TSTP Solution File: REL043-2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL043-2 : 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 : n017.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 : Unsatisfiable 0.20s 0.73s
% Output : Proof 3.15s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : REL043-2 : TPTP v8.1.2. Released v4.0.0.
% 0.06/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n017.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Fri Aug 25 18:24:25 EDT 2023
% 0.14/0.34 % CPUTime :
% 0.20/0.73 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.20/0.73
% 0.20/0.73 % SZS status Unsatisfiable
% 0.20/0.73
% 3.15/0.76 % SZS output start Proof
% 3.15/0.76 Axiom 1 (composition_identity_6): composition(X, one) = X.
% 3.15/0.76 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 3.15/0.76 Axiom 3 (converse_idempotence_8): converse(converse(X)) = X.
% 3.15/0.76 Axiom 4 (def_top_12): top = join(X, complement(X)).
% 3.15/0.76 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 3.15/0.76 Axiom 6 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 3.15/0.76 Axiom 7 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 3.15/0.76 Axiom 8 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 3.15/0.76 Axiom 9 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 3.15/0.76 Axiom 10 (goals_17): join(composition(sk1, converse(sk2)), sk3) = sk3.
% 3.15/0.76 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 3.15/0.76 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 3.15/0.76 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 3.15/0.76 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 3.15/0.76 Axiom 15 (modular_law_2_16): join(meet(composition(X, Y), Z), meet(composition(meet(X, composition(Z, converse(Y))), Y), Z)) = meet(composition(meet(X, composition(Z, converse(Y))), Y), Z).
% 3.15/0.76
% 3.15/0.76 Lemma 16: complement(top) = zero.
% 3.15/0.76 Proof:
% 3.15/0.76 complement(top)
% 3.15/0.76 = { by axiom 4 (def_top_12) }
% 3.15/0.76 complement(join(complement(X), complement(complement(X))))
% 3.15/0.76 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 3.15/0.76 meet(X, complement(X))
% 3.15/0.76 = { by axiom 5 (def_zero_13) R->L }
% 3.15/0.76 zero
% 3.15/0.76
% 3.15/0.76 Lemma 17: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 3.15/0.76 Proof:
% 3.15/0.76 join(meet(X, Y), complement(join(complement(X), Y)))
% 3.15/0.76 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 3.15/0.76 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 3.15/0.76 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 3.15/0.76 X
% 3.15/0.76
% 3.15/0.76 Lemma 18: join(zero, meet(X, X)) = X.
% 3.15/0.76 Proof:
% 3.15/0.76 join(zero, meet(X, X))
% 3.15/0.76 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 3.15/0.76 join(zero, complement(join(complement(X), complement(X))))
% 3.15/0.76 = { by axiom 5 (def_zero_13) }
% 3.15/0.76 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 3.15/0.76 = { by lemma 17 }
% 3.15/0.76 X
% 3.15/0.76
% 3.15/0.76 Lemma 19: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 3.15/0.76 Proof:
% 3.15/0.76 converse(composition(converse(X), Y))
% 3.15/0.76 = { by axiom 6 (converse_multiplicativity_10) }
% 3.15/0.76 composition(converse(Y), converse(converse(X)))
% 3.15/0.76 = { by axiom 3 (converse_idempotence_8) }
% 3.15/0.76 composition(converse(Y), X)
% 3.15/0.76
% 3.15/0.76 Lemma 20: composition(converse(one), X) = X.
% 3.15/0.76 Proof:
% 3.15/0.76 composition(converse(one), X)
% 3.15/0.76 = { by lemma 19 R->L }
% 3.15/0.76 converse(composition(converse(X), one))
% 3.15/0.76 = { by axiom 1 (composition_identity_6) }
% 3.15/0.76 converse(converse(X))
% 3.15/0.76 = { by axiom 3 (converse_idempotence_8) }
% 3.15/0.76 X
% 3.15/0.76
% 3.15/0.76 Lemma 21: composition(one, X) = X.
