TSTP Solution File: REL029+3 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL029+3 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n010.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:44:13 EDT 2023
% Result : Theorem 52.13s 7.00s
% Output : Proof 52.89s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL029+3 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n010.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Fri Aug 25 21:17:35 EDT 2023
% 0.14/0.35 % CPUTime :
% 52.13/7.00 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 52.13/7.00
% 52.13/7.00 % SZS status Theorem
% 52.13/7.00
% 52.89/7.17 % SZS output start Proof
% 52.89/7.17 Axiom 1 (composition_identity): composition(X, one) = X.
% 52.89/7.17 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 52.89/7.17 Axiom 3 (goals): join(x1, one) = one.
% 52.89/7.17 Axiom 4 (goals_1): join(x0, one) = one.
% 52.89/7.17 Axiom 5 (converse_idempotence): converse(converse(X)) = X.
% 52.89/7.17 Axiom 6 (def_zero): zero = meet(X, complement(X)).
% 52.89/7.17 Axiom 7 (def_top): top = join(X, complement(X)).
% 52.89/7.17 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 52.89/7.17 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 52.89/7.17 Axiom 10 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 52.89/7.17 Axiom 11 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 52.89/7.17 Axiom 12 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 52.89/7.17 Axiom 13 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 52.89/7.17 Axiom 14 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 52.89/7.17 Axiom 15 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 52.89/7.17 Axiom 16 (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))).
% 52.89/7.17
% 52.89/7.17 Lemma 17: complement(top) = zero.
% 52.89/7.17 Proof:
% 52.89/7.17 complement(top)
% 52.89/7.17 = { by axiom 7 (def_top) }
% 52.89/7.17 complement(join(complement(X), complement(complement(X))))
% 52.89/7.17 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 52.89/7.17 meet(X, complement(X))
% 52.89/7.17 = { by axiom 6 (def_zero) R->L }
% 52.89/7.17 zero
% 52.89/7.17
% 52.89/7.17 Lemma 18: join(X, join(Y, complement(X))) = join(Y, top).
% 52.89/7.17 Proof:
% 52.89/7.17 join(X, join(Y, complement(X)))
% 52.89/7.17 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.17 join(X, join(complement(X), Y))
% 52.89/7.17 = { by axiom 11 (maddux2_join_associativity) }
% 52.89/7.17 join(join(X, complement(X)), Y)
% 52.89/7.17 = { by axiom 7 (def_top) R->L }
% 52.89/7.17 join(top, Y)
% 52.89/7.17 = { by axiom 2 (maddux1_join_commutativity) }
% 52.89/7.17 join(Y, top)
% 52.89/7.17
% 52.89/7.17 Lemma 19: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 52.89/7.18 Proof:
% 52.89/7.18 converse(composition(converse(X), Y))
% 52.89/7.18 = { by axiom 8 (converse_multiplicativity) }
% 52.89/7.18 composition(converse(Y), converse(converse(X)))
% 52.89/7.18 = { by axiom 5 (converse_idempotence) }
% 52.89/7.18 composition(converse(Y), X)
% 52.89/7.18
% 52.89/7.18 Lemma 20: composition(converse(one), X) = X.
% 52.89/7.18 Proof:
% 52.89/7.18 composition(converse(one), X)
% 52.89/7.18 = { by lemma 19 R->L }
% 52.89/7.18 converse(composition(converse(X), one))
% 52.89/7.18 = { by axiom 1 (composition_identity) }
% 52.89/7.18 converse(converse(X))
% 52.89/7.18 = { by axiom 5 (converse_idempotence) }
% 52.89/7.18 X
% 52.89/7.18
% 52.89/7.18 Lemma 21: composition(X, composition(one, Y)) = composition(X, Y).
% 52.89/7.18 Proof:
% 52.89/7.18 composition(X, composition(one, Y))
% 52.89/7.18 = { by axiom 9 (composition_associativity) }
% 52.89/7.18 composition(composition(X, one), Y)
% 52.89/7.18 = { by axiom 1 (composition_identity) }
% 52.89/7.18 composition(X, Y)
% 52.89/7.18
% 52.89/7.18 Lemma 22: composition(one, X) = X.
% 52.89/7.18 Proof:
% 52.89/7.18 composition(one, X)
% 52.89/7.18 = { by lemma 20 R->L }
% 52.89/7.18 composition(converse(one), composition(one, X))
% 52.89/7.18 = { by lemma 21 }
% 52.89/7.18 composition(converse(one), X)
% 52.89/7.18 = { by lemma 20 }
% 52.89/7.18 X
% 52.89/7.18
% 52.89/7.18 Lemma 23: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 52.89/7.18 Proof:
% 52.89/7.18 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 52.89/7.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.18 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 52.89/7.18 = { by axiom 14 (converse_cancellativity) }
% 52.89/7.18 complement(X)
% 52.89/7.18
% 52.89/7.18 Lemma 24: join(complement(X), complement(X)) = complement(X).
% 52.89/7.18 Proof:
% 52.89/7.18 join(complement(X), complement(X))
% 52.89/7.18 = { by lemma 20 R->L }
% 52.89/7.18 join(complement(X), composition(converse(one), complement(X)))
% 52.89/7.18 = { by lemma 22 R->L }
% 52.89/7.18 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 52.89/7.18 = { by lemma 23 }
% 52.89/7.18 complement(X)
% 52.89/7.18
% 52.89/7.18 Lemma 25: join(top, complement(X)) = top.
% 52.89/7.18 Proof:
% 52.89/7.18 join(top, complement(X))
% 52.89/7.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.18 join(complement(X), top)
% 52.89/7.18 = { by lemma 18 R->L }
% 52.89/7.18 join(X, join(complement(X), complement(X)))
% 52.89/7.18 = { by lemma 24 }
% 52.89/7.18 join(X, complement(X))
% 52.89/7.18 = { by axiom 7 (def_top) R->L }
% 52.89/7.18 top
% 52.89/7.18
% 52.89/7.18 Lemma 26: join(Y, top) = join(X, top).
% 52.89/7.18 Proof:
% 52.89/7.18 join(Y, top)
% 52.89/7.18 = { by lemma 25 R->L }
% 52.89/7.18 join(Y, join(top, complement(Y)))
% 52.89/7.18 = { by lemma 18 }
% 52.89/7.18 join(top, top)
% 52.89/7.18 = { by lemma 18 R->L }
% 52.89/7.18 join(X, join(top, complement(X)))
% 52.89/7.18 = { by lemma 25 }
% 52.89/7.18 join(X, top)
% 52.89/7.18
% 52.89/7.18 Lemma 27: join(X, top) = top.
% 52.89/7.18 Proof:
% 52.89/7.18 join(X, top)
% 52.89/7.18 = { by lemma 26 }
% 52.89/7.18 join(x1, top)
% 52.89/7.18 = { by axiom 7 (def_top) }
% 52.89/7.18 join(x1, join(one, complement(one)))
% 52.89/7.18 = { by axiom 11 (maddux2_join_associativity) }
% 52.89/7.18 join(join(x1, one), complement(one))
% 52.89/7.18 = { by axiom 3 (goals) }
% 52.89/7.18 join(one, complement(one))
% 52.89/7.18 = { by axiom 7 (def_top) R->L }
% 52.89/7.18 top
% 52.89/7.18
% 52.89/7.18 Lemma 28: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 52.89/7.18 Proof:
% 52.89/7.18 join(meet(X, Y), complement(join(complement(X), Y)))
% 52.89/7.18 = { by axiom 12 (maddux4_definiton_of_meet) }
% 52.89/7.18 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 52.89/7.18 = { by axiom 15 (maddux3_a_kind_of_de_Morgan) R->L }
% 52.89/7.18 X
% 52.89/7.18
% 52.89/7.18 Lemma 29: join(zero, meet(X, X)) = X.
