TSTP Solution File: REL043+2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL043+2 : 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 : n016.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 13:44:29 EDT 2023
% Result : Theorem 0.19s 0.69s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : REL043+2 : 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 : n016.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 21:49:11 EDT 2023
% 0.12/0.33 % CPUTime :
% 0.19/0.69 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.19/0.69
% 0.19/0.69 % SZS status Theorem
% 0.19/0.69
% 0.19/0.72 % SZS output start Proof
% 0.19/0.72 Axiom 1 (composition_identity): composition(X, one) = X.
% 0.19/0.72 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 0.19/0.72 Axiom 3 (converse_idempotence): converse(converse(X)) = X.
% 0.19/0.72 Axiom 4 (def_top): top = join(X, complement(X)).
% 0.19/0.72 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 0.19/0.72 Axiom 6 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 0.19/0.72 Axiom 7 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 0.19/0.72 Axiom 8 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 0.19/0.72 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 0.19/0.72 Axiom 10 (goals): join(composition(x0, converse(x1)), x2) = x2.
% 0.19/0.72 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 0.19/0.72 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 0.19/0.72 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 0.19/0.72 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 0.19/0.72 Axiom 15 (modular_law_2): join(meet(composition(X, Y), Z), meet(composition(meet(X, composition(Z, converse(Y))), Y), Z)) = meet(composition(meet(X, composition(Z, converse(Y))), Y), Z).
% 0.19/0.72
% 0.19/0.72 Lemma 16: complement(top) = zero.
% 0.19/0.72 Proof:
% 0.19/0.72 complement(top)
% 0.19/0.72 = { by axiom 4 (def_top) }
% 0.19/0.72 complement(join(complement(X), complement(complement(X))))
% 0.19/0.72 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 0.19/0.72 meet(X, complement(X))
% 0.19/0.72 = { by axiom 5 (def_zero) R->L }
% 0.19/0.72 zero
% 0.19/0.72
% 0.19/0.72 Lemma 17: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 0.19/0.72 Proof:
% 0.19/0.72 join(meet(X, Y), complement(join(complement(X), Y)))
% 0.19/0.72 = { by axiom 11 (maddux4_definiton_of_meet) }
% 0.19/0.72 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 0.19/0.72 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 0.19/0.72 X
% 0.19/0.72
% 0.19/0.72 Lemma 18: join(zero, meet(X, X)) = X.
% 0.19/0.72 Proof:
% 0.19/0.72 join(zero, meet(X, X))
% 0.19/0.72 = { by axiom 11 (maddux4_definiton_of_meet) }
% 0.19/0.72 join(zero, complement(join(complement(X), complement(X))))
% 0.19/0.72 = { by axiom 5 (def_zero) }
% 0.19/0.72 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 0.19/0.72 = { by lemma 17 }
% 0.19/0.72 X
% 0.19/0.72
% 0.19/0.72 Lemma 19: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 0.19/0.72 Proof:
% 0.19/0.72 converse(composition(converse(X), Y))
% 0.19/0.72 = { by axiom 6 (converse_multiplicativity) }
% 0.19/0.72 composition(converse(Y), converse(converse(X)))
% 0.19/0.72 = { by axiom 3 (converse_idempotence) }
% 0.19/0.72 composition(converse(Y), X)
% 0.19/0.72
% 0.19/0.72 Lemma 20: composition(converse(one), X) = X.
% 0.19/0.72 Proof:
% 0.19/0.72 composition(converse(one), X)
% 0.19/0.72 = { by lemma 19 R->L }
% 0.19/0.72 converse(composition(converse(X), one))
% 0.19/0.72 = { by axiom 1 (composition_identity) }
% 0.19/0.72 converse(converse(X))
% 0.19/0.72 = { by axiom 3 (converse_idempotence) }
% 0.19/0.72 X
% 0.19/0.72
% 0.19/0.72 Lemma 21: composition(one, X) = X.
