TSTP Solution File: REL026+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL026+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 : n023.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 13:44:06 EDT 2023
% Result : Theorem 20.92s 2.97s
% Output : Proof 22.44s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.05/0.10 % Problem : REL026+1 : TPTP v8.1.2. Released v4.0.0.
% 0.05/0.11 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.10/0.31 % Computer : n023.cluster.edu
% 0.10/0.31 % Model : x86_64 x86_64
% 0.10/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.31 % Memory : 8042.1875MB
% 0.10/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.31 % CPULimit : 300
% 0.10/0.31 % WCLimit : 300
% 0.10/0.31 % DateTime : Fri Aug 25 19:37:26 EDT 2023
% 0.10/0.32 % CPUTime :
% 20.92/2.97 Command-line arguments: --ground-connectedness --complete-subsets
% 20.92/2.97
% 20.92/2.97 % SZS status Theorem
% 20.92/2.97
% 22.00/3.13 % SZS output start Proof
% 22.00/3.13 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 22.00/3.13 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 22.00/3.13 Axiom 3 (goals): join(x0, one) = one.
% 22.00/3.13 Axiom 4 (composition_identity): composition(X, one) = X.
% 22.00/3.13 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 22.00/3.13 Axiom 6 (def_top): top = join(X, complement(X)).
% 22.00/3.13 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 22.00/3.13 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 22.00/3.13 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 22.00/3.13 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 22.00/3.13 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 22.00/3.13 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 22.00/3.13 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 22.00/3.13 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 22.00/3.13
% 22.00/3.13 Lemma 15: complement(top) = zero.
% 22.00/3.13 Proof:
% 22.00/3.13 complement(top)
% 22.00/3.13 = { by axiom 6 (def_top) }
% 22.00/3.13 complement(join(complement(X), complement(complement(X))))
% 22.00/3.13 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 22.00/3.13 meet(X, complement(X))
% 22.00/3.13 = { by axiom 5 (def_zero) R->L }
% 22.00/3.13 zero
% 22.00/3.13
% 22.00/3.13 Lemma 16: join(X, join(Y, complement(X))) = join(Y, top).
% 22.00/3.13 Proof:
% 22.00/3.13 join(X, join(Y, complement(X)))
% 22.00/3.13 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.00/3.13 join(X, join(complement(X), Y))
% 22.00/3.13 = { by axiom 8 (maddux2_join_associativity) }
% 22.00/3.13 join(join(X, complement(X)), Y)
% 22.00/3.13 = { by axiom 6 (def_top) R->L }
% 22.00/3.13 join(top, Y)
% 22.00/3.13 = { by axiom 2 (maddux1_join_commutativity) }
% 22.00/3.13 join(Y, top)
% 22.00/3.13
% 22.00/3.13 Lemma 17: composition(converse(one), X) = X.
% 22.00/3.13 Proof:
% 22.00/3.13 composition(converse(one), X)
% 22.00/3.13 = { by axiom 1 (converse_idempotence) R->L }
% 22.00/3.13 composition(converse(one), converse(converse(X)))
% 22.00/3.13 = { by axiom 9 (converse_multiplicativity) R->L }
% 22.00/3.13 converse(composition(converse(X), one))
% 22.00/3.13 = { by axiom 4 (composition_identity) }
% 22.00/3.13 converse(converse(X))
% 22.00/3.13 = { by axiom 1 (converse_idempotence) }
% 22.00/3.13 X
% 22.00/3.13
% 22.00/3.13 Lemma 18: composition(one, X) = X.
% 22.00/3.13 Proof:
% 22.00/3.13 composition(one, X)
% 22.00/3.13 = { by lemma 17 R->L }
% 22.00/3.13 composition(converse(one), composition(one, X))
% 22.00/3.13 = { by axiom 10 (composition_associativity) }
% 22.00/3.13 composition(composition(converse(one), one), X)
% 22.00/3.13 = { by axiom 4 (composition_identity) }
% 22.00/3.13 composition(converse(one), X)
% 22.00/3.13 = { by lemma 17 }
% 22.00/3.13 X
% 22.00/3.13
% 22.00/3.13 Lemma 19: join(complement(X), complement(X)) = complement(X).
% 22.00/3.13 Proof:
% 22.00/3.13 join(complement(X), complement(X))
% 22.00/3.13 = { by lemma 17 R->L }
% 22.00/3.13 join(complement(X), composition(converse(one), complement(X)))
% 22.00/3.13 = { by lemma 18 R->L }
% 22.00/3.13 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 22.00/3.13 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.00/3.13 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 22.00/3.13 = { by axiom 13 (converse_cancellativity) }
% 22.00/3.13 complement(X)
% 22.00/3.13
% 22.00/3.13 Lemma 20: join(top, complement(X)) = top.
% 22.00/3.13 Proof:
% 22.00/3.13 join(top, complement(X))
% 22.00/3.13 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.00/3.13 join(complement(X), top)
% 22.00/3.13 = { by lemma 16 R->L }
% 22.00/3.13 join(X, join(complement(X), complement(X)))
% 22.00/3.13 = { by lemma 19 }
% 22.00/3.13 join(X, complement(X))
% 22.00/3.13 = { by axiom 6 (def_top) R->L }
% 22.00/3.13 top
% 22.00/3.13
% 22.00/3.13 Lemma 21: join(Y, top) = join(X, top).
% 22.00/3.13 Proof:
% 22.00/3.13 join(Y, top)
% 22.00/3.13 = { by lemma 20 R->L }
% 22.00/3.13 join(Y, join(top, complement(Y)))
% 22.00/3.13 = { by lemma 16 }
% 22.00/3.13 join(top, top)
% 22.00/3.13 = { by lemma 16 R->L }
% 22.00/3.13 join(X, join(top, complement(X)))
% 22.00/3.13 = { by lemma 20 }
% 22.00/3.13 join(X, top)
% 22.00/3.13
% 22.00/3.13 Lemma 22: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 22.00/3.13 Proof:
% 22.00/3.13 join(meet(X, Y), complement(join(complement(X), Y)))
% 22.00/3.13 = { by axiom 11 (maddux4_definiton_of_meet) }
% 22.00/3.13 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 22.00/3.13 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 22.00/3.13 X
% 22.00/3.13
% 22.00/3.13 Lemma 23: join(zero, meet(X, X)) = X.
% 22.00/3.13 Proof:
% 22.00/3.13 join(zero, meet(X, X))
% 22.00/3.13 = { by axiom 11 (maddux4_definiton_of_meet) }
% 22.00/3.13 join(zero, complement(join(complement(X), complement(X))))
% 22.00/3.13 = { by axiom 5 (def_zero) }
% 22.00/3.14 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 22.00/3.14 = { by lemma 22 }
% 22.00/3.14 X
% 22.00/3.14
% 22.00/3.14 Lemma 24: join(zero, join(X, meet(Y, Y))) = join(X, Y).
