TSTP Solution File: REL012+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL012+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 : n015.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 13:43:52 EDT 2023
% Result : Theorem 4.98s 1.01s
% Output : Proof 5.71s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL012+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
% 0.16/0.34 % Computer : n015.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Fri Aug 25 19:54:52 EDT 2023
% 0.16/0.34 % CPUTime :
% 4.98/1.01 Command-line arguments: --no-flatten-goal
% 4.98/1.01
% 4.98/1.01 % SZS status Theorem
% 4.98/1.01
% 4.98/1.06 % SZS output start Proof
% 4.98/1.06 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 4.98/1.06 Axiom 2 (composition_identity): composition(X, one) = X.
% 4.98/1.06 Axiom 3 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 4.98/1.06 Axiom 4 (def_zero): zero = meet(X, complement(X)).
% 4.98/1.06 Axiom 5 (def_top): top = join(X, complement(X)).
% 4.98/1.06 Axiom 6 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 4.98/1.06 Axiom 7 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 4.98/1.06 Axiom 8 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 4.98/1.06 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 4.98/1.06 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 4.98/1.06 Axiom 11 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 4.98/1.06 Axiom 12 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 4.98/1.06
% 4.98/1.06 Lemma 13: complement(top) = zero.
% 4.98/1.06 Proof:
% 4.98/1.06 complement(top)
% 4.98/1.06 = { by axiom 5 (def_top) }
% 4.98/1.06 complement(join(complement(X), complement(complement(X))))
% 4.98/1.06 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 4.98/1.06 meet(X, complement(X))
% 4.98/1.06 = { by axiom 4 (def_zero) R->L }
% 4.98/1.06 zero
% 4.98/1.06
% 4.98/1.06 Lemma 14: join(X, join(Y, complement(X))) = join(Y, top).
% 4.98/1.06 Proof:
% 4.98/1.06 join(X, join(Y, complement(X)))
% 4.98/1.06 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 4.98/1.06 join(X, join(complement(X), Y))
% 4.98/1.06 = { by axiom 9 (maddux2_join_associativity) }
% 4.98/1.06 join(join(X, complement(X)), Y)
% 4.98/1.06 = { by axiom 5 (def_top) R->L }
% 4.98/1.06 join(top, Y)
% 4.98/1.06 = { by axiom 3 (maddux1_join_commutativity) }
% 4.98/1.06 join(Y, top)
% 4.98/1.06
% 4.98/1.06 Lemma 15: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 4.98/1.06 Proof:
% 4.98/1.06 converse(composition(converse(X), Y))
% 4.98/1.06 = { by axiom 6 (converse_multiplicativity) }
% 4.98/1.06 composition(converse(Y), converse(converse(X)))
% 4.98/1.06 = { by axiom 1 (converse_idempotence) }
% 4.98/1.06 composition(converse(Y), X)
% 4.98/1.06
% 4.98/1.06 Lemma 16: composition(converse(one), X) = X.
% 4.98/1.06 Proof:
% 4.98/1.06 composition(converse(one), X)
% 4.98/1.06 = { by lemma 15 R->L }
% 4.98/1.06 converse(composition(converse(X), one))
% 4.98/1.06 = { by axiom 2 (composition_identity) }
% 4.98/1.06 converse(converse(X))
% 4.98/1.06 = { by axiom 1 (converse_idempotence) }
% 4.98/1.06 X
% 4.98/1.06
% 4.98/1.06 Lemma 17: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 4.98/1.06 Proof:
% 4.98/1.06 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 4.98/1.06 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 4.98/1.06 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 4.98/1.06 = { by axiom 11 (converse_cancellativity) }
% 4.98/1.06 complement(X)
% 4.98/1.06
% 4.98/1.06 Lemma 18: join(complement(X), complement(X)) = complement(X).
% 4.98/1.06 Proof:
% 4.98/1.06 join(complement(X), complement(X))
% 4.98/1.06 = { by lemma 16 R->L }
% 4.98/1.06 join(complement(X), composition(converse(one), complement(X)))
% 4.98/1.06 = { by lemma 16 R->L }
% 4.98/1.06 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 4.98/1.06 = { by axiom 2 (composition_identity) R->L }
% 4.98/1.06 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 4.98/1.06 = { by axiom 7 (composition_associativity) R->L }
% 4.98/1.06 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 4.98/1.06 = { by lemma 16 }
% 4.98/1.06 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 4.98/1.06 = { by lemma 17 }
% 4.98/1.06 complement(X)
% 4.98/1.06
% 4.98/1.06 Lemma 19: join(top, complement(X)) = top.
