TSTP Solution File: REL042+2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL042+2 : TPTP v8.1.2. Released v4.0.0.
% 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 : n004.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 13:44:28 EDT 2023
% Result : Theorem 26.08s 3.72s
% Output : Proof 26.89s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL042+2 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n004.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 20:32:37 EDT 2023
% 0.13/0.34 % CPUTime :
% 26.08/3.72 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 26.08/3.72
% 26.08/3.72 % SZS status Theorem
% 26.08/3.72
% 26.66/3.76 % SZS output start Proof
% 26.66/3.76 Axiom 1 (composition_identity): composition(X, one) = X.
% 26.66/3.76 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 26.66/3.76 Axiom 3 (converse_idempotence): converse(converse(X)) = X.
% 26.66/3.76 Axiom 4 (def_top): top = join(X, complement(X)).
% 26.66/3.76 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 26.66/3.76 Axiom 6 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 26.66/3.76 Axiom 7 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 26.66/3.76 Axiom 8 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 26.66/3.76 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 26.66/3.76 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 26.66/3.76 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 26.66/3.76 Axiom 12 (goals): meet(composition(x0, X), composition(x0, complement(X))) = zero.
% 26.66/3.76 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 26.66/3.76 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 26.66/3.76 Axiom 15 (modular_law_1): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
% 26.66/3.76
% 26.66/3.76 Lemma 16: complement(top) = zero.
% 26.66/3.76 Proof:
% 26.66/3.76 complement(top)
% 26.66/3.76 = { by axiom 4 (def_top) }
% 26.66/3.76 complement(join(complement(X), complement(complement(X))))
% 26.66/3.76 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 26.66/3.76 meet(X, complement(X))
% 26.66/3.76 = { by axiom 5 (def_zero) R->L }
% 26.66/3.76 zero
% 26.66/3.76
% 26.66/3.76 Lemma 17: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 26.66/3.76 Proof:
% 26.66/3.76 converse(composition(converse(X), Y))
% 26.66/3.76 = { by axiom 6 (converse_multiplicativity) }
% 26.66/3.76 composition(converse(Y), converse(converse(X)))
% 26.66/3.76 = { by axiom 3 (converse_idempotence) }
% 26.66/3.76 composition(converse(Y), X)
% 26.66/3.76
% 26.66/3.76 Lemma 18: composition(converse(one), X) = X.
% 26.66/3.76 Proof:
% 26.66/3.76 composition(converse(one), X)
% 26.66/3.76 = { by lemma 17 R->L }
% 26.66/3.76 converse(composition(converse(X), one))
% 26.66/3.76 = { by axiom 1 (composition_identity) }
% 26.66/3.76 converse(converse(X))
% 26.66/3.76 = { by axiom 3 (converse_idempotence) }
% 26.66/3.76 X
% 26.66/3.76
% 26.66/3.76 Lemma 19: converse(one) = one.
% 26.66/3.76 Proof:
% 26.66/3.76 converse(one)
% 26.66/3.76 = { by axiom 1 (composition_identity) R->L }
% 26.66/3.76 composition(converse(one), one)
% 26.66/3.76 = { by lemma 18 }
% 26.66/3.76 one
% 26.66/3.76
% 26.66/3.76 Lemma 20: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 26.66/3.76 Proof:
% 26.66/3.76 join(meet(X, Y), complement(join(complement(X), Y)))
% 26.66/3.76 = { by axiom 10 (maddux4_definiton_of_meet) }
% 26.66/3.76 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 26.66/3.76 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 26.66/3.76 X
% 26.66/3.76
% 26.66/3.76 Lemma 21: meet(Y, X) = meet(X, Y).
% 26.66/3.76 Proof:
% 26.66/3.76 meet(Y, X)
% 26.66/3.76 = { by axiom 10 (maddux4_definiton_of_meet) }
% 26.66/3.76 complement(join(complement(Y), complement(X)))
% 26.66/3.76 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 26.66/3.76 complement(join(complement(X), complement(Y)))
% 26.66/3.76 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 26.66/3.76 meet(X, Y)
% 26.66/3.76
% 26.66/3.76 Lemma 22: complement(complement(X)) = X.