% 52.89/7.18 Proof:
% 52.89/7.18 join(zero, meet(X, X))
% 52.89/7.18 = { by axiom 12 (maddux4_definiton_of_meet) }
% 52.89/7.18 join(zero, complement(join(complement(X), complement(X))))
% 52.89/7.18 = { by axiom 6 (def_zero) }
% 52.89/7.18 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 52.89/7.18 = { by lemma 28 }
% 52.89/7.18 X
% 52.89/7.18
% 52.89/7.18 Lemma 30: join(X, zero) = X.
% 52.89/7.18 Proof:
% 52.89/7.18 join(X, zero)
% 52.89/7.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.18 join(zero, X)
% 52.89/7.18 = { by lemma 29 R->L }
% 52.89/7.18 join(zero, join(zero, meet(X, X)))
% 52.89/7.18 = { by axiom 11 (maddux2_join_associativity) }
% 52.89/7.18 join(join(zero, zero), meet(X, X))
% 52.89/7.18 = { by lemma 17 R->L }
% 52.89/7.18 join(join(zero, complement(top)), meet(X, X))
% 52.89/7.18 = { by lemma 17 R->L }
% 52.89/7.18 join(join(complement(top), complement(top)), meet(X, X))
% 52.89/7.18 = { by lemma 24 }
% 52.89/7.18 join(complement(top), meet(X, X))
% 52.89/7.18 = { by lemma 17 }
% 52.89/7.18 join(zero, meet(X, X))
% 52.89/7.18 = { by lemma 29 }
% 52.89/7.18 X
% 52.89/7.18
% 52.89/7.18 Lemma 31: join(top, X) = top.
% 52.89/7.18 Proof:
% 52.89/7.18 join(top, X)
% 52.89/7.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.18 join(X, top)
% 52.89/7.18 = { by lemma 26 R->L }
% 52.89/7.18 join(Y, top)
% 52.89/7.18 = { by lemma 27 }
% 52.89/7.18 top
% 52.89/7.18
% 52.89/7.18 Lemma 32: join(zero, X) = X.
% 52.89/7.18 Proof:
% 52.89/7.18 join(zero, X)
% 52.89/7.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.18 join(X, zero)
% 52.89/7.18 = { by lemma 30 }
% 52.89/7.18 X
% 52.89/7.18
% 52.89/7.18 Lemma 33: complement(complement(X)) = meet(X, X).
% 52.89/7.18 Proof:
% 52.89/7.18 complement(complement(X))
% 52.89/7.18 = { by lemma 24 R->L }
% 52.89/7.18 complement(join(complement(X), complement(X)))
% 52.89/7.18 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 52.89/7.18 meet(X, X)
% 52.89/7.18
% 52.89/7.18 Lemma 34: complement(complement(X)) = X.
% 52.89/7.18 Proof:
% 52.89/7.18 complement(complement(X))
% 52.89/7.18 = { by lemma 32 R->L }
% 52.89/7.18 join(zero, complement(complement(X)))
% 52.89/7.18 = { by lemma 33 }
% 52.89/7.18 join(zero, meet(X, X))
% 52.89/7.18 = { by lemma 29 }
% 52.89/7.18 X
% 52.89/7.18
% 52.89/7.18 Lemma 35: meet(Y, X) = meet(X, Y).
% 52.89/7.18 Proof:
% 52.89/7.18 meet(Y, X)
% 52.89/7.18 = { by axiom 12 (maddux4_definiton_of_meet) }
% 52.89/7.18 complement(join(complement(Y), complement(X)))
% 52.89/7.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.18 complement(join(complement(X), complement(Y)))
% 52.89/7.18 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 52.89/7.18 meet(X, Y)
% 52.89/7.18
% 52.89/7.18 Lemma 36: complement(join(zero, complement(X))) = meet(X, top).
% 52.89/7.18 Proof:
% 52.89/7.18 complement(join(zero, complement(X)))
% 52.89/7.18 = { by lemma 17 R->L }
% 52.89/7.18 complement(join(complement(top), complement(X)))
% 52.89/7.18 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 52.89/7.18 meet(top, X)
% 52.89/7.18 = { by lemma 35 R->L }
% 52.89/7.18 meet(X, top)
% 52.89/7.18
% 52.89/7.18 Lemma 37: meet(X, top) = X.
% 52.89/7.18 Proof:
% 52.89/7.18 meet(X, top)
% 52.89/7.18 = { by lemma 36 R->L }
% 52.89/7.18 complement(join(zero, complement(X)))
% 52.89/7.18 = { by lemma 32 }
% 52.89/7.18 complement(complement(X))
% 52.89/7.18 = { by lemma 34 }
% 52.89/7.18 X
% 52.89/7.18
% 52.89/7.18 Lemma 38: meet(X, zero) = zero.
% 52.89/7.18 Proof:
% 52.89/7.18 meet(X, zero)
% 52.89/7.18 = { by lemma 35 }
% 52.89/7.18 meet(zero, X)
% 52.89/7.18 = { by axiom 12 (maddux4_definiton_of_meet) }
% 52.89/7.18 complement(join(complement(zero), complement(X)))
% 52.89/7.18 = { by lemma 32 R->L }
% 52.89/7.18 complement(join(join(zero, complement(zero)), complement(X)))
% 52.89/7.18 = { by axiom 7 (def_top) R->L }
% 52.89/7.18 complement(join(top, complement(X)))
% 52.89/7.18 = { by lemma 25 }
% 52.89/7.18 complement(top)
% 52.89/7.18 = { by lemma 17 }
% 52.89/7.18 zero
% 52.89/7.18
% 52.89/7.18 Lemma 39: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 52.89/7.18 Proof:
% 52.89/7.18 converse(join(X, converse(Y)))
% 52.89/7.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.18 converse(join(converse(Y), X))
% 52.89/7.18 = { by axiom 10 (converse_additivity) }
% 52.89/7.18 join(converse(converse(Y)), converse(X))
% 52.89/7.18 = { by axiom 5 (converse_idempotence) }
% 52.89/7.18 join(Y, converse(X))
% 52.89/7.18
% 52.89/7.18 Lemma 40: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 52.89/7.18 Proof:
% 52.89/7.18 join(X, composition(Y, X))
% 52.89/7.18 = { by lemma 22 R->L }
% 52.89/7.18 join(composition(one, X), composition(Y, X))
% 52.89/7.18 = { by axiom 13 (composition_distributivity) R->L }
% 52.89/7.18 composition(join(one, Y), X)
% 52.89/7.18 = { by axiom 2 (maddux1_join_commutativity) }
% 52.89/7.18 composition(join(Y, one), X)
% 52.89/7.18
% 52.89/7.18 Lemma 41: composition(top, zero) = zero.
% 52.89/7.18 Proof:
% 52.89/7.18 composition(top, zero)
% 52.89/7.18 = { by lemma 17 R->L }
% 52.89/7.18 composition(top, complement(top))
% 52.89/7.18 = { by lemma 31 R->L }
% 52.89/7.18 composition(join(top, one), complement(top))
% 52.89/7.18 = { by lemma 31 R->L }
% 52.89/7.18 composition(join(join(top, converse(top)), one), complement(top))
% 52.89/7.18 = { by lemma 39 R->L }
% 52.89/7.18 composition(join(converse(join(top, converse(top))), one), complement(top))
% 52.89/7.18 = { by lemma 31 }
% 52.89/7.18 composition(join(converse(top), one), complement(top))
% 52.89/7.18 = { by lemma 40 R->L }
% 52.89/7.18 join(complement(top), composition(converse(top), complement(top)))
% 52.89/7.18 = { by lemma 31 R->L }
% 52.89/7.18 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 52.89/7.18 = { by lemma 40 }
% 52.89/7.18 join(complement(top), composition(converse(top), complement(composition(join(top, one), top))))
% 52.89/7.18 = { by lemma 31 }
% 52.89/7.18 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 52.89/7.18 = { by lemma 23 }
% 52.89/7.18 complement(top)
% 52.89/7.18 = { by lemma 17 }
% 52.89/7.18 zero
% 52.89/7.18
% 52.89/7.18 Lemma 42: composition(X, zero) = zero.
