TSTP Solution File: REL017+3 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL017+3 : 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 : n010.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 13:43:58 EDT 2023
% Result : Theorem 11.83s 1.87s
% Output : Proof 11.83s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : REL017+3 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n010.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Aug 25 20:05:35 EDT 2023
% 0.12/0.34 % CPUTime :
% 11.83/1.87 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 11.83/1.87
% 11.83/1.87 % SZS status Theorem
% 11.83/1.87
% 11.83/1.92 % SZS output start Proof
% 11.83/1.92 Axiom 1 (composition_identity): composition(X, one) = X.
% 11.83/1.92 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 11.83/1.92 Axiom 3 (converse_idempotence): converse(converse(X)) = X.
% 11.83/1.92 Axiom 4 (def_top): top = join(X, complement(X)).
% 11.83/1.92 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 11.83/1.92 Axiom 6 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 11.83/1.92 Axiom 7 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 11.83/1.92 Axiom 8 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 11.83/1.92 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 11.83/1.92 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 11.83/1.92 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 11.83/1.92 Axiom 12 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 11.83/1.92 Axiom 13 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 11.83/1.92 Axiom 14 (modular_law_1): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
% 11.83/1.92
% 11.83/1.92 Lemma 15: complement(top) = zero.
% 11.83/1.92 Proof:
% 11.83/1.92 complement(top)
% 11.83/1.92 = { by axiom 4 (def_top) }
% 11.83/1.92 complement(join(complement(X), complement(complement(X))))
% 11.83/1.92 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 11.83/1.92 meet(X, complement(X))
% 11.83/1.92 = { by axiom 5 (def_zero) R->L }
% 11.83/1.92 zero
% 11.83/1.92
% 11.83/1.92 Lemma 16: join(complement(X), X) = top.
% 11.83/1.92 Proof:
% 11.83/1.92 join(complement(X), X)
% 11.83/1.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.92 join(X, complement(X))
% 11.83/1.92 = { by axiom 4 (def_top) R->L }
% 11.83/1.92 top
% 11.83/1.92
% 11.83/1.92 Lemma 17: meet(Y, X) = meet(X, Y).
% 11.83/1.92 Proof:
% 11.83/1.92 meet(Y, X)
% 11.83/1.92 = { by axiom 10 (maddux4_definiton_of_meet) }
% 11.83/1.92 complement(join(complement(Y), complement(X)))
% 11.83/1.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.92 complement(join(complement(X), complement(Y)))
% 11.83/1.92 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 11.83/1.92 meet(X, Y)
% 11.83/1.92
% 11.83/1.92 Lemma 18: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 11.83/1.92 Proof:
% 11.83/1.92 join(meet(X, Y), complement(join(complement(X), Y)))
% 11.83/1.92 = { by axiom 10 (maddux4_definiton_of_meet) }
% 11.83/1.92 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 11.83/1.92 = { by axiom 13 (maddux3_a_kind_of_de_Morgan) R->L }
% 11.83/1.92 X
% 11.83/1.92
% 11.83/1.92 Lemma 19: join(meet(X, Y), complement(join(X, complement(Y)))) = Y.
% 11.83/1.92 Proof:
% 11.83/1.92 join(meet(X, Y), complement(join(X, complement(Y))))
% 11.83/1.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.92 join(meet(X, Y), complement(join(complement(Y), X)))
% 11.83/1.92 = { by lemma 17 }
% 11.83/1.92 join(meet(Y, X), complement(join(complement(Y), X)))
% 11.83/1.92 = { by lemma 18 }
% 11.83/1.92 Y
% 11.83/1.92
% 11.83/1.92 Lemma 20: join(X, join(Y, complement(X))) = join(Y, top).
% 11.83/1.92 Proof:
% 11.83/1.92 join(X, join(Y, complement(X)))
% 11.83/1.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.92 join(X, join(complement(X), Y))
% 11.83/1.92 = { by axiom 9 (maddux2_join_associativity) }
% 11.83/1.92 join(join(X, complement(X)), Y)
% 11.83/1.92 = { by axiom 4 (def_top) R->L }
% 11.83/1.92 join(top, Y)
% 11.83/1.92 = { by axiom 2 (maddux1_join_commutativity) }
% 11.83/1.92 join(Y, top)
% 11.83/1.92
% 11.83/1.92 Lemma 21: join(X, join(Y, complement(join(X, Y)))) = top.
% 11.83/1.92 Proof:
% 11.83/1.92 join(X, join(Y, complement(join(X, Y))))
% 11.83/1.92 = { by axiom 9 (maddux2_join_associativity) }
% 11.83/1.92 join(join(X, Y), complement(join(X, Y)))
% 11.83/1.92 = { by axiom 4 (def_top) R->L }
% 11.83/1.92 top
% 11.83/1.92
% 11.83/1.92 Lemma 22: join(X, top) = top.
