TSTP Solution File: REL012+2 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL012+2 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n007.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:43:53 EDT 2023
% Result : Theorem 16.25s 2.50s
% Output : Proof 16.25s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : REL012+2 : TPTP v8.1.2. Released v4.0.0.
% 0.13/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.35 % Computer : n007.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Fri Aug 25 19:29:11 EDT 2023
% 0.15/0.35 % CPUTime :
% 16.25/2.50 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 16.25/2.50
% 16.25/2.50 % SZS status Theorem
% 16.25/2.50
% 16.25/2.54 % SZS output start Proof
% 16.25/2.54 Axiom 1 (composition_identity): composition(X, one) = X.
% 16.25/2.54 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 16.25/2.54 Axiom 3 (converse_idempotence): converse(converse(X)) = X.
% 16.25/2.54 Axiom 4 (def_top): top = join(X, complement(X)).
% 16.25/2.54 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 16.25/2.54 Axiom 6 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 16.25/2.54 Axiom 7 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 16.25/2.54 Axiom 8 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 16.25/2.54 Axiom 9 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 16.25/2.54 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 16.25/2.54 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 16.25/2.54 Axiom 12 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 16.25/2.54 Axiom 13 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 16.25/2.54 Axiom 14 (dedekind_law): join(meet(composition(X, Y), Z), composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z)))) = composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z))).
% 16.25/2.54
% 16.25/2.54 Lemma 15: complement(top) = zero.
% 16.25/2.54 Proof:
% 16.25/2.54 complement(top)
% 16.25/2.54 = { by axiom 4 (def_top) }
% 16.25/2.54 complement(join(complement(X), complement(complement(X))))
% 16.25/2.54 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 16.25/2.54 meet(X, complement(X))
% 16.25/2.54 = { by axiom 5 (def_zero) R->L }
% 16.25/2.54 zero
% 16.25/2.54
% 16.25/2.54 Lemma 16: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 16.25/2.54 Proof:
% 16.25/2.54 converse(join(X, converse(Y)))
% 16.25/2.54 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.54 converse(join(converse(Y), X))
% 16.25/2.54 = { by axiom 8 (converse_additivity) }
% 16.25/2.54 join(converse(converse(Y)), converse(X))
% 16.25/2.54 = { by axiom 3 (converse_idempotence) }
% 16.25/2.54 join(Y, converse(X))
% 16.25/2.54
% 16.25/2.54 Lemma 17: join(X, join(Y, complement(X))) = join(Y, top).
% 16.25/2.54 Proof:
% 16.25/2.54 join(X, join(Y, complement(X)))
% 16.25/2.54 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.54 join(X, join(complement(X), Y))
% 16.25/2.54 = { by axiom 9 (maddux2_join_associativity) }
% 16.25/2.54 join(join(X, complement(X)), Y)
% 16.25/2.54 = { by axiom 4 (def_top) R->L }
% 16.25/2.54 join(top, Y)
% 16.25/2.54 = { by axiom 2 (maddux1_join_commutativity) }
% 16.25/2.54 join(Y, top)
% 16.25/2.54
% 16.25/2.54 Lemma 18: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 16.25/2.54 Proof:
% 16.25/2.54 converse(composition(converse(X), Y))
% 16.25/2.54 = { by axiom 6 (converse_multiplicativity) }
% 16.25/2.54 composition(converse(Y), converse(converse(X)))
% 16.25/2.54 = { by axiom 3 (converse_idempotence) }
% 16.25/2.54 composition(converse(Y), X)
% 16.25/2.54
% 16.25/2.54 Lemma 19: composition(converse(one), X) = X.
% 16.25/2.54 Proof:
% 16.25/2.54 composition(converse(one), X)
% 16.25/2.54 = { by lemma 18 R->L }
% 16.25/2.54 converse(composition(converse(X), one))
% 16.25/2.54 = { by axiom 1 (composition_identity) }
% 16.25/2.54 converse(converse(X))
% 16.25/2.54 = { by axiom 3 (converse_idempotence) }
% 16.25/2.54 X
% 16.25/2.54
% 16.25/2.54 Lemma 20: composition(one, X) = X.
% 16.25/2.54 Proof:
% 16.25/2.54 composition(one, X)
% 16.25/2.54 = { by lemma 19 R->L }
% 16.25/2.54 composition(converse(one), composition(one, X))
% 16.25/2.54 = { by axiom 7 (composition_associativity) }
% 16.25/2.54 composition(composition(converse(one), one), X)
% 16.25/2.54 = { by axiom 1 (composition_identity) }
% 16.25/2.54 composition(converse(one), X)
% 16.25/2.54 = { by lemma 19 }
% 16.25/2.54 X
% 16.25/2.54
% 16.25/2.54 Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 16.25/2.54 Proof:
% 16.25/2.54 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 16.25/2.54 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.54 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 16.25/2.54 = { by axiom 12 (converse_cancellativity) }
% 16.25/2.54 complement(X)
% 16.25/2.54
% 16.25/2.54 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 16.25/2.54 Proof:
% 16.25/2.54 join(complement(X), complement(X))
% 16.25/2.54 = { by lemma 19 R->L }
% 16.25/2.54 join(complement(X), composition(converse(one), complement(X)))
% 16.25/2.54 = { by lemma 20 R->L }
% 16.25/2.54 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 16.25/2.54 = { by lemma 21 }
% 16.25/2.54 complement(X)
% 16.25/2.54
% 16.25/2.54 Lemma 23: join(top, complement(X)) = top.
% 16.25/2.54 Proof:
% 16.25/2.54 join(top, complement(X))
% 16.25/2.54 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.54 join(complement(X), top)
% 16.25/2.54 = { by lemma 17 R->L }
% 16.25/2.54 join(X, join(complement(X), complement(X)))
% 16.25/2.54 = { by lemma 22 }
% 16.25/2.54 join(X, complement(X))
% 16.25/2.54 = { by axiom 4 (def_top) R->L }
% 16.25/2.54 top
% 16.25/2.54
% 16.25/2.54 Lemma 24: join(Y, top) = join(X, top).
