TSTP Solution File: REL025+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL025+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 : n027.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 13:44:05 EDT 2023
% Result : Theorem 12.87s 2.18s
% Output : Proof 14.17s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : REL025+1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.36 % Computer : n027.cluster.edu
% 0.14/0.36 % Model : x86_64 x86_64
% 0.14/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.36 % Memory : 8042.1875MB
% 0.14/0.36 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.36 % CPULimit : 300
% 0.14/0.36 % WCLimit : 300
% 0.14/0.36 % DateTime : Fri Aug 25 22:44:34 EDT 2023
% 0.14/0.36 % CPUTime :
% 12.87/2.18 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 12.87/2.18
% 12.87/2.18 % SZS status Theorem
% 12.87/2.18
% 14.17/2.25 % SZS output start Proof
% 14.17/2.25 Axiom 1 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 14.17/2.25 Axiom 2 (goals): join(x0, one) = one.
% 14.17/2.25 Axiom 3 (composition_identity): composition(X, one) = X.
% 14.17/2.25 Axiom 4 (converse_idempotence): converse(converse(X)) = X.
% 14.17/2.25 Axiom 5 (def_top): top = join(X, complement(X)).
% 14.17/2.25 Axiom 6 (def_zero): zero = meet(X, complement(X)).
% 14.17/2.25 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 14.17/2.25 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 14.17/2.25 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 14.17/2.25 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 14.17/2.25 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 14.17/2.25 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 14.17/2.25 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 14.17/2.25 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 14.17/2.25
% 14.17/2.25 Lemma 15: complement(top) = zero.
% 14.17/2.25 Proof:
% 14.17/2.25 complement(top)
% 14.17/2.25 = { by axiom 5 (def_top) }
% 14.17/2.25 complement(join(complement(X), complement(complement(X))))
% 14.17/2.25 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.17/2.25 meet(X, complement(X))
% 14.17/2.25 = { by axiom 6 (def_zero) R->L }
% 14.17/2.25 zero
% 14.17/2.25
% 14.17/2.25 Lemma 16: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 14.17/2.25 Proof:
% 14.17/2.25 converse(composition(converse(X), Y))
% 14.17/2.25 = { by axiom 9 (converse_multiplicativity) }
% 14.17/2.25 composition(converse(Y), converse(converse(X)))
% 14.17/2.25 = { by axiom 4 (converse_idempotence) }
% 14.17/2.25 composition(converse(Y), X)
% 14.17/2.25
% 14.17/2.25 Lemma 17: composition(converse(one), X) = X.
% 14.17/2.25 Proof:
% 14.17/2.25 composition(converse(one), X)
% 14.17/2.25 = { by lemma 16 R->L }
% 14.17/2.25 converse(composition(converse(X), one))
% 14.17/2.25 = { by axiom 3 (composition_identity) }
% 14.17/2.25 converse(converse(X))
% 14.17/2.25 = { by axiom 4 (converse_idempotence) }
% 14.17/2.25 X
% 14.17/2.25
% 14.17/2.25 Lemma 18: converse(one) = one.
% 14.17/2.25 Proof:
% 14.17/2.25 converse(one)
% 14.17/2.25 = { by axiom 3 (composition_identity) R->L }
% 14.17/2.25 composition(converse(one), one)
% 14.17/2.25 = { by lemma 17 }
% 14.17/2.25 one
% 14.17/2.25
% 14.17/2.25 Lemma 19: join(X, join(Y, complement(X))) = join(Y, top).
% 14.17/2.25 Proof:
% 14.17/2.25 join(X, join(Y, complement(X)))
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.25 join(X, join(complement(X), Y))
% 14.17/2.25 = { by axiom 8 (maddux2_join_associativity) }
% 14.17/2.25 join(join(X, complement(X)), Y)
% 14.17/2.25 = { by axiom 5 (def_top) R->L }
% 14.17/2.25 join(top, Y)
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) }
% 14.17/2.25 join(Y, top)
% 14.17/2.25
% 14.17/2.25 Lemma 20: composition(one, X) = X.
% 14.17/2.25 Proof:
% 14.17/2.25 composition(one, X)
% 14.17/2.25 = { by lemma 17 R->L }
% 14.17/2.25 composition(converse(one), composition(one, X))
% 14.17/2.25 = { by axiom 10 (composition_associativity) }
% 14.17/2.25 composition(composition(converse(one), one), X)
% 14.17/2.25 = { by axiom 3 (composition_identity) }
% 14.17/2.25 composition(converse(one), X)
% 14.17/2.25 = { by lemma 17 }
% 14.17/2.25 X
% 14.17/2.25
% 14.17/2.25 Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 14.17/2.25 Proof:
% 14.17/2.25 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.25 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 14.17/2.25 = { by axiom 13 (converse_cancellativity) }
% 14.17/2.25 complement(X)
% 14.17/2.25
% 14.17/2.25 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 14.17/2.25 Proof:
% 14.17/2.25 join(complement(X), complement(X))
% 14.17/2.25 = { by lemma 17 R->L }
% 14.17/2.25 join(complement(X), composition(converse(one), complement(X)))
% 14.17/2.25 = { by lemma 20 R->L }
% 14.17/2.25 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 14.17/2.25 = { by lemma 21 }
% 14.17/2.25 complement(X)
% 14.17/2.25
% 14.17/2.25 Lemma 23: join(top, complement(X)) = top.
% 14.17/2.25 Proof:
% 14.17/2.25 join(top, complement(X))
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.25 join(complement(X), top)
% 14.17/2.25 = { by lemma 19 R->L }
% 14.17/2.25 join(X, join(complement(X), complement(X)))
% 14.17/2.25 = { by lemma 22 }
% 14.17/2.25 join(X, complement(X))
% 14.17/2.25 = { by axiom 5 (def_top) R->L }
% 14.17/2.25 top
% 14.17/2.25
% 14.17/2.25 Lemma 24: join(Y, top) = join(X, top).
% 14.17/2.25 Proof:
% 14.17/2.25 join(Y, top)
% 14.17/2.25 = { by lemma 23 R->L }
% 14.17/2.25 join(Y, join(top, complement(Y)))
% 14.17/2.25 = { by lemma 19 }
% 14.17/2.25 join(top, top)
% 14.17/2.25 = { by lemma 19 R->L }
% 14.17/2.25 join(X, join(top, complement(X)))
% 14.17/2.25 = { by lemma 23 }
% 14.17/2.25 join(X, top)
% 14.17/2.25
% 14.17/2.25 Lemma 25: join(x0, join(one, X)) = join(X, one).