% 3.15/0.76 Proof:
% 3.15/0.76 composition(one, X)
% 3.15/0.76 = { by lemma 20 R->L }
% 3.15/0.76 composition(converse(one), composition(one, X))
% 3.15/0.76 = { by axiom 7 (composition_associativity_5) }
% 3.15/0.76 composition(composition(converse(one), one), X)
% 3.15/0.76 = { by axiom 1 (composition_identity_6) }
% 3.15/0.76 composition(converse(one), X)
% 3.15/0.76 = { by lemma 20 }
% 3.15/0.76 X
% 3.15/0.76
% 3.15/0.76 Lemma 22: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 3.15/0.76 Proof:
% 3.15/0.76 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 3.15/0.76 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 3.15/0.76 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 3.15/0.76 = { by axiom 13 (converse_cancellativity_11) }
% 3.15/0.76 complement(X)
% 3.15/0.76
% 3.15/0.76 Lemma 23: join(complement(X), complement(X)) = complement(X).
% 3.15/0.76 Proof:
% 3.15/0.76 join(complement(X), complement(X))
% 3.15/0.76 = { by lemma 20 R->L }
% 3.15/0.76 join(complement(X), composition(converse(one), complement(X)))
% 3.15/0.76 = { by lemma 21 R->L }
% 3.15/0.76 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 3.15/0.76 = { by lemma 22 }
% 3.15/0.76 complement(X)
% 3.15/0.76
% 3.15/0.76 Lemma 24: join(zero, zero) = zero.
% 3.15/0.76 Proof:
% 3.15/0.76 join(zero, zero)
% 3.15/0.76 = { by lemma 16 R->L }
% 3.15/0.76 join(zero, complement(top))
% 3.15/0.76 = { by lemma 16 R->L }
% 3.15/0.76 join(complement(top), complement(top))
% 3.15/0.76 = { by lemma 23 }
% 3.15/0.76 complement(top)
% 3.15/0.76 = { by lemma 16 }
% 3.15/0.76 zero
% 3.15/0.76
% 3.15/0.76 Lemma 25: join(zero, join(zero, X)) = join(X, zero).
% 3.15/0.76 Proof:
% 3.15/0.76 join(zero, join(zero, X))
% 3.15/0.76 = { by axiom 9 (maddux2_join_associativity_2) }
% 3.15/0.76 join(join(zero, zero), X)
% 3.15/0.76 = { by lemma 24 }
% 3.15/0.76 join(zero, X)
% 3.15/0.76 = { by axiom 2 (maddux1_join_commutativity_1) }
% 3.15/0.76 join(X, zero)
% 3.15/0.76
% 3.15/0.76 Lemma 26: join(zero, complement(complement(X))) = X.
% 3.15/0.76 Proof:
% 3.15/0.76 join(zero, complement(complement(X)))
% 3.15/0.76 = { by axiom 5 (def_zero_13) }
% 3.15/0.76 join(meet(X, complement(X)), complement(complement(X)))
% 3.15/0.76 = { by lemma 23 R->L }
% 3.15/0.76 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 3.15/0.76 = { by lemma 17 }
% 3.15/0.76 X
% 3.15/0.76
% 3.15/0.76 Lemma 27: join(X, zero) = X.
