%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL025-2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n018.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 12:33:14 PM UTC 2026
% Result : Unsatisfiable 22.64s 3.32s
% Output : Proof 23.47s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : REL025-2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.10/0.37 % Computer : n018.cluster.edu
% 0.10/0.37 % Model : x86_64 x86_64
% 0.10/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.37 % Memory : 8046.5625MB
% 0.10/0.37 % OS : Linux 6.8.0-71-generic
% 0.10/0.37 % CPULimit : 300
% 0.10/0.37 % WCLimit : 300
% 0.10/0.37 % DateTime : Sun Sep 27 22:55:09 UTC 2026
% 0.10/0.37 % CPUTime :
% 0.10/0.37 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 22.64/3.32 Command-line arguments: --no-flatten-goal
% 22.64/3.32
% 22.64/3.32 % SZS status Unsatisfiable
% 22.64/3.32
% 23.47/3.40 % SZS output start Proof
% 23.47/3.40 Axiom 1 (def_zero_13): zero = meet(X, complement(X)).
% 23.47/3.40 Axiom 2 (composition_identity_6): composition(X, one) = X.
% 23.47/3.40 Axiom 3 (converse_idempotence_8): converse(converse(X)) = X.
% 23.47/3.40 Axiom 4 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 23.47/3.40 Axiom 5 (def_top_12): top = join(X, complement(X)).
% 23.47/3.40 Axiom 6 (goals_14): join(sk1, one) = one.
% 23.47/3.40 Axiom 7 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 23.47/3.40 Axiom 8 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 23.47/3.40 Axiom 9 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 23.47/3.40 Axiom 10 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 23.47/3.40 Axiom 11 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 23.47/3.40 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 23.47/3.40 Axiom 13 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 23.47/3.40 Axiom 14 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 23.47/3.40
% 23.47/3.40 Lemma 15: complement(top) = zero.
% 23.47/3.40 Proof:
% 23.47/3.40 complement(top)
% 23.47/3.40 = { by axiom 5 (def_top_12) }
% 23.47/3.40 complement(join(complement(X), complement(complement(X))))
% 23.47/3.40 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 23.47/3.40 meet(X, complement(X))
% 23.47/3.40 = { by axiom 1 (def_zero_13) R->L }
% 23.47/3.40 zero
% 23.47/3.40
% 23.47/3.40 Lemma 16: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 23.47/3.40 Proof:
% 23.47/3.41 join(meet(X, Y), complement(join(complement(X), Y)))
% 23.47/3.41 = { by axiom 7 (maddux4_definiton_of_meet_4) }
% 23.47/3.41 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 23.47/3.41 = { by axiom 13 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 23.47/3.41 X
% 23.47/3.41
% 23.47/3.41 Lemma 17: join(meet(X, Y), meet(X, complement(Y))) = X.
% 23.47/3.41 Proof:
% 23.47/3.41 join(meet(X, Y), meet(X, complement(Y)))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.41 join(meet(X, complement(Y)), meet(X, Y))
% 23.47/3.41 = { by axiom 7 (maddux4_definiton_of_meet_4) }
% 23.47/3.41 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 23.47/3.41 = { by lemma 16 }
% 23.47/3.41 X
% 23.47/3.41
% 23.47/3.41 Lemma 18: meet(Y, X) = meet(X, Y).
% 23.47/3.41 Proof:
% 23.47/3.41 meet(Y, X)
% 23.47/3.41 = { by axiom 7 (maddux4_definiton_of_meet_4) }
% 23.47/3.41 complement(join(complement(Y), complement(X)))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.41 complement(join(complement(X), complement(Y)))
% 23.47/3.41 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 23.47/3.41 meet(X, Y)
% 23.47/3.41
% 23.47/3.41 Lemma 19: complement(join(zero, complement(X))) = meet(X, top).
% 23.47/3.41 Proof:
% 23.47/3.41 complement(join(zero, complement(X)))
% 23.47/3.41 = { by lemma 15 R->L }
% 23.47/3.41 complement(join(complement(top), complement(X)))
% 23.47/3.41 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 23.47/3.41 meet(top, X)
% 23.47/3.41 = { by lemma 18 R->L }
% 23.47/3.41 meet(X, top)
% 23.47/3.41
% 23.47/3.41 Lemma 20: composition(converse(one), X) = X.
% 23.47/3.41 Proof:
% 23.47/3.41 composition(converse(one), X)
% 23.47/3.41 = { by axiom 3 (converse_idempotence_8) R->L }
% 23.47/3.41 composition(converse(one), converse(converse(X)))
% 23.47/3.41 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 23.47/3.41 converse(composition(converse(X), one))
% 23.47/3.41 = { by axiom 2 (composition_identity_6) }
% 23.47/3.41 converse(converse(X))
% 23.47/3.41 = { by axiom 3 (converse_idempotence_8) }
% 23.47/3.41 X
% 23.47/3.41
% 23.47/3.41 Lemma 21: composition(one, X) = X.
% 23.47/3.41 Proof:
% 23.47/3.41 composition(one, X)
% 23.47/3.41 = { by lemma 20 R->L }
% 23.47/3.41 composition(converse(one), composition(one, X))
% 23.47/3.41 = { by axiom 9 (composition_associativity_5) }
% 23.47/3.41 composition(composition(converse(one), one), X)
% 23.47/3.41 = { by axiom 2 (composition_identity_6) }
% 23.47/3.41 composition(converse(one), X)
% 23.47/3.41 = { by lemma 20 }
% 23.47/3.41 X
% 23.47/3.41
% 23.47/3.41 Lemma 22: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 23.47/3.41 Proof:
% 23.47/3.41 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.41 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 23.47/3.41 = { by axiom 14 (converse_cancellativity_11) }
% 23.47/3.41 complement(X)
% 23.47/3.41
% 23.47/3.41 Lemma 23: join(complement(X), complement(X)) = complement(X).
