TSTP Solution File: REL025-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL025-1 : 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 : n013.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 13:44:05 EDT 2023
% Result : Unsatisfiable 15.03s 2.36s
% Output : Proof 16.43s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : REL025-1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.18/0.35 % Computer : n013.cluster.edu
% 0.18/0.35 % Model : x86_64 x86_64
% 0.18/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.35 % Memory : 8042.1875MB
% 0.18/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.35 % CPULimit : 300
% 0.18/0.35 % WCLimit : 300
% 0.18/0.35 % DateTime : Fri Aug 25 18:54:32 EDT 2023
% 0.18/0.35 % CPUTime :
% 15.03/2.36 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 15.03/2.36
% 15.03/2.36 % SZS status Unsatisfiable
% 15.03/2.36
% 15.66/2.46 % SZS output start Proof
% 15.66/2.46 Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 15.66/2.46 Axiom 2 (goals_14): join(sk1, one) = one.
% 15.66/2.46 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 15.66/2.46 Axiom 4 (converse_idempotence_8): converse(converse(X)) = X.
% 15.66/2.46 Axiom 5 (def_top_12): top = join(X, complement(X)).
% 15.66/2.46 Axiom 6 (def_zero_13): zero = meet(X, complement(X)).
% 15.66/2.46 Axiom 7 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 15.66/2.46 Axiom 8 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 15.66/2.46 Axiom 9 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 15.66/2.46 Axiom 10 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 15.66/2.46 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 15.66/2.46 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 15.66/2.46 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 15.66/2.46 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 15.66/2.46
% 15.66/2.46 Lemma 15: complement(top) = zero.
% 15.66/2.46 Proof:
% 15.66/2.46 complement(top)
% 15.66/2.46 = { by axiom 5 (def_top_12) }
% 15.66/2.46 complement(join(complement(X), complement(complement(X))))
% 15.66/2.46 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.66/2.46 meet(X, complement(X))
% 15.66/2.46 = { by axiom 6 (def_zero_13) R->L }
% 15.66/2.46 zero
% 15.66/2.46
% 15.66/2.46 Lemma 16: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 15.66/2.46 Proof:
% 15.66/2.46 converse(composition(converse(X), Y))
% 15.66/2.46 = { by axiom 9 (converse_multiplicativity_10) }
% 15.66/2.46 composition(converse(Y), converse(converse(X)))
% 15.66/2.46 = { by axiom 4 (converse_idempotence_8) }
% 15.66/2.46 composition(converse(Y), X)
% 15.66/2.46
% 15.66/2.46 Lemma 17: composition(converse(one), X) = X.
% 15.66/2.46 Proof:
% 15.66/2.46 composition(converse(one), X)
% 15.66/2.46 = { by lemma 16 R->L }
% 15.66/2.46 converse(composition(converse(X), one))
% 15.66/2.46 = { by axiom 3 (composition_identity_6) }
% 15.66/2.46 converse(converse(X))
% 15.66/2.46 = { by axiom 4 (converse_idempotence_8) }
% 15.66/2.46 X
% 15.66/2.46
% 15.66/2.46 Lemma 18: converse(one) = one.
% 15.66/2.46 Proof:
% 15.66/2.46 converse(one)
% 15.66/2.46 = { by axiom 3 (composition_identity_6) R->L }
% 15.66/2.46 composition(converse(one), one)
% 15.66/2.46 = { by lemma 17 }
% 15.66/2.46 one
% 15.66/2.46
% 15.66/2.46 Lemma 19: join(X, join(Y, complement(X))) = join(Y, top).
% 15.66/2.46 Proof:
% 15.66/2.46 join(X, join(Y, complement(X)))
% 15.66/2.46 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 15.66/2.46 join(X, join(complement(X), Y))
% 15.66/2.46 = { by axiom 8 (maddux2_join_associativity_2) }
% 15.66/2.46 join(join(X, complement(X)), Y)
% 15.66/2.46 = { by axiom 5 (def_top_12) R->L }
% 15.66/2.46 join(top, Y)
% 15.66/2.46 = { by axiom 1 (maddux1_join_commutativity_1) }
% 15.66/2.47 join(Y, top)
% 15.66/2.47
% 15.66/2.47 Lemma 20: composition(one, X) = X.
% 15.66/2.47 Proof:
% 15.66/2.47 composition(one, X)
% 15.66/2.47 = { by lemma 17 R->L }
% 15.66/2.47 composition(converse(one), composition(one, X))
% 15.66/2.47 = { by axiom 10 (composition_associativity_5) }
% 15.66/2.47 composition(composition(converse(one), one), X)
% 15.66/2.47 = { by axiom 3 (composition_identity_6) }
% 15.66/2.47 composition(converse(one), X)
% 15.66/2.47 = { by lemma 17 }
% 15.66/2.47 X
% 15.66/2.47
% 15.66/2.47 Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 15.66/2.47 Proof:
% 15.66/2.47 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 15.66/2.47 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 15.66/2.47 = { by axiom 13 (converse_cancellativity_11) }
% 15.66/2.47 complement(X)
% 15.66/2.47
% 15.66/2.47 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 15.66/2.47 Proof:
% 15.66/2.47 join(complement(X), complement(X))
% 15.66/2.47 = { by lemma 17 R->L }
% 15.66/2.47 join(complement(X), composition(converse(one), complement(X)))
% 15.66/2.47 = { by lemma 20 R->L }
% 15.66/2.47 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 15.66/2.47 = { by lemma 21 }
% 15.66/2.47 complement(X)
% 15.66/2.47
% 15.66/2.47 Lemma 23: join(top, complement(X)) = top.
% 15.66/2.47 Proof:
% 15.66/2.47 join(top, complement(X))
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 15.66/2.47 join(complement(X), top)
% 15.66/2.47 = { by lemma 19 R->L }
% 15.66/2.47 join(X, join(complement(X), complement(X)))
% 15.66/2.47 = { by lemma 22 }
% 15.66/2.47 join(X, complement(X))
% 15.66/2.47 = { by axiom 5 (def_top_12) R->L }
% 15.66/2.47 top
% 15.66/2.47
% 15.66/2.47 Lemma 24: join(Y, top) = join(X, top).
% 15.66/2.47 Proof:
% 15.66/2.47 join(Y, top)
% 15.66/2.47 = { by lemma 23 R->L }
% 15.66/2.47 join(Y, join(top, complement(Y)))
% 15.66/2.47 = { by lemma 19 }
% 15.66/2.47 join(top, top)
% 15.66/2.47 = { by lemma 19 R->L }
% 15.66/2.47 join(X, join(top, complement(X)))
% 15.66/2.47 = { by lemma 23 }
% 15.66/2.47 join(X, top)
% 15.66/2.47
% 15.66/2.47 Lemma 25: join(sk1, join(one, X)) = join(X, one).