% 4.98/1.06 Proof:
% 4.98/1.06 join(top, complement(X))
% 4.98/1.06 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 4.98/1.06 join(complement(X), top)
% 4.98/1.06 = { by lemma 14 R->L }
% 4.98/1.06 join(X, join(complement(X), complement(X)))
% 4.98/1.06 = { by lemma 18 }
% 4.98/1.06 join(X, complement(X))
% 4.98/1.06 = { by axiom 5 (def_top) R->L }
% 4.98/1.06 top
% 4.98/1.06
% 4.98/1.06 Lemma 20: join(Y, top) = join(X, top).
% 4.98/1.06 Proof:
% 4.98/1.06 join(Y, top)
% 4.98/1.06 = { by lemma 19 R->L }
% 4.98/1.06 join(Y, join(top, complement(Y)))
% 4.98/1.06 = { by lemma 14 }
% 4.98/1.06 join(top, top)
% 4.98/1.06 = { by lemma 14 R->L }
% 4.98/1.06 join(X, join(top, complement(X)))
% 4.98/1.06 = { by lemma 19 }
% 4.98/1.06 join(X, top)
% 4.98/1.06
% 4.98/1.06 Lemma 21: join(X, top) = top.
% 4.98/1.06 Proof:
% 4.98/1.06 join(X, top)
% 4.98/1.06 = { by lemma 20 }
% 4.98/1.06 join(join(zero, zero), top)
% 4.98/1.06 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 4.98/1.06 join(top, join(zero, zero))
% 4.98/1.06 = { by lemma 13 R->L }
% 4.98/1.06 join(top, join(zero, complement(top)))
% 4.98/1.06 = { by lemma 13 R->L }
% 4.98/1.06 join(top, join(complement(top), complement(top)))
% 4.98/1.06 = { by lemma 18 }
% 4.98/1.06 join(top, complement(top))
% 4.98/1.06 = { by axiom 5 (def_top) R->L }
% 4.98/1.06 top
% 4.98/1.06
% 4.98/1.06 Lemma 22: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 4.98/1.06 Proof:
% 4.98/1.06 join(meet(X, Y), complement(join(complement(X), Y)))
% 4.98/1.06 = { by axiom 10 (maddux4_definiton_of_meet) }
% 4.98/1.06 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 4.98/1.06 = { by axiom 12 (maddux3_a_kind_of_de_Morgan) R->L }
% 4.98/1.06 X
% 4.98/1.06
% 4.98/1.06 Lemma 23: join(zero, meet(X, X)) = X.
% 4.98/1.06 Proof:
% 4.98/1.06 join(zero, meet(X, X))
% 4.98/1.06 = { by axiom 10 (maddux4_definiton_of_meet) }
% 4.98/1.06 join(zero, complement(join(complement(X), complement(X))))
% 4.98/1.06 = { by axiom 4 (def_zero) }
% 4.98/1.06 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 4.98/1.06 = { by lemma 22 }
% 4.98/1.06 X
% 4.98/1.06
% 4.98/1.06 Lemma 24: complement(complement(X)) = meet(X, X).
% 4.98/1.06 Proof:
% 4.98/1.06 complement(complement(X))
% 4.98/1.06 = { by lemma 18 R->L }
% 4.98/1.06 complement(join(complement(X), complement(X)))
% 4.98/1.06 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 4.98/1.06 meet(X, X)
% 4.98/1.06
% 4.98/1.06 Lemma 25: meet(Y, X) = meet(X, Y).
% 4.98/1.06 Proof:
% 4.98/1.06 meet(Y, X)
% 4.98/1.06 = { by axiom 10 (maddux4_definiton_of_meet) }
% 4.98/1.06 complement(join(complement(Y), complement(X)))
% 4.98/1.06 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 4.98/1.06 complement(join(complement(X), complement(Y)))
% 4.98/1.06 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 4.98/1.06 meet(X, Y)
% 4.98/1.06
% 4.98/1.06 Lemma 26: complement(join(zero, complement(X))) = meet(X, top).