% 52.89/7.18 Proof:
% 52.89/7.18 composition(X, zero)
% 52.89/7.18 = { by lemma 32 R->L }
% 52.89/7.18 join(zero, composition(X, zero))
% 52.89/7.18 = { by lemma 41 R->L }
% 52.89/7.18 join(composition(top, zero), composition(X, zero))
% 52.89/7.18 = { by axiom 13 (composition_distributivity) R->L }
% 52.89/7.18 composition(join(top, X), zero)
% 52.89/7.18 = { by lemma 31 }
% 52.89/7.18 composition(top, zero)
% 52.89/7.18 = { by lemma 41 }
% 52.89/7.19 zero
% 52.89/7.19
% 52.89/7.19 Lemma 43: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 52.89/7.19 Proof:
% 52.89/7.19 complement(join(complement(X), meet(Y, Z)))
% 52.89/7.19 = { by lemma 35 }
% 52.89/7.19 complement(join(complement(X), meet(Z, Y)))
% 52.89/7.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.19 complement(join(meet(Z, Y), complement(X)))
% 52.89/7.19 = { by axiom 12 (maddux4_definiton_of_meet) }
% 52.89/7.19 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 52.89/7.19 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 52.89/7.19 meet(join(complement(Z), complement(Y)), X)
% 52.89/7.19 = { by lemma 35 R->L }
% 52.89/7.19 meet(X, join(complement(Z), complement(Y)))
% 52.89/7.19 = { by axiom 2 (maddux1_join_commutativity) }
% 52.89/7.19 meet(X, join(complement(Y), complement(Z)))
% 52.89/7.19
% 52.89/7.19 Lemma 44: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 52.89/7.19 Proof:
% 52.89/7.19 join(complement(X), complement(Y))
% 52.89/7.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.19 join(complement(Y), complement(X))
% 52.89/7.19 = { by lemma 37 R->L }
% 52.89/7.19 meet(join(complement(Y), complement(X)), top)
% 52.89/7.19 = { by lemma 35 R->L }
% 52.89/7.19 meet(top, join(complement(Y), complement(X)))
% 52.89/7.19 = { by lemma 43 R->L }
% 52.89/7.19 complement(join(complement(top), meet(Y, X)))
% 52.89/7.19 = { by lemma 17 }
% 52.89/7.19 complement(join(zero, meet(Y, X)))
% 52.89/7.19 = { by lemma 32 }
% 52.89/7.19 complement(meet(Y, X))
% 52.89/7.19 = { by lemma 35 R->L }
% 52.89/7.19 complement(meet(X, Y))
% 52.89/7.19
% 52.89/7.19 Lemma 45: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 52.89/7.19 Proof:
% 52.89/7.19 complement(meet(X, complement(Y)))
% 52.89/7.19 = { by lemma 35 }
% 52.89/7.19 complement(meet(complement(Y), X))
% 52.89/7.19 = { by lemma 32 R->L }
% 52.89/7.19 complement(meet(join(zero, complement(Y)), X))
% 52.89/7.19 = { by lemma 44 R->L }
% 52.89/7.19 join(complement(join(zero, complement(Y))), complement(X))
% 52.89/7.19 = { by lemma 36 }
% 52.89/7.19 join(meet(Y, top), complement(X))
% 52.89/7.19 = { by lemma 37 }
% 52.89/7.19 join(Y, complement(X))
% 52.89/7.19
% 52.89/7.19 Lemma 46: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 52.89/7.19 Proof:
% 52.89/7.19 complement(meet(complement(X), Y))
% 52.89/7.19 = { by lemma 35 }
% 52.89/7.19 complement(meet(Y, complement(X)))
% 52.89/7.19 = { by lemma 45 }
% 52.89/7.19 join(X, complement(Y))
% 52.89/7.19
% 52.89/7.19 Lemma 47: join(X, complement(meet(X, Y))) = top.
% 52.89/7.19 Proof:
% 52.89/7.19 join(X, complement(meet(X, Y)))
% 52.89/7.19 = { by lemma 35 }
% 52.89/7.19 join(X, complement(meet(Y, X)))
% 52.89/7.19 = { by lemma 44 R->L }
% 52.89/7.19 join(X, join(complement(Y), complement(X)))
% 52.89/7.19 = { by lemma 18 }
% 52.89/7.19 join(complement(Y), top)
% 52.89/7.19 = { by lemma 27 }
% 52.89/7.19 top
% 52.89/7.19
% 52.89/7.19 Lemma 48: meet(X, join(X, complement(Y))) = X.
% 52.89/7.19 Proof:
% 52.89/7.19 meet(X, join(X, complement(Y)))
% 52.89/7.19 = { by lemma 30 R->L }
% 52.89/7.19 join(meet(X, join(X, complement(Y))), zero)
% 52.89/7.19 = { by lemma 17 R->L }
% 52.89/7.19 join(meet(X, join(X, complement(Y))), complement(top))
% 52.89/7.19 = { by lemma 46 R->L }
% 52.89/7.19 join(meet(X, complement(meet(complement(X), Y))), complement(top))
% 52.89/7.19 = { by lemma 47 R->L }
% 52.89/7.19 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(complement(X), Y)))))
% 52.89/7.19 = { by lemma 28 }
% 52.89/7.19 X
% 52.89/7.19
% 52.89/7.19 Lemma 49: join(X, meet(X, Y)) = X.
% 52.89/7.19 Proof:
% 52.89/7.19 join(X, meet(X, Y))
% 52.89/7.19 = { by axiom 12 (maddux4_definiton_of_meet) }
% 52.89/7.19 join(X, complement(join(complement(X), complement(Y))))
% 52.89/7.19 = { by lemma 46 R->L }
% 52.89/7.19 complement(meet(complement(X), join(complement(X), complement(Y))))
% 52.89/7.19 = { by lemma 48 }
% 52.89/7.19 complement(complement(X))
% 52.89/7.19 = { by lemma 34 }
% 52.89/7.19 X
% 52.89/7.19
% 52.89/7.19 Lemma 50: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 52.89/7.19 Proof:
% 52.89/7.19 converse(composition(X, converse(Y)))
% 52.89/7.19 = { by axiom 8 (converse_multiplicativity) }
% 52.89/7.19 composition(converse(converse(Y)), converse(X))
% 52.89/7.19 = { by axiom 5 (converse_idempotence) }
% 52.89/7.19 composition(Y, converse(X))
% 52.89/7.19
% 52.89/7.19 Lemma 51: join(X, composition(X, Y)) = composition(X, join(Y, one)).
