TSTP Solution File: REL034-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL034-1 : 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 : n028.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 13:44:19 EDT 2023
% Result : Unsatisfiable 30.15s 4.24s
% Output : Proof 30.15s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL034-1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % 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 : n028.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 19:36:06 EDT 2023
% 0.13/0.34 % CPUTime :
% 30.15/4.24 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 30.15/4.24
% 30.15/4.24 % SZS status Unsatisfiable
% 30.15/4.24
% 30.15/4.28 % SZS output start Proof
% 30.15/4.28 Axiom 1 (composition_identity_6): composition(X, one) = X.
% 30.15/4.28 Axiom 2 (goals_14): composition(sk1, top) = sk1.
% 30.15/4.28 Axiom 3 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 30.15/4.28 Axiom 4 (converse_idempotence_8): converse(converse(X)) = X.
% 30.15/4.28 Axiom 5 (def_top_12): top = join(X, complement(X)).
% 30.15/4.28 Axiom 6 (def_zero_13): zero = meet(X, complement(X)).
% 30.15/4.28 Axiom 7 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 30.15/4.28 Axiom 8 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 30.15/4.28 Axiom 9 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 30.15/4.28 Axiom 10 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 30.15/4.28 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 30.15/4.28 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 30.15/4.28 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 30.15/4.28 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 30.15/4.28
% 30.15/4.28 Lemma 15: complement(top) = zero.
% 30.15/4.28 Proof:
% 30.15/4.28 complement(top)
% 30.15/4.28 = { by axiom 5 (def_top_12) }
% 30.15/4.28 complement(join(complement(X), complement(complement(X))))
% 30.15/4.28 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 30.15/4.28 meet(X, complement(X))
% 30.15/4.28 = { by axiom 6 (def_zero_13) R->L }
% 30.15/4.28 zero
% 30.15/4.28
% 30.15/4.28 Lemma 16: join(X, join(Y, complement(X))) = join(Y, top).
% 30.15/4.28 Proof:
% 30.15/4.28 join(X, join(Y, complement(X)))
% 30.15/4.28 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 30.15/4.28 join(X, join(complement(X), Y))
% 30.15/4.28 = { by axiom 10 (maddux2_join_associativity_2) }
% 30.15/4.28 join(join(X, complement(X)), Y)
% 30.15/4.28 = { by axiom 5 (def_top_12) R->L }
% 30.15/4.28 join(top, Y)
% 30.15/4.28 = { by axiom 3 (maddux1_join_commutativity_1) }
% 30.15/4.28 join(Y, top)
% 30.15/4.28
% 30.15/4.28 Lemma 17: composition(converse(one), X) = X.
% 30.15/4.28 Proof:
% 30.15/4.28 composition(converse(one), X)
% 30.15/4.28 = { by axiom 4 (converse_idempotence_8) R->L }
% 30.15/4.28 composition(converse(one), converse(converse(X)))
% 30.15/4.28 = { by axiom 7 (converse_multiplicativity_10) R->L }
% 30.15/4.28 converse(composition(converse(X), one))
% 30.15/4.28 = { by axiom 1 (composition_identity_6) }
% 30.15/4.28 converse(converse(X))
% 30.15/4.28 = { by axiom 4 (converse_idempotence_8) }
% 30.15/4.28 X
% 30.15/4.28
% 30.15/4.28 Lemma 18: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 30.15/4.28 Proof:
% 30.15/4.28 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 30.15/4.28 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 30.15/4.28 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 30.15/4.28 = { by axiom 13 (converse_cancellativity_11) }
% 30.15/4.28 complement(X)
% 30.15/4.28
% 30.15/4.28 Lemma 19: join(complement(X), complement(X)) = complement(X).