% 23.47/3.41 Proof:
% 23.47/3.41 join(complement(X), complement(X))
% 23.47/3.41 = { by lemma 20 R->L }
% 23.47/3.41 join(complement(X), composition(converse(one), complement(X)))
% 23.47/3.41 = { by lemma 21 R->L }
% 23.47/3.41 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 23.47/3.41 = { by lemma 22 }
% 23.47/3.41 complement(X)
% 23.47/3.41
% 23.47/3.41 Lemma 24: join(zero, complement(complement(X))) = X.
% 23.47/3.41 Proof:
% 23.47/3.41 join(zero, complement(complement(X)))
% 23.47/3.41 = { by axiom 1 (def_zero_13) }
% 23.47/3.41 join(meet(X, complement(X)), complement(complement(X)))
% 23.47/3.41 = { by lemma 23 R->L }
% 23.47/3.41 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 23.47/3.41 = { by lemma 16 }
% 23.47/3.41 X
% 23.47/3.41
% 23.47/3.41 Lemma 25: meet(top, complement(X)) = complement(X).
% 23.47/3.41 Proof:
% 23.47/3.41 meet(top, complement(X))
% 23.47/3.41 = { by lemma 18 }
% 23.47/3.41 meet(complement(X), top)
% 23.47/3.41 = { by lemma 19 R->L }
% 23.47/3.41 complement(join(zero, complement(complement(X))))
% 23.47/3.41 = { by lemma 24 }
% 23.47/3.41 complement(X)
% 23.47/3.41
% 23.47/3.41 Lemma 26: complement(zero) = top.
% 23.47/3.41 Proof:
% 23.47/3.41 complement(zero)
% 23.47/3.41 = { by lemma 17 R->L }
% 23.47/3.41 join(meet(complement(zero), top), meet(complement(zero), complement(top)))
% 23.47/3.41 = { by lemma 15 }
% 23.47/3.41 join(meet(complement(zero), top), meet(complement(zero), zero))
% 23.47/3.41 = { by lemma 18 R->L }
% 23.47/3.41 join(meet(complement(zero), top), meet(zero, complement(zero)))
% 23.47/3.41 = { by axiom 1 (def_zero_13) R->L }
% 23.47/3.41 join(meet(complement(zero), top), zero)
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.41 join(zero, meet(complement(zero), top))
% 23.47/3.41 = { by lemma 18 R->L }
% 23.47/3.41 join(zero, meet(top, complement(zero)))
% 23.47/3.41 = { by lemma 25 }
% 23.47/3.41 join(zero, complement(zero))
% 23.47/3.41 = { by axiom 5 (def_top_12) R->L }
% 23.47/3.41 top
% 23.47/3.41
% 23.47/3.41 Lemma 27: join(X, join(Y, complement(X))) = join(Y, top).
% 23.47/3.41 Proof:
% 23.47/3.41 join(X, join(Y, complement(X)))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.41 join(X, join(complement(X), Y))
% 23.47/3.41 = { by axiom 11 (maddux2_join_associativity_2) }
% 23.47/3.41 join(join(X, complement(X)), Y)
% 23.47/3.41 = { by axiom 5 (def_top_12) R->L }
% 23.47/3.41 join(top, Y)
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.41 join(Y, top)
% 23.47/3.41
% 23.47/3.41 Lemma 28: join(top, complement(X)) = top.
% 23.47/3.41 Proof:
% 23.47/3.41 join(top, complement(X))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.41 join(complement(X), top)
% 23.47/3.41 = { by lemma 27 R->L }
% 23.47/3.41 join(X, join(complement(X), complement(X)))
% 23.47/3.41 = { by lemma 23 }
% 23.47/3.41 join(X, complement(X))
% 23.47/3.41 = { by axiom 5 (def_top_12) R->L }
% 23.47/3.41 top
% 23.47/3.41
% 23.47/3.41 Lemma 29: join(Y, top) = join(X, top).
% 23.47/3.41 Proof:
% 23.47/3.41 join(Y, top)
% 23.47/3.41 = { by lemma 28 R->L }
% 23.47/3.41 join(Y, join(top, complement(Y)))
% 23.47/3.41 = { by lemma 27 }
% 23.47/3.41 join(top, top)
% 23.47/3.41 = { by lemma 27 R->L }
% 23.47/3.41 join(X, join(top, complement(X)))
% 23.47/3.41 = { by lemma 28 }
% 23.47/3.41 join(X, top)
% 23.47/3.41
% 23.47/3.41 Lemma 30: join(X, top) = top.
% 23.47/3.41 Proof:
% 23.47/3.41 join(X, top)
% 23.47/3.41 = { by lemma 29 }
% 23.47/3.41 join(zero, top)
% 23.47/3.41 = { by lemma 27 R->L }
% 23.47/3.41 join(top, join(zero, complement(top)))
% 23.47/3.41 = { by lemma 15 R->L }
% 23.47/3.41 join(top, join(complement(top), complement(top)))
% 23.47/3.41 = { by lemma 23 }
% 23.47/3.41 join(top, complement(top))
% 23.47/3.41 = { by axiom 5 (def_top_12) R->L }
% 23.47/3.41 top
% 23.47/3.41
% 23.47/3.41 Lemma 31: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 23.47/3.41 Proof:
% 23.47/3.41 converse(join(X, converse(Y)))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.41 converse(join(converse(Y), X))
% 23.47/3.41 = { by axiom 10 (converse_additivity_9) }
% 23.47/3.41 join(converse(converse(Y)), converse(X))
% 23.47/3.41 = { by axiom 3 (converse_idempotence_8) }
% 23.47/3.41 join(Y, converse(X))
% 23.47/3.41
% 23.47/3.41 Lemma 32: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 23.47/3.41 Proof:
% 23.47/3.41 converse(join(converse(X), Y))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.41 converse(join(Y, converse(X)))
% 23.47/3.41 = { by lemma 31 }
% 23.47/3.41 join(X, converse(Y))
% 23.47/3.41
% 23.47/3.41 Lemma 33: join(X, converse(complement(converse(X)))) = converse(top).
