diff --git a/thys/SimplifiedOntologicalArgument/DisableKodkodScala.thy b/thys/SimplifiedOntologicalArgument/DisableKodkodScala.thy deleted file mode 100644 --- a/thys/SimplifiedOntologicalArgument/DisableKodkodScala.thy +++ /dev/null @@ -1,14 +0,0 @@ -theory DisableKodkodScala - imports Main -begin - -text \Some of the nitpick invocation within this AFP entry do not work, - if "Kodkod Scala" is enabled, i.e., if the box under - Plugin Options — Isabelle — General — Miscelleaneous Tools — Kodkod Scala - is activated. Therefore, in this theory we explicitly disable this configuration option.\ - -ML \ - Options.default_put_bool \<^system_option>\kodkod_scala\ false -\ - -end \ No newline at end of file diff --git a/thys/SimplifiedOntologicalArgument/SimpleVariant.thy b/thys/SimplifiedOntologicalArgument/SimpleVariant.thy --- a/thys/SimplifiedOntologicalArgument/SimpleVariant.thy +++ b/thys/SimplifiedOntologicalArgument/SimpleVariant.thy @@ -1,42 +1,41 @@ subsection\Simplified Variant (Fig.~6 in \cite{C85})\ theory SimpleVariant imports HOML MFilter BaseDefs - DisableKodkodScala begin text\Axiom's of new, simplified variant.\ axiomatization where A1': "\\<^bold>\(\

(\x.(x\<^bold>\x)))\" and A2': "\\<^bold>\X Y.(((\

X) \<^bold>\ ((X\<^bold>\Y) \<^bold>\ (X\Y))) \<^bold>\ (\

Y))\" and A3: "\\<^bold>\\.((\

\\ \) \<^bold>\ (\<^bold>\X.((X\\) \<^bold>\ (\

X))))\" lemma T2: "\\

\\" by (metis A3 G_def) \\From A3\ lemma L1: "\\

(\x.(x\<^bold>=x))\" by (metis A2' A3) text\Necessary existence of a Godlike entity.\ theorem T6: "\\<^bold>\(\<^bold>\\<^sup>E \)\" \\Proof found by sledgehammer\ proof - have T1: "\\<^bold>\X.((\

X) \<^bold>\ \<^bold>\(\<^bold>\\<^sup>E X))\" by (metis A1' A2') have T3: "\\<^bold>\(\<^bold>\\<^sup>E \)\" using T1 T2 by simp have T5: "\(\<^bold>\(\<^bold>\\<^sup>E \)) \<^bold>\ \<^bold>\(\<^bold>\\<^sup>E \)\" by (metis A1' A2' T2) thus ?thesis using T3 by blast qed lemma True nitpick[satisfy] oops \\Consistency: model found\ text\Modal collapse and monotheism: not implied.\ lemma MC: "\\<^bold>\\.(\ \<^bold>\ \<^bold>\\)\" nitpick oops \\Countermodel\ lemma MT: "\\<^bold>\x y.(((\ x) \<^bold>\ (\ y)) \<^bold>\ (x\<^bold>=y))\" nitpick oops \\Countermodel.\ text\Gödel's A1, A4, A5: not implied anymore.\ lemma A1: "\\<^bold>\X.((\<^bold>\(\

X))\<^bold>\(\

(\<^bold>\X)))\" nitpick oops \\Countermodel\ lemma A4: "\\<^bold>\X.((\

X) \<^bold>\ \<^bold>\(\

X))\" nitpick oops \\Countermodel\ lemma A5: "\\

\\\" nitpick oops \\Countermodel\ text\Checking filter and ultrafilter properties.\ theorem F1: "\Filter \

\" oops \\Proof found by sledgehammer, but reconstruction timeout\ theorem U1: "\UFilter \