% 26.66/3.76 Proof:
% 26.66/3.76 complement(complement(X))
% 26.66/3.76 = { by lemma 20 R->L }
% 26.66/3.76 join(meet(complement(complement(X)), complement(X)), complement(join(complement(complement(complement(X))), complement(X))))
% 26.66/3.76 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 26.66/3.76 join(meet(complement(complement(X)), complement(X)), meet(complement(complement(X)), X))
% 26.66/3.76 = { by axiom 2 (maddux1_join_commutativity) }
% 26.66/3.76 join(meet(complement(complement(X)), X), meet(complement(complement(X)), complement(X)))
% 26.66/3.76 = { by lemma 21 R->L }
% 26.66/3.76 join(meet(complement(complement(X)), X), meet(complement(X), complement(complement(X))))
% 26.66/3.76 = { by axiom 5 (def_zero) R->L }
% 26.66/3.76 join(meet(complement(complement(X)), X), zero)
% 26.66/3.76 = { by axiom 2 (maddux1_join_commutativity) }
% 26.66/3.76 join(zero, meet(complement(complement(X)), X))
% 26.66/3.76 = { by lemma 21 R->L }
% 26.66/3.76 join(zero, meet(X, complement(complement(X))))
% 26.66/3.77 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 26.66/3.77 join(meet(X, complement(complement(X))), zero)
% 26.66/3.77 = { by lemma 16 R->L }
% 26.66/3.77 join(meet(X, complement(complement(X))), complement(top))
% 26.66/3.77 = { by axiom 4 (def_top) }
% 26.66/3.77 join(meet(X, complement(complement(X))), complement(join(complement(X), complement(complement(X)))))
% 26.66/3.77 = { by lemma 20 }
% 26.66/3.77 X
% 26.66/3.77
% 26.66/3.77 Lemma 23: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 26.66/3.77 Proof:
% 26.66/3.77 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 26.66/3.77 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 26.66/3.77 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 26.66/3.77 = { by axiom 13 (converse_cancellativity) }
% 26.66/3.77 complement(X)
% 26.66/3.77
% 26.66/3.77 Lemma 24: join(X, X) = X.
% 26.66/3.77 Proof:
% 26.66/3.77 join(X, X)
% 26.66/3.77 = { by lemma 22 R->L }
% 26.66/3.77 join(X, complement(complement(X)))
% 26.66/3.77 = { by lemma 22 R->L }
% 26.66/3.77 join(complement(complement(X)), complement(complement(X)))
% 26.66/3.77 = { by lemma 18 R->L }
% 26.66/3.77 join(complement(complement(X)), complement(composition(converse(one), complement(X))))
% 26.66/3.77 = { by axiom 1 (composition_identity) R->L }
% 26.66/3.77 join(complement(complement(X)), complement(composition(composition(converse(one), one), complement(X))))
% 26.66/3.77 = { by axiom 7 (composition_associativity) R->L }
% 26.66/3.77 join(complement(complement(X)), complement(composition(converse(one), composition(one, complement(X)))))
% 26.66/3.77 = { by lemma 18 }
% 26.66/3.77 join(complement(complement(X)), complement(composition(one, complement(X))))
% 26.66/3.77 = { by axiom 3 (converse_idempotence) R->L }
% 26.66/3.77 join(complement(complement(X)), complement(composition(converse(converse(one)), complement(X))))
% 26.66/3.77 = { by lemma 18 R->L }
% 26.66/3.77 join(complement(complement(X)), composition(converse(one), complement(composition(converse(converse(one)), complement(X)))))
% 26.66/3.77 = { by axiom 3 (converse_idempotence) R->L }
% 26.66/3.77 join(complement(complement(X)), composition(converse(converse(converse(one))), complement(composition(converse(converse(one)), complement(X)))))
% 26.66/3.77 = { by lemma 23 }
% 26.66/3.77 complement(complement(X))
% 26.66/3.77 = { by lemma 22 }
% 26.66/3.77 X
% 26.66/3.77
% 26.66/3.77 Lemma 25: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 26.66/3.77 Proof:
% 26.66/3.77 join(X, composition(Y, X))
% 26.66/3.77 = { by lemma 18 R->L }
% 26.66/3.77 join(composition(converse(one), X), composition(Y, X))
% 26.66/3.77 = { by axiom 11 (composition_distributivity) R->L }
% 26.66/3.77 composition(join(converse(one), Y), X)
% 26.66/3.77 = { by lemma 19 }
% 26.66/3.77 composition(join(one, Y), X)
% 26.66/3.77 = { by axiom 2 (maddux1_join_commutativity) }
% 26.66/3.77 composition(join(Y, one), X)
% 26.66/3.77
% 26.66/3.77 Lemma 26: join(Y, join(Z, X)) = join(X, join(Y, Z)).