% 14.17/2.25 Proof:
% 14.17/2.25 join(x0, join(one, X))
% 14.17/2.25 = { by axiom 8 (maddux2_join_associativity) }
% 14.17/2.25 join(join(x0, one), X)
% 14.17/2.25 = { by axiom 2 (goals) }
% 14.17/2.25 join(one, X)
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) }
% 14.17/2.25 join(X, one)
% 14.17/2.25
% 14.17/2.25 Lemma 26: join(X, top) = top.
% 14.17/2.25 Proof:
% 14.17/2.25 join(X, top)
% 14.17/2.25 = { by lemma 24 }
% 14.17/2.25 join(x0, top)
% 14.17/2.25 = { by axiom 5 (def_top) }
% 14.17/2.25 join(x0, join(one, complement(one)))
% 14.17/2.25 = { by lemma 25 }
% 14.17/2.25 join(complement(one), one)
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) }
% 14.17/2.25 join(one, complement(one))
% 14.17/2.25 = { by axiom 5 (def_top) R->L }
% 14.17/2.25 top
% 14.17/2.25
% 14.17/2.25 Lemma 27: join(top, X) = top.
% 14.17/2.25 Proof:
% 14.17/2.25 join(top, X)
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.25 join(X, top)
% 14.17/2.25 = { by lemma 24 R->L }
% 14.17/2.25 join(Y, top)
% 14.17/2.25 = { by lemma 26 }
% 14.17/2.25 top
% 14.17/2.25
% 14.17/2.25 Lemma 28: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 14.17/2.25 Proof:
% 14.17/2.25 converse(join(X, converse(Y)))
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.25 converse(join(converse(Y), X))
% 14.17/2.25 = { by axiom 7 (converse_additivity) }
% 14.17/2.25 join(converse(converse(Y)), converse(X))
% 14.17/2.25 = { by axiom 4 (converse_idempotence) }
% 14.17/2.25 join(Y, converse(X))
% 14.17/2.25
% 14.17/2.25 Lemma 29: converse(top) = top.
% 14.17/2.25 Proof:
% 14.17/2.25 converse(top)
% 14.17/2.25 = { by lemma 27 R->L }
% 14.17/2.25 converse(join(top, converse(top)))
% 14.17/2.25 = { by lemma 28 }
% 14.17/2.25 join(top, converse(top))
% 14.17/2.25 = { by lemma 27 }
% 14.17/2.25 top
% 14.17/2.25
% 14.17/2.25 Lemma 30: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 14.17/2.25 Proof:
% 14.17/2.25 join(meet(X, Y), complement(join(complement(X), Y)))
% 14.17/2.25 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.17/2.25 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 14.17/2.25 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 14.17/2.25 X
% 14.17/2.25
% 14.17/2.25 Lemma 31: join(zero, meet(X, X)) = X.
% 14.17/2.25 Proof:
% 14.17/2.25 join(zero, meet(X, X))
% 14.17/2.25 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.17/2.25 join(zero, complement(join(complement(X), complement(X))))
% 14.17/2.25 = { by axiom 6 (def_zero) }
% 14.17/2.25 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 14.17/2.25 = { by lemma 30 }
% 14.17/2.25 X
% 14.17/2.25
% 14.17/2.25 Lemma 32: join(zero, complement(complement(X))) = X.
% 14.17/2.25 Proof:
% 14.17/2.25 join(zero, complement(complement(X)))
% 14.17/2.25 = { by axiom 6 (def_zero) }
% 14.17/2.25 join(meet(X, complement(X)), complement(complement(X)))
% 14.17/2.25 = { by lemma 22 R->L }
% 14.17/2.25 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 14.17/2.25 = { by lemma 30 }
% 14.17/2.25 X
% 14.17/2.25
% 14.17/2.25 Lemma 33: join(X, zero) = X.
% 14.17/2.25 Proof:
% 14.17/2.25 join(X, zero)
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.25 join(zero, X)
% 14.17/2.25 = { by lemma 31 R->L }
% 14.17/2.25 join(zero, join(zero, meet(X, X)))
% 14.17/2.25 = { by axiom 8 (maddux2_join_associativity) }
% 14.17/2.25 join(join(zero, zero), meet(X, X))
% 14.17/2.25 = { by lemma 15 R->L }
% 14.17/2.25 join(join(zero, complement(top)), meet(X, X))
% 14.17/2.25 = { by lemma 15 R->L }
% 14.17/2.25 join(join(complement(top), complement(top)), meet(X, X))
% 14.17/2.25 = { by lemma 22 }
% 14.17/2.25 join(complement(top), meet(X, X))
% 14.17/2.25 = { by lemma 15 }
% 14.17/2.25 join(zero, meet(X, X))
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) }
% 14.17/2.25 join(meet(X, X), zero)
% 14.17/2.25 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.17/2.25 join(complement(join(complement(X), complement(X))), zero)
% 14.17/2.25 = { by lemma 22 }
% 14.17/2.25 join(complement(complement(X)), zero)
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) }
% 14.17/2.25 join(zero, complement(complement(X)))
% 14.17/2.25 = { by lemma 32 }
% 14.17/2.25 X
% 14.17/2.25
% 14.17/2.25 Lemma 34: join(zero, X) = X.
% 14.17/2.25 Proof:
% 14.17/2.25 join(zero, X)
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.25 join(X, zero)
% 14.17/2.25 = { by lemma 33 }
% 14.17/2.25 X
% 14.17/2.25
% 14.17/2.25 Lemma 35: complement(complement(X)) = X.
% 14.17/2.25 Proof:
% 14.17/2.25 complement(complement(X))
% 14.17/2.25 = { by lemma 34 R->L }
% 14.17/2.25 join(zero, complement(complement(X)))
% 14.17/2.25 = { by lemma 32 }
% 14.17/2.25 X
% 14.17/2.25
% 14.17/2.25 Lemma 36: meet(Y, X) = meet(X, Y).
% 14.17/2.25 Proof:
% 14.17/2.25 meet(Y, X)
% 14.17/2.25 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.17/2.25 complement(join(complement(Y), complement(X)))
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.25 complement(join(complement(X), complement(Y)))
% 14.17/2.25 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.17/2.25 meet(X, Y)
% 14.17/2.25
% 14.17/2.25 Lemma 37: complement(join(zero, complement(X))) = meet(X, top).
