2 include "logic/equality.ma".
3 (* Inclusion of: GRP001-2.p *)
4 (* -------------------------------------------------------------------------- *)
5 (* File : GRP001-2 : TPTP v3.1.1. Released v1.0.0. *)
6 (* Domain : Group Theory *)
7 (* Problem : X^2 = identity => commutativity *)
8 (* Version : [MOW76] (equality) axioms : Augmented. *)
9 (* English : If the square of every element is the identity, the system *)
11 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
12 (* : [LO85] Lusk & Overbeek (1985), Reasoning about Equality *)
13 (* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
15 (* Names : GP1 [MOW76] *)
16 (* : Problem 1 [LO85] *)
18 (* : xsquared.ver2.in [ANL] *)
19 (* Status : Unsatisfiable *)
20 (* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
21 (* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 2 RR) *)
22 (* Number of atoms : 8 ( 8 equality) *)
23 (* Maximal clause size : 1 ( 1 average) *)
24 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
25 (* Number of functors : 6 ( 4 constant; 0-2 arity) *)
26 (* Number of variables : 8 ( 0 singleton) *)
27 (* Maximal term depth : 3 ( 2 average) *)
29 (* -------------------------------------------------------------------------- *)
30 (* ----Include equality group theory axioms *)
31 (* Inclusion of: Axioms/GRP004-0.ax *)
32 (* -------------------------------------------------------------------------- *)
33 (* File : GRP004-0 : TPTP v3.1.1. Released v1.0.0. *)
34 (* Domain : Group Theory *)
35 (* Axioms : Group theory (equality) axioms *)
36 (* Version : [MOW76] (equality) axioms : *)
37 (* Reduced > Complete. *)
39 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
40 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
44 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
45 (* Number of literals : 3 ( 3 equality) *)
46 (* Maximal clause size : 1 ( 1 average) *)
47 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
48 (* Number of functors : 3 ( 1 constant; 0-2 arity) *)
49 (* Number of variables : 5 ( 0 singleton) *)
50 (* Maximal term depth : 3 ( 2 average) *)
51 (* Comments : [MOW76] also contains redundant right_identity and *)
52 (* right_inverse axioms. *)
53 (* : These axioms are also used in [Wos88] p.186, also with *)
54 (* right_identity and right_inverse. *)
55 (* -------------------------------------------------------------------------- *)
56 (* ----For any x and y in the group x*y is also in the group. No clause *)
57 (* ----is needed here since this is an instance of reflexivity *)
58 (* ----There exists an identity element *)
59 (* ----For any x in the group, there exists an element y such that x*y = y*x *)
61 (* ----The operation '*' is associative *)
62 (* -------------------------------------------------------------------------- *)
63 (* -------------------------------------------------------------------------- *)
64 (* ----Redundant two axioms *)
65 theorem prove_b_times_a_is_c:
70 \forall identity:Univ.
71 \forall inverse:\forall _:Univ.Univ.
72 \forall multiply:\forall _:Univ.\forall _:Univ.Univ.
73 \forall H0:eq Univ (multiply a b) c.
74 \forall H1:\forall X:Univ.eq Univ (multiply X X) identity.
75 \forall H2:\forall X:Univ.eq Univ (multiply X (inverse X)) identity.
76 \forall H3:\forall X:Univ.eq Univ (multiply X identity) X.
77 \forall H4:\forall X:Univ.\forall Y:Univ.\forall Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
78 \forall H5:\forall X:Univ.eq Univ (multiply (inverse X) X) identity.
79 \forall H6:\forall X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply b a) c
82 autobatch paramodulation timeout=100;
86 (* -------------------------------------------------------------------------- *)