src/HOL/Transitive_Closure.thy
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(*  Title:      HOL/Transitive_Closure.thy
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    ID:         $Id$
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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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import Inductive
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files ("../Provers/trancl.ML")
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begin
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text {*
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  @{text rtrancl} is reflexive/transitive closure,
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  @{text trancl} is transitive closure,
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  @{text reflcl} is reflexive closure.
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  These postfix operators have \emph{maximum priority}, forcing their
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  operands to be atomic.
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*}
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consts
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  rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^*)" [1000] 999)
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inductive "r^*"
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  intros
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    rtrancl_refl [intro!, CPure.intro!, simp]: "(a, a) : r^*"
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    rtrancl_into_rtrancl [CPure.intro]: "(a, b) : r^* ==> (b, c) : r ==> (a, c) : r^*"
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consts
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  trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^+)" [1000] 999)
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inductive "r^+"
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  intros
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    r_into_trancl [intro, CPure.intro]: "(a, b) : r ==> (a, b) : r^+"
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    trancl_into_trancl [CPure.intro]: "(a, b) : r^+ ==> (b, c) : r ==> (a,c) : r^+"
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syntax
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  "_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_^=)" [1000] 999)
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translations
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  "r^=" == "r \<union> Id"
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syntax (xsymbols)
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  rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>*)" [1000] 999)
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  trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>+)" [1000] 999)
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  "_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>=)" [1000] 999)
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syntax (HTML output)
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  rtrancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>*)" [1000] 999)
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  trancl :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>+)" [1000] 999)
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  "_reflcl" :: "('a \<times> 'a) set => ('a \<times> 'a) set"    ("(_\<^sup>=)" [1000] 999)
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subsection {* Reflexive-transitive closure *}
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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 rtrancl_mono: "r \<subseteq> s ==> r^* \<subseteq> s^*"
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  -- {* monotonicity of @{text rtrancl} *}
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  apply (rule subsetI)
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  apply (simp only: split_tupled_all)
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  apply (erule rtrancl.induct)
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   apply (rule_tac [2] rtrancl_into_rtrancl, blast+)
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  done
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theorem rtrancl_induct [consumes 1, induct set: rtrancl]:
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  assumes a: "(a, b) : r^*"
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    and cases: "P a" "!!y z. [| (a, y) : r^*; (y, z) : r; P y |] ==> P z"
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  shows "P b"
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proof -
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  from a have "a = a --> P b"
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    by (induct "%x y. x = a --> P y" a b) (rules intro: cases)+
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  thus ?thesis by rules
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qed
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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 trans_rtrancl: "trans(r^*)"
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  -- {* transitivity of transitive closure!! -- by induction *}
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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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  thus "(x, z) \<in> r\<^sup>*" by induct (rules!)+
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qed
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lemmas rtrancl_trans = trans_rtrancl [THEN transD, standard]
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lemma rtranclE:
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  "[| (a::'a,b) : r^*;  (a = b) ==> P;
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      !!y.[| (a,y) : r^*; (y,b) : r |] ==> P
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   |] ==> P"
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  -- {* elimination of @{text rtrancl} -- by induction on a special formula *}
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proof -
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  assume major: "(a::'a,b) : r^*"
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  case rule_context
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  show ?thesis
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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 prems)+
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    done
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qed
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lemma converse_rtrancl_into_rtrancl:
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  "(a, b) \<in> r \<Longrightarrow> (b, c) \<in> r\<^sup>* \<Longrightarrow> (a, c) \<in> r\<^sup>*"
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  by (rule rtrancl_trans) rules+