% 11.83/1.92 Proof:
% 11.83/1.92 join(X, top)
% 11.83/1.92 = { by lemma 16 R->L }
% 11.83/1.92 join(X, join(complement(Y), Y))
% 11.83/1.92 = { by axiom 9 (maddux2_join_associativity) }
% 11.83/1.92 join(join(X, complement(Y)), Y)
% 11.83/1.92 = { by lemma 19 R->L }
% 11.83/1.92 join(join(X, complement(Y)), join(meet(X, Y), complement(join(X, complement(Y)))))
% 11.83/1.92 = { by lemma 20 }
% 11.83/1.92 join(meet(X, Y), top)
% 11.83/1.92 = { by axiom 2 (maddux1_join_commutativity) }
% 11.83/1.92 join(top, meet(X, Y))
% 11.83/1.92 = { by axiom 1 (composition_identity) R->L }
% 11.83/1.92 join(top, meet(X, composition(Y, one)))
% 11.83/1.92 = { by lemma 17 }
% 11.83/1.92 join(top, meet(composition(Y, one), X))
% 11.83/1.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.92 join(meet(composition(Y, one), X), top)
% 11.83/1.92 = { by axiom 4 (def_top) }
% 11.83/1.92 join(meet(composition(Y, one), X), join(meet(composition(Y, meet(one, composition(converse(Y), X))), X), complement(meet(composition(Y, meet(one, composition(converse(Y), X))), X))))
% 11.83/1.92 = { by axiom 14 (modular_law_1) R->L }
% 11.83/1.92 join(meet(composition(Y, one), X), join(meet(composition(Y, meet(one, composition(converse(Y), X))), X), complement(join(meet(composition(Y, one), X), meet(composition(Y, meet(one, composition(converse(Y), X))), X)))))
% 11.83/1.92 = { by lemma 21 }
% 11.83/1.92 top
% 11.83/1.92
% 11.83/1.92 Lemma 23: join(X, complement(zero)) = top.
% 11.83/1.92 Proof:
% 11.83/1.92 join(X, complement(zero))
% 11.83/1.92 = { by lemma 18 R->L }
% 11.83/1.92 join(join(meet(X, complement(X)), complement(join(complement(X), complement(X)))), complement(zero))
% 11.83/1.92 = { by axiom 5 (def_zero) R->L }
% 11.83/1.92 join(join(zero, complement(join(complement(X), complement(X)))), complement(zero))
% 11.83/1.92 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 11.83/1.92 join(join(zero, meet(X, X)), complement(zero))
% 11.83/1.92 = { by axiom 9 (maddux2_join_associativity) R->L }
% 11.83/1.92 join(zero, join(meet(X, X), complement(zero)))
% 11.83/1.92 = { by lemma 20 }
% 11.83/1.92 join(meet(X, X), top)
% 11.83/1.92 = { by lemma 22 }
% 11.83/1.92 top
% 11.83/1.92
% 11.83/1.92 Lemma 24: meet(X, zero) = zero.
% 11.83/1.92 Proof:
% 11.83/1.92 meet(X, zero)
% 11.83/1.92 = { by axiom 10 (maddux4_definiton_of_meet) }
% 11.83/1.92 complement(join(complement(X), complement(zero)))
% 11.83/1.92 = { by lemma 23 }
% 11.83/1.92 complement(top)
% 11.83/1.92 = { by lemma 15 }
% 11.83/1.92 zero
% 11.83/1.92
% 11.83/1.92 Lemma 25: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
% 11.83/1.92 Proof:
% 11.83/1.92 join(meet(X, Y), complement(join(Y, complement(X))))
% 11.83/1.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.92 join(meet(X, Y), complement(join(complement(X), Y)))
% 11.83/1.92 = { by lemma 18 }
% 11.83/1.92 X
% 11.83/1.92
% 11.83/1.92 Lemma 26: join(meet(X, Y), join(Z, complement(join(complement(X), Y)))) = join(X, Z).
% 11.83/1.92 Proof:
% 11.83/1.92 join(meet(X, Y), join(Z, complement(join(complement(X), Y))))
% 11.83/1.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.92 join(meet(X, Y), join(complement(join(complement(X), Y)), Z))
% 11.83/1.92 = { by axiom 9 (maddux2_join_associativity) }
% 11.83/1.92 join(join(meet(X, Y), complement(join(complement(X), Y))), Z)
% 11.83/1.92 = { by lemma 18 }
% 11.83/1.92 join(X, Z)
% 11.83/1.92
% 11.83/1.92 Lemma 27: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 11.83/1.92 Proof:
% 11.83/1.92 converse(composition(converse(X), Y))
% 11.83/1.92 = { by axiom 6 (converse_multiplicativity) }
% 11.83/1.92 composition(converse(Y), converse(converse(X)))
% 11.83/1.92 = { by axiom 3 (converse_idempotence) }
% 11.83/1.92 composition(converse(Y), X)
% 11.83/1.92
% 11.83/1.92 Lemma 28: composition(converse(one), X) = X.
