2 include "logic/equality.ma".
3 (* Inclusion of: LCL154-1.p *)
4 (* -------------------------------------------------------------------------- *)
5 (* File : LCL154-1 : TPTP v3.1.1. Released v1.0.0. *)
6 (* Domain : Logic Calculi (Wajsberg Algebra) *)
7 (* Problem : The 2nd alternative Wajsberg algebra axiom *)
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 2 [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 : 17 ( 0 non-Horn; 17 unit; 2 RR) *)
18 (* Number of atoms : 17 ( 17 equality) *)
19 (* Maximal clause size : 1 ( 1 average) *)
20 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
21 (* Number of functors : 9 ( 3 constant; 0-2 arity) *)
22 (* Number of variables : 33 ( 0 singleton) *)
23 (* Maximal term depth : 4 ( 2 average) *)
25 (* -------------------------------------------------------------------------- *)
26 (* ----Include Wajsberg algebra axioms *)
27 (* Inclusion of: Axioms/LCL001-0.ax *)
28 (* -------------------------------------------------------------------------- *)
29 (* File : LCL001-0 : TPTP v3.1.1. Released v1.0.0. *)
30 (* Domain : Logic Calculi (Wajsberg Algebras) *)
31 (* Axioms : Wajsberg algebra axioms *)
32 (* Version : [Bon91] (equality) axioms. *)
34 (* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
35 (* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
36 (* : [MW92] McCune & Wos (1992), Experiments in Automated Deductio *)
38 (* Names : MV Sentential Calculus [MW92] *)
40 (* Syntax : Number of clauses : 4 ( 0 non-Horn; 4 unit; 0 RR) *)
41 (* Number of literals : 4 ( 4 equality) *)
42 (* Maximal clause size : 1 ( 1 average) *)
43 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
44 (* Number of functors : 3 ( 1 constant; 0-2 arity) *)
45 (* Number of variables : 8 ( 0 singleton) *)
46 (* Maximal term depth : 4 ( 2 average) *)
48 (* -------------------------------------------------------------------------- *)
49 (* -------------------------------------------------------------------------- *)
50 (* ----Include Wajsberg algebra and and or definitions *)
51 (* Inclusion of: Axioms/LCL001-2.ax *)
52 (* -------------------------------------------------------------------------- *)
53 (* File : LCL001-2 : TPTP v3.1.1. Released v1.0.0. *)
54 (* Domain : Logic Calculi (Wajsberg Algebras) *)
55 (* Axioms : Wajsberg algebra AND and OR definitions *)
56 (* Version : [AB90] (equality) axioms. *)
58 (* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
59 (* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
60 (* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
61 (* Source : [Bon91] *)
64 (* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 0 RR) *)
65 (* Number of literals : 6 ( 6 equality) *)
66 (* Maximal clause size : 1 ( 1 average) *)
67 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
68 (* Number of functors : 4 ( 0 constant; 1-2 arity) *)
69 (* Number of variables : 14 ( 0 singleton) *)
70 (* Maximal term depth : 4 ( 3 average) *)
71 (* Comments : Requires LCL001-0.ax *)
72 (* -------------------------------------------------------------------------- *)
73 (* ----Definitions of or and and, which are AC *)
74 (* -------------------------------------------------------------------------- *)
75 (* ----Include Alternative Wajsberg algebra definitions *)
76 (* Inclusion of: Axioms/LCL002-1.ax *)
77 (* -------------------------------------------------------------------------- *)
78 (* File : LCL002-1 : TPTP v3.1.1. Released v1.0.0. *)
79 (* Domain : Logic Calculi (Wajsberg Algebras) *)
80 (* Axioms : Alternative Wajsberg algebra definitions *)
81 (* Version : [AB90] (equality) axioms. *)
83 (* Refs : [FRT84] Font et al. (1984), Wajsberg Algebras *)
84 (* : [AB90] Anantharaman & Bonacina (1990), An Application of the *)
85 (* : [Bon91] Bonacina (1991), Problems in Lukasiewicz Logic *)
86 (* Source : [Bon91] *)
89 (* Syntax : Number of clauses : 6 ( 0 non-Horn; 6 unit; 1 RR) *)
90 (* Number of literals : 6 ( 6 equality) *)
91 (* Maximal clause size : 1 ( 1 average) *)
92 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
93 (* Number of functors : 7 ( 2 constant; 0-2 arity) *)
94 (* Number of variables : 11 ( 0 singleton) *)
95 (* Maximal term depth : 4 ( 2 average) *)
96 (* Comments : Requires LCL001-0.ax LCL001-2.ax *)
97 (* -------------------------------------------------------------------------- *)
98 (* ----Definitions of and_star and xor, where and_star is AC and xor is C *)
99 (* ---I guess the next two can be derived from the AC of and *)
100 (* ----Definition of false in terms of truth *)
101 (* -------------------------------------------------------------------------- *)
102 (* -------------------------------------------------------------------------- *)
103 theorem prove_alternative_wajsberg_axiom:
105 \forall myand:\forall _:Univ.\forall _:Univ.Univ.
106 \forall and_star:\forall _:Univ.\forall _:Univ.Univ.
107 \forall falsehood:Univ.
108 \forall implies:\forall _:Univ.\forall _:Univ.Univ.
109 \forall not:\forall _:Univ.Univ.
110 \forall or:\forall _:Univ.\forall _:Univ.Univ.
113 \forall xor:\forall _:Univ.\forall _:Univ.Univ.
114 \forall H0:eq Univ (not truth) falsehood.
115 \forall H1:\forall X:Univ.\forall Y:Univ.eq Univ (and_star X Y) (and_star Y X).
116 \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)).
117 \forall H3:\forall X:Univ.\forall Y:Univ.eq Univ (and_star X Y) (not (or (not X) (not Y))).
118 \forall H4:\forall X:Univ.\forall Y:Univ.eq Univ (xor X Y) (xor Y X).
119 \forall H5:\forall X:Univ.\forall Y:Univ.eq Univ (xor X Y) (or (myand X (not Y)) (myand (not X) Y)).
120 \forall H6:\forall X:Univ.\forall Y:Univ.eq Univ (myand X Y) (myand Y X).
121 \forall H7:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (myand (myand X Y) Z) (myand X (myand Y Z)).
122 \forall H8:\forall X:Univ.\forall Y:Univ.eq Univ (myand X Y) (not (or (not X) (not Y))).
123 \forall H9:\forall X:Univ.\forall Y:Univ.eq Univ (or X Y) (or Y X).
124 \forall H10:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (or (or X Y) Z) (or X (or Y Z)).
125 \forall H11:\forall X:Univ.\forall Y:Univ.eq Univ (or X Y) (implies (not X) Y).
126 \forall H12:\forall X:Univ.\forall Y:Univ.eq Univ (implies (implies (not X) (not Y)) (implies Y X)) truth.
127 \forall H13:\forall X:Univ.\forall Y:Univ.eq Univ (implies (implies X Y) Y) (implies (implies Y X) X).
128 \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.
129 \forall H15:\forall X:Univ.eq Univ (implies truth X) X.eq Univ (xor x falsehood) x
132 autobatch paramodulation timeout=100;
136 (* -------------------------------------------------------------------------- *)