src/HOL/ATP.thy
 author paulson Sun, 14 Aug 2022 11:11:11 +0100 changeset 75857 86d60f4a10a7 parent 74899 b4beb55c574e permissions -rw-r--r--
merged
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(*  Title:      HOL/ATP.thy
Author:     Fabian Immler, TU Muenchen
Author:     Jasmin Blanchette, TU Muenchen
Author:     Martin Desharnais, UniBw Muenchen
*)

section \<open>Automatic Theorem Provers (ATPs)\<close>

theory ATP
imports Meson Hilbert_Choice
begin

subsection \<open>ATP problems and proofs\<close>

ML_file \<open>Tools/ATP/atp_util.ML\<close>
ML_file \<open>Tools/ATP/atp_problem.ML\<close>
ML_file \<open>Tools/ATP/atp_proof.ML\<close>
ML_file \<open>Tools/ATP/atp_proof_redirect.ML\<close>

subsection \<open>Higher-order reasoning helpers\<close>

definition fFalse :: bool where
"fFalse \<longleftrightarrow> False"

definition fTrue :: bool where
"fTrue \<longleftrightarrow> True"

definition fNot :: "bool \<Rightarrow> bool" where
"fNot P \<longleftrightarrow> \<not> P"

definition fComp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool" where
"fComp P = (\<lambda>x. \<not> P x)"

definition fconj :: "bool \<Rightarrow> bool \<Rightarrow> bool" where
"fconj P Q \<longleftrightarrow> P \<and> Q"

definition fdisj :: "bool \<Rightarrow> bool \<Rightarrow> bool" where
"fdisj P Q \<longleftrightarrow> P \<or> Q"

definition fimplies :: "bool \<Rightarrow> bool \<Rightarrow> bool" where
"fimplies P Q \<longleftrightarrow> (P \<longrightarrow> Q)"

definition fAll :: "('a \<Rightarrow> bool) \<Rightarrow> bool" where
"fAll P \<longleftrightarrow> All P"

definition fEx :: "('a \<Rightarrow> bool) \<Rightarrow> bool" where
"fEx P \<longleftrightarrow> Ex P"

definition fequal :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
"fequal x y \<longleftrightarrow> (x = y)"

definition fChoice :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" where
"fChoice \<equiv> Hilbert_Choice.Eps"

lemma fTrue_ne_fFalse: "fFalse \<noteq> 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) \<or> fAll P = fTrue"
"All P \<or> fAll P = fFalse"
unfolding fFalse_def fTrue_def fComp_def fAll_def by auto

lemma fEx_table:
"All (fComp P) \<or> fEx P = fTrue"
"Ex P \<or> fEx P = fFalse"
unfolding fFalse_def fTrue_def fComp_def fEx_def by auto

lemma fequal_table:
"fequal x x = fTrue"
"x = y \<or> fequal x y = fFalse"
unfolding fFalse_def fTrue_def fequal_def by auto

lemma fNot_law:
"fNot P \<noteq> P"
unfolding fNot_def by auto

lemma fComp_law:
"fComp P x \<longleftrightarrow> \<not> P x"
unfolding fComp_def ..

lemma fconj_laws:
"fconj P P \<longleftrightarrow> P"
"fconj P Q \<longleftrightarrow> fconj Q P"
"fNot (fconj P Q) \<longleftrightarrow> fdisj (fNot P) (fNot Q)"
unfolding fNot_def fconj_def fdisj_def by auto

lemma fdisj_laws:
"fdisj P P \<longleftrightarrow> P"
"fdisj P Q \<longleftrightarrow> fdisj Q P"
"fNot (fdisj P Q) \<longleftrightarrow> fconj (fNot P) (fNot Q)"
unfolding fNot_def fconj_def fdisj_def by auto

lemma fimplies_laws:
"fimplies P Q \<longleftrightarrow> fdisj (\<not> P) Q"
"fNot (fimplies P Q) \<longleftrightarrow> fconj P (fNot Q)"
unfolding fNot_def fconj_def fdisj_def fimplies_def by auto

lemma fAll_law:
"fNot (fAll R) \<longleftrightarrow> fEx (fComp R)"
unfolding fNot_def fComp_def fAll_def fEx_def by auto

lemma fEx_law:
"fNot (fEx R) \<longleftrightarrow> 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 \<or> fequal y z = fFalse \<or> fequal x z = fTrue"
"fequal x y = fFalse \<or> fequal (f x) (f y) = fTrue"
unfolding fFalse_def fTrue_def fequal_def by auto

lemma fChoice_iff_Ex: "P (fChoice P) \<longleftrightarrow> HOL.Ex P"
unfolding fChoice_def
by (fact some_eq_ex)

text \<open>
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 "\<exists>x. P x"}, as existentials are not skolimized by the metis proof
method but @{term "Ex P"} is eta-expanded if FOOL is supported.\<close>

subsection \<open>Basic connection between ATPs and HOL\<close>

ML_file \<open>Tools/lambda_lifting.ML\<close>
ML_file \<open>Tools/monomorph.ML\<close>
ML_file \<open>Tools/ATP/atp_problem_generate.ML\<close>
ML_file \<open>Tools/ATP/atp_proof_reconstruct.ML\<close>

end
```