TSTP Solution File: REL045+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL045+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 : n005.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 13:44:30 EDT 2023
% Result : Theorem 110.81s 14.56s
% Output : Proof 111.49s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : REL045+1 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n005.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Aug 25 23:01:38 EDT 2023
% 0.12/0.34 % CPUTime :
% 110.81/14.56 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 110.81/14.56
% 110.81/14.56 % SZS status Theorem
% 110.81/14.56
% 110.81/14.60 % SZS output start Proof
% 110.81/14.60 Axiom 1 (composition_identity): composition(X, one) = X.
% 110.81/14.60 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 110.81/14.60 Axiom 3 (converse_idempotence): converse(converse(X)) = X.
% 110.81/14.60 Axiom 4 (def_top): top = join(X, complement(X)).
% 110.81/14.60 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 110.81/14.60 Axiom 6 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 110.81/14.60 Axiom 7 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 110.81/14.60 Axiom 8 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 110.81/14.60 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 110.81/14.60 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 110.81/14.60 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 110.81/14.60 Axiom 12 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 110.81/14.60 Axiom 13 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 110.81/14.60
% 110.81/14.60 Lemma 14: complement(top) = zero.
% 110.81/14.60 Proof:
% 110.81/14.60 complement(top)
% 110.81/14.60 = { by axiom 4 (def_top) }
% 110.81/14.60 complement(join(complement(X), complement(complement(X))))
% 110.81/14.60 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 110.81/14.60 meet(X, complement(X))
% 110.81/14.60 = { by axiom 5 (def_zero) R->L }
% 110.81/14.60 zero
% 110.81/14.60
% 110.81/14.60 Lemma 15: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 110.81/14.60 Proof:
% 110.81/14.60 converse(join(X, converse(Y)))
% 110.81/14.60 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 110.81/14.60 converse(join(converse(Y), X))
% 110.81/14.60 = { by axiom 8 (converse_additivity) }
% 110.81/14.60 join(converse(converse(Y)), converse(X))
% 110.81/14.60 = { by axiom 3 (converse_idempotence) }
% 110.81/14.60 join(Y, converse(X))
% 110.81/14.60
% 110.81/14.60 Lemma 16: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 110.81/14.60 Proof:
% 110.81/14.60 converse(join(converse(X), Y))
% 110.81/14.60 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 110.81/14.60 converse(join(Y, converse(X)))
% 110.81/14.60 = { by lemma 15 }
% 110.81/14.60 join(X, converse(Y))
% 110.81/14.60
% 110.81/14.60 Lemma 17: join(X, converse(complement(converse(X)))) = converse(top).
% 110.81/14.60 Proof:
% 110.81/14.60 join(X, converse(complement(converse(X))))
% 110.81/14.60 = { by lemma 16 R->L }
% 110.81/14.60 converse(join(converse(X), complement(converse(X))))
% 110.81/14.60 = { by axiom 4 (def_top) R->L }
% 110.81/14.60 converse(top)
% 110.81/14.60
% 110.81/14.60 Lemma 18: join(X, join(Y, complement(X))) = join(Y, top).
% 110.81/14.60 Proof:
% 110.81/14.60 join(X, join(Y, complement(X)))
% 110.81/14.60 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 110.81/14.60 join(X, join(complement(X), Y))
% 110.81/14.60 = { by axiom 9 (maddux2_join_associativity) }
% 110.81/14.60 join(join(X, complement(X)), Y)
% 110.81/14.60 = { by axiom 4 (def_top) R->L }
% 110.81/14.60 join(top, Y)
% 110.81/14.60 = { by axiom 2 (maddux1_join_commutativity) }
% 110.81/14.60 join(Y, top)
% 110.81/14.60
% 110.81/14.60 Lemma 19: join(X, join(complement(X), Y)) = join(Y, top).