% 22.00/3.14 Proof:
% 22.00/3.14 join(zero, join(X, meet(Y, Y)))
% 22.00/3.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.00/3.14 join(zero, join(meet(Y, Y), X))
% 22.00/3.14 = { by axiom 8 (maddux2_join_associativity) }
% 22.00/3.14 join(join(zero, meet(Y, Y)), X)
% 22.00/3.14 = { by lemma 23 }
% 22.00/3.14 join(Y, X)
% 22.00/3.14 = { by axiom 2 (maddux1_join_commutativity) }
% 22.00/3.14 join(X, Y)
% 22.00/3.14
% 22.00/3.14 Lemma 25: join(X, zero) = join(X, X).
% 22.00/3.14 Proof:
% 22.00/3.14 join(X, zero)
% 22.00/3.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.00/3.14 join(zero, X)
% 22.00/3.14 = { by lemma 23 R->L }
% 22.00/3.14 join(zero, join(zero, meet(X, X)))
% 22.00/3.14 = { by axiom 11 (maddux4_definiton_of_meet) }
% 22.00/3.14 join(zero, join(zero, complement(join(complement(X), complement(X)))))
% 22.00/3.14 = { by lemma 19 R->L }
% 22.00/3.14 join(zero, join(zero, join(complement(join(complement(X), complement(X))), complement(join(complement(X), complement(X))))))
% 22.00/3.14 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 22.00/3.14 join(zero, join(zero, join(meet(X, X), complement(join(complement(X), complement(X))))))
% 22.00/3.14 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 22.00/3.14 join(zero, join(zero, join(meet(X, X), meet(X, X))))
% 22.00/3.14 = { by lemma 24 }
% 22.00/3.14 join(zero, join(meet(X, X), X))
% 22.00/3.14 = { by axiom 2 (maddux1_join_commutativity) }
% 22.00/3.14 join(zero, join(X, meet(X, X)))
% 22.00/3.14 = { by lemma 24 }
% 22.00/3.14 join(X, X)
% 22.00/3.14
% 22.00/3.14 Lemma 26: composition(join(X, one), Y) = join(Y, composition(X, Y)).
% 22.00/3.14 Proof:
% 22.00/3.14 composition(join(X, one), Y)
% 22.00/3.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.00/3.14 composition(join(one, X), Y)
% 22.00/3.14 = { by axiom 12 (composition_distributivity) }
% 22.00/3.14 join(composition(one, Y), composition(X, Y))
% 22.00/3.14 = { by lemma 18 }
% 22.00/3.14 join(Y, composition(X, Y))
% 22.00/3.14
% 22.00/3.14 Lemma 27: join(one, x0) = one.
% 22.00/3.14 Proof:
% 22.00/3.14 join(one, x0)
% 22.00/3.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.00/3.14 join(x0, one)
% 22.00/3.14 = { by axiom 3 (goals) }
% 22.00/3.14 one
% 22.00/3.14
% 22.00/3.14 Lemma 28: join(one, join(X, x0)) = join(X, one).
% 22.00/3.14 Proof:
% 22.00/3.14 join(one, join(X, x0))
% 22.00/3.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.00/3.14 join(one, join(x0, X))
% 22.00/3.14 = { by axiom 8 (maddux2_join_associativity) }
% 22.00/3.14 join(join(one, x0), X)
% 22.00/3.14 = { by lemma 27 }
% 22.00/3.14 join(one, X)
% 22.00/3.14 = { by axiom 2 (maddux1_join_commutativity) }
% 22.00/3.14 join(X, one)
% 22.00/3.14
% 22.00/3.14 Lemma 29: join(X, zero) = X.
% 22.00/3.14 Proof:
% 22.00/3.14 join(X, zero)
% 22.00/3.14 = { by lemma 25 }
% 22.00/3.14 join(X, X)
% 22.00/3.14 = { by lemma 17 R->L }
% 22.00/3.14 join(X, composition(converse(one), X))
% 22.00/3.14 = { by lemma 26 R->L }
% 22.00/3.14 composition(join(converse(one), one), X)
% 22.00/3.14 = { by axiom 4 (composition_identity) R->L }
% 22.00/3.14 composition(join(composition(converse(one), one), one), X)
% 22.00/3.14 = { by lemma 17 }
% 22.00/3.14 composition(join(one, one), X)
% 22.00/3.14 = { by lemma 25 R->L }
% 22.00/3.14 composition(join(one, zero), X)
% 22.00/3.14 = { by axiom 2 (maddux1_join_commutativity) }
% 22.00/3.14 composition(join(zero, one), X)
% 22.00/3.14 = { by lemma 28 R->L }
% 22.00/3.14 composition(join(one, join(zero, x0)), X)
% 22.00/3.14 = { by axiom 2 (maddux1_join_commutativity) }
% 22.00/3.14 composition(join(one, join(x0, zero)), X)
% 22.00/3.14 = { by lemma 25 }
% 22.00/3.14 composition(join(one, join(x0, x0)), X)
% 22.00/3.14 = { by lemma 28 }
% 22.00/3.14 composition(join(x0, one), X)
% 22.00/3.14 = { by axiom 2 (maddux1_join_commutativity) }
% 22.00/3.14 composition(join(one, x0), X)
% 22.00/3.14 = { by lemma 27 }
% 22.00/3.14 composition(one, X)
% 22.00/3.14 = { by lemma 18 }
% 22.00/3.14 X
% 22.00/3.14
% 22.00/3.14 Lemma 30: join(zero, X) = X.
% 22.00/3.14 Proof:
% 22.39/3.14 join(zero, X)
% 22.39/3.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.39/3.14 join(X, zero)
% 22.39/3.14 = { by lemma 29 }
% 22.39/3.14 X
% 22.39/3.14
% 22.39/3.14 Lemma 31: join(X, top) = top.
% 22.39/3.14 Proof:
% 22.39/3.14 join(X, top)
% 22.39/3.14 = { by lemma 21 }
% 22.39/3.14 join(zero, top)
% 22.39/3.14 = { by lemma 30 }
% 22.39/3.14 top
% 22.39/3.14
% 22.39/3.14 Lemma 32: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 22.39/3.14 Proof:
% 22.39/3.14 converse(join(X, converse(Y)))
% 22.39/3.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.39/3.14 converse(join(converse(Y), X))
% 22.39/3.14 = { by axiom 7 (converse_additivity) }
% 22.39/3.14 join(converse(converse(Y)), converse(X))
% 22.39/3.14 = { by axiom 1 (converse_idempotence) }
% 22.39/3.14 join(Y, converse(X))
% 22.39/3.14
% 22.39/3.14 Lemma 33: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 22.39/3.14 Proof:
% 22.39/3.14 converse(join(converse(X), Y))
% 22.39/3.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.39/3.14 converse(join(Y, converse(X)))
% 22.39/3.14 = { by lemma 32 }
% 22.39/3.14 join(X, converse(Y))
% 22.39/3.14
% 22.39/3.14 Lemma 34: join(X, converse(complement(converse(X)))) = converse(top).