% 14.17/2.25 Proof:
% 14.17/2.25 complement(join(zero, complement(X)))
% 14.17/2.25 = { by lemma 15 R->L }
% 14.17/2.25 complement(join(complement(top), complement(X)))
% 14.17/2.25 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.17/2.25 meet(top, X)
% 14.17/2.25 = { by lemma 36 R->L }
% 14.17/2.25 meet(X, top)
% 14.17/2.25
% 14.17/2.25 Lemma 38: meet(X, top) = X.
% 14.17/2.25 Proof:
% 14.17/2.25 meet(X, top)
% 14.17/2.25 = { by lemma 37 R->L }
% 14.17/2.25 complement(join(zero, complement(X)))
% 14.17/2.25 = { by lemma 34 }
% 14.17/2.25 complement(complement(X))
% 14.17/2.25 = { by lemma 35 }
% 14.17/2.25 X
% 14.17/2.25
% 14.17/2.25 Lemma 39: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 14.17/2.25 Proof:
% 14.17/2.25 meet(X, join(complement(Y), complement(Z)))
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.25 meet(X, join(complement(Z), complement(Y)))
% 14.17/2.25 = { by lemma 36 }
% 14.17/2.25 meet(join(complement(Z), complement(Y)), X)
% 14.17/2.25 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.17/2.25 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 14.17/2.25 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.17/2.25 complement(join(meet(Z, Y), complement(X)))
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) }
% 14.17/2.25 complement(join(complement(X), meet(Z, Y)))
% 14.17/2.25 = { by lemma 36 R->L }
% 14.17/2.25 complement(join(complement(X), meet(Y, Z)))
% 14.17/2.25
% 14.17/2.25 Lemma 40: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 14.17/2.25 Proof:
% 14.17/2.25 complement(join(X, complement(Y)))
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.25 complement(join(complement(Y), X))
% 14.17/2.25 = { by lemma 38 R->L }
% 14.17/2.25 complement(join(complement(Y), meet(X, top)))
% 14.17/2.25 = { by lemma 36 R->L }
% 14.17/2.25 complement(join(complement(Y), meet(top, X)))
% 14.17/2.25 = { by lemma 39 R->L }
% 14.17/2.25 meet(Y, join(complement(top), complement(X)))
% 14.17/2.25 = { by lemma 15 }
% 14.17/2.25 meet(Y, join(zero, complement(X)))
% 14.17/2.25 = { by lemma 34 }
% 14.17/2.25 meet(Y, complement(X))
% 14.17/2.25
% 14.17/2.25 Lemma 41: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 14.17/2.25 Proof:
% 14.17/2.25 complement(meet(X, complement(Y)))
% 14.17/2.25 = { by lemma 34 R->L }
% 14.17/2.25 complement(join(zero, meet(X, complement(Y))))
% 14.17/2.25 = { by lemma 40 R->L }
% 14.17/2.25 complement(join(zero, complement(join(Y, complement(X)))))
% 14.17/2.25 = { by lemma 37 }
% 14.17/2.25 meet(join(Y, complement(X)), top)
% 14.17/2.25 = { by lemma 38 }
% 14.17/2.25 join(Y, complement(X))
% 14.17/2.25
% 14.17/2.25 Lemma 42: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 14.17/2.25 Proof:
% 14.17/2.25 complement(join(complement(X), Y))
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.25 complement(join(Y, complement(X)))
% 14.17/2.25 = { by lemma 40 }
% 14.17/2.25 meet(X, complement(Y))
% 14.17/2.25
% 14.17/2.25 Lemma 43: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 14.17/2.25 Proof:
% 14.17/2.25 complement(meet(complement(X), Y))
% 14.17/2.25 = { by lemma 36 }
% 14.17/2.25 complement(meet(Y, complement(X)))
% 14.17/2.25 = { by lemma 41 }
% 14.17/2.25 join(X, complement(Y))
% 14.17/2.25
% 14.17/2.25 Lemma 44: meet(X, join(X, complement(Y))) = X.
% 14.17/2.25 Proof:
% 14.17/2.25 meet(X, join(X, complement(Y)))
% 14.17/2.25 = { by lemma 33 R->L }
% 14.17/2.25 join(meet(X, join(X, complement(Y))), zero)
% 14.17/2.25 = { by lemma 15 R->L }
% 14.17/2.25 join(meet(X, join(X, complement(Y))), complement(top))
% 14.17/2.25 = { by lemma 43 R->L }
% 14.17/2.25 join(meet(X, complement(meet(complement(X), Y))), complement(top))
% 14.17/2.25 = { by lemma 26 R->L }
% 14.17/2.25 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(Y), top)))
% 14.17/2.25 = { by lemma 19 R->L }
% 14.17/2.25 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), join(complement(Y), complement(complement(X))))))
% 14.17/2.25 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.25 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), join(complement(complement(X)), complement(Y)))))
% 14.17/2.25 = { by lemma 31 R->L }
% 14.17/2.26 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), join(zero, meet(join(complement(complement(X)), complement(Y)), join(complement(complement(X)), complement(Y)))))))
% 14.17/2.26 = { by lemma 39 }
% 14.17/2.26 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), join(zero, complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y)))))))
% 14.17/2.26 = { by lemma 34 }
% 14.17/2.26 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y))))))
% 14.17/2.26 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.17/2.26 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(join(meet(complement(X), Y), meet(complement(X), Y))))))
% 14.17/2.26 = { by lemma 36 }
% 14.17/2.26 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(join(meet(Y, complement(X)), meet(complement(X), Y))))))
% 14.17/2.26 = { by lemma 36 }
% 14.17/2.26 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(join(meet(Y, complement(X)), meet(Y, complement(X)))))))
% 14.17/2.26 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.17/2.26 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(join(meet(Y, complement(X)), complement(join(complement(Y), complement(complement(X)))))))))
% 14.17/2.26 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.17/2.26 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(join(complement(join(complement(Y), complement(complement(X)))), complement(join(complement(Y), complement(complement(X)))))))))