% 15.66/2.47 Proof:
% 15.66/2.47 join(sk1, join(one, X))
% 15.66/2.47 = { by axiom 8 (maddux2_join_associativity_2) }
% 15.66/2.47 join(join(sk1, one), X)
% 15.66/2.47 = { by axiom 2 (goals_14) }
% 15.66/2.47 join(one, X)
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) }
% 15.66/2.47 join(X, one)
% 15.66/2.47
% 15.66/2.47 Lemma 26: join(X, top) = top.
% 15.66/2.47 Proof:
% 15.66/2.47 join(X, top)
% 15.66/2.47 = { by lemma 24 }
% 15.66/2.47 join(sk1, top)
% 15.66/2.47 = { by axiom 5 (def_top_12) }
% 15.66/2.47 join(sk1, join(one, complement(one)))
% 15.66/2.47 = { by lemma 25 }
% 15.66/2.47 join(complement(one), one)
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) }
% 15.66/2.47 join(one, complement(one))
% 15.66/2.47 = { by axiom 5 (def_top_12) R->L }
% 15.66/2.47 top
% 15.66/2.47
% 15.66/2.47 Lemma 27: join(top, X) = top.
% 15.66/2.47 Proof:
% 15.66/2.47 join(top, X)
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 15.66/2.47 join(X, top)
% 15.66/2.47 = { by lemma 24 R->L }
% 15.66/2.47 join(Y, top)
% 15.66/2.47 = { by lemma 26 }
% 15.66/2.47 top
% 15.66/2.47
% 15.66/2.47 Lemma 28: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 15.66/2.47 Proof:
% 15.66/2.47 converse(join(X, converse(Y)))
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 15.66/2.47 converse(join(converse(Y), X))
% 15.66/2.47 = { by axiom 7 (converse_additivity_9) }
% 15.66/2.47 join(converse(converse(Y)), converse(X))
% 15.66/2.47 = { by axiom 4 (converse_idempotence_8) }
% 15.66/2.47 join(Y, converse(X))
% 15.66/2.47
% 15.66/2.47 Lemma 29: converse(top) = top.
% 15.66/2.47 Proof:
% 15.66/2.47 converse(top)
% 15.66/2.47 = { by lemma 27 R->L }
% 15.66/2.47 converse(join(top, converse(top)))
% 15.66/2.47 = { by lemma 28 }
% 15.66/2.47 join(top, converse(top))
% 15.66/2.47 = { by lemma 27 }
% 15.66/2.47 top
% 15.66/2.47
% 15.66/2.47 Lemma 30: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 15.66/2.47 Proof:
% 15.66/2.47 join(meet(X, Y), complement(join(complement(X), Y)))
% 15.66/2.47 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.66/2.47 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 15.66/2.47 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 15.66/2.47 X
% 15.66/2.47
% 15.66/2.47 Lemma 31: join(zero, meet(X, X)) = X.
% 15.66/2.47 Proof:
% 15.66/2.47 join(zero, meet(X, X))
% 15.66/2.47 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.66/2.47 join(zero, complement(join(complement(X), complement(X))))
% 15.66/2.47 = { by axiom 6 (def_zero_13) }
% 15.66/2.47 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 15.66/2.47 = { by lemma 30 }
% 15.66/2.47 X
% 15.66/2.47
% 15.66/2.47 Lemma 32: join(zero, complement(complement(X))) = X.
% 15.66/2.47 Proof:
% 15.66/2.47 join(zero, complement(complement(X)))
% 15.66/2.47 = { by axiom 6 (def_zero_13) }
% 15.66/2.47 join(meet(X, complement(X)), complement(complement(X)))
% 15.66/2.47 = { by lemma 22 R->L }
% 15.66/2.47 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 15.66/2.47 = { by lemma 30 }
% 15.66/2.47 X
% 15.66/2.47
% 15.66/2.47 Lemma 33: join(X, zero) = X.
% 15.66/2.47 Proof:
% 15.66/2.47 join(X, zero)
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 15.66/2.47 join(zero, X)
% 15.66/2.47 = { by lemma 31 R->L }
% 15.66/2.47 join(zero, join(zero, meet(X, X)))
% 15.66/2.47 = { by axiom 8 (maddux2_join_associativity_2) }
% 15.66/2.47 join(join(zero, zero), meet(X, X))
% 15.66/2.47 = { by lemma 15 R->L }
% 15.66/2.47 join(join(zero, complement(top)), meet(X, X))
% 15.66/2.47 = { by lemma 15 R->L }
% 15.66/2.47 join(join(complement(top), complement(top)), meet(X, X))
% 15.66/2.47 = { by lemma 22 }
% 15.66/2.47 join(complement(top), meet(X, X))
% 15.66/2.47 = { by lemma 15 }
% 15.66/2.47 join(zero, meet(X, X))
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) }
% 15.66/2.47 join(meet(X, X), zero)
% 15.66/2.47 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.66/2.47 join(complement(join(complement(X), complement(X))), zero)
% 15.66/2.47 = { by lemma 22 }
% 15.66/2.47 join(complement(complement(X)), zero)
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) }
% 15.66/2.47 join(zero, complement(complement(X)))
% 15.66/2.47 = { by lemma 32 }
% 15.66/2.47 X
% 15.66/2.47
% 15.66/2.47 Lemma 34: join(zero, X) = X.
% 15.66/2.47 Proof:
% 15.66/2.47 join(zero, X)
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 15.66/2.47 join(X, zero)
% 15.66/2.47 = { by lemma 33 }
% 15.66/2.47 X
% 15.66/2.47
% 15.66/2.47 Lemma 35: complement(complement(X)) = X.
% 15.66/2.47 Proof:
% 15.66/2.47 complement(complement(X))
% 15.66/2.47 = { by lemma 34 R->L }
% 15.66/2.47 join(zero, complement(complement(X)))
% 15.66/2.47 = { by lemma 32 }
% 15.66/2.47 X
% 15.66/2.47
% 15.66/2.47 Lemma 36: meet(Y, X) = meet(X, Y).
% 15.66/2.47 Proof:
% 15.66/2.47 meet(Y, X)
% 15.66/2.47 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.66/2.47 complement(join(complement(Y), complement(X)))
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 15.66/2.47 complement(join(complement(X), complement(Y)))
% 15.66/2.47 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.66/2.47 meet(X, Y)
% 15.66/2.47
% 15.66/2.47 Lemma 37: complement(join(zero, complement(X))) = meet(X, top).