% 11.83/1.92 Proof:
% 11.83/1.92 composition(converse(one), X)
% 11.83/1.92 = { by lemma 27 R->L }
% 11.83/1.92 converse(composition(converse(X), one))
% 11.83/1.92 = { by axiom 1 (composition_identity) }
% 11.83/1.92 converse(converse(X))
% 11.83/1.92 = { by axiom 3 (converse_idempotence) }
% 11.83/1.92 X
% 11.83/1.92
% 11.83/1.92 Lemma 29: join(complement(X), complement(X)) = complement(X).
% 11.83/1.92 Proof:
% 11.83/1.92 join(complement(X), complement(X))
% 11.83/1.92 = { by lemma 28 R->L }
% 11.83/1.92 join(complement(X), complement(composition(converse(one), X)))
% 11.83/1.92 = { by axiom 1 (composition_identity) R->L }
% 11.83/1.92 join(complement(X), complement(composition(composition(converse(one), one), X)))
% 11.83/1.92 = { by axiom 7 (composition_associativity) R->L }
% 11.83/1.92 join(complement(X), complement(composition(converse(one), composition(one, X))))
% 11.83/1.92 = { by lemma 28 }
% 11.83/1.92 join(complement(X), complement(composition(one, X)))
% 11.83/1.92 = { by axiom 3 (converse_idempotence) R->L }
% 11.83/1.92 join(complement(X), complement(composition(converse(converse(one)), X)))
% 11.83/1.92 = { by lemma 28 R->L }
% 11.83/1.92 join(complement(X), composition(converse(one), complement(composition(converse(converse(one)), X))))
% 11.83/1.92 = { by axiom 3 (converse_idempotence) R->L }
% 11.83/1.92 join(complement(X), composition(converse(converse(converse(one))), complement(composition(converse(converse(one)), X))))
% 11.83/1.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.92 join(composition(converse(converse(converse(one))), complement(composition(converse(converse(one)), X))), complement(X))
% 11.83/1.92 = { by axiom 12 (converse_cancellativity) }
% 11.83/1.92 complement(X)
% 11.83/1.92
% 11.83/1.92 Lemma 30: join(meet(X, top), meet(X, zero)) = X.
% 11.83/1.92 Proof:
% 11.83/1.92 join(meet(X, top), meet(X, zero))
% 11.83/1.92 = { by lemma 15 R->L }
% 11.83/1.92 join(meet(X, top), meet(X, complement(top)))
% 11.83/1.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.92 join(meet(X, complement(top)), meet(X, top))
% 11.83/1.92 = { by axiom 10 (maddux4_definiton_of_meet) }
% 11.83/1.92 join(meet(X, complement(top)), complement(join(complement(X), complement(top))))
% 11.83/1.92 = { by lemma 18 }
% 11.83/1.92 X
% 11.83/1.92
% 11.83/1.92 Lemma 31: join(X, zero) = X.
% 11.83/1.92 Proof:
% 11.83/1.92 join(X, zero)
% 11.83/1.92 = { by lemma 24 R->L }
% 11.83/1.92 join(X, meet(complement(X), zero))
% 11.83/1.92 = { by lemma 17 R->L }
% 11.83/1.92 join(X, meet(zero, complement(X)))
% 11.83/1.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.92 join(meet(zero, complement(X)), X)
% 11.83/1.92 = { by lemma 25 R->L }
% 11.83/1.92 join(meet(zero, complement(X)), join(meet(X, complement(zero)), complement(join(complement(zero), complement(X)))))
% 11.83/1.92 = { by lemma 26 }
% 11.83/1.92 join(zero, meet(X, complement(zero)))
% 11.83/1.92 = { by lemma 29 R->L }
% 11.83/1.92 join(zero, meet(X, join(complement(zero), complement(zero))))
% 11.83/1.92 = { by lemma 23 }
% 11.83/1.92 join(zero, meet(X, top))
% 11.83/1.92 = { by lemma 24 R->L }
% 11.83/1.92 join(meet(X, zero), meet(X, top))
% 11.83/1.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.92 join(meet(X, top), meet(X, zero))
% 11.83/1.92 = { by lemma 30 }
% 11.83/1.92 X
% 11.83/1.92
% 11.83/1.92 Lemma 32: meet(X, join(Y, complement(join(Y, complement(X))))) = X.
% 11.83/1.92 Proof:
% 11.83/1.92 meet(X, join(Y, complement(join(Y, complement(X)))))
% 11.83/1.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.92 meet(X, join(Y, complement(join(complement(X), Y))))
% 11.83/1.92 = { by lemma 31 R->L }
% 11.83/1.92 join(meet(X, join(Y, complement(join(complement(X), Y)))), zero)
% 11.83/1.92 = { by lemma 15 R->L }
% 11.83/1.92 join(meet(X, join(Y, complement(join(complement(X), Y)))), complement(top))
% 11.83/1.92 = { by lemma 21 R->L }
% 11.83/1.92 join(meet(X, join(Y, complement(join(complement(X), Y)))), complement(join(complement(X), join(Y, complement(join(complement(X), Y))))))
% 11.83/1.92 = { by lemma 18 }
% 11.83/1.92 X
% 11.83/1.92
% 11.83/1.92 Lemma 33: complement(complement(X)) = X.
