TSTP Solution File: REL029+1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL029+1 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n015.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 13:44:12 EDT 2023
% Result : Theorem 48.73s 6.73s
% Output : Proof 50.09s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : REL029+1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n015.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Aug 25 19:39:07 EDT 2023
% 0.14/0.35 % CPUTime :
% 48.73/6.73 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 48.73/6.73
% 48.73/6.73 % SZS status Theorem
% 48.73/6.73
% 50.09/6.85 % SZS output start Proof
% 50.09/6.85 Axiom 1 (composition_identity): composition(X, one) = X.
% 50.09/6.85 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 50.09/6.85 Axiom 3 (goals): join(x1, one) = one.
% 50.09/6.85 Axiom 4 (goals_1): join(x0, one) = one.
% 50.09/6.85 Axiom 5 (converse_idempotence): converse(converse(X)) = X.
% 50.09/6.85 Axiom 6 (def_top): top = join(X, complement(X)).
% 50.09/6.85 Axiom 7 (def_zero): zero = meet(X, complement(X)).
% 50.09/6.85 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 50.09/6.85 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 50.09/6.85 Axiom 10 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 50.09/6.85 Axiom 11 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 50.09/6.85 Axiom 12 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 50.09/6.85 Axiom 13 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 50.09/6.85 Axiom 14 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 50.09/6.85 Axiom 15 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 50.09/6.85
% 50.09/6.85 Lemma 16: complement(top) = zero.
% 50.09/6.85 Proof:
% 50.09/6.85 complement(top)
% 50.09/6.85 = { by axiom 6 (def_top) }
% 50.09/6.85 complement(join(complement(X), complement(complement(X))))
% 50.09/6.85 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 50.09/6.85 meet(X, complement(X))
% 50.09/6.85 = { by axiom 7 (def_zero) R->L }
% 50.09/6.85 zero
% 50.09/6.85
% 50.09/6.85 Lemma 17: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 50.09/6.85 Proof:
% 50.09/6.85 converse(composition(converse(X), Y))
% 50.09/6.85 = { by axiom 8 (converse_multiplicativity) }
% 50.09/6.85 composition(converse(Y), converse(converse(X)))
% 50.09/6.85 = { by axiom 5 (converse_idempotence) }
% 50.09/6.85 composition(converse(Y), X)
% 50.09/6.85
% 50.09/6.85 Lemma 18: composition(converse(one), X) = X.
% 50.09/6.85 Proof:
% 50.09/6.85 composition(converse(one), X)
% 50.09/6.85 = { by lemma 17 R->L }
% 50.09/6.85 converse(composition(converse(X), one))
% 50.09/6.86 = { by axiom 1 (composition_identity) }
% 50.09/6.86 converse(converse(X))
% 50.09/6.86 = { by axiom 5 (converse_idempotence) }
% 50.09/6.86 X
% 50.09/6.86
% 50.09/6.86 Lemma 19: converse(one) = one.
% 50.09/6.86 Proof:
% 50.09/6.86 converse(one)
% 50.09/6.86 = { by axiom 1 (composition_identity) R->L }
% 50.09/6.86 composition(converse(one), one)
% 50.09/6.86 = { by lemma 18 }
% 50.09/6.86 one
% 50.09/6.86
% 50.09/6.86 Lemma 20: join(X, join(Y, complement(X))) = join(Y, top).
% 50.09/6.86 Proof:
% 50.09/6.86 join(X, join(Y, complement(X)))
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.86 join(X, join(complement(X), Y))
% 50.09/6.86 = { by axiom 11 (maddux2_join_associativity) }
% 50.09/6.86 join(join(X, complement(X)), Y)
% 50.09/6.86 = { by axiom 6 (def_top) R->L }
% 50.09/6.86 join(top, Y)
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.86 join(Y, top)
% 50.09/6.86
% 50.09/6.86 Lemma 21: composition(one, X) = X.
% 50.09/6.86 Proof:
% 50.09/6.86 composition(one, X)
% 50.09/6.86 = { by lemma 18 R->L }
% 50.09/6.86 composition(converse(one), composition(one, X))
% 50.09/6.86 = { by axiom 9 (composition_associativity) }
% 50.09/6.86 composition(composition(converse(one), one), X)
% 50.09/6.86 = { by axiom 1 (composition_identity) }
% 50.09/6.86 composition(converse(one), X)
% 50.09/6.86 = { by lemma 18 }
% 50.09/6.86 X
% 50.09/6.86
% 50.09/6.86 Lemma 22: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 50.09/6.86 Proof:
% 50.09/6.86 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.86 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 50.09/6.86 = { by axiom 14 (converse_cancellativity) }
% 50.09/6.86 complement(X)
% 50.09/6.86
% 50.09/6.86 Lemma 23: join(complement(X), complement(X)) = complement(X).
% 50.09/6.86 Proof:
% 50.09/6.86 join(complement(X), complement(X))
% 50.09/6.86 = { by lemma 18 R->L }
% 50.09/6.86 join(complement(X), composition(converse(one), complement(X)))
% 50.09/6.86 = { by lemma 21 R->L }
% 50.09/6.86 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 50.09/6.86 = { by lemma 22 }
% 50.09/6.86 complement(X)
% 50.09/6.86
% 50.09/6.86 Lemma 24: join(top, complement(X)) = top.
% 50.09/6.86 Proof:
% 50.09/6.86 join(top, complement(X))
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.86 join(complement(X), top)
% 50.09/6.86 = { by lemma 20 R->L }
% 50.09/6.86 join(X, join(complement(X), complement(X)))
% 50.09/6.86 = { by lemma 23 }
% 50.09/6.86 join(X, complement(X))
% 50.09/6.86 = { by axiom 6 (def_top) R->L }
% 50.09/6.86 top
% 50.09/6.86
% 50.09/6.86 Lemma 25: join(Y, top) = join(X, top).
% 50.09/6.86 Proof:
% 50.09/6.86 join(Y, top)
% 50.09/6.86 = { by lemma 24 R->L }
% 50.09/6.86 join(Y, join(top, complement(Y)))
% 50.09/6.86 = { by lemma 20 }
% 50.09/6.86 join(top, top)
% 50.09/6.86 = { by lemma 20 R->L }
% 50.09/6.86 join(X, join(top, complement(X)))
% 50.09/6.86 = { by lemma 24 }
% 50.09/6.86 join(X, top)
% 50.09/6.86
% 50.09/6.86 Lemma 26: join(x1, join(one, X)) = join(X, one).
% 50.09/6.86 Proof:
% 50.09/6.86 join(x1, join(one, X))
% 50.09/6.86 = { by axiom 11 (maddux2_join_associativity) }
% 50.09/6.86 join(join(x1, one), X)
% 50.09/6.86 = { by axiom 3 (goals) }
% 50.09/6.86 join(one, X)
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.86 join(X, one)
% 50.09/6.86
% 50.09/6.86 Lemma 27: join(X, top) = top.
% 50.09/6.86 Proof:
% 50.09/6.86 join(X, top)
% 50.09/6.86 = { by lemma 25 }
% 50.09/6.86 join(x1, top)
% 50.09/6.86 = { by axiom 6 (def_top) }
% 50.09/6.86 join(x1, join(one, complement(one)))
% 50.09/6.86 = { by lemma 26 }
% 50.09/6.86 join(complement(one), one)
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.86 join(one, complement(one))
% 50.09/6.86 = { by axiom 6 (def_top) R->L }
% 50.09/6.86 top
% 50.09/6.86
% 50.09/6.86 Lemma 28: join(top, X) = top.
% 50.09/6.86 Proof:
% 50.09/6.86 join(top, X)
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.86 join(X, top)
% 50.09/6.86 = { by lemma 25 R->L }
% 50.09/6.86 join(Y, top)
% 50.09/6.86 = { by lemma 27 }
% 50.09/6.86 top
% 50.09/6.86
% 50.09/6.86 Lemma 29: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 50.09/6.86 Proof:
% 50.09/6.86 converse(join(X, converse(Y)))
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.86 converse(join(converse(Y), X))
% 50.09/6.86 = { by axiom 10 (converse_additivity) }
% 50.09/6.86 join(converse(converse(Y)), converse(X))
% 50.09/6.86 = { by axiom 5 (converse_idempotence) }
% 50.09/6.86 join(Y, converse(X))
% 50.09/6.86
% 50.09/6.86 Lemma 30: converse(top) = top.
% 50.09/6.86 Proof:
% 50.09/6.86 converse(top)
% 50.09/6.86 = { by lemma 28 R->L }
% 50.09/6.86 converse(join(top, converse(top)))
% 50.09/6.86 = { by lemma 29 }
% 50.09/6.86 join(top, converse(top))
% 50.09/6.86 = { by lemma 28 }
% 50.09/6.86 top
% 50.09/6.86
% 50.09/6.86 Lemma 31: meet(Y, X) = meet(X, Y).
% 50.09/6.86 Proof:
% 50.09/6.86 meet(Y, X)
% 50.09/6.86 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.86 complement(join(complement(Y), complement(X)))
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.86 complement(join(complement(X), complement(Y)))
% 50.09/6.86 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 50.09/6.86 meet(X, Y)
% 50.09/6.86
% 50.09/6.86 Lemma 32: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 50.09/6.86 Proof:
% 50.09/6.86 join(meet(X, Y), complement(join(complement(X), Y)))
% 50.09/6.86 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.86 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 50.09/6.86 = { by axiom 15 (maddux3_a_kind_of_de_Morgan) R->L }
% 50.09/6.86 X
% 50.09/6.86
% 50.09/6.86 Lemma 33: join(zero, meet(X, X)) = X.
% 50.09/6.86 Proof:
% 50.09/6.86 join(zero, meet(X, X))
% 50.09/6.86 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.86 join(zero, complement(join(complement(X), complement(X))))
% 50.09/6.86 = { by axiom 7 (def_zero) }
% 50.09/6.86 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 50.09/6.86 = { by lemma 32 }
% 50.09/6.86 X
% 50.09/6.86
% 50.09/6.86 Lemma 34: join(zero, complement(complement(X))) = X.