% 16.25/2.54 Proof:
% 16.25/2.54 join(Y, top)
% 16.25/2.54 = { by lemma 23 R->L }
% 16.25/2.54 join(Y, join(top, complement(Y)))
% 16.25/2.54 = { by lemma 17 }
% 16.25/2.54 join(top, top)
% 16.25/2.54 = { by lemma 17 R->L }
% 16.25/2.54 join(X, join(top, complement(X)))
% 16.25/2.54 = { by lemma 23 }
% 16.25/2.54 join(X, top)
% 16.25/2.54
% 16.25/2.54 Lemma 25: join(X, top) = top.
% 16.25/2.54 Proof:
% 16.25/2.54 join(X, top)
% 16.25/2.54 = { by lemma 24 }
% 16.25/2.54 join(complement(Y), top)
% 16.25/2.54 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.54 join(top, complement(Y))
% 16.25/2.54 = { by lemma 23 }
% 16.25/2.54 top
% 16.25/2.54
% 16.25/2.54 Lemma 26: join(top, X) = top.
% 16.25/2.54 Proof:
% 16.25/2.54 join(top, X)
% 16.25/2.54 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.54 join(X, top)
% 16.25/2.54 = { by lemma 24 R->L }
% 16.25/2.54 join(Y, top)
% 16.25/2.54 = { by lemma 25 }
% 16.25/2.54 top
% 16.25/2.54
% 16.25/2.54 Lemma 27: join(X, converse(top)) = converse(top).
% 16.25/2.54 Proof:
% 16.25/2.55 join(X, converse(top))
% 16.25/2.55 = { by lemma 16 R->L }
% 16.25/2.55 converse(join(top, converse(X)))
% 16.25/2.55 = { by lemma 26 }
% 16.25/2.55 converse(top)
% 16.25/2.55
% 16.25/2.55 Lemma 28: converse(top) = top.
% 16.25/2.55 Proof:
% 16.25/2.55 converse(top)
% 16.25/2.55 = { by lemma 27 R->L }
% 16.25/2.55 join(X, converse(top))
% 16.25/2.55 = { by lemma 27 R->L }
% 16.25/2.55 join(X, join(complement(X), converse(top)))
% 16.25/2.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.55 join(X, join(converse(top), complement(X)))
% 16.25/2.55 = { by lemma 17 }
% 16.25/2.55 join(converse(top), top)
% 16.25/2.55 = { by lemma 25 }
% 16.25/2.55 top
% 16.25/2.55
% 16.25/2.55 Lemma 29: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 16.25/2.55 Proof:
% 16.25/2.55 join(meet(X, Y), complement(join(complement(X), Y)))
% 16.25/2.55 = { by axiom 10 (maddux4_definiton_of_meet) }
% 16.25/2.55 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 16.25/2.55 = { by axiom 13 (maddux3_a_kind_of_de_Morgan) R->L }
% 16.25/2.55 X
% 16.25/2.55
% 16.25/2.55 Lemma 30: join(zero, meet(X, X)) = X.
% 16.25/2.55 Proof:
% 16.25/2.55 join(zero, meet(X, X))
% 16.25/2.55 = { by axiom 10 (maddux4_definiton_of_meet) }
% 16.25/2.55 join(zero, complement(join(complement(X), complement(X))))
% 16.25/2.55 = { by axiom 5 (def_zero) }
% 16.25/2.55 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 16.25/2.55 = { by lemma 29 }
% 16.25/2.55 X
% 16.25/2.55
% 16.25/2.55 Lemma 31: join(zero, zero) = zero.
% 16.25/2.55 Proof:
% 16.25/2.55 join(zero, zero)
% 16.25/2.55 = { by lemma 15 R->L }
% 16.25/2.55 join(zero, complement(top))
% 16.25/2.55 = { by lemma 15 R->L }
% 16.25/2.55 join(complement(top), complement(top))
% 16.25/2.55 = { by lemma 22 }
% 16.25/2.55 complement(top)
% 16.25/2.55 = { by lemma 15 }
% 16.25/2.55 zero
% 16.25/2.55
% 16.25/2.55 Lemma 32: join(zero, join(zero, X)) = join(X, zero).
% 16.25/2.55 Proof:
% 16.25/2.55 join(zero, join(zero, X))
% 16.25/2.55 = { by axiom 9 (maddux2_join_associativity) }
% 16.25/2.55 join(join(zero, zero), X)
% 16.25/2.55 = { by lemma 31 }
% 16.25/2.55 join(zero, X)
% 16.25/2.55 = { by axiom 2 (maddux1_join_commutativity) }
% 16.25/2.55 join(X, zero)
% 16.25/2.55
% 16.25/2.55 Lemma 33: join(zero, complement(complement(X))) = X.
% 16.25/2.55 Proof:
% 16.25/2.55 join(zero, complement(complement(X)))
% 16.25/2.55 = { by axiom 5 (def_zero) }
% 16.25/2.55 join(meet(X, complement(X)), complement(complement(X)))
% 16.25/2.55 = { by lemma 22 R->L }
% 16.25/2.55 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 16.25/2.55 = { by lemma 29 }
% 16.25/2.55 X
% 16.25/2.55
% 16.25/2.55 Lemma 34: join(X, zero) = X.