% 0.19/0.72 Proof:
% 0.19/0.72 composition(one, X)
% 0.19/0.72 = { by lemma 20 R->L }
% 0.19/0.72 composition(converse(one), composition(one, X))
% 0.19/0.72 = { by axiom 7 (composition_associativity) }
% 0.19/0.72 composition(composition(converse(one), one), X)
% 0.19/0.72 = { by axiom 1 (composition_identity) }
% 0.19/0.72 composition(converse(one), X)
% 0.19/0.72 = { by lemma 20 }
% 0.19/0.72 X
% 0.19/0.72
% 0.19/0.72 Lemma 22: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 0.19/0.72 Proof:
% 0.19/0.72 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.19/0.72 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 0.19/0.72 = { by axiom 13 (converse_cancellativity) }
% 0.19/0.72 complement(X)
% 0.19/0.72
% 0.19/0.72 Lemma 23: join(complement(X), complement(X)) = complement(X).
% 0.19/0.72 Proof:
% 0.19/0.72 join(complement(X), complement(X))
% 0.19/0.72 = { by lemma 20 R->L }
% 0.19/0.72 join(complement(X), composition(converse(one), complement(X)))
% 0.19/0.72 = { by lemma 21 R->L }
% 0.19/0.72 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 0.19/0.72 = { by lemma 22 }
% 0.19/0.72 complement(X)
% 0.19/0.72
% 0.19/0.72 Lemma 24: join(zero, zero) = zero.
% 0.19/0.72 Proof:
% 0.19/0.72 join(zero, zero)
% 0.19/0.72 = { by lemma 16 R->L }
% 0.19/0.72 join(zero, complement(top))
% 0.19/0.72 = { by lemma 16 R->L }
% 0.19/0.72 join(complement(top), complement(top))
% 0.19/0.72 = { by lemma 23 }
% 0.19/0.72 complement(top)
% 0.19/0.72 = { by lemma 16 }
% 0.19/0.72 zero
% 0.19/0.72
% 0.19/0.72 Lemma 25: join(zero, join(zero, X)) = join(X, zero).
% 0.19/0.72 Proof:
% 0.19/0.72 join(zero, join(zero, X))
% 0.19/0.72 = { by axiom 9 (maddux2_join_associativity) }
% 0.19/0.72 join(join(zero, zero), X)
% 0.19/0.72 = { by lemma 24 }
% 0.19/0.72 join(zero, X)
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) }
% 0.19/0.72 join(X, zero)
% 0.19/0.72
% 0.19/0.72 Lemma 26: join(zero, complement(complement(X))) = X.
% 0.19/0.72 Proof:
% 0.19/0.72 join(zero, complement(complement(X)))
% 0.19/0.72 = { by axiom 5 (def_zero) }
% 0.19/0.72 join(meet(X, complement(X)), complement(complement(X)))
% 0.19/0.72 = { by lemma 23 R->L }
% 0.19/0.72 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 0.19/0.72 = { by lemma 17 }
% 0.19/0.72 X
% 0.19/0.72
% 0.19/0.72 Lemma 27: join(X, zero) = X.
% 0.19/0.72 Proof:
% 0.19/0.72 join(X, zero)
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.19/0.72 join(zero, X)
% 0.19/0.72 = { by lemma 18 R->L }
% 0.19/0.72 join(zero, join(zero, meet(X, X)))
% 0.19/0.72 = { by lemma 25 }
% 0.19/0.72 join(meet(X, X), zero)
% 0.19/0.72 = { by axiom 11 (maddux4_definiton_of_meet) }
% 0.19/0.72 join(complement(join(complement(X), complement(X))), zero)
% 0.19/0.72 = { by lemma 23 }
% 0.19/0.72 join(complement(complement(X)), zero)
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) }
% 0.19/0.72 join(zero, complement(complement(X)))
% 0.19/0.72 = { by lemma 26 }
% 0.19/0.72 X
% 0.19/0.72
% 0.19/0.72 Lemma 28: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 0.19/0.72 Proof:
% 0.19/0.72 converse(join(X, converse(Y)))
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.19/0.72 converse(join(converse(Y), X))
% 0.19/0.72 = { by axiom 8 (converse_additivity) }
% 0.19/0.72 join(converse(converse(Y)), converse(X))
% 0.19/0.72 = { by axiom 3 (converse_idempotence) }
% 0.19/0.72 join(Y, converse(X))
% 0.19/0.72
% 0.19/0.72 Lemma 29: converse(zero) = zero.