% 52.89/7.19 Proof:
% 52.89/7.19 join(X, composition(X, Y))
% 52.89/7.19 = { by axiom 5 (converse_idempotence) R->L }
% 52.89/7.19 join(X, composition(X, converse(converse(Y))))
% 52.89/7.19 = { by lemma 50 R->L }
% 52.89/7.19 join(X, converse(composition(converse(Y), converse(X))))
% 52.89/7.19 = { by lemma 39 R->L }
% 52.89/7.19 converse(join(composition(converse(Y), converse(X)), converse(X)))
% 52.89/7.19 = { by axiom 2 (maddux1_join_commutativity) }
% 52.89/7.19 converse(join(converse(X), composition(converse(Y), converse(X))))
% 52.89/7.19 = { by lemma 40 }
% 52.89/7.19 converse(composition(join(converse(Y), one), converse(X)))
% 52.89/7.19 = { by lemma 50 }
% 52.89/7.19 composition(X, converse(join(converse(Y), one)))
% 52.89/7.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.19 composition(X, converse(join(one, converse(Y))))
% 52.89/7.19 = { by axiom 10 (converse_additivity) }
% 52.89/7.19 composition(X, join(converse(one), converse(converse(Y))))
% 52.89/7.19 = { by axiom 1 (composition_identity) R->L }
% 52.89/7.19 composition(X, join(composition(converse(one), one), converse(converse(Y))))
% 52.89/7.19 = { by lemma 20 }
% 52.89/7.19 composition(X, join(one, converse(converse(Y))))
% 52.89/7.19 = { by axiom 5 (converse_idempotence) }
% 52.89/7.19 composition(X, join(one, Y))
% 52.89/7.19 = { by axiom 2 (maddux1_join_commutativity) }
% 52.89/7.19 composition(X, join(Y, one))
% 52.89/7.19
% 52.89/7.19 Lemma 52: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 52.89/7.19 Proof:
% 52.89/7.19 complement(join(X, complement(Y)))
% 52.89/7.19 = { by lemma 32 R->L }
% 52.89/7.19 complement(join(zero, join(X, complement(Y))))
% 52.89/7.19 = { by lemma 45 R->L }
% 52.89/7.19 complement(join(zero, complement(meet(Y, complement(X)))))
% 52.89/7.19 = { by lemma 36 }
% 52.89/7.19 meet(meet(Y, complement(X)), top)
% 52.89/7.19 = { by lemma 37 }
% 52.89/7.19 meet(Y, complement(X))
% 52.89/7.19
% 52.89/7.19 Lemma 53: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 52.89/7.19 Proof:
% 52.89/7.19 complement(join(complement(X), Y))
% 52.89/7.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.19 complement(join(Y, complement(X)))
% 52.89/7.19 = { by lemma 52 }
% 52.89/7.19 meet(X, complement(Y))
% 52.89/7.19
% 52.89/7.19 Lemma 54: meet(X, join(X, Y)) = X.
% 52.89/7.19 Proof:
% 52.89/7.19 meet(X, join(X, Y))
% 52.89/7.19 = { by lemma 37 R->L }
% 52.89/7.19 meet(X, join(X, meet(Y, top)))
% 52.89/7.19 = { by lemma 36 R->L }
% 52.89/7.19 meet(X, join(X, complement(join(zero, complement(Y)))))
% 52.89/7.19 = { by lemma 48 }
% 52.89/7.19 X
% 52.89/7.19
% 52.89/7.19 Lemma 55: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 52.89/7.19 Proof:
% 52.89/7.19 meet(Y, meet(X, Z))
% 52.89/7.19 = { by lemma 35 }
% 52.89/7.19 meet(Y, meet(Z, X))
% 52.89/7.19 = { by lemma 37 R->L }
% 52.89/7.19 meet(meet(Y, meet(Z, X)), top)
% 52.89/7.19 = { by lemma 36 R->L }
% 52.89/7.19 complement(join(zero, complement(meet(Y, meet(Z, X)))))
% 52.89/7.19 = { by lemma 35 }
% 52.89/7.19 complement(join(zero, complement(meet(Y, meet(X, Z)))))
% 52.89/7.19 = { by axiom 12 (maddux4_definiton_of_meet) }
% 52.89/7.19 complement(join(zero, complement(meet(Y, complement(join(complement(X), complement(Z)))))))
% 52.89/7.19 = { by lemma 45 }
% 52.89/7.19 complement(join(zero, join(join(complement(X), complement(Z)), complement(Y))))
% 52.89/7.19 = { by axiom 11 (maddux2_join_associativity) R->L }
% 52.89/7.19 complement(join(zero, join(complement(X), join(complement(Z), complement(Y)))))
% 52.89/7.19 = { by lemma 44 }
% 52.89/7.19 complement(join(zero, join(complement(X), complement(meet(Z, Y)))))
% 52.89/7.19 = { by lemma 44 }
% 52.89/7.19 complement(join(zero, complement(meet(X, meet(Z, Y)))))
% 52.89/7.19 = { by lemma 35 R->L }
% 52.89/7.19 complement(join(zero, complement(meet(X, meet(Y, Z)))))
% 52.89/7.19 = { by lemma 36 }
% 52.89/7.19 meet(meet(X, meet(Y, Z)), top)
% 52.89/7.19 = { by lemma 37 }
% 52.89/7.19 meet(X, meet(Y, Z))
% 52.89/7.19
% 52.89/7.19 Lemma 56: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 52.89/7.19 Proof:
% 52.89/7.19 meet(complement(X), complement(Y))
% 52.89/7.19 = { by lemma 35 }
% 52.89/7.19 meet(complement(Y), complement(X))
% 52.89/7.19 = { by lemma 32 R->L }
% 52.89/7.19 meet(join(zero, complement(Y)), complement(X))
% 52.89/7.20 = { by lemma 52 R->L }
% 52.89/7.20 complement(join(X, complement(join(zero, complement(Y)))))
% 52.89/7.20 = { by lemma 36 }
% 52.89/7.20 complement(join(X, meet(Y, top)))
% 52.89/7.20 = { by lemma 37 }
% 52.89/7.20 complement(join(X, Y))
% 52.89/7.20
% 52.89/7.20 Lemma 57: meet(x0, composition(converse(X), complement(X))) = zero.
% 52.89/7.20 Proof:
% 52.89/7.20 meet(x0, composition(converse(X), complement(X)))
% 52.89/7.20 = { by lemma 35 }
% 52.89/7.20 meet(composition(converse(X), complement(X)), x0)
% 52.89/7.20 = { by lemma 28 R->L }
% 52.89/7.20 meet(composition(converse(X), complement(X)), join(meet(x0, complement(one)), complement(join(complement(x0), complement(one)))))
% 52.89/7.20 = { by lemma 52 R->L }
% 52.89/7.20 meet(composition(converse(X), complement(X)), join(complement(join(one, complement(x0))), complement(join(complement(x0), complement(one)))))
% 52.89/7.20 = { by axiom 4 (goals_1) R->L }
% 52.89/7.20 meet(composition(converse(X), complement(X)), join(complement(join(join(x0, one), complement(x0))), complement(join(complement(x0), complement(one)))))
% 52.89/7.20 = { by axiom 11 (maddux2_join_associativity) R->L }
% 52.89/7.20 meet(composition(converse(X), complement(X)), join(complement(join(x0, join(one, complement(x0)))), complement(join(complement(x0), complement(one)))))
% 52.89/7.20 = { by lemma 18 }
% 52.89/7.20 meet(composition(converse(X), complement(X)), join(complement(join(one, top)), complement(join(complement(x0), complement(one)))))
% 52.89/7.20 = { by lemma 27 }
% 52.89/7.20 meet(composition(converse(X), complement(X)), join(complement(top), complement(join(complement(x0), complement(one)))))
% 52.89/7.20 = { by lemma 17 }
% 52.89/7.20 meet(composition(converse(X), complement(X)), join(zero, complement(join(complement(x0), complement(one)))))
% 52.89/7.20 = { by lemma 32 }
% 52.89/7.20 meet(composition(converse(X), complement(X)), complement(join(complement(x0), complement(one))))
% 52.89/7.20 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 52.89/7.20 meet(composition(converse(X), complement(X)), meet(x0, one))
% 52.89/7.20 = { by lemma 55 R->L }
% 52.89/7.20 meet(x0, meet(composition(converse(X), complement(X)), one))
% 52.89/7.20 = { by lemma 34 R->L }
% 52.89/7.20 meet(x0, meet(composition(converse(X), complement(X)), complement(complement(one))))
% 52.89/7.20 = { by lemma 23 R->L }
% 52.89/7.20 meet(x0, meet(composition(converse(X), complement(X)), complement(join(complement(one), composition(converse(X), complement(composition(X, one)))))))
% 52.89/7.20 = { by axiom 1 (composition_identity) }
% 52.89/7.20 meet(x0, meet(composition(converse(X), complement(X)), complement(join(complement(one), composition(converse(X), complement(X))))))
% 52.89/7.20 = { by lemma 56 R->L }
% 52.89/7.20 meet(x0, meet(composition(converse(X), complement(X)), meet(complement(complement(one)), complement(composition(converse(X), complement(X))))))
% 52.89/7.20 = { by lemma 35 }
% 52.89/7.20 meet(x0, meet(composition(converse(X), complement(X)), meet(complement(composition(converse(X), complement(X))), complement(complement(one)))))
% 52.89/7.20 = { by axiom 12 (maddux4_definiton_of_meet) }
% 52.89/7.20 meet(x0, complement(join(complement(composition(converse(X), complement(X))), complement(meet(complement(composition(converse(X), complement(X))), complement(complement(one)))))))
% 52.89/7.20 = { by lemma 47 }
% 52.89/7.20 meet(x0, complement(top))
% 52.89/7.20 = { by lemma 17 }
% 52.89/7.20 meet(x0, zero)
% 52.89/7.20 = { by lemma 38 }
% 52.89/7.20 zero
% 52.89/7.20
% 52.89/7.20 Lemma 58: join(meet(X, Y), meet(X, complement(Y))) = X.