% 16.25/2.55 Proof:
% 16.25/2.55 join(X, zero)
% 16.25/2.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.55 join(zero, X)
% 16.25/2.55 = { by lemma 30 R->L }
% 16.25/2.55 join(zero, join(zero, meet(X, X)))
% 16.25/2.55 = { by lemma 32 }
% 16.25/2.55 join(meet(X, X), zero)
% 16.25/2.55 = { by axiom 10 (maddux4_definiton_of_meet) }
% 16.25/2.55 join(complement(join(complement(X), complement(X))), zero)
% 16.25/2.55 = { by lemma 22 }
% 16.25/2.55 join(complement(complement(X)), zero)
% 16.25/2.55 = { by axiom 2 (maddux1_join_commutativity) }
% 16.25/2.55 join(zero, complement(complement(X)))
% 16.25/2.55 = { by lemma 33 }
% 16.25/2.55 X
% 16.25/2.55
% 16.25/2.55 Lemma 35: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 16.25/2.55 Proof:
% 16.25/2.55 converse(join(converse(X), Y))
% 16.25/2.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.55 converse(join(Y, converse(X)))
% 16.25/2.55 = { by lemma 16 }
% 16.25/2.55 join(X, converse(Y))
% 16.25/2.55
% 16.25/2.55 Lemma 36: converse(zero) = zero.
% 16.25/2.55 Proof:
% 16.25/2.55 converse(zero)
% 16.25/2.55 = { by lemma 34 R->L }
% 16.25/2.55 join(converse(zero), zero)
% 16.25/2.55 = { by lemma 32 R->L }
% 16.25/2.55 join(zero, join(zero, converse(zero)))
% 16.25/2.55 = { by lemma 35 R->L }
% 16.25/2.55 join(zero, converse(join(converse(zero), zero)))
% 16.25/2.55 = { by lemma 34 }
% 16.25/2.55 join(zero, converse(converse(zero)))
% 16.25/2.55 = { by axiom 3 (converse_idempotence) }
% 16.25/2.55 join(zero, zero)
% 16.25/2.55 = { by lemma 31 }
% 16.25/2.55 zero
% 16.25/2.55
% 16.25/2.55 Lemma 37: join(zero, X) = X.
% 16.25/2.55 Proof:
% 16.25/2.55 join(zero, X)
% 16.25/2.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.55 join(X, zero)
% 16.25/2.55 = { by lemma 34 }
% 16.25/2.55 X
% 16.25/2.55
% 16.25/2.55 Lemma 38: complement(complement(X)) = X.
% 16.25/2.55 Proof:
% 16.25/2.55 complement(complement(X))
% 16.25/2.55 = { by lemma 37 R->L }
% 16.25/2.55 join(zero, complement(complement(X)))
% 16.25/2.55 = { by lemma 33 }
% 16.25/2.55 X
% 16.25/2.55
% 16.25/2.55 Lemma 39: meet(Y, X) = meet(X, Y).
% 16.25/2.55 Proof:
% 16.25/2.55 meet(Y, X)
% 16.25/2.55 = { by axiom 10 (maddux4_definiton_of_meet) }
% 16.25/2.55 complement(join(complement(Y), complement(X)))
% 16.25/2.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.55 complement(join(complement(X), complement(Y)))
% 16.25/2.55 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 16.25/2.55 meet(X, Y)
% 16.25/2.55
% 16.25/2.55 Lemma 40: complement(join(zero, complement(X))) = meet(X, top).
% 16.25/2.55 Proof:
% 16.25/2.55 complement(join(zero, complement(X)))
% 16.25/2.55 = { by lemma 15 R->L }
% 16.25/2.55 complement(join(complement(top), complement(X)))
% 16.25/2.55 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 16.25/2.55 meet(top, X)
% 16.25/2.55 = { by lemma 39 R->L }
% 16.25/2.55 meet(X, top)
% 16.25/2.55
% 16.25/2.55 Lemma 41: meet(X, top) = X.
% 16.25/2.55 Proof:
% 16.25/2.55 meet(X, top)
% 16.25/2.55 = { by lemma 40 R->L }
% 16.25/2.55 complement(join(zero, complement(X)))
% 16.25/2.55 = { by lemma 37 }
% 16.25/2.55 complement(complement(X))
% 16.25/2.55 = { by lemma 38 }
% 16.25/2.55 X
% 16.25/2.55
% 16.25/2.55 Lemma 42: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 16.25/2.55 Proof:
% 16.25/2.55 join(X, composition(Y, X))
% 16.25/2.55 = { by lemma 20 R->L }
% 16.25/2.55 join(composition(one, X), composition(Y, X))
% 16.25/2.55 = { by axiom 11 (composition_distributivity) R->L }
% 16.25/2.55 composition(join(one, Y), X)
% 16.25/2.55 = { by axiom 2 (maddux1_join_commutativity) }
% 16.25/2.55 composition(join(Y, one), X)
% 16.25/2.55
% 16.25/2.55 Lemma 43: composition(top, zero) = zero.
% 16.25/2.55 Proof:
% 16.25/2.55 composition(top, zero)
% 16.25/2.55 = { by lemma 15 R->L }
% 16.25/2.55 composition(top, complement(top))
% 16.25/2.55 = { by lemma 26 R->L }
% 16.25/2.55 composition(join(top, one), complement(top))
% 16.25/2.55 = { by lemma 28 R->L }
% 16.25/2.55 composition(join(converse(top), one), complement(top))
% 16.25/2.55 = { by lemma 42 R->L }
% 16.25/2.55 join(complement(top), composition(converse(top), complement(top)))
% 16.25/2.55 = { by lemma 26 R->L }
% 16.25/2.55 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 16.25/2.55 = { by lemma 42 }
% 16.25/2.55 join(complement(top), composition(converse(top), complement(composition(join(top, one), top))))
% 16.25/2.55 = { by lemma 26 }
% 16.25/2.55 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 16.25/2.55 = { by lemma 21 }
% 16.25/2.55 complement(top)
% 16.25/2.55 = { by lemma 15 }
% 16.25/2.55 zero
% 16.25/2.55
% 16.25/2.55 Lemma 44: composition(zero, X) = zero.