% 110.81/14.60 Proof:
% 110.81/14.60 join(X, join(complement(X), Y))
% 110.81/14.60 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 110.81/14.60 join(X, join(Y, complement(X)))
% 110.81/14.60 = { by lemma 18 }
% 110.81/14.60 join(Y, top)
% 110.81/14.60
% 110.81/14.60 Lemma 20: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 110.81/14.60 Proof:
% 110.81/14.60 converse(composition(converse(X), Y))
% 110.81/14.60 = { by axiom 6 (converse_multiplicativity) }
% 110.81/14.60 composition(converse(Y), converse(converse(X)))
% 110.81/14.60 = { by axiom 3 (converse_idempotence) }
% 110.81/14.60 composition(converse(Y), X)
% 110.81/14.60
% 110.81/14.60 Lemma 21: composition(converse(one), X) = X.
% 110.81/14.60 Proof:
% 110.81/14.60 composition(converse(one), X)
% 110.81/14.60 = { by lemma 20 R->L }
% 110.81/14.60 converse(composition(converse(X), one))
% 110.81/14.60 = { by axiom 1 (composition_identity) }
% 110.81/14.60 converse(converse(X))
% 110.81/14.60 = { by axiom 3 (converse_idempotence) }
% 110.81/14.60 X
% 110.81/14.60
% 110.81/14.60 Lemma 22: join(complement(X), composition(Y, complement(composition(converse(Y), X)))) = complement(X).
% 110.81/14.60 Proof:
% 110.81/14.60 join(complement(X), composition(Y, complement(composition(converse(Y), X))))
% 110.81/14.60 = { by axiom 3 (converse_idempotence) R->L }
% 110.81/14.60 join(complement(X), composition(converse(converse(Y)), complement(composition(converse(Y), X))))
% 110.81/14.60 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 110.81/14.60 join(composition(converse(converse(Y)), complement(composition(converse(Y), X))), complement(X))
% 110.81/14.60 = { by axiom 12 (converse_cancellativity) }
% 110.81/14.60 complement(X)
% 110.81/14.60
% 110.81/14.60 Lemma 23: join(complement(X), complement(X)) = complement(X).
% 110.81/14.60 Proof:
% 110.81/14.60 join(complement(X), complement(X))
% 110.81/14.60 = { by lemma 21 R->L }
% 110.81/14.60 join(complement(X), complement(composition(converse(one), X)))
% 110.81/14.61 = { by axiom 1 (composition_identity) R->L }
% 110.81/14.61 join(complement(X), complement(composition(composition(converse(one), one), X)))
% 110.81/14.61 = { by axiom 7 (composition_associativity) R->L }
% 110.81/14.61 join(complement(X), complement(composition(converse(one), composition(one, X))))
% 110.81/14.61 = { by lemma 21 }
% 110.81/14.61 join(complement(X), complement(composition(one, X)))
% 110.81/14.61 = { by axiom 3 (converse_idempotence) R->L }
% 110.81/14.61 join(complement(X), complement(composition(converse(converse(one)), X)))
% 110.81/14.61 = { by lemma 21 R->L }
% 110.81/14.61 join(complement(X), composition(converse(one), complement(composition(converse(converse(one)), X))))
% 110.81/14.61 = { by lemma 22 }
% 110.81/14.61 complement(X)
% 110.81/14.61
% 110.81/14.61 Lemma 24: join(X, top) = top.
