%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL008+2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 12:33:01 PM UTC 2026
% Result : Theorem 6.19s 1.25s
% Output : Proof 6.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : REL008+2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.35 % Computer : n011.cluster.edu
% 0.09/0.35 % Model : x86_64 x86_64
% 0.09/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.35 % Memory : 8046.5625MB
% 0.09/0.35 % OS : Linux 6.8.0-71-generic
% 0.09/0.35 % CPULimit : 300
% 0.09/0.35 % WCLimit : 300
% 0.09/0.35 % DateTime : Sun Sep 27 22:50:31 UTC 2026
% 0.09/0.35 % CPUTime :
% 0.09/0.35 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 6.19/1.25 Command-line arguments: --no-flatten-goal
% 6.19/1.25
% 6.19/1.25 % SZS status Theorem
% 6.19/1.25
% 6.19/1.29 % SZS output start Proof
% 6.19/1.29 Axiom 1 (def_zero): zero = meet(X, complement(X)).
% 6.19/1.29 Axiom 2 (converse_idempotence): converse(converse(X)) = X.
% 6.19/1.29 Axiom 3 (composition_identity): composition(X, one) = X.
% 6.19/1.29 Axiom 4 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 6.19/1.30 Axiom 5 (def_top): top = join(X, complement(X)).
% 6.19/1.30 Axiom 6 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 6.19/1.30 Axiom 7 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 6.19/1.30 Axiom 8 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 6.19/1.30 Axiom 9 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 6.19/1.30 Axiom 10 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 6.19/1.30 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 6.19/1.30 Axiom 12 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 6.19/1.30 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 6.19/1.30
% 6.19/1.30 Lemma 14: complement(top) = zero.
% 6.19/1.30 Proof:
% 6.19/1.30 complement(top)
% 6.19/1.30 = { by axiom 5 (def_top) }
% 6.19/1.30 complement(join(complement(X), complement(complement(X))))
% 6.19/1.30 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 6.19/1.30 meet(X, complement(X))
% 6.19/1.30 = { by axiom 1 (def_zero) R->L }
% 6.19/1.30 zero
% 6.19/1.30
% 6.19/1.30 Lemma 15: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 6.19/1.30 Proof:
% 6.19/1.30 join(meet(X, Y), complement(join(complement(X), Y)))
% 6.19/1.30 = { by axiom 6 (maddux4_definiton_of_meet) }
% 6.19/1.30 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 6.19/1.30 = { by axiom 12 (maddux3_a_kind_of_de_Morgan) R->L }
% 6.19/1.30 X
% 6.19/1.30
% 6.19/1.30 Lemma 16: join(meet(X, Y), meet(X, complement(Y))) = X.
% 6.19/1.30 Proof:
% 6.19/1.30 join(meet(X, Y), meet(X, complement(Y)))
% 6.19/1.30 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 6.19/1.30 join(meet(X, complement(Y)), meet(X, Y))
% 6.19/1.30 = { by axiom 6 (maddux4_definiton_of_meet) }
% 6.19/1.30 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 6.19/1.30 = { by lemma 15 }
% 6.19/1.30 X
% 6.19/1.30
% 6.19/1.30 Lemma 17: meet(Y, X) = meet(X, Y).
% 6.19/1.30 Proof:
% 6.19/1.30 meet(Y, X)
% 6.19/1.30 = { by axiom 6 (maddux4_definiton_of_meet) }
% 6.19/1.30 complement(join(complement(Y), complement(X)))
% 6.19/1.30 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 6.19/1.30 complement(join(complement(X), complement(Y)))
% 6.19/1.30 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 6.19/1.30 meet(X, Y)
% 6.19/1.30
% 6.19/1.30 Lemma 18: complement(join(zero, complement(X))) = meet(X, top).
% 6.19/1.30 Proof:
% 6.19/1.30 complement(join(zero, complement(X)))
% 6.19/1.30 = { by lemma 14 R->L }
% 6.19/1.30 complement(join(complement(top), complement(X)))
% 6.19/1.30 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 6.19/1.30 meet(top, X)
% 6.19/1.30 = { by lemma 17 R->L }
% 6.19/1.30 meet(X, top)
% 6.19/1.30
% 6.19/1.30 Lemma 19: composition(converse(one), X) = X.