% 16.25/2.55 Proof:
% 16.25/2.55 composition(zero, X)
% 16.25/2.55 = { by lemma 36 R->L }
% 16.25/2.55 composition(converse(zero), X)
% 16.25/2.55 = { by lemma 18 R->L }
% 16.25/2.55 converse(composition(converse(X), zero))
% 16.25/2.55 = { by lemma 37 R->L }
% 16.25/2.55 converse(join(zero, composition(converse(X), zero)))
% 16.25/2.55 = { by lemma 43 R->L }
% 16.25/2.55 converse(join(composition(top, zero), composition(converse(X), zero)))
% 16.25/2.55 = { by axiom 11 (composition_distributivity) R->L }
% 16.25/2.55 converse(composition(join(top, converse(X)), zero))
% 16.25/2.55 = { by lemma 26 }
% 16.25/2.55 converse(composition(top, zero))
% 16.25/2.55 = { by lemma 43 }
% 16.25/2.55 converse(zero)
% 16.25/2.55 = { by lemma 36 }
% 16.25/2.55 zero
% 16.25/2.55
% 16.25/2.55 Lemma 45: join(X, converse(complement(converse(X)))) = top.
% 16.25/2.55 Proof:
% 16.25/2.55 join(X, converse(complement(converse(X))))
% 16.25/2.55 = { by lemma 35 R->L }
% 16.25/2.55 converse(join(converse(X), complement(converse(X))))
% 16.25/2.55 = { by axiom 4 (def_top) R->L }
% 16.25/2.55 converse(top)
% 16.25/2.55 = { by lemma 28 }
% 16.25/2.55 top
% 16.25/2.55
% 16.25/2.55 Lemma 46: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 16.25/2.55 Proof:
% 16.25/2.55 meet(X, join(complement(Y), complement(Z)))
% 16.25/2.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.55 meet(X, join(complement(Z), complement(Y)))
% 16.25/2.55 = { by lemma 39 }
% 16.25/2.55 meet(join(complement(Z), complement(Y)), X)
% 16.25/2.55 = { by axiom 10 (maddux4_definiton_of_meet) }
% 16.25/2.55 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 16.25/2.55 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 16.25/2.55 complement(join(meet(Z, Y), complement(X)))
% 16.25/2.55 = { by axiom 2 (maddux1_join_commutativity) }
% 16.25/2.55 complement(join(complement(X), meet(Z, Y)))
% 16.25/2.55 = { by lemma 39 R->L }
% 16.25/2.55 complement(join(complement(X), meet(Y, Z)))
% 16.25/2.55
% 16.25/2.55 Lemma 47: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 16.25/2.55 Proof:
% 16.25/2.55 complement(join(X, complement(Y)))
% 16.25/2.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.55 complement(join(complement(Y), X))
% 16.25/2.55 = { by lemma 41 R->L }
% 16.25/2.55 complement(join(complement(Y), meet(X, top)))
% 16.25/2.55 = { by lemma 39 R->L }
% 16.25/2.55 complement(join(complement(Y), meet(top, X)))
% 16.25/2.55 = { by lemma 46 R->L }
% 16.25/2.55 meet(Y, join(complement(top), complement(X)))
% 16.25/2.55 = { by lemma 15 }
% 16.25/2.55 meet(Y, join(zero, complement(X)))
% 16.25/2.55 = { by lemma 37 }
% 16.25/2.55 meet(Y, complement(X))
% 16.25/2.55
% 16.25/2.55 Lemma 48: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 16.25/2.55 Proof:
% 16.25/2.55 meet(complement(X), complement(Y))
% 16.25/2.55 = { by lemma 39 }
% 16.25/2.55 meet(complement(Y), complement(X))
% 16.25/2.55 = { by lemma 37 R->L }
% 16.25/2.55 meet(join(zero, complement(Y)), complement(X))
% 16.25/2.55 = { by lemma 47 R->L }
% 16.25/2.55 complement(join(X, complement(join(zero, complement(Y)))))
% 16.25/2.55 = { by lemma 40 }
% 16.25/2.55 complement(join(X, meet(Y, top)))
% 16.25/2.55 = { by lemma 41 }
% 16.25/2.55 complement(join(X, Y))
% 16.25/2.55
% 16.25/2.55 Lemma 49: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 16.25/2.55 Proof:
% 16.25/2.55 complement(join(complement(X), Y))
% 16.25/2.55 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.55 complement(join(Y, complement(X)))
% 16.25/2.55 = { by lemma 47 }
% 16.25/2.55 meet(X, complement(Y))
% 16.25/2.55
% 16.25/2.55 Lemma 50: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 16.25/2.55 Proof:
% 16.25/2.55 complement(meet(complement(X), Y))
% 16.25/2.55 = { by lemma 39 }
% 16.25/2.55 complement(meet(Y, complement(X)))
% 16.25/2.55 = { by lemma 37 R->L }
% 16.25/2.55 complement(join(zero, meet(Y, complement(X))))
% 16.25/2.55 = { by lemma 47 R->L }
% 16.25/2.55 complement(join(zero, complement(join(X, complement(Y)))))
% 16.25/2.55 = { by lemma 40 }
% 16.25/2.55 meet(join(X, complement(Y)), top)
% 16.25/2.55 = { by lemma 41 }
% 16.25/2.55 join(X, complement(Y))
% 16.25/2.55
% 16.25/2.55 Lemma 51: join(X, complement(meet(X, Y))) = top.