% 52.89/7.20 Proof:
% 52.89/7.20 join(meet(X, Y), meet(X, complement(Y)))
% 52.89/7.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.20 join(meet(X, complement(Y)), meet(X, Y))
% 52.89/7.20 = { by axiom 12 (maddux4_definiton_of_meet) }
% 52.89/7.20 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 52.89/7.20 = { by lemma 28 }
% 52.89/7.20 X
% 52.89/7.20
% 52.89/7.20 Lemma 59: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 52.89/7.20 Proof:
% 52.89/7.20 join(meet(X, Y), meet(Y, complement(X)))
% 52.89/7.20 = { by lemma 35 }
% 52.89/7.20 join(meet(Y, X), meet(Y, complement(X)))
% 52.89/7.20 = { by lemma 58 }
% 52.89/7.20 Y
% 52.89/7.20
% 52.89/7.20 Lemma 60: meet(X, composition(X, x0)) = composition(X, x0).
% 52.89/7.20 Proof:
% 52.89/7.20 meet(X, composition(X, x0))
% 52.89/7.20 = { by lemma 35 }
% 52.89/7.20 meet(composition(X, x0), X)
% 52.89/7.20 = { by lemma 34 R->L }
% 52.89/7.20 meet(composition(X, x0), complement(complement(X)))
% 52.89/7.20 = { by lemma 32 R->L }
% 52.89/7.20 join(zero, meet(composition(X, x0), complement(complement(X))))
% 52.89/7.20 = { by lemma 42 R->L }
% 52.89/7.20 join(composition(meet(X, composition(complement(X), converse(x0))), zero), meet(composition(X, x0), complement(complement(X))))
% 52.89/7.20 = { by lemma 57 R->L }
% 52.89/7.20 join(composition(meet(X, composition(complement(X), converse(x0))), meet(x0, composition(converse(X), complement(X)))), meet(composition(X, x0), complement(complement(X))))
% 52.89/7.20 = { by axiom 16 (dedekind_law) R->L }
% 52.89/7.20 join(join(meet(composition(X, x0), complement(X)), composition(meet(X, composition(complement(X), converse(x0))), meet(x0, composition(converse(X), complement(X))))), meet(composition(X, x0), complement(complement(X))))
% 52.89/7.20 = { by lemma 57 }
% 52.89/7.20 join(join(meet(composition(X, x0), complement(X)), composition(meet(X, composition(complement(X), converse(x0))), zero)), meet(composition(X, x0), complement(complement(X))))
% 52.89/7.20 = { by lemma 42 }
% 52.89/7.20 join(join(meet(composition(X, x0), complement(X)), zero), meet(composition(X, x0), complement(complement(X))))
% 52.89/7.20 = { by lemma 30 }
% 52.89/7.20 join(meet(composition(X, x0), complement(X)), meet(composition(X, x0), complement(complement(X))))
% 52.89/7.20 = { by lemma 35 R->L }
% 52.89/7.20 join(meet(complement(X), composition(X, x0)), meet(composition(X, x0), complement(complement(X))))
% 52.89/7.20 = { by lemma 59 }
% 52.89/7.20 composition(X, x0)
% 52.89/7.20
% 52.89/7.20 Lemma 61: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
% 52.89/7.20 Proof:
% 52.89/7.20 join(complement(X), meet(X, Y))
% 52.89/7.20 = { by lemma 35 }
% 52.89/7.20 join(complement(X), meet(Y, X))
% 52.89/7.20 = { by lemma 49 R->L }
% 52.89/7.20 join(join(complement(X), meet(complement(X), Y)), meet(Y, X))
% 52.89/7.20 = { by axiom 11 (maddux2_join_associativity) R->L }
% 52.89/7.20 join(complement(X), join(meet(complement(X), Y), meet(Y, X)))
% 52.89/7.20 = { by axiom 2 (maddux1_join_commutativity) }
% 52.89/7.20 join(complement(X), join(meet(Y, X), meet(complement(X), Y)))
% 52.89/7.20 = { by lemma 35 }
% 52.89/7.20 join(complement(X), join(meet(Y, X), meet(Y, complement(X))))
% 52.89/7.20 = { by lemma 58 }
% 52.89/7.20 join(complement(X), Y)
% 52.89/7.20 = { by axiom 2 (maddux1_join_commutativity) }
% 52.89/7.20 join(Y, complement(X))
% 52.89/7.20
% 52.89/7.20 Lemma 62: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 52.89/7.20 Proof:
% 52.89/7.20 meet(X, complement(meet(X, Y)))
% 52.89/7.20 = { by lemma 44 R->L }
% 52.89/7.20 meet(X, join(complement(X), complement(Y)))
% 52.89/7.20 = { by lemma 43 R->L }
% 52.89/7.20 complement(join(complement(X), meet(X, Y)))
% 52.89/7.20 = { by lemma 61 }
% 52.89/7.20 complement(join(Y, complement(X)))
% 52.89/7.20 = { by lemma 52 }
% 52.89/7.20 meet(X, complement(Y))
% 52.89/7.20
% 52.89/7.20 Lemma 63: meet(X, composition(x1, X)) = composition(x1, X).