% 14.17/2.26 = { by lemma 22 }
% 14.17/2.26 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(complement(join(complement(Y), complement(complement(X))))))))
% 14.17/2.26 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.17/2.26 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(Y, complement(X))))))
% 14.17/2.26 = { by lemma 36 R->L }
% 14.17/2.26 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(complement(X), Y)))))
% 14.17/2.26 = { by lemma 30 }
% 14.17/2.26 X
% 14.17/2.26
% 14.17/2.26 Lemma 45: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 14.17/2.26 Proof:
% 14.17/2.26 meet(X, complement(meet(X, Y)))
% 14.17/2.26 = { by lemma 42 R->L }
% 14.17/2.26 complement(join(complement(X), meet(X, Y)))
% 14.17/2.26 = { by lemma 36 }
% 14.17/2.26 complement(join(complement(X), meet(Y, X)))
% 14.17/2.26 = { by lemma 35 R->L }
% 14.17/2.26 complement(join(complement(complement(complement(X))), meet(Y, X)))
% 14.17/2.26 = { by lemma 44 R->L }
% 14.17/2.26 complement(join(complement(meet(complement(complement(X)), join(complement(complement(X)), complement(Y)))), meet(Y, X)))
% 14.17/2.26 = { by lemma 43 }
% 14.17/2.26 complement(join(join(complement(X), complement(join(complement(complement(X)), complement(Y)))), meet(Y, X)))
% 14.17/2.26 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.17/2.26 complement(join(join(complement(X), meet(complement(X), Y)), meet(Y, X)))
% 14.17/2.26 = { by axiom 8 (maddux2_join_associativity) R->L }
% 14.17/2.26 complement(join(complement(X), join(meet(complement(X), Y), meet(Y, X))))
% 14.17/2.26 = { by axiom 1 (maddux1_join_commutativity) }
% 14.17/2.26 complement(join(complement(X), join(meet(Y, X), meet(complement(X), Y))))
% 14.17/2.26 = { by lemma 36 }
% 14.17/2.26 complement(join(complement(X), join(meet(Y, X), meet(Y, complement(X)))))
% 14.17/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.26 complement(join(complement(X), join(meet(Y, complement(X)), meet(Y, X))))
% 14.17/2.26 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.17/2.26 complement(join(complement(X), join(meet(Y, complement(X)), complement(join(complement(Y), complement(X))))))
% 14.17/2.26 = { by lemma 30 }
% 14.17/2.26 complement(join(complement(X), Y))
% 14.17/2.26 = { by axiom 1 (maddux1_join_commutativity) }
% 14.17/2.26 complement(join(Y, complement(X)))
% 14.17/2.26 = { by lemma 40 }
% 14.17/2.26 meet(X, complement(Y))
% 14.17/2.26
% 14.17/2.26 Lemma 46: meet(X, join(complement(X), Y)) = meet(X, Y).
% 14.17/2.26 Proof:
% 14.17/2.26 meet(X, join(complement(X), Y))
% 14.17/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.26 meet(X, join(Y, complement(X)))
% 14.17/2.26 = { by lemma 41 R->L }
% 14.17/2.26 meet(X, complement(meet(X, complement(Y))))
% 14.17/2.26 = { by lemma 45 }
% 14.17/2.26 meet(X, complement(complement(Y)))
% 14.17/2.26 = { by lemma 35 }
% 14.17/2.26 meet(X, Y)
% 14.17/2.26
% 14.17/2.26 Lemma 47: composition(join(X, one), Y) = join(Y, composition(X, Y)).
% 14.17/2.26 Proof:
% 14.17/2.26 composition(join(X, one), Y)
% 14.17/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.26 composition(join(one, X), Y)
% 14.17/2.26 = { by axiom 12 (composition_distributivity) }
% 14.17/2.26 join(composition(one, Y), composition(X, Y))
% 14.17/2.26 = { by lemma 20 }
% 14.17/2.26 join(Y, composition(X, Y))
% 14.17/2.26
% 14.17/2.26 Lemma 48: join(X, composition(x0, complement(X))) = join(X, composition(x0, top)).
% 14.17/2.26 Proof:
% 14.17/2.26 join(X, composition(x0, complement(X)))
% 14.17/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.26 join(composition(x0, complement(X)), X)
% 14.17/2.26 = { by lemma 20 R->L }
% 14.17/2.26 join(composition(x0, complement(X)), composition(one, X))
% 14.17/2.26 = { by axiom 2 (goals) R->L }
% 14.17/2.26 join(composition(x0, complement(X)), composition(join(x0, one), X))
% 14.17/2.26 = { by lemma 47 }
% 14.17/2.26 join(composition(x0, complement(X)), join(X, composition(x0, X)))
% 14.17/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.26 join(join(X, composition(x0, X)), composition(x0, complement(X)))
% 14.17/2.26 = { by axiom 8 (maddux2_join_associativity) R->L }
% 14.17/2.26 join(X, join(composition(x0, X), composition(x0, complement(X))))
% 14.17/2.26 = { by axiom 4 (converse_idempotence) R->L }
% 14.17/2.26 join(X, converse(converse(join(composition(x0, X), composition(x0, complement(X))))))
% 14.17/2.26 = { by axiom 7 (converse_additivity) }
% 14.17/2.26 join(X, converse(join(converse(composition(x0, X)), converse(composition(x0, complement(X))))))
% 14.17/2.26 = { by axiom 9 (converse_multiplicativity) }
% 14.17/2.26 join(X, converse(join(composition(converse(X), converse(x0)), converse(composition(x0, complement(X))))))
% 14.17/2.26 = { by axiom 9 (converse_multiplicativity) }
% 14.17/2.26 join(X, converse(join(composition(converse(X), converse(x0)), composition(converse(complement(X)), converse(x0)))))
% 14.17/2.26 = { by axiom 12 (composition_distributivity) R->L }
% 14.17/2.26 join(X, converse(composition(join(converse(X), converse(complement(X))), converse(x0))))
% 14.17/2.26 = { by axiom 7 (converse_additivity) R->L }
% 14.17/2.26 join(X, converse(composition(converse(join(X, complement(X))), converse(x0))))
% 14.17/2.26 = { by lemma 16 }
% 14.17/2.26 join(X, composition(converse(converse(x0)), join(X, complement(X))))
% 14.17/2.26 = { by axiom 4 (converse_idempotence) }
% 14.17/2.26 join(X, composition(x0, join(X, complement(X))))
% 14.17/2.26 = { by axiom 5 (def_top) R->L }
% 14.17/2.26 join(X, composition(x0, top))
% 14.17/2.26
% 14.17/2.26 Lemma 49: join(X, composition(x0, X)) = X.