% 22.39/3.14 Proof:
% 22.39/3.14 join(X, converse(complement(converse(X))))
% 22.39/3.14 = { by lemma 33 R->L }
% 22.39/3.14 converse(join(converse(X), complement(converse(X))))
% 22.39/3.14 = { by axiom 6 (def_top) R->L }
% 22.39/3.14 converse(top)
% 22.39/3.14
% 22.39/3.14 Lemma 35: join(X, join(complement(X), Y)) = top.
% 22.39/3.14 Proof:
% 22.39/3.14 join(X, join(complement(X), Y))
% 22.39/3.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.39/3.14 join(X, join(Y, complement(X)))
% 22.39/3.14 = { by lemma 16 }
% 22.39/3.14 join(Y, top)
% 22.39/3.14 = { by lemma 21 R->L }
% 22.39/3.14 join(Z, top)
% 22.39/3.14 = { by lemma 31 }
% 22.39/3.14 top
% 22.39/3.14
% 22.39/3.14 Lemma 36: converse(top) = top.
% 22.39/3.14 Proof:
% 22.39/3.14 converse(top)
% 22.39/3.14 = { by lemma 31 R->L }
% 22.39/3.14 converse(join(X, top))
% 22.39/3.14 = { by axiom 7 (converse_additivity) }
% 22.39/3.14 join(converse(X), converse(top))
% 22.39/3.14 = { by lemma 34 R->L }
% 22.39/3.14 join(converse(X), join(complement(converse(X)), converse(complement(converse(complement(converse(X)))))))
% 22.39/3.14 = { by lemma 35 }
% 22.39/3.14 top
% 22.39/3.14
% 22.39/3.14 Lemma 37: complement(complement(X)) = meet(X, X).
% 22.39/3.14 Proof:
% 22.39/3.14 complement(complement(X))
% 22.39/3.14 = { by lemma 19 R->L }
% 22.39/3.14 complement(join(complement(X), complement(X)))
% 22.39/3.14 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 22.39/3.14 meet(X, X)
% 22.39/3.14
% 22.39/3.14 Lemma 38: complement(complement(X)) = X.
% 22.39/3.14 Proof:
% 22.39/3.14 complement(complement(X))
% 22.39/3.14 = { by lemma 30 R->L }
% 22.39/3.14 join(zero, complement(complement(X)))
% 22.39/3.14 = { by lemma 37 }
% 22.39/3.14 join(zero, meet(X, X))
% 22.39/3.14 = { by lemma 23 }
% 22.39/3.14 X
% 22.39/3.14
% 22.39/3.14 Lemma 39: meet(X, X) = X.
% 22.39/3.14 Proof:
% 22.39/3.14 meet(X, X)
% 22.39/3.14 = { by lemma 37 R->L }
% 22.39/3.14 complement(complement(X))
% 22.39/3.14 = { by lemma 38 }
% 22.39/3.14 X
% 22.39/3.14
% 22.39/3.14 Lemma 40: meet(Y, X) = meet(X, Y).
% 22.39/3.14 Proof:
% 22.39/3.14 meet(Y, X)
% 22.39/3.14 = { by axiom 11 (maddux4_definiton_of_meet) }
% 22.39/3.14 complement(join(complement(Y), complement(X)))
% 22.39/3.14 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.39/3.14 complement(join(complement(X), complement(Y)))
% 22.39/3.14 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 22.39/3.14 meet(X, Y)
% 22.39/3.14
% 22.39/3.14 Lemma 41: complement(join(zero, complement(X))) = meet(X, top).
% 22.39/3.14 Proof:
% 22.39/3.14 complement(join(zero, complement(X)))
% 22.39/3.14 = { by lemma 15 R->L }
% 22.39/3.14 complement(join(complement(top), complement(X)))
% 22.39/3.14 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 22.39/3.14 meet(top, X)
% 22.39/3.14 = { by lemma 40 R->L }
% 22.39/3.14 meet(X, top)
% 22.39/3.14
% 22.39/3.14 Lemma 42: meet(X, top) = X.
% 22.39/3.15 Proof:
% 22.39/3.15 meet(X, top)
% 22.39/3.15 = { by lemma 40 }
% 22.39/3.15 meet(top, X)
% 22.39/3.15 = { by lemma 38 R->L }
% 22.39/3.15 meet(top, complement(complement(X)))
% 22.39/3.15 = { by lemma 40 }
% 22.39/3.15 meet(complement(complement(X)), top)
% 22.39/3.15 = { by lemma 41 R->L }
% 22.39/3.15 complement(join(zero, complement(complement(complement(X)))))
% 22.39/3.15 = { by lemma 37 }
% 22.39/3.15 complement(join(zero, meet(complement(X), complement(X))))
% 22.39/3.15 = { by lemma 23 }
% 22.39/3.15 complement(complement(X))
% 22.39/3.15 = { by lemma 38 }
% 22.39/3.15 X
% 22.39/3.15
% 22.39/3.15 Lemma 43: meet(top, X) = X.
% 22.39/3.15 Proof:
% 22.39/3.15 meet(top, X)
% 22.39/3.15 = { by lemma 40 }
% 22.39/3.15 meet(X, top)
% 22.39/3.15 = { by lemma 42 }
% 22.39/3.15 X
% 22.39/3.15
% 22.39/3.15 Lemma 44: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 22.39/3.15 Proof:
% 22.39/3.15 complement(join(complement(X), meet(Y, Z)))
% 22.44/3.15 = { by lemma 40 }
% 22.44/3.15 complement(join(complement(X), meet(Z, Y)))
% 22.44/3.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.44/3.15 complement(join(meet(Z, Y), complement(X)))
% 22.44/3.15 = { by axiom 11 (maddux4_definiton_of_meet) }
% 22.44/3.15 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 22.44/3.15 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 22.44/3.15 meet(join(complement(Z), complement(Y)), X)
% 22.44/3.15 = { by lemma 40 R->L }
% 22.44/3.15 meet(X, join(complement(Z), complement(Y)))
% 22.44/3.15 = { by axiom 2 (maddux1_join_commutativity) }
% 22.44/3.15 meet(X, join(complement(Y), complement(Z)))
% 22.44/3.15
% 22.44/3.15 Lemma 45: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 22.44/3.15 Proof:
% 22.44/3.15 join(complement(X), complement(Y))
% 22.44/3.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.44/3.15 join(complement(Y), complement(X))
% 22.44/3.15 = { by lemma 43 R->L }
% 22.44/3.15 meet(top, join(complement(Y), complement(X)))
% 22.44/3.15 = { by lemma 44 R->L }
% 22.44/3.15 complement(join(complement(top), meet(Y, X)))
% 22.44/3.15 = { by lemma 15 }
% 22.44/3.15 complement(join(zero, meet(Y, X)))
% 22.44/3.15 = { by lemma 30 }
% 22.44/3.15 complement(meet(Y, X))
% 22.44/3.15 = { by lemma 40 R->L }
% 22.44/3.15 complement(meet(X, Y))
% 22.44/3.15
% 22.44/3.15 Lemma 46: meet(X, meet(X, X)) = X.