% 23.47/3.41 Proof:
% 23.47/3.41 join(X, converse(complement(converse(X))))
% 23.47/3.41 = { by lemma 32 R->L }
% 23.47/3.41 converse(join(converse(X), complement(converse(X))))
% 23.47/3.41 = { by axiom 5 (def_top_12) R->L }
% 23.47/3.41 converse(top)
% 23.47/3.41
% 23.47/3.41 Lemma 34: join(X, join(complement(X), Y)) = top.
% 23.47/3.41 Proof:
% 23.47/3.41 join(X, join(complement(X), Y))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.41 join(X, join(Y, complement(X)))
% 23.47/3.41 = { by lemma 27 }
% 23.47/3.41 join(Y, top)
% 23.47/3.41 = { by lemma 29 R->L }
% 23.47/3.41 join(Z, top)
% 23.47/3.41 = { by lemma 30 }
% 23.47/3.41 top
% 23.47/3.41
% 23.47/3.41 Lemma 35: converse(top) = top.
% 23.47/3.41 Proof:
% 23.47/3.41 converse(top)
% 23.47/3.41 = { by lemma 30 R->L }
% 23.47/3.41 converse(join(X, top))
% 23.47/3.41 = { by axiom 10 (converse_additivity_9) }
% 23.47/3.41 join(converse(X), converse(top))
% 23.47/3.41 = { by lemma 33 R->L }
% 23.47/3.41 join(converse(X), join(complement(converse(X)), converse(complement(converse(complement(converse(X)))))))
% 23.47/3.41 = { by lemma 34 }
% 23.47/3.41 top
% 23.47/3.41
% 23.47/3.41 Lemma 36: join(zero, meet(X, top)) = X.
% 23.47/3.41 Proof:
% 23.47/3.41 join(zero, meet(X, top))
% 23.47/3.41 = { by lemma 26 R->L }
% 23.47/3.41 join(zero, meet(X, complement(zero)))
% 23.47/3.41 = { by lemma 15 R->L }
% 23.47/3.41 join(complement(top), meet(X, complement(zero)))
% 23.47/3.41 = { by lemma 28 R->L }
% 23.47/3.41 join(complement(join(top, complement(X))), meet(X, complement(zero)))
% 23.47/3.41 = { by lemma 26 R->L }
% 23.47/3.41 join(complement(join(complement(zero), complement(X))), meet(X, complement(zero)))
% 23.47/3.41 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 23.47/3.41 join(meet(zero, X), meet(X, complement(zero)))
% 23.47/3.41 = { by lemma 18 R->L }
% 23.47/3.41 join(meet(X, zero), meet(X, complement(zero)))
% 23.47/3.41 = { by lemma 17 }
% 23.47/3.41 X
% 23.47/3.41
% 23.47/3.41 Lemma 37: join(zero, complement(X)) = complement(X).
% 23.47/3.41 Proof:
% 23.47/3.41 join(zero, complement(X))
% 23.47/3.41 = { by lemma 25 R->L }
% 23.47/3.41 join(zero, meet(top, complement(X)))
% 23.47/3.41 = { by lemma 18 }
% 23.47/3.41 join(zero, meet(complement(X), top))
% 23.47/3.41 = { by lemma 36 }
% 23.47/3.41 complement(X)
% 23.47/3.41
% 23.47/3.41 Lemma 38: meet(X, top) = X.
% 23.47/3.41 Proof:
% 23.47/3.41 meet(X, top)
% 23.47/3.41 = { by lemma 19 R->L }
% 23.47/3.41 complement(join(zero, complement(X)))
% 23.47/3.41 = { by lemma 37 R->L }
% 23.47/3.41 join(zero, complement(join(zero, complement(X))))
% 23.47/3.41 = { by lemma 19 }
% 23.47/3.41 join(zero, meet(X, top))
% 23.47/3.41 = { by lemma 36 }
% 23.47/3.41 X
% 23.47/3.41
% 23.47/3.41 Lemma 39: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
% 23.47/3.41 Proof:
% 23.47/3.41 join(meet(X, Y), complement(join(Y, complement(X))))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.41 join(meet(X, Y), complement(join(complement(X), Y)))
% 23.47/3.41 = { by lemma 16 }
% 23.47/3.41 X
% 23.47/3.41
% 23.47/3.41 Lemma 40: meet(X, join(Y, complement(X))) = meet(X, Y).