% 16.25/2.55 Proof:
% 16.25/2.55 join(X, complement(meet(X, Y)))
% 16.25/2.55 = { by lemma 39 }
% 16.25/2.55 join(X, complement(meet(Y, X)))
% 16.25/2.55 = { by axiom 10 (maddux4_definiton_of_meet) }
% 16.25/2.55 join(X, complement(complement(join(complement(Y), complement(X)))))
% 16.25/2.55 = { by lemma 22 R->L }
% 16.25/2.55 join(X, complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))))
% 16.25/2.55 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 16.25/2.55 join(X, complement(join(meet(Y, X), complement(join(complement(Y), complement(X))))))
% 16.25/2.55 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 16.25/2.55 join(X, complement(join(meet(Y, X), meet(Y, X))))
% 16.25/2.55 = { by lemma 39 R->L }
% 16.25/2.55 join(X, complement(join(meet(Y, X), meet(X, Y))))
% 16.25/2.56 = { by lemma 39 R->L }
% 16.25/2.56 join(X, complement(join(meet(X, Y), meet(X, Y))))
% 16.25/2.56 = { by axiom 10 (maddux4_definiton_of_meet) }
% 16.25/2.56 join(X, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y))))
% 16.25/2.56 = { by lemma 37 R->L }
% 16.25/2.56 join(X, join(zero, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y)))))
% 16.25/2.56 = { by lemma 46 R->L }
% 16.25/2.56 join(X, join(zero, meet(join(complement(X), complement(Y)), join(complement(X), complement(Y)))))
% 16.25/2.56 = { by lemma 30 }
% 16.25/2.56 join(X, join(complement(X), complement(Y)))
% 16.25/2.56 = { by axiom 2 (maddux1_join_commutativity) }
% 16.25/2.56 join(X, join(complement(Y), complement(X)))
% 16.25/2.56 = { by lemma 17 }
% 16.25/2.56 join(complement(Y), top)
% 16.25/2.56 = { by lemma 25 }
% 16.25/2.56 top
% 16.25/2.56
% 16.25/2.56 Lemma 52: meet(X, join(X, complement(Y))) = X.
% 16.25/2.56 Proof:
% 16.25/2.56 meet(X, join(X, complement(Y)))
% 16.25/2.56 = { by lemma 34 R->L }
% 16.25/2.56 join(meet(X, join(X, complement(Y))), zero)
% 16.25/2.56 = { by lemma 15 R->L }
% 16.25/2.56 join(meet(X, join(X, complement(Y))), complement(top))
% 16.25/2.56 = { by lemma 50 R->L }
% 16.25/2.56 join(meet(X, complement(meet(complement(X), Y))), complement(top))
% 16.25/2.56 = { by lemma 51 R->L }
% 16.25/2.56 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(complement(X), Y)))))
% 16.25/2.56 = { by lemma 29 }
% 16.25/2.56 X
% 16.25/2.56
% 16.25/2.56 Lemma 53: join(X, meet(X, Y)) = X.
% 16.25/2.56 Proof:
% 16.25/2.56 join(X, meet(X, Y))
% 16.25/2.56 = { by axiom 10 (maddux4_definiton_of_meet) }
% 16.25/2.56 join(X, complement(join(complement(X), complement(Y))))
% 16.25/2.56 = { by lemma 50 R->L }
% 16.25/2.56 complement(meet(complement(X), join(complement(X), complement(Y))))
% 16.25/2.56 = { by lemma 52 }
% 16.25/2.56 complement(complement(X))
% 16.25/2.56 = { by lemma 38 }
% 16.25/2.56 X
% 16.25/2.56
% 16.25/2.56 Lemma 54: join(meet(X, Y), meet(X, complement(Y))) = X.
% 16.25/2.56 Proof:
% 16.25/2.56 join(meet(X, Y), meet(X, complement(Y)))
% 16.25/2.56 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.56 join(meet(X, complement(Y)), meet(X, Y))
% 16.25/2.56 = { by axiom 10 (maddux4_definiton_of_meet) }
% 16.25/2.56 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 16.25/2.56 = { by lemma 29 }
% 16.25/2.56 X
% 16.25/2.56
% 16.25/2.56 Lemma 55: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
% 16.25/2.56 Proof:
% 16.25/2.56 join(complement(X), meet(X, Y))
% 16.25/2.56 = { by lemma 39 }
% 16.25/2.56 join(complement(X), meet(Y, X))
% 16.25/2.56 = { by lemma 53 R->L }
% 16.25/2.56 join(join(complement(X), meet(complement(X), Y)), meet(Y, X))
% 16.25/2.56 = { by axiom 9 (maddux2_join_associativity) R->L }
% 16.25/2.56 join(complement(X), join(meet(complement(X), Y), meet(Y, X)))
% 16.25/2.56 = { by axiom 2 (maddux1_join_commutativity) }
% 16.25/2.56 join(complement(X), join(meet(Y, X), meet(complement(X), Y)))
% 16.25/2.56 = { by lemma 39 }
% 16.25/2.56 join(complement(X), join(meet(Y, X), meet(Y, complement(X))))
% 16.25/2.56 = { by lemma 54 }
% 16.25/2.56 join(complement(X), Y)
% 16.25/2.56 = { by axiom 2 (maddux1_join_commutativity) }
% 16.25/2.56 join(Y, complement(X))
% 16.25/2.56
% 16.25/2.56 Lemma 56: join(complement(one), converse(complement(one))) = complement(one).