% 110.81/14.61 Proof:
% 110.81/14.61 join(X, top)
% 110.81/14.61 = { by axiom 4 (def_top) }
% 110.81/14.61 join(X, join(complement(X), complement(complement(X))))
% 110.81/14.61 = { by lemma 19 }
% 110.81/14.61 join(complement(complement(X)), top)
% 110.81/14.61 = { by lemma 18 R->L }
% 110.81/14.61 join(complement(complement(X)), join(complement(complement(X)), complement(complement(complement(X)))))
% 110.81/14.61 = { by lemma 23 R->L }
% 110.81/14.61 join(complement(complement(X)), join(complement(complement(X)), complement(join(complement(complement(X)), complement(complement(X))))))
% 110.81/14.61 = { by axiom 9 (maddux2_join_associativity) }
% 110.81/14.61 join(join(complement(complement(X)), complement(complement(X))), complement(join(complement(complement(X)), complement(complement(X)))))
% 110.81/14.61 = { by axiom 4 (def_top) R->L }
% 110.81/14.61 top
% 110.81/14.61
% 110.81/14.61 Lemma 25: join(top, X) = top.
% 110.81/14.61 Proof:
% 110.81/14.61 join(top, X)
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 110.81/14.61 join(X, top)
% 110.81/14.61 = { by lemma 24 }
% 110.81/14.61 top
% 110.81/14.61
% 110.81/14.61 Lemma 26: converse(top) = top.
% 110.81/14.61 Proof:
% 110.81/14.61 converse(top)
% 110.81/14.61 = { by lemma 17 R->L }
% 110.81/14.61 join(top, converse(complement(converse(top))))
% 110.81/14.61 = { by lemma 25 }
% 110.81/14.61 top
% 110.81/14.61
% 110.81/14.61 Lemma 27: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 110.81/14.61 Proof:
% 110.81/14.61 join(meet(X, Y), complement(join(complement(X), Y)))
% 110.81/14.61 = { by axiom 10 (maddux4_definiton_of_meet) }
% 110.81/14.61 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 110.81/14.61 = { by axiom 13 (maddux3_a_kind_of_de_Morgan) R->L }
% 110.81/14.61 X
% 110.81/14.61
% 110.81/14.61 Lemma 28: meet(Y, X) = meet(X, Y).
% 110.81/14.61 Proof:
% 110.81/14.61 meet(Y, X)
% 110.81/14.61 = { by axiom 10 (maddux4_definiton_of_meet) }
% 110.81/14.61 complement(join(complement(Y), complement(X)))
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 110.81/14.61 complement(join(complement(X), complement(Y)))
% 110.81/14.61 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 110.81/14.61 meet(X, Y)
% 110.81/14.61
% 110.81/14.61 Lemma 29: complement(complement(X)) = X.
% 110.81/14.61 Proof:
% 110.81/14.61 complement(complement(X))
% 110.81/14.61 = { by lemma 27 R->L }
% 110.81/14.61 join(meet(complement(complement(X)), complement(X)), complement(join(complement(complement(complement(X))), complement(X))))
% 110.81/14.61 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 110.81/14.61 join(meet(complement(complement(X)), complement(X)), meet(complement(complement(X)), X))
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) }
% 110.81/14.61 join(meet(complement(complement(X)), X), meet(complement(complement(X)), complement(X)))
% 110.81/14.61 = { by lemma 28 R->L }
% 110.81/14.61 join(meet(complement(complement(X)), X), meet(complement(X), complement(complement(X))))
% 110.81/14.61 = { by axiom 5 (def_zero) R->L }
% 110.81/14.61 join(meet(complement(complement(X)), X), zero)
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) }
% 110.81/14.61 join(zero, meet(complement(complement(X)), X))
% 110.81/14.61 = { by lemma 28 R->L }
% 110.81/14.61 join(zero, meet(X, complement(complement(X))))
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 110.81/14.61 join(meet(X, complement(complement(X))), zero)
% 110.81/14.61 = { by lemma 14 R->L }
% 110.81/14.61 join(meet(X, complement(complement(X))), complement(top))
% 110.81/14.61 = { by axiom 4 (def_top) }
% 110.81/14.61 join(meet(X, complement(complement(X))), complement(join(complement(X), complement(complement(X)))))
% 110.81/14.61 = { by lemma 27 }
% 110.81/14.61 X
% 110.81/14.61
% 110.81/14.61 Lemma 30: join(zero, complement(complement(X))) = X.
