(* Title: HOL/Proofs/ex/Hilbert_Classical.thy
Author: Stefan Berghofer and Markus Wenzel, TU Muenchen
*)
section \<open>Hilbert's choice and classical logic\<close>
theory Hilbert_Classical
imports Main
begin
text \<open>
Derivation of the classical law of tertium-non-datur by means of
Hilbert's choice operator (due to M. J. Beeson and J. Harrison).
\<close>
subsection \<open>Proof text\<close>
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 (rule ext)
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 \<open>Proof term of text\<close>
prf tnd
subsection \<open>Proof script\<close>
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 \<open>Proof term of script\<close>
prf tnd'
end