% 26.66/3.77 Proof:
% 26.66/3.77 join(Y, join(Z, X))
% 26.66/3.77 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 26.66/3.77 join(join(Z, X), Y)
% 26.66/3.77 = { by axiom 2 (maddux1_join_commutativity) }
% 26.66/3.77 join(join(X, Z), Y)
% 26.66/3.77 = { by axiom 9 (maddux2_join_associativity) R->L }
% 26.66/3.77 join(X, join(Z, Y))
% 26.66/3.77 = { by axiom 2 (maddux1_join_commutativity) }
% 26.66/3.77 join(X, join(Y, Z))
% 26.66/3.77
% 26.66/3.77 Lemma 27: join(complement(one), composition(converse(X), complement(X))) = complement(one).
% 26.66/3.77 Proof:
% 26.66/3.77 join(complement(one), composition(converse(X), complement(X)))
% 26.66/3.77 = { by axiom 1 (composition_identity) R->L }
% 26.66/3.77 join(complement(one), composition(converse(X), complement(composition(X, one))))
% 26.66/3.77 = { by lemma 23 }
% 26.89/3.77 complement(one)
% 26.89/3.77
% 26.89/3.77 Goal 1 (goals_1): join(composition(converse(x0), x0), one) = one.
% 26.89/3.77 Proof:
% 26.89/3.77 join(composition(converse(x0), x0), one)
% 26.89/3.77 = { by lemma 24 R->L }
% 26.89/3.77 join(join(composition(converse(x0), x0), one), join(composition(converse(x0), x0), one))
% 26.89/3.77 = { by lemma 26 }
% 26.89/3.77 join(one, join(join(composition(converse(x0), x0), one), composition(converse(x0), x0)))
% 26.89/3.77 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 26.89/3.77 join(one, join(composition(converse(x0), x0), join(composition(converse(x0), x0), one)))
% 26.89/3.77 = { by lemma 26 }
% 26.89/3.77 join(one, join(one, join(composition(converse(x0), x0), composition(converse(x0), x0))))
% 26.89/3.77 = { by lemma 24 }
% 26.89/3.77 join(one, join(one, composition(converse(x0), x0)))
% 26.89/3.77 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 26.89/3.77 join(one, join(composition(converse(x0), x0), one))
% 26.89/3.77 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 26.89/3.77 join(join(composition(converse(x0), x0), one), one)
% 26.89/3.77 = { by axiom 1 (composition_identity) R->L }
% 26.89/3.77 composition(join(join(composition(converse(x0), x0), one), one), one)
% 26.89/3.77 = { by lemma 22 R->L }
% 26.89/3.77 composition(join(join(composition(converse(x0), x0), one), one), complement(complement(one)))
% 26.89/3.77 = { by lemma 19 R->L }
% 26.89/3.77 composition(join(join(composition(converse(x0), x0), converse(one)), one), complement(complement(one)))
% 26.89/3.77 = { by lemma 17 R->L }
% 26.89/3.77 composition(join(join(converse(composition(converse(x0), x0)), converse(one)), one), complement(complement(one)))
% 26.89/3.77 = { by axiom 8 (converse_additivity) R->L }
% 26.89/3.77 composition(join(converse(join(composition(converse(x0), x0), one)), one), complement(complement(one)))
% 26.89/3.77 = { by lemma 25 R->L }
% 26.89/3.77 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(complement(one))))
% 26.89/3.77 = { by lemma 27 R->L }
% 26.89/3.77 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(converse(join(complement(composition(x0, complement(one))), x0)), complement(join(complement(composition(x0, complement(one))), x0)))))))
% 26.89/3.77 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 26.89/3.77 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(converse(join(x0, complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), x0)))))))
% 26.89/3.77 = { by axiom 8 (converse_additivity) }
% 26.89/3.77 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), x0)))))))
% 26.89/3.77 = { by lemma 20 R->L }
% 26.89/3.77 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(meet(join(complement(composition(x0, complement(one))), x0), meet(composition(x0, complement(one)), x0)), complement(join(complement(join(complement(composition(x0, complement(one))), x0)), meet(composition(x0, complement(one)), x0))))))))))
% 26.89/3.77 = { by axiom 2 (maddux1_join_commutativity) }
% 26.89/3.77 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(meet(join(complement(composition(x0, complement(one))), x0), meet(composition(x0, complement(one)), x0)), complement(join(meet(composition(x0, complement(one)), x0), complement(join(complement(composition(x0, complement(one))), x0)))))))))))
% 26.89/3.77 = { by lemma 20 }
% 26.89/3.77 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(meet(join(complement(composition(x0, complement(one))), x0), meet(composition(x0, complement(one)), x0)), complement(composition(x0, complement(one))))))))))
% 26.89/3.77 = { by axiom 2 (maddux1_join_commutativity) }
% 26.89/3.77 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), meet(join(complement(composition(x0, complement(one))), x0), meet(composition(x0, complement(one)), x0)))))))))
% 26.89/3.77 = { by lemma 21 R->L }
% 26.89/3.77 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), meet(meet(composition(x0, complement(one)), x0), join(complement(composition(x0, complement(one))), x0)))))))))
% 26.89/3.77 = { by axiom 2 (maddux1_join_commutativity) }
% 26.89/3.77 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), meet(meet(composition(x0, complement(one)), x0), join(x0, complement(composition(x0, complement(one))))))))))))
% 26.89/3.77 = { by lemma 21 R->L }
% 26.89/3.77 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), meet(meet(x0, composition(x0, complement(one))), join(x0, complement(composition(x0, complement(one))))))))))))
% 26.89/3.77 = { by axiom 1 (composition_identity) R->L }
% 26.89/3.77 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), meet(meet(composition(x0, one), composition(x0, complement(one))), join(x0, complement(composition(x0, complement(one))))))))))))
% 26.89/3.77 = { by axiom 12 (goals) }
