Quellcode-Bibliothek

^© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: Quantifiers_Int.thy Sprache: Latech

Original von: Isabelle^©

(*  Title:      FOL/ex/Quantifiers_Int.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1991  University of Cambridge
*)

section \<open>First-Order Logic: quantifier examples (intuitionistic version)\<close>

theory Quantifiers_Int
imports IFOL
begin

lemma \<open>(\<forall>x y. P(x,y)) \<longrightarrow> (\<forall>y x. P(x,y))\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

lemma \<open>(\<exists>x y. P(x,y)) \<longrightarrow> (\<exists>y x. P(x,y))\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

\<comment> \<open>Converse is false\<close>
lemma \<open>(\<forall>x. P(x)) \<or> (\<forall>x. Q(x)) \<longrightarrow> (\<forall>x. P(x) \<or> Q(x))\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

lemma \<open>(\<forall>x. P \<longrightarrow> Q(x)) \<longleftrightarrow> (P \<longrightarrow> (\<forall>x. Q(x)))\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

lemma \<open>(\<forall>x. P(x) \<longrightarrow> Q) \<longleftrightarrow> ((\<exists>x. P(x)) \<longrightarrow> Q)\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

text \<open>Some harder ones\<close>

lemma \<open>(\<exists>x. P(x) \<or> Q(x)) \<longleftrightarrow> (\<exists>x. P(x)) \<or> (\<exists>x. Q(x))\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

\<comment> \<open>Converse is false\<close>
lemma \<open>(\<exists>x. P(x) \<and> Q(x)) \<longrightarrow> (\<exists>x. P(x)) \<and> (\<exists>x. Q(x))\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

text \<open>Basic test of quantifier reasoning\<close>

\<comment> \<open>TRUE\<close>
lemma \<open>(\<exists>y. \<forall>x. Q(x,y)) \<longrightarrow> (\<forall>x. \<exists>y. Q(x,y))\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

lemma \<open>(\<forall>x. Q(x)) \<longrightarrow> (\<exists>x. Q(x))\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

text \<open>The following should fail, as they are false!\<close>

lemma \<open>(\<forall>x. \<exists>y. Q(x,y)) \<longrightarrow> (\<exists>y. \<forall>x. Q(x,y))\<close>
  apply (tactic "IntPr.fast_tac \<^context> 1")?
  oops

lemma \<open>(\<exists>x. Q(x)) \<longrightarrow> (\<forall>x. Q(x))\<close>
  apply (tactic "IntPr.fast_tac \<^context> 1")?
  oops

schematic_goal \<open>P(?a) \<longrightarrow> (\<forall>x. P(x))\<close>
  apply (tactic "IntPr.fast_tac \<^context> 1")?
  oops

schematic_goal \<open>(P(?a) \<longrightarrow> (\<forall>x. Q(x))) \<longrightarrow> (\<forall>x. P(x) \<longrightarrow> Q(x))\<close>
  apply (tactic "IntPr.fast_tac \<^context> 1")?
  oops

text \<open>Back to things that are provable \dots\<close>

lemma \<open>(\<forall>x. P(x) \<longrightarrow> Q(x)) \<and> (\<exists>x. P(x)) \<longrightarrow> (\<exists>x. Q(x))\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

\<comment> \<open>An example of why exI should be delayed as long as possible\<close>
lemma \<open>(P \<longrightarrow> (\<exists>x. Q(x))) \<and> P \<longrightarrow> (\<exists>x. Q(x))\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

schematic_goal \<open>(\<forall>x. P(x) \<longrightarrow> Q(f(x))) \<and> (\<forall>x. Q(x) \<longrightarrow> R(g(x))) \<and> P(d) \<longrightarrow> R(?a)\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

lemma \<open>(\<forall>x. Q(x)) \<longrightarrow> (\<exists>x. Q(x))\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

text \<open>Some slow ones\<close>

\<comment> \<open>Principia Mathematica *11.53\<close>
lemma \<open>(\<forall>x y. P(x) \<longrightarrow> Q(y)) \<longleftrightarrow> ((\<exists>x. P(x)) \<longrightarrow> (\<forall>y. Q(y)))\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

(*Principia Mathematica *11.55  *)
lemma \<open>(\<exists>x y. P(x) \<and> Q(x,y)) \<longleftrightarrow> (\<exists>x. P(x) \<and> (\<exists>y. Q(x,y)))\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

(*Principia Mathematica *11.61  *)
lemma \<open>(\<exists>y. \<forall>x. P(x) \<longrightarrow> Q(x,y)) \<longrightarrow> (\<forall>x. P(x) \<longrightarrow> (\<exists>y. Q(x,y)))\<close>
  by (tactic "IntPr.fast_tac \<^context> 1")

end

¤ Dauer der Verarbeitung: 0.22 Sekunden (vorverarbeitet) ¤

Download des Quellennavigators
Download des sprechenden Kalenders
in der Quellcodebibliothek suchen

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.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.