% 52.89/7.20 Proof:
% 52.89/7.20 meet(X, composition(x1, X))
% 52.89/7.20 = { by lemma 35 }
% 52.89/7.20 meet(composition(x1, X), X)
% 52.89/7.20 = { by lemma 34 R->L }
% 52.89/7.20 meet(composition(x1, X), complement(complement(X)))
% 52.89/7.20 = { by lemma 62 R->L }
% 52.89/7.20 meet(composition(x1, X), complement(meet(composition(x1, X), complement(X))))
% 52.89/7.20 = { by lemma 45 }
% 52.89/7.20 meet(composition(x1, X), join(X, complement(composition(x1, X))))
% 52.89/7.20 = { by lemma 34 R->L }
% 52.89/7.20 meet(composition(x1, X), join(X, complement(composition(x1, complement(complement(X))))))
% 52.89/7.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.20 meet(composition(x1, X), join(complement(composition(x1, complement(complement(X)))), X))
% 52.89/7.20 = { by lemma 34 R->L }
% 52.89/7.20 meet(composition(x1, X), join(complement(composition(x1, complement(complement(X)))), complement(complement(X))))
% 52.89/7.20 = { by lemma 61 R->L }
% 52.89/7.20 meet(composition(x1, X), join(complement(complement(X)), meet(complement(X), complement(composition(x1, complement(complement(X)))))))
% 52.89/7.20 = { by lemma 33 }
% 52.89/7.21 meet(composition(x1, X), join(meet(X, X), meet(complement(X), complement(composition(x1, complement(complement(X)))))))
% 52.89/7.21 = { by lemma 32 R->L }
% 52.89/7.21 meet(composition(x1, X), join(join(zero, meet(X, X)), meet(complement(X), complement(composition(x1, complement(complement(X)))))))
% 52.89/7.21 = { by lemma 29 }
% 52.89/7.21 meet(composition(x1, X), join(X, meet(complement(X), complement(composition(x1, complement(complement(X)))))))
% 52.89/7.21 = { by lemma 53 R->L }
% 52.89/7.21 meet(composition(x1, X), join(X, complement(join(complement(complement(X)), composition(x1, complement(complement(X)))))))
% 52.89/7.21 = { by lemma 40 }
% 52.89/7.21 meet(composition(x1, X), join(X, complement(composition(join(x1, one), complement(complement(X))))))
% 52.89/7.21 = { by axiom 3 (goals) }
% 52.89/7.21 meet(composition(x1, X), join(X, complement(composition(one, complement(complement(X))))))
% 52.89/7.21 = { by lemma 22 }
% 52.89/7.21 meet(composition(x1, X), join(X, complement(complement(complement(X)))))
% 52.89/7.21 = { by lemma 34 }
% 52.89/7.21 meet(composition(x1, X), join(X, complement(X)))
% 52.89/7.21 = { by axiom 7 (def_top) R->L }
% 52.89/7.21 meet(composition(x1, X), top)
% 52.89/7.21 = { by lemma 37 }
% 52.89/7.21 composition(x1, X)
% 52.89/7.21
% 52.89/7.21 Lemma 64: meet(Z, meet(Y, X)) = meet(X, meet(Y, Z)).
% 52.89/7.21 Proof:
% 52.89/7.21 meet(Z, meet(Y, X))
% 52.89/7.21 = { by lemma 35 R->L }
% 52.89/7.21 meet(Z, meet(X, Y))
% 52.89/7.21 = { by lemma 55 R->L }
% 52.89/7.21 meet(X, meet(Z, Y))
% 52.89/7.21 = { by lemma 35 R->L }
% 52.89/7.21 meet(X, meet(Y, Z))
% 52.89/7.21
% 52.89/7.21 Lemma 65: meet(meet(X, Y), Y) = meet(X, Y).
% 52.89/7.21 Proof:
% 52.89/7.21 meet(meet(X, Y), Y)
% 52.89/7.21 = { by axiom 12 (maddux4_definiton_of_meet) }
% 52.89/7.21 complement(join(complement(meet(X, Y)), complement(Y)))
% 52.89/7.21 = { by lemma 32 R->L }
% 52.89/7.21 join(zero, complement(join(complement(meet(X, Y)), complement(Y))))
% 52.89/7.21 = { by lemma 17 R->L }
% 52.89/7.21 join(complement(top), complement(join(complement(meet(X, Y)), complement(Y))))
% 52.89/7.21 = { by lemma 47 R->L }
% 52.89/7.21 join(complement(join(Y, complement(meet(Y, X)))), complement(join(complement(meet(X, Y)), complement(Y))))
% 52.89/7.21 = { by lemma 35 }
% 52.89/7.21 join(complement(join(Y, complement(meet(X, Y)))), complement(join(complement(meet(X, Y)), complement(Y))))
% 52.89/7.21 = { by lemma 52 }
% 52.89/7.21 join(meet(meet(X, Y), complement(Y)), complement(join(complement(meet(X, Y)), complement(Y))))
% 52.89/7.21 = { by lemma 28 }
% 52.89/7.21 meet(X, Y)
% 52.89/7.21
% 52.89/7.21 Lemma 66: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 52.89/7.21 Proof:
% 52.89/7.21 meet(complement(X), join(X, Y))
% 52.89/7.21 = { by lemma 35 }
% 52.89/7.21 meet(join(X, Y), complement(X))
% 52.89/7.21 = { by lemma 52 R->L }
% 52.89/7.21 complement(join(X, complement(join(X, Y))))
% 52.89/7.21 = { by lemma 56 R->L }
% 52.89/7.21 meet(complement(X), complement(complement(join(X, Y))))
% 52.89/7.21 = { by lemma 56 R->L }
% 52.89/7.21 meet(complement(X), complement(meet(complement(X), complement(Y))))
% 52.89/7.21 = { by lemma 62 }
% 52.89/7.21 meet(complement(X), complement(complement(Y)))
% 52.89/7.21 = { by lemma 56 }
% 52.89/7.21 complement(join(X, complement(Y)))
% 52.89/7.21 = { by lemma 52 }
% 52.89/7.21 meet(Y, complement(X))
% 52.89/7.21
% 52.89/7.21 Lemma 67: composition(converse(X), complement(composition(X, top))) = zero.
% 52.89/7.21 Proof:
% 52.89/7.21 composition(converse(X), complement(composition(X, top)))
% 52.89/7.21 = { by lemma 32 R->L }
% 52.89/7.21 join(zero, composition(converse(X), complement(composition(X, top))))
% 52.89/7.21 = { by lemma 17 R->L }
% 52.89/7.21 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 52.89/7.21 = { by lemma 23 }
% 52.89/7.21 complement(top)
% 52.89/7.21 = { by lemma 17 }
% 52.89/7.21 zero
% 52.89/7.21
% 52.89/7.21 Lemma 68: meet(meet(X, Y), complement(meet(X, Z))) = meet(meet(X, Y), complement(Z)).
% 52.89/7.21 Proof:
% 52.89/7.21 meet(meet(X, Y), complement(meet(X, Z)))
% 52.89/7.21 = { by lemma 35 }
% 52.89/7.21 meet(meet(X, Y), complement(meet(Z, X)))
% 52.89/7.21 = { by lemma 35 }
% 52.89/7.21 meet(complement(meet(Z, X)), meet(X, Y))
% 52.89/7.21 = { by lemma 64 }
% 52.89/7.21 meet(Y, meet(X, complement(meet(Z, X))))
% 52.89/7.21 = { by lemma 35 }
% 52.89/7.21 meet(Y, meet(X, complement(meet(X, Z))))
% 52.89/7.21 = { by lemma 62 }
% 52.89/7.21 meet(Y, meet(X, complement(Z)))
% 52.89/7.21 = { by lemma 64 R->L }
% 52.89/7.21 meet(complement(Z), meet(X, Y))
% 52.89/7.21 = { by lemma 35 R->L }
% 52.89/7.21 meet(meet(X, Y), complement(Z))
% 52.89/7.21
% 52.89/7.21 Goal 1 (goals_2): meet(composition(x0, x2), composition(x1, x2)) = composition(meet(x0, x1), x2).