% 11.83/1.92 Proof:
% 11.83/1.92 complement(complement(X))
% 11.83/1.92 = { by lemma 29 R->L }
% 11.83/1.92 complement(join(complement(X), complement(X)))
% 11.83/1.92 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 11.83/1.92 meet(X, X)
% 11.83/1.92 = { by lemma 31 R->L }
% 11.83/1.92 meet(X, join(X, zero))
% 11.83/1.92 = { by lemma 15 R->L }
% 11.83/1.92 meet(X, join(X, complement(top)))
% 11.83/1.92 = { by axiom 4 (def_top) }
% 11.83/1.92 meet(X, join(X, complement(join(X, complement(X)))))
% 11.83/1.92 = { by lemma 32 }
% 11.83/1.92 X
% 11.83/1.92
% 11.83/1.92 Lemma 34: meet(X, top) = X.
% 11.83/1.92 Proof:
% 11.83/1.92 meet(X, top)
% 11.83/1.92 = { by lemma 31 R->L }
% 11.83/1.92 meet(join(X, zero), top)
% 11.83/1.92 = { by lemma 15 R->L }
% 11.83/1.92 meet(join(X, complement(top)), top)
% 11.83/1.92 = { by lemma 23 R->L }
% 11.83/1.92 meet(join(X, complement(join(X, complement(zero)))), top)
% 11.83/1.92 = { by lemma 31 R->L }
% 11.83/1.92 join(meet(join(X, complement(join(X, complement(zero)))), top), zero)
% 11.83/1.92 = { by lemma 32 R->L }
% 11.83/1.92 join(meet(join(X, complement(join(X, complement(zero)))), top), meet(zero, join(X, complement(join(X, complement(zero))))))
% 11.83/1.92 = { by lemma 17 }
% 11.83/1.92 join(meet(join(X, complement(join(X, complement(zero)))), top), meet(join(X, complement(join(X, complement(zero)))), zero))
% 11.83/1.92 = { by lemma 30 }
% 11.83/1.92 join(X, complement(join(X, complement(zero))))
% 11.83/1.92 = { by lemma 23 }
% 11.83/1.92 join(X, complement(top))
% 11.83/1.92 = { by lemma 15 }
% 11.83/1.92 join(X, zero)
% 11.83/1.92 = { by lemma 31 }
% 11.83/1.92 X
% 11.83/1.92
% 11.83/1.92 Lemma 35: meet(top, X) = X.
% 11.83/1.92 Proof:
% 11.83/1.92 meet(top, X)
% 11.83/1.92 = { by lemma 17 }
% 11.83/1.92 meet(X, top)
% 11.83/1.92 = { by lemma 34 }
% 11.83/1.92 X
% 11.83/1.92
% 11.83/1.92 Lemma 36: join(zero, complement(X)) = complement(X).
% 11.83/1.92 Proof:
% 11.83/1.92 join(zero, complement(X))
% 11.83/1.92 = { by lemma 33 R->L }
% 11.83/1.92 join(zero, complement(complement(complement(X))))
% 11.83/1.92 = { by lemma 29 R->L }
% 11.83/1.92 join(zero, complement(join(complement(complement(X)), complement(complement(X)))))
% 11.83/1.92 = { by lemma 33 }
% 11.83/1.92 join(zero, complement(join(X, complement(complement(X)))))
% 11.83/1.92 = { by lemma 33 }
% 11.83/1.92 join(zero, complement(join(X, X)))
% 11.83/1.92 = { by lemma 33 R->L }
% 11.83/1.92 join(zero, complement(join(complement(complement(X)), X)))
% 11.83/1.92 = { by axiom 5 (def_zero) }
% 11.83/1.92 join(meet(X, complement(X)), complement(join(complement(complement(X)), X)))
% 11.83/1.92 = { by lemma 17 }
% 11.83/1.92 join(meet(complement(X), X), complement(join(complement(complement(X)), X)))
% 11.83/1.92 = { by lemma 18 }
% 11.83/1.93 complement(X)
% 11.83/1.93
% 11.83/1.93 Lemma 37: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 11.83/1.93 Proof:
% 11.83/1.93 join(Y, join(X, Z))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.93 join(join(X, Z), Y)
% 11.83/1.93 = { by axiom 9 (maddux2_join_associativity) R->L }
% 11.83/1.93 join(X, join(Z, Y))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) }
% 11.83/1.93 join(X, join(Y, Z))
% 11.83/1.93
% 11.83/1.93 Lemma 38: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 11.83/1.93 Proof:
% 11.83/1.93 complement(join(X, complement(Y)))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.93 complement(join(complement(Y), X))
% 11.83/1.93 = { by lemma 35 R->L }
% 11.83/1.93 complement(join(complement(Y), meet(top, X)))
% 11.83/1.93 = { by lemma 17 }
% 11.83/1.93 complement(join(complement(Y), meet(X, top)))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.93 complement(join(meet(X, top), complement(Y)))
% 11.83/1.93 = { by axiom 10 (maddux4_definiton_of_meet) }
% 11.83/1.93 complement(join(complement(join(complement(X), complement(top))), complement(Y)))
% 11.83/1.93 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 11.83/1.93 meet(join(complement(X), complement(top)), Y)
% 11.83/1.93 = { by lemma 17 R->L }
% 11.83/1.93 meet(Y, join(complement(X), complement(top)))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) }
% 11.83/1.93 meet(Y, join(complement(top), complement(X)))
% 11.83/1.93 = { by lemma 15 }
% 11.83/1.93 meet(Y, join(zero, complement(X)))
% 11.83/1.93 = { by lemma 36 }
% 11.83/1.93 meet(Y, complement(X))
% 11.83/1.93
% 11.83/1.93 Lemma 39: complement(join(zero, complement(X))) = meet(X, top).