% 22.44/3.15 Proof:
% 22.44/3.15 meet(X, meet(X, X))
% 22.44/3.15 = { by lemma 37 R->L }
% 22.44/3.15 meet(X, complement(complement(X)))
% 22.44/3.15 = { by lemma 29 R->L }
% 22.44/3.15 join(meet(X, complement(complement(X))), zero)
% 22.44/3.15 = { by lemma 15 R->L }
% 22.44/3.15 join(meet(X, complement(complement(X))), complement(top))
% 22.44/3.15 = { by axiom 6 (def_top) }
% 22.44/3.15 join(meet(X, complement(complement(X))), complement(join(complement(X), complement(complement(X)))))
% 22.44/3.15 = { by lemma 22 }
% 22.44/3.15 X
% 22.44/3.15
% 22.44/3.15 Lemma 47: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 22.44/3.15 Proof:
% 22.44/3.15 complement(join(X, complement(Y)))
% 22.44/3.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.44/3.15 complement(join(complement(Y), X))
% 22.44/3.15 = { by lemma 46 R->L }
% 22.44/3.15 complement(join(complement(Y), meet(X, meet(X, X))))
% 22.44/3.15 = { by lemma 44 }
% 22.44/3.15 meet(Y, join(complement(X), complement(meet(X, X))))
% 22.44/3.15 = { by lemma 45 }
% 22.44/3.15 meet(Y, complement(meet(X, meet(X, X))))
% 22.44/3.15 = { by lemma 46 }
% 22.44/3.15 meet(Y, complement(X))
% 22.44/3.15
% 22.44/3.15 Lemma 48: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 22.44/3.15 Proof:
% 22.44/3.15 complement(meet(X, complement(Y)))
% 22.44/3.15 = { by lemma 40 }
% 22.44/3.15 complement(meet(complement(Y), X))
% 22.44/3.15 = { by lemma 30 R->L }
% 22.44/3.15 complement(meet(join(zero, complement(Y)), X))
% 22.44/3.15 = { by lemma 45 R->L }
% 22.44/3.15 join(complement(join(zero, complement(Y))), complement(X))
% 22.44/3.15 = { by lemma 41 }
% 22.44/3.15 join(meet(Y, top), complement(X))
% 22.44/3.15 = { by lemma 42 }
% 22.44/3.15 join(Y, complement(X))
% 22.44/3.15
% 22.44/3.15 Lemma 49: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 22.44/3.15 Proof:
% 22.44/3.15 meet(Y, meet(X, Z))
% 22.44/3.15 = { by lemma 40 }
% 22.44/3.15 meet(Y, meet(Z, X))
% 22.44/3.15 = { by lemma 42 R->L }
% 22.44/3.15 meet(meet(Y, meet(Z, X)), top)
% 22.44/3.15 = { by lemma 41 R->L }
% 22.44/3.15 complement(join(zero, complement(meet(Y, meet(Z, X)))))
% 22.44/3.15 = { by lemma 40 }
% 22.44/3.15 complement(join(zero, complement(meet(Y, meet(X, Z)))))
% 22.44/3.15 = { by axiom 11 (maddux4_definiton_of_meet) }
% 22.44/3.15 complement(join(zero, complement(meet(Y, complement(join(complement(X), complement(Z)))))))
% 22.44/3.15 = { by lemma 48 }
% 22.44/3.15 complement(join(zero, join(join(complement(X), complement(Z)), complement(Y))))
% 22.44/3.15 = { by axiom 8 (maddux2_join_associativity) R->L }
% 22.44/3.15 complement(join(zero, join(complement(X), join(complement(Z), complement(Y)))))
% 22.44/3.15 = { by lemma 45 }
% 22.44/3.15 complement(join(zero, join(complement(X), complement(meet(Z, Y)))))
% 22.44/3.15 = { by lemma 45 }
% 22.44/3.15 complement(join(zero, complement(meet(X, meet(Z, Y)))))
% 22.44/3.15 = { by lemma 40 R->L }
% 22.44/3.15 complement(join(zero, complement(meet(X, meet(Y, Z)))))
% 22.44/3.15 = { by lemma 41 }
% 22.44/3.15 meet(meet(X, meet(Y, Z)), top)
% 22.44/3.15 = { by lemma 42 }
% 22.44/3.15 meet(X, meet(Y, Z))
% 22.44/3.15
% 22.44/3.15 Lemma 50: meet(meet(X, Y), Z) = meet(X, meet(Z, Y)).
% 22.44/3.15 Proof:
% 22.44/3.15 meet(meet(X, Y), Z)
% 22.44/3.15 = { by lemma 40 }
% 22.44/3.15 meet(Z, meet(X, Y))
% 22.44/3.15 = { by lemma 49 R->L }
% 22.44/3.15 meet(X, meet(Z, Y))
% 22.44/3.15
% 22.44/3.15 Lemma 51: join(X, complement(meet(X, Y))) = top.
% 22.44/3.15 Proof:
% 22.44/3.15 join(X, complement(meet(X, Y)))
% 22.44/3.15 = { by lemma 40 }
% 22.44/3.15 join(X, complement(meet(Y, X)))
% 22.44/3.15 = { by lemma 45 R->L }
% 22.44/3.15 join(X, join(complement(Y), complement(X)))
% 22.44/3.15 = { by lemma 16 }
% 22.44/3.15 join(complement(Y), top)
% 22.44/3.15 = { by lemma 31 }
% 22.44/3.15 top
% 22.44/3.15
% 22.44/3.15 Lemma 52: join(meet(X, Y), meet(X, complement(Y))) = X.
% 22.44/3.15 Proof:
% 22.44/3.15 join(meet(X, Y), meet(X, complement(Y)))
% 22.44/3.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.44/3.15 join(meet(X, complement(Y)), meet(X, Y))
% 22.44/3.15 = { by axiom 11 (maddux4_definiton_of_meet) }
% 22.44/3.15 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 22.44/3.15 = { by lemma 22 }
% 22.44/3.15 X
% 22.44/3.15
% 22.44/3.15 Lemma 53: join(meet(X, Y), meet(complement(Y), X)) = X.