\" nitpick oops \\Countermodel\ end diff --git a/thys/SimplifiedOntologicalArgument/SimpleVariantHF.thy b/thys/SimplifiedOntologicalArgument/SimpleVariantHF.thy --- a/thys/SimplifiedOntologicalArgument/SimpleVariantHF.thy +++ b/thys/SimplifiedOntologicalArgument/SimpleVariantHF.thy @@ -1,39 +1,38 @@ subsection\Hauptfiltervariant (Fig.~10 in \cite{C85})\ theory SimpleVariantHF imports HOML MFilter BaseDefs - DisableKodkodScala begin text\Definition: Set of supersets of \X\, we call this \\\ X\.\ abbreviation HF::"\\(\\\)" where "HF X \ \Y.(X\<^bold>\Y)" text\Postulate: \\\ \\ is a filter; i.e., Hauptfilter of \\\.\ axiomatization where F1: "\Filter (HF \)\" text\Necessary existence of a Godlike entity.\ theorem T6: "\\<^bold>\(\<^bold>\\<^sup>E \)\" using F1 by auto \\Proof found\ theorem T6again: "\\<^bold>\(\<^bold>\\<^sup>E \)\" proof - have T3': "\\<^bold>\\<^sup>E \\" using F1 by auto have T6: "\\<^bold>\(\<^bold>\\<^sup>E \)\" using T3' by blast thus ?thesis by simp qed text\Possible existence of Godlike entity not implied.\ lemma T3: "\\<^bold>\(\<^bold>\\<^sup>E \)\" nitpick oops \\Countermodel\ text\Axiom T enforces possible existence of Godlike entity.\ axiomatization lemma T3: assumes T: "\\<^bold>\\.((\<^bold>\\) \<^bold>\ \)\" shows "\\<^bold>\(\<^bold>\\<^sup>E \)\" using F1 T by auto lemma True nitpick[satisfy] oops \\Consistency: model found\ text\Modal collapse: not implied anymore.\ lemma MC: "\\<^bold>\\.(\ \<^bold>\ \<^bold>\\)\" nitpick oops \\Countermodel\ lemma MT: "\\<^bold>\x y.(((\ x) \<^bold>\ (\ y)) \<^bold>\ (x\<^bold>=y))\" nitpick oops \\Countermodel\ end diff --git a/thys/SimplifiedOntologicalArgument/SimpleVariantPG.thy b/thys/SimplifiedOntologicalArgument/SimpleVariantPG.thy --- a/thys/SimplifiedOntologicalArgument/SimpleVariantPG.thy +++ b/thys/SimplifiedOntologicalArgument/SimpleVariantPG.thy @@ -1,28 +1,27 @@ subsection\Simplified Variant with Axiom T2 (Fig.~7 in \cite{C85})\ theory SimpleVariantPG imports HOML MFilter BaseDefs - DisableKodkodScala begin text\Axiom's of simplified variant with A3 replaced.\ axiomatization where A1': "\\<^bold>\(\

(\x.(x\<^bold>\x)))\" and A2': "\\<^bold>\X Y.(((\

X) \<^bold>\ ((X\<^bold>\Y)\<^bold>\(X\Y))) \<^bold>\ (\

Y))\" and T2: "\\

\\" text\Necessary existence of a Godlike entity.\ theorem T6: "\\<^bold>\(\<^bold>\\<^sup>E \)\" \\Proof found by sledgehammer\ proof - have T1: "\\<^bold>\X.((\