% 15.66/2.47 Proof:
% 15.66/2.47 complement(join(zero, complement(X)))
% 15.66/2.47 = { by lemma 15 R->L }
% 15.66/2.47 complement(join(complement(top), complement(X)))
% 15.66/2.47 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.66/2.47 meet(top, X)
% 15.66/2.47 = { by lemma 36 R->L }
% 15.66/2.47 meet(X, top)
% 15.66/2.47
% 15.66/2.47 Lemma 38: meet(X, top) = X.
% 15.66/2.47 Proof:
% 15.66/2.47 meet(X, top)
% 15.66/2.47 = { by lemma 37 R->L }
% 15.66/2.47 complement(join(zero, complement(X)))
% 15.66/2.47 = { by lemma 34 }
% 15.66/2.47 complement(complement(X))
% 15.66/2.47 = { by lemma 35 }
% 15.66/2.47 X
% 15.66/2.47
% 15.66/2.47 Lemma 39: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 15.66/2.47 Proof:
% 15.66/2.47 meet(X, join(complement(Y), complement(Z)))
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 15.66/2.47 meet(X, join(complement(Z), complement(Y)))
% 15.66/2.47 = { by lemma 36 }
% 15.66/2.47 meet(join(complement(Z), complement(Y)), X)
% 15.66/2.47 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.66/2.47 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 15.66/2.47 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.66/2.47 complement(join(meet(Z, Y), complement(X)))
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) }
% 15.66/2.47 complement(join(complement(X), meet(Z, Y)))
% 15.66/2.47 = { by lemma 36 R->L }
% 15.66/2.47 complement(join(complement(X), meet(Y, Z)))
% 15.66/2.47
% 15.66/2.47 Lemma 40: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 15.66/2.47 Proof:
% 15.66/2.47 complement(join(X, complement(Y)))
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 15.66/2.47 complement(join(complement(Y), X))
% 15.66/2.47 = { by lemma 38 R->L }
% 15.66/2.47 complement(join(complement(Y), meet(X, top)))
% 15.66/2.47 = { by lemma 36 R->L }
% 15.66/2.47 complement(join(complement(Y), meet(top, X)))
% 15.66/2.47 = { by lemma 39 R->L }
% 15.66/2.47 meet(Y, join(complement(top), complement(X)))
% 15.66/2.47 = { by lemma 15 }
% 15.66/2.47 meet(Y, join(zero, complement(X)))
% 15.66/2.47 = { by lemma 34 }
% 15.66/2.47 meet(Y, complement(X))
% 15.66/2.47
% 15.66/2.47 Lemma 41: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 15.66/2.47 Proof:
% 15.66/2.47 complement(meet(X, complement(Y)))
% 15.66/2.47 = { by lemma 34 R->L }
% 15.66/2.47 complement(join(zero, meet(X, complement(Y))))
% 15.66/2.47 = { by lemma 40 R->L }
% 15.66/2.47 complement(join(zero, complement(join(Y, complement(X)))))
% 15.66/2.47 = { by lemma 37 }
% 15.66/2.47 meet(join(Y, complement(X)), top)
% 15.66/2.47 = { by lemma 38 }
% 15.66/2.47 join(Y, complement(X))
% 15.66/2.47
% 15.66/2.47 Lemma 42: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 15.66/2.47 Proof:
% 15.66/2.47 complement(join(complement(X), Y))
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 15.66/2.47 complement(join(Y, complement(X)))
% 15.66/2.47 = { by lemma 40 }
% 15.66/2.47 meet(X, complement(Y))
% 15.66/2.47
% 15.66/2.47 Lemma 43: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 15.66/2.47 Proof:
% 15.66/2.47 complement(meet(complement(X), Y))
% 15.66/2.47 = { by lemma 36 }
% 15.66/2.47 complement(meet(Y, complement(X)))
% 15.66/2.47 = { by lemma 41 }
% 15.66/2.47 join(X, complement(Y))
% 15.66/2.47
% 15.66/2.47 Lemma 44: meet(X, join(X, complement(Y))) = X.
% 15.66/2.47 Proof:
% 15.66/2.47 meet(X, join(X, complement(Y)))
% 15.66/2.47 = { by lemma 33 R->L }
% 15.66/2.47 join(meet(X, join(X, complement(Y))), zero)
% 15.66/2.47 = { by lemma 15 R->L }
% 15.66/2.47 join(meet(X, join(X, complement(Y))), complement(top))
% 15.66/2.47 = { by lemma 43 R->L }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(top))
% 15.66/2.47 = { by lemma 26 R->L }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(Y), top)))
% 15.66/2.47 = { by lemma 19 R->L }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), join(complement(Y), complement(complement(X))))))
% 15.66/2.47 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), join(complement(complement(X)), complement(Y)))))
% 15.66/2.47 = { by lemma 31 R->L }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), join(zero, meet(join(complement(complement(X)), complement(Y)), join(complement(complement(X)), complement(Y)))))))
% 15.66/2.47 = { by lemma 39 }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), join(zero, complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y)))))))
% 15.66/2.47 = { by lemma 34 }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y))))))
% 15.66/2.47 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(join(meet(complement(X), Y), meet(complement(X), Y))))))
% 15.66/2.47 = { by lemma 36 }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(join(meet(Y, complement(X)), meet(complement(X), Y))))))
% 15.66/2.47 = { by lemma 36 }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(join(meet(Y, complement(X)), meet(Y, complement(X)))))))
% 15.66/2.47 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(join(meet(Y, complement(X)), complement(join(complement(Y), complement(complement(X)))))))))
% 15.66/2.47 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(join(complement(join(complement(Y), complement(complement(X)))), complement(join(complement(Y), complement(complement(X)))))))))
% 15.66/2.47 = { by lemma 22 }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(complement(join(complement(Y), complement(complement(X))))))))
% 15.66/2.47 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(Y, complement(X))))))
% 15.66/2.47 = { by lemma 36 R->L }
% 15.66/2.47 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(complement(X), Y)))))
% 15.66/2.47 = { by lemma 30 }
% 15.66/2.48 X
% 15.66/2.48
% 15.66/2.48 Lemma 45: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 15.66/2.48 Proof:
% 15.66/2.48 meet(X, complement(meet(X, Y)))
% 15.66/2.48 = { by lemma 42 R->L }
% 15.66/2.48 complement(join(complement(X), meet(X, Y)))
% 15.66/2.48 = { by lemma 36 }
% 15.66/2.48 complement(join(complement(X), meet(Y, X)))
% 15.66/2.48 = { by lemma 35 R->L }
% 15.66/2.48 complement(join(complement(complement(complement(X))), meet(Y, X)))
% 15.66/2.48 = { by lemma 44 R->L }
% 15.66/2.48 complement(join(complement(meet(complement(complement(X)), join(complement(complement(X)), complement(Y)))), meet(Y, X)))
% 15.66/2.48 = { by lemma 43 }
% 15.66/2.48 complement(join(join(complement(X), complement(join(complement(complement(X)), complement(Y)))), meet(Y, X)))
% 15.66/2.48 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.66/2.48 complement(join(join(complement(X), meet(complement(X), Y)), meet(Y, X)))
% 15.66/2.48 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 15.66/2.48 complement(join(complement(X), join(meet(complement(X), Y), meet(Y, X))))
% 15.66/2.48 = { by axiom 1 (maddux1_join_commutativity_1) }
% 15.66/2.48 complement(join(complement(X), join(meet(Y, X), meet(complement(X), Y))))
% 15.66/2.48 = { by lemma 36 }
% 15.66/2.48 complement(join(complement(X), join(meet(Y, X), meet(Y, complement(X)))))
% 15.66/2.48 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 15.66/2.48 complement(join(complement(X), join(meet(Y, complement(X)), meet(Y, X))))
% 15.66/2.48 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.66/2.48 complement(join(complement(X), join(meet(Y, complement(X)), complement(join(complement(Y), complement(X))))))
% 15.66/2.48 = { by lemma 30 }
% 15.66/2.48 complement(join(complement(X), Y))
% 15.66/2.48 = { by axiom 1 (maddux1_join_commutativity_1) }
% 15.66/2.48 complement(join(Y, complement(X)))
% 15.66/2.48 = { by lemma 40 }
% 15.66/2.48 meet(X, complement(Y))
% 16.43/2.48
% 16.43/2.48 Lemma 46: meet(X, join(complement(X), Y)) = meet(X, Y).