% 3.15/0.76 Proof:
% 3.15/0.76 join(X, zero)
% 3.15/0.76 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 3.15/0.76 join(zero, X)
% 3.15/0.76 = { by lemma 18 R->L }
% 3.15/0.76 join(zero, join(zero, meet(X, X)))
% 3.15/0.76 = { by lemma 25 }
% 3.15/0.76 join(meet(X, X), zero)
% 3.15/0.76 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 3.15/0.76 join(complement(join(complement(X), complement(X))), zero)
% 3.15/0.76 = { by lemma 23 }
% 3.15/0.76 join(complement(complement(X)), zero)
% 3.15/0.76 = { by axiom 2 (maddux1_join_commutativity_1) }
% 3.15/0.76 join(zero, complement(complement(X)))
% 3.15/0.76 = { by lemma 26 }
% 3.15/0.76 X
% 3.15/0.76
% 3.15/0.76 Lemma 28: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 3.15/0.76 Proof:
% 3.15/0.76 converse(join(X, converse(Y)))
% 3.15/0.76 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 3.15/0.76 converse(join(converse(Y), X))
% 3.15/0.76 = { by axiom 8 (converse_additivity_9) }
% 3.15/0.76 join(converse(converse(Y)), converse(X))
% 3.15/0.76 = { by axiom 3 (converse_idempotence_8) }
% 3.15/0.76 join(Y, converse(X))
% 3.15/0.76
% 3.15/0.76 Lemma 29: converse(zero) = zero.
% 3.15/0.76 Proof:
% 3.15/0.76 converse(zero)
% 3.15/0.76 = { by lemma 27 R->L }
% 3.15/0.76 join(converse(zero), zero)
% 3.15/0.76 = { by lemma 25 R->L }
% 3.15/0.76 join(zero, join(zero, converse(zero)))
% 3.15/0.76 = { by lemma 28 R->L }
% 3.15/0.76 join(zero, converse(join(zero, converse(zero))))
% 3.15/0.76 = { by axiom 2 (maddux1_join_commutativity_1) }
% 3.15/0.77 join(zero, converse(join(converse(zero), zero)))
% 3.15/0.77 = { by lemma 27 }
% 3.15/0.77 join(zero, converse(converse(zero)))
% 3.15/0.77 = { by axiom 3 (converse_idempotence_8) }
% 3.15/0.77 join(zero, zero)
% 3.15/0.77 = { by lemma 24 }
% 3.15/0.77 zero
% 3.15/0.77
% 3.15/0.77 Lemma 30: join(X, join(Y, complement(X))) = join(Y, top).
% 3.15/0.77 Proof:
% 3.15/0.77 join(X, join(Y, complement(X)))
% 3.15/0.77 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 3.15/0.77 join(X, join(complement(X), Y))
% 3.15/0.77 = { by axiom 9 (maddux2_join_associativity_2) }
% 3.15/0.77 join(join(X, complement(X)), Y)
% 3.15/0.77 = { by axiom 4 (def_top_12) R->L }
% 3.15/0.77 join(top, Y)
% 3.15/0.77 = { by axiom 2 (maddux1_join_commutativity_1) }
% 3.15/0.77 join(Y, top)
% 3.15/0.77
% 3.15/0.77 Lemma 31: join(X, join(complement(X), Y)) = join(Y, top).
% 3.15/0.77 Proof:
% 3.15/0.77 join(X, join(complement(X), Y))
% 3.15/0.77 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 3.15/0.77 join(X, join(Y, complement(X)))
% 3.15/0.77 = { by lemma 30 }
% 3.15/0.77 join(Y, top)
% 3.15/0.77
% 3.15/0.77 Lemma 32: join(top, complement(X)) = top.
% 3.15/0.77 Proof:
% 3.15/0.77 join(top, complement(X))
% 3.15/0.77 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 3.15/0.77 join(complement(X), top)
% 3.15/0.77 = { by lemma 30 R->L }
% 3.15/0.77 join(X, join(complement(X), complement(X)))
% 3.15/0.77 = { by lemma 23 }
% 3.15/0.77 join(X, complement(X))
% 3.15/0.77 = { by axiom 4 (def_top_12) R->L }
% 3.15/0.77 top
% 3.15/0.77
% 3.15/0.77 Lemma 33: join(X, top) = top.