% 14.17/2.26 Proof:
% 14.17/2.26 join(X, composition(x0, X))
% 14.17/2.26 = { by lemma 47 R->L }
% 14.17/2.26 composition(join(x0, one), X)
% 14.17/2.26 = { by axiom 2 (goals) }
% 14.17/2.26 composition(one, X)
% 14.17/2.26 = { by lemma 20 }
% 14.17/2.26 X
% 14.17/2.26
% 14.17/2.26 Lemma 50: meet(X, join(X, Y)) = X.
% 14.17/2.26 Proof:
% 14.17/2.26 meet(X, join(X, Y))
% 14.17/2.26 = { by lemma 38 R->L }
% 14.17/2.26 meet(X, join(X, meet(Y, top)))
% 14.17/2.26 = { by lemma 37 R->L }
% 14.17/2.26 meet(X, join(X, complement(join(zero, complement(Y)))))
% 14.17/2.26 = { by lemma 44 }
% 14.17/2.26 X
% 14.17/2.26
% 14.17/2.26 Lemma 51: meet(X, composition(x0, top)) = composition(x0, X).
% 14.17/2.26 Proof:
% 14.17/2.26 meet(X, composition(x0, top))
% 14.17/2.26 = { by lemma 46 R->L }
% 14.17/2.26 meet(X, join(complement(X), composition(x0, top)))
% 14.17/2.26 = { by lemma 48 R->L }
% 14.17/2.26 meet(X, join(complement(X), composition(x0, complement(complement(X)))))
% 14.17/2.26 = { by lemma 46 }
% 14.17/2.26 meet(X, composition(x0, complement(complement(X))))
% 14.17/2.26 = { by lemma 35 }
% 14.17/2.26 meet(X, composition(x0, X))
% 14.17/2.26 = { by lemma 36 }
% 14.17/2.26 meet(composition(x0, X), X)
% 14.17/2.26 = { by lemma 49 R->L }
% 14.17/2.26 meet(composition(x0, X), join(X, composition(x0, X)))
% 14.17/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.26 meet(composition(x0, X), join(composition(x0, X), X))
% 14.17/2.26 = { by lemma 50 }
% 14.17/2.26 composition(x0, X)
% 14.17/2.26
% 14.17/2.26 Lemma 52: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 14.17/2.26 Proof:
% 14.17/2.26 meet(complement(X), complement(Y))
% 14.17/2.26 = { by lemma 36 }
% 14.17/2.26 meet(complement(Y), complement(X))
% 14.17/2.26 = { by lemma 34 R->L }
% 14.17/2.26 meet(join(zero, complement(Y)), complement(X))
% 14.17/2.26 = { by lemma 40 R->L }
% 14.17/2.26 complement(join(X, complement(join(zero, complement(Y)))))
% 14.17/2.26 = { by lemma 37 }
% 14.17/2.26 complement(join(X, meet(Y, top)))
% 14.17/2.26 = { by lemma 38 }
% 14.17/2.26 complement(join(X, Y))
% 14.17/2.26
% 14.17/2.26 Lemma 53: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 14.17/2.26 Proof:
% 14.17/2.26 meet(complement(X), join(X, Y))
% 14.17/2.26 = { by lemma 36 }
% 14.17/2.26 meet(join(X, Y), complement(X))
% 14.17/2.26 = { by lemma 40 R->L }
% 14.17/2.26 complement(join(X, complement(join(X, Y))))
% 14.17/2.26 = { by lemma 52 R->L }
% 14.17/2.26 meet(complement(X), complement(complement(join(X, Y))))
% 14.17/2.26 = { by lemma 52 R->L }
% 14.17/2.26 meet(complement(X), complement(meet(complement(X), complement(Y))))
% 14.17/2.26 = { by lemma 45 }
% 14.17/2.26 meet(complement(X), complement(complement(Y)))
% 14.17/2.26 = { by lemma 52 }
% 14.17/2.26 complement(join(X, complement(Y)))
% 14.17/2.26 = { by lemma 40 }
% 14.17/2.26 meet(Y, complement(X))
% 14.17/2.26
% 14.17/2.26 Lemma 54: join(one, complement(converse(x0))) = top.
% 14.17/2.26 Proof:
% 14.17/2.26 join(one, complement(converse(x0)))
% 14.17/2.26 = { by lemma 18 R->L }
% 14.17/2.26 join(converse(one), complement(converse(x0)))
% 14.17/2.26 = { by axiom 2 (goals) R->L }
% 14.17/2.26 join(converse(join(x0, one)), complement(converse(x0)))
% 14.17/2.26 = { by axiom 7 (converse_additivity) }
% 14.17/2.26 join(join(converse(x0), converse(one)), complement(converse(x0)))
% 14.17/2.26 = { by lemma 18 }
% 14.17/2.26 join(join(converse(x0), one), complement(converse(x0)))
% 14.17/2.26 = { by axiom 1 (maddux1_join_commutativity) }
% 14.17/2.26 join(join(one, converse(x0)), complement(converse(x0)))
% 14.17/2.26 = { by axiom 8 (maddux2_join_associativity) R->L }
% 14.17/2.26 join(one, join(converse(x0), complement(converse(x0))))
% 14.17/2.26 = { by axiom 5 (def_top) R->L }
% 14.17/2.26 join(one, top)
% 14.17/2.26 = { by lemma 26 }
% 14.17/2.26 top
% 14.17/2.26
% 14.17/2.26 Lemma 55: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 14.17/2.26 Proof:
% 14.17/2.26 converse(composition(X, converse(Y)))
% 14.17/2.26 = { by axiom 9 (converse_multiplicativity) }
% 14.17/2.26 composition(converse(converse(Y)), converse(X))
% 14.17/2.26 = { by axiom 4 (converse_idempotence) }
% 14.17/2.26 composition(Y, converse(X))
% 14.17/2.26
% 14.17/2.26 Lemma 56: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 14.17/2.26 Proof:
% 14.17/2.26 converse(join(converse(X), Y))
% 14.17/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.26 converse(join(Y, converse(X)))
% 14.17/2.26 = { by lemma 28 }
% 14.17/2.26 join(X, converse(Y))
% 14.17/2.26
% 14.17/2.26 Lemma 57: meet(complement(X), join(Y, complement(Z))) = complement(join(X, meet(Z, complement(Y)))).