% 26.89/3.77 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), meet(zero, join(x0, complement(composition(x0, complement(one))))))))))))
% 26.89/3.78 = { by lemma 21 }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), meet(join(x0, complement(composition(x0, complement(one)))), zero))))))))
% 26.89/3.78 = { by axiom 10 (maddux4_definiton_of_meet) }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(complement(join(x0, complement(composition(x0, complement(one))))), complement(zero))))))))))
% 26.89/3.78 = { by axiom 5 (def_zero) }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(complement(join(x0, complement(composition(x0, complement(one))))), complement(meet(complement(join(x0, complement(composition(x0, complement(one))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))))))))))))
% 26.89/3.78 = { by lemma 20 R->L }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(join(meet(complement(join(x0, complement(composition(x0, complement(one))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), complement(meet(complement(join(x0, complement(composition(x0, complement(one))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))))))))))))
% 26.89/3.78 = { by axiom 9 (maddux2_join_associativity) R->L }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(meet(complement(join(x0, complement(composition(x0, complement(one))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))), join(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), complement(meet(complement(join(x0, complement(composition(x0, complement(one))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))))))))))))
% 26.89/3.78 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(meet(complement(join(x0, complement(composition(x0, complement(one))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))), join(complement(meet(complement(join(x0, complement(composition(x0, complement(one))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))))))))))))
% 26.89/3.78 = { by axiom 9 (maddux2_join_associativity) }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(join(meet(complement(join(x0, complement(composition(x0, complement(one))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))), complement(meet(complement(join(x0, complement(composition(x0, complement(one))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))))))))))))
% 26.89/3.78 = { by axiom 4 (def_top) R->L }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(top, complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))))))))))))
% 26.89/3.78 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), top)))))))))
% 26.89/3.78 = { by lemma 20 R->L }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(join(meet(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), complement(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))))), complement(join(complement(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), complement(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))))))), top)))))))))
% 26.89/3.78 = { by axiom 5 (def_zero) R->L }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(join(zero, complement(join(complement(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), complement(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))))))), top)))))))))
% 26.89/3.78 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(join(zero, meet(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))))), top)))))))))
% 26.89/3.78 = { by axiom 9 (maddux2_join_associativity) R->L }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(zero, join(meet(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), top))))))))))
% 26.89/3.78 = { by axiom 2 (maddux1_join_commutativity) }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(zero, join(top, meet(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))))))))))))))
% 26.89/3.78 = { by axiom 1 (composition_identity) R->L }
% 26.89/3.78 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(zero, join(top, meet(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), composition(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), one))))))))))))
% 26.89/3.79 = { by lemma 21 }
% 26.89/3.79 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(zero, join(top, meet(composition(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), one), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))))))))))))))
% 26.89/3.79 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 26.89/3.79 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(zero, join(meet(composition(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), one), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), top))))))))))
% 26.89/3.79 = { by axiom 4 (def_top) }
% 26.89/3.79 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(zero, join(meet(composition(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), one), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), join(meet(composition(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), meet(one, composition(converse(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), complement(meet(composition(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), meet(one, composition(converse(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))))))))))))))))
% 26.89/3.79 = { by axiom 15 (modular_law_1) R->L }