% 30.15/4.28 Proof:
% 30.15/4.28 join(complement(X), complement(X))
% 30.15/4.28 = { by lemma 17 R->L }
% 30.15/4.28 join(complement(X), composition(converse(one), complement(X)))
% 30.15/4.28 = { by lemma 17 R->L }
% 30.15/4.28 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 30.15/4.28 = { by axiom 1 (composition_identity_6) R->L }
% 30.15/4.28 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 30.15/4.28 = { by axiom 8 (composition_associativity_5) R->L }
% 30.15/4.28 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 30.15/4.28 = { by lemma 17 }
% 30.15/4.28 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 30.15/4.28 = { by lemma 18 }
% 30.15/4.28 complement(X)
% 30.15/4.28
% 30.15/4.28 Lemma 20: join(top, complement(X)) = top.
% 30.15/4.28 Proof:
% 30.15/4.28 join(top, complement(X))
% 30.15/4.28 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 30.15/4.28 join(complement(X), top)
% 30.15/4.28 = { by lemma 16 R->L }
% 30.15/4.28 join(X, join(complement(X), complement(X)))
% 30.15/4.28 = { by lemma 19 }
% 30.15/4.28 join(X, complement(X))
% 30.15/4.28 = { by axiom 5 (def_top_12) R->L }
% 30.15/4.28 top
% 30.15/4.28
% 30.15/4.28 Lemma 21: join(top, X) = join(Y, top).
% 30.15/4.28 Proof:
% 30.15/4.28 join(top, X)
% 30.15/4.28 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 30.15/4.28 join(X, top)
% 30.15/4.28 = { by lemma 20 R->L }
% 30.15/4.28 join(X, join(top, complement(X)))
% 30.15/4.28 = { by lemma 16 }
% 30.15/4.28 join(top, top)
% 30.15/4.28 = { by lemma 16 R->L }
% 30.15/4.28 join(Y, join(top, complement(Y)))
% 30.15/4.28 = { by lemma 20 }
% 30.15/4.28 join(Y, top)
% 30.15/4.28
% 30.15/4.28 Lemma 22: join(X, top) = top.
% 30.15/4.28 Proof:
% 30.15/4.28 join(X, top)
% 30.15/4.28 = { by lemma 21 R->L }
% 30.15/4.28 join(top, complement(top))
% 30.15/4.28 = { by axiom 5 (def_top_12) R->L }
% 30.15/4.28 top
% 30.15/4.28
% 30.15/4.28 Lemma 23: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 30.15/4.28 Proof:
% 30.15/4.28 converse(join(converse(X), Y))
% 30.15/4.28 = { by axiom 9 (converse_additivity_9) }
% 30.15/4.28 join(converse(converse(X)), converse(Y))
% 30.15/4.28 = { by axiom 4 (converse_idempotence_8) }
% 30.15/4.28 join(X, converse(Y))
% 30.15/4.28
% 30.15/4.28 Lemma 24: converse(top) = top.
% 30.15/4.28 Proof:
% 30.15/4.28 converse(top)
% 30.15/4.28 = { by lemma 22 R->L }
% 30.15/4.28 converse(join(converse(top), top))
% 30.15/4.28 = { by lemma 23 }
% 30.15/4.28 join(top, converse(top))
% 30.15/4.28 = { by lemma 21 }
% 30.15/4.28 join(X, top)
% 30.15/4.28 = { by lemma 22 }
% 30.15/4.28 top
% 30.15/4.28
% 30.15/4.28 Lemma 25: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 30.15/4.28 Proof:
% 30.15/4.28 join(meet(X, Y), complement(join(complement(X), Y)))
% 30.15/4.28 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 30.15/4.28 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 30.15/4.28 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 30.15/4.28 X
% 30.15/4.28
% 30.15/4.28 Lemma 26: join(zero, meet(X, X)) = X.
% 30.15/4.28 Proof:
% 30.15/4.28 join(zero, meet(X, X))
% 30.15/4.28 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 30.15/4.28 join(zero, complement(join(complement(X), complement(X))))
% 30.15/4.28 = { by axiom 6 (def_zero_13) }
% 30.15/4.28 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 30.15/4.28 = { by lemma 25 }
% 30.15/4.28 X
% 30.15/4.28
% 30.15/4.28 Lemma 27: join(zero, zero) = zero.