% 50.09/6.86 Proof:
% 50.09/6.86 join(zero, complement(complement(X)))
% 50.09/6.86 = { by axiom 7 (def_zero) }
% 50.09/6.86 join(meet(X, complement(X)), complement(complement(X)))
% 50.09/6.86 = { by lemma 23 R->L }
% 50.09/6.86 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 50.09/6.86 = { by lemma 32 }
% 50.09/6.86 X
% 50.09/6.86
% 50.09/6.86 Lemma 35: join(X, zero) = X.
% 50.09/6.86 Proof:
% 50.09/6.86 join(X, zero)
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.86 join(zero, X)
% 50.09/6.86 = { by lemma 33 R->L }
% 50.09/6.86 join(zero, join(zero, meet(X, X)))
% 50.09/6.86 = { by axiom 11 (maddux2_join_associativity) }
% 50.09/6.86 join(join(zero, zero), meet(X, X))
% 50.09/6.86 = { by lemma 16 R->L }
% 50.09/6.86 join(join(zero, complement(top)), meet(X, X))
% 50.09/6.86 = { by lemma 16 R->L }
% 50.09/6.86 join(join(complement(top), complement(top)), meet(X, X))
% 50.09/6.86 = { by lemma 23 }
% 50.09/6.86 join(complement(top), meet(X, X))
% 50.09/6.86 = { by lemma 16 }
% 50.09/6.86 join(zero, meet(X, X))
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.86 join(meet(X, X), zero)
% 50.09/6.86 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.86 join(complement(join(complement(X), complement(X))), zero)
% 50.09/6.86 = { by lemma 23 }
% 50.09/6.86 join(complement(complement(X)), zero)
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.86 join(zero, complement(complement(X)))
% 50.09/6.86 = { by lemma 34 }
% 50.09/6.86 X
% 50.09/6.86
% 50.09/6.86 Lemma 36: join(zero, X) = X.
% 50.09/6.86 Proof:
% 50.09/6.86 join(zero, X)
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.86 join(X, zero)
% 50.09/6.86 = { by lemma 35 }
% 50.09/6.86 X
% 50.09/6.86
% 50.09/6.86 Lemma 37: meet(X, zero) = zero.
% 50.09/6.86 Proof:
% 50.09/6.86 meet(X, zero)
% 50.09/6.86 = { by lemma 31 }
% 50.09/6.86 meet(zero, X)
% 50.09/6.86 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.86 complement(join(complement(zero), complement(X)))
% 50.09/6.86 = { by lemma 36 R->L }
% 50.09/6.86 complement(join(join(zero, complement(zero)), complement(X)))
% 50.09/6.86 = { by axiom 6 (def_top) R->L }
% 50.09/6.86 complement(join(top, complement(X)))
% 50.09/6.86 = { by lemma 24 }
% 50.09/6.86 complement(top)
% 50.09/6.86 = { by lemma 16 }
% 50.09/6.86 zero
% 50.09/6.86
% 50.09/6.86 Lemma 38: complement(complement(X)) = X.
% 50.09/6.86 Proof:
% 50.09/6.86 complement(complement(X))
% 50.09/6.86 = { by lemma 36 R->L }
% 50.09/6.86 join(zero, complement(complement(X)))
% 50.09/6.86 = { by lemma 34 }
% 50.09/6.86 X
% 50.09/6.86
% 50.09/6.86 Lemma 39: complement(join(zero, complement(X))) = meet(X, top).
% 50.09/6.86 Proof:
% 50.09/6.86 complement(join(zero, complement(X)))
% 50.09/6.86 = { by lemma 16 R->L }
% 50.09/6.86 complement(join(complement(top), complement(X)))
% 50.09/6.86 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 50.09/6.86 meet(top, X)
% 50.09/6.86 = { by lemma 31 R->L }
% 50.09/6.86 meet(X, top)
% 50.09/6.86
% 50.09/6.86 Lemma 40: meet(X, top) = X.
% 50.09/6.86 Proof:
% 50.09/6.86 meet(X, top)
% 50.09/6.86 = { by lemma 39 R->L }
% 50.09/6.86 complement(join(zero, complement(X)))
% 50.09/6.86 = { by lemma 36 }
% 50.09/6.86 complement(complement(X))
% 50.09/6.86 = { by lemma 38 }
% 50.09/6.86 X
% 50.09/6.86
% 50.09/6.86 Lemma 41: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 50.09/6.86 Proof:
% 50.09/6.86 converse(join(converse(X), Y))
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.86 converse(join(Y, converse(X)))
% 50.09/6.86 = { by lemma 29 }
% 50.09/6.86 join(X, converse(Y))
% 50.09/6.86
% 50.09/6.86 Lemma 42: join(X, converse(complement(converse(X)))) = top.
% 50.09/6.86 Proof:
% 50.09/6.86 join(X, converse(complement(converse(X))))
% 50.09/6.86 = { by lemma 41 R->L }
% 50.09/6.86 converse(join(converse(X), complement(converse(X))))
% 50.09/6.86 = { by axiom 6 (def_top) R->L }
% 50.09/6.86 converse(top)
% 50.09/6.86 = { by lemma 30 }
% 50.09/6.86 top
% 50.09/6.86
% 50.09/6.86 Lemma 43: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 50.09/6.86 Proof:
% 50.09/6.86 meet(X, join(complement(Y), complement(Z)))
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.86 meet(X, join(complement(Z), complement(Y)))
% 50.09/6.86 = { by lemma 31 }
% 50.09/6.86 meet(join(complement(Z), complement(Y)), X)
% 50.09/6.86 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.86 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 50.09/6.86 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 50.09/6.86 complement(join(meet(Z, Y), complement(X)))
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.86 complement(join(complement(X), meet(Z, Y)))
% 50.09/6.86 = { by lemma 31 R->L }
% 50.09/6.86 complement(join(complement(X), meet(Y, Z)))
% 50.09/6.86
% 50.09/6.86 Lemma 44: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 50.09/6.86 Proof:
% 50.09/6.86 complement(join(X, complement(Y)))
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.86 complement(join(complement(Y), X))
% 50.09/6.86 = { by lemma 40 R->L }
% 50.09/6.86 complement(join(complement(Y), meet(X, top)))
% 50.09/6.86 = { by lemma 31 R->L }
% 50.09/6.86 complement(join(complement(Y), meet(top, X)))
% 50.09/6.86 = { by lemma 43 R->L }
% 50.09/6.86 meet(Y, join(complement(top), complement(X)))
% 50.09/6.86 = { by lemma 16 }
% 50.09/6.86 meet(Y, join(zero, complement(X)))
% 50.09/6.86 = { by lemma 36 }
% 50.09/6.86 meet(Y, complement(X))
% 50.09/6.86
% 50.09/6.86 Lemma 45: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 50.09/6.86 Proof:
% 50.09/6.86 meet(complement(X), complement(Y))
% 50.09/6.86 = { by lemma 31 }
% 50.09/6.86 meet(complement(Y), complement(X))
% 50.09/6.86 = { by lemma 36 R->L }
% 50.09/6.86 meet(join(zero, complement(Y)), complement(X))
% 50.09/6.86 = { by lemma 44 R->L }
% 50.09/6.86 complement(join(X, complement(join(zero, complement(Y)))))
% 50.09/6.86 = { by lemma 39 }
% 50.09/6.86 complement(join(X, meet(Y, top)))
% 50.09/6.86 = { by lemma 40 }
% 50.09/6.86 complement(join(X, Y))
% 50.09/6.86
% 50.09/6.86 Lemma 46: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 50.09/6.86 Proof:
% 50.09/6.86 complement(join(complement(X), Y))
% 50.09/6.86 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.86 complement(join(Y, complement(X)))
% 50.09/6.86 = { by lemma 44 }
% 50.09/6.86 meet(X, complement(Y))
% 50.09/6.86
% 50.09/6.86 Lemma 47: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 50.09/6.86 Proof:
% 50.09/6.86 complement(meet(X, complement(Y)))
% 50.09/6.86 = { by lemma 36 R->L }
% 50.09/6.86 complement(join(zero, meet(X, complement(Y))))
% 50.09/6.86 = { by lemma 44 R->L }
% 50.09/6.86 complement(join(zero, complement(join(Y, complement(X)))))
% 50.09/6.86 = { by lemma 39 }
% 50.09/6.86 meet(join(Y, complement(X)), top)
% 50.09/6.86 = { by lemma 40 }
% 50.09/6.86 join(Y, complement(X))
% 50.09/6.86
% 50.09/6.86 Lemma 48: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 50.09/6.86 Proof:
% 50.09/6.86 complement(meet(complement(X), Y))
% 50.09/6.86 = { by lemma 31 }
% 50.09/6.86 complement(meet(Y, complement(X)))
% 50.09/6.86 = { by lemma 47 }
% 50.09/6.87 join(X, complement(Y))
% 50.09/6.87
% 50.09/6.87 Lemma 49: join(X, complement(meet(X, Y))) = top.
% 50.09/6.87 Proof:
% 50.09/6.87 join(X, complement(meet(X, Y)))
% 50.09/6.87 = { by lemma 31 }
% 50.09/6.87 join(X, complement(meet(Y, X)))
% 50.09/6.87 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.87 join(X, complement(complement(join(complement(Y), complement(X)))))
% 50.09/6.87 = { by lemma 23 R->L }
% 50.09/6.87 join(X, complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))))
% 50.09/6.87 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 50.09/6.87 join(X, complement(join(meet(Y, X), complement(join(complement(Y), complement(X))))))
% 50.09/6.87 = { by axiom 12 (maddux4_definiton_of_meet) R->L }
% 50.09/6.87 join(X, complement(join(meet(Y, X), meet(Y, X))))
% 50.09/6.87 = { by lemma 31 R->L }
% 50.09/6.87 join(X, complement(join(meet(Y, X), meet(X, Y))))
% 50.09/6.87 = { by lemma 31 R->L }
% 50.09/6.87 join(X, complement(join(meet(X, Y), meet(X, Y))))
% 50.09/6.87 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.87 join(X, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y))))
% 50.09/6.87 = { by lemma 36 R->L }
% 50.09/6.87 join(X, join(zero, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y)))))
% 50.09/6.87 = { by lemma 43 R->L }
% 50.09/6.87 join(X, join(zero, meet(join(complement(X), complement(Y)), join(complement(X), complement(Y)))))
% 50.09/6.87 = { by lemma 33 }
% 50.09/6.87 join(X, join(complement(X), complement(Y)))
% 50.09/6.87 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.87 join(X, join(complement(Y), complement(X)))
% 50.09/6.87 = { by lemma 20 }
% 50.09/6.87 join(complement(Y), top)
% 50.09/6.87 = { by lemma 27 }
% 50.09/6.87 top
% 50.09/6.87
% 50.09/6.87 Lemma 50: meet(X, join(X, complement(Y))) = X.