% 23.47/3.41 Proof:
% 23.47/3.41 meet(X, join(Y, complement(X)))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.41 meet(X, join(complement(X), Y))
% 23.47/3.41 = { by axiom 7 (maddux4_definiton_of_meet_4) }
% 23.47/3.41 complement(join(complement(X), complement(join(complement(X), Y))))
% 23.47/3.41 = { by lemma 16 R->L }
% 23.47/3.41 complement(join(complement(X), complement(join(complement(X), join(meet(Y, complement(X)), complement(join(complement(Y), complement(X))))))))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.41 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), meet(Y, complement(X)))))))
% 23.47/3.41 = { by lemma 23 R->L }
% 23.47/3.41 complement(join(complement(X), complement(join(complement(X), join(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X)))), meet(Y, complement(X)))))))
% 23.47/3.41 = { by axiom 11 (maddux2_join_associativity_2) R->L }
% 23.47/3.41 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), join(complement(join(complement(Y), complement(X))), meet(Y, complement(X))))))))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.41 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), join(meet(Y, complement(X)), complement(join(complement(Y), complement(X)))))))))
% 23.47/3.41 = { by lemma 16 }
% 23.47/3.41 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), Y)))))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.41 complement(join(complement(X), complement(join(complement(X), join(Y, complement(join(complement(Y), complement(X))))))))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.41 complement(join(complement(X), complement(join(complement(X), join(Y, complement(join(complement(X), complement(Y))))))))
% 23.47/3.41 = { by axiom 11 (maddux2_join_associativity_2) }
% 23.47/3.41 complement(join(complement(X), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.41 = { by lemma 38 R->L }
% 23.47/3.41 complement(join(meet(complement(X), top), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.41 = { by lemma 19 R->L }
% 23.47/3.41 complement(join(complement(join(zero, complement(complement(X)))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.41 = { by lemma 16 R->L }
% 23.47/3.41 complement(join(complement(join(meet(join(zero, complement(complement(X))), Y), complement(join(complement(join(zero, complement(complement(X)))), Y)))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.41 = { by axiom 7 (maddux4_definiton_of_meet_4) }
% 23.47/3.41 complement(join(complement(join(complement(join(complement(join(zero, complement(complement(X)))), complement(Y))), complement(join(complement(join(zero, complement(complement(X)))), Y)))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.41 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 23.47/3.41 complement(join(meet(join(complement(join(zero, complement(complement(X)))), complement(Y)), join(complement(join(zero, complement(complement(X)))), Y)), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.41 = { by lemma 18 R->L }
% 23.47/3.41 complement(join(meet(join(complement(join(zero, complement(complement(X)))), Y), join(complement(join(zero, complement(complement(X)))), complement(Y))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.41 complement(join(meet(join(Y, complement(join(zero, complement(complement(X))))), join(complement(join(zero, complement(complement(X)))), complement(Y))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.41 complement(join(meet(join(Y, complement(join(zero, complement(complement(X))))), join(complement(Y), complement(join(zero, complement(complement(X)))))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.41 = { by lemma 19 }
% 23.47/3.41 complement(join(meet(join(Y, meet(complement(X), top)), join(complement(Y), complement(join(zero, complement(complement(X)))))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.41 = { by lemma 38 }
% 23.47/3.41 complement(join(meet(join(Y, complement(X)), join(complement(Y), complement(join(zero, complement(complement(X)))))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.41 = { by lemma 19 }
% 23.47/3.41 complement(join(meet(join(Y, complement(X)), join(complement(Y), meet(complement(X), top))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.41 = { by lemma 38 }
% 23.47/3.41 complement(join(meet(join(Y, complement(X)), join(complement(Y), complement(X))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.41 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.42 complement(join(meet(join(complement(X), Y), join(complement(Y), complement(X))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.42 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.42 complement(join(meet(join(complement(X), Y), join(complement(X), complement(Y))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.42 = { by lemma 18 }
% 23.47/3.42 complement(join(meet(join(complement(X), complement(Y)), join(complement(X), Y)), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 23.47/3.42 = { by lemma 39 }
% 23.47/3.42 complement(join(complement(X), complement(Y)))
% 23.47/3.42 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 23.47/3.42 meet(X, Y)
% 23.47/3.42
% 23.47/3.42 Lemma 41: meet(X, join(complement(X), Y)) = meet(X, Y).
% 23.47/3.42 Proof:
% 23.47/3.42 meet(X, join(complement(X), Y))
% 23.47/3.42 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.42 meet(X, join(Y, complement(X)))
% 23.47/3.42 = { by lemma 40 }
% 23.47/3.42 meet(X, Y)
% 23.47/3.42
% 23.47/3.42 Lemma 42: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 23.47/3.42 Proof:
% 23.47/3.42 converse(composition(X, converse(Y)))
% 23.47/3.42 = { by axiom 8 (converse_multiplicativity_10) }
% 23.47/3.42 composition(converse(converse(Y)), converse(X))
% 23.47/3.42 = { by axiom 3 (converse_idempotence_8) }
% 23.47/3.42 composition(Y, converse(X))
% 23.47/3.42
% 23.47/3.42 Lemma 43: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 23.47/3.42 Proof:
% 23.47/3.42 join(composition(X, Y), composition(X, Z))
% 23.47/3.42 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.42 join(composition(X, Z), composition(X, Y))
% 23.47/3.42 = { by axiom 3 (converse_idempotence_8) R->L }
% 23.47/3.42 join(composition(X, Z), composition(X, converse(converse(Y))))
% 23.47/3.42 = { by axiom 3 (converse_idempotence_8) R->L }
% 23.47/3.42 converse(converse(join(composition(X, Z), composition(X, converse(converse(Y))))))
% 23.47/3.42 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.42 converse(converse(join(composition(X, converse(converse(Y))), composition(X, Z))))
% 23.47/3.42 = { by axiom 10 (converse_additivity_9) }
% 23.47/3.42 converse(join(converse(composition(X, converse(converse(Y)))), converse(composition(X, Z))))
% 23.47/3.42 = { by lemma 42 }
% 23.47/3.42 converse(join(composition(converse(Y), converse(X)), converse(composition(X, Z))))
% 23.47/3.42 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.42 converse(join(converse(composition(X, Z)), composition(converse(Y), converse(X))))
% 23.47/3.42 = { by axiom 8 (converse_multiplicativity_10) }
% 23.47/3.42 converse(join(composition(converse(Z), converse(X)), composition(converse(Y), converse(X))))
% 23.47/3.42 = { by axiom 12 (composition_distributivity_7) R->L }
% 23.47/3.42 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 23.47/3.42 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.42 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 23.47/3.42 = { by axiom 8 (converse_multiplicativity_10) }
% 23.47/3.42 composition(converse(converse(X)), converse(join(converse(Y), converse(Z))))
% 23.47/3.42 = { by lemma 31 }
% 23.47/3.42 composition(converse(converse(X)), join(Z, converse(converse(Y))))
% 23.47/3.42 = { by axiom 3 (converse_idempotence_8) }
% 23.47/3.42 composition(X, join(Z, converse(converse(Y))))
% 23.47/3.42 = { by axiom 3 (converse_idempotence_8) }
% 23.47/3.42 composition(X, join(Z, Y))
% 23.47/3.42 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.42 composition(X, join(Y, Z))
% 23.47/3.42
% 23.47/3.42 Lemma 44: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 23.47/3.42 Proof:
% 23.47/3.42 join(X, composition(Y, X))
% 23.47/3.42 = { by lemma 21 R->L }
% 23.47/3.42 join(composition(one, X), composition(Y, X))
% 23.47/3.42 = { by axiom 12 (composition_distributivity_7) R->L }
% 23.47/3.42 composition(join(one, Y), X)
% 23.47/3.42 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.42 composition(join(Y, one), X)
% 23.47/3.42
% 23.47/3.42 Lemma 45: join(one, sk1) = one.