% 3.15/0.77 Proof:
% 3.15/0.77 join(X, top)
% 3.15/0.77 = { by axiom 4 (def_top_12) }
% 3.15/0.77 join(X, join(complement(X), complement(complement(X))))
% 3.15/0.77 = { by lemma 31 }
% 3.15/0.77 join(complement(complement(X)), top)
% 3.15/0.77 = { by axiom 2 (maddux1_join_commutativity_1) }
% 3.15/0.77 join(top, complement(complement(X)))
% 3.15/0.77 = { by lemma 32 }
% 3.15/0.77 top
% 3.15/0.77
% 3.15/0.77 Lemma 34: join(top, X) = top.
% 3.15/0.77 Proof:
% 3.15/0.77 join(top, X)
% 3.15/0.77 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 3.15/0.77 join(X, top)
% 3.15/0.77 = { by lemma 32 R->L }
% 3.15/0.77 join(X, join(top, complement(X)))
% 3.15/0.77 = { by lemma 30 }
% 3.15/0.77 join(top, top)
% 3.15/0.77 = { by lemma 33 }
% 3.15/0.77 top
% 3.15/0.77
% 3.15/0.77 Lemma 35: join(zero, X) = X.
% 3.15/0.77 Proof:
% 3.15/0.77 join(zero, X)
% 3.15/0.77 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 3.15/0.77 join(X, zero)
% 3.15/0.77 = { by lemma 27 }
% 3.15/0.77 X
% 3.15/0.77
% 3.15/0.77 Lemma 36: complement(complement(X)) = X.
% 3.15/0.77 Proof:
% 3.15/0.77 complement(complement(X))
% 3.15/0.77 = { by lemma 35 R->L }
% 3.15/0.77 join(zero, complement(complement(X)))
% 3.15/0.77 = { by lemma 26 }
% 3.15/0.77 X
% 3.15/0.77
% 3.15/0.77 Lemma 37: meet(Y, X) = meet(X, Y).
% 3.15/0.77 Proof:
% 3.15/0.77 meet(Y, X)
% 3.15/0.77 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 3.15/0.77 complement(join(complement(Y), complement(X)))
% 3.15/0.77 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 3.15/0.77 complement(join(complement(X), complement(Y)))
% 3.15/0.77 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 3.15/0.77 meet(X, Y)
% 3.15/0.77
% 3.15/0.77 Lemma 38: complement(join(zero, complement(X))) = meet(X, top).
% 3.15/0.77 Proof:
% 3.15/0.77 complement(join(zero, complement(X)))
% 3.15/0.77 = { by lemma 16 R->L }
% 3.15/0.77 complement(join(complement(top), complement(X)))
% 3.15/0.77 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 3.15/0.77 meet(top, X)
% 3.15/0.77 = { by lemma 37 R->L }
% 3.15/0.77 meet(X, top)
% 3.15/0.77
% 3.15/0.77 Lemma 39: meet(X, top) = X.
% 3.15/0.77 Proof:
% 3.15/0.77 meet(X, top)
% 3.15/0.77 = { by lemma 38 R->L }
% 3.15/0.77 complement(join(zero, complement(X)))
% 3.15/0.77 = { by lemma 35 }
% 3.15/0.77 complement(complement(X))
% 3.15/0.77 = { by lemma 36 }
% 3.15/0.77 X
% 3.15/0.77
% 3.15/0.77 Lemma 40: meet(zero, X) = zero.
% 3.15/0.77 Proof:
% 3.15/0.77 meet(zero, X)
% 3.15/0.77 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 3.15/0.77 complement(join(complement(zero), complement(X)))
% 3.15/0.77 = { by lemma 35 R->L }
% 3.15/0.77 complement(join(join(zero, complement(zero)), complement(X)))
% 3.15/0.77 = { by axiom 4 (def_top_12) R->L }
% 3.15/0.77 complement(join(top, complement(X)))
% 3.15/0.77 = { by lemma 32 }
% 3.15/0.77 complement(top)
% 3.15/0.77 = { by lemma 16 }
% 3.15/0.77 zero
% 3.15/0.77
% 3.15/0.77 Lemma 41: join(X, converse(top)) = converse(top).