% 0.19/0.72 Proof:
% 0.19/0.72 converse(zero)
% 0.19/0.72 = { by lemma 27 R->L }
% 0.19/0.72 join(converse(zero), zero)
% 0.19/0.72 = { by lemma 25 R->L }
% 0.19/0.72 join(zero, join(zero, converse(zero)))
% 0.19/0.72 = { by lemma 28 R->L }
% 0.19/0.72 join(zero, converse(join(zero, converse(zero))))
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) }
% 0.19/0.72 join(zero, converse(join(converse(zero), zero)))
% 0.19/0.72 = { by lemma 27 }
% 0.19/0.72 join(zero, converse(converse(zero)))
% 0.19/0.72 = { by axiom 3 (converse_idempotence) }
% 0.19/0.72 join(zero, zero)
% 0.19/0.72 = { by lemma 24 }
% 0.19/0.72 zero
% 0.19/0.72
% 0.19/0.72 Lemma 30: join(X, join(Y, complement(X))) = join(Y, top).
% 0.19/0.72 Proof:
% 0.19/0.72 join(X, join(Y, complement(X)))
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.19/0.72 join(X, join(complement(X), Y))
% 0.19/0.72 = { by axiom 9 (maddux2_join_associativity) }
% 0.19/0.72 join(join(X, complement(X)), Y)
% 0.19/0.72 = { by axiom 4 (def_top) R->L }
% 0.19/0.72 join(top, Y)
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) }
% 0.19/0.72 join(Y, top)
% 0.19/0.72
% 0.19/0.72 Lemma 31: join(X, join(complement(X), Y)) = join(Y, top).
% 0.19/0.72 Proof:
% 0.19/0.72 join(X, join(complement(X), Y))
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.19/0.72 join(X, join(Y, complement(X)))
% 0.19/0.72 = { by lemma 30 }
% 0.19/0.72 join(Y, top)
% 0.19/0.72
% 0.19/0.72 Lemma 32: join(top, complement(X)) = top.
% 0.19/0.72 Proof:
% 0.19/0.72 join(top, complement(X))
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.19/0.72 join(complement(X), top)
% 0.19/0.72 = { by lemma 30 R->L }
% 0.19/0.72 join(X, join(complement(X), complement(X)))
% 0.19/0.72 = { by lemma 23 }
% 0.19/0.72 join(X, complement(X))
% 0.19/0.72 = { by axiom 4 (def_top) R->L }
% 0.19/0.72 top
% 0.19/0.72
% 0.19/0.72 Lemma 33: join(X, top) = top.
% 0.19/0.72 Proof:
% 0.19/0.72 join(X, top)
% 0.19/0.72 = { by axiom 4 (def_top) }
% 0.19/0.72 join(X, join(complement(X), complement(complement(X))))
% 0.19/0.72 = { by lemma 31 }
% 0.19/0.72 join(complement(complement(X)), top)
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) }
% 0.19/0.72 join(top, complement(complement(X)))
% 0.19/0.72 = { by lemma 32 }
% 0.19/0.72 top
% 0.19/0.72
% 0.19/0.72 Lemma 34: join(top, X) = top.
% 0.19/0.72 Proof:
% 0.19/0.72 join(top, X)
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.19/0.72 join(X, top)
% 0.19/0.72 = { by lemma 32 R->L }
% 0.19/0.72 join(X, join(top, complement(X)))
% 0.19/0.72 = { by lemma 30 }
% 0.19/0.72 join(top, top)
% 0.19/0.72 = { by lemma 33 }
% 0.19/0.72 top
% 0.19/0.72
% 0.19/0.72 Lemma 35: join(zero, X) = X.
% 0.19/0.72 Proof:
% 0.19/0.72 join(zero, X)
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.19/0.72 join(X, zero)
% 0.19/0.72 = { by lemma 27 }
% 0.19/0.72 X
% 0.19/0.72
% 0.19/0.72 Lemma 36: complement(complement(X)) = X.
% 0.19/0.72 Proof:
% 0.19/0.72 complement(complement(X))
% 0.19/0.72 = { by lemma 35 R->L }
% 0.19/0.72 join(zero, complement(complement(X)))
% 0.19/0.72 = { by lemma 26 }
% 0.19/0.72 X
% 0.19/0.72
% 0.19/0.72 Lemma 37: meet(Y, X) = meet(X, Y).
% 0.19/0.72 Proof:
% 0.19/0.72 meet(Y, X)
% 0.19/0.72 = { by axiom 11 (maddux4_definiton_of_meet) }
% 0.19/0.72 complement(join(complement(Y), complement(X)))
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.19/0.72 complement(join(complement(X), complement(Y)))
% 0.19/0.72 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 0.19/0.72 meet(X, Y)
% 0.19/0.72
% 0.19/0.72 Lemma 38: complement(join(zero, complement(X))) = meet(X, top).