% 50.09/6.87 Proof:
% 50.09/6.87 meet(X, join(X, complement(Y)))
% 50.09/6.87 = { by lemma 35 R->L }
% 50.09/6.87 join(meet(X, join(X, complement(Y))), zero)
% 50.09/6.87 = { by lemma 16 R->L }
% 50.09/6.87 join(meet(X, join(X, complement(Y))), complement(top))
% 50.09/6.87 = { by lemma 48 R->L }
% 50.09/6.87 join(meet(X, complement(meet(complement(X), Y))), complement(top))
% 50.09/6.87 = { by lemma 49 R->L }
% 50.09/6.87 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(complement(X), Y)))))
% 50.09/6.87 = { by lemma 32 }
% 50.09/6.87 X
% 50.09/6.87
% 50.09/6.87 Lemma 51: join(X, meet(X, Y)) = X.
% 50.09/6.87 Proof:
% 50.09/6.87 join(X, meet(X, Y))
% 50.09/6.87 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.87 join(X, complement(join(complement(X), complement(Y))))
% 50.09/6.87 = { by lemma 48 R->L }
% 50.09/6.87 complement(meet(complement(X), join(complement(X), complement(Y))))
% 50.09/6.87 = { by lemma 50 }
% 50.09/6.87 complement(complement(X))
% 50.09/6.87 = { by lemma 38 }
% 50.09/6.87 X
% 50.09/6.87
% 50.09/6.87 Lemma 52: join(meet(X, Y), meet(X, complement(Y))) = X.
% 50.09/6.87 Proof:
% 50.09/6.87 join(meet(X, Y), meet(X, complement(Y)))
% 50.09/6.87 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.87 join(meet(X, complement(Y)), meet(X, Y))
% 50.09/6.87 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.87 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 50.09/6.87 = { by lemma 32 }
% 50.09/6.87 X
% 50.09/6.87
% 50.09/6.87 Lemma 53: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 50.09/6.87 Proof:
% 50.09/6.87 meet(X, complement(meet(X, Y)))
% 50.09/6.87 = { by lemma 46 R->L }
% 50.09/6.87 complement(join(complement(X), meet(X, Y)))
% 50.09/6.87 = { by lemma 31 }
% 50.09/6.87 complement(join(complement(X), meet(Y, X)))
% 50.09/6.87 = { by lemma 51 R->L }
% 50.09/6.87 complement(join(join(complement(X), meet(complement(X), Y)), meet(Y, X)))
% 50.09/6.87 = { by axiom 11 (maddux2_join_associativity) R->L }
% 50.09/6.87 complement(join(complement(X), join(meet(complement(X), Y), meet(Y, X))))
% 50.09/6.87 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.87 complement(join(complement(X), join(meet(Y, X), meet(complement(X), Y))))
% 50.09/6.87 = { by lemma 31 }
% 50.09/6.87 complement(join(complement(X), join(meet(Y, X), meet(Y, complement(X)))))
% 50.09/6.87 = { by lemma 52 }
% 50.09/6.87 complement(join(complement(X), Y))
% 50.09/6.87 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.87 complement(join(Y, complement(X)))
% 50.09/6.87 = { by lemma 44 }
% 50.09/6.87 meet(X, complement(Y))
% 50.09/6.87
% 50.09/6.87 Lemma 54: join(X, converse(meet(Y, converse(X)))) = X.
% 50.09/6.87 Proof:
% 50.09/6.87 join(X, converse(meet(Y, converse(X))))
% 50.09/6.87 = { by lemma 31 }
% 50.09/6.87 join(X, converse(meet(converse(X), Y)))
% 50.09/6.87 = { by lemma 41 R->L }
% 50.09/6.87 converse(join(converse(X), meet(converse(X), Y)))
% 50.09/6.87 = { by lemma 51 }
% 50.09/6.87 converse(converse(X))
% 50.09/6.87 = { by axiom 5 (converse_idempotence) }
% 50.09/6.87 X
% 50.09/6.87
% 50.09/6.87 Lemma 55: join(complement(one), converse(complement(one))) = complement(one).
% 50.09/6.87 Proof:
% 50.09/6.87 join(complement(one), converse(complement(one)))
% 50.09/6.87 = { by lemma 40 R->L }
% 50.09/6.87 join(complement(one), converse(meet(complement(one), top)))
% 50.09/6.87 = { by lemma 42 R->L }
% 50.09/6.87 join(complement(one), converse(meet(complement(one), join(one, converse(complement(converse(one)))))))
% 50.09/6.87 = { by lemma 19 }
% 50.09/6.87 join(complement(one), converse(meet(complement(one), join(one, converse(complement(one))))))
% 50.09/6.87 = { by lemma 31 }
% 50.09/6.87 join(complement(one), converse(meet(join(one, converse(complement(one))), complement(one))))
% 50.09/6.87 = { by lemma 44 R->L }
% 50.09/6.87 join(complement(one), converse(complement(join(one, complement(join(one, converse(complement(one))))))))
% 50.09/6.87 = { by lemma 45 R->L }
% 50.09/6.87 join(complement(one), converse(meet(complement(one), complement(complement(join(one, converse(complement(one))))))))
% 50.09/6.87 = { by lemma 45 R->L }
% 50.09/6.87 join(complement(one), converse(meet(complement(one), complement(meet(complement(one), complement(converse(complement(one))))))))
% 50.09/6.87 = { by lemma 53 }
% 50.09/6.87 join(complement(one), converse(meet(complement(one), complement(complement(converse(complement(one)))))))
% 50.09/6.87 = { by lemma 45 }
% 50.09/6.87 join(complement(one), converse(complement(join(one, complement(converse(complement(one)))))))
% 50.09/6.87 = { by lemma 44 }
% 50.09/6.87 join(complement(one), converse(meet(converse(complement(one)), complement(one))))
% 50.09/6.87 = { by lemma 31 R->L }
% 50.09/6.87 join(complement(one), converse(meet(complement(one), converse(complement(one)))))
% 50.09/6.87 = { by lemma 54 }
% 50.09/6.87 complement(one)
% 50.09/6.87
% 50.09/6.87 Lemma 56: converse(complement(one)) = complement(one).
% 50.09/6.87 Proof:
% 50.09/6.87 converse(complement(one))
% 50.09/6.87 = { by lemma 55 R->L }
% 50.09/6.87 converse(join(complement(one), converse(complement(one))))
% 50.09/6.87 = { by lemma 29 }
% 50.09/6.87 join(complement(one), converse(complement(one)))
% 50.09/6.87 = { by lemma 55 }
% 50.09/6.87 complement(one)
% 50.09/6.87
% 50.09/6.87 Lemma 57: converse(join(X, composition(converse(Y), Z))) = join(converse(X), composition(converse(Z), Y)).
% 50.09/6.87 Proof:
% 50.09/6.87 converse(join(X, composition(converse(Y), Z)))
% 50.09/6.87 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.87 converse(join(composition(converse(Y), Z), X))
% 50.09/6.87 = { by axiom 10 (converse_additivity) }
% 50.09/6.87 join(converse(composition(converse(Y), Z)), converse(X))
% 50.09/6.87 = { by lemma 17 }
% 50.09/6.87 join(composition(converse(Z), Y), converse(X))
% 50.09/6.87 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.87 join(converse(X), composition(converse(Z), Y))
% 50.09/6.87
% 50.09/6.87 Lemma 58: meet(one, composition(converse(complement(X)), X)) = zero.