% 3.15/0.77 Proof:
% 3.15/0.77 join(X, converse(top))
% 3.15/0.77 = { by lemma 28 R->L }
% 3.15/0.77 converse(join(top, converse(X)))
% 3.15/0.77 = { by lemma 34 }
% 3.15/0.77 converse(top)
% 3.15/0.77
% 3.15/0.77 Lemma 42: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 3.15/0.77 Proof:
% 3.15/0.77 join(X, composition(Y, X))
% 3.15/0.77 = { by lemma 21 R->L }
% 3.15/0.77 join(composition(one, X), composition(Y, X))
% 3.15/0.77 = { by axiom 12 (composition_distributivity_7) R->L }
% 3.15/0.77 composition(join(one, Y), X)
% 3.15/0.77 = { by axiom 2 (maddux1_join_commutativity_1) }
% 3.15/0.77 composition(join(Y, one), X)
% 3.15/0.77
% 3.15/0.77 Lemma 43: composition(top, zero) = zero.
% 3.15/0.77 Proof:
% 3.15/0.77 composition(top, zero)
% 3.15/0.77 = { by lemma 16 R->L }
% 3.15/0.77 composition(top, complement(top))
% 3.15/0.77 = { by lemma 34 R->L }
% 3.15/0.77 composition(join(top, one), complement(top))
% 3.15/0.77 = { by lemma 33 R->L }
% 3.15/0.77 composition(join(join(converse(top), top), one), complement(top))
% 3.15/0.77 = { by lemma 31 R->L }
% 3.15/0.77 composition(join(join(X, join(complement(X), converse(top))), one), complement(top))
% 3.15/0.77 = { by lemma 41 }
% 3.15/0.77 composition(join(join(X, converse(top)), one), complement(top))
% 3.15/0.77 = { by lemma 41 }
% 3.15/0.77 composition(join(converse(top), one), complement(top))
% 3.15/0.77 = { by lemma 42 R->L }
% 3.15/0.77 join(complement(top), composition(converse(top), complement(top)))
% 3.15/0.77 = { by lemma 34 R->L }
% 3.15/0.77 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 3.15/0.77 = { by lemma 42 }
% 3.15/0.77 join(complement(top), composition(converse(top), complement(composition(join(top, one), top))))
% 3.15/0.77 = { by lemma 34 }
% 3.15/0.77 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 3.15/0.77 = { by lemma 22 }
% 3.15/0.77 complement(top)
% 3.15/0.77 = { by lemma 16 }
% 3.15/0.77 zero
% 3.15/0.77
% 3.15/0.77 Lemma 44: composition(zero, X) = zero.
% 3.15/0.77 Proof:
% 3.15/0.77 composition(zero, X)
% 3.15/0.77 = { by lemma 29 R->L }
% 3.15/0.77 composition(converse(zero), X)
% 3.15/0.77 = { by lemma 19 R->L }
% 3.15/0.77 converse(composition(converse(X), zero))
% 3.15/0.77 = { by lemma 35 R->L }
% 3.15/0.77 converse(join(zero, composition(converse(X), zero)))
% 3.15/0.77 = { by lemma 43 R->L }
% 3.15/0.77 converse(join(composition(top, zero), composition(converse(X), zero)))
% 3.15/0.77 = { by axiom 12 (composition_distributivity_7) R->L }
% 3.15/0.77 converse(composition(join(top, converse(X)), zero))
% 3.15/0.77 = { by lemma 34 }
% 3.15/0.77 converse(composition(top, zero))
% 3.15/0.77 = { by lemma 43 }
% 3.15/0.77 converse(zero)
% 3.15/0.77 = { by lemma 29 }
% 3.15/0.77 zero
% 3.15/0.77
% 3.15/0.77 Lemma 45: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 3.15/0.77 Proof:
% 3.15/0.77 meet(X, join(complement(Y), complement(Z)))
% 3.15/0.77 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 3.15/0.77 meet(X, join(complement(Z), complement(Y)))
% 3.15/0.77 = { by lemma 37 }
% 3.15/0.77 meet(join(complement(Z), complement(Y)), X)
% 3.15/0.77 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 3.15/0.77 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 3.15/0.77 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 3.15/0.77 complement(join(meet(Z, Y), complement(X)))