% 23.47/3.42 Proof:
% 23.47/3.42 join(one, sk1)
% 23.47/3.42 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.42 join(sk1, one)
% 23.47/3.42 = { by axiom 6 (goals_14) }
% 23.47/3.42 one
% 23.47/3.42
% 23.47/3.42 Lemma 46: join(X, join(Y, composition(sk1, X))) = join(X, Y).
% 23.47/3.42 Proof:
% 23.47/3.42 join(X, join(Y, composition(sk1, X)))
% 23.47/3.42 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.42 join(X, join(composition(sk1, X), Y))
% 23.47/3.42 = { by axiom 11 (maddux2_join_associativity_2) }
% 23.47/3.42 join(join(X, composition(sk1, X)), Y)
% 23.47/3.42 = { by lemma 44 }
% 23.47/3.42 join(composition(join(sk1, one), X), Y)
% 23.47/3.42 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.42 join(composition(join(one, sk1), X), Y)
% 23.47/3.42 = { by lemma 45 }
% 23.47/3.42 join(composition(one, X), Y)
% 23.47/3.42 = { by lemma 21 }
% 23.47/3.42 join(X, Y)
% 23.47/3.42
% 23.47/3.42 Lemma 47: complement(complement(X)) = X.
% 23.47/3.42 Proof:
% 23.47/3.42 complement(complement(X))
% 23.47/3.42 = { by lemma 37 R->L }
% 23.47/3.42 join(zero, complement(complement(X)))
% 23.47/3.42 = { by lemma 24 }
% 23.47/3.42 X
% 23.47/3.42
% 23.47/3.42 Lemma 48: meet(X, composition(sk1, top)) = composition(sk1, X).
% 23.47/3.42 Proof:
% 23.47/3.42 meet(X, composition(sk1, top))
% 23.47/3.42 = { by lemma 41 R->L }
% 23.47/3.42 meet(X, join(complement(X), composition(sk1, top)))
% 23.47/3.42 = { by axiom 5 (def_top_12) }
% 23.47/3.42 meet(X, join(complement(X), composition(sk1, join(complement(X), complement(complement(X))))))
% 23.47/3.42 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.42 meet(X, join(complement(X), composition(sk1, join(complement(complement(X)), complement(X)))))
% 23.47/3.42 = { by lemma 43 R->L }
% 23.47/3.42 meet(X, join(complement(X), join(composition(sk1, complement(complement(X))), composition(sk1, complement(X)))))
% 23.47/3.42 = { by lemma 46 }
% 23.47/3.42 meet(X, join(complement(X), composition(sk1, complement(complement(X)))))
% 23.47/3.42 = { by lemma 41 }
% 23.47/3.42 meet(X, composition(sk1, complement(complement(X))))
% 23.47/3.42 = { by lemma 47 }
% 23.47/3.42 meet(X, composition(sk1, X))
% 23.47/3.42 = { by lemma 18 }
% 23.47/3.42 meet(composition(sk1, X), X)
% 23.47/3.42 = { by lemma 40 R->L }
% 23.47/3.42 meet(composition(sk1, X), join(X, complement(composition(sk1, X))))
% 23.47/3.42 = { by lemma 46 R->L }
% 23.47/3.42 meet(composition(sk1, X), join(X, join(complement(composition(sk1, X)), composition(sk1, X))))
% 23.47/3.43 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.43 meet(composition(sk1, X), join(X, join(composition(sk1, X), complement(composition(sk1, X)))))
% 23.47/3.43 = { by axiom 5 (def_top_12) R->L }
% 23.47/3.43 meet(composition(sk1, X), join(X, top))
% 23.47/3.43 = { by lemma 29 R->L }
% 23.47/3.43 meet(composition(sk1, X), join(Y, top))
% 23.47/3.43 = { by lemma 30 }
% 23.47/3.43 meet(composition(sk1, X), top)
% 23.47/3.43 = { by lemma 38 }
% 23.47/3.43 composition(sk1, X)
% 23.47/3.43
% 23.47/3.43 Lemma 49: join(zero, X) = X.