% 30.15/4.28 Proof:
% 30.15/4.28 join(zero, zero)
% 30.15/4.28 = { by lemma 15 R->L }
% 30.15/4.28 join(zero, complement(top))
% 30.15/4.28 = { by lemma 15 R->L }
% 30.15/4.28 join(complement(top), complement(top))
% 30.15/4.28 = { by lemma 19 }
% 30.15/4.28 complement(top)
% 30.15/4.28 = { by lemma 15 }
% 30.15/4.28 zero
% 30.15/4.28
% 30.15/4.28 Lemma 28: join(zero, join(zero, X)) = join(X, zero).
% 30.15/4.28 Proof:
% 30.15/4.28 join(zero, join(zero, X))
% 30.15/4.28 = { by axiom 10 (maddux2_join_associativity_2) }
% 30.15/4.28 join(join(zero, zero), X)
% 30.15/4.28 = { by lemma 27 }
% 30.15/4.28 join(zero, X)
% 30.15/4.28 = { by axiom 3 (maddux1_join_commutativity_1) }
% 30.15/4.28 join(X, zero)
% 30.15/4.28
% 30.15/4.28 Lemma 29: join(X, zero) = X.
% 30.15/4.28 Proof:
% 30.15/4.28 join(X, zero)
% 30.15/4.28 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 30.15/4.28 join(zero, X)
% 30.15/4.28 = { by lemma 26 R->L }
% 30.15/4.28 join(zero, join(zero, meet(X, X)))
% 30.15/4.28 = { by lemma 28 }
% 30.15/4.28 join(meet(X, X), zero)
% 30.15/4.28 = { by axiom 3 (maddux1_join_commutativity_1) }
% 30.15/4.28 join(zero, meet(X, X))
% 30.15/4.28 = { by lemma 26 }
% 30.15/4.28 X
% 30.15/4.28
% 30.15/4.28 Lemma 30: join(zero, X) = X.
% 30.15/4.28 Proof:
% 30.15/4.28 join(zero, X)
% 30.15/4.28 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 30.15/4.28 join(X, zero)
% 30.15/4.28 = { by lemma 29 }
% 30.15/4.28 X
% 30.15/4.28
% 30.15/4.28 Lemma 31: complement(complement(X)) = X.
% 30.15/4.28 Proof:
% 30.15/4.28 complement(complement(X))
% 30.15/4.28 = { by lemma 30 R->L }
% 30.15/4.28 join(zero, complement(complement(X)))
% 30.15/4.28 = { by lemma 19 R->L }
% 30.15/4.28 join(zero, complement(join(complement(X), complement(X))))
% 30.15/4.28 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 30.15/4.28 join(zero, meet(X, X))
% 30.15/4.28 = { by lemma 26 }
% 30.15/4.28 X
% 30.15/4.28
% 30.15/4.28 Lemma 32: meet(Y, X) = meet(X, Y).
% 30.15/4.28 Proof:
% 30.15/4.28 meet(Y, X)
% 30.15/4.28 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 30.15/4.28 complement(join(complement(Y), complement(X)))
% 30.15/4.28 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 30.15/4.28 complement(join(complement(X), complement(Y)))
% 30.15/4.28 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 30.15/4.28 meet(X, Y)
% 30.15/4.28
% 30.15/4.28 Lemma 33: complement(join(zero, complement(X))) = meet(X, top).
% 30.15/4.28 Proof:
% 30.15/4.28 complement(join(zero, complement(X)))
% 30.15/4.28 = { by lemma 15 R->L }
% 30.15/4.28 complement(join(complement(top), complement(X)))
% 30.15/4.28 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 30.15/4.28 meet(top, X)
% 30.15/4.28 = { by lemma 32 R->L }
% 30.15/4.28 meet(X, top)
% 30.15/4.28
% 30.15/4.28 Lemma 34: meet(X, top) = X.