% 16.43/2.48 Proof:
% 16.43/2.48 meet(X, join(complement(X), Y))
% 16.43/2.48 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 16.43/2.48 meet(X, join(Y, complement(X)))
% 16.43/2.48 = { by lemma 41 R->L }
% 16.43/2.48 meet(X, complement(meet(X, complement(Y))))
% 16.43/2.48 = { by lemma 45 }
% 16.43/2.48 meet(X, complement(complement(Y)))
% 16.43/2.48 = { by lemma 35 }
% 16.43/2.48 meet(X, Y)
% 16.43/2.48
% 16.43/2.48 Lemma 47: composition(join(X, one), Y) = join(Y, composition(X, Y)).
% 16.43/2.48 Proof:
% 16.43/2.48 composition(join(X, one), Y)
% 16.43/2.48 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 16.43/2.48 composition(join(one, X), Y)
% 16.43/2.48 = { by axiom 12 (composition_distributivity_7) }
% 16.43/2.48 join(composition(one, Y), composition(X, Y))
% 16.43/2.48 = { by lemma 20 }
% 16.43/2.48 join(Y, composition(X, Y))
% 16.43/2.48
% 16.43/2.48 Lemma 48: join(X, composition(sk1, complement(X))) = join(X, composition(sk1, top)).
% 16.43/2.48 Proof:
% 16.43/2.48 join(X, composition(sk1, complement(X)))
% 16.43/2.48 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 16.43/2.48 join(composition(sk1, complement(X)), X)
% 16.43/2.48 = { by lemma 20 R->L }
% 16.43/2.48 join(composition(sk1, complement(X)), composition(one, X))
% 16.43/2.48 = { by axiom 2 (goals_14) R->L }
% 16.43/2.48 join(composition(sk1, complement(X)), composition(join(sk1, one), X))
% 16.43/2.48 = { by lemma 47 }
% 16.43/2.48 join(composition(sk1, complement(X)), join(X, composition(sk1, X)))
% 16.43/2.48 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 16.43/2.48 join(join(X, composition(sk1, X)), composition(sk1, complement(X)))
% 16.43/2.48 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 16.43/2.48 join(X, join(composition(sk1, X), composition(sk1, complement(X))))
% 16.43/2.48 = { by axiom 4 (converse_idempotence_8) R->L }
% 16.43/2.48 join(X, converse(converse(join(composition(sk1, X), composition(sk1, complement(X))))))
% 16.43/2.48 = { by axiom 7 (converse_additivity_9) }
% 16.43/2.48 join(X, converse(join(converse(composition(sk1, X)), converse(composition(sk1, complement(X))))))
% 16.43/2.48 = { by axiom 9 (converse_multiplicativity_10) }
% 16.43/2.48 join(X, converse(join(composition(converse(X), converse(sk1)), converse(composition(sk1, complement(X))))))
% 16.43/2.48 = { by axiom 9 (converse_multiplicativity_10) }
% 16.43/2.48 join(X, converse(join(composition(converse(X), converse(sk1)), composition(converse(complement(X)), converse(sk1)))))
% 16.43/2.48 = { by axiom 12 (composition_distributivity_7) R->L }
% 16.43/2.48 join(X, converse(composition(join(converse(X), converse(complement(X))), converse(sk1))))
% 16.43/2.48 = { by axiom 7 (converse_additivity_9) R->L }
% 16.43/2.48 join(X, converse(composition(converse(join(X, complement(X))), converse(sk1))))
% 16.43/2.48 = { by lemma 16 }
% 16.43/2.48 join(X, composition(converse(converse(sk1)), join(X, complement(X))))
% 16.43/2.48 = { by axiom 4 (converse_idempotence_8) }
% 16.43/2.48 join(X, composition(sk1, join(X, complement(X))))
% 16.43/2.48 = { by axiom 5 (def_top_12) R->L }
% 16.43/2.48 join(X, composition(sk1, top))
% 16.43/2.48
% 16.43/2.48 Lemma 49: join(X, composition(sk1, X)) = X.
% 16.43/2.48 Proof:
% 16.43/2.48 join(X, composition(sk1, X))
% 16.43/2.48 = { by lemma 47 R->L }
% 16.43/2.48 composition(join(sk1, one), X)
% 16.43/2.48 = { by axiom 2 (goals_14) }
% 16.43/2.48 composition(one, X)
% 16.43/2.48 = { by lemma 20 }
% 16.43/2.48 X
% 16.43/2.48
% 16.43/2.48 Lemma 50: meet(X, join(X, Y)) = X.