% 110.81/14.61 Proof:
% 110.81/14.61 join(zero, complement(complement(X)))
% 110.81/14.61 = { by axiom 5 (def_zero) }
% 110.81/14.61 join(meet(X, complement(X)), complement(complement(X)))
% 110.81/14.61 = { by lemma 23 R->L }
% 110.81/14.61 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 110.81/14.61 = { by lemma 27 }
% 110.81/14.61 X
% 110.81/14.61
% 110.81/14.61 Lemma 31: join(X, zero) = X.
% 110.81/14.61 Proof:
% 110.81/14.61 join(X, zero)
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 110.81/14.61 join(zero, X)
% 110.81/14.61 = { by lemma 29 R->L }
% 110.81/14.61 join(zero, complement(complement(X)))
% 110.81/14.61 = { by lemma 30 }
% 110.81/14.61 X
% 110.81/14.61
% 110.81/14.61 Lemma 32: complement(join(zero, complement(X))) = meet(X, top).
% 110.81/14.61 Proof:
% 110.81/14.61 complement(join(zero, complement(X)))
% 110.81/14.61 = { by lemma 14 R->L }
% 110.81/14.61 complement(join(complement(top), complement(X)))
% 110.81/14.61 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 110.81/14.61 meet(top, X)
% 110.81/14.61 = { by lemma 28 R->L }
% 110.81/14.61 meet(X, top)
% 110.81/14.61
% 110.81/14.61 Lemma 33: join(zero, complement(X)) = complement(X).
% 110.81/14.61 Proof:
% 110.81/14.61 join(zero, complement(X))
% 110.81/14.61 = { by lemma 30 R->L }
% 110.81/14.61 join(zero, complement(join(zero, complement(complement(X)))))
% 110.81/14.61 = { by lemma 23 R->L }
% 110.81/14.61 join(zero, complement(join(zero, join(complement(complement(X)), complement(complement(X))))))
% 110.81/14.61 = { by lemma 23 R->L }
% 110.81/14.61 join(zero, complement(join(zero, join(complement(join(complement(X), complement(X))), complement(complement(X))))))
% 110.81/14.61 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 110.81/14.61 join(zero, complement(join(zero, join(meet(X, X), complement(complement(X))))))
% 110.81/14.61 = { by axiom 9 (maddux2_join_associativity) }
% 110.81/14.61 join(zero, complement(join(join(zero, meet(X, X)), complement(complement(X)))))
% 110.81/14.61 = { by axiom 10 (maddux4_definiton_of_meet) }
% 110.81/14.61 join(zero, complement(join(join(zero, complement(join(complement(X), complement(X)))), complement(complement(X)))))
% 110.81/14.61 = { by axiom 5 (def_zero) }
% 110.81/14.61 join(zero, complement(join(join(meet(X, complement(X)), complement(join(complement(X), complement(X)))), complement(complement(X)))))
% 110.81/14.61 = { by lemma 27 }
% 110.81/14.61 join(zero, complement(join(X, complement(complement(X)))))
% 110.81/14.61 = { by lemma 29 }
% 110.81/14.61 join(zero, complement(join(X, X)))
% 110.81/14.61 = { by lemma 29 R->L }
% 110.81/14.61 join(zero, complement(join(complement(complement(X)), X)))
% 110.81/14.61 = { by axiom 5 (def_zero) }
% 110.81/14.61 join(meet(X, complement(X)), complement(join(complement(complement(X)), X)))
% 110.81/14.61 = { by lemma 28 }
% 110.81/14.61 join(meet(complement(X), X), complement(join(complement(complement(X)), X)))
% 110.81/14.61 = { by lemma 27 }
% 110.81/14.61 complement(X)
% 110.81/14.61
% 110.81/14.61 Lemma 34: meet(X, top) = X.