% 50.09/6.87 Proof:
% 50.09/6.87 meet(one, composition(converse(complement(X)), X))
% 50.09/6.87 = { by lemma 31 }
% 50.09/6.87 meet(composition(converse(complement(X)), X), one)
% 50.09/6.87 = { by lemma 38 R->L }
% 50.09/6.87 meet(composition(converse(complement(X)), X), complement(complement(one)))
% 50.09/6.87 = { by lemma 56 R->L }
% 50.09/6.87 meet(composition(converse(complement(X)), X), complement(converse(complement(one))))
% 50.09/6.87 = { by lemma 22 R->L }
% 50.09/6.87 meet(composition(converse(complement(X)), X), complement(converse(join(complement(one), composition(converse(converse(converse(X))), complement(composition(converse(converse(X)), one)))))))
% 50.09/6.87 = { by axiom 1 (composition_identity) }
% 50.09/6.87 meet(composition(converse(complement(X)), X), complement(converse(join(complement(one), composition(converse(converse(converse(X))), complement(converse(converse(X))))))))
% 50.09/6.87 = { by axiom 5 (converse_idempotence) }
% 50.09/6.87 meet(composition(converse(complement(X)), X), complement(converse(join(complement(one), composition(converse(X), complement(converse(converse(X))))))))
% 50.09/6.87 = { by lemma 57 }
% 50.09/6.87 meet(composition(converse(complement(X)), X), complement(join(converse(complement(one)), composition(converse(complement(converse(converse(X)))), X))))
% 50.09/6.87 = { by lemma 56 }
% 50.09/6.87 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(converse(converse(X)))), X))))
% 50.09/6.87 = { by axiom 5 (converse_idempotence) }
% 50.09/6.87 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), X))))
% 50.09/6.87 = { by lemma 45 R->L }
% 50.09/6.87 meet(composition(converse(complement(X)), X), meet(complement(complement(one)), complement(composition(converse(complement(X)), X))))
% 50.09/6.87 = { by lemma 31 }
% 50.09/6.87 meet(composition(converse(complement(X)), X), meet(complement(composition(converse(complement(X)), X)), complement(complement(one))))
% 50.09/6.87 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.87 complement(join(complement(composition(converse(complement(X)), X)), complement(meet(complement(composition(converse(complement(X)), X)), complement(complement(one))))))
% 50.09/6.87 = { by lemma 49 }
% 50.09/6.87 complement(top)
% 50.09/6.87 = { by lemma 16 }
% 50.09/6.87 zero
% 50.09/6.87
% 50.09/6.87 Lemma 59: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 50.09/6.87 Proof:
% 50.09/6.87 meet(Y, meet(Z, X))
% 50.09/6.87 = { by lemma 40 R->L }
% 50.09/6.87 meet(meet(Y, top), meet(Z, X))
% 50.09/6.87 = { by lemma 39 R->L }
% 50.09/6.87 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 50.09/6.87 = { by lemma 31 }
% 50.09/6.87 meet(complement(join(zero, complement(Y))), meet(X, Z))
% 50.09/6.87 = { by lemma 31 }
% 50.09/6.87 meet(meet(X, Z), complement(join(zero, complement(Y))))
% 50.09/6.87 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.87 meet(complement(join(complement(X), complement(Z))), complement(join(zero, complement(Y))))
% 50.09/6.87 = { by lemma 45 }
% 50.09/6.87 complement(join(join(complement(X), complement(Z)), join(zero, complement(Y))))
% 50.09/6.87 = { by axiom 11 (maddux2_join_associativity) R->L }
% 50.09/6.87 complement(join(complement(X), join(complement(Z), join(zero, complement(Y)))))
% 50.09/6.87 = { by lemma 46 }
% 50.09/6.87 meet(X, complement(join(complement(Z), join(zero, complement(Y)))))
% 50.09/6.87 = { by lemma 46 }
% 50.09/6.87 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 50.09/6.87 = { by lemma 39 }
% 50.09/6.87 meet(X, meet(Z, meet(Y, top)))
% 50.09/6.87 = { by lemma 40 }
% 50.09/6.87 meet(X, meet(Z, Y))
% 50.09/6.87 = { by lemma 31 R->L }
% 50.09/6.87 meet(X, meet(Y, Z))
% 50.09/6.87
% 50.09/6.87 Lemma 60: meet(X, complement(converse(complement(converse(complement(X)))))) = zero.
% 50.09/6.87 Proof:
% 50.09/6.87 meet(X, complement(converse(complement(converse(complement(X))))))
% 50.09/6.87 = { by lemma 46 R->L }
% 50.09/6.87 complement(join(complement(X), converse(complement(converse(complement(X))))))
% 50.09/6.87 = { by lemma 42 }
% 50.09/6.87 complement(top)
% 50.09/6.87 = { by lemma 16 }
% 50.09/6.87 zero
% 50.09/6.87
% 50.09/6.87 Lemma 61: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 50.09/6.87 Proof:
% 50.09/6.87 join(meet(X, Y), meet(Y, complement(X)))
% 50.09/6.87 = { by lemma 31 R->L }
% 50.09/6.87 join(meet(Y, X), meet(Y, complement(X)))
% 50.09/6.87 = { by lemma 52 }
% 50.09/6.87 Y
% 50.09/6.87
% 50.09/6.87 Lemma 62: meet(X, complement(converse(complement(converse(X))))) = complement(converse(complement(converse(X)))).
% 50.09/6.87 Proof:
% 50.09/6.87 meet(X, complement(converse(complement(converse(X)))))
% 50.09/6.87 = { by lemma 38 R->L }
% 50.09/6.87 meet(X, complement(converse(complement(converse(complement(complement(X)))))))
% 50.09/6.87 = { by lemma 35 R->L }
% 50.09/6.87 join(meet(X, complement(converse(complement(converse(complement(complement(X))))))), zero)
% 50.09/6.87 = { by lemma 60 R->L }
% 50.09/6.87 join(meet(X, complement(converse(complement(converse(complement(complement(X))))))), meet(complement(X), complement(converse(complement(converse(complement(complement(X))))))))
% 50.09/6.87 = { by lemma 31 }
% 50.09/6.87 join(meet(X, complement(converse(complement(converse(complement(complement(X))))))), meet(complement(converse(complement(converse(complement(complement(X)))))), complement(X)))
% 50.09/6.87 = { by lemma 61 }
% 50.09/6.87 complement(converse(complement(converse(complement(complement(X))))))
% 50.09/6.87 = { by lemma 38 }
% 50.09/6.87 complement(converse(complement(converse(X))))
% 50.09/6.88
% 50.09/6.88 Lemma 63: meet(X, join(X, Y)) = X.
% 50.09/6.88 Proof:
% 50.09/6.88 meet(X, join(X, Y))
% 50.09/6.88 = { by lemma 40 R->L }
% 50.09/6.88 meet(X, join(X, meet(Y, top)))
% 50.09/6.88 = { by lemma 39 R->L }
% 50.09/6.88 meet(X, join(X, complement(join(zero, complement(Y)))))
% 50.09/6.88 = { by lemma 50 }
% 50.09/6.88 X
% 50.09/6.88
% 50.09/6.88 Lemma 64: meet(converse(X), converse(join(X, Y))) = converse(X).
% 50.09/6.88 Proof:
% 50.09/6.88 meet(converse(X), converse(join(X, Y)))
% 50.09/6.88 = { by axiom 10 (converse_additivity) }
% 50.09/6.88 meet(converse(X), join(converse(X), converse(Y)))
% 50.09/6.88 = { by lemma 63 }
% 50.09/6.88 converse(X)
% 50.09/6.88
% 50.09/6.88 Lemma 65: converse(complement(X)) = complement(converse(X)).
% 50.09/6.88 Proof:
% 50.09/6.88 converse(complement(X))
% 50.09/6.88 = { by lemma 38 R->L }
% 50.09/6.88 complement(complement(converse(complement(X))))
% 50.09/6.88 = { by axiom 5 (converse_idempotence) R->L }
% 50.09/6.88 complement(complement(converse(complement(converse(converse(X))))))
% 50.09/6.88 = { by axiom 5 (converse_idempotence) R->L }
% 50.09/6.88 complement(converse(converse(complement(converse(complement(converse(converse(X))))))))
% 50.09/6.88 = { by lemma 62 R->L }
% 50.09/6.88 complement(converse(converse(meet(converse(X), complement(converse(complement(converse(converse(X)))))))))
% 50.09/6.88 = { by lemma 31 }
% 50.09/6.88 complement(converse(converse(meet(complement(converse(complement(converse(converse(X))))), converse(X)))))
% 50.09/6.88 = { by lemma 64 R->L }
% 50.09/6.88 complement(converse(meet(converse(meet(complement(converse(complement(converse(converse(X))))), converse(X))), converse(join(meet(complement(converse(complement(converse(converse(X))))), converse(X)), meet(converse(X), complement(complement(converse(complement(converse(converse(X))))))))))))
% 50.09/6.88 = { by lemma 61 }
% 50.09/6.88 complement(converse(meet(converse(meet(complement(converse(complement(converse(converse(X))))), converse(X))), converse(converse(X)))))
% 50.09/6.88 = { by lemma 31 R->L }
% 50.09/6.88 complement(converse(meet(converse(converse(X)), converse(meet(complement(converse(complement(converse(converse(X))))), converse(X))))))
% 50.09/6.88 = { by lemma 31 R->L }
% 50.09/6.88 complement(converse(meet(converse(converse(X)), converse(meet(converse(X), complement(converse(complement(converse(converse(X))))))))))
% 50.09/6.88 = { by lemma 62 }
% 50.09/6.88 complement(converse(meet(converse(converse(X)), converse(complement(converse(complement(converse(converse(X)))))))))
% 50.09/6.88 = { by lemma 31 }
% 50.09/6.88 complement(converse(meet(converse(complement(converse(complement(converse(converse(X)))))), converse(converse(X)))))
% 50.09/6.88 = { by lemma 35 R->L }
% 50.09/6.88 complement(converse(join(meet(converse(complement(converse(complement(converse(converse(X)))))), converse(converse(X))), zero)))
% 50.09/6.88 = { by lemma 60 R->L }
% 50.09/6.88 complement(converse(join(meet(converse(complement(converse(complement(converse(converse(X)))))), converse(converse(X))), meet(converse(converse(X)), complement(converse(complement(converse(complement(converse(converse(X)))))))))))
% 50.09/6.88 = { by lemma 61 }
% 50.09/6.88 complement(converse(converse(converse(X))))
% 50.09/6.88 = { by axiom 5 (converse_idempotence) }
% 50.09/6.88 complement(converse(X))
% 50.09/6.88
% 50.09/6.88 Lemma 66: meet(X, join(complement(X), Y)) = meet(X, Y).
% 50.09/6.88 Proof:
% 50.09/6.88 meet(X, join(complement(X), Y))
% 50.09/6.88 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.88 meet(X, join(Y, complement(X)))
% 50.09/6.88 = { by lemma 47 R->L }
% 50.09/6.88 meet(X, complement(meet(X, complement(Y))))
% 50.09/6.88 = { by lemma 53 }
% 50.09/6.88 meet(X, complement(complement(Y)))
% 50.09/6.88 = { by lemma 38 }
% 50.09/6.88 meet(X, Y)
% 50.09/6.88
% 50.09/6.88 Lemma 67: converse(join(X, one)) = join(one, converse(X)).
% 50.09/6.88 Proof:
% 50.09/6.88 converse(join(X, one))
% 50.09/6.88 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.88 converse(join(one, X))
% 50.09/6.88 = { by axiom 10 (converse_additivity) }
% 50.09/6.88 join(converse(one), converse(X))
% 50.09/6.88 = { by lemma 19 }
% 50.09/6.88 join(one, converse(X))
% 50.09/6.88
% 50.09/6.88 Lemma 68: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 50.09/6.88 Proof:
% 50.09/6.88 converse(composition(X, converse(Y)))
% 50.09/6.88 = { by axiom 8 (converse_multiplicativity) }
% 50.09/6.88 composition(converse(converse(Y)), converse(X))
% 50.09/6.88 = { by axiom 5 (converse_idempotence) }
% 50.09/6.88 composition(Y, converse(X))
% 50.09/6.88
% 50.09/6.88 Lemma 69: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 50.09/6.88 Proof:
% 50.09/6.88 join(X, composition(Y, X))
% 50.09/6.88 = { by lemma 21 R->L }
% 50.09/6.88 join(composition(one, X), composition(Y, X))
% 50.09/6.88 = { by axiom 13 (composition_distributivity) R->L }
% 50.09/6.88 composition(join(one, Y), X)
% 50.09/6.88 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.88 composition(join(Y, one), X)
% 50.09/6.88
% 50.09/6.88 Lemma 70: join(composition(X, join(Z, one)), Y) = join(X, join(Y, composition(X, Z))).