% 30.15/4.28 Proof:
% 30.15/4.28 meet(X, top)
% 30.15/4.28 = { by lemma 33 R->L }
% 30.15/4.28 complement(join(zero, complement(X)))
% 30.15/4.28 = { by lemma 30 }
% 30.15/4.28 complement(complement(X))
% 30.15/4.28 = { by lemma 31 }
% 30.15/4.28 X
% 30.15/4.28
% 30.15/4.28 Lemma 35: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 30.15/4.28 Proof:
% 30.15/4.28 complement(join(complement(X), meet(Y, Z)))
% 30.15/4.28 = { by lemma 32 }
% 30.15/4.28 complement(join(complement(X), meet(Z, Y)))
% 30.15/4.28 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 30.15/4.28 complement(join(meet(Z, Y), complement(X)))
% 30.15/4.28 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 30.15/4.28 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 30.15/4.28 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 30.15/4.28 meet(join(complement(Z), complement(Y)), X)
% 30.15/4.28 = { by lemma 32 R->L }
% 30.15/4.28 meet(X, join(complement(Z), complement(Y)))
% 30.15/4.28 = { by axiom 3 (maddux1_join_commutativity_1) }
% 30.15/4.28 meet(X, join(complement(Y), complement(Z)))
% 30.15/4.28
% 30.15/4.28 Lemma 36: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 30.15/4.28 Proof:
% 30.15/4.28 join(complement(X), complement(Y))
% 30.15/4.28 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 30.15/4.28 join(complement(Y), complement(X))
% 30.15/4.28 = { by lemma 34 R->L }
% 30.15/4.28 meet(join(complement(Y), complement(X)), top)
% 30.15/4.28 = { by lemma 32 R->L }
% 30.15/4.28 meet(top, join(complement(Y), complement(X)))
% 30.15/4.28 = { by lemma 35 R->L }
% 30.15/4.28 complement(join(complement(top), meet(Y, X)))
% 30.15/4.28 = { by lemma 15 }
% 30.15/4.28 complement(join(zero, meet(Y, X)))
% 30.15/4.28 = { by lemma 30 }
% 30.15/4.28 complement(meet(Y, X))
% 30.15/4.28 = { by lemma 32 R->L }
% 30.15/4.28 complement(meet(X, Y))
% 30.15/4.28
% 30.15/4.28 Lemma 37: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 30.15/4.28 Proof:
% 30.15/4.28 complement(meet(X, complement(Y)))
% 30.15/4.28 = { by lemma 32 }
% 30.15/4.28 complement(meet(complement(Y), X))
% 30.15/4.28 = { by lemma 30 R->L }
% 30.15/4.28 complement(meet(join(zero, complement(Y)), X))
% 30.15/4.28 = { by lemma 36 R->L }
% 30.15/4.28 join(complement(join(zero, complement(Y))), complement(X))
% 30.15/4.28 = { by lemma 33 }
% 30.15/4.28 join(meet(Y, top), complement(X))
% 30.15/4.28 = { by lemma 34 }
% 30.15/4.28 join(Y, complement(X))
% 30.15/4.28
% 30.15/4.28 Lemma 38: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 30.15/4.28 Proof:
% 30.15/4.28 complement(meet(complement(X), Y))
% 30.15/4.28 = { by lemma 32 }
% 30.15/4.28 complement(meet(Y, complement(X)))
% 30.15/4.28 = { by lemma 37 }
% 30.15/4.28 join(X, complement(Y))
% 30.15/4.28
% 30.15/4.28 Lemma 39: meet(X, join(X, complement(Y))) = X.
% 30.15/4.28 Proof:
% 30.15/4.28 meet(X, join(X, complement(Y)))
% 30.15/4.28 = { by lemma 29 R->L }
% 30.15/4.28 join(meet(X, join(X, complement(Y))), zero)
% 30.15/4.28 = { by lemma 15 R->L }
% 30.15/4.28 join(meet(X, join(X, complement(Y))), complement(top))
% 30.15/4.28 = { by lemma 38 R->L }
% 30.15/4.28 join(meet(X, complement(meet(complement(X), Y))), complement(top))
% 30.15/4.28 = { by lemma 22 R->L }
% 30.15/4.28 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(Y), top)))
% 30.15/4.29 = { by lemma 16 R->L }
% 30.15/4.29 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), join(complement(Y), complement(complement(X))))))
% 30.15/4.29 = { by lemma 36 }
% 30.15/4.29 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(Y, complement(X))))))
% 30.15/4.29 = { by lemma 32 R->L }
% 30.15/4.29 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(complement(X), Y)))))
% 30.15/4.29 = { by lemma 25 }
% 30.15/4.29 X
% 30.15/4.29
% 30.15/4.29 Lemma 40: join(X, meet(X, Y)) = X.
