Ziele Untersuchung
mit Columbo Integrität von
Datenbanken Interaktion und
Portierbarkeit Ergonomie der
Schnittstellen

Angebot Produkte Projekt Beratung

Mittel Analytik Modellierung Sprachen Algebra Logik Hardware Denken Kreativität

Zusammenhänge Gesellschaft Wirtschaft Branche Firma


products/sources/formale Sprachen/Isabelle/HOL/Proofs/Lambda/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 8 kB

Quelle Commutation.thy Sprache: Isabelle

(*  Title:      HOL/Proofs/Lambda/Commutation.thy
    Author:     Tobias Nipkow
    Copyright   1995  TU Muenchen
*)

section \<open>Abstract commutation and confluence notions\<close>

theory Commutation
imports Main
begin

declare [[syntax_ambiguity_warning = false]]

subsection \<open>Basic definitions\<close>

definition
  square :: "['a => 'a => bool, 'a => 'a => bool, 'a => 'a => bool, 'a => 'a => bool] => bool" where
  "square R S T U =
    (\<forall>x y. R x y --> (\<forall>z. S x z --> (\<exists>u. T y u \<and> U z u)))"

definition
  commute :: "['a => 'a => bool, 'a => 'a => bool] => bool" where
  "commute R S = square R S S R"

definition
  diamond :: "('a => 'a => bool) => bool" where
  "diamond R = commute R R"

definition
  Church_Rosser :: "('a => 'a => bool) => bool" where
  "Church_Rosser R =
    (\<forall>x y. (sup R (R\<inverse>\<inverse>))\<^sup>*\<^sup>* x y \<longrightarrow> (\<exists>z. R\<^sup>*\<^sup>* x z \<and> R\<^sup>*\<^sup>* y z))"

abbreviation
  confluent :: "('a => 'a => bool) => bool" where
  "confluent R == diamond (R\<^sup>*\<^sup>*)"

subsection \<open>Basic lemmas\<close>

subsubsection \<open>\<open>square\<close>\<close>

lemma square_sym: "square R S T U ==> square S R U T"
  apply (unfold square_def)
  apply blast
  done

lemma square_subset:
    "[| square R S T U; T \ T' |] ==> square R S T' U"
  apply (unfold square_def)
  apply (blast dest: predicate2D)
  done

lemma square_reflcl:
    "[| square R S T (R\<^sup>=\<^sup>=); S \ T |] ==> square (R\<^sup>=\<^sup>=) S T (R\<^sup>=\<^sup>=)"
  apply (unfold square_def)
  apply (blast dest: predicate2D)
  done

lemma square_rtrancl:
    "square R S S T ==> square (R\<^sup>*\<^sup>*) S S (T\<^sup>*\<^sup>*)"
  apply (unfold square_def)
  apply (intro strip)
  apply (erule rtranclp_induct)
   apply blast
  apply (blast intro: rtranclp.rtrancl_into_rtrancl)
  done

lemma square_rtrancl_reflcl_commute:
    "square R S (S\<^sup>*\<^sup>*) (R\<^sup>=\<^sup>=) ==> commute (R\<^sup>*\<^sup>*) (S\<^sup>*\<^sup>*)"
  apply (unfold commute_def)
  apply (fastforce dest: square_reflcl square_sym [THEN square_rtrancl])
  done

subsubsection \<open>\<open>commute\<close>\<close>

lemma commute_sym: "commute R S ==> commute S R"
  apply (unfold commute_def)
  apply (blast intro: square_sym)
  done

lemma commute_rtrancl: "commute R S ==> commute (R\<^sup>*\<^sup>*) (S\<^sup>*\<^sup>*)"
  apply (unfold commute_def)
  apply (blast intro: square_rtrancl square_sym)
  done

lemma commute_Un:
    "[| commute R T; commute S T |] ==> commute (sup R S) T"
  apply (unfold commute_def square_def)
  apply blast
  done

