--- /dev/null
+
+include "logic/equality.ma".
+(* Inclusion of: LCL153-1.p *)
+(* -------------------------------------------------------------------------- *)
+(* File : LCL153-1 : TPTP v3.1.1. Released v1.0.0. *)
+(* Domain : Logic Calculi (Wajsberg Algebra) *)
+(* Problem : The 1st alternative Wajsberg algebra axiom *)
+(* Version : [Bon91] (equality) axioms. *)
+(* English : *)
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+(* Source : [Bon91] *)
+(* Names : W' axiom 1 [Bon91] *)
+(* Status : Unsatisfiable *)
+(* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
+(* Syntax : Number of clauses : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
+(* Number of atoms : 17 ( 17 equality) *)
+(* Maximal clause size : 1 ( 1 average) *)
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+(* Number of functors : 9 ( 3 constant; 0-2 arity) *)
+(* Number of variables : 33 ( 0 singleton) *)
+(* Maximal term depth : 4 ( 2 average) *)
+(* Comments : *)
+(* -------------------------------------------------------------------------- *)
+(* ----Include Wajsberg algebra axioms *)
+(* Inclusion of: Axioms/LCL001-0.ax *)
+(* -------------------------------------------------------------------------- *)
+(* File : LCL001-0 : TPTP v3.1.1. Released v1.0.0. *)
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+(* Axioms : Wajsberg algebra axioms *)
+(* Version : [Bon91] (equality) axioms. *)
+(* English : *)
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+(* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
+(* Source : [MW92] *)
+(* Names : MV Sentential Calculus [MW92] *)
+(* Status : *)
+(* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
+(* Number of literals : 4 ( 4 equality) *)
+(* Maximal clause size : 1 ( 1 average) *)
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+(* Number of functors : 3 ( 1 constant; 0-2 arity) *)
+(* Number of variables : 8 ( 0 singleton) *)
+(* Maximal term depth : 4 ( 2 average) *)
+(* Comments : *)
+(* -------------------------------------------------------------------------- *)
+(* -------------------------------------------------------------------------- *)
+(* ----Include Wajsberg algebra and and or definitions *)
+(* Inclusion of: Axioms/LCL001-2.ax *)
+(* -------------------------------------------------------------------------- *)
+(* File : LCL001-2 : TPTP v3.1.1. Released v1.0.0. *)
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+(* Axioms : Wajsberg algebra AND and OR definitions *)
+(* Version : [AB90] (equality) axioms. *)
+(* English : *)
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+(* Source : [Bon91] *)
+(* Names : *)
+(* Status : *)
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
+(* Number of literals : 6 ( 6 equality) *)
+(* Maximal clause size : 1 ( 1 average) *)
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+(* Number of functors : 4 ( 0 constant; 1-2 arity) *)
+(* Number of variables : 14 ( 0 singleton) *)
+(* Maximal term depth : 4 ( 3 average) *)
+(* Comments : Requires LCL001-0.ax *)
+(* -------------------------------------------------------------------------- *)
+(* ----Definitions of or and and, which are AC *)
+(* -------------------------------------------------------------------------- *)
+(* ----Include Alternative Wajsberg algebra definitions *)
+(* Inclusion of: Axioms/LCL002-1.ax *)
+(* -------------------------------------------------------------------------- *)
+(* File : LCL002-1 : TPTP v3.1.1. Released v1.0.0. *)
+(* Domain : Logic Calculi (Wajsberg Algebras) *)
+(* Axioms : Alternative Wajsberg algebra definitions *)
+(* Version : [AB90] (equality) axioms. *)
+(* English : *)
+(* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
+(* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
+(* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
+(* Source : [Bon91] *)
+(* Names : *)
+(* Status : *)
+(* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
+(* Number of literals : 6 ( 6 equality) *)
+(* Maximal clause size : 1 ( 1 average) *)
+(* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
+(* Number of functors : 7 ( 2 constant; 0-2 arity) *)
+(* Number of variables : 11 ( 0 singleton) *)
+(* Maximal term depth : 4 ( 2 average) *)
+(* Comments : Requires LCL001-0.ax LCL001-2.ax *)
+(* -------------------------------------------------------------------------- *)
+(* ----Definitions of and_star and xor, where and_star is AC and xor is C *)
+(* ---I guess the next two can be derived from the AC of and *)
+(* ----Definition of false in terms of truth *)
+(* -------------------------------------------------------------------------- *)
+(* -------------------------------------------------------------------------- *)
+theorem prove_alternative_wajsberg_axiom:
+ \forall Univ:Set.
+\forall myand:\forall _:Univ.\forall _:Univ.Univ.
+\forall and_star:\forall _:Univ.\forall _:Univ.Univ.
+\forall falsehood:Univ.
+\forall implies:\forall _:Univ.\forall _:Univ.Univ.
+\forall not:\forall _:Univ.Univ.
+\forall or:\forall _:Univ.\forall _:Univ.Univ.
+\forall truth:Univ.
+\forall x:Univ.
+\forall xor:\forall _:Univ.\forall _:Univ.Univ.
+\forall H0:eq Univ (not truth) falsehood.
+\forall H1:\forall X:Univ.\forall Y:Univ.eq Univ (and_star X Y) (and_star Y X).
+\forall H2:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (and_star (and_star X Y) Z) (and_star X (and_star Y Z)).
+\forall H3:\forall X:Univ.\forall Y:Univ.eq Univ (and_star X Y) (not (or (not X) (not Y))).
+\forall H4:\forall X:Univ.\forall Y:Univ.eq Univ (xor X Y) (xor Y X).
+\forall H5:\forall X:Univ.\forall Y:Univ.eq Univ (xor X Y) (or (myand X (not Y)) (myand (not X) Y)).
+\forall H6:\forall X:Univ.\forall Y:Univ.eq Univ (myand X Y) (myand Y X).
+\forall H7:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (myand (myand X Y) Z) (myand X (myand Y Z)).
+\forall H8:\forall X:Univ.\forall Y:Univ.eq Univ (myand X Y) (not (or (not X) (not Y))).
+\forall H9:\forall X:Univ.\forall Y:Univ.eq Univ (or X Y) (or Y X).
+\forall H10:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (or (or X Y) Z) (or X (or Y Z)).
+\forall H11:\forall X:Univ.\forall Y:Univ.eq Univ (or X Y) (implies (not X) Y).
+\forall H12:\forall X:Univ.\forall Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
+\forall H13:\forall X:Univ.\forall Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
+\forall H14:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (implies (implies X Y) (implies (implies Y Z) (implies X Z))) truth.
+\forall H15:\forall X:Univ.eq Univ (implies truth X) X.eq Univ (not x) (xor x truth)
+.
+intros.
+autobatch paramodulation timeout=100;
+try assumption.
+print proofterm.
+qed.
+(* -------------------------------------------------------------------------- *)
projects / helm.git / blobdiff