset
  1161
45141
b2eb87bd541b avoid very specific code equation for card; corrected spelling
haftmann
parents: 45140
diff changeset
  1162
lemma finite_trancl_ntranl:
45140
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1163
  "finite R \<Longrightarrow> trancl R = ntrancl (card R - 1) R"
46362
b2878f059f91 renaming all lemmas with name rel_pow to relpow
bulwahn
parents: 46360
diff changeset
  1164
  by (cases "card R") (auto simp add: trancl_finite_eq_relpow relpow_empty ntrancl_def)
45140
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1165
339a8b3c4791 bouned transitive closure
haftmann
parents: 45139
diff changeset
  1166
45139
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1167
subsection {* Acyclic relations *}
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1168
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1169
definition acyclic :: "('a * 'a) set => bool" where
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1170
  "acyclic r \<longleftrightarrow> (!x. (x,x) ~: r^+)"
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1171
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1172
abbreviation acyclicP :: "('a => 'a => bool) => bool" where
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1173
  "acyclicP r \<equiv> acyclic {(x, y). r x y}"
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1174
46127
af3b95160b59 cleanup of code declarations
haftmann
parents: 45976
diff changeset
  1175
lemma acyclic_irrefl [code]:
45139
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1176
  "acyclic r \<longleftrightarrow> irrefl (r^+)"
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1177
  by (simp add: acyclic_def irrefl_def)
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1178
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1179
lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1180
  by (simp add: acyclic_def)
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1181
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1182
lemma acyclic_insert [iff]:
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1183
     "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1184
apply (simp add: acyclic_def trancl_insert)
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1185
apply (blast intro: rtrancl_trans)
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1186
done
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1187
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1188
lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1189
by (simp add: acyclic_def trancl_converse)
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1190
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1191
lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1192
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1193
lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1194
apply (simp add: acyclic_def antisym_def)
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1195
apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1196
done
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1197
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1198
(* Other direction:
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1199
acyclic = no loops
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1200
antisym = only self loops
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1201
Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1202
==> antisym( r^* ) = acyclic(r - Id)";
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1203
*)
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1204
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1205
lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1206
apply (simp add: acyclic_def)
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1207
apply (blast intro: trancl_mono)
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1208
done
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1209
bdcaa3f3a2f4 moved acyclic predicate up in hierarchy
haftmann
parents: 45137
diff changeset
  1210
15076
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
  1211
subsection {* Setup of transitivity reasoner *}
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
  1212
26340
a85fe32e7b2f more antiquotations;
wenzelm
parents: 26271
diff changeset
  1213
ML {*
15076
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
  1214
32215
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31970
diff changeset
  1215
structure Trancl_Tac = Trancl_Tac
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31970
diff changeset
  1216
(
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31970
diff changeset
  1217
  val r_into_trancl = @{thm trancl.r_into_trancl};
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31970
diff changeset
  1218
  val trancl_trans  = @{thm trancl_trans};
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31970
diff changeset
  1219
  val rtrancl_refl = @{thm rtrancl.rtrancl_refl};
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31970
diff changeset
  1220
  val r_into_rtrancl = @{thm r_into_rtrancl};
87806301a813 replaced old METAHYPS by FOCUS;
wenzelm
parents: 31970
diff changeset
  1221
  val trancl_into_rtrancl = @{thm trancl_into_rtrancl};