% 110.81/14.61 Proof:
% 110.81/14.61 meet(X, top)
% 110.81/14.61 = { by lemma 32 R->L }
% 110.81/14.61 complement(join(zero, complement(X)))
% 110.81/14.61 = { by lemma 33 }
% 110.81/14.61 complement(complement(X))
% 110.81/14.61 = { by lemma 29 }
% 110.81/14.61 X
% 110.81/14.61
% 110.81/14.61 Lemma 35: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 110.81/14.61 Proof:
% 110.81/14.61 complement(join(X, complement(Y)))
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 110.81/14.61 complement(join(complement(Y), X))
% 110.81/14.61 = { by lemma 34 R->L }
% 110.81/14.61 complement(join(complement(Y), meet(X, top)))
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 110.81/14.61 complement(join(meet(X, top), complement(Y)))
% 110.81/14.61 = { by axiom 10 (maddux4_definiton_of_meet) }
% 110.81/14.61 complement(join(complement(join(complement(X), complement(top))), complement(Y)))
% 110.81/14.61 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 110.81/14.61 meet(join(complement(X), complement(top)), Y)
% 110.81/14.61 = { by lemma 28 R->L }
% 110.81/14.61 meet(Y, join(complement(X), complement(top)))
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) }
% 110.81/14.61 meet(Y, join(complement(top), complement(X)))
% 110.81/14.61 = { by lemma 14 }
% 110.81/14.61 meet(Y, join(zero, complement(X)))
% 110.81/14.61 = { by lemma 33 }
% 110.81/14.61 meet(Y, complement(X))
% 110.81/14.61
% 110.81/14.61 Lemma 36: join(composition(X, converse(Y)), converse(composition(Y, Z))) = composition(join(X, converse(Z)), converse(Y)).
% 110.81/14.61 Proof:
% 110.81/14.61 join(composition(X, converse(Y)), converse(composition(Y, Z)))
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 110.81/14.61 join(converse(composition(Y, Z)), composition(X, converse(Y)))
% 110.81/14.61 = { by axiom 6 (converse_multiplicativity) }
% 110.81/14.61 join(composition(converse(Z), converse(Y)), composition(X, converse(Y)))
% 110.81/14.61 = { by axiom 11 (composition_distributivity) R->L }
% 110.81/14.61 composition(join(converse(Z), X), converse(Y))
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) }
% 110.81/14.61 composition(join(X, converse(Z)), converse(Y))
% 110.81/14.61
% 110.81/14.61 Lemma 37: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 110.81/14.61 Proof:
% 110.81/14.61 join(composition(X, Y), composition(X, Z))
% 110.81/14.61 = { by axiom 3 (converse_idempotence) R->L }
% 110.81/14.61 converse(converse(join(composition(X, Y), composition(X, Z))))
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 110.81/14.61 converse(converse(join(composition(X, Z), composition(X, Y))))
% 110.81/14.61 = { by axiom 8 (converse_additivity) }
% 110.81/14.61 converse(join(converse(composition(X, Z)), converse(composition(X, Y))))
% 110.81/14.61 = { by axiom 6 (converse_multiplicativity) }
% 110.81/14.61 converse(join(composition(converse(Z), converse(X)), converse(composition(X, Y))))
% 110.81/14.61 = { by lemma 36 }
% 110.81/14.61 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 110.81/14.61 = { by axiom 8 (converse_additivity) R->L }
% 110.81/14.61 converse(composition(converse(join(Z, Y)), converse(X)))
% 110.81/14.61 = { by axiom 6 (converse_multiplicativity) R->L }
% 110.81/14.61 converse(converse(composition(X, join(Z, Y))))
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) }
% 110.81/14.61 converse(converse(composition(X, join(Y, Z))))
% 110.81/14.61 = { by axiom 3 (converse_idempotence) }
% 110.81/14.61 composition(X, join(Y, Z))
% 110.81/14.61
% 110.81/14.61 Lemma 38: composition(X, composition(converse(X), top)) = composition(X, top).