% 22.44/3.15 Proof:
% 22.44/3.15 join(meet(X, Y), meet(complement(Y), X))
% 22.44/3.15 = { by lemma 40 }
% 22.44/3.15 join(meet(X, Y), meet(X, complement(Y)))
% 22.44/3.15 = { by lemma 52 }
% 22.44/3.15 X
% 22.44/3.15
% 22.44/3.15 Lemma 54: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 22.44/3.15 Proof:
% 22.44/3.15 meet(complement(X), join(X, Y))
% 22.44/3.15 = { by lemma 38 R->L }
% 22.44/3.15 meet(complement(X), join(X, complement(complement(Y))))
% 22.44/3.15 = { by lemma 53 R->L }
% 22.44/3.15 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(complement(X), join(X, complement(complement(Y))))))
% 22.44/3.15 = { by lemma 40 }
% 22.44/3.15 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(join(X, complement(complement(Y))), complement(X))))
% 22.44/3.15 = { by lemma 49 }
% 22.44/3.15 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), meet(complement(Y), complement(X))))
% 22.44/3.15 = { by lemma 47 R->L }
% 22.44/3.15 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), complement(join(X, complement(complement(Y))))))
% 22.44/3.16 = { by axiom 5 (def_zero) R->L }
% 22.44/3.16 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), zero)
% 22.44/3.16 = { by lemma 29 }
% 22.44/3.16 meet(meet(complement(X), join(X, complement(complement(Y)))), Y)
% 22.44/3.16 = { by lemma 50 }
% 22.44/3.16 meet(complement(X), meet(Y, join(X, complement(complement(Y)))))
% 22.44/3.16 = { by lemma 38 }
% 22.44/3.16 meet(complement(X), meet(Y, join(X, Y)))
% 22.44/3.16 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.44/3.16 meet(complement(X), meet(Y, join(Y, X)))
% 22.44/3.16 = { by lemma 39 R->L }
% 22.44/3.16 meet(complement(X), meet(Y, join(Y, meet(X, X))))
% 22.44/3.16 = { by lemma 37 R->L }
% 22.44/3.16 meet(complement(X), meet(Y, join(Y, complement(complement(X)))))
% 22.44/3.16 = { by lemma 48 R->L }
% 22.44/3.16 meet(complement(X), meet(Y, complement(meet(complement(X), complement(Y)))))
% 22.44/3.16 = { by lemma 45 R->L }
% 22.44/3.16 meet(complement(X), meet(Y, join(complement(complement(X)), complement(complement(Y)))))
% 22.44/3.16 = { by lemma 44 R->L }
% 22.44/3.16 meet(complement(X), complement(join(complement(Y), meet(complement(X), complement(Y)))))
% 22.44/3.16 = { by lemma 30 R->L }
% 22.44/3.16 meet(complement(X), join(zero, complement(join(complement(Y), meet(complement(X), complement(Y))))))
% 22.44/3.16 = { by lemma 15 R->L }
% 22.44/3.16 meet(complement(X), join(complement(top), complement(join(complement(Y), meet(complement(X), complement(Y))))))
% 22.44/3.16 = { by lemma 51 R->L }
% 22.44/3.16 meet(complement(X), join(complement(join(complement(Y), complement(meet(complement(Y), complement(X))))), complement(join(complement(Y), meet(complement(X), complement(Y))))))
% 22.44/3.16 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 22.44/3.16 meet(complement(X), join(meet(Y, meet(complement(Y), complement(X))), complement(join(complement(Y), meet(complement(X), complement(Y))))))
% 22.44/3.16 = { by lemma 40 R->L }
% 22.44/3.16 meet(complement(X), join(meet(Y, meet(complement(X), complement(Y))), complement(join(complement(Y), meet(complement(X), complement(Y))))))
% 22.44/3.16 = { by lemma 22 }
% 22.44/3.16 meet(complement(X), Y)
% 22.44/3.16 = { by lemma 40 R->L }
% 22.44/3.16 meet(Y, complement(X))
% 22.44/3.16
% 22.44/3.16 Lemma 55: meet(complement(X), join(Y, X)) = meet(Y, complement(X)).
% 22.44/3.16 Proof:
% 22.44/3.16 meet(complement(X), join(Y, X))
% 22.44/3.16 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.44/3.16 meet(complement(X), join(X, Y))
% 22.44/3.16 = { by lemma 54 }
% 22.44/3.16 meet(Y, complement(X))
% 22.44/3.16
% 22.44/3.16 Lemma 56: complement(join(X, meet(Y, complement(Z)))) = meet(complement(X), join(Z, complement(Y))).
% 22.44/3.16 Proof:
% 22.44/3.16 complement(join(X, meet(Y, complement(Z))))
% 22.44/3.16 = { by lemma 47 R->L }
% 22.44/3.16 complement(join(X, complement(join(Z, complement(Y)))))
% 22.44/3.16 = { by lemma 47 }
% 22.44/3.16 meet(join(Z, complement(Y)), complement(X))
% 22.44/3.16 = { by lemma 40 R->L }
% 22.44/3.16 meet(complement(X), join(Z, complement(Y)))
% 22.44/3.16
% 22.44/3.16 Lemma 57: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 22.44/3.16 Proof:
% 22.44/3.16 join(meet(X, Y), meet(Y, complement(X)))
% 22.44/3.16 = { by lemma 40 }
% 22.44/3.16 join(meet(Y, X), meet(Y, complement(X)))
% 22.44/3.16 = { by lemma 52 }
% 22.44/3.16 Y
% 22.44/3.16
% 22.44/3.16 Goal 1 (goals_1): meet(composition(x0, top), x1) = composition(x0, x1).