% 16.25/2.56 Proof:
% 16.25/2.56 join(complement(one), converse(complement(one)))
% 16.25/2.56 = { by lemma 41 R->L }
% 16.25/2.56 join(complement(one), converse(meet(complement(one), top)))
% 16.25/2.56 = { by lemma 45 R->L }
% 16.25/2.56 join(complement(one), converse(meet(complement(one), join(one, converse(complement(converse(one)))))))
% 16.25/2.56 = { by axiom 1 (composition_identity) R->L }
% 16.25/2.56 join(complement(one), converse(meet(complement(one), join(one, converse(complement(composition(converse(one), one)))))))
% 16.25/2.56 = { by lemma 19 }
% 16.25/2.56 join(complement(one), converse(meet(complement(one), join(one, converse(complement(one))))))
% 16.25/2.56 = { by lemma 39 }
% 16.25/2.56 join(complement(one), converse(meet(join(one, converse(complement(one))), complement(one))))
% 16.25/2.56 = { by lemma 47 R->L }
% 16.25/2.56 join(complement(one), converse(complement(join(one, complement(join(one, converse(complement(one))))))))
% 16.25/2.56 = { by lemma 48 R->L }
% 16.25/2.56 join(complement(one), converse(meet(complement(one), complement(complement(join(one, converse(complement(one))))))))
% 16.25/2.56 = { by lemma 48 R->L }
% 16.25/2.56 join(complement(one), converse(meet(complement(one), complement(meet(complement(one), complement(converse(complement(one))))))))
% 16.25/2.56 = { by lemma 49 R->L }
% 16.25/2.56 join(complement(one), converse(complement(join(complement(complement(one)), meet(complement(one), complement(converse(complement(one))))))))
% 16.25/2.56 = { by lemma 55 }
% 16.25/2.56 join(complement(one), converse(complement(join(complement(converse(complement(one))), complement(complement(one))))))
% 16.25/2.56 = { by lemma 47 }
% 16.25/2.56 join(complement(one), converse(meet(complement(one), complement(complement(converse(complement(one)))))))
% 16.25/2.56 = { by lemma 48 }
% 16.25/2.56 join(complement(one), converse(complement(join(one, complement(converse(complement(one)))))))
% 16.25/2.56 = { by lemma 47 }
% 16.25/2.56 join(complement(one), converse(meet(converse(complement(one)), complement(one))))
% 16.25/2.56 = { by lemma 35 R->L }
% 16.25/2.56 converse(join(converse(complement(one)), meet(converse(complement(one)), complement(one))))
% 16.25/2.56 = { by lemma 53 }
% 16.25/2.56 converse(converse(complement(one)))
% 16.25/2.56 = { by axiom 3 (converse_idempotence) }
% 16.25/2.56 complement(one)
% 16.25/2.56
% 16.25/2.56 Lemma 57: converse(complement(one)) = complement(one).
% 16.25/2.56 Proof:
% 16.25/2.56 converse(complement(one))
% 16.25/2.56 = { by lemma 56 R->L }
% 16.25/2.56 converse(join(complement(one), converse(complement(one))))
% 16.25/2.56 = { by lemma 16 }
% 16.25/2.56 join(complement(one), converse(complement(one)))
% 16.25/2.56 = { by lemma 56 }
% 16.25/2.56 complement(one)
% 16.25/2.56
% 16.25/2.56 Lemma 58: meet(X, complement(join(Y, X))) = zero.
% 16.25/2.56 Proof:
% 16.25/2.56 meet(X, complement(join(Y, X)))
% 16.25/2.56 = { by lemma 48 R->L }
% 16.25/2.56 meet(X, meet(complement(Y), complement(X)))
% 16.25/2.56 = { by lemma 39 }
% 16.25/2.56 meet(X, meet(complement(X), complement(Y)))
% 16.25/2.56 = { by axiom 10 (maddux4_definiton_of_meet) }
% 16.25/2.56 complement(join(complement(X), complement(meet(complement(X), complement(Y)))))
% 16.25/2.56 = { by lemma 51 }
% 16.25/2.56 complement(top)
% 16.25/2.56 = { by lemma 15 }
% 16.25/2.56 zero
% 16.25/2.56
% 16.25/2.56 Lemma 59: join(complement(one), composition(converse(X), complement(X))) = complement(one).
% 16.25/2.56 Proof:
% 16.25/2.56 join(complement(one), composition(converse(X), complement(X)))
% 16.25/2.56 = { by axiom 1 (composition_identity) R->L }
% 16.25/2.56 join(complement(one), composition(converse(X), complement(composition(X, one))))
% 16.25/2.56 = { by lemma 21 }
% 16.25/2.56 complement(one)
% 16.25/2.56
% 16.25/2.56 Lemma 60: meet(one, composition(converse(complement(X)), X)) = zero.
% 16.25/2.56 Proof:
% 16.25/2.56 meet(one, composition(converse(complement(X)), X))
% 16.25/2.56 = { by lemma 39 }
% 16.25/2.56 meet(composition(converse(complement(X)), X), one)
% 16.25/2.56 = { by lemma 38 R->L }
% 16.25/2.56 meet(composition(converse(complement(X)), X), complement(complement(one)))
% 16.25/2.56 = { by lemma 57 R->L }
% 16.25/2.56 meet(composition(converse(complement(X)), X), complement(converse(complement(one))))
% 16.25/2.56 = { by lemma 59 R->L }
% 16.25/2.56 meet(composition(converse(complement(X)), X), complement(converse(join(complement(one), composition(converse(converse(converse(X))), complement(converse(converse(X))))))))
% 16.25/2.56 = { by axiom 3 (converse_idempotence) }
% 16.25/2.56 meet(composition(converse(complement(X)), X), complement(converse(join(complement(one), composition(converse(X), complement(converse(converse(X))))))))
% 16.25/2.56 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.56 meet(composition(converse(complement(X)), X), complement(converse(join(composition(converse(X), complement(converse(converse(X)))), complement(one)))))
% 16.25/2.56 = { by axiom 8 (converse_additivity) }
% 16.25/2.56 meet(composition(converse(complement(X)), X), complement(join(converse(composition(converse(X), complement(converse(converse(X))))), converse(complement(one)))))
% 16.25/2.56 = { by lemma 18 }
% 16.25/2.56 meet(composition(converse(complement(X)), X), complement(join(composition(converse(complement(converse(converse(X)))), X), converse(complement(one)))))
% 16.25/2.56 = { by axiom 2 (maddux1_join_commutativity) }
% 16.25/2.56 meet(composition(converse(complement(X)), X), complement(join(converse(complement(one)), composition(converse(complement(converse(converse(X)))), X))))
% 16.25/2.56 = { by lemma 57 }
% 16.25/2.56 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(converse(converse(X)))), X))))
% 16.25/2.56 = { by axiom 3 (converse_idempotence) }
% 16.25/2.56 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), X))))
% 16.25/2.56 = { by lemma 58 }
% 16.25/2.56 zero
% 16.25/2.56
% 16.25/2.56 Lemma 61: meet(one, composition(X, converse(composition(complement(composition(X, Y)), converse(Y))))) = zero.