% 110.81/14.61 Proof:
% 110.81/14.61 composition(X, composition(converse(X), top))
% 110.81/14.61 = { by lemma 31 R->L }
% 110.81/14.61 join(composition(X, composition(converse(X), top)), zero)
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) }
% 110.81/14.61 join(zero, composition(X, composition(converse(X), top)))
% 110.81/14.61 = { by lemma 14 R->L }
% 110.81/14.61 join(complement(top), composition(X, composition(converse(X), top)))
% 110.81/14.61 = { by lemma 22 R->L }
% 110.81/14.61 join(join(complement(top), composition(X, complement(composition(converse(X), top)))), composition(X, composition(converse(X), top)))
% 110.81/14.61 = { by lemma 14 }
% 110.81/14.61 join(join(zero, composition(X, complement(composition(converse(X), top)))), composition(X, composition(converse(X), top)))
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) }
% 110.81/14.61 join(join(composition(X, complement(composition(converse(X), top))), zero), composition(X, composition(converse(X), top)))
% 110.81/14.61 = { by lemma 31 }
% 110.81/14.61 join(composition(X, complement(composition(converse(X), top))), composition(X, composition(converse(X), top)))
% 110.81/14.61 = { by lemma 37 }
% 110.81/14.61 composition(X, join(complement(composition(converse(X), top)), composition(converse(X), top)))
% 110.81/14.61 = { by axiom 2 (maddux1_join_commutativity) }
% 110.81/14.61 composition(X, join(composition(converse(X), top), complement(composition(converse(X), top))))
% 110.81/14.61 = { by axiom 4 (def_top) R->L }
% 110.81/14.61 composition(X, top)
% 110.81/14.61
% 110.81/14.61 Goal 1 (goals): join(x0, composition(composition(x0, converse(x0)), x0)) = composition(composition(x0, converse(x0)), x0).
% 110.81/14.61 Proof:
% 110.81/14.61 join(x0, composition(composition(x0, converse(x0)), x0))
% 110.81/14.61 = { by lemma 34 R->L }
% 110.81/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), top)
% 110.81/14.61 = { by lemma 24 R->L }
% 110.81/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(composition(complement(complement(x0)), converse(complement(converse(one)))), top))
% 110.81/14.61 = { by lemma 19 R->L }
% 110.81/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), join(complement(complement(x0)), composition(complement(complement(x0)), converse(complement(converse(one)))))))
% 110.81/14.61 = { by axiom 1 (composition_identity) R->L }
% 110.81/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), join(composition(complement(complement(x0)), one), composition(complement(complement(x0)), converse(complement(converse(one)))))))
% 110.81/14.61 = { by lemma 37 }
% 110.81/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), composition(complement(complement(x0)), join(one, converse(complement(converse(one)))))))
% 110.81/14.61 = { by lemma 17 }
% 110.81/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), composition(complement(complement(x0)), converse(top))))
% 110.81/14.61 = { by lemma 26 }
% 110.81/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), composition(complement(complement(x0)), top)))
% 110.81/14.61 = { by lemma 29 }
% 110.81/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), composition(x0, top)))
% 110.81/14.61 = { by lemma 38 R->L }
% 110.81/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), composition(x0, composition(converse(x0), top))))
% 110.81/14.61 = { by axiom 7 (composition_associativity) }
% 110.81/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), composition(composition(x0, converse(x0)), top)))
% 110.81/14.61 = { by axiom 3 (converse_idempotence) R->L }
% 110.81/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), composition(composition(converse(converse(x0)), converse(x0)), top)))
% 110.81/14.61 = { by axiom 6 (converse_multiplicativity) R->L }
% 110.81/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), composition(converse(composition(x0, converse(x0))), top)))
% 110.81/14.61 = { by lemma 24 R->L }
% 110.81/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), composition(converse(composition(x0, converse(x0))), join(X, top))))
% 110.81/14.61 = { by lemma 26 R->L }
% 110.81/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), composition(converse(composition(x0, converse(x0))), join(X, converse(top)))))