% 11.83/1.93 Proof:
% 11.83/1.93 complement(join(zero, complement(X)))
% 11.83/1.93 = { by lemma 15 R->L }
% 11.83/1.93 complement(join(complement(top), complement(X)))
% 11.83/1.93 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 11.83/1.93 meet(top, X)
% 11.83/1.93 = { by lemma 17 R->L }
% 11.83/1.93 meet(X, top)
% 11.83/1.93
% 11.83/1.93 Lemma 40: join(complement(X), join(X, Y)) = top.
% 11.83/1.93 Proof:
% 11.83/1.93 join(complement(X), join(X, Y))
% 11.83/1.93 = { by axiom 9 (maddux2_join_associativity) }
% 11.83/1.93 join(join(complement(X), X), Y)
% 11.83/1.93 = { by lemma 16 }
% 11.83/1.93 join(top, Y)
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) }
% 11.83/1.93 join(Y, top)
% 11.83/1.93 = { by lemma 22 }
% 11.83/1.93 top
% 11.83/1.93
% 11.83/1.93 Lemma 41: join(composition(X, Y), composition(X, Z)) = composition(X, join(Z, Y)).
% 11.83/1.93 Proof:
% 11.83/1.93 join(composition(X, Y), composition(X, Z))
% 11.83/1.93 = { by axiom 3 (converse_idempotence) R->L }
% 11.83/1.93 join(composition(X, Y), composition(converse(converse(X)), Z))
% 11.83/1.93 = { by axiom 3 (converse_idempotence) R->L }
% 11.83/1.93 join(converse(converse(composition(X, Y))), composition(converse(converse(X)), Z))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.93 join(composition(converse(converse(X)), Z), converse(converse(composition(X, Y))))
% 11.83/1.93 = { by lemma 27 R->L }
% 11.83/1.93 join(converse(composition(converse(Z), converse(X))), converse(converse(composition(X, Y))))
% 11.83/1.93 = { by axiom 8 (converse_additivity) R->L }
% 11.83/1.93 converse(join(composition(converse(Z), converse(X)), converse(composition(X, Y))))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.93 converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
% 11.83/1.93 = { by axiom 6 (converse_multiplicativity) }
% 11.83/1.93 converse(join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X))))
% 11.83/1.93 = { by axiom 11 (composition_distributivity) R->L }
% 11.83/1.93 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) }
% 11.83/1.93 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 11.83/1.93 = { by axiom 6 (converse_multiplicativity) }
% 11.83/1.93 composition(converse(converse(X)), converse(join(converse(Z), converse(Y))))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.93 composition(converse(converse(X)), converse(join(converse(Y), converse(Z))))
% 11.83/1.93 = { by axiom 8 (converse_additivity) }
% 11.83/1.93 composition(converse(converse(X)), join(converse(converse(Y)), converse(converse(Z))))
% 11.83/1.93 = { by axiom 3 (converse_idempotence) }
% 11.83/1.93 composition(converse(converse(X)), join(Y, converse(converse(Z))))
% 11.83/1.93 = { by axiom 3 (converse_idempotence) }
% 11.83/1.93 composition(X, join(Y, converse(converse(Z))))
% 11.83/1.93 = { by axiom 3 (converse_idempotence) }
% 11.83/1.93 composition(X, join(Y, Z))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) }
% 11.83/1.93 composition(X, join(Z, Y))
% 11.83/1.93
% 11.83/1.93 Lemma 42: meet(join(X, Y), join(X, complement(Y))) = X.