% 0.19/0.72 Proof:
% 0.19/0.72 complement(join(zero, complement(X)))
% 0.19/0.72 = { by lemma 16 R->L }
% 0.19/0.72 complement(join(complement(top), complement(X)))
% 0.19/0.72 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 0.19/0.72 meet(top, X)
% 0.19/0.72 = { by lemma 37 R->L }
% 0.19/0.72 meet(X, top)
% 0.19/0.72
% 0.19/0.72 Lemma 39: meet(X, top) = X.
% 0.19/0.72 Proof:
% 0.19/0.72 meet(X, top)
% 0.19/0.72 = { by lemma 38 R->L }
% 0.19/0.72 complement(join(zero, complement(X)))
% 0.19/0.72 = { by lemma 35 }
% 0.19/0.72 complement(complement(X))
% 0.19/0.72 = { by lemma 36 }
% 0.19/0.72 X
% 0.19/0.72
% 0.19/0.72 Lemma 40: meet(zero, X) = zero.
% 0.19/0.72 Proof:
% 0.19/0.72 meet(zero, X)
% 0.19/0.72 = { by axiom 11 (maddux4_definiton_of_meet) }
% 0.19/0.72 complement(join(complement(zero), complement(X)))
% 0.19/0.72 = { by lemma 35 R->L }
% 0.19/0.72 complement(join(join(zero, complement(zero)), complement(X)))
% 0.19/0.72 = { by axiom 4 (def_top) R->L }
% 0.19/0.72 complement(join(top, complement(X)))
% 0.19/0.72 = { by lemma 32 }
% 0.19/0.72 complement(top)
% 0.19/0.72 = { by lemma 16 }
% 0.19/0.72 zero
% 0.19/0.72
% 0.19/0.72 Lemma 41: join(X, converse(top)) = converse(top).
% 0.19/0.72 Proof:
% 0.19/0.72 join(X, converse(top))
% 0.19/0.72 = { by lemma 28 R->L }
% 0.19/0.72 converse(join(top, converse(X)))
% 0.19/0.72 = { by lemma 34 }
% 0.19/0.72 converse(top)
% 0.19/0.72
% 0.19/0.72 Lemma 42: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 0.19/0.72 Proof:
% 0.19/0.72 join(X, composition(Y, X))
% 0.19/0.72 = { by lemma 21 R->L }
% 0.19/0.72 join(composition(one, X), composition(Y, X))
% 0.19/0.72 = { by axiom 12 (composition_distributivity) R->L }
% 0.19/0.72 composition(join(one, Y), X)
% 0.19/0.72 = { by axiom 2 (maddux1_join_commutativity) }
% 0.19/0.72 composition(join(Y, one), X)
% 0.19/0.72
% 0.19/0.72 Lemma 43: composition(top, zero) = zero.
% 0.19/0.72 Proof:
% 0.19/0.72 composition(top, zero)
% 0.19/0.72 = { by lemma 16 R->L }
% 0.19/0.72 composition(top, complement(top))
% 0.19/0.72 = { by lemma 34 R->L }
% 0.19/0.72 composition(join(top, one), complement(top))
% 0.19/0.72 = { by lemma 33 R->L }
% 0.19/0.73 composition(join(join(converse(top), top), one), complement(top))
% 0.19/0.73 = { by lemma 31 R->L }
% 0.19/0.73 composition(join(join(X, join(complement(X), converse(top))), one), complement(top))
% 0.19/0.73 = { by lemma 41 }
% 0.19/0.73 composition(join(join(X, converse(top)), one), complement(top))
% 0.19/0.73 = { by lemma 41 }
% 0.19/0.73 composition(join(converse(top), one), complement(top))
% 0.19/0.73 = { by lemma 42 R->L }
% 0.19/0.73 join(complement(top), composition(converse(top), complement(top)))
% 0.19/0.73 = { by lemma 34 R->L }
% 0.19/0.73 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 0.19/0.73 = { by lemma 42 }
% 0.19/0.73 join(complement(top), composition(converse(top), complement(composition(join(top, one), top))))
% 0.19/0.73 = { by lemma 34 }
% 0.19/0.73 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 0.19/0.73 = { by lemma 22 }
% 0.19/0.73 complement(top)
% 0.19/0.73 = { by lemma 16 }
% 0.19/0.73 zero
% 0.19/0.73
% 0.19/0.73 Lemma 44: composition(zero, X) = zero.
