Theory ATP
section ‹Automatic Theorem Provers (ATPs)›
theory ATP
imports Meson Hilbert_Choice
begin
subsection ‹ATP problems and proofs›
ML_file ‹Tools/ATP/atp_util.ML›
ML_file ‹Tools/ATP/atp_problem.ML›
ML_file ‹Tools/ATP/atp_proof.ML›
ML_file ‹Tools/ATP/atp_proof_redirect.ML›
subsection ‹Higher-order reasoning helpers›
definition fFalse :: bool where
"fFalse ⟷ False"
definition fTrue :: bool where
"fTrue ⟷ True"
definition fNot :: "bool ⇒ bool" where
"fNot P ⟷ ¬ P"
definition fComp :: "('a ⇒ bool) ⇒ 'a ⇒ bool" where
"fComp P = (λx. ¬ P x)"
definition fconj :: "bool ⇒ bool ⇒ bool" where
"fconj P Q ⟷ P ∧ Q"
definition fdisj :: "bool ⇒ bool ⇒ bool" where
"fdisj P Q ⟷ P ∨ Q"
definition fimplies :: "bool ⇒ bool ⇒ bool" where
"fimplies P Q ⟷ (P ⟶ Q)"
definition fAll :: "('a ⇒ bool) ⇒ bool" where
"fAll P ⟷ All P"
definition fEx :: "('a ⇒ bool) ⇒ bool" where
"fEx P ⟷ Ex P"
definition fequal :: "'a ⇒ 'a ⇒ bool" where
"fequal x y ⟷ (x = y)"
definition fChoice :: "('a ⇒ bool) ⇒ 'a" where
"fChoice ≡ Hilbert_Choice.Eps"
lemma fTrue_ne_fFalse: "fFalse ≠ fTrue"
unfolding fFalse_def fTrue_def by simp
lemma fNot_table:
"fNot fFalse = fTrue"
"fNot fTrue = fFalse"
unfolding fFalse_def fTrue_def fNot_def by auto
lemma fconj_table:
"fconj fFalse P = fFalse"
"fconj P fFalse = fFalse"
"fconj fTrue fTrue = fTrue"
unfolding fFalse_def fTrue_def fconj_def by auto
lemma fdisj_table:
"fdisj fTrue P = fTrue"
"fdisj P fTrue = fTrue"
"fdisj fFalse fFalse = fFalse"
unfolding fFalse_def fTrue_def fdisj_def by auto
lemma fimplies_table:
"fimplies P fTrue = fTrue"
"fimplies fFalse P = fTrue"
"fimplies fTrue fFalse = fFalse"
unfolding fFalse_def fTrue_def fimplies_def by auto
lemma fAll_table:
"Ex (fComp P) ∨ fAll P = fTrue"
"All P ∨ fAll P = fFalse"
unfolding fFalse_def fTrue_def fComp_def fAll_def by auto
lemma fEx_table:
"All (fComp P) ∨ fEx P = fTrue"
"Ex P ∨ fEx P = fFalse"
unfolding fFalse_def fTrue_def fComp_def fEx_def by auto
lemma fequal_table:
"fequal x x = fTrue"
"x = y ∨ fequal x y = fFalse"
unfolding fFalse_def fTrue_def fequal_def by auto
lemma fNot_law:
"fNot P ≠ P"
unfolding fNot_def by auto
lemma fComp_law:
"fComp P x ⟷ ¬ P x"
unfolding fComp_def ..
lemma fconj_laws:
"fconj P P ⟷ P"
"fconj P Q ⟷ fconj Q P"
"fNot (fconj P Q) ⟷ fdisj (fNot P) (fNot Q)"
unfolding fNot_def fconj_def fdisj_def by auto
lemma fdisj_laws:
"fdisj P P ⟷ P"
"fdisj P Q ⟷ fdisj Q P"
"fNot (fdisj P Q) ⟷ fconj (fNot P) (fNot Q)"
unfolding fNot_def fconj_def fdisj_def by auto
lemma fimplies_laws:
"fimplies P Q ⟷ fdisj (¬ P) Q"
"fNot (fimplies P Q) ⟷ fconj P (fNot Q)"
unfolding fNot_def fconj_def fdisj_def fimplies_def by auto
lemma fAll_law:
"fNot (fAll R) ⟷ fEx (fComp R)"
unfolding fNot_def fComp_def fAll_def fEx_def by auto
lemma fEx_law:
"fNot (fEx R) ⟷ fAll (fComp R)"
unfolding fNot_def fComp_def fAll_def fEx_def by auto
lemma fequal_laws:
"fequal x y = fequal y x"
"fequal x y = fFalse ∨ fequal y z = fFalse ∨ fequal x z = fTrue"
"fequal x y = fFalse ∨ fequal (f x) (f y) = fTrue"
unfolding fFalse_def fTrue_def fequal_def by auto
lemma fChoice_iff_Ex: "P (fChoice P) ⟷ HOL.Ex P"
unfolding fChoice_def
by (fact some_eq_ex)
text ‹
We use the @{const HOL.Ex} constant on the right-hand side of @{thm [source] fChoice_iff_Ex} because
we want to use the TPTP-native version if fChoice is introduced in a logic that supports FOOL.
In logics that don't support it, it gets replaced by fEx during processing.
Notice that we cannot use @{term "∃x. P x"}, as existentials are not skolimized by the metis proof
method but @{term "Ex P"} is eta-expanded if FOOL is supported.›
subsection ‹Basic connection between ATPs and HOL›
ML_file ‹Tools/lambda_lifting.ML›
ML_file ‹Tools/monomorph.ML›
ML_file ‹Tools/ATP/atp_problem_generate.ML›
ML_file ‹Tools/ATP/atp_proof_reconstruct.ML›
end