2 include "logic/equality.ma".
3 (* Inclusion of: RNG023-7.p *)
4 (* -------------------------------------------------------------------------- *)
5 (* File : RNG023-7 : TPTP v3.1.1. Released v1.0.0. *)
6 (* Domain : Ring Theory (Alternative) *)
7 (* Problem : Left alternative *)
8 (* Version : [Ste87] (equality) axioms : Augmented. *)
9 (* Theorem formulation : In terms of associators *)
11 (* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
12 (* : [Ste92] Stevens (1992), Unpublished Note *)
15 (* Status : Unsatisfiable *)
16 (* Rating : 0.00 v2.2.1, 0.11 v2.2.0, 0.00 v2.1.0, 0.13 v2.0.0 *)
17 (* Syntax : Number of clauses : 23 ( 0 non-Horn; 23 unit; 1 RR) *)
18 (* Number of atoms : 23 ( 23 equality) *)
19 (* Maximal clause size : 1 ( 1 average) *)
20 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
21 (* Number of functors : 8 ( 3 constant; 0-3 arity) *)
22 (* Number of variables : 45 ( 2 singleton) *)
23 (* Maximal term depth : 5 ( 3 average) *)
25 (* -------------------------------------------------------------------------- *)
26 (* ----Include nonassociative ring axioms *)
27 (* Inclusion of: Axioms/RNG003-0.ax *)
28 (* -------------------------------------------------------------------------- *)
29 (* File : RNG003-0 : TPTP v3.1.1. Released v1.0.0. *)
30 (* Domain : Ring Theory (Alternative) *)
31 (* Axioms : Alternative ring theory (equality) axioms *)
32 (* Version : [Ste87] (equality) axioms. *)
34 (* Refs : [Ste87] Stevens (1987), Some Experiments in Nonassociative Rin *)
35 (* Source : [Ste87] *)
38 (* Syntax : Number of clauses : 15 ( 0 non-Horn; 15 unit; 0 RR) *)
39 (* Number of literals : 15 ( 15 equality) *)
40 (* Maximal clause size : 1 ( 1 average) *)
41 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
42 (* Number of functors : 6 ( 1 constant; 0-3 arity) *)
43 (* Number of variables : 27 ( 2 singleton) *)
44 (* Maximal term depth : 5 ( 2 average) *)
46 (* -------------------------------------------------------------------------- *)
47 (* ----There exists an additive identity element *)
48 (* ----Multiplicative zero *)
49 (* ----Existence of left additive additive_inverse *)
50 (* ----Inverse of additive_inverse of X is X *)
51 (* ----Distributive property of product over sum *)
52 (* ----Commutativity for addition *)
53 (* ----Associativity for addition *)
54 (* ----Right alternative law *)
55 (* ----Left alternative law *)
58 (* -------------------------------------------------------------------------- *)
59 (* -------------------------------------------------------------------------- *)
60 (* ----The next 7 clause are extra lemmas which Stevens found useful *)
61 theorem prove_left_alternative:
63 \forall add:\forall _:Univ.\forall _:Univ.Univ.
64 \forall additive_identity:Univ.
65 \forall additive_inverse:\forall _:Univ.Univ.
66 \forall associator:\forall _:Univ.\forall _:Univ.\forall _:Univ.Univ.
67 \forall commutator:\forall _:Univ.\forall _:Univ.Univ.
68 \forall multiply:\forall _:Univ.\forall _:Univ.Univ.
71 \forall H0:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (add X Y) (additive_inverse Z)) (add (additive_inverse (multiply X Z)) (additive_inverse (multiply Y Z))).
72 \forall H1:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (additive_inverse X) (add Y Z)) (add (additive_inverse (multiply X Y)) (additive_inverse (multiply X Z))).
73 \forall H2:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (add X (additive_inverse Y)) Z) (add (multiply X Z) (additive_inverse (multiply Y Z))).
74 \forall H3:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply X (add Y (additive_inverse Z))) (add (multiply X Y) (additive_inverse (multiply X Z))).
75 \forall H4:\forall X:Univ.\forall Y:Univ.eq Univ (multiply X (additive_inverse Y)) (additive_inverse (multiply X Y)).
76 \forall H5:\forall X:Univ.\forall Y:Univ.eq Univ (multiply (additive_inverse X) Y) (additive_inverse (multiply X Y)).
77 \forall H6:\forall X:Univ.\forall Y:Univ.eq Univ (multiply (additive_inverse X) (additive_inverse Y)) (multiply X Y).
78 \forall H7:\forall X:Univ.\forall Y:Univ.eq Univ (commutator X Y) (add (multiply Y X) (additive_inverse (multiply X Y))).
79 \forall H8:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (associator X Y Z) (add (multiply (multiply X Y) Z) (additive_inverse (multiply X (multiply Y Z)))).
80 \forall H9:\forall X:Univ.\forall Y:Univ.eq Univ (multiply (multiply X X) Y) (multiply X (multiply X Y)).
81 \forall H10:\forall X:Univ.\forall Y:Univ.eq Univ (multiply (multiply X Y) Y) (multiply X (multiply Y Y)).
82 \forall H11:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (add X (add Y Z)) (add (add X Y) Z).
83 \forall H12:\forall X:Univ.\forall Y:Univ.eq Univ (add X Y) (add Y X).
84 \forall H13:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (add X Y) Z) (add (multiply X Z) (multiply Y Z)).
85 \forall H14:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply X (add Y Z)) (add (multiply X Y) (multiply X Z)).
86 \forall H15:\forall X:Univ.eq Univ (additive_inverse (additive_inverse X)) X.
87 \forall H16:\forall X:Univ.eq Univ (add X (additive_inverse X)) additive_identity.
88 \forall H17:\forall X:Univ.eq Univ (add (additive_inverse X) X) additive_identity.
89 \forall H18:\forall X:Univ.eq Univ (multiply X additive_identity) additive_identity.
90 \forall H19:\forall X:Univ.eq Univ (multiply additive_identity X) additive_identity.
91 \forall H20:\forall X:Univ.eq Univ (add X additive_identity) X.
92 \forall H21:\forall X:Univ.eq Univ (add additive_identity X) X.eq Univ (associator x x y) additive_identity
95 autobatch paramodulation timeout=100;
99 (* -------------------------------------------------------------------------- *)