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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 rtrancl_idemp [simp]: "(r^*)^* = r^*"
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  apply auto
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  apply (erule rtrancl_induct)
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   apply (rule rtrancl_refl)
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  apply (blast intro: rtrancl_trans)
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  done
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lemma rtrancl_idemp_self_comp [simp]: "R^* O R^* = R^*"
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  apply (rule set_ext)
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  apply (simp only: split_tupled_all)
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  apply (blast intro: rtrancl_trans)
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  done
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lemma rtrancl_subset_rtrancl: "r \<subseteq> s^* ==> r^* \<subseteq> s^*"
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by (drule rtrancl_mono, simp)
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lemma rtrancl_subset: "R \<subseteq> S ==> S \<subseteq> R^* ==> S^* = R^*"
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  apply (drule rtrancl_mono)
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  apply (drule rtrancl_mono, simp)
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  done
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lemma rtrancl_Un_rtrancl: "(R^* \<union> S^*)^* = (R \<union> S)^*"
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  by (blast intro!: rtrancl_subset intro: r_into_rtrancl rtrancl_mono [THEN subsetD])
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lemma rtrancl_reflcl [simp]: "(R^=)^* = R^*"
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  by (blast intro!: rtrancl_subset intro: r_into_rtrancl)
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lemma rtrancl_r_diff_Id: "(r - Id)^* = r^*"
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  apply (rule sym)
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  apply (rule rtrancl_subset, blast, clarify)
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  apply (rename_tac a b)
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  apply (case_tac "a = b", blast)
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  apply (blast intro!: r_into_rtrancl)
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   154
  done
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   155
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   156
theorem rtrancl_converseD:
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  assumes r: "(x, y) \<in> (r^-1)^*"
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  shows "(y, x) \<in> r^*"
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   159
proof -
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   160
  from r show ?thesis
9d3f5056296b Made some proofs constructive.
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    by induct (rules intro: rtrancl_trans dest!: converseD)+
9d3f5056296b Made some proofs constructive.
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   162
qed
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   163
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theorem rtrancl_converseI:
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  assumes r: "(y, x) \<in> r^*"
0c4fd7529467 clarified syntax of ``long'' statements: fixes/assumes/shows;
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   166
  shows "(x, y) \<in> (r^-1)^*"
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   167
proof -
9d3f5056296b Made some proofs constructive.
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   168
  from r show ?thesis
9d3f5056296b Made some proofs constructive.
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    by induct (rules intro: rtrancl_trans converseI)+
9d3f5056296b Made some proofs constructive.
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qed
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d21db58bcdc2 converted theory Transitive_Closure;
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lemma rtrancl_converse: "(r^-1)^* = (r^*)^-1"
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  by (fast dest!: rtrancl_converseD intro!: rtrancl_converseI)
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theorem converse_rtrancl_induct[consumes 1]:
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  assumes major: "(a, b) : r^*"
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    and cases: "P b" "!!y z. [| (y, z) : r; (z, b) : r^*; P z |] ==> P y"
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  shows "P a"
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   179
proof -
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   180
  from rtrancl_converseI [OF major]
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  show ?thesis
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    by induct (rules intro: cases dest!: converseD rtrancl_converseD)+
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   183
qed
d21db58bcdc2 converted theory Transitive_Closure;
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   184
14404
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   185
lemmas converse_rtrancl_induct2 =
4952c5a92e04 Transitive_Closure: added consumes and case_names attributes
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   186
  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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   188
d21db58bcdc2 converted theory Transitive_Closure;
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   189
lemma converse_rtranclE:
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  "[| (x,z):r^*;
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   191
      x=z ==> P;
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      !!y. [| (x,y):r; (y,z):r^* |] ==> P
d21db58bcdc2 converted theory Transitive_Closure;
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   193
   |] ==> P"
d21db58bcdc2 converted theory Transitive_Closure;
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   194
proof -
d21db58bcdc2 converted theory Transitive_Closure;
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   195
  assume major: "(x,z):r^*"
d21db58bcdc2 converted theory Transitive_Closure;
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   196
  case rule_context
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   197
  show ?thesis
d21db58bcdc2 converted theory Transitive_Closure;
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   198
    apply (subgoal_tac "x = z | (EX y. (x,y) : r & (y,z) : r^*)")
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   199
     apply (rule_tac [2] major [THEN converse_rtrancl_induct])
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      prefer 2 apply rules
9550a6f4ed4a Replaced some blasts by rules.