X) \<^bold>\ \<^bold>\(\<^bold>\\<^sup>E X))\" by (metis A1' A2') have T3: "\\<^bold>\(\<^bold>\\<^sup>E \)\" using T1 T2 by simp have T5: "\(\<^bold>\(\<^bold>\\<^sup>E \)) \<^bold>\ \<^bold>\(\<^bold>\\<^sup>E \)\" by (metis A1' A2' T2) thus ?thesis using T3 by blast qed lemma True nitpick[satisfy] oops \\Consistency: model found\ text\Modal collapse and Monotheism: not implied.\ lemma MC: "\\<^bold>\\.(\ \<^bold>\ \<^bold>\\)\" nitpick oops \\Countermodel\ lemma MT: "\\<^bold>\x y.(((\ x)\<^bold>\(\ y))\<^bold>\(x\<^bold>=y))\" nitpick oops \\Countermodel\ end diff --git a/thys/SimplifiedOntologicalArgument/SimplifiedOntologicalArgument.thy b/thys/SimplifiedOntologicalArgument/SimplifiedOntologicalArgument.thy --- a/thys/SimplifiedOntologicalArgument/SimplifiedOntologicalArgument.thy +++ b/thys/SimplifiedOntologicalArgument/SimplifiedOntologicalArgument.thy @@ -1,72 +1,71 @@ section \Selected Simplified Ontological Argument \label{sec:selected}\ theory SimplifiedOntologicalArgument imports HOML - DisableKodkodScala begin text \Positive properties:\ consts posProp::"\\\" ("\

") text \An entity x is God-like if it possesses all positive properties.\ definition G ("\") where "\(x) \ \<^bold>\\.(\

(\) \<^bold>\ \(x))" text \The axiom's of the simplified variant are presented next; these axioms are further motivated in \cite{C85,J52}).\ text \Self-difference is not a positive property (possible alternative: the empty property is not a positive property).\ axiomatization where CORO1: "\\<^bold>\(\

(\x.(x\<^bold>\x)))\" text \A property entailed by a positive property is positive.\ axiomatization where CORO2: "\\<^bold>\\ \. \

(\) \<^bold>\ (\<^bold>\x. \(x) \<^bold>\ \(x)) \<^bold>\ \

(\)\" text \Being Godlike is a positive property.\ axiomatization where AXIOM3: "\\

\\" subsection\Verifying the Selected Simplified Ontological Argument (version 1)\ text \The existence of a non-exemplified positive property implies that self-difference (or, alternatively, the empty property) is a positive property.\ lemma LEMMA1: "\(\<^bold>\\.(\

(\) \<^bold>\ \<^bold>\(\<^bold>\x. \(x)))) \<^bold>\ \

(\x.(x\<^bold>\x))\" using CORO2 by meson text \A non-exemplified positive property does not exist.\ lemma LEMMA2: "\\<^bold>\(\<^bold>\\.(\

(\) \<^bold>\ \<^bold>\(\<^bold>\x. \(x))))\" using CORO1 LEMMA1 by blast text \Positive properties are exemplified.\ lemma LEMMA3: "\\<^bold>\\.(\

(\) \<^bold>\ (\<^bold>\x. \(x)))\" using LEMMA2 by blast text \There exists a God-like entity.\ theorem THEOREM3': "\\<^bold>\x. \(x)\" using AXIOM3 LEMMA3 by auto text \Necessarily, there exists a God-like entity.\ theorem THEOREM3: "\\<^bold>\(\<^bold>\x. \(x))\" using THEOREM3' by simp text \However, the possible existence of Godlike entity is not implied.\ theorem CORO: "\\<^bold>\(\<^bold>\x. \(x))\" nitpick oops (*Countermodel*) subsection\Verifying the Selected Simplified Ontological Argument (version 2)\ text \We switch to logic T.\ axiomatization where T: "\\<^bold>\\. \<^bold>\\ \<^bold>\ \\" lemma T': "\\<^bold>\\. \ \<^bold>\ \<^bold>\\\" using T by metis text \Positive properties are possibly exemplified.\ theorem THEOREM1: "\\<^bold>\\. \