% 50.09/6.88 Proof:
% 50.09/6.88 join(composition(X, join(Z, one)), Y)
% 50.09/6.88 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.88 join(composition(X, join(one, Z)), Y)
% 50.09/6.88 = { by axiom 5 (converse_idempotence) R->L }
% 50.09/6.88 join(composition(X, join(one, converse(converse(Z)))), Y)
% 50.09/6.88 = { by lemma 67 R->L }
% 50.09/6.88 join(composition(X, converse(join(converse(Z), one))), Y)
% 50.09/6.88 = { by lemma 68 R->L }
% 50.09/6.88 join(converse(composition(join(converse(Z), one), converse(X))), Y)
% 50.09/6.88 = { by lemma 69 R->L }
% 50.09/6.88 join(converse(join(converse(X), composition(converse(Z), converse(X)))), Y)
% 50.09/6.88 = { by lemma 41 }
% 50.09/6.88 join(join(X, converse(composition(converse(Z), converse(X)))), Y)
% 50.09/6.88 = { by lemma 68 }
% 50.09/6.88 join(join(X, composition(X, converse(converse(Z)))), Y)
% 50.09/6.88 = { by axiom 5 (converse_idempotence) }
% 50.09/6.88 join(join(X, composition(X, Z)), Y)
% 50.09/6.88 = { by axiom 11 (maddux2_join_associativity) R->L }
% 50.09/6.88 join(X, join(composition(X, Z), Y))
% 50.09/6.88 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.88 join(X, join(Y, composition(X, Z)))
% 50.09/6.88
% 50.09/6.88 Lemma 71: join(X, join(Y, composition(X, x1))) = join(X, Y).
% 50.09/6.88 Proof:
% 50.09/6.88 join(X, join(Y, composition(X, x1)))
% 50.09/6.88 = { by lemma 70 R->L }
% 50.09/6.88 join(composition(X, join(x1, one)), Y)
% 50.09/6.88 = { by axiom 3 (goals) }
% 50.09/6.88 join(composition(X, one), Y)
% 50.09/6.88 = { by axiom 1 (composition_identity) }
% 50.09/6.88 join(X, Y)
% 50.09/6.88
% 50.09/6.88 Lemma 72: join(X, composition(complement(X), x1)) = join(X, composition(top, x1)).
% 50.09/6.88 Proof:
% 50.09/6.88 join(X, composition(complement(X), x1))
% 50.09/6.88 = { by lemma 71 R->L }
% 50.09/6.88 join(X, join(composition(complement(X), x1), composition(X, x1)))
% 50.09/6.88 = { by axiom 13 (composition_distributivity) R->L }
% 50.09/6.88 join(X, composition(join(complement(X), X), x1))
% 50.09/6.88 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.88 join(X, composition(join(X, complement(X)), x1))
% 50.09/6.88 = { by axiom 6 (def_top) R->L }
% 50.09/6.88 join(X, composition(top, x1))
% 50.09/6.88
% 50.09/6.88 Lemma 73: meet(X, join(Y, join(Z, X))) = X.
% 50.09/6.88 Proof:
% 50.09/6.88 meet(X, join(Y, join(Z, X)))
% 50.09/6.88 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.88 meet(X, join(Y, join(X, Z)))
% 50.09/6.88 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.88 meet(X, join(join(X, Z), Y))
% 50.09/6.88 = { by axiom 11 (maddux2_join_associativity) R->L }
% 50.09/6.88 meet(X, join(X, join(Z, Y)))
% 50.09/6.88 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.88 meet(X, join(X, join(Y, Z)))
% 50.09/6.88 = { by lemma 63 }
% 50.09/6.88 X
% 50.09/6.88
% 50.09/6.88 Lemma 74: meet(X, composition(top, x1)) = composition(X, x1).
% 50.09/6.88 Proof:
% 50.09/6.88 meet(X, composition(top, x1))
% 50.09/6.88 = { by lemma 66 R->L }
% 50.09/6.88 meet(X, join(complement(X), composition(top, x1)))
% 50.09/6.88 = { by lemma 72 R->L }
% 50.09/6.88 meet(X, join(complement(X), composition(complement(complement(X)), x1)))
% 50.09/6.88 = { by lemma 66 }
% 50.09/6.88 meet(X, composition(complement(complement(X)), x1))
% 50.09/6.88 = { by lemma 38 }
% 50.09/6.88 meet(X, composition(X, x1))
% 50.09/6.88 = { by lemma 31 }
% 50.09/6.88 meet(composition(X, x1), X)
% 50.09/6.88 = { by lemma 35 R->L }
% 50.09/6.88 meet(composition(X, x1), join(X, zero))
% 50.09/6.88 = { by lemma 71 R->L }
% 50.09/6.88 meet(composition(X, x1), join(X, join(zero, composition(X, x1))))
% 50.09/6.88 = { by lemma 73 }
% 50.09/6.88 composition(X, x1)
% 50.09/6.88
% 50.09/6.88 Lemma 75: meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 50.09/6.88 Proof:
% 50.09/6.88 meet(meet(X, Y), Z)
% 50.09/6.88 = { by lemma 31 }
% 50.09/6.88 meet(Z, meet(X, Y))
% 50.09/6.88 = { by lemma 59 R->L }
% 50.09/6.88 meet(X, meet(Y, Z))
% 50.09/6.88
% 50.09/6.88 Lemma 76: meet(X, composition(top, X)) = X.
% 50.09/6.88 Proof:
% 50.09/6.88 meet(X, composition(top, X))
% 50.09/6.88 = { by lemma 28 R->L }
% 50.09/6.88 meet(X, composition(join(top, one), X))
% 50.09/6.88 = { by lemma 69 R->L }
% 50.09/6.88 meet(X, join(X, composition(top, X)))
% 50.09/6.88 = { by lemma 63 }
% 50.09/6.88 X
% 50.09/6.88
% 50.09/6.88 Lemma 77: meet(X, meet(Y, composition(top, X))) = meet(X, Y).
% 50.09/6.88 Proof:
% 50.09/6.88 meet(X, meet(Y, composition(top, X)))
% 50.09/6.88 = { by lemma 31 }
% 50.09/6.88 meet(X, meet(composition(top, X), Y))
% 50.09/6.88 = { by lemma 75 R->L }
% 50.09/6.88 meet(meet(X, composition(top, X)), Y)
% 50.09/6.88 = { by lemma 76 }
% 50.09/6.88 meet(X, Y)
% 50.09/6.88
% 50.09/6.88 Lemma 78: meet(x1, converse(x1)) = x1.
% 50.09/6.88 Proof:
% 50.09/6.88 meet(x1, converse(x1))
% 50.09/6.88 = { by lemma 31 }
% 50.09/6.88 meet(converse(x1), x1)
% 50.09/6.88 = { by lemma 35 R->L }
% 50.09/6.88 join(meet(converse(x1), x1), zero)
% 50.09/6.88 = { by lemma 37 R->L }
% 50.09/6.88 join(meet(converse(x1), x1), meet(x1, zero))
% 50.09/6.88 = { by lemma 58 R->L }
% 50.09/6.88 join(meet(converse(x1), x1), meet(x1, meet(one, composition(converse(complement(x1)), x1))))
% 50.09/6.88 = { by lemma 59 }
% 50.09/6.88 join(meet(converse(x1), x1), meet(composition(converse(complement(x1)), x1), meet(x1, one)))
% 50.09/6.88 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.88 join(meet(converse(x1), x1), meet(composition(converse(complement(x1)), x1), complement(join(complement(x1), complement(one)))))
% 50.09/6.88 = { by lemma 36 R->L }
% 50.09/6.88 join(meet(converse(x1), x1), meet(composition(converse(complement(x1)), x1), join(zero, complement(join(complement(x1), complement(one))))))
% 50.09/6.88 = { by lemma 16 R->L }
% 50.09/6.88 join(meet(converse(x1), x1), meet(composition(converse(complement(x1)), x1), join(complement(top), complement(join(complement(x1), complement(one))))))
% 50.09/6.88 = { by lemma 27 R->L }
% 50.09/6.88 join(meet(converse(x1), x1), meet(composition(converse(complement(x1)), x1), join(complement(join(one, top)), complement(join(complement(x1), complement(one))))))
% 50.09/6.88 = { by lemma 20 R->L }
% 50.09/6.88 join(meet(converse(x1), x1), meet(composition(converse(complement(x1)), x1), join(complement(join(x1, join(one, complement(x1)))), complement(join(complement(x1), complement(one))))))
% 50.09/6.88 = { by lemma 26 }
% 50.09/6.88 join(meet(converse(x1), x1), meet(composition(converse(complement(x1)), x1), join(complement(join(complement(x1), one)), complement(join(complement(x1), complement(one))))))
% 50.09/6.88 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.88 join(meet(converse(x1), x1), meet(composition(converse(complement(x1)), x1), join(complement(join(one, complement(x1))), complement(join(complement(x1), complement(one))))))
% 50.09/6.88 = { by lemma 44 }
% 50.09/6.88 join(meet(converse(x1), x1), meet(composition(converse(complement(x1)), x1), join(meet(x1, complement(one)), complement(join(complement(x1), complement(one))))))
% 50.09/6.88 = { by lemma 32 }
% 50.09/6.88 join(meet(converse(x1), x1), meet(composition(converse(complement(x1)), x1), x1))
% 50.09/6.88 = { by lemma 65 }
% 50.09/6.88 join(meet(converse(x1), x1), meet(composition(complement(converse(x1)), x1), x1))
% 50.09/6.88 = { by lemma 31 R->L }
% 50.09/6.88 join(meet(converse(x1), x1), meet(x1, composition(complement(converse(x1)), x1)))
% 50.09/6.88 = { by lemma 74 R->L }
% 50.09/6.88 join(meet(converse(x1), x1), meet(x1, meet(complement(converse(x1)), composition(top, x1))))
% 50.09/6.88 = { by lemma 77 }
% 50.09/6.88 join(meet(converse(x1), x1), meet(x1, complement(converse(x1))))
% 50.09/6.88 = { by lemma 61 }
% 50.09/6.88 x1
% 50.09/6.88
% 50.09/6.88 Lemma 79: meet(X, join(Y, X)) = X.