% 30.15/4.29 Proof:
% 30.15/4.29 join(X, meet(X, Y))
% 30.15/4.29 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 30.15/4.29 join(X, complement(join(complement(X), complement(Y))))
% 30.15/4.29 = { by lemma 38 R->L }
% 30.15/4.29 complement(meet(complement(X), join(complement(X), complement(Y))))
% 30.15/4.29 = { by lemma 39 }
% 30.15/4.29 complement(complement(X))
% 30.15/4.29 = { by lemma 31 }
% 30.15/4.29 X
% 30.15/4.29
% 30.15/4.29 Lemma 41: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 30.15/4.29 Proof:
% 30.15/4.29 complement(join(X, complement(Y)))
% 30.15/4.29 = { by lemma 30 R->L }
% 30.15/4.29 complement(join(zero, join(X, complement(Y))))
% 30.15/4.29 = { by lemma 37 R->L }
% 30.15/4.29 complement(join(zero, complement(meet(Y, complement(X)))))
% 30.15/4.29 = { by lemma 33 }
% 30.15/4.29 meet(meet(Y, complement(X)), top)
% 30.15/4.29 = { by lemma 34 }
% 30.15/4.29 meet(Y, complement(X))
% 30.15/4.29
% 30.15/4.29 Lemma 42: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 30.15/4.29 Proof:
% 30.15/4.29 meet(complement(X), complement(Y))
% 30.15/4.29 = { by lemma 32 }
% 30.15/4.29 meet(complement(Y), complement(X))
% 30.15/4.29 = { by lemma 30 R->L }
% 30.15/4.29 meet(join(zero, complement(Y)), complement(X))
% 30.15/4.29 = { by lemma 41 R->L }
% 30.15/4.29 complement(join(X, complement(join(zero, complement(Y)))))
% 30.15/4.29 = { by lemma 33 }
% 30.15/4.29 complement(join(X, meet(Y, top)))
% 30.15/4.29 = { by lemma 34 }
% 30.15/4.29 complement(join(X, Y))
% 30.15/4.29
% 30.15/4.29 Lemma 43: composition(converse(sk1), complement(sk1)) = zero.
% 30.15/4.29 Proof:
% 30.15/4.29 composition(converse(sk1), complement(sk1))
% 30.15/4.29 = { by lemma 30 R->L }
% 30.15/4.29 join(zero, composition(converse(sk1), complement(sk1)))
% 30.15/4.29 = { by lemma 15 R->L }
% 30.15/4.29 join(complement(top), composition(converse(sk1), complement(sk1)))
% 30.15/4.29 = { by axiom 2 (goals_14) R->L }
% 30.15/4.29 join(complement(top), composition(converse(sk1), complement(composition(sk1, top))))
% 30.15/4.29 = { by lemma 18 }
% 30.15/4.29 complement(top)
% 30.15/4.29 = { by lemma 15 }
% 30.15/4.29 zero
% 30.15/4.29
% 30.15/4.29 Lemma 44: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
% 30.15/4.29 Proof:
% 30.15/4.29 join(complement(X), meet(X, Y))
% 30.15/4.29 = { by lemma 32 }
% 30.15/4.29 join(complement(X), meet(Y, X))
% 30.15/4.29 = { by lemma 40 R->L }
% 30.15/4.29 join(join(complement(X), meet(complement(X), Y)), meet(Y, X))
% 30.15/4.29 = { by axiom 10 (maddux2_join_associativity_2) R->L }
% 30.15/4.29 join(complement(X), join(meet(complement(X), Y), meet(Y, X)))
% 30.15/4.29 = { by axiom 3 (maddux1_join_commutativity_1) }