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     prefer 2 apply rules
12691
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    apply (erule asm_rl exE disjE conjE prems)+
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   203
    done
d21db58bcdc2 converted theory Transitive_Closure;
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   204
qed
d21db58bcdc2 converted theory Transitive_Closure;
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   205
d21db58bcdc2 converted theory Transitive_Closure;
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   206
ML_setup {*
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  bind_thm ("converse_rtranclE2", split_rule
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    (read_instantiate [("x","(xa,xb)"), ("z","(za,zb)")] (thm "converse_rtranclE")));
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   209
*}
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   210
d21db58bcdc2 converted theory Transitive_Closure;
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   211
lemma r_comp_rtrancl_eq: "r O r^* = r^* O r"
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   212
  by (blast elim: rtranclE converse_rtranclE
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   213
    intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
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   214
d21db58bcdc2 converted theory Transitive_Closure;
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   215
d21db58bcdc2 converted theory Transitive_Closure;
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subsection {* Transitive closure *}
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lemma trancl_mono: "!!p. p \<in> r^+ ==> r \<subseteq> s ==> p \<in> s^+"
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  apply (simp only: split_tupled_all)
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   220
  apply (erule trancl.induct)
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   221
  apply (rules dest: subsetD)+
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   222
  done
d21db58bcdc2 converted theory Transitive_Closure;
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   223
13704
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   224
lemma r_into_trancl': "!!p. p : r ==> p : r^+"
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   225
  by (simp only: split_tupled_all) (erule r_into_trancl)
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   226
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text {*
d21db58bcdc2 converted theory Transitive_Closure;
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   228
  \medskip Conversions between @{text trancl} and @{text rtrancl}.
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   229
*}
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   230
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   231
lemma trancl_into_rtrancl: "(a, b) \<in> r^+ ==> (a, b) \<in> r^*"
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   232
  by (erule trancl.induct) rules+
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   233
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   234
lemma rtrancl_into_trancl1: assumes r: "(a, b) \<in> r^*"
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  shows "!!c. (b, c) \<in> r ==> (a, c) \<in> r^+" using r
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   236
  by induct rules+
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   237
d21db58bcdc2 converted theory Transitive_Closure;
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   238
lemma rtrancl_into_trancl2: "[| (a,b) : r;  (b,c) : r^* |]   ==>  (a,c) : r^+"
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  -- {* intro rule from @{text r} and @{text rtrancl} *}
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   240
  apply (erule rtranclE, rules)
12691
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   241
  apply (rule rtrancl_trans [THEN rtrancl_into_trancl1])
d21db58bcdc2 converted theory Transitive_Closure;
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   242
   apply (assumption | rule r_into_rtrancl)+
d21db58bcdc2 converted theory Transitive_Closure;
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   243
  done
d21db58bcdc2 converted theory Transitive_Closure;
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   244
13704
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   245
lemma trancl_induct [consumes 1, induct set: trancl]:
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   246
  assumes a: "(a,b) : r^+"
854501b1e957 Transitive closure is now defined inductively as well.
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   247
  and cases: "!!y. (a, y) : r ==> P y"
854501b1e957 Transitive closure is now defined inductively as well.
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   248
    "!!y z. (a,y) : r^+ ==> (y, z) : r ==> P y ==> P z"
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   249
  shows "P b"
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   250
  -- {* Nice induction rule for @{text trancl} *}
d21db58bcdc2 converted theory Transitive_Closure;
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diff changeset
   251
proof -
13704
854501b1e957 Transitive closure is now defined inductively as well.
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   252
  from a have "a = a --> P b"
854501b1e957 Transitive closure is now defined inductively as well.
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   253
    by (induct "%x y. x = a --> P y" a b) (rules intro: cases)+
854501b1e957 Transitive closure is now defined inductively as well.