(\) \<^bold>\ \<^bold>\(\<^bold>\x. \(x))\" using CORO1 CORO2 T' by metis text \Possibly there exists a God-like entity.\ theorem CORO: "\\<^bold>\(\<^bold>\x. \(x))\" using AXIOM3 THEOREM1 by auto text \The possible existence of a God-like entity impplies the necessary existence of a God-like entity.\ theorem THEOREM2: "\\<^bold>\(\<^bold>\x. \(x)) \<^bold>\ \<^bold>\(\<^bold>\x. \(x))\" using AXIOM3 CORO1 CORO2 by metis text \Necessarily, there exists a God-like entity.\ theorem THEO3: "\\<^bold>\(\<^bold>\x. \(x))\" using CORO THEOREM2 by blast text \There exists a God-like entity.\ theorem THEO3': "\\<^bold>\x. \(x)\" using T THEO3 by metis text \Modal collapse is not implied; nitpick reports a countermodel.\ lemma MC: "\\<^bold>\\. \ \<^bold>\ \<^bold>\\\" nitpick oops text \Consistency of the theory; nitpick reports a model.\ lemma True nitpick[satisfy] oops (*Model found*) end diff --git a/thys/SimplifiedOntologicalArgument/UFilterVariant.thy b/thys/SimplifiedOntologicalArgument/UFilterVariant.thy --- a/thys/SimplifiedOntologicalArgument/UFilterVariant.thy +++ b/thys/SimplifiedOntologicalArgument/UFilterVariant.thy @@ -1,46 +1,45 @@ subsection\Ultrafilter Variant (Fig.~5 in \cite{C85})\ theory UFilterVariant imports HOML MFilter BaseDefs - DisableKodkodScala begin text\Axiom's of ultrafilter variant.\ axiomatization where U1: "\UFilter \

\" and A2: "\\<^bold>\X Y.(((\

X) \<^bold>\ (X\Y)) \<^bold>\ (\

Y))\" and A3: "\\<^bold>\\.((\

\\ \) \<^bold>\ (\<^bold>\X.((X\\) \<^bold>\ (\

X))))\" text\Necessary existence of a Godlike entity.\ theorem T6: "\\<^bold>\(\<^bold>\\<^sup>E \)\" \\Proof also found by sledgehammer\ proof - have T1: "\\<^bold>\X.((\

X) \<^bold>\ \<^bold>\(\<^bold>\\<^sup>E X))\" by (metis A2 U1) have T2: "\\

\\" by (metis A3 G_def) have T3: "\\<^bold>\(\<^bold>\\<^sup>E \)\" using T1 T2 by simp have T5: "\(\<^bold>\(\<^bold>\\<^sup>E \)) \<^bold>\ \<^bold>\(\<^bold>\\<^sup>E \)\" by (metis A2 G_def T2 U1) thus ?thesis using T3 by blast qed text\Checking for consistency.\ lemma True nitpick[satisfy] oops \\Model found\ text\Checking for modal collapse.\ lemma MC: "\\<^bold>\\.(\ \<^bold>\ \<^bold>\\)\" nitpick oops \\Countermodel\ end (* definition ess ("\") where "\ Y x \ Y x \<^bold>\ (\<^bold>\Z.(Z x \<^bold>\ (Y\Z)))" definition NE ("NE") where "NE x \ \w.((\<^bold>\Y.(\ Y x \<^bold>\ \<^bold>\\<^bold>\\<^sup>E Y)) w)" consts Godlike::\ UltraFilter::"(\\\)\\" NeEx::\ axiomatization where 1: "Godlike = G" and 2: "UltraFilter = Ultrafilter" and 3: "NeEx = NE" lemma True nitpick[satisfy] oops lemma MC: "\\<^bold>\\. \ \<^bold>\ \<^bold>\\\" nitpick[format=8] oops (*Countermodel*) lemma A1: "\\<^bold>\X. \<^bold>\(\

X) \<^bold>\ \

(\<^bold>\X)\" using U1 by fastforce lemma A4: "\(\

X \<^bold>\ \<^bold>\(\

X))\" nitpick [format=4] oops (*Countermodel*) lemma A5: "\\

NE\" nitpick oops (*Countermodel*) *) \ No newline at end of file