subsubsection \<open>\<open>diamond\<close>, \<open>confluence\<close>, and \<open>union\<close>\<close>

lemma diamond_Un:
    "[| diamond R; diamond S; commute R S |] ==> diamond (sup R S)"
  apply (unfold diamond_def)
  apply (blast intro: commute_Un commute_sym)
  done

lemma diamond_confluent: "diamond R ==> confluent R"
  apply (unfold diamond_def)
  apply (erule commute_rtrancl)
  done

lemma square_reflcl_confluent:
    "square R R (R\<^sup>=\<^sup>=) (R\<^sup>=\<^sup>=) ==> confluent R"
  apply (unfold diamond_def)
  apply (fast intro: square_rtrancl_reflcl_commute elim: square_subset)
  done

lemma confluent_Un:
    "[| confluent R; confluent S; commute (R\<^sup>*\<^sup>*) (S\<^sup>*\<^sup>*) |] ==> confluent (sup R S)"
  apply (rule rtranclp_sup_rtranclp [THEN subst])
  apply (blast dest: diamond_Un intro: diamond_confluent)
  done

lemma diamond_to_confluence:
    "[| diamond R; T \ R; R \ T\<^sup>*\<^sup>* |] ==> confluent T"
  apply (force intro: diamond_confluent
    dest: rtranclp_subset [symmetric])
  done

subsection \<open>Church-Rosser\<close>

lemma Church_Rosser_confluent: "Church_Rosser R = confluent R"
  apply (unfold square_def commute_def diamond_def Church_Rosser_def)
  apply (tactic \<open>safe_tac (put_claset HOL_cs \<^context>)\<close>)
   apply (tactic \<open>
     blast_tac (put_claset HOL_cs \<^context> addIs
       [@{thm sup_ge2} RS @{thm rtranclp_mono} RS @{thm predicate2D} RS @{thm rtranclp_trans},
        @{thm rtranclp_converseI}, @{thm conversepI},
        @{thm sup_ge1} RS @{thm rtranclp_mono} RS @{thm predicate2D}]) 1\<close>)
  apply (erule rtranclp_induct)
   apply blast
  apply (blast del: rtranclp.rtrancl_refl intro: rtranclp_trans)
  done

subsection \<open>Newman's lemma\<close>

text \<open>Proof by Stefan Berghofer\<close>

theorem newman:
  assumes wf: "wfP (R\\)"
  and lc: "\a b c. R a b \ R a c \
    \<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
  shows "\b c. R\<^sup>*\<^sup>* a b \ R\<^sup>*\<^sup>* a c \
    \<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
  using wf
proof induct
  case (less x b c)
  have xc: "R\<^sup>*\<^sup>* x c" by fact
  have xb: "R\<^sup>*\<^sup>* x b" by fact thus ?case
  proof (rule converse_rtranclpE)
    assume "x = b"
    with xc have "R\<^sup>*\<^sup>* b c" by simp
    thus ?thesis by iprover
  next
    fix y
    assume xy: "R x y"
    assume yb: "R\<^sup>*\<^sup>* y b"
    from xc show ?thesis
    proof (rule converse_rtranclpE)
      assume "x = c"
      with xb have "R\<^sup>*\<^sup>* c b" by simp
      thus ?thesis by iprover
    next
      fix y'
      assume y'c: "R\<^sup>*\<^sup>* y' c"
      assume xy': "R x y'"
      with xy have "\u. R\<^sup>*\<^sup>* y u \ R\<^sup>*\<^sup>* y' u" by (rule lc)
      then obtain u where yu: "R\<^sup>*\<^sup>* y u" and y'u: "R\<^sup>*\<^sup>* y' u" by iprover
      from xy have "R\\ y x" ..
      from this and yb yu have "\d. R\<^sup>*\<^sup>* b d \ R\<^sup>*\<^sup>* u d" by (rule less)
      then obtain v where bv: "R\<^sup>*\<^sup>* b v" and uv: "R\<^sup>*\<^sup>* u v" by iprover
      from xy' have "R\\ y' x" ..
      moreover from y'u and uv have "R\<^sup>*\<^sup>* y' v" by (rule rtranclp_trans)
      moreover note y'c
      ultimately have "\d. R\<^sup>*\<^sup>* v d \ R\<^sup>*\<^sup>* c d" by (rule less)
      then obtain w where vw: "R\<^sup>*\<^sup>* v w" and cw: "R\<^sup>*\<^sup>* c w" by iprover
      from bv vw have "R\<^sup>*\<^sup>* b w" by (rule rtranclp_trans)
      with cw show ?thesis by iprover
    qed
  qed
qed