% 0.19/0.73 Proof:
% 0.19/0.73 composition(zero, X)
% 0.19/0.73 = { by lemma 29 R->L }
% 0.19/0.73 composition(converse(zero), X)
% 0.19/0.73 = { by lemma 19 R->L }
% 0.19/0.73 converse(composition(converse(X), zero))
% 0.19/0.73 = { by lemma 35 R->L }
% 0.19/0.73 converse(join(zero, composition(converse(X), zero)))
% 0.19/0.73 = { by lemma 43 R->L }
% 0.19/0.73 converse(join(composition(top, zero), composition(converse(X), zero)))
% 0.19/0.73 = { by axiom 12 (composition_distributivity) R->L }
% 0.19/0.73 converse(composition(join(top, converse(X)), zero))
% 0.19/0.73 = { by lemma 34 }
% 0.19/0.73 converse(composition(top, zero))
% 0.19/0.73 = { by lemma 43 }
% 0.19/0.73 converse(zero)
% 0.19/0.73 = { by lemma 29 }
% 0.19/0.73 zero
% 0.19/0.73
% 0.19/0.73 Lemma 45: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 0.19/0.73 Proof:
% 0.19/0.73 meet(X, join(complement(Y), complement(Z)))
% 0.19/0.73 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.19/0.73 meet(X, join(complement(Z), complement(Y)))
% 0.19/0.73 = { by lemma 37 }
% 0.19/0.73 meet(join(complement(Z), complement(Y)), X)
% 0.19/0.73 = { by axiom 11 (maddux4_definiton_of_meet) }
% 0.19/0.73 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 0.19/0.73 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 0.19/0.73 complement(join(meet(Z, Y), complement(X)))
% 0.19/0.73 = { by axiom 2 (maddux1_join_commutativity) }
% 0.19/0.73 complement(join(complement(X), meet(Z, Y)))
% 0.19/0.73 = { by lemma 37 R->L }
% 0.19/0.73 complement(join(complement(X), meet(Y, Z)))
% 0.19/0.73
% 0.19/0.73 Lemma 46: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 0.19/0.73 Proof:
% 0.19/0.73 complement(join(X, complement(Y)))
% 0.19/0.73 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.19/0.73 complement(join(complement(Y), X))
% 0.19/0.73 = { by lemma 39 R->L }
% 0.19/0.73 complement(join(complement(Y), meet(X, top)))
% 0.19/0.73 = { by lemma 37 R->L }
% 0.19/0.73 complement(join(complement(Y), meet(top, X)))
% 0.19/0.73 = { by lemma 45 R->L }
% 0.19/0.73 meet(Y, join(complement(top), complement(X)))
% 0.19/0.73 = { by lemma 16 }
% 0.19/0.73 meet(Y, join(zero, complement(X)))
% 0.19/0.73 = { by lemma 35 }
% 0.19/0.73 meet(Y, complement(X))
% 0.19/0.73
% 0.19/0.73 Lemma 47: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 0.19/0.73 Proof:
% 0.19/0.73 complement(meet(complement(X), Y))
% 0.19/0.73 = { by lemma 37 }
% 0.19/0.73 complement(meet(Y, complement(X)))
% 0.19/0.73 = { by lemma 35 R->L }
% 0.19/0.73 complement(join(zero, meet(Y, complement(X))))
% 0.19/0.73 = { by lemma 46 R->L }
% 0.19/0.73 complement(join(zero, complement(join(X, complement(Y)))))
% 0.19/0.73 = { by lemma 38 }
% 0.19/0.73 meet(join(X, complement(Y)), top)
% 0.19/0.73 = { by lemma 39 }
% 0.19/0.73 join(X, complement(Y))
% 0.19/0.73
% 0.19/0.73 Lemma 48: join(X, complement(meet(X, Y))) = top.