% 14.17/2.26 Proof:
% 14.17/2.26 meet(complement(X), join(Y, complement(Z)))
% 14.17/2.26 = { by lemma 36 }
% 14.17/2.26 meet(join(Y, complement(Z)), complement(X))
% 14.17/2.26 = { by lemma 40 R->L }
% 14.17/2.26 complement(join(X, complement(join(Y, complement(Z)))))
% 14.17/2.26 = { by lemma 40 }
% 14.17/2.26 complement(join(X, meet(Z, complement(Y))))
% 14.17/2.26
% 14.17/2.26 Lemma 58: composition(join(one, Y), X) = join(X, composition(Y, X)).
% 14.17/2.26 Proof:
% 14.17/2.26 composition(join(one, Y), X)
% 14.17/2.26 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.26 composition(join(Y, one), X)
% 14.17/2.26 = { by lemma 47 }
% 14.17/2.26 join(X, composition(Y, X))
% 14.17/2.26
% 14.17/2.26 Lemma 59: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 14.17/2.26 Proof:
% 14.17/2.26 meet(Y, meet(Z, X))
% 14.17/2.26 = { by lemma 38 R->L }
% 14.17/2.26 meet(meet(Y, top), meet(Z, X))
% 14.17/2.26 = { by lemma 37 R->L }
% 14.17/2.26 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 14.17/2.26 = { by lemma 36 }
% 14.17/2.26 meet(complement(join(zero, complement(Y))), meet(X, Z))
% 14.17/2.26 = { by lemma 36 }
% 14.17/2.26 meet(meet(X, Z), complement(join(zero, complement(Y))))
% 14.17/2.26 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.17/2.26 meet(complement(join(complement(X), complement(Z))), complement(join(zero, complement(Y))))
% 14.17/2.26 = { by lemma 52 }
% 14.17/2.26 complement(join(join(complement(X), complement(Z)), join(zero, complement(Y))))
% 14.17/2.26 = { by axiom 8 (maddux2_join_associativity) R->L }
% 14.17/2.26 complement(join(complement(X), join(complement(Z), join(zero, complement(Y)))))
% 14.17/2.26 = { by lemma 42 }
% 14.17/2.26 meet(X, complement(join(complement(Z), join(zero, complement(Y)))))
% 14.17/2.26 = { by lemma 42 }
% 14.17/2.26 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 14.17/2.26 = { by lemma 37 }
% 14.17/2.26 meet(X, meet(Z, meet(Y, top)))
% 14.17/2.26 = { by lemma 38 }
% 14.17/2.26 meet(X, meet(Z, Y))
% 14.17/2.26 = { by lemma 36 R->L }
% 14.17/2.26 meet(X, meet(Y, Z))
% 14.17/2.26
% 14.17/2.26 Lemma 60: composition(x0, converse(x0)) = x0.
% 14.17/2.26 Proof:
% 14.17/2.26 composition(x0, converse(x0))
% 14.17/2.26 = { by lemma 51 R->L }
% 14.17/2.26 meet(converse(x0), composition(x0, top))
% 14.17/2.26 = { by lemma 36 }
% 14.17/2.26 meet(composition(x0, top), converse(x0))
% 14.17/2.26 = { by lemma 35 R->L }
% 14.17/2.26 meet(composition(x0, top), complement(complement(converse(x0))))
% 14.17/2.26 = { by lemma 53 R->L }
% 14.17/2.26 meet(complement(complement(converse(x0))), join(complement(converse(x0)), composition(x0, top)))
% 14.17/2.26 = { by lemma 54 R->L }
% 14.17/2.26 meet(complement(complement(converse(x0))), join(complement(converse(x0)), composition(x0, join(one, complement(converse(x0))))))
% 14.17/2.26 = { by axiom 4 (converse_idempotence) R->L }
% 14.17/2.26 meet(complement(complement(converse(x0))), join(complement(converse(x0)), composition(x0, join(one, converse(converse(complement(converse(x0))))))))
% 14.17/2.26 = { by lemma 18 R->L }
% 14.17/2.26 meet(complement(complement(converse(x0))), join(complement(converse(x0)), composition(x0, join(converse(one), converse(converse(complement(converse(x0))))))))
% 14.17/2.26 = { by axiom 7 (converse_additivity) R->L }
% 14.17/2.26 meet(complement(complement(converse(x0))), join(complement(converse(x0)), composition(x0, converse(join(one, converse(complement(converse(x0))))))))
% 14.17/2.26 = { by axiom 1 (maddux1_join_commutativity) }
% 14.17/2.26 meet(complement(complement(converse(x0))), join(complement(converse(x0)), composition(x0, converse(join(converse(complement(converse(x0))), one)))))
% 14.17/2.27 = { by lemma 55 R->L }
% 14.17/2.27 meet(complement(complement(converse(x0))), join(complement(converse(x0)), converse(composition(join(converse(complement(converse(x0))), one), converse(x0)))))
% 14.17/2.27 = { by lemma 47 }
% 14.17/2.27 meet(complement(complement(converse(x0))), join(complement(converse(x0)), converse(join(converse(x0), composition(converse(complement(converse(x0))), converse(x0))))))
% 14.17/2.27 = { by lemma 56 }
% 14.17/2.27 meet(complement(complement(converse(x0))), join(complement(converse(x0)), join(x0, converse(composition(converse(complement(converse(x0))), converse(x0))))))
% 14.17/2.27 = { by lemma 55 }
% 14.17/2.27 meet(complement(complement(converse(x0))), join(complement(converse(x0)), join(x0, composition(x0, converse(converse(complement(converse(x0))))))))
% 14.17/2.27 = { by axiom 4 (converse_idempotence) }
% 14.17/2.27 meet(complement(complement(converse(x0))), join(complement(converse(x0)), join(x0, composition(x0, complement(converse(x0))))))
% 14.17/2.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.27 meet(complement(complement(converse(x0))), join(complement(converse(x0)), join(composition(x0, complement(converse(x0))), x0)))
% 14.17/2.27 = { by axiom 8 (maddux2_join_associativity) }
% 14.17/2.27 meet(complement(complement(converse(x0))), join(join(complement(converse(x0)), composition(x0, complement(converse(x0)))), x0))
% 14.17/2.27 = { by lemma 49 }
% 14.17/2.27 meet(complement(complement(converse(x0))), join(complement(converse(x0)), x0))
% 14.17/2.27 = { by axiom 1 (maddux1_join_commutativity) }
% 14.17/2.27 meet(complement(complement(converse(x0))), join(x0, complement(converse(x0))))