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   254
  thus ?thesis by rules
12691
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   255
qed
d21db58bcdc2 converted theory Transitive_Closure;
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   256
d21db58bcdc2 converted theory Transitive_Closure;
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   257
lemma trancl_trans_induct:
d21db58bcdc2 converted theory Transitive_Closure;
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   258
  "[| (x,y) : r^+;
d21db58bcdc2 converted theory Transitive_Closure;
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   259
      !!x y. (x,y) : r ==> P x y;
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      !!x y z. [| (x,y) : r^+; P x y; (y,z) : r^+; P y z |] ==> P x z
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   261
   |] ==> P x y"
d21db58bcdc2 converted theory Transitive_Closure;
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   262
  -- {* Another induction rule for trancl, incorporating transitivity *}
d21db58bcdc2 converted theory Transitive_Closure;
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diff changeset
   263
proof -
d21db58bcdc2 converted theory Transitive_Closure;
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   264
  assume major: "(x,y) : r^+"
d21db58bcdc2 converted theory Transitive_Closure;
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   265
  case rule_context
d21db58bcdc2 converted theory Transitive_Closure;
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   266
  show ?thesis
13704
854501b1e957 Transitive closure is now defined inductively as well.
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   267
    by (rules intro: r_into_trancl major [THEN trancl_induct] prems)
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   268
qed
d21db58bcdc2 converted theory Transitive_Closure;
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   269
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   270
inductive_cases tranclE: "(a, b) : r^+"
10980
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   271
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   272
lemma trans_trancl: "trans(r^+)"
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   273
  -- {* Transitivity of @{term "r^+"} *}
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   274
proof (rule transI)
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   275
  fix x y z
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   276
  assume "(x, y) \<in> r^+"
854501b1e957 Transitive closure is now defined inductively as well.
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   277
  assume "(y, z) \<in> r^+"
854501b1e957 Transitive closure is now defined inductively as well.
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   278
  thus "(x, z) \<in> r^+" by induct (rules!)+
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   279
qed
12691
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   280
d21db58bcdc2 converted theory Transitive_Closure;
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   281
lemmas trancl_trans = trans_trancl [THEN transD, standard]
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   282
13704
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   283
lemma rtrancl_trancl_trancl: assumes r: "(x, y) \<in> r^*"
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   284
  shows "!!z. (y, z) \<in> r^+ ==> (x, z) \<in> r^+" using r
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   285
  by induct (rules intro: trancl_trans)+
12691
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   286
d21db58bcdc2 converted theory Transitive_Closure;
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   287
lemma trancl_into_trancl2: "(a, b) \<in> r ==> (b, c) \<in> r^+ ==> (a, c) \<in> r^+"
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diff changeset
   288
  by (erule transD [OF trans_trancl r_into_trancl])
d21db58bcdc2 converted theory Transitive_Closure;
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   289
d21db58bcdc2 converted theory Transitive_Closure;
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   290
lemma trancl_insert:
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   291
  "(insert (y, x) r)^+ = r^+ \<union> {(a, b). (a, y) \<in> r^* \<and> (x, b) \<in> r^*}"
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   292
  -- {* primitive recursion for @{text trancl} over finite relations *}
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   293
  apply (rule equalityI)
d21db58bcdc2 converted theory Transitive_Closure;
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diff changeset
   294
   apply (rule subsetI)
d21db58bcdc2 converted theory Transitive_Closure;
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   295
   apply (simp only: split_tupled_all)
14208
144f45277d5a misc tidying
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diff changeset
   296
   apply (erule trancl_induct, blast)
12691
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   297
   apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl r_into_trancl trancl_trans)
d21db58bcdc2 converted theory Transitive_Closure;