% 6.19/1.30 Proof:
% 6.19/1.30 composition(converse(one), X)
% 6.19/1.30 = { by axiom 2 (converse_idempotence) R->L }
% 6.19/1.30 composition(converse(one), converse(converse(X)))
% 6.19/1.30 = { by axiom 7 (converse_multiplicativity) R->L }
% 6.19/1.30 converse(composition(converse(X), one))
% 6.19/1.30 = { by axiom 3 (composition_identity) }
% 6.19/1.30 converse(converse(X))
% 6.19/1.30 = { by axiom 2 (converse_idempotence) }
% 6.19/1.30 X
% 6.19/1.30
% 6.19/1.30 Lemma 20: join(complement(X), complement(X)) = complement(X).
% 6.19/1.30 Proof:
% 6.19/1.30 join(complement(X), complement(X))
% 6.19/1.30 = { by lemma 19 R->L }
% 6.19/1.30 join(complement(X), composition(converse(one), complement(X)))
% 6.19/1.30 = { by lemma 19 R->L }
% 6.19/1.30 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 6.19/1.30 = { by axiom 3 (composition_identity) R->L }
% 6.19/1.30 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 6.19/1.30 = { by axiom 8 (composition_associativity) R->L }
% 6.19/1.30 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 6.19/1.30 = { by lemma 19 }
% 6.19/1.30 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 6.19/1.30 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 6.19/1.30 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 6.19/1.30 = { by axiom 13 (converse_cancellativity) }
% 6.19/1.30 complement(X)
% 6.19/1.30
% 6.19/1.30 Lemma 21: meet(top, complement(X)) = complement(X).
% 6.19/1.30 Proof:
% 6.19/1.30 meet(top, complement(X))
% 6.19/1.30 = { by lemma 17 }
% 6.19/1.30 meet(complement(X), top)
% 6.19/1.30 = { by lemma 18 R->L }
% 6.19/1.30 complement(join(zero, complement(complement(X))))
% 6.19/1.30 = { by axiom 1 (def_zero) }
% 6.19/1.30 complement(join(meet(X, complement(X)), complement(complement(X))))
% 6.19/1.30 = { by lemma 20 R->L }
% 6.19/1.30 complement(join(meet(X, complement(X)), complement(join(complement(X), complement(X)))))
% 6.19/1.30 = { by lemma 15 }
% 6.19/1.30 complement(X)
% 6.19/1.30
% 6.19/1.30 Lemma 22: complement(zero) = top.
% 6.19/1.30 Proof:
% 6.19/1.30 complement(zero)
% 6.19/1.30 = { by lemma 16 R->L }
% 6.19/1.30 join(meet(complement(zero), top), meet(complement(zero), complement(top)))
% 6.19/1.30 = { by lemma 14 }
% 6.19/1.30 join(meet(complement(zero), top), meet(complement(zero), zero))
% 6.19/1.30 = { by lemma 17 R->L }
% 6.19/1.30 join(meet(complement(zero), top), meet(zero, complement(zero)))
% 6.19/1.30 = { by axiom 1 (def_zero) R->L }
% 6.19/1.30 join(meet(complement(zero), top), zero)
% 6.19/1.30 = { by axiom 4 (maddux1_join_commutativity) }
% 6.19/1.30 join(zero, meet(complement(zero), top))
% 6.19/1.30 = { by lemma 17 R->L }
% 6.19/1.30 join(zero, meet(top, complement(zero)))
% 6.19/1.30 = { by lemma 21 }
% 6.19/1.30 join(zero, complement(zero))
% 6.19/1.30 = { by axiom 5 (def_top) R->L }
% 6.19/1.30 top
% 6.19/1.30
% 6.19/1.30 Lemma 23: join(zero, meet(X, top)) = X.