% 22.44/3.16 Proof:
% 22.44/3.16 meet(composition(x0, top), x1)
% 22.44/3.16 = { by lemma 22 R->L }
% 22.44/3.16 join(meet(meet(composition(x0, top), x1), complement(x1)), complement(join(complement(meet(composition(x0, top), x1)), complement(x1))))
% 22.44/3.16 = { by lemma 47 R->L }
% 22.44/3.16 join(complement(join(x1, complement(meet(composition(x0, top), x1)))), complement(join(complement(meet(composition(x0, top), x1)), complement(x1))))
% 22.44/3.16 = { by lemma 40 }
% 22.44/3.16 join(complement(join(x1, complement(meet(x1, composition(x0, top))))), complement(join(complement(meet(composition(x0, top), x1)), complement(x1))))
% 22.44/3.16 = { by lemma 51 }
% 22.44/3.16 join(complement(top), complement(join(complement(meet(composition(x0, top), x1)), complement(x1))))
% 22.44/3.16 = { by lemma 15 }
% 22.44/3.16 join(zero, complement(join(complement(meet(composition(x0, top), x1)), complement(x1))))
% 22.44/3.16 = { by lemma 30 }
% 22.44/3.16 complement(join(complement(meet(composition(x0, top), x1)), complement(x1)))
% 22.44/3.16 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 22.44/3.16 meet(meet(composition(x0, top), x1), x1)
% 22.44/3.16 = { by lemma 38 R->L }
% 22.44/3.16 meet(meet(composition(x0, top), x1), complement(complement(x1)))
% 22.44/3.16 = { by lemma 55 R->L }
% 22.44/3.16 meet(complement(complement(x1)), join(meet(composition(x0, top), x1), complement(x1)))
% 22.44/3.16 = { by lemma 22 R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(join(meet(meet(composition(x0, top), x1), complement(join(complement(x1), composition(x0, top)))), complement(join(complement(meet(composition(x0, top), x1)), complement(join(complement(x1), composition(x0, top)))))), complement(x1)))
% 22.44/3.17 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(join(meet(meet(composition(x0, top), x1), complement(join(composition(x0, top), complement(x1)))), complement(join(complement(meet(composition(x0, top), x1)), complement(join(complement(x1), composition(x0, top)))))), complement(x1)))
% 22.44/3.17 = { by lemma 47 R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(join(complement(join(join(composition(x0, top), complement(x1)), complement(meet(composition(x0, top), x1)))), complement(join(complement(meet(composition(x0, top), x1)), complement(join(complement(x1), composition(x0, top)))))), complement(x1)))
% 22.44/3.17 = { by axiom 8 (maddux2_join_associativity) R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(join(complement(join(composition(x0, top), join(complement(x1), complement(meet(composition(x0, top), x1))))), complement(join(complement(meet(composition(x0, top), x1)), complement(join(complement(x1), composition(x0, top)))))), complement(x1)))
% 22.44/3.17 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(join(complement(join(composition(x0, top), join(complement(meet(composition(x0, top), x1)), complement(x1)))), complement(join(complement(meet(composition(x0, top), x1)), complement(join(complement(x1), composition(x0, top)))))), complement(x1)))
% 22.44/3.17 = { by axiom 8 (maddux2_join_associativity) }
% 22.44/3.17 meet(complement(complement(x1)), join(join(complement(join(join(composition(x0, top), complement(meet(composition(x0, top), x1))), complement(x1))), complement(join(complement(meet(composition(x0, top), x1)), complement(join(complement(x1), composition(x0, top)))))), complement(x1)))
% 22.44/3.17 = { by axiom 2 (maddux1_join_commutativity) }
% 22.44/3.17 meet(complement(complement(x1)), join(join(complement(join(complement(x1), join(composition(x0, top), complement(meet(composition(x0, top), x1))))), complement(join(complement(meet(composition(x0, top), x1)), complement(join(complement(x1), composition(x0, top)))))), complement(x1)))
% 22.44/3.17 = { by lemma 51 }
% 22.44/3.17 meet(complement(complement(x1)), join(join(complement(join(complement(x1), top)), complement(join(complement(meet(composition(x0, top), x1)), complement(join(complement(x1), composition(x0, top)))))), complement(x1)))
% 22.44/3.17 = { by lemma 31 }
% 22.44/3.17 meet(complement(complement(x1)), join(join(complement(top), complement(join(complement(meet(composition(x0, top), x1)), complement(join(complement(x1), composition(x0, top)))))), complement(x1)))
% 22.44/3.17 = { by lemma 15 }
% 22.44/3.17 meet(complement(complement(x1)), join(join(zero, complement(join(complement(meet(composition(x0, top), x1)), complement(join(complement(x1), composition(x0, top)))))), complement(x1)))
% 22.44/3.17 = { by lemma 30 }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(join(complement(meet(composition(x0, top), x1)), complement(join(complement(x1), composition(x0, top))))), complement(x1)))
% 22.44/3.17 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(meet(meet(composition(x0, top), x1), join(complement(x1), composition(x0, top))), complement(x1)))
% 22.44/3.17 = { by axiom 2 (maddux1_join_commutativity) }
% 22.44/3.17 meet(complement(complement(x1)), join(meet(meet(composition(x0, top), x1), join(composition(x0, top), complement(x1))), complement(x1)))
% 22.44/3.17 = { by lemma 48 R->L }
% 22.44/3.17 meet(complement(complement(x1)), complement(meet(x1, complement(meet(meet(composition(x0, top), x1), join(composition(x0, top), complement(x1)))))))
% 22.44/3.17 = { by lemma 45 R->L }
% 22.44/3.17 meet(complement(complement(x1)), complement(meet(x1, join(complement(meet(composition(x0, top), x1)), complement(join(composition(x0, top), complement(x1)))))))
% 22.44/3.17 = { by lemma 40 }
% 22.44/3.17 meet(complement(complement(x1)), complement(meet(join(complement(meet(composition(x0, top), x1)), complement(join(composition(x0, top), complement(x1)))), x1)))
% 22.44/3.17 = { by lemma 45 R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(join(complement(meet(composition(x0, top), x1)), complement(join(composition(x0, top), complement(x1))))), complement(x1)))
% 22.44/3.17 = { by lemma 47 }
% 22.44/3.17 meet(complement(complement(x1)), join(meet(join(composition(x0, top), complement(x1)), complement(complement(meet(composition(x0, top), x1)))), complement(x1)))
% 22.44/3.17 = { by axiom 2 (maddux1_join_commutativity) }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), meet(join(composition(x0, top), complement(x1)), complement(complement(meet(composition(x0, top), x1))))))
% 22.44/3.17 = { by lemma 57 R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(join(meet(composition(x0, top), x1), meet(x1, complement(composition(x0, top))))), meet(join(composition(x0, top), complement(x1)), complement(complement(meet(composition(x0, top), x1))))))
% 22.44/3.17 = { by lemma 56 }
% 22.44/3.17 meet(complement(complement(x1)), join(meet(complement(meet(composition(x0, top), x1)), join(composition(x0, top), complement(x1))), meet(join(composition(x0, top), complement(x1)), complement(complement(meet(composition(x0, top), x1))))))
% 22.44/3.17 = { by lemma 57 }
% 22.44/3.17 meet(complement(complement(x1)), join(composition(x0, top), complement(x1)))
% 22.44/3.17 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), composition(x0, top)))
% 22.44/3.17 = { by lemma 36 R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), composition(x0, converse(top))))
% 22.44/3.17 = { by lemma 34 R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), composition(x0, join(x1, converse(complement(converse(x1)))))))
% 22.44/3.17 = { by axiom 1 (converse_idempotence) R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), converse(converse(composition(x0, join(x1, converse(complement(converse(x1)))))))))