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   298
  apply (rule subsetI)
d21db58bcdc2 converted theory Transitive_Closure;
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   299
  apply (blast intro: trancl_mono rtrancl_mono
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   300
    [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
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   301
  done
d21db58bcdc2 converted theory Transitive_Closure;
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   302
13704
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diff changeset
   303
lemma trancl_converseI: "(x, y) \<in> (r^+)^-1 ==> (x, y) \<in> (r^-1)^+"
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   304
  apply (drule converseD)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   305
  apply (erule trancl.induct)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   306
  apply (rules intro: converseI trancl_trans)+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   307
  done
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   308
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   309
lemma trancl_converseD: "(x, y) \<in> (r^-1)^+ ==> (x, y) \<in> (r^+)^-1"
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   310
  apply (rule converseI)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   311
  apply (erule trancl.induct)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   312
  apply (rules dest: converseD intro: trancl_trans)+
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   313
  done
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   314
13704
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   315
lemma trancl_converse: "(r^-1)^+ = (r^+)^-1"
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   316
  by (fastsimp simp add: split_tupled_all
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   317
    intro!: trancl_converseI trancl_converseD)
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   318
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   319
lemma converse_trancl_induct:
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   320
  "[| (a,b) : r^+; !!y. (y,b) : r ==> P(y);
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   321
      !!y z.[| (y,z) : r;  (z,b) : r^+;  P(z) |] ==> P(y) |]
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   322
    ==> P(a)"
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   323
proof -
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   324
  assume major: "(a,b) : r^+"
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   325
  case rule_context
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   326
  show ?thesis
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   327
    apply (rule major [THEN converseI, THEN trancl_converseI [THEN trancl_induct]])
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   328
     apply (rule prems)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   329
     apply (erule converseD)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   330
    apply (blast intro: prems dest!: trancl_converseD)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   331
    done
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   332
qed
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   333
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   334
lemma tranclD: "(x, y) \<in> R^+ ==> EX z. (x, z) \<in> R \<and> (z, y) \<in> R^*"
14208
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   335
  apply (erule converse_trancl_induct, auto)
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   336
  apply (blast intro: rtrancl_trans)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   337
  done
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   338
13867
1fdecd15437f just a few mods to a few thms
nipkow
parents: 13726
diff changeset
   339
lemma irrefl_tranclI: "r^-1 \<inter> r^* = {} ==> (x, x) \<notin> r^+"
1fdecd15437f just a few mods to a few thms
nipkow
parents: 13726
diff changeset
   340
by(blast elim: tranclE dest: trancl_into_rtrancl)
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   341
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   342
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
   343
  by (blast dest: r_into_trancl)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   344
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   345
lemma trancl_subset_Sigma_aux:
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   346
    "(a, b) \<in> r^* ==> r \<subseteq> A \<times> A ==> a = b \<or> a \<in> A"
14208
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   347
  apply (erule rtrancl_induct, auto)
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   348
  done
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   349
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   350
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
   351
  apply (rule subsetI)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   352
  apply (simp only: split_tupled_all)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   353
  apply (erule tranclE)
854501b1e957 Transitive closure is now defined inductively as well.