% 23.47/3.43 Proof:
% 23.47/3.43 join(zero, X)
% 23.47/3.43 = { by lemma 47 R->L }
% 23.47/3.43 join(zero, complement(complement(X)))
% 23.47/3.43 = { by lemma 24 }
% 23.47/3.43 X
% 23.47/3.43
% 23.47/3.43 Lemma 50: composition(converse(X), composition(X, top)) = composition(converse(X), top).
% 23.47/3.43 Proof:
% 23.47/3.43 composition(converse(X), composition(X, top))
% 23.47/3.43 = { by lemma 49 R->L }
% 23.47/3.43 join(zero, composition(converse(X), composition(X, top)))
% 23.47/3.43 = { by lemma 15 R->L }
% 23.47/3.43 join(complement(top), composition(converse(X), composition(X, top)))
% 23.47/3.43 = { by lemma 22 R->L }
% 23.47/3.43 join(join(complement(top), composition(converse(X), complement(composition(X, top)))), composition(converse(X), composition(X, top)))
% 23.47/3.43 = { by lemma 15 }
% 23.47/3.43 join(join(zero, composition(converse(X), complement(composition(X, top)))), composition(converse(X), composition(X, top)))
% 23.47/3.43 = { by lemma 49 }
% 23.47/3.43 join(composition(converse(X), complement(composition(X, top))), composition(converse(X), composition(X, top)))
% 23.47/3.43 = { by lemma 43 }
% 23.47/3.43 composition(converse(X), join(complement(composition(X, top)), composition(X, top)))
% 23.47/3.43 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.43 composition(converse(X), join(composition(X, top), complement(composition(X, top))))
% 23.47/3.43 = { by axiom 5 (def_top_12) R->L }
% 23.47/3.43 composition(converse(X), top)
% 23.47/3.43
% 23.47/3.43 Lemma 51: join(one, converse(X)) = converse(join(X, one)).
% 23.47/3.43 Proof:
% 23.47/3.43 join(one, converse(X))
% 23.47/3.43 = { by lemma 20 R->L }
% 23.47/3.43 join(composition(converse(one), one), converse(X))
% 23.47/3.43 = { by axiom 2 (composition_identity_6) }
% 23.47/3.43 join(converse(one), converse(X))
% 23.47/3.43 = { by axiom 10 (converse_additivity_9) R->L }
% 23.47/3.43 converse(join(one, X))
% 23.47/3.43 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.43 converse(join(X, one))
% 23.47/3.43
% 23.47/3.43 Lemma 52: join(top, X) = top.
% 23.47/3.43 Proof:
% 23.47/3.43 join(top, X)
% 23.47/3.43 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.43 join(X, top)
% 23.47/3.43 = { by lemma 29 R->L }
% 23.47/3.43 join(Y, top)
% 23.47/3.43 = { by lemma 30 }
% 23.47/3.43 top
% 23.47/3.43
% 23.47/3.43 Lemma 53: join(X, zero) = X.
% 23.47/3.43 Proof:
% 23.47/3.43 join(X, zero)
% 23.47/3.43 = { by lemma 15 R->L }
% 23.47/3.43 join(X, complement(top))
% 23.47/3.43 = { by lemma 30 R->L }
% 23.47/3.43 join(X, complement(join(complement(X), top)))
% 23.47/3.43 = { by lemma 38 R->L }
% 23.47/3.43 join(meet(X, top), complement(join(complement(X), top)))
% 23.47/3.43 = { by lemma 16 }
% 23.47/3.43 X
% 23.47/3.43
% 23.47/3.43 Lemma 54: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 23.47/3.43 Proof:
% 23.47/3.43 join(X, join(Y, Z))
% 23.47/3.43 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.43 join(join(Y, Z), X)
% 23.47/3.43 = { by axiom 11 (maddux2_join_associativity_2) R->L }
% 23.47/3.43 join(Y, join(Z, X))
% 23.47/3.43 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.43 join(Y, join(X, Z))
% 23.47/3.43
% 23.47/3.43 Lemma 55: composition(sk1, converse(sk1)) = converse(sk1).