% 0.19/0.73 Proof:
% 0.19/0.73 join(X, complement(meet(X, Y)))
% 0.19/0.73 = { by lemma 37 }
% 0.19/0.73 join(X, complement(meet(Y, X)))
% 0.19/0.73 = { by axiom 11 (maddux4_definiton_of_meet) }
% 0.19/0.73 join(X, complement(complement(join(complement(Y), complement(X)))))
% 0.19/0.73 = { by lemma 23 R->L }
% 0.19/0.73 join(X, complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))))
% 0.19/0.73 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 0.19/0.73 join(X, complement(join(meet(Y, X), complement(join(complement(Y), complement(X))))))
% 0.19/0.73 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 0.19/0.73 join(X, complement(join(meet(Y, X), meet(Y, X))))
% 0.19/0.73 = { by lemma 37 R->L }
% 0.19/0.73 join(X, complement(join(meet(Y, X), meet(X, Y))))
% 0.19/0.73 = { by lemma 37 R->L }
% 0.19/0.73 join(X, complement(join(meet(X, Y), meet(X, Y))))
% 0.19/0.73 = { by axiom 11 (maddux4_definiton_of_meet) }
% 0.19/0.73 join(X, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y))))
% 0.19/0.73 = { by lemma 35 R->L }
% 0.19/0.73 join(X, join(zero, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y)))))
% 0.19/0.73 = { by lemma 45 R->L }
% 0.19/0.73 join(X, join(zero, meet(join(complement(X), complement(Y)), join(complement(X), complement(Y)))))
% 0.19/0.73 = { by lemma 18 }
% 0.19/0.73 join(X, join(complement(X), complement(Y)))
% 0.19/0.73 = { by axiom 2 (maddux1_join_commutativity) }
% 0.19/0.73 join(X, join(complement(Y), complement(X)))
% 0.19/0.73 = { by lemma 30 }
% 0.19/0.73 join(complement(Y), top)
% 0.19/0.73 = { by lemma 33 }
% 0.19/0.73 top
% 0.19/0.73
% 0.19/0.73 Lemma 49: meet(complement(x2), composition(x0, converse(x1))) = zero.
% 0.19/0.73 Proof:
% 0.19/0.73 meet(complement(x2), composition(x0, converse(x1)))
% 0.19/0.73 = { by lemma 37 }
% 0.19/0.73 meet(composition(x0, converse(x1)), complement(x2))
% 0.19/0.73 = { by axiom 10 (goals) R->L }
% 0.19/0.73 meet(composition(x0, converse(x1)), complement(join(composition(x0, converse(x1)), x2)))
% 0.19/0.73 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 0.19/0.73 meet(composition(x0, converse(x1)), complement(join(x2, composition(x0, converse(x1)))))
% 0.19/0.73 = { by lemma 39 R->L }
% 0.19/0.73 meet(composition(x0, converse(x1)), complement(join(x2, meet(composition(x0, converse(x1)), top))))
% 0.19/0.73 = { by lemma 38 R->L }
% 0.19/0.73 meet(composition(x0, converse(x1)), complement(join(x2, complement(join(zero, complement(composition(x0, converse(x1))))))))
% 0.19/0.73 = { by lemma 46 }
% 0.19/0.73 meet(composition(x0, converse(x1)), meet(join(zero, complement(composition(x0, converse(x1)))), complement(x2)))
% 0.19/0.73 = { by lemma 35 }
% 0.19/0.73 meet(composition(x0, converse(x1)), meet(complement(composition(x0, converse(x1))), complement(x2)))
% 0.19/0.73 = { by axiom 11 (maddux4_definiton_of_meet) }
% 0.19/0.73 complement(join(complement(composition(x0, converse(x1))), complement(meet(complement(composition(x0, converse(x1))), complement(x2)))))
% 0.19/0.73 = { by lemma 48 }
% 0.19/0.73 complement(top)
% 0.19/0.73 = { by lemma 16 }
% 0.19/0.73 zero
% 0.19/0.73
% 0.19/0.73 Goal 1 (goals_1): join(composition(complement(x2), x1), complement(x0)) = complement(x0).
