;--------------------------------------------------------------------------
; File     : GRP001-1 : TPTP v2.2.0. Released v1.0.0.
; Domain   : Group Theory
; Problem  : X^2 = identity => commutativity
; Version  : [MOW76] axioms.
; English  : If the square of every element is the identity, the system 
;            is commutative.

; Refs     : [Rob63] Robinson (1963), Theorem Proving on the Computer
;          : [Wos65] Wos (1965), Unpublished Note
;          : [MOW76] McCharen et al. (1976), Problems and Experiments for a
;          : [WM76]  Wilson & Minker (1976), Resolution, Refinements, and S
;          : [Ove90] Overbeek (1990), ATP competition announced at CADE-10
;          : [Ove93] Overbeek (1993), The CADE-11 Competitions: A Personal 
;          : [LM93]  Lusk & McCune (1993), Uniform Strategies: The CADE-11 
; Source   : [MOW76]
; Names    : - [Rob63]
;          : wos10 [WM76]
;          : G1 [MOW76]
;          : CADE-11 Competition 1 [Ove90]
;          : THEOREM 1 [LM93]
;          : xsquared.ver1.in [ANL]

; Status   : unsatisfiable
; Rating   : 0.00 v2.0.0
; Syntax   : Number of clauses    :   11 (   0 non-Horn;   8 unit;   5 RR)
;            Number of literals   :   19 (   1 equality)
;            Maximal clause size  :    4 (   1 average)
;            Number of predicates :    2 (   0 propositional; 2-3 arity)
;            Number of functors   :    6 (   4 constant; 0-2 arity)
;            Number of variables  :   23 (   0 singleton)
;            Maximal term depth   :    2 (   1 average)

; Comments : 
;          : tptp2X -f kif -t rm_equality:rstfp GRP001-1.p 
;--------------------------------------------------------------------------
; left_identity, axiom.
(or (product identity ?A ?A))

; right_identity, axiom.
(or (product ?A identity ?A))

; left_inverse, axiom.
(or (product (inverse ?A) ?A identity))

; right_inverse, axiom.
(or (product ?A (inverse ?A) identity))

; total_function1, axiom.
(or (product ?A ?B (multiply ?A ?B)))

; total_function2, axiom.
(or (not (product ?A ?B ?C))
    (not (product ?A ?B ?D))
    (= ?C ?D))

; associativity1, axiom.
(or (not (product ?A ?B ?C))
    (not (product ?B ?D ?E))
    (not (product ?C ?D ?F))
    (product ?A ?E ?F))

; associativity2, axiom.
(or (not (product ?A ?B ?C))
    (not (product ?B ?D ?E))
    (not (product ?A ?E ?F))
    (product ?C ?D ?F))

; square_element, hypothesis.
(or (product ?A ?A identity))

; a_times_b_is_c, hypothesis.
(or (product a b c))

; prove_b_times_a_is_c, conjecture.
(or (not (product b a c)))

;--------------------------------------------------------------------------