% 50.09/6.88 Proof:
% 50.09/6.88 meet(X, join(Y, X))
% 50.09/6.88 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.88 meet(X, join(X, Y))
% 50.09/6.88 = { by lemma 63 }
% 50.09/6.88 X
% 50.09/6.88
% 50.09/6.88 Lemma 80: meet(X, converse(meet(Y, converse(X)))) = converse(meet(Y, converse(X))).
% 50.09/6.88 Proof:
% 50.09/6.88 meet(X, converse(meet(Y, converse(X))))
% 50.09/6.88 = { by lemma 31 }
% 50.09/6.88 meet(converse(meet(Y, converse(X))), X)
% 50.09/6.88 = { by lemma 54 R->L }
% 50.09/6.88 meet(converse(meet(Y, converse(X))), join(X, converse(meet(Y, converse(X)))))
% 50.09/6.88 = { by lemma 79 }
% 50.09/6.88 converse(meet(Y, converse(X)))
% 50.09/6.88
% 50.09/6.88 Lemma 81: converse(x1) = x1.
% 50.09/6.88 Proof:
% 50.09/6.88 converse(x1)
% 50.09/6.88 = { by lemma 78 R->L }
% 50.09/6.88 converse(meet(x1, converse(x1)))
% 50.09/6.88 = { by lemma 80 R->L }
% 50.09/6.88 meet(x1, converse(meet(x1, converse(x1))))
% 50.09/6.88 = { by lemma 78 }
% 50.09/6.88 meet(x1, converse(x1))
% 50.09/6.88 = { by lemma 78 }
% 50.09/6.88 x1
% 50.09/6.88
% 50.09/6.88 Lemma 82: join(x0, join(one, X)) = join(X, one).
% 50.09/6.88 Proof:
% 50.09/6.88 join(x0, join(one, X))
% 50.09/6.88 = { by axiom 11 (maddux2_join_associativity) }
% 50.09/6.88 join(join(x0, one), X)
% 50.09/6.88 = { by axiom 4 (goals_1) }
% 50.09/6.88 join(one, X)
% 50.09/6.88 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.88 join(X, one)
% 50.09/6.88
% 50.09/6.88 Lemma 83: join(X, join(Y, composition(X, x0))) = join(X, Y).
% 50.09/6.88 Proof:
% 50.09/6.88 join(X, join(Y, composition(X, x0)))
% 50.09/6.88 = { by lemma 70 R->L }
% 50.09/6.88 join(composition(X, join(x0, one)), Y)
% 50.09/6.89 = { by axiom 4 (goals_1) }
% 50.09/6.89 join(composition(X, one), Y)
% 50.09/6.89 = { by axiom 1 (composition_identity) }
% 50.09/6.89 join(X, Y)
% 50.09/6.89
% 50.09/6.89 Lemma 84: join(X, composition(join(X, Y), x0)) = join(X, composition(Y, x0)).
% 50.09/6.89 Proof:
% 50.09/6.89 join(X, composition(join(X, Y), x0))
% 50.09/6.89 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.89 join(X, composition(join(Y, X), x0))
% 50.09/6.89 = { by axiom 13 (composition_distributivity) }
% 50.09/6.89 join(X, join(composition(Y, x0), composition(X, x0)))
% 50.09/6.89 = { by lemma 83 }
% 50.09/6.89 join(X, composition(Y, x0))
% 50.09/6.89
% 50.09/6.89 Lemma 85: meet(X, composition(X, x0)) = composition(X, x0).
% 50.09/6.89 Proof:
% 50.09/6.89 meet(X, composition(X, x0))
% 50.09/6.89 = { by lemma 31 }
% 50.09/6.89 meet(composition(X, x0), X)
% 50.09/6.89 = { by lemma 35 R->L }
% 50.09/6.89 meet(composition(X, x0), join(X, zero))
% 50.09/6.89 = { by lemma 83 R->L }
% 50.09/6.89 meet(composition(X, x0), join(X, join(zero, composition(X, x0))))
% 50.09/6.89 = { by lemma 73 }
% 50.09/6.89 composition(X, x0)
% 50.09/6.89
% 50.09/6.89 Lemma 86: meet(X, composition(top, x0)) = composition(X, x0).
% 50.09/6.89 Proof:
% 50.09/6.89 meet(X, composition(top, x0))
% 50.09/6.89 = { by lemma 66 R->L }
% 50.09/6.89 meet(X, join(complement(X), composition(top, x0)))
% 50.09/6.89 = { by axiom 6 (def_top) }
% 50.09/6.89 meet(X, join(complement(X), composition(join(complement(X), complement(complement(X))), x0)))
% 50.09/6.89 = { by lemma 84 }
% 50.09/6.89 meet(X, join(complement(X), composition(complement(complement(X)), x0)))
% 50.09/6.89 = { by lemma 66 }
% 50.09/6.89 meet(X, composition(complement(complement(X)), x0))
% 50.09/6.89 = { by lemma 38 }
% 50.09/6.89 meet(X, composition(X, x0))
% 50.09/6.89 = { by lemma 85 }
% 50.09/6.89 composition(X, x0)
% 50.09/6.89
% 50.09/6.89 Lemma 87: meet(x0, converse(x0)) = x0.
% 50.09/6.89 Proof:
% 50.09/6.89 meet(x0, converse(x0))
% 50.09/6.89 = { by lemma 31 }
% 50.09/6.89 meet(converse(x0), x0)
% 50.09/6.89 = { by lemma 35 R->L }
% 50.09/6.89 join(meet(converse(x0), x0), zero)
% 50.09/6.89 = { by lemma 37 R->L }
% 50.09/6.89 join(meet(converse(x0), x0), meet(x0, zero))
% 50.09/6.89 = { by lemma 58 R->L }
% 50.09/6.89 join(meet(converse(x0), x0), meet(x0, meet(one, composition(converse(complement(x0)), x0))))
% 50.09/6.89 = { by lemma 59 }
% 50.09/6.89 join(meet(converse(x0), x0), meet(composition(converse(complement(x0)), x0), meet(x0, one)))
% 50.09/6.89 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.89 join(meet(converse(x0), x0), meet(composition(converse(complement(x0)), x0), complement(join(complement(x0), complement(one)))))
% 50.09/6.89 = { by lemma 36 R->L }
% 50.09/6.89 join(meet(converse(x0), x0), meet(composition(converse(complement(x0)), x0), join(zero, complement(join(complement(x0), complement(one))))))
% 50.09/6.89 = { by lemma 16 R->L }
% 50.09/6.89 join(meet(converse(x0), x0), meet(composition(converse(complement(x0)), x0), join(complement(top), complement(join(complement(x0), complement(one))))))
% 50.09/6.89 = { by lemma 27 R->L }
% 50.09/6.89 join(meet(converse(x0), x0), meet(composition(converse(complement(x0)), x0), join(complement(join(one, top)), complement(join(complement(x0), complement(one))))))
% 50.09/6.89 = { by lemma 20 R->L }
% 50.09/6.89 join(meet(converse(x0), x0), meet(composition(converse(complement(x0)), x0), join(complement(join(x0, join(one, complement(x0)))), complement(join(complement(x0), complement(one))))))
% 50.09/6.89 = { by lemma 82 }
% 50.09/6.89 join(meet(converse(x0), x0), meet(composition(converse(complement(x0)), x0), join(complement(join(complement(x0), one)), complement(join(complement(x0), complement(one))))))
% 50.09/6.89 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.89 join(meet(converse(x0), x0), meet(composition(converse(complement(x0)), x0), join(complement(join(one, complement(x0))), complement(join(complement(x0), complement(one))))))
% 50.09/6.89 = { by lemma 44 }
% 50.09/6.89 join(meet(converse(x0), x0), meet(composition(converse(complement(x0)), x0), join(meet(x0, complement(one)), complement(join(complement(x0), complement(one))))))
% 50.09/6.89 = { by lemma 32 }
% 50.09/6.89 join(meet(converse(x0), x0), meet(composition(converse(complement(x0)), x0), x0))
% 50.09/6.89 = { by lemma 65 }
% 50.09/6.89 join(meet(converse(x0), x0), meet(composition(complement(converse(x0)), x0), x0))
% 50.09/6.89 = { by lemma 31 R->L }
% 50.09/6.89 join(meet(converse(x0), x0), meet(x0, composition(complement(converse(x0)), x0)))
% 50.09/6.89 = { by lemma 86 R->L }
% 50.09/6.89 join(meet(converse(x0), x0), meet(x0, meet(complement(converse(x0)), composition(top, x0))))
% 50.09/6.89 = { by lemma 77 }
% 50.09/6.89 join(meet(converse(x0), x0), meet(x0, complement(converse(x0))))
% 50.09/6.89 = { by lemma 61 }
% 50.09/6.89 x0
% 50.09/6.89
% 50.09/6.89 Lemma 88: converse(x0) = x0.
% 50.09/6.89 Proof:
% 50.09/6.89 converse(x0)
% 50.09/6.89 = { by lemma 87 R->L }
% 50.09/6.89 converse(meet(x0, converse(x0)))
% 50.09/6.89 = { by lemma 80 R->L }
% 50.09/6.89 meet(x0, converse(meet(x0, converse(x0))))
% 50.09/6.89 = { by lemma 87 }
% 50.09/6.89 meet(x0, converse(x0))
% 50.09/6.89 = { by lemma 87 }
% 50.09/6.89 x0
% 50.09/6.89
% 50.09/6.89 Lemma 89: meet(x1, converse(meet(X, x1))) = converse(meet(X, x1)).