% 111.49/14.61 = { by lemma 15 R->L }
% 111.49/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), composition(converse(composition(x0, converse(x0))), converse(join(top, converse(X))))))
% 111.49/14.61 = { by axiom 6 (converse_multiplicativity) R->L }
% 111.49/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(composition(join(top, converse(X)), composition(x0, converse(x0))))))
% 111.49/14.61 = { by lemma 25 }
% 111.49/14.61 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(composition(top, composition(x0, converse(x0))))))
% 111.49/14.61 = { by lemma 26 R->L }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(composition(converse(top), composition(x0, converse(x0))))))
% 111.49/14.62 = { by axiom 7 (composition_associativity) }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(composition(composition(converse(top), x0), converse(x0)))))
% 111.49/14.62 = { by lemma 20 R->L }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(composition(converse(composition(converse(x0), top)), converse(x0)))))
% 111.49/14.62 = { by axiom 6 (converse_multiplicativity) R->L }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(converse(composition(x0, composition(converse(x0), top))))))
% 111.49/14.62 = { by lemma 38 }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(converse(composition(x0, top)))))
% 111.49/14.62 = { by axiom 6 (converse_multiplicativity) }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(composition(converse(top), converse(x0)))))
% 111.49/14.62 = { by axiom 4 (def_top) }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(composition(converse(join(converse(composition(converse(x0), x0)), complement(converse(composition(converse(x0), x0))))), converse(x0)))))
% 111.49/14.62 = { by lemma 16 }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(composition(join(composition(converse(x0), x0), converse(complement(converse(composition(converse(x0), x0))))), converse(x0)))))
% 111.49/14.62 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(composition(join(converse(complement(converse(composition(converse(x0), x0)))), composition(converse(x0), x0)), converse(x0)))))
% 111.49/14.62 = { by lemma 20 R->L }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(composition(join(converse(complement(converse(composition(converse(x0), x0)))), converse(composition(converse(x0), x0))), converse(x0)))))
% 111.49/14.62 = { by lemma 36 R->L }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(join(composition(converse(complement(converse(composition(converse(x0), x0)))), converse(x0)), converse(composition(x0, composition(converse(x0), x0)))))))
% 111.49/14.62 = { by axiom 7 (composition_associativity) }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(join(composition(converse(complement(converse(composition(converse(x0), x0)))), converse(x0)), converse(composition(composition(x0, converse(x0)), x0))))))
% 111.49/14.62 = { by axiom 2 (maddux1_join_commutativity) }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(join(converse(composition(composition(x0, converse(x0)), x0)), composition(converse(complement(converse(composition(converse(x0), x0)))), converse(x0))))))
% 111.49/14.62 = { by axiom 6 (converse_multiplicativity) R->L }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(join(converse(composition(composition(x0, converse(x0)), x0)), converse(composition(x0, complement(converse(composition(converse(x0), x0)))))))))
% 111.49/14.62 = { by axiom 8 (converse_additivity) R->L }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(converse(join(composition(composition(x0, converse(x0)), x0), composition(x0, complement(converse(composition(converse(x0), x0)))))))))
% 111.49/14.62 = { by lemma 20 }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), converse(converse(join(composition(composition(x0, converse(x0)), x0), composition(x0, complement(composition(converse(x0), x0))))))))
% 111.49/14.62 = { by axiom 3 (converse_idempotence) }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), join(composition(composition(x0, converse(x0)), x0), composition(x0, complement(composition(converse(x0), x0))))))