% 22.44/3.17 = { by axiom 9 (converse_multiplicativity) }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), converse(composition(converse(join(x1, converse(complement(converse(x1))))), converse(x0)))))
% 22.44/3.17 = { by lemma 32 }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), converse(composition(join(complement(converse(x1)), converse(x1)), converse(x0)))))
% 22.44/3.17 = { by axiom 2 (maddux1_join_commutativity) }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), converse(composition(join(converse(x1), complement(converse(x1))), converse(x0)))))
% 22.44/3.17 = { by axiom 12 (composition_distributivity) }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), converse(join(composition(converse(x1), converse(x0)), composition(complement(converse(x1)), converse(x0))))))
% 22.44/3.17 = { by axiom 9 (converse_multiplicativity) R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), converse(join(converse(composition(x0, x1)), composition(complement(converse(x1)), converse(x0))))))
% 22.44/3.17 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), converse(join(composition(complement(converse(x1)), converse(x0)), converse(composition(x0, x1))))))
% 22.44/3.17 = { by axiom 1 (converse_idempotence) R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), converse(join(composition(converse(converse(complement(converse(x1)))), converse(x0)), converse(composition(x0, x1))))))
% 22.44/3.17 = { by axiom 9 (converse_multiplicativity) R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), converse(join(converse(composition(x0, converse(complement(converse(x1))))), converse(composition(x0, x1))))))
% 22.44/3.17 = { by axiom 7 (converse_additivity) R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), converse(converse(join(composition(x0, converse(complement(converse(x1)))), composition(x0, x1))))))
% 22.44/3.17 = { by axiom 2 (maddux1_join_commutativity) }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), converse(converse(join(composition(x0, x1), composition(x0, converse(complement(converse(x1)))))))))
% 22.44/3.17 = { by axiom 1 (converse_idempotence) }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(x1)))))))
% 22.44/3.17 = { by lemma 22 R->L }
% 22.44/3.17 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, complement(converse(complement(converse(complement(x1)))))), complement(join(complement(x1), complement(converse(complement(converse(complement(x1)))))))))))))))
% 22.44/3.18 = { by lemma 47 R->L }
% 22.44/3.18 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(complement(join(converse(complement(converse(complement(x1)))), complement(x1))), complement(join(complement(x1), complement(converse(complement(converse(complement(x1)))))))))))))))
% 22.44/3.18 = { by axiom 2 (maddux1_join_commutativity) }
% 22.44/3.18 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(complement(join(complement(x1), converse(complement(converse(complement(x1)))))), complement(join(complement(x1), complement(converse(complement(converse(complement(x1)))))))))))))))
% 22.44/3.18 = { by lemma 34 }
% 22.44/3.18 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(complement(converse(top)), complement(join(complement(x1), complement(converse(complement(converse(complement(x1)))))))))))))))
% 22.44/3.18 = { by lemma 36 }
% 22.44/3.18 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(complement(top), complement(join(complement(x1), complement(converse(complement(converse(complement(x1)))))))))))))))
% 22.44/3.18 = { by lemma 15 }
% 22.44/3.18 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(zero, complement(join(complement(x1), complement(converse(complement(converse(complement(x1)))))))))))))))
% 22.44/3.18 = { by lemma 30 }
% 22.44/3.18 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(complement(join(complement(x1), complement(converse(complement(converse(complement(x1))))))))))))))
% 22.44/3.18 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 22.44/3.18 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(meet(x1, converse(complement(converse(complement(x1))))))))))))
% 22.44/3.18 = { by lemma 29 R->L }
% 22.44/3.18 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), zero))))))))
% 22.44/3.18 = { by axiom 5 (def_zero) }
% 22.44/3.18 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), complement(complement(x1)))))))))))
% 22.44/3.18 = { by axiom 1 (converse_idempotence) R->L }
% 22.44/3.18 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), converse(converse(complement(complement(x1)))))))))))))
% 22.44/3.18 = { by lemma 38 R->L }
% 22.44/3.18 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), converse(complement(complement(converse(complement(complement(x1)))))))))))))))
% 22.44/3.18 = { by lemma 53 R->L }
% 22.44/3.18 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), converse(complement(join(meet(complement(converse(complement(complement(x1)))), converse(complement(converse(converse(complement(complement(x1))))))), meet(complement(converse(complement(converse(converse(complement(complement(x1))))))), complement(converse(complement(complement(x1)))))))))))))))))
% 22.44/3.18 = { by lemma 56 }
% 22.44/3.18 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), converse(meet(complement(meet(complement(converse(complement(complement(x1)))), converse(complement(converse(converse(complement(complement(x1)))))))), join(converse(complement(complement(x1))), complement(complement(converse(complement(converse(converse(complement(complement(x1))))))))))))))))))))
% 22.44/3.18 = { by lemma 40 }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), converse(meet(complement(meet(converse(complement(converse(converse(complement(complement(x1)))))), complement(converse(complement(complement(x1)))))), join(converse(complement(complement(x1))), complement(complement(converse(complement(converse(converse(complement(complement(x1))))))))))))))))))))
% 22.44/3.19 = { by lemma 48 }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), converse(meet(join(converse(complement(complement(x1))), complement(converse(complement(converse(converse(complement(complement(x1)))))))), join(converse(complement(complement(x1))), complement(complement(converse(complement(converse(converse(complement(complement(x1))))))))))))))))))))
% 22.44/3.19 = { by lemma 38 }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), converse(meet(join(converse(complement(complement(x1))), complement(converse(complement(converse(converse(complement(complement(x1)))))))), join(converse(complement(complement(x1))), converse(complement(converse(converse(complement(complement(x1))))))))))))))))))
% 22.44/3.19 = { by lemma 40 R->L }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), converse(meet(join(converse(complement(complement(x1))), converse(complement(converse(converse(complement(complement(x1))))))), join(converse(complement(complement(x1))), complement(converse(complement(converse(converse(complement(complement(x1)))))))))))))))))))
% 22.44/3.19 = { by lemma 34 }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), converse(meet(converse(top), join(converse(complement(complement(x1))), complement(converse(complement(converse(converse(complement(complement(x1)))))))))))))))))))
% 22.44/3.19 = { by lemma 36 }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), converse(meet(top, join(converse(complement(complement(x1))), complement(converse(complement(converse(converse(complement(complement(x1)))))))))))))))))))