berghofe
parents: 12937
diff changeset
   354
  apply (blast dest!: trancl_into_rtrancl trancl_subset_Sigma_aux)+
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   355
  done
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   356
11090
wenzelm
parents: 11084
diff changeset
   357
lemma reflcl_trancl [simp]: "(r^+)^= = r^*"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   358
  apply safe
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   359
   apply (erule trancl_into_rtrancl)
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   360
  apply (blast elim: rtranclE dest: rtrancl_into_trancl1)
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   361
  done
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   362
11090
wenzelm
parents: 11084
diff changeset
   363
lemma trancl_reflcl [simp]: "(r^=)^+ = r^*"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   364
  apply safe
14208
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   365
   apply (drule trancl_into_rtrancl, simp)
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   366
  apply (erule rtranclE, safe)
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   367
   apply (rule r_into_trancl, simp)
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   368
  apply (rule rtrancl_into_trancl1)
14208
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   369
   apply (erule rtrancl_reflcl [THEN equalityD2, THEN subsetD], fast)
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   370
  done
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   371
11090
wenzelm
parents: 11084
diff changeset
   372
lemma trancl_empty [simp]: "{}^+ = {}"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   373
  by (auto elim: trancl_induct)
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   374
11090
wenzelm
parents: 11084
diff changeset
   375
lemma rtrancl_empty [simp]: "{}^* = Id"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   376
  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
   377
11090
wenzelm
parents: 11084
diff changeset
   378
lemma rtranclD: "(a, b) \<in> R^* ==> a = b \<or> a \<noteq> b \<and> (a, b) \<in> R^+"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   379
  by (force simp add: reflcl_trancl [symmetric] simp del: reflcl_trancl)
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   380
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   381
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   382
text {* @{text Domain} and @{text Range} *}
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   383
11090
wenzelm
parents: 11084
diff changeset
   384
lemma Domain_rtrancl [simp]: "Domain (R^*) = UNIV"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   385
  by blast
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   386
11090
wenzelm
parents: 11084
diff changeset
   387
lemma Range_rtrancl [simp]: "Range (R^*) = UNIV"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   388
  by blast
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   389
11090
wenzelm
parents: 11084
diff changeset
   390
lemma rtrancl_Un_subset: "(R^* \<union> S^*) \<subseteq> (R Un S)^*"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   391
  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
   392
11090
wenzelm
parents: 11084
diff changeset
   393
lemma in_rtrancl_UnI: "x \<in> R^* \<or> x \<in> S^* ==> x \<in> (R \<union> S)^*"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   394
  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
   395
11090
wenzelm
parents: 11084
diff changeset
   396
lemma trancl_domain [simp]: "Domain (r^+) = Domain r"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   397
  by (unfold Domain_def) (blast dest: tranclD)
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   398
11090
wenzelm
parents: 11084
diff changeset
   399
lemma trancl_range [simp]: "Range (r^+) = Range r"
11084
32c1deea5bcd unsymbolized;
wenzelm
parents: 10996
diff changeset
   400
  by (simp add: Range_def trancl_converse [symmetric])
10996
74e970389def Moved some thms from Transitive_ClosureTr.ML to Transitive_Closure.thy
nipkow
parents: 10980
diff changeset
   401
11115
285b31e9e026 a new theorem from Bryan Ford
paulson
parents: 11090
diff changeset
   402
lemma Not_Domain_rtrancl:
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   403
    "x ~: Domain R ==> ((x, y) : R^*) = (x = y)"
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   404
  apply auto
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   405
  by (erule rev_mp, erule rtrancl_induct, auto)
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   406
11327
cd2c27a23df1 Transitive closure is now defined via "inductive".
berghofe
parents: 11115
diff changeset
   407
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   408
text {* More about converse @{text rtrancl} and @{text trancl}, should
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   409
  be merged with main body. *}
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   410
14337
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   411
lemma single_valued_confluent:
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   412
  "\<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
   413
  \<Longrightarrow> (y,z) \<in> r^* \<or> (z,y) \<in> r^*"
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   414
apply(erule rtrancl_induct)
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   415
 apply simp
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   416
apply(erule disjE)
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   417
 apply(blast elim:converse_rtranclE dest:single_valuedD)
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   418
apply(blast intro:rtrancl_trans)
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   419
done
e13731554e50 undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents: 14208
diff changeset
   420
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   421
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
   422
  by (fast intro: trancl_trans)
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   423
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   424
lemma trancl_into_trancl [rule_format]:
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   425
    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r --> (a,c) \<in> r\<^sup>+"
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   426
  apply (erule trancl_induct)
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   427
   apply (fast intro: r_r_into_trancl)
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   428
  apply (fast intro: r_r_into_trancl trancl_trans)
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   429
  done
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   430
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   431
lemma trancl_rtrancl_trancl:
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   432
    "(a, b) \<in> r\<^sup>+ ==> (b, c) \<in> r\<^sup>* ==> (a, c) \<in> r\<^sup>+"
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   433
  apply (drule tranclD)
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   434
  apply (erule exE, erule conjE)
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   435
  apply (drule rtrancl_trans, assumption)
14208
144f45277d5a misc tidying
paulson
parents: 13867
diff changeset
   436
  apply (drule rtrancl_into_trancl2, assumption, assumption)
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   437
  done
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   438
12691
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   439
lemmas transitive_closure_trans [trans] =
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   440
  r_r_into_trancl trancl_trans rtrancl_trans
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   441
  trancl_into_trancl trancl_into_trancl2
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   442
  rtrancl_into_rtrancl converse_rtrancl_into_rtrancl
d21db58bcdc2 converted theory Transitive_Closure;
wenzelm
parents: 12566
diff changeset
   443
  rtrancl_trancl_trancl trancl_rtrancl_trancl
12428
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   444
f3033eed309a setup [trans] rules for calculational Isar reasoning
kleing
parents: 11327
diff changeset
   445
declare trancl_into_rtrancl [elim]
11327
cd2c27a23df1 Transitive closure is now defined via "inductive".
