#--------------------------------------------------------------------------
# File     : GRP080-1 : TPTP v2.1.0. Released v1.0.0.
# Domain   : Group Theory
# Problem  : Single axiom for group theory, in double division and identity
# Version  : [McC93] (equality) axioms.
# English  : This is a single axiom for group theory, in terms of double 
#            division and identity.

# Refs     : [McC93] McCune (1993), Single Axioms for Groups and Abelian Gr
# Source   : [McC93]
# Names    : Axiom 3.5.5 [McC93]

# Status   : unsatisfiable
# Rating   : 0.33 v2.1.0, 0.67 v2.0.0
# Syntax   : Number of clauses    :    4 (   0 non-Horn;   3 unit;   1 RR)
#            Number of literals   :    6 (   6 equality)
#            Maximal clause size  :    3 (   1 average)
#            Number of predicates :    1 (   0 propositional; 2-2 arity)
#            Number of functors   :   11 (   8 constant; 0-2 arity)
#            Number of variables  :    6 (   0 singleton)
#            Maximal term depth   :    5 (   2 average)

# Comments : 
#          : tptp2X -f setheo:sign -t rm_equality:rstfp GRP080-1.p 
#--------------------------------------------------------------------------
# single_axiom, axiom.
equal(double_divide(double_divide(identity, double_divide(X, double_divide(Y, identity))), double_divide(double_divide(Y, double_divide(Z, X)), identity)), Z) <- .

# double_division, axiom.
equal(double_divide(X, Y), multiply(inverse(X), inverse(Y))) <- .

# identity, axiom.
equal(identity, double_divide(X, inverse(X))) <- .

# prove_these_axioms, conjecture.
 <- equal(multiply(inverse(a1), a1), multiply(inverse(b1), b1)),
    equal(multiply(multiply(inverse(b2), b2), a2), a2),
    equal(multiply(multiply(a3, b3), c3), multiply(a3, multiply(b3, c3))).

#--------------------------------------------------------------------------