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src/HOL/ex/Hilbert_Classical.thy

author | wenzelm |

Fri, 28 Sep 2001 20:08:05 +0200 | |

changeset 11635 | fd242f857508 |

parent 11591 | 4b171ad4ff65 |

child 14981 | e73f8140af78 |

permissions | -rw-r--r-- |

tuned;

(* Title: HOL/ex/Hilbert_Classical.thy ID: $Id$ Author: Stefan Berghofer and Markus Wenzel, TU Muenchen License: GPL (GNU GENERAL PUBLIC LICENSE) *) header {* Hilbert's choice and classical logic *} theory Hilbert_Classical = Main: text {* Derivation of the classical law of tertium-non-datur by means of Hilbert's choice operator (due to M. J. Beeson and J. Harrison). *} subsection {* Proof text *} theorem tnd: "A \<or> \<not> A" proof - let ?P = "\<lambda>X. X = False \<or> X = True \<and> A" let ?Q = "\<lambda>X. X = False \<and> A \<or> X = True" have a: "?P (Eps ?P)" proof (rule someI) have "False = False" .. thus "?P False" .. qed have b: "?Q (Eps ?Q)" proof (rule someI) have "True = True" .. thus "?Q True" .. qed from a show ?thesis proof assume "Eps ?P = True \<and> A" hence A .. thus ?thesis .. next assume P: "Eps ?P = False" from b show ?thesis proof assume "Eps ?Q = False \<and> A" hence A .. thus ?thesis .. next assume Q: "Eps ?Q = True" have neq: "?P \<noteq> ?Q" proof assume "?P = ?Q" hence "Eps ?P = Eps ?Q" by (rule arg_cong) also note P also note Q finally show False by (rule False_neq_True) qed have "\<not> A" proof assume a: A have "?P = ?Q" proof fix x show "?P x = ?Q x" proof assume "?P x" thus "?Q x" proof assume "x = False" from this and a have "x = False \<and> A" .. thus "?Q x" .. next assume "x = True \<and> A" hence "x = True" .. thus "?Q x" .. qed next assume "?Q x" thus "?P x" proof assume "x = False \<and> A" hence "x = False" .. thus "?P x" .. next assume "x = True" from this and a have "x = True \<and> A" .. thus "?P x" .. qed qed qed with neq show False by contradiction qed thus ?thesis .. qed qed qed subsection {* Proof term of text *} text {* {\small @{prf [display, margin = 80] tnd}} *} subsection {* Proof script *} theorem tnd': "A \<or> \<not> A" apply (subgoal_tac "(((SOME x. x = False \<or> x = True \<and> A) = False) \<or> ((SOME x. x = False \<or> x = True \<and> A) = True) \<and> A) \<and> (((SOME x. x = False \<and> A \<or> x = True) = False) \<and> A \<or> ((SOME x. x = False \<and> A \<or> x = True) = True))") prefer 2 apply (rule conjI) apply (rule someI) apply (rule disjI1) apply (rule refl) apply (rule someI) apply (rule disjI2) apply (rule refl) apply (erule conjE) apply (erule disjE) apply (erule disjE) apply (erule conjE) apply (erule disjI1) prefer 2 apply (erule conjE) apply (erule disjI1) apply (subgoal_tac "(\<lambda>x. (x = False) \<or> (x = True) \<and> A) \<noteq> (\<lambda>x. (x = False) \<and> A \<or> (x = True))") prefer 2 apply (rule notI) apply (drule_tac f = "\<lambda>y. SOME x. y x" in arg_cong) apply (drule trans, assumption) apply (drule sym) apply (drule trans, assumption) apply (erule False_neq_True) apply (rule disjI2) apply (rule notI) apply (erule notE) apply (rule ext) apply (rule iffI) apply (erule disjE) apply (rule disjI1) apply (erule conjI) apply assumption apply (erule conjE) apply (erule disjI2) apply (erule disjE) apply (erule conjE) apply (erule disjI1) apply (rule disjI2) apply (erule conjI) apply assumption done subsection {* Proof term of script *} text {* {\small @{prf [display, margin = 80] tnd'}} *} end