% 52.89/7.21 Proof:
% 52.89/7.21 meet(composition(x0, x2), composition(x1, x2))
% 52.89/7.21 = { by lemma 65 R->L }
% 52.89/7.21 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x1, x2))
% 52.89/7.21 = { by lemma 59 R->L }
% 52.89/7.21 meet(meet(composition(x0, x2), composition(x1, x2)), join(meet(composition(x1, top), composition(x1, x2)), meet(composition(x1, x2), complement(composition(x1, top)))))
% 52.89/7.21 = { by lemma 30 R->L }
% 52.89/7.21 meet(meet(composition(x0, x2), composition(x1, x2)), join(meet(composition(x1, top), composition(x1, x2)), join(meet(composition(x1, x2), complement(composition(x1, top))), zero)))
% 52.89/7.21 = { by lemma 42 R->L }
% 52.89/7.21 meet(meet(composition(x0, x2), composition(x1, x2)), join(meet(composition(x1, top), composition(x1, x2)), join(meet(composition(x1, x2), complement(composition(x1, top))), composition(meet(x1, composition(complement(composition(x1, top)), converse(x2))), zero))))
% 52.89/7.21 = { by lemma 38 R->L }
% 52.89/7.21 meet(meet(composition(x0, x2), composition(x1, x2)), join(meet(composition(x1, top), composition(x1, x2)), join(meet(composition(x1, x2), complement(composition(x1, top))), composition(meet(x1, composition(complement(composition(x1, top)), converse(x2))), meet(x2, zero)))))
% 52.89/7.21 = { by lemma 67 R->L }
% 52.89/7.21 meet(meet(composition(x0, x2), composition(x1, x2)), join(meet(composition(x1, top), composition(x1, x2)), join(meet(composition(x1, x2), complement(composition(x1, top))), composition(meet(x1, composition(complement(composition(x1, top)), converse(x2))), meet(x2, composition(converse(x1), complement(composition(x1, top))))))))
% 52.89/7.21 = { by axiom 16 (dedekind_law) }
% 52.89/7.21 meet(meet(composition(x0, x2), composition(x1, x2)), join(meet(composition(x1, top), composition(x1, x2)), composition(meet(x1, composition(complement(composition(x1, top)), converse(x2))), meet(x2, composition(converse(x1), complement(composition(x1, top)))))))
% 52.89/7.21 = { by lemma 67 }
% 52.89/7.21 meet(meet(composition(x0, x2), composition(x1, x2)), join(meet(composition(x1, top), composition(x1, x2)), composition(meet(x1, composition(complement(composition(x1, top)), converse(x2))), meet(x2, zero))))
% 52.89/7.21 = { by lemma 38 }
% 52.89/7.21 meet(meet(composition(x0, x2), composition(x1, x2)), join(meet(composition(x1, top), composition(x1, x2)), composition(meet(x1, composition(complement(composition(x1, top)), converse(x2))), zero)))
% 52.89/7.21 = { by lemma 42 }
% 52.89/7.21 meet(meet(composition(x0, x2), composition(x1, x2)), join(meet(composition(x1, top), composition(x1, x2)), zero))
% 52.89/7.21 = { by lemma 30 }
% 52.89/7.21 meet(meet(composition(x0, x2), composition(x1, x2)), meet(composition(x1, top), composition(x1, x2)))
% 52.89/7.21 = { by lemma 55 }
% 52.89/7.21 meet(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), composition(x1, x2)))
% 52.89/7.21 = { by lemma 65 }
% 52.89/7.21 meet(composition(x1, top), meet(composition(x0, x2), composition(x1, x2)))
% 52.89/7.21 = { by lemma 35 R->L }
% 52.89/7.21 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x1, top))
% 52.89/7.21 = { by lemma 34 R->L }
% 52.89/7.21 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(composition(x1, top))))
% 52.89/7.22 = { by lemma 68 R->L }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(meet(composition(x0, x2), complement(composition(x1, top)))))
% 52.89/7.22 = { by lemma 53 R->L }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), composition(x1, top)))))
% 52.89/7.22 = { by axiom 7 (def_top) }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), composition(x1, join(complement(composition(x0, x2)), complement(complement(composition(x0, x2)))))))))
% 52.89/7.22 = { by axiom 5 (converse_idempotence) R->L }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), composition(x1, join(complement(composition(x0, x2)), converse(converse(complement(complement(composition(x0, x2)))))))))))
% 52.89/7.22 = { by axiom 5 (converse_idempotence) R->L }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), composition(converse(converse(x1)), join(complement(composition(x0, x2)), converse(converse(complement(complement(composition(x0, x2)))))))))))
% 52.89/7.22 = { by lemma 39 R->L }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), composition(converse(converse(x1)), converse(join(converse(complement(complement(composition(x0, x2)))), converse(complement(composition(x0, x2))))))))))
% 52.89/7.22 = { by axiom 8 (converse_multiplicativity) R->L }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), converse(composition(join(converse(complement(complement(composition(x0, x2)))), converse(complement(composition(x0, x2)))), converse(x1)))))))
% 52.89/7.22 = { by axiom 13 (composition_distributivity) }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), converse(join(composition(converse(complement(complement(composition(x0, x2)))), converse(x1)), composition(converse(complement(composition(x0, x2))), converse(x1))))))))
% 52.89/7.22 = { by axiom 8 (converse_multiplicativity) R->L }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), converse(join(converse(composition(x1, complement(complement(composition(x0, x2))))), composition(converse(complement(composition(x0, x2))), converse(x1))))))))
% 52.89/7.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), converse(join(composition(converse(complement(composition(x0, x2))), converse(x1)), converse(composition(x1, complement(complement(composition(x0, x2)))))))))))
% 52.89/7.22 = { by axiom 10 (converse_additivity) }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), join(converse(composition(converse(complement(composition(x0, x2))), converse(x1))), converse(converse(composition(x1, complement(complement(composition(x0, x2)))))))))))
% 52.89/7.22 = { by lemma 19 }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), join(composition(converse(converse(x1)), complement(composition(x0, x2))), converse(converse(composition(x1, complement(complement(composition(x0, x2)))))))))))
% 52.89/7.22 = { by axiom 2 (maddux1_join_commutativity) }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), join(converse(converse(composition(x1, complement(complement(composition(x0, x2)))))), composition(converse(converse(x1)), complement(composition(x0, x2))))))))
% 52.89/7.22 = { by axiom 5 (converse_idempotence) }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), join(composition(x1, complement(complement(composition(x0, x2)))), composition(converse(converse(x1)), complement(composition(x0, x2))))))))
% 52.89/7.22 = { by axiom 5 (converse_idempotence) }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), join(composition(x1, complement(complement(composition(x0, x2)))), composition(x1, complement(composition(x0, x2))))))))
% 52.89/7.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), join(composition(x1, complement(composition(x0, x2))), composition(x1, complement(complement(composition(x0, x2)))))))))
% 52.89/7.22 = { by axiom 11 (maddux2_join_associativity) }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(join(complement(composition(x0, x2)), composition(x1, complement(composition(x0, x2)))), composition(x1, complement(complement(composition(x0, x2))))))))
% 52.89/7.22 = { by lemma 40 }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(composition(join(x1, one), complement(composition(x0, x2))), composition(x1, complement(complement(composition(x0, x2))))))))
% 52.89/7.22 = { by axiom 3 (goals) }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(composition(one, complement(composition(x0, x2))), composition(x1, complement(complement(composition(x0, x2))))))))