% 11.83/1.93 Proof:
% 11.83/1.93 meet(join(X, Y), join(X, complement(Y)))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.93 meet(join(Y, X), join(X, complement(Y)))
% 11.83/1.93 = { by lemma 17 }
% 11.83/1.93 meet(join(X, complement(Y)), join(Y, X))
% 11.83/1.93 = { by lemma 34 R->L }
% 11.83/1.93 meet(join(X, complement(Y)), join(Y, meet(X, top)))
% 11.83/1.93 = { by lemma 34 R->L }
% 11.83/1.93 meet(meet(join(X, complement(Y)), top), join(Y, meet(X, top)))
% 11.83/1.93 = { by lemma 39 R->L }
% 11.83/1.93 meet(complement(join(zero, complement(join(X, complement(Y))))), join(Y, meet(X, top)))
% 11.83/1.93 = { by lemma 38 }
% 11.83/1.93 meet(complement(join(zero, meet(Y, complement(X)))), join(Y, meet(X, top)))
% 11.83/1.93 = { by lemma 17 }
% 11.83/1.93 meet(complement(join(zero, meet(complement(X), Y))), join(Y, meet(X, top)))
% 11.83/1.93 = { by axiom 10 (maddux4_definiton_of_meet) }
% 11.83/1.93 meet(complement(join(zero, complement(join(complement(complement(X)), complement(Y))))), join(Y, meet(X, top)))
% 11.83/1.93 = { by lemma 36 }
% 11.83/1.93 meet(complement(complement(join(complement(complement(X)), complement(Y)))), join(Y, meet(X, top)))
% 11.83/1.93 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 11.83/1.93 meet(complement(meet(complement(X), Y)), join(Y, meet(X, top)))
% 11.83/1.93 = { by lemma 17 R->L }
% 11.83/1.93 meet(complement(meet(Y, complement(X))), join(Y, meet(X, top)))
% 11.83/1.93 = { by lemma 36 R->L }
% 11.83/1.93 meet(complement(meet(Y, join(zero, complement(X)))), join(Y, meet(X, top)))
% 11.83/1.93 = { by lemma 39 R->L }
% 11.83/1.93 meet(complement(meet(Y, join(zero, complement(X)))), join(Y, complement(join(zero, complement(X)))))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.93 meet(complement(meet(Y, join(zero, complement(X)))), join(complement(join(zero, complement(X))), Y))
% 11.83/1.93 = { by lemma 17 }
% 11.83/1.93 meet(complement(meet(join(zero, complement(X)), Y)), join(complement(join(zero, complement(X))), Y))
% 11.83/1.93 = { by lemma 17 }
% 11.83/1.93 meet(join(complement(join(zero, complement(X))), Y), complement(meet(join(zero, complement(X)), Y)))
% 11.83/1.93 = { by lemma 38 R->L }
% 11.83/1.93 complement(join(meet(join(zero, complement(X)), Y), complement(join(complement(join(zero, complement(X))), Y))))
% 11.83/1.93 = { by lemma 18 }
% 11.83/1.93 complement(join(zero, complement(X)))
% 11.83/1.93 = { by lemma 39 }
% 11.83/1.93 meet(X, top)
% 11.83/1.93 = { by lemma 34 }
% 11.83/1.93 X
% 11.83/1.93
% 11.83/1.93 Lemma 43: join(composition(X, meet(Y, complement(Z))), composition(X, Z)) = join(composition(X, Y), composition(X, Z)).
% 11.83/1.93 Proof:
% 11.83/1.93 join(composition(X, meet(Y, complement(Z))), composition(X, Z))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.93 join(composition(X, Z), composition(X, meet(Y, complement(Z))))
% 11.83/1.93 = { by lemma 41 }
% 11.83/1.93 composition(X, join(meet(Y, complement(Z)), Z))
% 11.83/1.93 = { by lemma 32 R->L }
% 11.83/1.93 composition(X, join(meet(Y, complement(Z)), meet(Z, join(join(join(Y, meet(Y, complement(Z))), complement(complement(Z))), complement(join(join(join(Y, meet(Y, complement(Z))), complement(complement(Z))), complement(Z)))))))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) }
% 11.83/1.93 composition(X, join(meet(Y, complement(Z)), meet(Z, join(join(join(Y, meet(Y, complement(Z))), complement(complement(Z))), complement(join(complement(Z), join(join(Y, meet(Y, complement(Z))), complement(complement(Z)))))))))
% 11.83/1.93 = { by lemma 20 }
% 11.83/1.93 composition(X, join(meet(Y, complement(Z)), meet(Z, join(join(join(Y, meet(Y, complement(Z))), complement(complement(Z))), complement(join(join(Y, meet(Y, complement(Z))), top))))))
% 11.83/1.93 = { by axiom 9 (maddux2_join_associativity) R->L }
% 11.83/1.93 composition(X, join(meet(Y, complement(Z)), meet(Z, join(join(Y, meet(Y, complement(Z))), join(complement(complement(Z)), complement(join(join(Y, meet(Y, complement(Z))), top)))))))
% 11.83/1.93 = { by lemma 33 }
% 11.83/1.93 composition(X, join(meet(Y, complement(Z)), meet(Z, join(join(Y, meet(Y, complement(Z))), join(Z, complement(join(join(Y, meet(Y, complement(Z))), top)))))))
% 11.83/1.93 = { by lemma 22 }
% 11.83/1.93 composition(X, join(meet(Y, complement(Z)), meet(Z, join(join(Y, meet(Y, complement(Z))), join(Z, complement(top))))))
% 11.83/1.93 = { by lemma 15 }
% 11.83/1.93 composition(X, join(meet(Y, complement(Z)), meet(Z, join(join(Y, meet(Y, complement(Z))), join(Z, zero)))))
% 11.83/1.93 = { by lemma 31 }
% 11.83/1.93 composition(X, join(meet(Y, complement(Z)), meet(Z, join(join(Y, meet(Y, complement(Z))), Z))))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) }
% 11.83/1.93 composition(X, join(meet(Y, complement(Z)), meet(Z, join(Z, join(Y, meet(Y, complement(Z)))))))