% 4.98/1.06 Proof:
% 4.98/1.06 complement(join(zero, complement(X)))
% 4.98/1.06 = { by lemma 13 R->L }
% 4.98/1.06 complement(join(complement(top), complement(X)))
% 4.98/1.06 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 4.98/1.06 meet(top, X)
% 4.98/1.06 = { by lemma 25 R->L }
% 4.98/1.06 meet(X, top)
% 4.98/1.06
% 4.98/1.06 Lemma 27: join(X, join(complement(X), Y)) = top.
% 4.98/1.06 Proof:
% 4.98/1.06 join(X, join(complement(X), Y))
% 4.98/1.06 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 4.98/1.06 join(X, join(Y, complement(X)))
% 4.98/1.06 = { by lemma 14 }
% 4.98/1.06 join(Y, top)
% 4.98/1.06 = { by lemma 20 R->L }
% 4.98/1.06 join(Z, top)
% 4.98/1.06 = { by lemma 21 }
% 4.98/1.06 top
% 4.98/1.06
% 4.98/1.06 Lemma 28: join(X, complement(zero)) = top.
% 4.98/1.06 Proof:
% 4.98/1.06 join(X, complement(zero))
% 4.98/1.06 = { by lemma 23 R->L }
% 4.98/1.06 join(join(zero, meet(X, X)), complement(zero))
% 4.98/1.06 = { by axiom 9 (maddux2_join_associativity) R->L }
% 4.98/1.06 join(zero, join(meet(X, X), complement(zero)))
% 4.98/1.06 = { by axiom 3 (maddux1_join_commutativity) }
% 4.98/1.06 join(zero, join(complement(zero), meet(X, X)))
% 4.98/1.06 = { by lemma 27 }
% 4.98/1.06 top
% 4.98/1.06
% 4.98/1.06 Lemma 29: join(meet(X, Y), meet(X, complement(Y))) = X.
% 4.98/1.06 Proof:
% 4.98/1.06 join(meet(X, Y), meet(X, complement(Y)))
% 4.98/1.06 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 4.98/1.06 join(meet(X, complement(Y)), meet(X, Y))
% 4.98/1.06 = { by axiom 10 (maddux4_definiton_of_meet) }
% 4.98/1.06 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 4.98/1.06 = { by lemma 22 }
% 4.98/1.06 X
% 4.98/1.06
% 4.98/1.06 Lemma 30: join(zero, meet(X, top)) = X.
% 4.98/1.06 Proof:
% 4.98/1.06 join(zero, meet(X, top))
% 4.98/1.06 = { by lemma 28 R->L }
% 4.98/1.06 join(zero, meet(X, join(complement(zero), complement(zero))))
% 4.98/1.06 = { by lemma 18 }
% 4.98/1.06 join(zero, meet(X, complement(zero)))
% 4.98/1.06 = { by lemma 13 R->L }
% 4.98/1.06 join(complement(top), meet(X, complement(zero)))
% 4.98/1.06 = { by lemma 28 R->L }
% 4.98/1.06 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 4.98/1.06 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 4.98/1.06 join(meet(X, zero), meet(X, complement(zero)))
% 4.98/1.06 = { by lemma 29 }
% 4.98/1.06 X
% 4.98/1.06
% 4.98/1.06 Lemma 31: join(zero, complement(X)) = complement(X).
% 4.98/1.06 Proof:
% 4.98/1.06 join(zero, complement(X))
% 4.98/1.06 = { by lemma 23 R->L }
% 4.98/1.06 join(zero, complement(join(zero, meet(X, X))))
% 4.98/1.06 = { by lemma 24 R->L }
% 4.98/1.06 join(zero, complement(join(zero, complement(complement(X)))))
% 4.98/1.06 = { by lemma 26 }
% 4.98/1.06 join(zero, meet(complement(X), top))
% 4.98/1.07 = { by lemma 30 }
% 4.98/1.07 complement(X)
% 4.98/1.07
% 4.98/1.07 Lemma 32: complement(complement(X)) = X.
% 4.98/1.07 Proof:
% 4.98/1.07 complement(complement(X))
% 4.98/1.07 = { by lemma 31 R->L }
% 4.98/1.07 join(zero, complement(complement(X)))
% 4.98/1.07 = { by lemma 24 }
% 4.98/1.07 join(zero, meet(X, X))
% 4.98/1.07 = { by lemma 23 }
% 4.98/1.07 X
% 4.98/1.07
% 4.98/1.07 Lemma 33: join(X, zero) = X.