% 0.19/0.73 Proof:
% 0.19/0.73 join(composition(complement(x2), x1), complement(x0))
% 0.19/0.73 = { by axiom 2 (maddux1_join_commutativity) }
% 0.19/0.73 join(complement(x0), composition(complement(x2), x1))
% 0.19/0.73 = { by lemma 17 R->L }
% 0.19/0.73 join(complement(x0), join(meet(composition(complement(x2), x1), complement(x0)), complement(join(complement(composition(complement(x2), x1)), complement(x0)))))
% 0.19/0.73 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 0.19/0.73 join(complement(x0), join(meet(composition(complement(x2), x1), complement(x0)), meet(composition(complement(x2), x1), x0)))
% 0.19/0.73 = { by axiom 2 (maddux1_join_commutativity) }
% 0.19/0.73 join(complement(x0), join(meet(composition(complement(x2), x1), x0), meet(composition(complement(x2), x1), complement(x0))))
% 0.19/0.73 = { by lemma 27 R->L }
% 0.19/0.73 join(complement(x0), join(join(meet(composition(complement(x2), x1), x0), zero), meet(composition(complement(x2), x1), complement(x0))))
% 0.19/0.73 = { by lemma 40 R->L }
% 0.19/0.73 join(complement(x0), join(join(meet(composition(complement(x2), x1), x0), meet(zero, x0)), meet(composition(complement(x2), x1), complement(x0))))
% 0.19/0.73 = { by lemma 44 R->L }
% 0.19/0.73 join(complement(x0), join(join(meet(composition(complement(x2), x1), x0), meet(composition(zero, x1), x0)), meet(composition(complement(x2), x1), complement(x0))))
% 0.19/0.73 = { by lemma 49 R->L }
% 0.19/0.73 join(complement(x0), join(join(meet(composition(complement(x2), x1), x0), meet(composition(meet(complement(x2), composition(x0, converse(x1))), x1), x0)), meet(composition(complement(x2), x1), complement(x0))))
% 0.19/0.73 = { by axiom 15 (modular_law_2) }
% 0.19/0.73 join(complement(x0), join(meet(composition(meet(complement(x2), composition(x0, converse(x1))), x1), x0), meet(composition(complement(x2), x1), complement(x0))))
% 0.19/0.73 = { by lemma 49 }
% 0.19/0.73 join(complement(x0), join(meet(composition(zero, x1), x0), meet(composition(complement(x2), x1), complement(x0))))
% 0.19/0.73 = { by lemma 44 }
% 0.19/0.73 join(complement(x0), join(meet(zero, x0), meet(composition(complement(x2), x1), complement(x0))))
% 0.19/0.73 = { by lemma 40 }
% 0.19/0.73 join(complement(x0), join(zero, meet(composition(complement(x2), x1), complement(x0))))
% 0.19/0.73 = { by lemma 35 }
% 0.19/0.73 join(complement(x0), meet(composition(complement(x2), x1), complement(x0)))
% 0.19/0.73 = { by lemma 37 R->L }
% 0.19/0.73 join(complement(x0), meet(complement(x0), composition(complement(x2), x1)))
% 0.19/0.73 = { by axiom 11 (maddux4_definiton_of_meet) }
% 0.19/0.73 join(complement(x0), complement(join(complement(complement(x0)), complement(composition(complement(x2), x1)))))
% 0.19/0.73 = { by lemma 47 R->L }
% 0.19/0.73 complement(meet(complement(complement(x0)), join(complement(complement(x0)), complement(composition(complement(x2), x1)))))
% 0.19/0.73 = { by lemma 27 R->L }
% 0.19/0.73 complement(join(meet(complement(complement(x0)), join(complement(complement(x0)), complement(composition(complement(x2), x1)))), zero))
% 0.19/0.73 = { by lemma 16 R->L }
% 0.19/0.73 complement(join(meet(complement(complement(x0)), join(complement(complement(x0)), complement(composition(complement(x2), x1)))), complement(top)))
% 0.19/0.73 = { by lemma 47 R->L }
% 0.19/0.73 complement(join(meet(complement(complement(x0)), complement(meet(complement(complement(complement(x0))), composition(complement(x2), x1)))), complement(top)))
% 0.19/0.73 = { by lemma 48 R->L }
% 0.19/0.73 complement(join(meet(complement(complement(x0)), complement(meet(complement(complement(complement(x0))), composition(complement(x2), x1)))), complement(join(complement(complement(complement(x0))), complement(meet(complement(complement(complement(x0))), composition(complement(x2), x1)))))))
% 0.19/0.73 = { by lemma 17 }
% 0.19/0.73 complement(complement(complement(x0)))
% 0.19/0.73 = { by lemma 36 }
% 0.19/0.73 complement(x0)
% 0.19/0.73 % SZS output end Proof
% 0.19/0.73
% 0.19/0.73 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------