% 3.15/0.77 = { by axiom 2 (maddux1_join_commutativity_1) }
% 3.15/0.77 complement(join(complement(X), meet(Z, Y)))
% 3.15/0.77 = { by lemma 37 R->L }
% 3.15/0.77 complement(join(complement(X), meet(Y, Z)))
% 3.15/0.77
% 3.15/0.77 Lemma 46: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 3.15/0.77 Proof:
% 3.15/0.77 complement(join(X, complement(Y)))
% 3.15/0.77 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 3.15/0.77 complement(join(complement(Y), X))
% 3.15/0.77 = { by lemma 39 R->L }
% 3.15/0.77 complement(join(complement(Y), meet(X, top)))
% 3.15/0.77 = { by lemma 37 R->L }
% 3.15/0.77 complement(join(complement(Y), meet(top, X)))
% 3.15/0.77 = { by lemma 45 R->L }
% 3.15/0.77 meet(Y, join(complement(top), complement(X)))
% 3.15/0.77 = { by lemma 16 }
% 3.15/0.77 meet(Y, join(zero, complement(X)))
% 3.15/0.77 = { by lemma 35 }
% 3.15/0.77 meet(Y, complement(X))
% 3.15/0.77
% 3.15/0.77 Lemma 47: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 3.15/0.77 Proof:
% 3.15/0.77 complement(meet(complement(X), Y))
% 3.15/0.77 = { by lemma 37 }
% 3.15/0.77 complement(meet(Y, complement(X)))
% 3.15/0.77 = { by lemma 35 R->L }
% 3.15/0.77 complement(join(zero, meet(Y, complement(X))))
% 3.15/0.77 = { by lemma 46 R->L }
% 3.15/0.77 complement(join(zero, complement(join(X, complement(Y)))))
% 3.15/0.77 = { by lemma 38 }
% 3.15/0.77 meet(join(X, complement(Y)), top)
% 3.15/0.77 = { by lemma 39 }
% 3.15/0.77 join(X, complement(Y))
% 3.15/0.77
% 3.15/0.77 Lemma 48: join(X, complement(meet(X, Y))) = top.
% 3.15/0.77 Proof:
% 3.15/0.77 join(X, complement(meet(X, Y)))
% 3.15/0.77 = { by lemma 37 }
% 3.15/0.77 join(X, complement(meet(Y, X)))
% 3.15/0.77 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 3.15/0.77 join(X, complement(complement(join(complement(Y), complement(X)))))
% 3.15/0.77 = { by lemma 23 R->L }
% 3.15/0.77 join(X, complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))))
% 3.15/0.77 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 3.15/0.77 join(X, complement(join(meet(Y, X), complement(join(complement(Y), complement(X))))))
% 3.15/0.77 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 3.15/0.77 join(X, complement(join(meet(Y, X), meet(Y, X))))
% 3.15/0.77 = { by lemma 37 R->L }
% 3.15/0.77 join(X, complement(join(meet(Y, X), meet(X, Y))))
% 3.15/0.77 = { by lemma 37 R->L }
% 3.15/0.77 join(X, complement(join(meet(X, Y), meet(X, Y))))
% 3.15/0.77 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 3.15/0.77 join(X, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y))))
% 3.15/0.77 = { by lemma 35 R->L }
% 3.15/0.77 join(X, join(zero, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y)))))
% 3.15/0.77 = { by lemma 45 R->L }
% 3.15/0.77 join(X, join(zero, meet(join(complement(X), complement(Y)), join(complement(X), complement(Y)))))
% 3.15/0.77 = { by lemma 18 }
% 3.15/0.77 join(X, join(complement(X), complement(Y)))
% 3.15/0.77 = { by axiom 2 (maddux1_join_commutativity_1) }
% 3.15/0.77 join(X, join(complement(Y), complement(X)))
% 3.15/0.77 = { by lemma 30 }
% 3.15/0.77 join(complement(Y), top)
% 3.15/0.77 = { by lemma 33 }
% 3.15/0.77 top
% 3.15/0.77
% 3.15/0.77 Lemma 49: meet(complement(sk3), composition(sk1, converse(sk2))) = zero.