berghofe
parents: 11115
diff changeset
   446
cd2c27a23df1 Transitive closure is now defined via "inductive".
berghofe
parents: 11115
diff changeset
   447
declare rtranclE [cases set: rtrancl]
cd2c27a23df1 Transitive closure is now defined via "inductive".
berghofe
parents: 11115
diff changeset
   448
declare tranclE [cases set: trancl]
cd2c27a23df1 Transitive closure is now defined via "inductive".
berghofe
parents: 11115
diff changeset
   449
15076
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   450
subsection {* Setup of transitivity reasoner *}
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   451
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
ballarin
parents: 14565
diff changeset
   452
use "../Provers/trancl.ML";
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ML_setup {*
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4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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structure Trancl_Tac = Trancl_Tac_Fun (
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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  struct
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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    val r_into_trancl = thm "r_into_trancl";
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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    val trancl_trans  = thm "trancl_trans";
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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    val rtrancl_refl = thm "rtrancl_refl";
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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    val r_into_rtrancl = thm "r_into_rtrancl";
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    val trancl_into_rtrancl = thm "trancl_into_rtrancl";
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    val rtrancl_trancl_trancl = thm "rtrancl_trancl_trancl";
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    val trancl_rtrancl_trancl = thm "trancl_rtrancl_trancl";
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    val rtrancl_trans = thm "rtrancl_trans";
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be1d3b8cfbd5 Documentation added; minor improvements.
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15076
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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  fun decomp (Trueprop $ t) = 
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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    let fun dec (Const ("op :", _) $ (Const ("Pair", _) $ a $ b) $ rel ) = 
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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	let fun decr (Const ("Transitive_Closure.rtrancl", _ ) $ r) = (r,"r*")
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	      | decr (Const ("Transitive_Closure.trancl", _ ) $ r)  = (r,"r+")
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	      | decr r = (r,"r");
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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	    val (rel,r) = decr rel;
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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	in Some (a,b,rel,r) end
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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      | dec _ =  None 
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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    in dec t end;
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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  end); (* struct *)
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simpset_ref() := simpset ()
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    addSolver (mk_solver "Trancl" (fn _ => Trancl_Tac.trancl_tac))
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    addSolver (mk_solver "Rtrancl" (fn _ => Trancl_Tac.rtrancl_tac));
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4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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*}
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4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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(* Optional methods
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method_setup trancl =
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  {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (trancl_tac)) *}
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  {* simple transitivity reasoner *}	    
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method_setup rtrancl =
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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  {* Method.no_args (Method.SIMPLE_METHOD' HEADGOAL (rtrancl_tac)) *}
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  {* simple transitivity reasoner *}
4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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4b3d280ef06a New prover for transitive and reflexive-transitive closure of relations.
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*)
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10213
01c2744a3786 *** empty log message ***
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end