% 23.47/3.43 Proof:
% 23.47/3.43 composition(sk1, converse(sk1))
% 23.47/3.43 = { by lemma 48 R->L }
% 23.47/3.43 meet(converse(sk1), composition(sk1, top))
% 23.47/3.43 = { by axiom 3 (converse_idempotence_8) R->L }
% 23.47/3.43 meet(converse(sk1), composition(converse(converse(sk1)), top))
% 23.47/3.43 = { by lemma 50 R->L }
% 23.47/3.43 meet(converse(sk1), composition(converse(converse(sk1)), composition(converse(sk1), top)))
% 23.47/3.43 = { by axiom 3 (converse_idempotence_8) }
% 23.47/3.43 meet(converse(sk1), composition(sk1, composition(converse(sk1), top)))
% 23.47/3.43 = { by lemma 50 R->L }
% 23.47/3.43 meet(converse(sk1), composition(sk1, composition(converse(sk1), composition(sk1, top))))
% 23.47/3.43 = { by lemma 48 R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), composition(sk1, top)))
% 23.47/3.43 = { by lemma 40 R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(sk1, top), complement(composition(converse(sk1), composition(sk1, top))))))
% 23.47/3.43 = { by lemma 21 R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(converse(sk1), composition(sk1, top))))))
% 23.47/3.43 = { by lemma 39 R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(join(meet(converse(sk1), one), complement(join(one, complement(converse(sk1))))), composition(sk1, top))))))
% 23.47/3.43 = { by axiom 3 (converse_idempotence_8) R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(join(meet(converse(sk1), one), complement(converse(converse(join(one, complement(converse(sk1))))))), composition(sk1, top))))))
% 23.47/3.43 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(join(meet(converse(sk1), one), complement(converse(converse(join(complement(converse(sk1)), one))))), composition(sk1, top))))))
% 23.47/3.43 = { by lemma 51 R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(join(meet(converse(sk1), one), complement(converse(join(one, converse(complement(converse(sk1))))))), composition(sk1, top))))))
% 23.47/3.43 = { by lemma 45 R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(join(meet(converse(sk1), one), complement(converse(join(join(one, sk1), converse(complement(converse(sk1))))))), composition(sk1, top))))))
% 23.47/3.43 = { by axiom 11 (maddux2_join_associativity_2) R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(join(meet(converse(sk1), one), complement(converse(join(one, join(sk1, converse(complement(converse(sk1)))))))), composition(sk1, top))))))
% 23.47/3.43 = { by lemma 33 }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(join(meet(converse(sk1), one), complement(converse(join(one, converse(top))))), composition(sk1, top))))))
% 23.47/3.43 = { by lemma 51 }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(join(meet(converse(sk1), one), complement(converse(converse(join(top, one))))), composition(sk1, top))))))
% 23.47/3.43 = { by lemma 52 }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(join(meet(converse(sk1), one), complement(converse(converse(top)))), composition(sk1, top))))))
% 23.47/3.43 = { by lemma 35 }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(join(meet(converse(sk1), one), complement(converse(top))), composition(sk1, top))))))
% 23.47/3.43 = { by lemma 35 }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(join(meet(converse(sk1), one), complement(top)), composition(sk1, top))))))
% 23.47/3.43 = { by lemma 15 }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(join(meet(converse(sk1), one), zero), composition(sk1, top))))))
% 23.47/3.43 = { by lemma 53 }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(meet(converse(sk1), one), composition(sk1, top))))))
% 23.47/3.43 = { by lemma 18 R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(one, composition(sk1, top)), complement(composition(meet(one, converse(sk1)), composition(sk1, top))))))
% 23.47/3.43 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(complement(composition(meet(one, converse(sk1)), composition(sk1, top))), composition(one, composition(sk1, top)))))
% 23.47/3.43 = { by lemma 17 R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(complement(composition(meet(one, converse(sk1)), composition(sk1, top))), composition(join(meet(one, converse(sk1)), meet(one, complement(converse(sk1)))), composition(sk1, top)))))
% 23.47/3.43 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(complement(composition(meet(one, converse(sk1)), composition(sk1, top))), composition(join(meet(one, complement(converse(sk1))), meet(one, converse(sk1))), composition(sk1, top)))))
% 23.47/3.43 = { by axiom 4 (maddux1_join_commutativity_1) R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(join(meet(one, complement(converse(sk1))), meet(one, converse(sk1))), composition(sk1, top)), complement(composition(meet(one, converse(sk1)), composition(sk1, top))))))
% 23.47/3.43 = { by axiom 12 (composition_distributivity_7) }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(join(composition(meet(one, complement(converse(sk1))), composition(sk1, top)), composition(meet(one, converse(sk1)), composition(sk1, top))), complement(composition(meet(one, converse(sk1)), composition(sk1, top))))))
% 23.47/3.43 = { by axiom 11 (maddux2_join_associativity_2) R->L }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(meet(one, complement(converse(sk1))), composition(sk1, top)), join(composition(meet(one, converse(sk1)), composition(sk1, top)), complement(composition(meet(one, converse(sk1)), composition(sk1, top)))))))
% 23.47/3.43 = { by lemma 54 }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(meet(one, converse(sk1)), composition(sk1, top)), join(composition(meet(one, complement(converse(sk1))), composition(sk1, top)), complement(composition(meet(one, converse(sk1)), composition(sk1, top)))))))