% 16.43/2.48 Proof:
% 16.43/2.48 meet(X, join(X, Y))
% 16.43/2.48 = { by lemma 38 R->L }
% 16.43/2.48 meet(X, join(X, meet(Y, top)))
% 16.43/2.48 = { by lemma 37 R->L }
% 16.43/2.48 meet(X, join(X, complement(join(zero, complement(Y)))))
% 16.43/2.48 = { by lemma 44 }
% 16.43/2.48 X
% 16.43/2.48
% 16.43/2.48 Lemma 51: meet(X, composition(sk1, top)) = composition(sk1, X).
% 16.43/2.48 Proof:
% 16.43/2.48 meet(X, composition(sk1, top))
% 16.43/2.48 = { by lemma 46 R->L }
% 16.43/2.48 meet(X, join(complement(X), composition(sk1, top)))
% 16.43/2.48 = { by lemma 48 R->L }
% 16.43/2.48 meet(X, join(complement(X), composition(sk1, complement(complement(X)))))
% 16.43/2.48 = { by lemma 46 }
% 16.43/2.48 meet(X, composition(sk1, complement(complement(X))))
% 16.43/2.48 = { by lemma 35 }
% 16.43/2.48 meet(X, composition(sk1, X))
% 16.43/2.48 = { by lemma 36 }
% 16.43/2.48 meet(composition(sk1, X), X)
% 16.43/2.48 = { by lemma 49 R->L }
% 16.43/2.48 meet(composition(sk1, X), join(X, composition(sk1, X)))
% 16.43/2.48 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 16.43/2.48 meet(composition(sk1, X), join(composition(sk1, X), X))
% 16.43/2.48 = { by lemma 50 }
% 16.43/2.48 composition(sk1, X)
% 16.43/2.48
% 16.43/2.48 Lemma 52: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 16.43/2.48 Proof:
% 16.43/2.48 meet(complement(X), complement(Y))
% 16.43/2.48 = { by lemma 36 }
% 16.43/2.48 meet(complement(Y), complement(X))
% 16.43/2.48 = { by lemma 34 R->L }
% 16.43/2.48 meet(join(zero, complement(Y)), complement(X))
% 16.43/2.48 = { by lemma 40 R->L }
% 16.43/2.48 complement(join(X, complement(join(zero, complement(Y)))))
% 16.43/2.48 = { by lemma 37 }
% 16.43/2.48 complement(join(X, meet(Y, top)))
% 16.43/2.48 = { by lemma 38 }
% 16.43/2.48 complement(join(X, Y))
% 16.43/2.48
% 16.43/2.48 Lemma 53: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 16.43/2.48 Proof:
% 16.43/2.48 meet(complement(X), join(X, Y))
% 16.43/2.48 = { by lemma 36 }
% 16.43/2.48 meet(join(X, Y), complement(X))
% 16.43/2.48 = { by lemma 40 R->L }
% 16.43/2.48 complement(join(X, complement(join(X, Y))))
% 16.43/2.48 = { by lemma 52 R->L }
% 16.43/2.48 meet(complement(X), complement(complement(join(X, Y))))
% 16.43/2.48 = { by lemma 52 R->L }
% 16.43/2.48 meet(complement(X), complement(meet(complement(X), complement(Y))))
% 16.43/2.48 = { by lemma 45 }
% 16.43/2.48 meet(complement(X), complement(complement(Y)))
% 16.43/2.48 = { by lemma 52 }
% 16.43/2.48 complement(join(X, complement(Y)))
% 16.43/2.48 = { by lemma 40 }
% 16.43/2.48 meet(Y, complement(X))
% 16.43/2.48
% 16.43/2.48 Lemma 54: join(one, complement(converse(sk1))) = top.
% 16.43/2.48 Proof:
% 16.43/2.48 join(one, complement(converse(sk1)))
% 16.43/2.48 = { by lemma 18 R->L }
% 16.43/2.48 join(converse(one), complement(converse(sk1)))
% 16.43/2.48 = { by axiom 2 (goals_14) R->L }
% 16.43/2.48 join(converse(join(sk1, one)), complement(converse(sk1)))
% 16.43/2.48 = { by axiom 7 (converse_additivity_9) }
% 16.43/2.48 join(join(converse(sk1), converse(one)), complement(converse(sk1)))
% 16.43/2.48 = { by lemma 18 }
% 16.43/2.48 join(join(converse(sk1), one), complement(converse(sk1)))
% 16.43/2.48 = { by axiom 1 (maddux1_join_commutativity_1) }
% 16.43/2.48 join(join(one, converse(sk1)), complement(converse(sk1)))
% 16.43/2.48 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 16.43/2.48 join(one, join(converse(sk1), complement(converse(sk1))))
% 16.43/2.48 = { by axiom 5 (def_top_12) R->L }
% 16.43/2.48 join(one, top)
% 16.43/2.48 = { by lemma 26 }
% 16.43/2.48 top
% 16.43/2.48
% 16.43/2.48 Lemma 55: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 16.43/2.48 Proof:
% 16.43/2.48 converse(composition(X, converse(Y)))
% 16.43/2.48 = { by axiom 9 (converse_multiplicativity_10) }
% 16.43/2.48 composition(converse(converse(Y)), converse(X))
% 16.43/2.48 = { by axiom 4 (converse_idempotence_8) }
% 16.43/2.48 composition(Y, converse(X))
% 16.43/2.48
% 16.43/2.48 Lemma 56: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 16.43/2.48 Proof:
% 16.43/2.48 converse(join(converse(X), Y))
% 16.43/2.48 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 16.43/2.48 converse(join(Y, converse(X)))
% 16.43/2.48 = { by lemma 28 }
% 16.43/2.48 join(X, converse(Y))
% 16.43/2.48
% 16.43/2.48 Lemma 57: meet(complement(X), join(Y, complement(Z))) = complement(join(X, meet(Z, complement(Y)))).
% 16.43/2.48 Proof:
% 16.43/2.48 meet(complement(X), join(Y, complement(Z)))
% 16.43/2.48 = { by lemma 36 }
% 16.43/2.48 meet(join(Y, complement(Z)), complement(X))
% 16.43/2.48 = { by lemma 40 R->L }
% 16.43/2.48 complement(join(X, complement(join(Y, complement(Z)))))
% 16.43/2.48 = { by lemma 40 }
% 16.43/2.48 complement(join(X, meet(Z, complement(Y))))
% 16.43/2.48
% 16.43/2.48 Lemma 58: composition(join(one, Y), X) = join(X, composition(Y, X)).