% 22.44/3.19 = { by lemma 43 }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), converse(join(converse(complement(complement(x1))), complement(converse(complement(converse(converse(complement(complement(x1))))))))))))))))))
% 22.44/3.19 = { by lemma 33 }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), join(complement(complement(x1)), converse(complement(converse(complement(converse(converse(complement(complement(x1))))))))))))))))))
% 22.44/3.19 = { by axiom 1 (converse_idempotence) }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), join(complement(complement(x1)), converse(complement(converse(complement(complement(complement(x1))))))))))))))))
% 22.44/3.19 = { by lemma 30 R->L }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), join(zero, join(complement(complement(x1)), converse(complement(converse(complement(complement(complement(x1)))))))))))))))))
% 22.44/3.19 = { by axiom 8 (maddux2_join_associativity) }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), join(join(zero, complement(complement(x1))), converse(complement(converse(complement(complement(complement(x1))))))))))))))))
% 22.44/3.19 = { by lemma 42 R->L }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), meet(join(join(zero, complement(complement(x1))), converse(complement(converse(complement(complement(complement(x1))))))), top))))))))))
% 22.44/3.19 = { by lemma 50 R->L }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(meet(complement(x1), top), join(join(zero, complement(complement(x1))), converse(complement(converse(complement(complement(complement(x1))))))))))))))))
% 22.44/3.19 = { by lemma 41 R->L }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(join(zero, complement(complement(x1)))), join(join(zero, complement(complement(x1))), converse(complement(converse(complement(complement(complement(x1))))))))))))))))
% 22.44/3.19 = { by lemma 54 }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(converse(complement(converse(complement(complement(complement(x1)))))), complement(join(zero, complement(complement(x1)))))))))))))
% 22.44/3.19 = { by lemma 41 }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(converse(complement(converse(complement(complement(complement(x1)))))), meet(complement(x1), top))))))))))
% 22.44/3.19 = { by lemma 42 }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(converse(complement(converse(complement(complement(complement(x1)))))), complement(x1))))))))))
% 22.44/3.19 = { by lemma 40 R->L }
% 22.44/3.19 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), converse(complement(converse(complement(complement(complement(x1)))))))))))))))
% 22.44/3.19 = { by lemma 38 }
% 22.44/3.20 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(complement(x1), converse(complement(converse(complement(x1)))))))))))))
% 22.44/3.20 = { by lemma 40 }
% 22.44/3.20 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(join(meet(x1, converse(complement(converse(complement(x1))))), meet(converse(complement(converse(complement(x1)))), complement(x1))))))))))
% 22.44/3.20 = { by lemma 57 }
% 22.44/3.20 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(converse(complement(converse(complement(x1)))))))))))
% 22.44/3.20 = { by axiom 1 (converse_idempotence) }
% 22.44/3.20 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(complement(converse(complement(x1)))))))))
% 22.44/3.20 = { by lemma 37 }
% 22.44/3.20 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(meet(converse(complement(x1)), converse(complement(x1))))))))
% 22.44/3.20 = { by lemma 39 }
% 22.44/3.20 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, converse(converse(complement(x1)))))))
% 22.44/3.20 = { by axiom 1 (converse_idempotence) }
% 22.44/3.20 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, x1), composition(x0, complement(x1)))))
% 22.44/3.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.44/3.20 meet(complement(complement(x1)), join(complement(x1), join(composition(x0, complement(x1)), composition(x0, x1))))
% 22.44/3.20 = { by axiom 8 (maddux2_join_associativity) }
% 22.44/3.20 meet(complement(complement(x1)), join(join(complement(x1), composition(x0, complement(x1))), composition(x0, x1)))
% 22.44/3.20 = { by lemma 26 R->L }
% 22.44/3.20 meet(complement(complement(x1)), join(composition(join(x0, one), complement(x1)), composition(x0, x1)))
% 22.44/3.20 = { by axiom 2 (maddux1_join_commutativity) }
% 22.44/3.20 meet(complement(complement(x1)), join(composition(join(one, x0), complement(x1)), composition(x0, x1)))
% 22.44/3.20 = { by lemma 27 }
% 22.44/3.20 meet(complement(complement(x1)), join(composition(one, complement(x1)), composition(x0, x1)))
% 22.44/3.20 = { by lemma 18 }
% 22.44/3.20 meet(complement(complement(x1)), join(complement(x1), composition(x0, x1)))
% 22.44/3.20 = { by axiom 2 (maddux1_join_commutativity) }
% 22.44/3.20 meet(complement(complement(x1)), join(composition(x0, x1), complement(x1)))
% 22.44/3.20 = { by lemma 55 }
% 22.44/3.20 meet(composition(x0, x1), complement(complement(x1)))
% 22.44/3.20 = { by lemma 38 }
% 22.44/3.20 meet(composition(x0, x1), x1)
% 22.44/3.20 = { by axiom 11 (maddux4_definiton_of_meet) }
% 22.44/3.20 complement(join(complement(composition(x0, x1)), complement(x1)))
% 22.44/3.20 = { by lemma 30 R->L }
% 22.44/3.20 join(zero, complement(join(complement(composition(x0, x1)), complement(x1))))
% 22.44/3.20 = { by lemma 15 R->L }
% 22.44/3.20 join(complement(top), complement(join(complement(composition(x0, x1)), complement(x1))))
% 22.44/3.20 = { by lemma 35 R->L }
% 22.44/3.20 join(complement(join(composition(x0, x1), join(complement(composition(x0, x1)), x1))), complement(join(complement(composition(x0, x1)), complement(x1))))
% 22.44/3.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 22.44/3.20 join(complement(join(composition(x0, x1), join(x1, complement(composition(x0, x1))))), complement(join(complement(composition(x0, x1)), complement(x1))))
% 22.44/3.20 = { by axiom 8 (maddux2_join_associativity) }
% 22.44/3.20 join(complement(join(join(composition(x0, x1), x1), complement(composition(x0, x1)))), complement(join(complement(composition(x0, x1)), complement(x1))))
% 22.44/3.20 = { by lemma 18 R->L }
% 22.44/3.20 join(complement(join(join(composition(x0, x1), composition(one, x1)), complement(composition(x0, x1)))), complement(join(complement(composition(x0, x1)), complement(x1))))
% 22.44/3.20 = { by axiom 12 (composition_distributivity) R->L }
% 22.44/3.20 join(complement(join(composition(join(x0, one), x1), complement(composition(x0, x1)))), complement(join(complement(composition(x0, x1)), complement(x1))))
% 22.44/3.20 = { by axiom 2 (maddux1_join_commutativity) }
% 22.44/3.20 join(complement(join(composition(join(one, x0), x1), complement(composition(x0, x1)))), complement(join(complement(composition(x0, x1)), complement(x1))))
% 22.44/3.20 = { by lemma 27 }
% 22.44/3.20 join(complement(join(composition(one, x1), complement(composition(x0, x1)))), complement(join(complement(composition(x0, x1)), complement(x1))))
% 22.44/3.20 = { by lemma 18 }
% 22.44/3.20 join(complement(join(x1, complement(composition(x0, x1)))), complement(join(complement(composition(x0, x1)), complement(x1))))
% 22.44/3.20 = { by lemma 47 }
% 22.44/3.20 join(meet(composition(x0, x1), complement(x1)), complement(join(complement(composition(x0, x1)), complement(x1))))
% 22.44/3.20 = { by lemma 22 }
% 22.44/3.20 composition(x0, x1)
% 22.44/3.20 % SZS output end Proof
% 22.44/3.20
% 22.44/3.20 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------