text \<open>
  Alternative version.  Partly automated by Tobias
  Nipkow. Takes 2 minutes (2002).

  This is the maximal amount of automation possible using \<open>blast\<close>.
\<close>

theorem newman':
  assumes wf: "wfP (R\\)"
  and lc: "\a b c. R a b \ R a c \
    \<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
  shows "\b c. R\<^sup>*\<^sup>* a b \ R\<^sup>*\<^sup>* a c \
    \<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
  using wf
proof induct
  case (less x b c)
  note IH = \<open>\<And>y b c. \<lbrakk>R\<inverse>\<inverse> y x; R\<^sup>*\<^sup>* y b; R\<^sup>*\<^sup>* y c\<rbrakk>
                     \<Longrightarrow> \<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d\<close>
  have xc: "R\<^sup>*\<^sup>* x c" by fact
  have xb: "R\<^sup>*\<^sup>* x b" by fact
  thus ?case
  proof (rule converse_rtranclpE)
    assume "x = b"
    with xc have "R\<^sup>*\<^sup>* b c" by simp
    thus ?thesis by iprover
  next
    fix y
    assume xy: "R x y"
    assume yb: "R\<^sup>*\<^sup>* y b"
    from xc show ?thesis
    proof (rule converse_rtranclpE)
      assume "x = c"
      with xb have "R\<^sup>*\<^sup>* c b" by simp
      thus ?thesis by iprover
    next
      fix y'
      assume y'c: "R\<^sup>*\<^sup>* y' c"
      assume xy': "R x y'"
      with xy obtain u where u: "R\<^sup>*\<^sup>* y u" "R\<^sup>*\<^sup>* y' u"
        by (blast dest: lc)
      from yb u y'c show ?thesis
        by (blast del: rtranclp.rtrancl_refl
            intro: rtranclp_trans
            dest: IH [OF conversepI, OF xy] IH [OF conversepI, OF xy'])
    qed
  qed
qed

text \<open>
  Using the coherent logic prover, the proof of the induction step
  is completely automatic.
\<close>

lemma eq_imp_rtranclp: "x = y \ r\<^sup>*\<^sup>* x y"
  by simp

theorem newman'':
  assumes wf: "wfP (R\\)"
  and lc: "\a b c. R a b \ R a c \
    \<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
  shows "\b c. R\<^sup>*\<^sup>* a b \ R\<^sup>*\<^sup>* a c \
    \<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
  using wf
proof induct
  case (less x b c)
  note IH = \<open>\<And>y b c. \<lbrakk>R\<inverse>\<inverse> y x; R\<^sup>*\<^sup>* y b; R\<^sup>*\<^sup>* y c\<rbrakk>
                     \<Longrightarrow> \<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d\<close>
  show ?case
    by (coherent
      \<open>R\<^sup>*\<^sup>* x c\<close> \<open>R\<^sup>*\<^sup>* x b\<close>
      refl [where 'a='a] sym
      eq_imp_rtranclp
      r_into_rtranclp [of R]
      rtranclp_trans
      lc IH [OF conversepI]
      converse_rtranclpE)
qed

end

quality100%

¤ Dauer der Verarbeitung: 0.17 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.