% 16.25/2.56 Proof:
% 16.25/2.56 meet(one, composition(X, converse(composition(complement(composition(X, Y)), converse(Y)))))
% 16.25/2.56 = { by axiom 6 (converse_multiplicativity) }
% 16.25/2.56 meet(one, composition(X, composition(converse(converse(Y)), converse(complement(composition(X, Y))))))
% 16.25/2.56 = { by axiom 3 (converse_idempotence) }
% 16.25/2.56 meet(one, composition(X, composition(Y, converse(complement(composition(X, Y))))))
% 16.25/2.56 = { by axiom 7 (composition_associativity) }
% 16.25/2.56 meet(one, composition(composition(X, Y), converse(complement(composition(X, Y)))))
% 16.25/2.56 = { by lemma 38 R->L }
% 16.25/2.56 meet(one, composition(composition(X, Y), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.56 = { by lemma 29 R->L }
% 16.25/2.56 meet(one, composition(join(meet(composition(X, Y), converse(complement(converse(complement(composition(X, Y)))))), complement(join(complement(composition(X, Y)), converse(complement(converse(complement(composition(X, Y)))))))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.56 = { by lemma 39 R->L }
% 16.25/2.56 meet(one, composition(join(meet(converse(complement(converse(complement(composition(X, Y))))), composition(X, Y)), complement(join(complement(composition(X, Y)), converse(complement(converse(complement(composition(X, Y)))))))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.56 = { by lemma 45 }
% 16.25/2.56 meet(one, composition(join(meet(converse(complement(converse(complement(composition(X, Y))))), composition(X, Y)), complement(top)), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.56 = { by lemma 15 }
% 16.25/2.56 meet(one, composition(join(meet(converse(complement(converse(complement(composition(X, Y))))), composition(X, Y)), zero), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.56 = { by lemma 34 }
% 16.25/2.56 meet(one, composition(meet(converse(complement(converse(complement(composition(X, Y))))), composition(X, Y)), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.56 = { by lemma 39 R->L }
% 16.25/2.56 meet(one, composition(meet(composition(X, Y), converse(complement(converse(complement(composition(X, Y)))))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.56 = { by lemma 37 R->L }
% 16.25/2.56 meet(one, composition(join(zero, meet(composition(X, Y), converse(complement(converse(complement(composition(X, Y))))))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.56 = { by lemma 44 R->L }
% 16.25/2.56 meet(one, composition(join(composition(zero, meet(complement(composition(X, Y)), composition(converse(one), converse(complement(converse(complement(composition(X, Y)))))))), meet(composition(X, Y), converse(complement(converse(complement(composition(X, Y))))))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.56 = { by lemma 60 R->L }
% 16.25/2.56 meet(one, composition(join(composition(meet(one, composition(converse(complement(converse(complement(composition(X, Y))))), converse(complement(composition(X, Y))))), meet(complement(composition(X, Y)), composition(converse(one), converse(complement(converse(complement(composition(X, Y)))))))), meet(composition(X, Y), converse(complement(converse(complement(composition(X, Y))))))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.56 = { by axiom 14 (dedekind_law) R->L }
% 16.25/2.57 meet(one, composition(join(join(meet(composition(one, complement(composition(X, Y))), converse(complement(converse(complement(composition(X, Y)))))), composition(meet(one, composition(converse(complement(converse(complement(composition(X, Y))))), converse(complement(composition(X, Y))))), meet(complement(composition(X, Y)), composition(converse(one), converse(complement(converse(complement(composition(X, Y))))))))), meet(composition(X, Y), converse(complement(converse(complement(composition(X, Y))))))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.57 = { by lemma 60 }
% 16.25/2.57 meet(one, composition(join(join(meet(composition(one, complement(composition(X, Y))), converse(complement(converse(complement(composition(X, Y)))))), composition(zero, meet(complement(composition(X, Y)), composition(converse(one), converse(complement(converse(complement(composition(X, Y))))))))), meet(composition(X, Y), converse(complement(converse(complement(composition(X, Y))))))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.57 = { by lemma 20 }
% 16.25/2.57 meet(one, composition(join(join(meet(complement(composition(X, Y)), converse(complement(converse(complement(composition(X, Y)))))), composition(zero, meet(complement(composition(X, Y)), composition(converse(one), converse(complement(converse(complement(composition(X, Y))))))))), meet(composition(X, Y), converse(complement(converse(complement(composition(X, Y))))))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.57 = { by lemma 44 }
% 16.25/2.57 meet(one, composition(join(join(meet(complement(composition(X, Y)), converse(complement(converse(complement(composition(X, Y)))))), zero), meet(composition(X, Y), converse(complement(converse(complement(composition(X, Y))))))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.57 = { by lemma 34 }
% 16.25/2.57 meet(one, composition(join(meet(complement(composition(X, Y)), converse(complement(converse(complement(composition(X, Y)))))), meet(composition(X, Y), converse(complement(converse(complement(composition(X, Y))))))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.57 = { by lemma 39 }
% 16.25/2.57 meet(one, composition(join(meet(complement(composition(X, Y)), converse(complement(converse(complement(composition(X, Y)))))), meet(converse(complement(converse(complement(composition(X, Y))))), composition(X, Y))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.57 = { by lemma 39 }
% 16.25/2.57 meet(one, composition(join(meet(converse(complement(converse(complement(composition(X, Y))))), complement(composition(X, Y))), meet(converse(complement(converse(complement(composition(X, Y))))), composition(X, Y))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.57 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 16.25/2.57 meet(one, composition(join(meet(converse(complement(converse(complement(composition(X, Y))))), composition(X, Y)), meet(converse(complement(converse(complement(composition(X, Y))))), complement(composition(X, Y)))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.57 = { by lemma 54 }
% 16.25/2.57 meet(one, composition(converse(complement(converse(complement(composition(X, Y))))), complement(complement(converse(complement(composition(X, Y)))))))
% 16.25/2.57 = { by lemma 39 }
% 16.25/2.57 meet(composition(converse(complement(converse(complement(composition(X, Y))))), complement(complement(converse(complement(composition(X, Y)))))), one)
% 16.25/2.57 = { by lemma 38 R->L }
% 16.25/2.57 meet(composition(converse(complement(converse(complement(composition(X, Y))))), complement(complement(converse(complement(composition(X, Y)))))), complement(complement(one)))
% 16.25/2.57 = { by lemma 59 R->L }
% 16.25/2.57 meet(composition(converse(complement(converse(complement(composition(X, Y))))), complement(complement(converse(complement(composition(X, Y)))))), complement(join(complement(one), composition(converse(complement(converse(complement(composition(X, Y))))), complement(complement(converse(complement(composition(X, Y)))))))))
% 16.25/2.57 = { by lemma 58 }
% 16.25/2.57 zero
% 16.25/2.57
% 16.25/2.57 Goal 1 (goals): join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)) = complement(x0).