% 16.43/2.48 Proof:
% 16.43/2.48 composition(join(one, Y), X)
% 16.43/2.48 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 16.43/2.48 composition(join(Y, one), X)
% 16.43/2.48 = { by lemma 47 }
% 16.43/2.48 join(X, composition(Y, X))
% 16.43/2.48
% 16.43/2.48 Lemma 59: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 16.43/2.48 Proof:
% 16.43/2.48 meet(Y, meet(Z, X))
% 16.43/2.48 = { by lemma 38 R->L }
% 16.43/2.48 meet(meet(Y, top), meet(Z, X))
% 16.43/2.48 = { by lemma 37 R->L }
% 16.43/2.48 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 16.43/2.48 = { by lemma 36 }
% 16.43/2.48 meet(complement(join(zero, complement(Y))), meet(X, Z))
% 16.43/2.48 = { by lemma 36 }
% 16.43/2.48 meet(meet(X, Z), complement(join(zero, complement(Y))))
% 16.43/2.48 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 16.43/2.48 meet(complement(join(complement(X), complement(Z))), complement(join(zero, complement(Y))))
% 16.43/2.48 = { by lemma 52 }
% 16.43/2.48 complement(join(join(complement(X), complement(Z)), join(zero, complement(Y))))
% 16.43/2.48 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 16.43/2.48 complement(join(complement(X), join(complement(Z), join(zero, complement(Y)))))
% 16.43/2.48 = { by lemma 42 }
% 16.43/2.48 meet(X, complement(join(complement(Z), join(zero, complement(Y)))))
% 16.43/2.48 = { by lemma 42 }
% 16.43/2.48 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 16.43/2.48 = { by lemma 37 }
% 16.43/2.48 meet(X, meet(Z, meet(Y, top)))
% 16.43/2.48 = { by lemma 38 }
% 16.43/2.48 meet(X, meet(Z, Y))
% 16.43/2.48 = { by lemma 36 R->L }
% 16.43/2.48 meet(X, meet(Y, Z))
% 16.43/2.48
% 16.43/2.48 Lemma 60: composition(sk1, converse(sk1)) = sk1.
% 16.43/2.48 Proof:
% 16.43/2.48 composition(sk1, converse(sk1))
% 16.43/2.48 = { by lemma 51 R->L }
% 16.43/2.48 meet(converse(sk1), composition(sk1, top))
% 16.43/2.48 = { by lemma 36 }
% 16.43/2.48 meet(composition(sk1, top), converse(sk1))
% 16.43/2.48 = { by lemma 35 R->L }
% 16.43/2.48 meet(composition(sk1, top), complement(complement(converse(sk1))))
% 16.43/2.48 = { by lemma 53 R->L }
% 16.43/2.48 meet(complement(complement(converse(sk1))), join(complement(converse(sk1)), composition(sk1, top)))
% 16.43/2.48 = { by lemma 54 R->L }
% 16.43/2.48 meet(complement(complement(converse(sk1))), join(complement(converse(sk1)), composition(sk1, join(one, complement(converse(sk1))))))
% 16.43/2.48 = { by axiom 4 (converse_idempotence_8) R->L }
% 16.43/2.48 meet(complement(complement(converse(sk1))), join(complement(converse(sk1)), composition(sk1, join(one, converse(converse(complement(converse(sk1))))))))
% 16.43/2.48 = { by lemma 18 R->L }
% 16.43/2.48 meet(complement(complement(converse(sk1))), join(complement(converse(sk1)), composition(sk1, join(converse(one), converse(converse(complement(converse(sk1))))))))
% 16.43/2.49 = { by axiom 7 (converse_additivity_9) R->L }
% 16.43/2.49 meet(complement(complement(converse(sk1))), join(complement(converse(sk1)), composition(sk1, converse(join(one, converse(complement(converse(sk1))))))))
% 16.43/2.49 = { by axiom 1 (maddux1_join_commutativity_1) }
% 16.43/2.49 meet(complement(complement(converse(sk1))), join(complement(converse(sk1)), composition(sk1, converse(join(converse(complement(converse(sk1))), one)))))
% 16.43/2.49 = { by lemma 55 R->L }
% 16.43/2.49 meet(complement(complement(converse(sk1))), join(complement(converse(sk1)), converse(composition(join(converse(complement(converse(sk1))), one), converse(sk1)))))
% 16.43/2.49 = { by lemma 47 }
% 16.43/2.49 meet(complement(complement(converse(sk1))), join(complement(converse(sk1)), converse(join(converse(sk1), composition(converse(complement(converse(sk1))), converse(sk1))))))
% 16.43/2.49 = { by lemma 56 }
% 16.43/2.49 meet(complement(complement(converse(sk1))), join(complement(converse(sk1)), join(sk1, converse(composition(converse(complement(converse(sk1))), converse(sk1))))))
% 16.43/2.49 = { by lemma 55 }
% 16.43/2.49 meet(complement(complement(converse(sk1))), join(complement(converse(sk1)), join(sk1, composition(sk1, converse(converse(complement(converse(sk1))))))))
% 16.43/2.49 = { by axiom 4 (converse_idempotence_8) }
% 16.43/2.49 meet(complement(complement(converse(sk1))), join(complement(converse(sk1)), join(sk1, composition(sk1, complement(converse(sk1))))))
% 16.43/2.49 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 16.43/2.49 meet(complement(complement(converse(sk1))), join(complement(converse(sk1)), join(composition(sk1, complement(converse(sk1))), sk1)))
% 16.43/2.49 = { by axiom 8 (maddux2_join_associativity_2) }
% 16.43/2.49 meet(complement(complement(converse(sk1))), join(join(complement(converse(sk1)), composition(sk1, complement(converse(sk1)))), sk1))
% 16.43/2.49 = { by lemma 49 }
% 16.43/2.49 meet(complement(complement(converse(sk1))), join(complement(converse(sk1)), sk1))
% 16.43/2.49 = { by axiom 1 (maddux1_join_commutativity_1) }
% 16.43/2.49 meet(complement(complement(converse(sk1))), join(sk1, complement(converse(sk1))))
% 16.43/2.49 = { by lemma 57 }
% 16.43/2.49 complement(join(complement(converse(sk1)), meet(converse(sk1), complement(sk1))))
% 16.43/2.49 = { by lemma 42 }
% 16.43/2.49 meet(converse(sk1), complement(meet(converse(sk1), complement(sk1))))
% 16.43/2.49 = { by lemma 45 }
% 16.43/2.49 meet(converse(sk1), complement(complement(sk1)))
% 16.43/2.49 = { by lemma 35 }
% 16.43/2.49 meet(converse(sk1), sk1)
% 16.43/2.49 = { by lemma 36 R->L }