% 6.19/1.30 Proof:
% 6.19/1.30 join(zero, meet(X, top))
% 6.19/1.30 = { by lemma 22 R->L }
% 6.19/1.30 join(zero, meet(X, complement(zero)))
% 6.19/1.30 = { by lemma 14 R->L }
% 6.19/1.30 join(complement(top), meet(X, complement(zero)))
% 6.19/1.30 = { by axiom 5 (def_top) }
% 6.19/1.31 join(complement(join(X, complement(X))), meet(X, complement(zero)))
% 6.19/1.31 = { by lemma 20 R->L }
% 6.19/1.31 join(complement(join(X, join(complement(X), complement(X)))), meet(X, complement(zero)))
% 6.19/1.31 = { by axiom 10 (maddux2_join_associativity) }
% 6.19/1.31 join(complement(join(join(X, complement(X)), complement(X))), meet(X, complement(zero)))
% 6.19/1.31 = { by axiom 5 (def_top) R->L }
% 6.19/1.31 join(complement(join(top, complement(X))), meet(X, complement(zero)))
% 6.19/1.31 = { by lemma 22 R->L }
% 6.19/1.31 join(complement(join(complement(zero), complement(X))), meet(X, complement(zero)))
% 6.19/1.31 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 6.19/1.31 join(meet(zero, X), meet(X, complement(zero)))
% 6.19/1.31 = { by lemma 17 R->L }
% 6.19/1.31 join(meet(X, zero), meet(X, complement(zero)))
% 6.19/1.31 = { by lemma 16 }
% 6.19/1.31 X
% 6.19/1.31
% 6.19/1.31 Lemma 24: meet(X, top) = X.
% 6.19/1.31 Proof:
% 6.19/1.31 meet(X, top)
% 6.19/1.31 = { by lemma 18 R->L }
% 6.19/1.31 complement(join(zero, complement(X)))
% 6.19/1.31 = { by lemma 23 R->L }
% 6.19/1.31 join(zero, meet(complement(join(zero, complement(X))), top))
% 6.19/1.31 = { by lemma 17 R->L }
% 6.19/1.31 join(zero, meet(top, complement(join(zero, complement(X)))))
% 6.19/1.31 = { by lemma 21 }
% 6.19/1.31 join(zero, complement(join(zero, complement(X))))
% 6.19/1.31 = { by lemma 18 }
% 6.19/1.31 join(zero, meet(X, top))
% 6.19/1.31 = { by lemma 23 }
% 6.19/1.31 X
% 6.19/1.31
% 6.19/1.31 Lemma 25: join(X, join(Y, X)) = join(X, Y).
% 6.19/1.31 Proof:
% 6.19/1.31 join(X, join(Y, X))
% 6.19/1.31 = { by lemma 24 R->L }
% 6.19/1.31 join(X, join(Y, meet(X, top)))
% 6.19/1.31 = { by lemma 18 R->L }
% 6.19/1.31 join(X, join(Y, complement(join(zero, complement(X)))))
% 6.19/1.31 = { by lemma 24 R->L }
% 6.19/1.31 join(meet(X, top), join(Y, complement(join(zero, complement(X)))))
% 6.19/1.31 = { by lemma 18 R->L }
% 6.19/1.31 join(complement(join(zero, complement(X))), join(Y, complement(join(zero, complement(X)))))
% 6.19/1.31 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 6.19/1.31 join(complement(join(zero, complement(X))), join(complement(join(zero, complement(X))), Y))
% 6.19/1.31 = { by axiom 10 (maddux2_join_associativity) }
% 6.19/1.31 join(join(complement(join(zero, complement(X))), complement(join(zero, complement(X)))), Y)
% 6.19/1.31 = { by lemma 20 }
% 6.19/1.31 join(complement(join(zero, complement(X))), Y)
% 6.19/1.31 = { by axiom 4 (maddux1_join_commutativity) }
% 6.19/1.31 join(Y, complement(join(zero, complement(X))))
% 6.19/1.31 = { by lemma 18 }
% 6.19/1.31 join(Y, meet(X, top))
% 6.19/1.31 = { by lemma 24 }
% 6.19/1.31 join(Y, X)
% 6.19/1.31 = { by axiom 4 (maddux1_join_commutativity) }
% 6.19/1.31 join(X, Y)
% 6.19/1.31
% 6.19/1.31 Lemma 26: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 6.19/1.31 Proof:
% 6.19/1.31 join(X, join(Y, Z))