% 4.98/1.07 Proof:
% 4.98/1.07 join(X, zero)
% 4.98/1.07 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 4.98/1.07 join(zero, X)
% 4.98/1.07 = { by lemma 32 R->L }
% 4.98/1.07 join(zero, complement(complement(X)))
% 4.98/1.07 = { by lemma 24 }
% 4.98/1.07 join(zero, meet(X, X))
% 4.98/1.07 = { by lemma 23 }
% 4.98/1.07 X
% 4.98/1.07
% 4.98/1.07 Lemma 34: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 4.98/1.07 Proof:
% 4.98/1.07 converse(join(X, converse(Y)))
% 4.98/1.07 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 4.98/1.07 converse(join(converse(Y), X))
% 4.98/1.07 = { by axiom 8 (converse_additivity) }
% 4.98/1.07 join(converse(converse(Y)), converse(X))
% 4.98/1.07 = { by axiom 1 (converse_idempotence) }
% 4.98/1.07 join(Y, converse(X))
% 4.98/1.07
% 4.98/1.07 Lemma 35: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 4.98/1.07 Proof:
% 4.98/1.07 converse(join(converse(X), Y))
% 4.98/1.07 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 4.98/1.07 converse(join(Y, converse(X)))
% 4.98/1.07 = { by lemma 34 }
% 4.98/1.07 join(X, converse(Y))
% 4.98/1.07
% 4.98/1.07 Lemma 36: meet(X, converse(complement(converse(complement(X))))) = X.
% 4.98/1.07 Proof:
% 4.98/1.07 meet(X, converse(complement(converse(complement(X)))))
% 4.98/1.07 = { by lemma 18 R->L }
% 4.98/1.07 meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X))))))
% 4.98/1.07 = { by lemma 33 R->L }
% 4.98/1.07 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), zero)
% 4.98/1.07 = { by lemma 13 R->L }
% 4.98/1.07 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(top))
% 4.98/1.07 = { by lemma 27 R->L }
% 4.98/1.07 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(join(converse(Y), join(complement(converse(Y)), converse(complement(converse(complement(converse(Y)))))))))
% 4.98/1.07 = { by lemma 35 R->L }
% 4.98/1.07 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(join(converse(Y), converse(join(converse(complement(converse(Y))), complement(converse(complement(converse(Y)))))))))
% 4.98/1.07 = { by axiom 5 (def_top) R->L }
% 4.98/1.07 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(join(converse(Y), converse(top))))
% 4.98/1.07 = { by axiom 8 (converse_additivity) R->L }
% 4.98/1.07 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(converse(join(Y, top))))
% 4.98/1.07 = { by lemma 21 }
% 4.98/1.07 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(converse(top)))
% 4.98/1.07 = { by lemma 27 R->L }
% 4.98/1.07 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(converse(join(converse(complement(X)), join(complement(converse(complement(X))), complement(converse(complement(X))))))))
% 4.98/1.07 = { by lemma 35 }
% 4.98/1.07 join(meet(X, converse(join(complement(converse(complement(X))), complement(converse(complement(X)))))), complement(join(complement(X), converse(join(complement(converse(complement(X))), complement(converse(complement(X))))))))
% 4.98/1.07 = { by lemma 22 }
% 4.98/1.07 X
% 4.98/1.07
% 4.98/1.07 Lemma 37: meet(X, top) = X.
% 4.98/1.07 Proof:
% 4.98/1.07 meet(X, top)
% 4.98/1.07 = { by lemma 26 R->L }
% 4.98/1.07 complement(join(zero, complement(X)))
% 4.98/1.07 = { by lemma 31 R->L }
% 4.98/1.07 join(zero, complement(join(zero, complement(X))))
% 4.98/1.07 = { by lemma 26 }
% 4.98/1.07 join(zero, meet(X, top))
% 4.98/1.07 = { by lemma 30 }
% 4.98/1.07 X
% 4.98/1.07
% 4.98/1.07 Lemma 38: meet(top, X) = X.