% 50.09/6.89 Proof:
% 50.09/6.89 meet(x1, converse(meet(X, x1)))
% 50.09/6.89 = { by lemma 81 R->L }
% 50.09/6.89 meet(converse(x1), converse(meet(X, x1)))
% 50.09/6.89 = { by lemma 31 }
% 50.09/6.89 meet(converse(meet(X, x1)), converse(x1))
% 50.09/6.89 = { by lemma 61 R->L }
% 50.09/6.89 meet(converse(meet(X, x1)), converse(join(meet(X, x1), meet(x1, complement(X)))))
% 50.09/6.89 = { by lemma 64 }
% 50.09/6.89 converse(meet(X, x1))
% 50.09/6.89
% 50.09/6.89 Lemma 90: meet(meet(X, Y), complement(Y)) = zero.
% 50.09/6.89 Proof:
% 50.09/6.89 meet(meet(X, Y), complement(Y))
% 50.09/6.89 = { by lemma 44 R->L }
% 50.09/6.89 complement(join(Y, complement(meet(X, Y))))
% 50.09/6.89 = { by lemma 31 R->L }
% 50.09/6.89 complement(join(Y, complement(meet(Y, X))))
% 50.09/6.89 = { by lemma 49 }
% 50.09/6.89 complement(top)
% 50.09/6.89 = { by lemma 16 }
% 50.09/6.89 zero
% 50.09/6.89
% 50.09/6.89 Lemma 91: composition(meet(X, x1), x1) = meet(X, x1).
% 50.09/6.89 Proof:
% 50.09/6.89 composition(meet(X, x1), x1)
% 50.09/6.89 = { by lemma 74 R->L }
% 50.09/6.89 meet(meet(X, x1), composition(top, x1))
% 50.09/6.89 = { by lemma 61 R->L }
% 50.09/6.89 meet(meet(X, x1), composition(top, join(meet(X, x1), meet(x1, complement(X)))))
% 50.09/6.89 = { by lemma 31 }
% 50.09/6.89 meet(composition(top, join(meet(X, x1), meet(x1, complement(X)))), meet(X, x1))
% 50.09/6.89 = { by lemma 63 R->L }
% 50.09/6.89 meet(composition(top, join(meet(X, x1), meet(x1, complement(X)))), meet(meet(X, x1), join(meet(X, x1), meet(x1, complement(X)))))
% 50.09/6.89 = { by lemma 59 R->L }
% 50.09/6.89 meet(meet(X, x1), meet(join(meet(X, x1), meet(x1, complement(X))), composition(top, join(meet(X, x1), meet(x1, complement(X))))))
% 50.09/6.89 = { by lemma 76 }
% 50.09/6.89 meet(meet(X, x1), join(meet(X, x1), meet(x1, complement(X))))
% 50.09/6.89 = { by lemma 61 }
% 50.09/6.89 meet(meet(X, x1), x1)
% 50.09/6.89 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.89 complement(join(complement(meet(X, x1)), complement(x1)))
% 50.09/6.89 = { by lemma 36 R->L }
% 50.09/6.89 join(zero, complement(join(complement(meet(X, x1)), complement(x1))))
% 50.09/6.89 = { by lemma 90 R->L }
% 50.09/6.89 join(meet(meet(X, x1), complement(x1)), complement(join(complement(meet(X, x1)), complement(x1))))
% 50.09/6.89 = { by lemma 32 }
% 50.09/6.89 meet(X, x1)
% 50.09/6.89
% 50.09/6.89 Lemma 92: meet(composition(top, x1), X) = composition(X, x1).
% 50.09/6.89 Proof:
% 50.09/6.89 meet(composition(top, x1), X)
% 50.09/6.89 = { by lemma 31 }
% 50.09/6.89 meet(X, composition(top, x1))
% 50.09/6.89 = { by lemma 74 }
% 50.09/6.89 composition(X, x1)
% 50.09/6.89
% 50.09/6.89 Lemma 93: join(composition(join(Z, one), X), Y) = join(X, join(Y, composition(Z, X))).
% 50.09/6.89 Proof:
% 50.09/6.89 join(composition(join(Z, one), X), Y)
% 50.09/6.89 = { by lemma 69 R->L }
% 50.09/6.89 join(join(X, composition(Z, X)), Y)
% 50.09/6.89 = { by axiom 11 (maddux2_join_associativity) R->L }
% 50.09/6.89 join(X, join(composition(Z, X), Y))
% 50.09/6.89 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.89 join(X, join(Y, composition(Z, X)))
% 50.09/6.89
% 50.09/6.89 Lemma 94: meet(X, composition(x0, X)) = composition(x0, X).
% 50.09/6.89 Proof:
% 50.09/6.89 meet(X, composition(x0, X))
% 50.09/6.89 = { by lemma 31 }
% 50.09/6.89 meet(composition(x0, X), X)
% 50.09/6.89 = { by lemma 35 R->L }
% 50.09/6.89 meet(composition(x0, X), join(X, zero))
% 50.09/6.89 = { by lemma 21 R->L }
% 50.09/6.89 meet(composition(x0, X), join(composition(one, X), zero))
% 50.09/6.89 = { by axiom 4 (goals_1) R->L }
% 50.09/6.89 meet(composition(x0, X), join(composition(join(x0, one), X), zero))
% 50.09/6.89 = { by lemma 93 }
% 50.09/6.89 meet(composition(x0, X), join(X, join(zero, composition(x0, X))))
% 50.09/6.89 = { by lemma 73 }
% 50.09/6.89 composition(x0, X)
% 50.09/6.89
% 50.09/6.89 Lemma 95: meet(x0, x1) = composition(x0, x1).
% 50.09/6.89 Proof:
% 50.09/6.89 meet(x0, x1)
% 50.09/6.89 = { by lemma 91 R->L }
% 50.09/6.89 composition(meet(x0, x1), x1)
% 50.09/6.89 = { by lemma 92 R->L }
% 50.09/6.89 meet(composition(top, x1), meet(x0, x1))
% 50.09/6.89 = { by lemma 59 }
% 50.09/6.89 meet(x1, meet(composition(top, x1), x0))
% 50.09/6.89 = { by lemma 92 }
% 50.09/6.89 meet(x1, composition(x0, x1))
% 50.09/6.89 = { by lemma 94 }
% 50.09/6.89 composition(x0, x1)
% 50.09/6.89
% 50.09/6.89 Lemma 96: join(X, join(Y, composition(x1, X))) = join(X, Y).
% 50.09/6.89 Proof:
% 50.09/6.89 join(X, join(Y, composition(x1, X)))
% 50.09/6.89 = { by lemma 93 R->L }
% 50.09/6.89 join(composition(join(x1, one), X), Y)
% 50.09/6.89 = { by axiom 3 (goals) }
% 50.09/6.89 join(composition(one, X), Y)
% 50.09/6.89 = { by lemma 21 }
% 50.09/6.89 join(X, Y)
% 50.09/6.89
% 50.09/6.89 Lemma 97: meet(one, composition(x1, x0)) = composition(x1, x0).
% 50.09/6.89 Proof:
% 50.09/6.89 meet(one, composition(x1, x0))
% 50.09/6.89 = { by lemma 31 }
% 50.09/6.89 meet(composition(x1, x0), one)
% 50.09/6.89 = { by axiom 4 (goals_1) R->L }
% 50.09/6.89 meet(composition(x1, x0), join(x0, one))
% 50.09/6.89 = { by lemma 96 R->L }
% 50.09/6.89 meet(composition(x1, x0), join(x0, join(one, composition(x1, x0))))
% 50.09/6.89 = { by lemma 82 }
% 50.09/6.89 meet(composition(x1, x0), join(composition(x1, x0), one))
% 50.09/6.89 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.89 meet(composition(x1, x0), join(one, composition(x1, x0)))
% 50.09/6.89 = { by lemma 79 }
% 50.09/6.89 composition(x1, x0)
% 50.09/6.89
% 50.09/6.89 Lemma 98: meet(X, composition(Y, x1)) = composition(meet(X, Y), x1).
% 50.09/6.89 Proof:
% 50.09/6.89 meet(X, composition(Y, x1))
% 50.09/6.89 = { by lemma 74 R->L }
% 50.09/6.89 meet(X, meet(Y, composition(top, x1)))
% 50.09/6.89 = { by lemma 75 R->L }
% 50.09/6.89 meet(meet(X, Y), composition(top, x1))
% 50.09/6.89 = { by lemma 74 }
% 50.09/6.89 composition(meet(X, Y), x1)
% 50.09/6.89
% 50.09/6.89 Lemma 99: meet(x0, x1) = composition(x1, x0).
% 50.09/6.89 Proof:
% 50.09/6.89 meet(x0, x1)
% 50.09/6.89 = { by axiom 5 (converse_idempotence) R->L }
% 50.09/6.89 converse(converse(meet(x0, x1)))
% 50.09/6.89 = { by lemma 89 R->L }
% 50.09/6.89 converse(meet(x1, converse(meet(x0, x1))))
% 50.09/6.89 = { by lemma 31 }
% 50.09/6.89 converse(meet(converse(meet(x0, x1)), x1))
% 50.09/6.89 = { by lemma 91 R->L }
% 50.09/6.89 converse(composition(meet(converse(meet(x0, x1)), x1), x1))
% 50.09/6.89 = { by lemma 31 R->L }
% 50.09/6.89 converse(composition(meet(x1, converse(meet(x0, x1))), x1))
% 50.09/6.89 = { by lemma 89 }
% 50.09/6.89 converse(composition(converse(meet(x0, x1)), x1))
% 50.09/6.89 = { by lemma 17 }
% 50.09/6.89 composition(converse(x1), meet(x0, x1))
% 50.09/6.89 = { by lemma 81 }
% 50.09/6.89 composition(x1, meet(x0, x1))
% 50.09/6.89 = { by lemma 95 }
% 50.09/6.89 composition(x1, composition(x0, x1))
% 50.09/6.89 = { by axiom 9 (composition_associativity) }
% 50.09/6.89 composition(composition(x1, x0), x1)
% 50.09/6.89 = { by lemma 97 R->L }
% 50.09/6.89 composition(meet(one, composition(x1, x0)), x1)
% 50.09/6.89 = { by lemma 31 }
% 50.09/6.89 composition(meet(composition(x1, x0), one), x1)
% 50.09/6.89 = { by lemma 98 R->L }
% 50.09/6.89 meet(composition(x1, x0), composition(one, x1))
% 50.09/6.89 = { by lemma 21 }
% 50.09/6.89 meet(composition(x1, x0), x1)
% 50.09/6.89 = { by lemma 31 R->L }
% 50.09/6.89 meet(x1, composition(x1, x0))
% 50.09/6.89 = { by lemma 85 }
% 50.09/6.89 composition(x1, x0)
% 50.09/6.89
% 50.09/6.89 Lemma 100: meet(X, composition(x1, X)) = composition(x1, X).