% 16.43/2.49 meet(sk1, converse(sk1))
% 16.43/2.49 = { by lemma 30 R->L }
% 16.43/2.49 meet(sk1, join(meet(converse(sk1), complement(one)), complement(join(complement(converse(sk1)), complement(one)))))
% 16.43/2.49 = { by lemma 40 R->L }
% 16.43/2.49 meet(sk1, join(complement(join(one, complement(converse(sk1)))), complement(join(complement(converse(sk1)), complement(one)))))
% 16.43/2.49 = { by lemma 54 }
% 16.43/2.49 meet(sk1, join(complement(top), complement(join(complement(converse(sk1)), complement(one)))))
% 16.43/2.49 = { by lemma 15 }
% 16.43/2.49 meet(sk1, join(zero, complement(join(complement(converse(sk1)), complement(one)))))
% 16.43/2.49 = { by lemma 34 }
% 16.43/2.49 meet(sk1, complement(join(complement(converse(sk1)), complement(one))))
% 16.43/2.49 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 16.43/2.49 meet(sk1, meet(converse(sk1), one))
% 16.43/2.49 = { by lemma 36 R->L }
% 16.43/2.49 meet(sk1, meet(one, converse(sk1)))
% 16.43/2.49 = { by lemma 35 R->L }
% 16.43/2.49 meet(sk1, meet(one, complement(complement(converse(sk1)))))
% 16.43/2.49 = { by lemma 45 R->L }
% 16.43/2.49 meet(sk1, meet(one, complement(meet(one, complement(converse(sk1))))))
% 16.43/2.49 = { by lemma 42 R->L }
% 16.43/2.49 meet(sk1, complement(join(complement(one), meet(one, complement(converse(sk1))))))
% 16.43/2.49 = { by lemma 57 R->L }
% 16.43/2.49 meet(sk1, meet(complement(complement(one)), join(converse(sk1), complement(one))))
% 16.43/2.49 = { by axiom 1 (maddux1_join_commutativity_1) R->L }
% 16.43/2.49 meet(sk1, meet(complement(complement(one)), join(complement(one), converse(sk1))))
% 16.43/2.49 = { by axiom 4 (converse_idempotence_8) R->L }
% 16.43/2.49 meet(sk1, meet(complement(complement(one)), join(converse(converse(complement(one))), converse(sk1))))
% 16.43/2.49 = { by lemma 49 R->L }
% 16.43/2.49 meet(sk1, meet(complement(complement(one)), join(converse(join(converse(complement(one)), composition(sk1, converse(complement(one))))), converse(sk1))))
% 16.43/2.49 = { by lemma 56 }
% 16.43/2.49 meet(sk1, meet(complement(complement(one)), join(join(complement(one), converse(composition(sk1, converse(complement(one))))), converse(sk1))))
% 16.43/2.49 = { by axiom 9 (converse_multiplicativity_10) }
% 16.43/2.49 meet(sk1, meet(complement(complement(one)), join(join(complement(one), composition(converse(converse(complement(one))), converse(sk1))), converse(sk1))))
% 16.43/2.49 = { by axiom 4 (converse_idempotence_8) }
% 16.43/2.49 meet(sk1, meet(complement(complement(one)), join(join(complement(one), composition(complement(one), converse(sk1))), converse(sk1))))
% 16.43/2.49 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 16.43/2.49 meet(sk1, meet(complement(complement(one)), join(complement(one), join(composition(complement(one), converse(sk1)), converse(sk1)))))
% 16.43/2.49 = { by axiom 1 (maddux1_join_commutativity_1) }
% 16.43/2.49 meet(sk1, meet(complement(complement(one)), join(complement(one), join(converse(sk1), composition(complement(one), converse(sk1))))))
% 16.43/2.49 = { by lemma 58 R->L }
% 16.43/2.49 meet(sk1, meet(complement(complement(one)), join(complement(one), composition(join(one, complement(one)), converse(sk1)))))
% 16.43/2.49 = { by axiom 5 (def_top_12) R->L }
% 16.43/2.49 meet(sk1, meet(complement(complement(one)), join(complement(one), composition(top, converse(sk1)))))
% 16.43/2.49 = { by lemma 53 }
% 16.43/2.49 meet(sk1, meet(composition(top, converse(sk1)), complement(complement(one))))
% 16.43/2.49 = { by lemma 35 }
% 16.43/2.49 meet(sk1, meet(composition(top, converse(sk1)), one))
% 16.43/2.49 = { by lemma 36 R->L }
% 16.43/2.49 meet(sk1, meet(one, composition(top, converse(sk1))))
% 16.43/2.49 = { by lemma 59 }
% 16.43/2.49 meet(composition(top, converse(sk1)), meet(sk1, one))
% 16.43/2.49 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 16.43/2.49 meet(composition(top, converse(sk1)), complement(join(complement(sk1), complement(one))))
% 16.43/2.49 = { by lemma 34 R->L }
% 16.43/2.49 meet(composition(top, converse(sk1)), join(zero, complement(join(complement(sk1), complement(one)))))
% 16.43/2.49 = { by lemma 15 R->L }
% 16.43/2.49 meet(composition(top, converse(sk1)), join(complement(top), complement(join(complement(sk1), complement(one)))))
% 16.43/2.49 = { by lemma 26 R->L }
% 16.43/2.49 meet(composition(top, converse(sk1)), join(complement(join(one, top)), complement(join(complement(sk1), complement(one)))))
% 16.43/2.49 = { by lemma 19 R->L }
% 16.43/2.49 meet(composition(top, converse(sk1)), join(complement(join(sk1, join(one, complement(sk1)))), complement(join(complement(sk1), complement(one)))))
% 16.43/2.49 = { by lemma 25 }
% 16.43/2.49 meet(composition(top, converse(sk1)), join(complement(join(complement(sk1), one)), complement(join(complement(sk1), complement(one)))))
% 16.43/2.49 = { by axiom 1 (maddux1_join_commutativity_1) }
% 16.43/2.49 meet(composition(top, converse(sk1)), join(complement(join(one, complement(sk1))), complement(join(complement(sk1), complement(one)))))
% 16.43/2.49 = { by lemma 40 }
% 16.43/2.49 meet(composition(top, converse(sk1)), join(meet(sk1, complement(one)), complement(join(complement(sk1), complement(one)))))
% 16.43/2.49 = { by lemma 30 }
% 16.43/2.49 meet(composition(top, converse(sk1)), sk1)
% 16.43/2.49 = { by lemma 36 R->L }
% 16.43/2.49 meet(sk1, composition(top, converse(sk1)))
% 16.43/2.49 = { by lemma 29 R->L }
% 16.43/2.49 meet(sk1, composition(converse(top), converse(sk1)))
