%------------------------------------------------------------------------------
% File     : Twee---2.4.2
% Problem  : GRP005-1 : TPTP v8.1.2. Released v1.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof

% Computer : n023.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 01:16:31 EDT 2023

% Result   : Unsatisfiable 0.13s 0.39s
% Output   : Proof 0.13s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12  % Problem  : GRP005-1 : TPTP v8.1.2. Released v1.0.0.
% 0.07/0.13  % Command  : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35  % Computer : n023.cluster.edu
% 0.13/0.35  % Model    : x86_64 x86_64
% 0.13/0.35  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35  % Memory   : 8042.1875MB
% 0.13/0.35  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35  % CPULimit : 300
% 0.13/0.35  % WCLimit  : 300
% 0.13/0.35  % DateTime : Mon Aug 28 21:26:40 EDT 2023
% 0.13/0.35  % CPUTime  : 
% 0.13/0.39  Command-line arguments: --no-flatten-goal
% 0.13/0.39  
% 0.13/0.39  % SZS status Unsatisfiable
% 0.13/0.39  
% 0.13/0.39  % SZS output start Proof
% 0.13/0.39  Take the following subset of the input axioms:
% 0.13/0.39    fof(condition, axiom, ![X, Y, Z]: (~an_element(X) | (~an_element(Y) | (~product(X, inverse(Y), Z) | an_element(Z))))).
% 0.13/0.39    fof(element_of_set, axiom, an_element(a)).
% 0.13/0.39    fof(prove_identity_is_an_element, negated_conjecture, ~an_element(identity)).
% 0.13/0.39    fof(right_inverse, axiom, ![X2]: product(X2, inverse(X2), identity)).
% 0.13/0.39  
% 0.13/0.39  Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.13/0.39  http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.13/0.39  We repeatedly replace C & s=t => u=v by the two clauses:
% 0.13/0.39    fresh(y, y, x1...xn) = u
% 0.13/0.39    C => fresh(s, t, x1...xn) = v
% 0.13/0.39  where fresh is a fresh function symbol and x1..xn are the free
% 0.13/0.39  variables of u and v.
% 0.13/0.39  A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.13/0.39  input problem has no model of domain size 1).
% 0.13/0.39  
% 0.13/0.39  The encoding turns the above axioms into the following unit equations and goals:
% 0.13/0.39  
% 0.13/0.39  Axiom 1 (element_of_set): an_element(a) = true.
% 0.13/0.39  Axiom 2 (condition): fresh(X, X, Y) = true.
% 0.13/0.39  Axiom 3 (right_inverse): product(X, inverse(X), identity) = true.
% 0.13/0.39  Axiom 4 (condition): fresh9(X, X, Y, Z, W) = an_element(W).
% 0.13/0.39  Axiom 5 (condition): fresh8(X, X, Y, Z, W) = fresh9(an_element(Y), true, Y, Z, W).
% 0.13/0.39  Axiom 6 (condition): fresh8(an_element(X), true, Y, X, Z) = fresh(product(Y, inverse(X), Z), true, Z).
% 0.13/0.39  
% 0.13/0.39  Goal 1 (prove_identity_is_an_element): an_element(identity) = true.
% 0.13/0.39  Proof:
% 0.13/0.39    an_element(identity)
% 0.13/0.39  = { by axiom 4 (condition) R->L }
% 0.13/0.39    fresh9(true, true, a, a, identity)
% 0.13/0.39  = { by axiom 1 (element_of_set) R->L }
% 0.13/0.39    fresh9(an_element(a), true, a, a, identity)
% 0.13/0.39  = { by axiom 5 (condition) R->L }
% 0.13/0.39    fresh8(true, true, a, a, identity)
% 0.13/0.39  = { by axiom 1 (element_of_set) R->L }
% 0.13/0.39    fresh8(an_element(a), true, a, a, identity)
% 0.13/0.39  = { by axiom 6 (condition) }
% 0.13/0.39    fresh(product(a, inverse(a), identity), true, identity)
% 0.13/0.39  = { by axiom 3 (right_inverse) }
% 0.13/0.39    fresh(true, true, identity)
% 0.13/0.39  = { by axiom 2 (condition) }
% 0.13/0.39    true
% 0.13/0.39  % SZS output end Proof
% 0.13/0.39  
% 0.13/0.39  RESULT: Unsatisfiable (the axioms are contradictory).
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