1 include "logic/equality.ma".
3 (* Inclusion of: GRP001-2.p *)
5 (* -------------------------------------------------------------------------- *)
7 (* File : GRP001-2 : TPTP v3.7.0. Released v1.0.0. *)
9 (* Domain : Group Theory *)
11 (* Problem : X^2 = identity => commutativity *)
13 (* Version : [MOW76] (equality) axioms : Augmented. *)
15 (* English : If the square of every element is the identity, the system *)
19 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
21 (* : [LO85] Lusk & Overbeek (1985), Reasoning about Equality *)
23 (* : [LW92] Lusk & Wos (1992), Benchmark Problems in Which Equalit *)
27 (* Names : GP1 [MOW76] *)
29 (* : Problem 1 [LO85] *)
33 (* : xsquared.ver2.in [ANL] *)
35 (* Status : Unsatisfiable *)
37 (* Rating : 0.00 v2.2.1, 0.22 v2.2.0, 0.29 v2.1.0, 0.25 v2.0.0 *)
39 (* Syntax : Number of clauses : 8 ( 0 non-Horn; 8 unit; 2 RR) *)
41 (* Number of atoms : 8 ( 8 equality) *)
43 (* Maximal clause size : 1 ( 1 average) *)
45 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
47 (* Number of functors : 6 ( 4 constant; 0-2 arity) *)
49 (* Number of variables : 8 ( 0 singleton) *)
51 (* Maximal term depth : 3 ( 2 average) *)
55 (* -------------------------------------------------------------------------- *)
57 (* ----Include equality group theory axioms *)
59 (* Inclusion of: Axioms/GRP004-0.ax *)
61 (* -------------------------------------------------------------------------- *)
63 (* File : GRP004-0 : TPTP v3.7.0. Released v1.0.0. *)
65 (* Domain : Group Theory *)
67 (* Axioms : Group theory (equality) axioms *)
69 (* Version : [MOW76] (equality) axioms : *)
71 (* Reduced > Complete. *)
75 (* Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a *)
77 (* : [Wos88] Wos (1988), Automated Reasoning - 33 Basic Research Pr *)
85 (* Syntax : Number of clauses : 3 ( 0 non-Horn; 3 unit; 0 RR) *)
87 (* Number of atoms : 3 ( 3 equality) *)
89 (* Maximal clause size : 1 ( 1 average) *)
91 (* Number of predicates : 1 ( 0 propositional; 2-2 arity) *)
93 (* Number of functors : 3 ( 1 constant; 0-2 arity) *)
95 (* Number of variables : 5 ( 0 singleton) *)
97 (* Maximal term depth : 3 ( 2 average) *)
99 (* Comments : [MOW76] also contains redundant right_identity and *)
101 (* right_inverse axioms. *)
103 (* : These axioms are also used in [Wos88] p.186, also with *)
105 (* right_identity and right_inverse. *)
107 (* -------------------------------------------------------------------------- *)
109 (* ----For any x and y in the group x*y is also in the group. No clause *)
111 (* ----is needed here since this is an instance of reflexivity *)
113 (* ----There exists an identity element *)
115 (* ----For any x in the group, there exists an element y such that x*y = y*x *)
117 (* ----= identity. *)
119 (* ----The operation '*' is associative *)
121 (* -------------------------------------------------------------------------- *)
123 (* -------------------------------------------------------------------------- *)
125 (* ----Redundant two axioms *)
126 ntheorem prove_b_times_a_is_c:
127 (∀Univ:Type.∀X:Univ.∀Y:Univ.∀Z:Univ.
132 ∀inverse:∀_:Univ.Univ.
133 ∀multiply:∀_:Univ.∀_:Univ.Univ.
134 ∀H0:eq Univ (multiply a b) c.
135 ∀H1:∀X:Univ.eq Univ (multiply X X) identity.
136 ∀H2:∀X:Univ.eq Univ (multiply X (inverse X)) identity.
137 ∀H3:∀X:Univ.eq Univ (multiply X identity) X.
138 ∀H4:∀X:Univ.∀Y:Univ.∀Z:Univ.eq Univ (multiply (multiply X Y) Z) (multiply X (multiply Y Z)).
139 ∀H5:∀X:Univ.eq Univ (multiply (inverse X) X) identity.
140 ∀H6:∀X:Univ.eq Univ (multiply identity X) X.eq Univ (multiply b a) c)
159 nauto by H0,H1,H2,H3,H4,H5,H6 ##;
160 ntry (nassumption) ##;
163 (* -------------------------------------------------------------------------- *)