% 11.83/1.93 = { by axiom 9 (maddux2_join_associativity) }
% 11.83/1.93 composition(X, join(meet(Y, complement(Z)), meet(Z, join(join(Z, Y), meet(Y, complement(Z))))))
% 11.83/1.93 = { by lemma 33 R->L }
% 11.83/1.93 composition(X, join(complement(complement(meet(Y, complement(Z)))), meet(Z, join(join(Z, Y), meet(Y, complement(Z))))))
% 11.83/1.93 = { by lemma 33 R->L }
% 11.83/1.93 composition(X, join(complement(complement(meet(Y, complement(Z)))), meet(Z, join(join(Z, Y), complement(complement(meet(Y, complement(Z))))))))
% 11.83/1.93 = { by lemma 33 R->L }
% 11.83/1.93 composition(X, join(complement(complement(meet(Y, complement(Z)))), meet(complement(complement(Z)), join(join(Z, Y), complement(complement(meet(Y, complement(Z))))))))
% 11.83/1.93 = { by lemma 19 R->L }
% 11.83/1.93 composition(X, join(complement(complement(meet(Y, complement(Z)))), meet(complement(join(meet(Y, complement(Z)), complement(join(Y, complement(complement(Z)))))), join(join(Z, Y), complement(complement(meet(Y, complement(Z))))))))
% 11.83/1.93 = { by lemma 38 }
% 11.83/1.93 composition(X, join(complement(complement(meet(Y, complement(Z)))), meet(meet(join(Y, complement(complement(Z))), complement(meet(Y, complement(Z)))), join(join(Z, Y), complement(complement(meet(Y, complement(Z))))))))
% 11.83/1.93 = { by lemma 33 }
% 11.83/1.93 composition(X, join(complement(complement(meet(Y, complement(Z)))), meet(meet(join(Y, Z), complement(meet(Y, complement(Z)))), join(join(Z, Y), complement(complement(meet(Y, complement(Z))))))))
% 11.83/1.93 = { by lemma 17 R->L }
% 11.83/1.93 composition(X, join(complement(complement(meet(Y, complement(Z)))), meet(meet(complement(meet(Y, complement(Z))), join(Y, Z)), join(join(Z, Y), complement(complement(meet(Y, complement(Z))))))))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) }
% 11.83/1.93 composition(X, join(complement(complement(meet(Y, complement(Z)))), meet(meet(complement(meet(Y, complement(Z))), join(Z, Y)), join(join(Z, Y), complement(complement(meet(Y, complement(Z))))))))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.93 composition(X, join(complement(complement(meet(Y, complement(Z)))), meet(meet(complement(meet(Y, complement(Z))), join(Z, Y)), join(complement(complement(meet(Y, complement(Z)))), join(Z, Y)))))
% 11.83/1.93 = { by lemma 17 }
% 11.83/1.93 composition(X, join(complement(complement(meet(Y, complement(Z)))), meet(join(complement(complement(meet(Y, complement(Z)))), join(Z, Y)), meet(complement(meet(Y, complement(Z))), join(Z, Y)))))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.93 composition(X, join(meet(join(complement(complement(meet(Y, complement(Z)))), join(Z, Y)), meet(complement(meet(Y, complement(Z))), join(Z, Y))), complement(complement(meet(Y, complement(Z))))))
% 11.83/1.93 = { by lemma 18 R->L }
% 11.83/1.93 composition(X, join(meet(join(complement(complement(meet(Y, complement(Z)))), join(Z, Y)), meet(complement(meet(Y, complement(Z))), join(Z, Y))), complement(join(meet(complement(meet(Y, complement(Z))), join(Z, Y)), complement(join(complement(complement(meet(Y, complement(Z)))), join(Z, Y)))))))
% 11.83/1.93 = { by lemma 25 }
% 11.83/1.93 composition(X, join(complement(complement(meet(Y, complement(Z)))), join(Z, Y)))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) }
% 11.83/1.93 composition(X, join(join(Z, Y), complement(complement(meet(Y, complement(Z))))))
% 11.83/1.93 = { by axiom 9 (maddux2_join_associativity) R->L }
% 11.83/1.93 composition(X, join(Z, join(Y, complement(complement(meet(Y, complement(Z)))))))
% 11.83/1.93 = { by lemma 33 }
% 11.83/1.93 composition(X, join(Z, join(Y, meet(Y, complement(Z)))))
% 11.83/1.93 = { by lemma 17 }
% 11.83/1.93 composition(X, join(Z, join(Y, meet(complement(Z), Y))))
% 11.83/1.93 = { by axiom 10 (maddux4_definiton_of_meet) }
% 11.83/1.93 composition(X, join(Z, join(Y, complement(join(complement(complement(Z)), complement(Y))))))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.93 composition(X, join(Z, join(Y, complement(join(complement(Y), complement(complement(Z)))))))
% 11.83/1.93 = { by lemma 26 R->L }
% 11.83/1.93 composition(X, join(Z, join(meet(Y, complement(complement(Z))), join(complement(join(complement(Y), complement(complement(Z)))), complement(join(complement(Y), complement(complement(Z))))))))
% 11.83/1.93 = { by lemma 29 }
% 11.83/1.93 composition(X, join(Z, join(meet(Y, complement(complement(Z))), complement(join(complement(Y), complement(complement(Z)))))))
% 11.83/1.93 = { by lemma 18 }
% 11.83/1.93 composition(X, join(Z, Y))
% 11.83/1.93 = { by lemma 41 R->L }
% 11.83/1.93 join(composition(X, Y), composition(X, Z))
% 11.83/1.93
% 11.83/1.93 Goal 1 (goals): join(complement(composition(x0, x1)), composition(x0, x2)) = join(complement(composition(x0, meet(x1, complement(x2)))), composition(x0, x2)).