% 4.98/1.07 Proof:
% 4.98/1.07 meet(top, X)
% 4.98/1.07 = { by lemma 25 }
% 4.98/1.07 meet(X, top)
% 4.98/1.07 = { by lemma 37 }
% 4.98/1.07 X
% 4.98/1.07
% 4.98/1.07 Lemma 39: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 4.98/1.07 Proof:
% 4.98/1.07 join(complement(X), complement(Y))
% 4.98/1.07 = { by lemma 38 R->L }
% 4.98/1.07 meet(top, join(complement(X), complement(Y)))
% 4.98/1.07 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 4.98/1.07 meet(top, join(complement(Y), complement(X)))
% 4.98/1.07 = { by lemma 25 }
% 4.98/1.07 meet(join(complement(Y), complement(X)), top)
% 4.98/1.07 = { by axiom 10 (maddux4_definiton_of_meet) }
% 4.98/1.07 complement(join(complement(join(complement(Y), complement(X))), complement(top)))
% 4.98/1.07 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 5.64/1.07 complement(join(meet(Y, X), complement(top)))
% 5.64/1.07 = { by axiom 3 (maddux1_join_commutativity) }
% 5.64/1.07 complement(join(complement(top), meet(Y, X)))
% 5.64/1.07 = { by lemma 25 R->L }
% 5.64/1.07 complement(join(complement(top), meet(X, Y)))
% 5.64/1.07 = { by lemma 13 }
% 5.64/1.07 complement(join(zero, meet(X, Y)))
% 5.64/1.07 = { by lemma 25 R->L }
% 5.64/1.07 complement(join(zero, meet(Y, X)))
% 5.64/1.07 = { by axiom 3 (maddux1_join_commutativity) }
% 5.64/1.07 complement(join(meet(Y, X), zero))
% 5.64/1.07 = { by lemma 33 }
% 5.64/1.07 complement(meet(Y, X))
% 5.64/1.07 = { by lemma 25 R->L }
% 5.64/1.07 complement(meet(X, Y))
% 5.64/1.07
% 5.64/1.07 Lemma 40: meet(X, X) = X.
% 5.64/1.07 Proof:
% 5.64/1.07 meet(X, X)
% 5.64/1.07 = { by lemma 24 R->L }
% 5.64/1.07 complement(complement(X))
% 5.64/1.07 = { by lemma 32 }
% 5.64/1.07 X
% 5.64/1.07
% 5.64/1.07 Lemma 41: complement(converse(X)) = converse(complement(X)).
% 5.64/1.07 Proof:
% 5.64/1.07 complement(converse(X))
% 5.64/1.07 = { by lemma 36 R->L }
% 5.64/1.07 complement(converse(meet(X, converse(complement(converse(complement(X)))))))
% 5.64/1.07 = { by lemma 33 R->L }
% 5.64/1.07 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), zero)))
% 5.64/1.07 = { by axiom 4 (def_zero) }
% 5.64/1.07 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), complement(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))))))))
% 5.64/1.07 = { by lemma 25 }
% 5.64/1.07 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), complement(meet(converse(complement(converse(complement(X)))), converse(converse(complement(X)))))))))
% 5.64/1.07 = { by lemma 39 R->L }
% 5.64/1.07 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(complement(converse(complement(X))))), complement(converse(converse(complement(X)))))))))
% 5.64/1.07 = { by lemma 36 R->L }
% 5.64/1.07 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(complement(converse(complement(X))))), complement(converse(meet(converse(complement(X)), converse(complement(converse(complement(converse(complement(X))))))))))))))
% 5.64/1.07 = { by lemma 25 }
% 5.64/1.07 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(complement(converse(complement(X))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))))))))
% 5.64/1.07 = { by axiom 1 (converse_idempotence) R->L }
% 5.64/1.07 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(converse(converse(complement(converse(complement(converse(complement(X))))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))))))))
% 5.64/1.07 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 5.64/1.07 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))), converse(converse(complement(converse(complement(converse(complement(X))))))))))))
% 5.64/1.07 = { by lemma 22 R->L }
% 5.64/1.07 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))), converse(join(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))), complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X)))))))))))
% 5.64/1.07 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 5.64/1.07 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(converse(join(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))), complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X)))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))))))))
% 5.64/1.07 = { by axiom 8 (converse_additivity) }
% 5.64/1.07 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(join(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))), converse(complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X)))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))))))))
% 5.64/1.07 = { by axiom 9 (maddux2_join_associativity) R->L }
% 5.64/1.07 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))), join(converse(complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X))))), complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))))))))))
% 5.64/1.07 = { by axiom 3 (maddux1_join_commutativity) }
% 5.64/1.07 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), join(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X)))), join(complement(converse(meet(converse(complement(converse(complement(converse(complement(X)))))), converse(complement(X))))), converse(complement(join(complement(converse(complement(converse(complement(converse(complement(X))))))), converse(complement(X)))))))))))