% 3.15/0.77 Proof:
% 3.15/0.77 meet(complement(sk3), composition(sk1, converse(sk2)))
% 3.15/0.77 = { by lemma 37 }
% 3.15/0.77 meet(composition(sk1, converse(sk2)), complement(sk3))
% 3.15/0.77 = { by axiom 10 (goals_17) R->L }
% 3.15/0.77 meet(composition(sk1, converse(sk2)), complement(join(composition(sk1, converse(sk2)), sk3)))
% 3.15/0.77 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 3.15/0.77 meet(composition(sk1, converse(sk2)), complement(join(sk3, composition(sk1, converse(sk2)))))
% 3.15/0.77 = { by lemma 39 R->L }
% 3.15/0.77 meet(composition(sk1, converse(sk2)), complement(join(sk3, meet(composition(sk1, converse(sk2)), top))))
% 3.15/0.77 = { by lemma 38 R->L }
% 3.15/0.77 meet(composition(sk1, converse(sk2)), complement(join(sk3, complement(join(zero, complement(composition(sk1, converse(sk2))))))))
% 3.15/0.77 = { by lemma 46 }
% 3.15/0.77 meet(composition(sk1, converse(sk2)), meet(join(zero, complement(composition(sk1, converse(sk2)))), complement(sk3)))
% 3.15/0.77 = { by lemma 35 }
% 3.15/0.77 meet(composition(sk1, converse(sk2)), meet(complement(composition(sk1, converse(sk2))), complement(sk3)))
% 3.15/0.77 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 3.15/0.77 complement(join(complement(composition(sk1, converse(sk2))), complement(meet(complement(composition(sk1, converse(sk2))), complement(sk3)))))
% 3.15/0.77 = { by lemma 48 }
% 3.15/0.77 complement(top)
% 3.15/0.77 = { by lemma 16 }
% 3.15/0.78 zero
% 3.15/0.78
% 3.15/0.78 Goal 1 (goals_18): join(composition(complement(sk3), sk2), complement(sk1)) = complement(sk1).
% 3.15/0.78 Proof:
% 3.15/0.78 join(composition(complement(sk3), sk2), complement(sk1))
% 3.15/0.78 = { by axiom 2 (maddux1_join_commutativity_1) }
% 3.15/0.78 join(complement(sk1), composition(complement(sk3), sk2))
% 3.15/0.78 = { by lemma 17 R->L }
% 3.15/0.78 join(complement(sk1), join(meet(composition(complement(sk3), sk2), complement(sk1)), complement(join(complement(composition(complement(sk3), sk2)), complement(sk1)))))
% 3.15/0.78 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 3.15/0.78 join(complement(sk1), join(meet(composition(complement(sk3), sk2), complement(sk1)), meet(composition(complement(sk3), sk2), sk1)))
% 3.15/0.78 = { by axiom 2 (maddux1_join_commutativity_1) }
% 3.15/0.78 join(complement(sk1), join(meet(composition(complement(sk3), sk2), sk1), meet(composition(complement(sk3), sk2), complement(sk1))))
% 3.15/0.78 = { by lemma 27 R->L }
% 3.15/0.78 join(complement(sk1), join(join(meet(composition(complement(sk3), sk2), sk1), zero), meet(composition(complement(sk3), sk2), complement(sk1))))
% 3.15/0.78 = { by lemma 40 R->L }
% 3.15/0.78 join(complement(sk1), join(join(meet(composition(complement(sk3), sk2), sk1), meet(zero, sk1)), meet(composition(complement(sk3), sk2), complement(sk1))))
% 3.15/0.78 = { by lemma 44 R->L }