% 30.15/4.29 join(complement(X), join(meet(Y, X), meet(complement(X), Y)))
% 30.15/4.29 = { by lemma 32 }
% 30.15/4.29 join(complement(X), join(meet(Y, X), meet(Y, complement(X))))
% 30.15/4.29 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 30.15/4.29 join(complement(X), join(meet(Y, complement(X)), meet(Y, X)))
% 30.15/4.29 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 30.15/4.29 join(complement(X), join(meet(Y, complement(X)), complement(join(complement(Y), complement(X)))))
% 30.15/4.29 = { by lemma 25 }
% 30.15/4.29 join(complement(X), Y)
% 30.15/4.29 = { by axiom 3 (maddux1_join_commutativity_1) }
% 30.15/4.29 join(Y, complement(X))
% 30.15/4.29
% 30.15/4.29 Lemma 45: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 30.15/4.29 Proof:
% 30.15/4.29 meet(complement(X), join(X, Y))
% 30.15/4.29 = { by lemma 32 }
% 30.15/4.29 meet(join(X, Y), complement(X))
% 30.15/4.29 = { by lemma 41 R->L }
% 30.15/4.29 complement(join(X, complement(join(X, Y))))
% 30.15/4.29 = { by lemma 42 R->L }
% 30.15/4.29 meet(complement(X), complement(complement(join(X, Y))))
% 30.15/4.29 = { by lemma 42 R->L }
% 30.15/4.29 meet(complement(X), complement(meet(complement(X), complement(Y))))
% 30.15/4.29 = { by lemma 36 R->L }
% 30.15/4.29 meet(complement(X), join(complement(complement(X)), complement(complement(Y))))
% 30.15/4.29 = { by lemma 35 R->L }
% 30.15/4.29 complement(join(complement(complement(X)), meet(complement(X), complement(Y))))
% 30.15/4.29 = { by lemma 44 }
% 30.15/4.29 complement(join(complement(Y), complement(complement(X))))
% 30.15/4.29 = { by lemma 41 }
% 30.15/4.29 meet(complement(X), complement(complement(Y)))
% 30.15/4.29 = { by lemma 42 }
% 30.15/4.29 complement(join(X, complement(Y)))
% 30.15/4.29 = { by lemma 41 }
% 30.15/4.29 meet(Y, complement(X))
% 30.15/4.29
% 30.15/4.29 Lemma 46: meet(complement(X), converse(complement(converse(X)))) = complement(X).
% 30.15/4.29 Proof:
% 30.15/4.29 meet(complement(X), converse(complement(converse(X))))
% 30.15/4.29 = { by lemma 32 }
% 30.15/4.29 meet(converse(complement(converse(X))), complement(X))
% 30.15/4.29 = { by lemma 45 R->L }
% 30.15/4.29 meet(complement(X), join(X, converse(complement(converse(X)))))
% 30.15/4.29 = { by lemma 23 R->L }
% 30.15/4.29 meet(complement(X), converse(join(converse(X), complement(converse(X)))))
% 30.15/4.29 = { by axiom 5 (def_top_12) R->L }
% 30.15/4.29 meet(complement(X), converse(top))
% 30.15/4.29 = { by lemma 24 }
% 30.15/4.29 meet(complement(X), top)
% 30.15/4.29 = { by lemma 34 }
% 30.15/4.29 complement(X)
% 30.15/4.29
% 30.15/4.29 Lemma 47: composition(join(X, complement(converse(sk1))), meet(sk1, Y)) = composition(X, meet(sk1, Y)).