% 52.89/7.22 = { by lemma 22 }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(join(complement(composition(x0, x2)), composition(x1, complement(complement(composition(x0, x2))))))))
% 52.89/7.22 = { by lemma 53 }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(meet(composition(x0, x2), complement(composition(x1, complement(complement(composition(x0, x2))))))))
% 52.89/7.22 = { by lemma 34 }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(meet(composition(x0, x2), complement(composition(x1, composition(x0, x2))))))
% 52.89/7.22 = { by lemma 68 }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), complement(complement(composition(x1, composition(x0, x2)))))
% 52.89/7.22 = { by lemma 34 }
% 52.89/7.22 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x1, composition(x0, x2)))
% 52.89/7.22 = { by lemma 35 }
% 52.89/7.22 meet(composition(x1, composition(x0, x2)), meet(composition(x0, x2), composition(x1, x2)))
% 52.89/7.22 = { by lemma 64 }
% 52.89/7.22 meet(composition(x1, x2), meet(composition(x0, x2), composition(x1, composition(x0, x2))))
% 52.89/7.22 = { by lemma 63 }
% 52.89/7.22 meet(composition(x1, x2), composition(x1, composition(x0, x2)))
% 52.89/7.22 = { by lemma 35 }
% 52.89/7.22 meet(composition(x1, composition(x0, x2)), composition(x1, x2))
% 52.89/7.22 = { by lemma 21 R->L }
% 52.89/7.22 meet(composition(x1, composition(x0, x2)), composition(x1, composition(one, x2)))
% 52.89/7.22 = { by axiom 4 (goals_1) R->L }
% 52.89/7.22 meet(composition(x1, composition(x0, x2)), composition(x1, composition(join(x0, one), x2)))
% 52.89/7.22 = { by axiom 9 (composition_associativity) }
% 52.89/7.22 meet(composition(x1, composition(x0, x2)), composition(composition(x1, join(x0, one)), x2))
% 52.89/7.22 = { by lemma 51 R->L }
% 52.89/7.22 meet(composition(x1, composition(x0, x2)), composition(join(x1, composition(x1, x0)), x2))
% 52.89/7.22 = { by axiom 13 (composition_distributivity) }
% 52.89/7.22 meet(composition(x1, composition(x0, x2)), join(composition(x1, x2), composition(composition(x1, x0), x2)))
% 52.89/7.22 = { by axiom 9 (composition_associativity) R->L }
% 52.89/7.22 meet(composition(x1, composition(x0, x2)), join(composition(x1, x2), composition(x1, composition(x0, x2))))
% 52.89/7.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.22 meet(composition(x1, composition(x0, x2)), join(composition(x1, composition(x0, x2)), composition(x1, x2)))
% 52.89/7.22 = { by lemma 54 }
% 52.89/7.22 composition(x1, composition(x0, x2))
% 52.89/7.22 = { by lemma 49 R->L }
% 52.89/7.22 composition(join(x1, meet(x1, x0)), composition(x0, x2))
% 52.89/7.22 = { by lemma 35 }
% 52.89/7.22 composition(join(x1, meet(x0, x1)), composition(x0, x2))
% 52.89/7.22 = { by axiom 9 (composition_associativity) }
% 52.89/7.22 composition(composition(join(x1, meet(x0, x1)), x0), x2)
% 52.89/7.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 52.89/7.22 composition(composition(join(meet(x0, x1), x1), x0), x2)
% 52.89/7.22 = { by axiom 13 (composition_distributivity) }
% 52.89/7.22 composition(join(composition(meet(x0, x1), x0), composition(x1, x0)), x2)
% 52.89/7.22 = { by lemma 60 R->L }
% 52.89/7.22 composition(join(meet(meet(x0, x1), composition(meet(x0, x1), x0)), composition(x1, x0)), x2)
% 52.89/7.22 = { by lemma 35 }
% 52.89/7.22 composition(join(meet(composition(meet(x0, x1), x0), meet(x0, x1)), composition(x1, x0)), x2)
% 52.89/7.22 = { by lemma 34 R->L }
% 52.89/7.22 composition(join(meet(composition(meet(x0, x1), x0), complement(complement(meet(x0, x1)))), composition(x1, x0)), x2)
% 52.89/7.22 = { by lemma 66 R->L }
% 52.89/7.23 composition(join(meet(complement(complement(meet(x0, x1))), join(complement(meet(x0, x1)), composition(meet(x0, x1), x0))), composition(x1, x0)), x2)
% 52.89/7.23 = { by lemma 34 R->L }
% 52.89/7.23 composition(join(meet(complement(complement(meet(x0, x1))), join(complement(meet(x0, x1)), composition(complement(complement(meet(x0, x1))), x0))), composition(x1, x0)), x2)
% 52.89/7.23 = { by axiom 1 (composition_identity) R->L }
% 52.89/7.23 composition(join(meet(complement(complement(meet(x0, x1))), join(composition(complement(meet(x0, x1)), one), composition(complement(complement(meet(x0, x1))), x0))), composition(x1, x0)), x2)
% 52.89/7.23 = { by axiom 4 (goals_1) R->L }
% 52.89/7.23 composition(join(meet(complement(complement(meet(x0, x1))), join(composition(complement(meet(x0, x1)), join(x0, one)), composition(complement(complement(meet(x0, x1))), x0))), composition(x1, x0)), x2)
% 52.89/7.23 = { by lemma 51 R->L }
% 52.89/7.23 composition(join(meet(complement(complement(meet(x0, x1))), join(join(complement(meet(x0, x1)), composition(complement(meet(x0, x1)), x0)), composition(complement(complement(meet(x0, x1))), x0))), composition(x1, x0)), x2)
% 52.89/7.23 = { by axiom 11 (maddux2_join_associativity) R->L }
% 52.89/7.23 composition(join(meet(complement(complement(meet(x0, x1))), join(complement(meet(x0, x1)), join(composition(complement(meet(x0, x1)), x0), composition(complement(complement(meet(x0, x1))), x0)))), composition(x1, x0)), x2)
% 52.89/7.23 = { by axiom 2 (maddux1_join_commutativity) }
% 52.89/7.23 composition(join(meet(complement(complement(meet(x0, x1))), join(complement(meet(x0, x1)), join(composition(complement(complement(meet(x0, x1))), x0), composition(complement(meet(x0, x1)), x0)))), composition(x1, x0)), x2)
% 52.89/7.23 = { by axiom 13 (composition_distributivity) R->L }
% 52.89/7.23 composition(join(meet(complement(complement(meet(x0, x1))), join(complement(meet(x0, x1)), composition(join(complement(complement(meet(x0, x1))), complement(meet(x0, x1))), x0))), composition(x1, x0)), x2)
% 52.89/7.23 = { by axiom 2 (maddux1_join_commutativity) }
% 52.89/7.23 composition(join(meet(complement(complement(meet(x0, x1))), join(complement(meet(x0, x1)), composition(join(complement(meet(x0, x1)), complement(complement(meet(x0, x1)))), x0))), composition(x1, x0)), x2)
% 52.89/7.23 = { by axiom 7 (def_top) R->L }
% 52.89/7.23 composition(join(meet(complement(complement(meet(x0, x1))), join(complement(meet(x0, x1)), composition(top, x0))), composition(x1, x0)), x2)
% 52.89/7.23 = { by lemma 66 }
% 52.89/7.23 composition(join(meet(composition(top, x0), complement(complement(meet(x0, x1)))), composition(x1, x0)), x2)
% 52.89/7.23 = { by lemma 34 }
% 52.89/7.23 composition(join(meet(composition(top, x0), meet(x0, x1)), composition(x1, x0)), x2)
% 52.89/7.23 = { by lemma 64 }
% 52.89/7.23 composition(join(meet(x1, meet(x0, composition(top, x0))), composition(x1, x0)), x2)
% 52.89/7.23 = { by lemma 31 R->L }
% 52.89/7.23 composition(join(meet(x1, meet(x0, composition(join(top, one), x0))), composition(x1, x0)), x2)
% 52.89/7.23 = { by lemma 40 R->L }
% 52.89/7.23 composition(join(meet(x1, meet(x0, join(x0, composition(top, x0)))), composition(x1, x0)), x2)
% 52.89/7.23 = { by lemma 54 }
% 52.89/7.23 composition(join(meet(x1, x0), composition(x1, x0)), x2)
% 52.89/7.23 = { by lemma 35 }
% 52.89/7.23 composition(join(meet(x0, x1), composition(x1, x0)), x2)
% 52.89/7.23 = { by lemma 60 R->L }
% 52.89/7.23 composition(join(meet(x0, x1), meet(x1, composition(x1, x0))), x2)
% 52.89/7.23 = { by lemma 63 R->L }
% 52.89/7.23 composition(join(meet(x0, x1), meet(x1, meet(x0, composition(x1, x0)))), x2)
% 52.89/7.23 = { by lemma 64 R->L }
% 52.89/7.23 composition(join(meet(x0, x1), meet(composition(x1, x0), meet(x0, x1))), x2)
% 52.89/7.23 = { by lemma 35 R->L }
% 52.89/7.23 composition(join(meet(x0, x1), meet(meet(x0, x1), composition(x1, x0))), x2)
% 52.89/7.23 = { by lemma 49 }
% 52.89/7.23 composition(meet(x0, x1), x2)
% 52.89/7.23 % SZS output end Proof
% 52.89/7.23
% 52.89/7.23 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------