% 23.47/3.43 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), join(composition(meet(one, converse(sk1)), composition(sk1, top)), join(complement(composition(meet(one, converse(sk1)), composition(sk1, top))), composition(meet(one, complement(converse(sk1))), composition(sk1, top))))))
% 23.47/3.43 = { by lemma 34 }
% 23.47/3.43 meet(converse(sk1), meet(composition(converse(sk1), composition(sk1, top)), top))
% 23.47/3.43 = { by lemma 38 }
% 23.47/3.43 meet(converse(sk1), composition(converse(sk1), composition(sk1, top)))
% 23.47/3.43 = { by lemma 50 }
% 23.47/3.43 meet(converse(sk1), composition(converse(sk1), top))
% 23.47/3.43 = { by lemma 35 R->L }
% 23.47/3.43 meet(converse(sk1), composition(converse(sk1), converse(top)))
% 23.47/3.43 = { by lemma 52 R->L }
% 23.47/3.43 meet(converse(sk1), composition(converse(sk1), converse(join(top, one))))
% 23.47/3.43 = { by lemma 42 R->L }
% 23.47/3.43 meet(converse(sk1), converse(composition(join(top, one), converse(converse(sk1)))))
% 23.47/3.44 = { by lemma 44 R->L }
% 23.47/3.44 meet(converse(sk1), converse(join(converse(converse(sk1)), composition(top, converse(converse(sk1))))))
% 23.47/3.44 = { by lemma 32 }
% 23.47/3.44 meet(converse(sk1), join(converse(sk1), converse(composition(top, converse(converse(sk1))))))
% 23.47/3.44 = { by lemma 42 }
% 23.47/3.44 meet(converse(sk1), join(converse(sk1), composition(converse(sk1), converse(top))))
% 23.47/3.44 = { by lemma 35 }
% 23.47/3.44 meet(converse(sk1), join(converse(sk1), composition(converse(sk1), top)))
% 23.47/3.44 = { by lemma 16 R->L }
% 23.47/3.44 meet(converse(sk1), join(converse(sk1), join(meet(composition(converse(sk1), top), X), complement(join(complement(composition(converse(sk1), top)), X)))))
% 23.47/3.44 = { by lemma 54 R->L }
% 23.47/3.44 meet(converse(sk1), join(meet(composition(converse(sk1), top), X), join(converse(sk1), complement(join(complement(composition(converse(sk1), top)), X)))))
% 23.47/3.44 = { by lemma 53 R->L }
% 23.47/3.44 join(meet(converse(sk1), join(meet(composition(converse(sk1), top), X), join(converse(sk1), complement(join(complement(composition(converse(sk1), top)), X))))), zero)
% 23.47/3.44 = { by lemma 15 R->L }
% 23.47/3.44 join(meet(converse(sk1), join(meet(composition(converse(sk1), top), X), join(converse(sk1), complement(join(complement(composition(converse(sk1), top)), X))))), complement(top))
% 23.47/3.44 = { by lemma 47 R->L }
% 23.47/3.44 join(meet(converse(sk1), join(meet(composition(converse(sk1), top), X), join(complement(complement(converse(sk1))), complement(join(complement(composition(converse(sk1), top)), X))))), complement(top))
% 23.47/3.44 = { by lemma 34 R->L }
% 23.47/3.44 join(meet(converse(sk1), join(meet(composition(converse(sk1), top), X), join(complement(complement(converse(sk1))), complement(join(complement(composition(converse(sk1), top)), X))))), complement(join(complement(converse(sk1)), join(complement(complement(converse(sk1))), join(meet(composition(converse(sk1), top), X), complement(join(complement(composition(converse(sk1), top)), X)))))))
% 23.47/3.44 = { by lemma 54 }
% 23.47/3.44 join(meet(converse(sk1), join(meet(composition(converse(sk1), top), X), join(complement(complement(converse(sk1))), complement(join(complement(composition(converse(sk1), top)), X))))), complement(join(complement(converse(sk1)), join(meet(composition(converse(sk1), top), X), join(complement(complement(converse(sk1))), complement(join(complement(composition(converse(sk1), top)), X)))))))
% 23.47/3.44 = { by lemma 16 }
% 23.47/3.44 converse(sk1)
% 23.47/3.44
% 23.47/3.44 Lemma 56: converse(sk1) = sk1.
% 23.47/3.44 Proof:
% 23.47/3.44 converse(sk1)
% 23.47/3.44 = { by lemma 55 R->L }
% 23.47/3.44 composition(sk1, converse(sk1))
% 23.47/3.44 = { by lemma 42 R->L }
% 23.47/3.44 converse(composition(sk1, converse(sk1)))
% 23.47/3.44 = { by lemma 55 }
% 23.47/3.44 converse(converse(sk1))
% 23.47/3.44 = { by axiom 3 (converse_idempotence_8) }
% 23.47/3.44 sk1
% 23.47/3.44
% 23.47/3.44 Lemma 57: join(X, X) = X.
% 23.47/3.44 Proof:
% 23.47/3.44 join(X, X)
% 23.47/3.44 = { by lemma 38 R->L }
% 23.47/3.44 join(X, meet(X, top))
% 23.47/3.44 = { by lemma 38 R->L }
% 23.47/3.44 join(meet(X, top), meet(X, top))
% 23.47/3.44 = { by lemma 18 }
% 23.47/3.44 join(meet(top, X), meet(X, top))
% 23.47/3.44 = { by lemma 18 }
% 23.47/3.44 join(meet(top, X), meet(top, X))
% 23.47/3.44 = { by axiom 7 (maddux4_definiton_of_meet_4) }
% 23.47/3.44 join(meet(top, X), complement(join(complement(top), complement(X))))
% 23.47/3.44 = { by axiom 7 (maddux4_definiton_of_meet_4) }
% 23.47/3.44 join(complement(join(complement(top), complement(X))), complement(join(complement(top), complement(X))))
% 23.47/3.44 = { by lemma 23 }
% 23.47/3.44 complement(join(complement(top), complement(X)))
% 23.47/3.44 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 23.47/3.44 meet(top, X)
% 23.47/3.44 = { by lemma 18 R->L }
% 23.47/3.44 meet(X, top)
% 23.47/3.44 = { by lemma 38 }
% 23.47/3.44 X
% 23.47/3.44
% 23.47/3.44 Goal 1 (goals_17): tuple(join(converse(sk1), sk1), join(sk1, converse(sk1))) = tuple(sk1, converse(sk1)).
% 23.47/3.44 Proof:
% 23.47/3.44 tuple(join(converse(sk1), sk1), join(sk1, converse(sk1)))
% 23.47/3.44 = { by axiom 4 (maddux1_join_commutativity_1) }
% 23.47/3.44 tuple(join(sk1, converse(sk1)), join(sk1, converse(sk1)))
% 23.47/3.44 = { by lemma 56 }
% 23.47/3.44 tuple(join(sk1, sk1), join(sk1, converse(sk1)))
% 23.47/3.44 = { by lemma 56 }
% 23.47/3.44 tuple(join(sk1, sk1), join(sk1, sk1))
% 23.47/3.44 = { by lemma 57 }
% 23.47/3.44 tuple(sk1, join(sk1, sk1))
% 23.47/3.44 = { by lemma 57 }
% 23.47/3.44 tuple(sk1, sk1)
% 23.47/3.44 = { by lemma 56 R->L }
% 23.47/3.44 tuple(sk1, converse(sk1))
% 23.47/3.44 % SZS output end Proof
% 23.47/3.44
% 23.47/3.44 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------