% 14.17/2.27 = { by lemma 57 }
% 14.17/2.27 complement(join(complement(converse(x0)), meet(converse(x0), complement(x0))))
% 14.17/2.27 = { by lemma 42 }
% 14.17/2.27 meet(converse(x0), complement(meet(converse(x0), complement(x0))))
% 14.17/2.27 = { by lemma 45 }
% 14.17/2.27 meet(converse(x0), complement(complement(x0)))
% 14.17/2.27 = { by lemma 35 }
% 14.17/2.27 meet(converse(x0), x0)
% 14.17/2.27 = { by lemma 36 R->L }
% 14.17/2.27 meet(x0, converse(x0))
% 14.17/2.27 = { by lemma 30 R->L }
% 14.17/2.27 meet(x0, join(meet(converse(x0), complement(one)), complement(join(complement(converse(x0)), complement(one)))))
% 14.17/2.27 = { by lemma 40 R->L }
% 14.17/2.27 meet(x0, join(complement(join(one, complement(converse(x0)))), complement(join(complement(converse(x0)), complement(one)))))
% 14.17/2.27 = { by lemma 54 }
% 14.17/2.27 meet(x0, join(complement(top), complement(join(complement(converse(x0)), complement(one)))))
% 14.17/2.27 = { by lemma 15 }
% 14.17/2.27 meet(x0, join(zero, complement(join(complement(converse(x0)), complement(one)))))
% 14.17/2.27 = { by lemma 34 }
% 14.17/2.27 meet(x0, complement(join(complement(converse(x0)), complement(one))))
% 14.17/2.27 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 14.17/2.27 meet(x0, meet(converse(x0), one))
% 14.17/2.27 = { by lemma 36 R->L }
% 14.17/2.27 meet(x0, meet(one, converse(x0)))
% 14.17/2.27 = { by lemma 35 R->L }
% 14.17/2.27 meet(x0, meet(one, complement(complement(converse(x0)))))
% 14.17/2.27 = { by lemma 45 R->L }
% 14.17/2.27 meet(x0, meet(one, complement(meet(one, complement(converse(x0))))))
% 14.17/2.27 = { by lemma 42 R->L }
% 14.17/2.27 meet(x0, complement(join(complement(one), meet(one, complement(converse(x0))))))
% 14.17/2.27 = { by lemma 57 R->L }
% 14.17/2.27 meet(x0, meet(complement(complement(one)), join(converse(x0), complement(one))))
% 14.17/2.27 = { by axiom 1 (maddux1_join_commutativity) R->L }
% 14.17/2.27 meet(x0, meet(complement(complement(one)), join(complement(one), converse(x0))))
% 14.17/2.27 = { by axiom 4 (converse_idempotence) R->L }
% 14.17/2.27 meet(x0, meet(complement(complement(one)), join(converse(converse(complement(one))), converse(x0))))
% 14.17/2.27 = { by lemma 49 R->L }
% 14.17/2.27 meet(x0, meet(complement(complement(one)), join(converse(join(converse(complement(one)), composition(x0, converse(complement(one))))), converse(x0))))
% 14.17/2.27 = { by lemma 56 }
% 14.17/2.27 meet(x0, meet(complement(complement(one)), join(join(complement(one), converse(composition(x0, converse(complement(one))))), converse(x0))))
% 14.17/2.27 = { by axiom 9 (converse_multiplicativity) }
% 14.17/2.27 meet(x0, meet(complement(complement(one)), join(join(complement(one), composition(converse(converse(complement(one))), converse(x0))), converse(x0))))
% 14.17/2.27 = { by axiom 4 (converse_idempotence) }
% 14.17/2.27 meet(x0, meet(complement(complement(one)), join(join(complement(one), composition(complement(one), converse(x0))), converse(x0))))
% 14.17/2.27 = { by axiom 8 (maddux2_join_associativity) R->L }
% 14.17/2.27 meet(x0, meet(complement(complement(one)), join(complement(one), join(composition(complement(one), converse(x0)), converse(x0)))))
% 14.17/2.27 = { by axiom 1 (maddux1_join_commutativity) }
% 14.17/2.27 meet(x0, meet(complement(complement(one)), join(complement(one), join(converse(x0), composition(complement(one), converse(x0))))))
% 14.17/2.27 = { by lemma 58 R->L }
% 14.17/2.27 meet(x0, meet(complement(complement(one)), join(complement(one), composition(join(one, complement(one)), converse(x0)))))
% 14.17/2.27 = { by axiom 5 (def_top) R->L }
% 14.17/2.27 meet(x0, meet(complement(complement(one)), join(complement(one), composition(top, converse(x0)))))
% 14.17/2.27 = { by lemma 53 }
% 14.17/2.27 meet(x0, meet(composition(top, converse(x0)), complement(complement(one))))
% 14.17/2.27 = { by lemma 35 }
% 14.17/2.27 meet(x0, meet(composition(top, converse(x0)), one))
% 14.17/2.27 = { by lemma 36 R->L }
% 14.17/2.27 meet(x0, meet(one, composition(top, converse(x0))))
% 14.17/2.27 = { by lemma 59 }
% 14.17/2.27 meet(composition(top, converse(x0)), meet(x0, one))
% 14.17/2.27 = { by axiom 11 (maddux4_definiton_of_meet) }
% 14.17/2.27 meet(composition(top, converse(x0)), complement(join(complement(x0), complement(one))))
% 14.17/2.27 = { by lemma 34 R->L }
% 14.17/2.27 meet(composition(top, converse(x0)), join(zero, complement(join(complement(x0), complement(one)))))
% 14.17/2.27 = { by lemma 15 R->L }
% 14.17/2.27 meet(composition(top, converse(x0)), join(complement(top), complement(join(complement(x0), complement(one)))))
% 14.17/2.27 = { by lemma 26 R->L }
% 14.17/2.27 meet(composition(top, converse(x0)), join(complement(join(one, top)), complement(join(complement(x0), complement(one)))))
% 14.17/2.27 = { by lemma 19 R->L }
% 14.17/2.27 meet(composition(top, converse(x0)), join(complement(join(x0, join(one, complement(x0)))), complement(join(complement(x0), complement(one)))))
% 14.17/2.27 = { by lemma 25 }
% 14.17/2.27 meet(composition(top, converse(x0)), join(complement(join(complement(x0), one)), complement(join(complement(x0), complement(one)))))
% 14.17/2.27 = { by axiom 1 (maddux1_join_commutativity) }
% 14.17/2.27 meet(composition(top, converse(x0)), join(complement(join(one, complement(x0))), complement(join(complement(x0), complement(one)))))