% 30.15/4.29 Proof:
% 30.15/4.29 composition(join(X, complement(converse(sk1))), meet(sk1, Y))
% 30.15/4.29 = { by axiom 4 (converse_idempotence_8) R->L }
% 30.15/4.29 composition(join(X, complement(converse(sk1))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by lemma 34 R->L }
% 30.15/4.29 composition(join(X, meet(complement(converse(sk1)), top)), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by lemma 24 R->L }
% 30.15/4.29 composition(join(X, meet(complement(converse(sk1)), converse(top))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by axiom 5 (def_top_12) }
% 30.15/4.29 composition(join(X, meet(complement(converse(sk1)), converse(join(sk1, complement(sk1))))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by axiom 9 (converse_additivity_9) }
% 30.15/4.29 composition(join(X, meet(complement(converse(sk1)), join(converse(sk1), converse(complement(sk1))))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by lemma 45 }
% 30.15/4.29 composition(join(X, meet(converse(complement(sk1)), complement(converse(sk1)))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by lemma 32 R->L }
% 30.15/4.29 composition(join(X, meet(complement(converse(sk1)), converse(complement(sk1)))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by lemma 46 R->L }
% 30.15/4.29 composition(join(X, meet(complement(converse(sk1)), converse(meet(complement(sk1), converse(complement(converse(sk1))))))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by lemma 32 }
% 30.15/4.29 composition(join(X, meet(converse(meet(complement(sk1), converse(complement(converse(sk1))))), complement(converse(sk1)))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by axiom 4 (converse_idempotence_8) R->L }
% 30.15/4.29 composition(join(X, meet(converse(meet(complement(sk1), converse(complement(converse(sk1))))), converse(converse(complement(converse(sk1)))))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by lemma 40 R->L }
% 30.15/4.29 composition(join(X, meet(converse(meet(complement(sk1), converse(complement(converse(sk1))))), converse(join(converse(complement(converse(sk1))), meet(converse(complement(converse(sk1))), complement(sk1)))))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by lemma 23 }
% 30.15/4.29 composition(join(X, meet(converse(meet(complement(sk1), converse(complement(converse(sk1))))), join(complement(converse(sk1)), converse(meet(converse(complement(converse(sk1))), complement(sk1)))))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by lemma 32 R->L }
% 30.15/4.29 composition(join(X, meet(converse(meet(complement(sk1), converse(complement(converse(sk1))))), join(complement(converse(sk1)), converse(meet(complement(sk1), converse(complement(converse(sk1)))))))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 30.15/4.29 composition(join(X, meet(converse(meet(complement(sk1), converse(complement(converse(sk1))))), join(converse(meet(complement(sk1), converse(complement(converse(sk1))))), complement(converse(sk1))))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by lemma 34 R->L }
% 30.15/4.29 composition(join(X, meet(converse(meet(complement(sk1), converse(complement(converse(sk1))))), join(converse(meet(complement(sk1), converse(complement(converse(sk1))))), meet(complement(converse(sk1)), top)))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by lemma 33 R->L }
% 30.15/4.29 composition(join(X, meet(converse(meet(complement(sk1), converse(complement(converse(sk1))))), join(converse(meet(complement(sk1), converse(complement(converse(sk1))))), complement(join(zero, complement(complement(converse(sk1)))))))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by lemma 39 }
% 30.15/4.29 composition(join(X, converse(meet(complement(sk1), converse(complement(converse(sk1)))))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by lemma 46 }
% 30.15/4.29 composition(join(X, converse(complement(sk1))), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 30.15/4.29 composition(join(converse(complement(sk1)), X), converse(converse(meet(sk1, Y))))