% 111.49/14.62 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), join(composition(x0, complement(composition(converse(x0), x0))), composition(composition(x0, converse(x0)), x0))))
% 111.49/14.62 = { by axiom 9 (maddux2_join_associativity) }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(join(complement(x0), composition(x0, complement(composition(converse(x0), x0)))), composition(composition(x0, converse(x0)), x0)))
% 111.49/14.62 = { by lemma 22 }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(complement(x0), composition(composition(x0, converse(x0)), x0)))
% 111.49/14.62 = { by axiom 2 (maddux1_join_commutativity) }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), join(composition(composition(x0, converse(x0)), x0), complement(x0)))
% 111.49/14.62 = { by lemma 34 R->L }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), meet(join(composition(composition(x0, converse(x0)), x0), complement(x0)), top))
% 111.49/14.62 = { by lemma 32 R->L }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), complement(join(zero, complement(join(composition(composition(x0, converse(x0)), x0), complement(x0))))))
% 111.49/14.62 = { by lemma 35 }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), complement(join(zero, meet(x0, complement(composition(composition(x0, converse(x0)), x0))))))
% 111.49/14.62 = { by lemma 28 }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), complement(join(zero, meet(complement(composition(composition(x0, converse(x0)), x0)), x0))))
% 111.49/14.62 = { by axiom 10 (maddux4_definiton_of_meet) }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), complement(join(zero, complement(join(complement(complement(composition(composition(x0, converse(x0)), x0))), complement(x0))))))
% 111.49/14.62 = { by lemma 33 }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), complement(complement(join(complement(complement(composition(composition(x0, converse(x0)), x0))), complement(x0)))))
% 111.49/14.62 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), complement(meet(complement(composition(composition(x0, converse(x0)), x0)), x0)))
% 111.49/14.62 = { by lemma 28 R->L }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), complement(meet(x0, complement(composition(composition(x0, converse(x0)), x0)))))
% 111.49/14.62 = { by lemma 33 R->L }
% 111.49/14.62 meet(join(x0, composition(composition(x0, converse(x0)), x0)), complement(meet(x0, join(zero, complement(composition(composition(x0, converse(x0)), x0))))))
% 111.49/14.62 = { by lemma 34 R->L }
% 111.49/14.62 meet(join(x0, meet(composition(composition(x0, converse(x0)), x0), top)), complement(meet(x0, join(zero, complement(composition(composition(x0, converse(x0)), x0))))))
% 111.49/14.62 = { by lemma 32 R->L }
% 111.49/14.62 meet(join(x0, complement(join(zero, complement(composition(composition(x0, converse(x0)), x0))))), complement(meet(x0, join(zero, complement(composition(composition(x0, converse(x0)), x0))))))
% 111.49/14.62 = { by lemma 28 }
% 111.49/14.62 meet(join(x0, complement(join(zero, complement(composition(composition(x0, converse(x0)), x0))))), complement(meet(join(zero, complement(composition(composition(x0, converse(x0)), x0))), x0)))
% 111.49/14.62 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 111.49/14.62 meet(join(complement(join(zero, complement(composition(composition(x0, converse(x0)), x0)))), x0), complement(meet(join(zero, complement(composition(composition(x0, converse(x0)), x0))), x0)))
% 111.49/14.62 = { by lemma 35 R->L }
% 111.49/14.62 complement(join(meet(join(zero, complement(composition(composition(x0, converse(x0)), x0))), x0), complement(join(complement(join(zero, complement(composition(composition(x0, converse(x0)), x0)))), x0))))
% 111.49/14.62 = { by lemma 27 }
% 111.49/14.62 complement(join(zero, complement(composition(composition(x0, converse(x0)), x0))))
% 111.49/14.62 = { by lemma 32 }
% 111.49/14.62 meet(composition(composition(x0, converse(x0)), x0), top)
% 111.49/14.62 = { by lemma 34 }
% 111.49/14.62 composition(composition(x0, converse(x0)), x0)
% 111.49/14.62 % SZS output end Proof
% 111.49/14.62
% 111.49/14.62 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------