% 6.19/1.31 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 6.19/1.31 join(join(Y, Z), X)
% 6.19/1.31 = { by axiom 10 (maddux2_join_associativity) R->L }
% 6.19/1.31 join(Y, join(Z, X))
% 6.19/1.31 = { by axiom 4 (maddux1_join_commutativity) }
% 6.19/1.31 join(Y, join(X, Z))
% 6.19/1.31
% 6.19/1.31 Lemma 27: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 6.19/1.31 Proof:
% 6.19/1.31 join(composition(X, Y), composition(X, Z))
% 6.19/1.31 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 6.19/1.31 join(composition(X, Z), composition(X, Y))
% 6.19/1.31 = { by axiom 2 (converse_idempotence) R->L }
% 6.19/1.31 join(composition(X, Z), composition(X, converse(converse(Y))))
% 6.19/1.31 = { by axiom 2 (converse_idempotence) R->L }
% 6.19/1.31 converse(converse(join(composition(X, Z), composition(X, converse(converse(Y))))))
% 6.19/1.31 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 6.19/1.31 converse(converse(join(composition(X, converse(converse(Y))), composition(X, Z))))
% 6.19/1.31 = { by axiom 9 (converse_additivity) }
% 6.19/1.31 converse(join(converse(composition(X, converse(converse(Y)))), converse(composition(X, Z))))
% 6.19/1.31 = { by axiom 7 (converse_multiplicativity) }
% 6.19/1.31 converse(join(composition(converse(converse(converse(Y))), converse(X)), converse(composition(X, Z))))
% 6.19/1.31 = { by axiom 2 (converse_idempotence) }
% 6.19/1.31 converse(join(composition(converse(Y), converse(X)), converse(composition(X, Z))))
% 6.19/1.31 = { by axiom 4 (maddux1_join_commutativity) }
% 6.19/1.31 converse(join(converse(composition(X, Z)), composition(converse(Y), converse(X))))
% 6.19/1.31 = { by axiom 7 (converse_multiplicativity) }
% 6.19/1.31 converse(join(composition(converse(Z), converse(X)), composition(converse(Y), converse(X))))
% 6.19/1.31 = { by axiom 11 (composition_distributivity) R->L }
% 6.19/1.31 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 6.19/1.31 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 6.19/1.31 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 6.19/1.31 = { by axiom 2 (converse_idempotence) R->L }
% 6.19/1.31 converse(composition(join(converse(converse(converse(Y))), converse(Z)), converse(X)))
% 6.19/1.31 = { by axiom 9 (converse_additivity) R->L }
% 6.19/1.31 converse(composition(converse(join(converse(converse(Y)), Z)), converse(X)))
% 6.19/1.31 = { by axiom 4 (maddux1_join_commutativity) }
% 6.19/1.31 converse(composition(converse(join(Z, converse(converse(Y)))), converse(X)))
% 6.19/1.31 = { by axiom 7 (converse_multiplicativity) R->L }
% 6.19/1.31 converse(converse(composition(X, join(Z, converse(converse(Y))))))
% 6.19/1.31 = { by axiom 2 (converse_idempotence) }
% 6.19/1.32 composition(X, join(Z, converse(converse(Y))))
% 6.19/1.32 = { by axiom 2 (converse_idempotence) }
% 6.19/1.32 composition(X, join(Z, Y))
% 6.19/1.32 = { by axiom 4 (maddux1_join_commutativity) }
% 6.19/1.32 composition(X, join(Y, Z))
% 6.19/1.32
% 6.19/1.32 Goal 1 (goals): tuple(join(join(composition(x0_2, join(x1_2, x2_2)), composition(x0_2, x1_2)), composition(x0_2, x2_2)), join(join(composition(x0, x1), composition(x0, x2)), composition(x0, join(x1, x2)))) = tuple(join(composition(x0_2, x1_2), composition(x0_2, x2_2)), composition(x0, join(x1, x2))).