% 11.83/1.93 Proof:
% 11.83/1.93 join(complement(composition(x0, x1)), composition(x0, x2))
% 11.83/1.93 = { by lemma 42 R->L }
% 11.83/1.93 meet(join(join(complement(composition(x0, x1)), composition(x0, x2)), composition(x0, meet(x1, complement(x2)))), join(join(complement(composition(x0, x1)), composition(x0, x2)), complement(composition(x0, meet(x1, complement(x2))))))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.93 meet(join(composition(x0, meet(x1, complement(x2))), join(complement(composition(x0, x1)), composition(x0, x2))), join(join(complement(composition(x0, x1)), composition(x0, x2)), complement(composition(x0, meet(x1, complement(x2))))))
% 11.83/1.93 = { by lemma 37 }
% 11.83/1.93 meet(join(complement(composition(x0, x1)), join(composition(x0, meet(x1, complement(x2))), composition(x0, x2))), join(join(complement(composition(x0, x1)), composition(x0, x2)), complement(composition(x0, meet(x1, complement(x2))))))
% 11.83/1.93 = { by lemma 43 }
% 11.83/1.93 meet(join(complement(composition(x0, x1)), join(composition(x0, x1), composition(x0, x2))), join(join(complement(composition(x0, x1)), composition(x0, x2)), complement(composition(x0, meet(x1, complement(x2))))))
% 11.83/1.93 = { by lemma 40 }
% 11.83/1.93 meet(top, join(join(complement(composition(x0, x1)), composition(x0, x2)), complement(composition(x0, meet(x1, complement(x2))))))
% 11.83/1.93 = { by lemma 35 }
% 11.83/1.93 join(join(complement(composition(x0, x1)), composition(x0, x2)), complement(composition(x0, meet(x1, complement(x2)))))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 11.83/1.93 join(complement(composition(x0, meet(x1, complement(x2)))), join(complement(composition(x0, x1)), composition(x0, x2)))
% 11.83/1.93 = { by lemma 37 }
% 11.83/1.93 join(complement(composition(x0, x1)), join(complement(composition(x0, meet(x1, complement(x2)))), composition(x0, x2)))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) }
% 11.83/1.93 join(join(complement(composition(x0, meet(x1, complement(x2)))), composition(x0, x2)), complement(composition(x0, x1)))
% 11.83/1.93 = { by lemma 35 R->L }
% 11.83/1.93 meet(top, join(join(complement(composition(x0, meet(x1, complement(x2)))), composition(x0, x2)), complement(composition(x0, x1))))
% 11.83/1.93 = { by lemma 40 R->L }
% 11.83/1.93 meet(join(complement(composition(x0, meet(x1, complement(x2)))), join(composition(x0, meet(x1, complement(x2))), composition(x0, x2))), join(join(complement(composition(x0, meet(x1, complement(x2)))), composition(x0, x2)), complement(composition(x0, x1))))
% 11.83/1.93 = { by lemma 43 }
% 11.83/1.93 meet(join(complement(composition(x0, meet(x1, complement(x2)))), join(composition(x0, x1), composition(x0, x2))), join(join(complement(composition(x0, meet(x1, complement(x2)))), composition(x0, x2)), complement(composition(x0, x1))))
% 11.83/1.93 = { by lemma 37 R->L }
% 11.83/1.93 meet(join(composition(x0, x1), join(complement(composition(x0, meet(x1, complement(x2)))), composition(x0, x2))), join(join(complement(composition(x0, meet(x1, complement(x2)))), composition(x0, x2)), complement(composition(x0, x1))))
% 11.83/1.93 = { by axiom 2 (maddux1_join_commutativity) }
% 11.83/1.93 meet(join(join(complement(composition(x0, meet(x1, complement(x2)))), composition(x0, x2)), composition(x0, x1)), join(join(complement(composition(x0, meet(x1, complement(x2)))), composition(x0, x2)), complement(composition(x0, x1))))
% 11.83/1.93 = { by lemma 42 }
% 11.83/1.93 join(complement(composition(x0, meet(x1, complement(x2)))), composition(x0, x2))
% 11.83/1.93 % SZS output end Proof
% 11.83/1.93
% 11.83/1.93 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------