2 include "logic/equality.ma".
3 (* Inclusion of: LCL164-1.p *)
4 (* -------------------------------------------------------------------------- *)
5 (* File : LCL164-1 : TPTP v3.1.1. Released v1.0.0. *)
6 (* Domain : Logic Calculi (Wajsberg Algebra) *)
7 (* Problem : The 4th Wajsberg algebra axiom, from the alternative axioms *)
8 (* Version : [Bon91] (equality) axioms. *)
10 (* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
11 (* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
12 (* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
13 (* Source : [Bon91] *)
14 (* Names : W axiom 4 [Bon91] *)
15 (* Status : Unsatisfiable *)
16 (* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.38 v2.0.0 *)
17 (* Syntax : Number of clauses : 14 ( 0 non-Horn; 14 unit; 2 RR) *)
18 (* Number of atoms : 14 ( 14 equality) *)
19 (* Maximal clause size : 1 ( 1 average) *)
20 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
21 (* Number of functors : 8 ( 4 constant; 0-2 arity) *)
22 (* Number of variables : 19 ( 1 singleton) *)
23 (* Maximal term depth : 5 ( 2 average) *)
25 (* -------------------------------------------------------------------------- *)
26 (* ----Include Alternative Wajsberg algebra axioms *)
27 (* Inclusion of: Axioms/LCL002-0.ax *)
28 (* -------------------------------------------------------------------------- *)
29 (* File : LCL002-0 : TPTP v3.1.1. Released v1.0.0. *)
30 (* Domain : Logic Calculi (Wajsberg Algebras) *)
31 (* Axioms : Alternative Wajsberg algebra axioms *)
32 (* Version : [AB90] (equality) axioms. *)
34 (* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
35 (* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
36 (* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
37 (* Source : [Bon91] *)
40 (* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 0 RR) *)
41 (* Number of literals : 8 ( 8 equality) *)
42 (* Maximal clause size : 1 ( 1 average) *)
43 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
44 (* Number of functors : 5 ( 2 constant; 0-2 arity) *)
45 (* Number of variables : 10 ( 1 singleton) *)
46 (* Maximal term depth : 5 ( 2 average) *)
47 (* Comments : To be used in conjunction with the LAT003 alternative *)
48 (* Wajsberg algebra definitions. *)
49 (* -------------------------------------------------------------------------- *)
50 (* -------------------------------------------------------------------------- *)
51 (* -------------------------------------------------------------------------- *)
52 (* ----Include some Alternative Wajsberg algebra definitions *)
53 (* include('Axioms/LCL002-1.ax'). *)
54 (* ----Definition that and_star is AC and xor is C *)
55 (* ----Definition of false in terms of true *)
56 (* ----Include the definition of implies in terms of xor and and_star *)
57 theorem prove_wajsberg_axiom:
59 \forall and_star:\forall _:Univ.\forall _:Univ.Univ.
60 \forall falsehood:Univ.
61 \forall implies:\forall _:Univ.\forall _:Univ.Univ.
62 \forall not:\forall _:Univ.Univ.
65 \forall xor:\forall _:Univ.\forall _:Univ.Univ.
67 \forall H0:\forall X:Univ.\forall Y:Univ.eq Univ (implies X Y) (xor truth (and_star X (xor truth Y))).
68 \forall H1:eq Univ (not truth) falsehood.
69 \forall H2:\forall X:Univ.\forall Y:Univ.eq Univ (and_star X Y) (and_star Y X).
70 \forall H3:\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)).
71 \forall H4:\forall X:Univ.\forall Y:Univ.eq Univ (xor X Y) (xor Y X).
72 \forall H5:\forall X:Univ.\forall Y:Univ.eq Univ (and_star (xor (and_star (xor truth X) Y) truth) Y) (and_star (xor (and_star (xor truth Y) X) truth) X).
73 \forall H6:\forall X:Univ.\forall Y:Univ.eq Univ (xor X (xor truth Y)) (xor (xor X truth) Y).
74 \forall H7:\forall X:Univ.eq Univ (and_star (xor truth X) X) falsehood.
75 \forall H8:\forall X:Univ.eq Univ (and_star X falsehood) falsehood.
76 \forall H9:\forall X:Univ.eq Univ (and_star X truth) X.
77 \forall H10:\forall X:Univ.eq Univ (xor X X) falsehood.
78 \forall H11:\forall X:Univ.eq Univ (xor X falsehood) X.
79 \forall H12:\forall X:Univ.eq Univ (not X) (xor X truth).eq Univ (implies (implies (not x) (not y)) (implies y x)) truth
82 autobatch paramodulation timeout=100;
86 (* -------------------------------------------------------------------------- *)