% 3.15/0.78 join(complement(sk1), join(join(meet(composition(complement(sk3), sk2), sk1), meet(composition(zero, sk2), sk1)), meet(composition(complement(sk3), sk2), complement(sk1))))
% 3.15/0.78 = { by lemma 49 R->L }
% 3.15/0.78 join(complement(sk1), join(join(meet(composition(complement(sk3), sk2), sk1), meet(composition(meet(complement(sk3), composition(sk1, converse(sk2))), sk2), sk1)), meet(composition(complement(sk3), sk2), complement(sk1))))
% 3.15/0.78 = { by axiom 15 (modular_law_2_16) }
% 3.15/0.78 join(complement(sk1), join(meet(composition(meet(complement(sk3), composition(sk1, converse(sk2))), sk2), sk1), meet(composition(complement(sk3), sk2), complement(sk1))))
% 3.15/0.78 = { by lemma 49 }
% 3.15/0.78 join(complement(sk1), join(meet(composition(zero, sk2), sk1), meet(composition(complement(sk3), sk2), complement(sk1))))
% 3.15/0.78 = { by lemma 44 }
% 3.15/0.78 join(complement(sk1), join(meet(zero, sk1), meet(composition(complement(sk3), sk2), complement(sk1))))
% 3.15/0.78 = { by lemma 40 }
% 3.15/0.78 join(complement(sk1), join(zero, meet(composition(complement(sk3), sk2), complement(sk1))))
% 3.15/0.78 = { by lemma 35 }
% 3.15/0.78 join(complement(sk1), meet(composition(complement(sk3), sk2), complement(sk1)))
% 3.15/0.78 = { by lemma 37 R->L }
% 3.15/0.78 join(complement(sk1), meet(complement(sk1), composition(complement(sk3), sk2)))
% 3.15/0.78 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 3.15/0.78 join(complement(sk1), complement(join(complement(complement(sk1)), complement(composition(complement(sk3), sk2)))))
% 3.15/0.78 = { by lemma 47 R->L }
% 3.15/0.78 complement(meet(complement(complement(sk1)), join(complement(complement(sk1)), complement(composition(complement(sk3), sk2)))))
% 3.15/0.78 = { by lemma 27 R->L }
% 3.15/0.78 complement(join(meet(complement(complement(sk1)), join(complement(complement(sk1)), complement(composition(complement(sk3), sk2)))), zero))
% 3.15/0.78 = { by lemma 16 R->L }
% 3.15/0.78 complement(join(meet(complement(complement(sk1)), join(complement(complement(sk1)), complement(composition(complement(sk3), sk2)))), complement(top)))
% 3.15/0.78 = { by lemma 47 R->L }
% 3.15/0.78 complement(join(meet(complement(complement(sk1)), complement(meet(complement(complement(complement(sk1))), composition(complement(sk3), sk2)))), complement(top)))
% 3.15/0.78 = { by lemma 48 R->L }
% 3.15/0.78 complement(join(meet(complement(complement(sk1)), complement(meet(complement(complement(complement(sk1))), composition(complement(sk3), sk2)))), complement(join(complement(complement(complement(sk1))), complement(meet(complement(complement(complement(sk1))), composition(complement(sk3), sk2)))))))
% 3.15/0.78 = { by lemma 17 }
% 3.15/0.78 complement(complement(complement(sk1)))
% 3.15/0.78 = { by lemma 36 }
% 3.15/0.78 complement(sk1)
% 3.15/0.78 % SZS output end Proof
% 3.15/0.78
% 3.15/0.78 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------