% 16.43/2.49 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 16.43/2.49 meet(sk1, converse(composition(sk1, top)))
% 16.43/2.49 = { by lemma 35 R->L }
% 16.43/2.49 meet(sk1, converse(complement(complement(composition(sk1, top)))))
% 16.43/2.49 = { by lemma 21 R->L }
% 16.43/2.49 meet(sk1, converse(join(complement(complement(composition(sk1, top))), composition(converse(sk1), complement(composition(sk1, complement(composition(sk1, top))))))))
% 16.43/2.49 = { by lemma 31 R->L }
% 16.43/2.49 meet(sk1, converse(join(complement(complement(composition(sk1, top))), composition(converse(sk1), complement(composition(sk1, complement(join(zero, meet(composition(sk1, top), composition(sk1, top))))))))))
% 16.43/2.49 = { by lemma 34 }
% 16.43/2.49 meet(sk1, converse(join(complement(complement(composition(sk1, top))), composition(converse(sk1), complement(composition(sk1, complement(meet(composition(sk1, top), composition(sk1, top)))))))))
% 16.43/2.49 = { by lemma 51 }
% 16.43/2.49 meet(sk1, converse(join(complement(complement(composition(sk1, top))), composition(converse(sk1), complement(composition(sk1, complement(composition(sk1, composition(sk1, top)))))))))
% 16.43/2.49 = { by lemma 51 R->L }
% 16.43/2.49 meet(sk1, converse(join(complement(complement(composition(sk1, top))), composition(converse(sk1), complement(meet(complement(composition(sk1, composition(sk1, top))), composition(sk1, top)))))))
% 16.43/2.49 = { by lemma 36 R->L }
% 16.43/2.49 meet(sk1, converse(join(complement(complement(composition(sk1, top))), composition(converse(sk1), complement(meet(composition(sk1, top), complement(composition(sk1, composition(sk1, top)))))))))
% 16.43/2.49 = { by lemma 35 R->L }
% 16.43/2.49 meet(sk1, converse(join(complement(complement(composition(sk1, top))), composition(converse(sk1), complement(meet(composition(sk1, top), complement(composition(sk1, complement(complement(composition(sk1, top)))))))))))
% 16.43/2.49 = { by lemma 42 R->L }
% 16.43/2.49 meet(sk1, converse(join(complement(complement(composition(sk1, top))), composition(converse(sk1), complement(complement(join(complement(composition(sk1, top)), composition(sk1, complement(complement(composition(sk1, top)))))))))))
% 16.43/2.49 = { by lemma 48 }
% 16.43/2.49 meet(sk1, converse(join(complement(complement(composition(sk1, top))), composition(converse(sk1), complement(complement(join(complement(composition(sk1, top)), composition(sk1, top))))))))
% 16.43/2.49 = { by lemma 42 }
% 16.43/2.49 meet(sk1, converse(join(complement(complement(composition(sk1, top))), composition(converse(sk1), complement(meet(composition(sk1, top), complement(composition(sk1, top))))))))
% 16.43/2.49 = { by axiom 6 (def_zero_13) R->L }
% 16.43/2.49 meet(sk1, converse(join(complement(complement(composition(sk1, top))), composition(converse(sk1), complement(zero)))))
% 16.43/2.49 = { by lemma 35 }
% 16.43/2.49 meet(sk1, converse(join(composition(sk1, top), composition(converse(sk1), complement(zero)))))
% 16.43/2.49 = { by lemma 34 R->L }
% 16.43/2.49 meet(sk1, converse(join(composition(sk1, top), composition(converse(sk1), join(zero, complement(zero))))))
% 16.43/2.49 = { by axiom 5 (def_top_12) R->L }
% 16.43/2.49 meet(sk1, converse(join(composition(sk1, top), composition(converse(sk1), top))))
% 16.43/2.49 = { by axiom 12 (composition_distributivity_7) R->L }
% 16.43/2.49 meet(sk1, converse(composition(join(sk1, converse(sk1)), top)))
% 16.43/2.49 = { by axiom 9 (converse_multiplicativity_10) }
% 16.43/2.49 meet(sk1, composition(converse(top), converse(join(sk1, converse(sk1)))))
% 16.43/2.49 = { by lemma 29 }
% 16.43/2.49 meet(sk1, composition(top, converse(join(sk1, converse(sk1)))))
% 16.43/2.49 = { by axiom 7 (converse_additivity_9) }
% 16.43/2.49 meet(sk1, composition(top, join(converse(sk1), converse(converse(sk1)))))
% 16.43/2.49 = { by axiom 4 (converse_idempotence_8) }
% 16.43/2.49 meet(sk1, composition(top, join(converse(sk1), sk1)))
% 16.43/2.49 = { by axiom 1 (maddux1_join_commutativity_1) }
% 16.43/2.49 meet(sk1, composition(top, join(sk1, converse(sk1))))
% 16.43/2.49 = { by lemma 36 }
% 16.43/2.49 meet(composition(top, join(sk1, converse(sk1))), sk1)
% 16.43/2.49 = { by lemma 50 R->L }
% 16.43/2.49 meet(composition(top, join(sk1, converse(sk1))), meet(sk1, join(sk1, converse(sk1))))
% 16.43/2.49 = { by lemma 59 R->L }
% 16.43/2.49 meet(sk1, meet(join(sk1, converse(sk1)), composition(top, join(sk1, converse(sk1)))))
% 16.43/2.49 = { by lemma 26 R->L }
% 16.43/2.49 meet(sk1, meet(join(sk1, converse(sk1)), composition(join(one, top), join(sk1, converse(sk1)))))
% 16.43/2.49 = { by lemma 58 }
% 16.43/2.49 meet(sk1, meet(join(sk1, converse(sk1)), join(join(sk1, converse(sk1)), composition(top, join(sk1, converse(sk1))))))
% 16.43/2.49 = { by lemma 50 }
% 16.43/2.49 meet(sk1, join(sk1, converse(sk1)))
% 16.43/2.49 = { by lemma 50 }
% 16.43/2.49 sk1
% 16.43/2.49
% 16.43/2.49 Goal 1 (goals_15): converse(sk1) = sk1.
% 16.43/2.49 Proof:
% 16.43/2.49 converse(sk1)
% 16.43/2.49 = { by lemma 60 R->L }
% 16.43/2.49 converse(composition(sk1, converse(sk1)))
% 16.43/2.49 = { by axiom 9 (converse_multiplicativity_10) }
% 16.43/2.49 composition(converse(converse(sk1)), converse(sk1))
% 16.43/2.49 = { by axiom 4 (converse_idempotence_8) }
% 16.43/2.49 composition(sk1, converse(sk1))
% 16.43/2.49 = { by lemma 60 }
% 16.43/2.49 sk1
% 16.43/2.49 % SZS output end Proof
% 16.43/2.49
% 16.43/2.49 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------