% 30.15/4.29 = { by axiom 12 (composition_distributivity_7) }
% 30.15/4.29 join(composition(converse(complement(sk1)), converse(converse(meet(sk1, Y)))), composition(X, converse(converse(meet(sk1, Y)))))
% 30.15/4.29 = { by axiom 7 (converse_multiplicativity_10) R->L }
% 30.15/4.29 join(converse(composition(converse(meet(sk1, Y)), complement(sk1))), composition(X, converse(converse(meet(sk1, Y)))))
% 30.15/4.29 = { by axiom 3 (maddux1_join_commutativity_1) }
% 30.15/4.29 join(composition(X, converse(converse(meet(sk1, Y)))), converse(composition(converse(meet(sk1, Y)), complement(sk1))))
% 30.15/4.29 = { by lemma 30 R->L }
% 30.15/4.29 join(composition(X, converse(converse(meet(sk1, Y)))), converse(join(zero, composition(converse(meet(sk1, Y)), complement(sk1)))))
% 30.15/4.29 = { by lemma 43 R->L }
% 30.15/4.29 join(composition(X, converse(converse(meet(sk1, Y)))), converse(join(composition(converse(sk1), complement(sk1)), composition(converse(meet(sk1, Y)), complement(sk1)))))
% 30.15/4.29 = { by axiom 12 (composition_distributivity_7) R->L }
% 30.15/4.29 join(composition(X, converse(converse(meet(sk1, Y)))), converse(composition(join(converse(sk1), converse(meet(sk1, Y))), complement(sk1))))
% 30.15/4.29 = { by axiom 9 (converse_additivity_9) R->L }
% 30.15/4.29 join(composition(X, converse(converse(meet(sk1, Y)))), converse(composition(converse(join(sk1, meet(sk1, Y))), complement(sk1))))
% 30.15/4.29 = { by lemma 40 }
% 30.15/4.29 join(composition(X, converse(converse(meet(sk1, Y)))), converse(composition(converse(sk1), complement(sk1))))
% 30.15/4.29 = { by lemma 43 }
% 30.15/4.29 join(composition(X, converse(converse(meet(sk1, Y)))), converse(zero))
% 30.15/4.29 = { by axiom 4 (converse_idempotence_8) }
% 30.15/4.29 join(composition(X, meet(sk1, Y)), converse(zero))
% 30.15/4.29 = { by lemma 29 R->L }
% 30.15/4.29 join(composition(X, meet(sk1, Y)), join(converse(zero), zero))
% 30.15/4.29 = { by lemma 28 R->L }
% 30.15/4.29 join(composition(X, meet(sk1, Y)), join(zero, join(zero, converse(zero))))
% 30.15/4.29 = { by lemma 23 R->L }
% 30.15/4.29 join(composition(X, meet(sk1, Y)), join(zero, converse(join(converse(zero), zero))))
% 30.15/4.29 = { by lemma 29 }
% 30.15/4.29 join(composition(X, meet(sk1, Y)), join(zero, converse(converse(zero))))
% 30.15/4.29 = { by axiom 4 (converse_idempotence_8) }
% 30.15/4.29 join(composition(X, meet(sk1, Y)), join(zero, zero))
% 30.15/4.29 = { by lemma 27 }
% 30.15/4.29 join(composition(X, meet(sk1, Y)), zero)
% 30.15/4.29 = { by lemma 29 }
% 30.15/4.29 composition(X, meet(sk1, Y))
% 30.15/4.29
% 30.15/4.29 Goal 1 (goals_15): join(composition(sk2, meet(sk1, sk3)), composition(meet(sk2, converse(sk1)), meet(sk1, sk3))) = composition(meet(sk2, converse(sk1)), meet(sk1, sk3)).
% 30.15/4.29 Proof:
% 30.15/4.29 join(composition(sk2, meet(sk1, sk3)), composition(meet(sk2, converse(sk1)), meet(sk1, sk3)))
% 30.15/4.29 = { by axiom 12 (composition_distributivity_7) R->L }
% 30.15/4.29 composition(join(sk2, meet(sk2, converse(sk1))), meet(sk1, sk3))
% 30.15/4.29 = { by lemma 40 }
% 30.15/4.29 composition(sk2, meet(sk1, sk3))
% 30.15/4.29 = { by lemma 47 R->L }
% 30.15/4.29 composition(join(sk2, complement(converse(sk1))), meet(sk1, sk3))
% 30.15/4.29 = { by lemma 44 R->L }
% 30.15/4.29 composition(join(complement(converse(sk1)), meet(converse(sk1), sk2)), meet(sk1, sk3))
% 30.15/4.29 = { by lemma 32 R->L }
% 30.15/4.29 composition(join(complement(converse(sk1)), meet(sk2, converse(sk1))), meet(sk1, sk3))
% 30.15/4.29 = { by axiom 3 (maddux1_join_commutativity_1) }
% 30.15/4.29 composition(join(meet(sk2, converse(sk1)), complement(converse(sk1))), meet(sk1, sk3))
% 30.15/4.29 = { by lemma 47 }
% 30.15/4.29 composition(meet(sk2, converse(sk1)), meet(sk1, sk3))
% 30.15/4.29 % SZS output end Proof
% 30.15/4.29
% 30.15/4.29 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------