% 5.64/1.07 = { by lemma 27 }
% 5.64/1.07 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))), top))))
% 5.64/1.08 = { by lemma 25 }
% 5.64/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(top, meet(converse(converse(complement(X))), converse(complement(converse(complement(X)))))))))
% 5.64/1.08 = { by lemma 40 R->L }
% 5.64/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(meet(top, meet(converse(converse(complement(X))), converse(complement(converse(complement(X)))))), meet(top, meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))))))))
% 5.64/1.08 = { by lemma 24 R->L }
% 5.64/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), complement(complement(meet(top, meet(converse(converse(complement(X))), converse(complement(converse(complement(X)))))))))))
% 5.64/1.08 = { by lemma 25 }
% 5.64/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), complement(complement(meet(top, meet(converse(complement(converse(complement(X)))), converse(converse(complement(X))))))))))
% 5.64/1.08 = { by lemma 39 R->L }
% 5.64/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), complement(join(complement(top), complement(meet(converse(complement(converse(complement(X)))), converse(converse(complement(X))))))))))
% 5.64/1.08 = { by lemma 39 R->L }
% 5.64/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), complement(join(complement(top), join(complement(converse(complement(converse(complement(X))))), complement(converse(converse(complement(X))))))))))
% 5.64/1.08 = { by axiom 9 (maddux2_join_associativity) }
% 5.64/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), complement(join(join(complement(top), complement(converse(complement(converse(complement(X)))))), complement(converse(converse(complement(X)))))))))
% 5.64/1.08 = { by lemma 37 R->L }
% 5.64/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), complement(join(meet(join(complement(top), complement(converse(complement(converse(complement(X)))))), top), complement(converse(converse(complement(X)))))))))
% 5.64/1.08 = { by lemma 26 R->L }
% 5.64/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), complement(join(complement(join(zero, complement(join(complement(top), complement(converse(complement(converse(complement(X))))))))), complement(converse(converse(complement(X)))))))))
% 5.64/1.08 = { by lemma 39 }
% 5.64/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), complement(complement(meet(join(zero, complement(join(complement(top), complement(converse(complement(converse(complement(X)))))))), converse(converse(complement(X)))))))))
% 5.71/1.08 = { by lemma 31 }
% 5.71/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), complement(complement(meet(complement(join(complement(top), complement(converse(complement(converse(complement(X))))))), converse(converse(complement(X)))))))))
% 5.71/1.08 = { by lemma 25 R->L }
% 5.71/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), complement(complement(meet(converse(converse(complement(X))), complement(join(complement(top), complement(converse(complement(converse(complement(X)))))))))))))
% 5.71/1.08 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 5.71/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), complement(complement(meet(converse(converse(complement(X))), meet(top, converse(complement(converse(complement(X)))))))))))
% 5.71/1.08 = { by lemma 25 R->L }
% 5.71/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), complement(complement(meet(converse(converse(complement(X))), meet(converse(complement(converse(complement(X)))), top)))))))
% 5.71/1.08 = { by lemma 32 }
% 5.71/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(converse(converse(complement(X))), meet(converse(complement(converse(complement(X)))), top)))))
% 5.71/1.08 = { by lemma 25 R->L }
% 5.71/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(converse(converse(complement(X))), meet(top, converse(complement(converse(complement(X)))))))))
% 5.71/1.08 = { by lemma 38 }
% 5.71/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(converse(converse(complement(X))), converse(complement(converse(complement(X))))))))
% 5.71/1.08 = { by axiom 1 (converse_idempotence) }
% 5.71/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(converse(complement(X))))))))
% 5.71/1.08 = { by lemma 25 }
% 5.71/1.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(converse(complement(converse(complement(X)))), complement(X)))))
% 5.71/1.08 = { by lemma 25 }
% 5.71/1.08 complement(converse(join(meet(converse(complement(converse(complement(X)))), X), meet(converse(complement(converse(complement(X)))), complement(X)))))
% 5.71/1.08 = { by lemma 29 }
% 5.71/1.08 complement(converse(converse(complement(converse(complement(X))))))
% 5.71/1.08 = { by axiom 1 (converse_idempotence) }
% 5.71/1.08 complement(complement(converse(complement(X))))
% 5.71/1.08 = { by lemma 24 }
% 5.71/1.08 meet(converse(complement(X)), converse(complement(X)))
% 5.71/1.08 = { by lemma 40 }
% 5.71/1.08 converse(complement(X))
% 5.71/1.08
% 5.71/1.08 Goal 1 (goals): join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)) = complement(x0).