% 26.89/3.79 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(zero, join(meet(composition(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), one), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), join(meet(composition(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), meet(one, composition(converse(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), complement(join(meet(composition(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), one), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), meet(composition(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), meet(one, composition(converse(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))))))))))))))))
% 26.89/3.79 = { by axiom 9 (maddux2_join_associativity) }
% 26.89/3.79 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(zero, join(join(meet(composition(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), one), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), meet(composition(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), meet(one, composition(converse(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))))), complement(join(meet(composition(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), one), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), meet(composition(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))), meet(one, composition(converse(complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one))))))))))), complement(join(complement(complement(join(x0, complement(composition(x0, complement(one)))))), complement(complement(join(x0, complement(composition(x0, complement(one)))))))))))))))))))))
% 26.89/3.79 = { by axiom 4 (def_top) R->L }
% 26.89/3.79 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(zero, top)))))))))
% 26.89/3.79 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 26.89/3.79 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(top, zero)))))))))
% 26.89/3.79 = { by lemma 16 R->L }
% 26.89/3.79 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(join(top, complement(top))))))))))
% 26.89/3.79 = { by axiom 4 (def_top) R->L }
% 26.89/3.79 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), complement(top))))))))
% 26.89/3.80 = { by lemma 16 }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(complement(composition(x0, complement(one))), zero)))))))
% 26.89/3.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(zero, complement(composition(x0, complement(one))))))))))
% 26.89/3.80 = { by lemma 22 R->L }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(zero, complement(complement(complement(composition(x0, complement(one))))))))))))
% 26.89/3.80 = { by axiom 5 (def_zero) }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(meet(complement(composition(x0, complement(one))), complement(complement(composition(x0, complement(one))))), complement(complement(complement(composition(x0, complement(one))))))))))))
% 26.89/3.80 = { by lemma 24 R->L }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(join(meet(complement(composition(x0, complement(one))), complement(complement(composition(x0, complement(one))))), complement(join(complement(complement(composition(x0, complement(one)))), complement(complement(composition(x0, complement(one)))))))))))))
% 26.89/3.80 = { by lemma 20 }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(complement(composition(x0, complement(one)))))))))
% 26.89/3.80 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(composition(join(converse(x0), converse(complement(composition(x0, complement(one))))), complement(complement(composition(x0, complement(one))))), complement(one)))))
% 26.89/3.80 = { by axiom 11 (composition_distributivity) }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(join(composition(converse(x0), complement(complement(composition(x0, complement(one))))), composition(converse(complement(composition(x0, complement(one)))), complement(complement(composition(x0, complement(one)))))), complement(one)))))
% 26.89/3.80 = { by axiom 9 (maddux2_join_associativity) R->L }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(composition(converse(x0), complement(complement(composition(x0, complement(one))))), join(composition(converse(complement(composition(x0, complement(one)))), complement(complement(composition(x0, complement(one))))), complement(one))))))
% 26.89/3.80 = { by axiom 2 (maddux1_join_commutativity) }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(composition(converse(x0), complement(complement(composition(x0, complement(one))))), join(complement(one), composition(converse(complement(composition(x0, complement(one)))), complement(complement(composition(x0, complement(one))))))))))
% 26.89/3.80 = { by lemma 27 }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(composition(converse(x0), complement(complement(composition(x0, complement(one))))), complement(one)))))
% 26.89/3.80 = { by axiom 2 (maddux1_join_commutativity) }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(converse(x0), complement(complement(composition(x0, complement(one)))))))))
% 26.89/3.80 = { by lemma 22 }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(converse(x0), composition(x0, complement(one)))))))
% 26.89/3.80 = { by axiom 7 (composition_associativity) }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(join(complement(one), composition(composition(converse(x0), x0), complement(one))))))
% 26.89/3.80 = { by lemma 25 }
% 26.89/3.80 join(complement(complement(one)), composition(converse(join(composition(converse(x0), x0), one)), complement(composition(join(composition(converse(x0), x0), one), complement(one)))))
% 26.89/3.80 = { by lemma 23 }
% 26.89/3.80 complement(complement(one))
% 26.89/3.80 = { by lemma 22 }
% 26.89/3.80 one
% 26.89/3.80 % SZS output end Proof
% 26.89/3.80
% 26.89/3.80 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------