% 16.25/2.57 Proof:
% 16.25/2.57 join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0))
% 16.25/2.57 = { by lemma 54 R->L }
% 16.25/2.57 join(meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), x0), meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0)))
% 16.25/2.57 = { by lemma 39 }
% 16.25/2.57 join(meet(x0, join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0))), meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0)))
% 16.25/2.57 = { by axiom 10 (maddux4_definiton_of_meet) }
% 16.25/2.57 join(complement(join(complement(x0), complement(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0))))), meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0)))
% 16.25/2.57 = { by lemma 47 }
% 16.25/2.57 join(complement(join(complement(x0), meet(x0, complement(composition(complement(composition(x0, x1)), converse(x1)))))), meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0)))
% 16.25/2.57 = { by lemma 55 }
% 16.25/2.57 join(complement(join(complement(composition(complement(composition(x0, x1)), converse(x1))), complement(x0))), meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0)))
% 16.25/2.57 = { by lemma 49 }
% 16.25/2.57 join(meet(composition(complement(composition(x0, x1)), converse(x1)), complement(complement(x0))), meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0)))
% 16.25/2.57 = { by lemma 38 }
% 16.25/2.57 join(meet(composition(complement(composition(x0, x1)), converse(x1)), x0), meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0)))
% 16.25/2.57 = { by lemma 34 R->L }
% 16.25/2.57 join(join(meet(composition(complement(composition(x0, x1)), converse(x1)), x0), zero), meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0)))
% 16.25/2.57 = { by lemma 44 R->L }
% 16.25/2.57 join(join(meet(composition(complement(composition(x0, x1)), converse(x1)), x0), composition(zero, meet(composition(complement(composition(x0, x1)), converse(x1)), composition(converse(one), x0)))), meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0)))
% 16.25/2.57 = { by lemma 20 R->L }
% 16.25/2.57 join(join(meet(composition(one, composition(complement(composition(x0, x1)), converse(x1))), x0), composition(zero, meet(composition(complement(composition(x0, x1)), converse(x1)), composition(converse(one), x0)))), meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0)))
% 16.25/2.57 = { by lemma 61 R->L }
% 16.25/2.57 join(join(meet(composition(one, composition(complement(composition(x0, x1)), converse(x1))), x0), composition(meet(one, composition(x0, converse(composition(complement(composition(x0, x1)), converse(x1))))), meet(composition(complement(composition(x0, x1)), converse(x1)), composition(converse(one), x0)))), meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0)))
% 16.25/2.57 = { by axiom 14 (dedekind_law) }
% 16.25/2.57 join(composition(meet(one, composition(x0, converse(composition(complement(composition(x0, x1)), converse(x1))))), meet(composition(complement(composition(x0, x1)), converse(x1)), composition(converse(one), x0))), meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0)))
% 16.25/2.57 = { by lemma 61 }
% 16.25/2.57 join(composition(zero, meet(composition(complement(composition(x0, x1)), converse(x1)), composition(converse(one), x0))), meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0)))
% 16.25/2.57 = { by lemma 44 }
% 16.25/2.57 join(zero, meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0)))
% 16.25/2.57 = { by lemma 37 }
% 16.25/2.57 meet(join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)), complement(x0))
% 16.25/2.57 = { by lemma 39 }
% 16.25/2.57 meet(complement(x0), join(composition(complement(composition(x0, x1)), converse(x1)), complement(x0)))
% 16.25/2.57 = { by axiom 2 (maddux1_join_commutativity) }
% 16.25/2.57 meet(complement(x0), join(complement(x0), composition(complement(composition(x0, x1)), converse(x1))))
% 16.25/2.57 = { by lemma 41 R->L }
% 16.25/2.57 meet(complement(x0), join(complement(x0), meet(composition(complement(composition(x0, x1)), converse(x1)), top)))
% 16.25/2.57 = { by lemma 40 R->L }
% 16.25/2.57 meet(complement(x0), join(complement(x0), complement(join(zero, complement(composition(complement(composition(x0, x1)), converse(x1)))))))
% 16.25/2.57 = { by lemma 52 }
% 16.25/2.57 complement(x0)
% 16.25/2.57 % SZS output end Proof
% 16.25/2.57
% 16.25/2.57 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------