% 5.71/1.08 Proof:
% 5.71/1.08 join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0))
% 5.71/1.08 = { by axiom 3 (maddux1_join_commutativity) }
% 5.71/1.08 join(complement(x0), composition(complement(composition(x0, x1)), converse(x1)))
% 5.71/1.08 = { by axiom 1 (converse_idempotence) R->L }
% 5.71/1.08 converse(converse(join(complement(x0), composition(complement(composition(x0, x1)), converse(x1)))))
% 5.71/1.08 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 5.71/1.08 converse(converse(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0))))
% 5.71/1.08 = { by axiom 8 (converse_additivity) }
% 5.71/1.08 converse(join(converse(composition(complement(composition(x0, x1)), converse(x1))), converse(complement(x0))))
% 5.71/1.08 = { by axiom 6 (converse_multiplicativity) }
% 5.71/1.08 converse(join(composition(converse(converse(x1)), converse(complement(composition(x0, x1)))), converse(complement(x0))))
% 5.71/1.08 = { by axiom 1 (converse_idempotence) }
% 5.71/1.08 converse(join(composition(x1, converse(complement(composition(x0, x1)))), converse(complement(x0))))
% 5.71/1.08 = { by lemma 41 R->L }
% 5.71/1.08 converse(join(composition(x1, complement(converse(composition(x0, x1)))), converse(complement(x0))))
% 5.71/1.08 = { by lemma 34 R->L }
% 5.71/1.08 converse(converse(join(complement(x0), converse(composition(x1, complement(converse(composition(x0, x1))))))))
% 5.71/1.08 = { by axiom 6 (converse_multiplicativity) }
% 5.71/1.08 converse(converse(join(complement(x0), composition(converse(complement(converse(composition(x0, x1)))), converse(x1)))))
% 5.71/1.08 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 5.71/1.08 converse(converse(join(composition(converse(complement(converse(composition(x0, x1)))), converse(x1)), complement(x0))))
% 5.71/1.08 = { by axiom 8 (converse_additivity) }
% 5.71/1.08 converse(join(converse(composition(converse(complement(converse(composition(x0, x1)))), converse(x1))), converse(complement(x0))))
% 5.71/1.08 = { by lemma 15 }
% 5.71/1.08 converse(join(composition(converse(converse(x1)), complement(converse(composition(x0, x1)))), converse(complement(x0))))
% 5.71/1.08 = { by axiom 3 (maddux1_join_commutativity) }
% 5.71/1.08 converse(join(converse(complement(x0)), composition(converse(converse(x1)), complement(converse(composition(x0, x1))))))
% 5.71/1.08 = { by lemma 41 R->L }
% 5.71/1.08 converse(join(complement(converse(x0)), composition(converse(converse(x1)), complement(converse(composition(x0, x1))))))
% 5.71/1.08 = { by axiom 6 (converse_multiplicativity) }
% 5.71/1.08 converse(join(complement(converse(x0)), composition(converse(converse(x1)), complement(composition(converse(x1), converse(x0))))))
% 5.71/1.08 = { by lemma 17 }
% 5.71/1.08 converse(complement(converse(x0)))
% 5.71/1.08 = { by lemma 41 }
% 5.71/1.08 converse(converse(complement(x0)))
% 5.71/1.08 = { by axiom 1 (converse_idempotence) }
% 5.71/1.08 complement(x0)
% 5.71/1.08 % SZS output end Proof
% 5.71/1.08
% 5.71/1.08 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------