% 6.19/1.32 Proof:
% 6.19/1.32 tuple(join(join(composition(x0_2, join(x1_2, x2_2)), composition(x0_2, x1_2)), composition(x0_2, x2_2)), join(join(composition(x0, x1), composition(x0, x2)), composition(x0, join(x1, x2))))
% 6.19/1.32 = { by axiom 4 (maddux1_join_commutativity) }
% 6.19/1.32 tuple(join(composition(x0_2, x2_2), join(composition(x0_2, join(x1_2, x2_2)), composition(x0_2, x1_2))), join(join(composition(x0, x1), composition(x0, x2)), composition(x0, join(x1, x2))))
% 6.19/1.32 = { by axiom 4 (maddux1_join_commutativity) }
% 6.19/1.32 tuple(join(composition(x0_2, x2_2), join(composition(x0_2, x1_2), composition(x0_2, join(x1_2, x2_2)))), join(join(composition(x0, x1), composition(x0, x2)), composition(x0, join(x1, x2))))
% 6.19/1.32 = { by axiom 4 (maddux1_join_commutativity) }
% 6.19/1.32 tuple(join(composition(x0_2, x2_2), join(composition(x0_2, x1_2), composition(x0_2, join(x1_2, x2_2)))), join(composition(x0, join(x1, x2)), join(composition(x0, x1), composition(x0, x2))))
% 6.19/1.32 = { by lemma 26 }
% 6.19/1.32 tuple(join(composition(x0_2, x1_2), join(composition(x0_2, x2_2), composition(x0_2, join(x1_2, x2_2)))), join(composition(x0, join(x1, x2)), join(composition(x0, x1), composition(x0, x2))))
% 6.19/1.32 = { by lemma 26 }
% 6.19/1.32 tuple(join(composition(x0_2, x1_2), join(composition(x0_2, x2_2), composition(x0_2, join(x1_2, x2_2)))), join(composition(x0, x1), join(composition(x0, join(x1, x2)), composition(x0, x2))))
% 6.19/1.32 = { by axiom 4 (maddux1_join_commutativity) }
% 6.19/1.32 tuple(join(composition(x0_2, x1_2), join(composition(x0_2, x2_2), composition(x0_2, join(x1_2, x2_2)))), join(composition(x0, x1), join(composition(x0, x2), composition(x0, join(x1, x2)))))
% 6.19/1.32 = { by lemma 27 }
% 6.19/1.32 tuple(join(composition(x0_2, x1_2), composition(x0_2, join(x2_2, join(x1_2, x2_2)))), join(composition(x0, x1), join(composition(x0, x2), composition(x0, join(x1, x2)))))
% 6.19/1.32 = { by lemma 25 }
% 6.19/1.32 tuple(join(composition(x0_2, x1_2), composition(x0_2, join(x2_2, x1_2))), join(composition(x0, x1), join(composition(x0, x2), composition(x0, join(x1, x2)))))
% 6.19/1.32 = { by lemma 27 }
% 6.19/1.32 tuple(composition(x0_2, join(x1_2, join(x2_2, x1_2))), join(composition(x0, x1), join(composition(x0, x2), composition(x0, join(x1, x2)))))
% 6.19/1.32 = { by lemma 25 }
% 6.19/1.32 tuple(composition(x0_2, join(x1_2, x2_2)), join(composition(x0, x1), join(composition(x0, x2), composition(x0, join(x1, x2)))))
% 6.19/1.32 = { by lemma 27 }
% 6.19/1.32 tuple(composition(x0_2, join(x1_2, x2_2)), join(composition(x0, x1), composition(x0, join(x2, join(x1, x2)))))
% 6.19/1.32 = { by lemma 25 }
% 6.19/1.32 tuple(composition(x0_2, join(x1_2, x2_2)), join(composition(x0, x1), composition(x0, join(x2, x1))))
% 6.19/1.32 = { by lemma 27 }
% 6.19/1.32 tuple(composition(x0_2, join(x1_2, x2_2)), composition(x0, join(x1, join(x2, x1))))
% 6.19/1.32 = { by lemma 25 }
% 6.19/1.32 tuple(composition(x0_2, join(x1_2, x2_2)), composition(x0, join(x1, x2)))
% 6.19/1.32 = { by lemma 27 R->L }
% 6.19/1.32 tuple(join(composition(x0_2, x1_2), composition(x0_2, x2_2)), composition(x0, join(x1, x2)))
% 6.19/1.32 % SZS output end Proof
% 6.19/1.32
% 6.19/1.32 RESULT: Theorem (the conjecture is true).
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