%------------------------------------------------------------------------------
% File     : Toma---0.4
% Problem  : GRP022-2 : TPTP v8.1.2. Released v1.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : toma --casc %s

% Computer : n012.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 300s
% DateTime : Thu Aug 31 01:13:36 EDT 2023

% Result   : Unsatisfiable 0.21s 0.64s
% Output   : CNFRefutation 0.21s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.14  % Problem    : GRP022-2 : TPTP v8.1.2. Released v1.0.0.
% 0.00/0.14  % Command    : toma --casc %s
% 0.15/0.36  % Computer : n012.cluster.edu
% 0.15/0.36  % Model    : x86_64 x86_64
% 0.15/0.36  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.36  % Memory   : 8042.1875MB
% 0.15/0.36  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.15/0.36  % CPULimit   : 300
% 0.15/0.36  % WCLimit    : 300
% 0.15/0.36  % DateTime   : Tue Aug 29 02:37:54 EDT 2023
% 0.15/0.36  % CPUTime    : 
% 0.21/0.64  % SZS status Unsatisfiable
% 0.21/0.64  % SZS output start Proof
% 0.21/0.64  original problem:
% 0.21/0.64  axioms:
% 0.21/0.64  multiply(identity(), X) = X
% 0.21/0.64  multiply(inverse(X), X) = identity()
% 0.21/0.64  multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z))
% 0.21/0.64  multiply(X, identity()) = X
% 0.21/0.64  multiply(X, inverse(X)) = identity()
% 0.21/0.64  goal:
% 0.21/0.64  inverse(inverse(a())) != a()
% 0.21/0.64  To show the unsatisfiability of the original goal,
% 0.21/0.64  it suffices to show that inverse(inverse(a())) = a() (skolemized goal) is valid under the axioms.
% 0.21/0.64  Here is an equational proof:
% 0.21/0.64  0: multiply(identity(), X0) = X0.
% 0.21/0.64  Proof: Axiom.
% 0.21/0.64  
% 0.21/0.64  2: multiply(multiply(X0, X1), X2) = multiply(X0, multiply(X1, X2)).
% 0.21/0.64  Proof: Axiom.
% 0.21/0.64  
% 0.21/0.64  3: multiply(X0, identity()) = X0.
% 0.21/0.64  Proof: Axiom.
% 0.21/0.64  
% 0.21/0.64  4: multiply(X0, inverse(X0)) = identity().
% 0.21/0.64  Proof: Axiom.
% 0.21/0.64  
% 0.21/0.64  7: multiply(X3, multiply(inverse(X3), X2)) = multiply(identity(), X2).
% 0.21/0.64  Proof: A critical pair between equations 2 and 4.
% 0.21/0.64  
% 0.21/0.64  9: multiply(X3, multiply(inverse(X3), X2)) = X2.
% 0.21/0.64  Proof: Rewrite equation 7,
% 0.21/0.64                 lhs with equations []
% 0.21/0.64                 rhs with equations [0].
% 0.21/0.64  
% 0.21/0.64  11: inverse(inverse(X3)) = multiply(X3, identity()).
% 0.21/0.64  Proof: A critical pair between equations 9 and 4.
% 0.21/0.64  
% 0.21/0.64  23: inverse(inverse(X3)) = X3.
% 0.21/0.64  Proof: Rewrite equation 11,
% 0.21/0.64                 lhs with equations []
% 0.21/0.64                 rhs with equations [3].
% 0.21/0.64  
% 0.21/0.64  24: inverse(inverse(a())) = a().
% 0.21/0.64  Proof: Rewrite lhs with equations [23]
% 0.21/0.64                 rhs with equations [].
% 0.21/0.64  
% 0.21/0.64  % SZS output end Proof
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