% 14.17/2.27 = { by lemma 40 }
% 14.17/2.27 meet(composition(top, converse(x0)), join(meet(x0, complement(one)), complement(join(complement(x0), complement(one)))))
% 14.17/2.27 = { by lemma 30 }
% 14.17/2.27 meet(composition(top, converse(x0)), x0)
% 14.17/2.27 = { by lemma 36 R->L }
% 14.17/2.27 meet(x0, composition(top, converse(x0)))
% 14.17/2.27 = { by lemma 29 R->L }
% 14.17/2.27 meet(x0, composition(converse(top), converse(x0)))
% 14.17/2.27 = { by axiom 9 (converse_multiplicativity) R->L }
% 14.17/2.27 meet(x0, converse(composition(x0, top)))
% 14.17/2.27 = { by lemma 35 R->L }
% 14.17/2.27 meet(x0, converse(complement(complement(composition(x0, top)))))
% 14.17/2.27 = { by lemma 21 R->L }
% 14.17/2.27 meet(x0, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(composition(x0, complement(composition(x0, top))))))))
% 14.17/2.27 = { by lemma 31 R->L }
% 14.17/2.27 meet(x0, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(composition(x0, complement(join(zero, meet(composition(x0, top), composition(x0, top))))))))))
% 14.17/2.27 = { by lemma 34 }
% 14.17/2.27 meet(x0, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(composition(x0, complement(meet(composition(x0, top), composition(x0, top)))))))))
% 14.17/2.27 = { by lemma 51 }
% 14.17/2.27 meet(x0, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(composition(x0, complement(composition(x0, composition(x0, top)))))))))
% 14.17/2.27 = { by lemma 51 R->L }
% 14.17/2.27 meet(x0, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(meet(complement(composition(x0, composition(x0, top))), composition(x0, top)))))))
% 14.17/2.27 = { by lemma 36 R->L }
% 14.17/2.27 meet(x0, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(meet(composition(x0, top), complement(composition(x0, composition(x0, top)))))))))
% 14.17/2.27 = { by lemma 35 R->L }
% 14.17/2.27 meet(x0, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(meet(composition(x0, top), complement(composition(x0, complement(complement(composition(x0, top)))))))))))
% 14.17/2.27 = { by lemma 42 R->L }
% 14.17/2.27 meet(x0, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, top)), composition(x0, complement(complement(composition(x0, top)))))))))))
% 14.17/2.27 = { by lemma 48 }
% 14.17/2.27 meet(x0, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(complement(join(complement(composition(x0, top)), composition(x0, top))))))))
% 14.17/2.27 = { by lemma 42 }
% 14.17/2.27 meet(x0, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(meet(composition(x0, top), complement(composition(x0, top))))))))
% 14.17/2.27 = { by axiom 6 (def_zero) R->L }
% 14.17/2.27 meet(x0, converse(join(complement(complement(composition(x0, top))), composition(converse(x0), complement(zero)))))
% 14.17/2.27 = { by lemma 35 }
% 14.17/2.27 meet(x0, converse(join(composition(x0, top), composition(converse(x0), complement(zero)))))
% 14.17/2.27 = { by lemma 34 R->L }
% 14.17/2.27 meet(x0, converse(join(composition(x0, top), composition(converse(x0), join(zero, complement(zero))))))
% 14.17/2.27 = { by axiom 5 (def_top) R->L }
% 14.17/2.27 meet(x0, converse(join(composition(x0, top), composition(converse(x0), top))))
% 14.17/2.27 = { by axiom 12 (composition_distributivity) R->L }
% 14.17/2.27 meet(x0, converse(composition(join(x0, converse(x0)), top)))
% 14.17/2.27 = { by axiom 9 (converse_multiplicativity) }
% 14.17/2.27 meet(x0, composition(converse(top), converse(join(x0, converse(x0)))))
% 14.17/2.27 = { by lemma 29 }
% 14.17/2.27 meet(x0, composition(top, converse(join(x0, converse(x0)))))
% 14.17/2.27 = { by axiom 7 (converse_additivity) }
% 14.17/2.27 meet(x0, composition(top, join(converse(x0), converse(converse(x0)))))
% 14.17/2.27 = { by axiom 4 (converse_idempotence) }
% 14.17/2.27 meet(x0, composition(top, join(converse(x0), x0)))
% 14.17/2.27 = { by axiom 1 (maddux1_join_commutativity) }
% 14.17/2.27 meet(x0, composition(top, join(x0, converse(x0))))
% 14.17/2.27 = { by lemma 36 }
% 14.17/2.27 meet(composition(top, join(x0, converse(x0))), x0)
% 14.17/2.27 = { by lemma 50 R->L }
% 14.17/2.27 meet(composition(top, join(x0, converse(x0))), meet(x0, join(x0, converse(x0))))
% 14.17/2.27 = { by lemma 59 R->L }
% 14.17/2.27 meet(x0, meet(join(x0, converse(x0)), composition(top, join(x0, converse(x0)))))
% 14.17/2.27 = { by lemma 26 R->L }
% 14.17/2.27 meet(x0, meet(join(x0, converse(x0)), composition(join(one, top), join(x0, converse(x0)))))
% 14.17/2.27 = { by lemma 58 }
% 14.17/2.27 meet(x0, meet(join(x0, converse(x0)), join(join(x0, converse(x0)), composition(top, join(x0, converse(x0))))))
% 14.17/2.27 = { by lemma 50 }
% 14.17/2.27 meet(x0, join(x0, converse(x0)))
% 14.17/2.27 = { by lemma 50 }
% 14.17/2.27 x0
% 14.17/2.27
% 14.17/2.27 Goal 1 (goals_1): converse(x0) = x0.
% 14.17/2.27 Proof:
% 14.17/2.27 converse(x0)
% 14.17/2.27 = { by lemma 60 R->L }
% 14.17/2.27 converse(composition(x0, converse(x0)))
% 14.17/2.27 = { by axiom 9 (converse_multiplicativity) }
% 14.17/2.27 composition(converse(converse(x0)), converse(x0))
% 14.17/2.27 = { by axiom 4 (converse_idempotence) }
% 14.17/2.27 composition(x0, converse(x0))
% 14.17/2.27 = { by lemma 60 }
% 14.17/2.27 x0
% 14.17/2.27 % SZS output end Proof
% 14.17/2.27
% 14.17/2.27 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------