% 50.09/6.89 Proof:
% 50.09/6.89 meet(X, composition(x1, X))
% 50.09/6.89 = { by lemma 31 }
% 50.09/6.89 meet(composition(x1, X), X)
% 50.09/6.89 = { by lemma 35 R->L }
% 50.09/6.89 meet(composition(x1, X), join(X, zero))
% 50.09/6.89 = { by lemma 96 R->L }
% 50.09/6.89 meet(composition(x1, X), join(X, join(zero, composition(x1, X))))
% 50.09/6.89 = { by lemma 73 }
% 50.09/6.89 composition(x1, X)
% 50.09/6.89
% 50.09/6.89 Lemma 101: converse(meet(x0, x1)) = meet(x0, x1).
% 50.09/6.89 Proof:
% 50.09/6.89 converse(meet(x0, x1))
% 50.09/6.89 = { by lemma 95 }
% 50.09/6.89 converse(composition(x0, x1))
% 50.09/6.89 = { by axiom 8 (converse_multiplicativity) }
% 50.09/6.89 composition(converse(x1), converse(x0))
% 50.09/6.89 = { by lemma 81 }
% 50.09/6.89 composition(x1, converse(x0))
% 50.09/6.89 = { by lemma 100 R->L }
% 50.09/6.89 meet(converse(x0), composition(x1, converse(x0)))
% 50.09/6.89 = { by lemma 31 }
% 50.09/6.89 meet(composition(x1, converse(x0)), converse(x0))
% 50.09/6.89 = { by lemma 79 R->L }
% 50.09/6.89 meet(composition(x1, converse(x0)), meet(converse(x0), join(one, converse(x0))))
% 50.09/6.90 = { by lemma 67 R->L }
% 50.09/6.90 meet(composition(x1, converse(x0)), meet(converse(x0), converse(join(x0, one))))
% 50.09/6.90 = { by axiom 4 (goals_1) }
% 50.09/6.90 meet(composition(x1, converse(x0)), meet(converse(x0), converse(one)))
% 50.09/6.90 = { by lemma 19 }
% 50.09/6.90 meet(composition(x1, converse(x0)), meet(converse(x0), one))
% 50.09/6.90 = { by lemma 31 R->L }
% 50.09/6.90 meet(composition(x1, converse(x0)), meet(one, converse(x0)))
% 50.09/6.90 = { by lemma 59 R->L }
% 50.09/6.90 meet(one, meet(converse(x0), composition(x1, converse(x0))))
% 50.09/6.90 = { by lemma 100 }
% 50.09/6.90 meet(one, composition(x1, converse(x0)))
% 50.09/6.90 = { by lemma 88 }
% 50.09/6.90 meet(one, composition(x1, x0))
% 50.09/6.90 = { by lemma 97 }
% 50.09/6.90 composition(x1, x0)
% 50.09/6.90 = { by lemma 99 R->L }
% 50.09/6.90 meet(x0, x1)
% 50.09/6.90
% 50.09/6.90 Lemma 102: join(X, meet(Y, X)) = X.
% 50.09/6.90 Proof:
% 50.09/6.90 join(X, meet(Y, X))
% 50.09/6.90 = { by lemma 31 }
% 50.09/6.90 join(X, meet(X, Y))
% 50.09/6.90 = { by lemma 51 }
% 50.09/6.90 X
% 50.09/6.90
% 50.09/6.90 Lemma 103: meet(X, meet(Y, Z)) = meet(Z, meet(Y, X)).
% 50.09/6.90 Proof:
% 50.09/6.90 meet(X, meet(Y, Z))
% 50.09/6.90 = { by lemma 59 }
% 50.09/6.90 meet(Z, meet(X, Y))
% 50.09/6.90 = { by lemma 31 }
% 50.09/6.90 meet(Z, meet(Y, X))
% 50.09/6.90
% 50.09/6.90 Lemma 104: meet(meet(X, Y), Y) = meet(X, Y).
% 50.09/6.90 Proof:
% 50.09/6.90 meet(meet(X, Y), Y)
% 50.09/6.90 = { by axiom 12 (maddux4_definiton_of_meet) }
% 50.09/6.90 complement(join(complement(meet(X, Y)), complement(Y)))
% 50.09/6.90 = { by lemma 36 R->L }
% 50.09/6.90 join(zero, complement(join(complement(meet(X, Y)), complement(Y))))
% 50.09/6.90 = { by lemma 90 R->L }
% 50.09/6.90 join(meet(meet(X, Y), complement(Y)), complement(join(complement(meet(X, Y)), complement(Y))))
% 50.09/6.90 = { by lemma 32 }
% 50.09/6.90 meet(X, Y)
% 50.09/6.90
% 50.09/6.90 Lemma 105: composition(meet(x0, x1), X) = composition(x1, composition(x0, X)).
% 50.09/6.90 Proof:
% 50.09/6.90 composition(meet(x0, x1), X)
% 50.09/6.90 = { by lemma 99 }
% 50.09/6.90 composition(composition(x1, x0), X)
% 50.09/6.90 = { by axiom 9 (composition_associativity) R->L }
% 50.09/6.90 composition(x1, composition(x0, X))
% 50.09/6.90
% 50.09/6.90 Lemma 106: join(composition(X, Y), composition(Z, Y)) = composition(join(Z, X), Y).
% 50.09/6.90 Proof:
% 50.09/6.90 join(composition(X, Y), composition(Z, Y))
% 50.09/6.90 = { by axiom 13 (composition_distributivity) R->L }
% 50.09/6.90 composition(join(X, Z), Y)
% 50.09/6.90 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.90 composition(join(Z, X), Y)
% 50.09/6.90
% 50.09/6.90 Lemma 107: meet(X, composition(meet(x0, x1), top)) = meet(X, composition(meet(x0, x1), X)).
% 50.09/6.90 Proof:
% 50.09/6.90 meet(X, composition(meet(x0, x1), top))
% 50.09/6.90 = { by lemma 66 R->L }
% 50.09/6.90 meet(X, join(complement(X), composition(meet(x0, x1), top)))
% 50.09/6.90 = { by lemma 101 R->L }
% 50.09/6.90 meet(X, join(complement(X), composition(converse(meet(x0, x1)), top)))
% 50.09/6.90 = { by lemma 30 R->L }
% 50.09/6.90 meet(X, join(complement(X), composition(converse(meet(x0, x1)), converse(top))))
% 50.09/6.90 = { by axiom 8 (converse_multiplicativity) R->L }
% 50.09/6.90 meet(X, join(complement(X), converse(composition(top, meet(x0, x1)))))
% 50.09/6.90 = { by lemma 41 R->L }
% 50.09/6.90 meet(X, converse(join(converse(complement(X)), composition(top, meet(x0, x1)))))
% 50.09/6.90 = { by lemma 99 }
% 50.09/6.90 meet(X, converse(join(converse(complement(X)), composition(top, composition(x1, x0)))))
% 50.09/6.90 = { by axiom 9 (composition_associativity) }
% 50.09/6.90 meet(X, converse(join(converse(complement(X)), composition(composition(top, x1), x0))))
% 50.09/6.90 = { by lemma 84 R->L }
% 50.09/6.90 meet(X, converse(join(converse(complement(X)), composition(join(converse(complement(X)), composition(top, x1)), x0))))
% 50.09/6.90 = { by lemma 72 R->L }
% 50.09/6.90 meet(X, converse(join(converse(complement(X)), composition(join(converse(complement(X)), composition(complement(converse(complement(X))), x1)), x0))))
% 50.09/6.90 = { by lemma 84 }
% 50.09/6.90 meet(X, converse(join(converse(complement(X)), composition(composition(complement(converse(complement(X))), x1), x0))))
% 50.09/6.90 = { by axiom 9 (composition_associativity) R->L }
% 50.09/6.90 meet(X, converse(join(converse(complement(X)), composition(complement(converse(complement(X))), composition(x1, x0)))))
% 50.09/6.90 = { by lemma 99 R->L }
% 50.09/6.90 meet(X, converse(join(converse(complement(X)), composition(complement(converse(complement(X))), meet(x0, x1)))))
% 50.09/6.90 = { by lemma 41 }
% 50.09/6.90 meet(X, join(complement(X), converse(composition(complement(converse(complement(X))), meet(x0, x1)))))
% 50.09/6.90 = { by lemma 65 R->L }
% 50.09/6.90 meet(X, join(complement(X), converse(composition(converse(complement(complement(X))), meet(x0, x1)))))
% 50.09/6.90 = { by lemma 17 }
% 50.09/6.90 meet(X, join(complement(X), composition(converse(meet(x0, x1)), complement(complement(X)))))
% 50.09/6.90 = { by lemma 101 }
% 50.09/6.90 meet(X, join(complement(X), composition(meet(x0, x1), complement(complement(X)))))
% 50.09/6.90 = { by lemma 66 }
% 50.09/6.90 meet(X, composition(meet(x0, x1), complement(complement(X))))
% 50.09/6.90 = { by lemma 38 }
% 50.09/6.90 meet(X, composition(meet(x0, x1), X))
% 50.09/6.90
% 50.09/6.90 Lemma 108: converse(composition(join(converse(X), Y), Z)) = composition(converse(Z), join(X, converse(Y))).
% 50.09/6.90 Proof:
% 50.09/6.90 converse(composition(join(converse(X), Y), Z))
% 50.09/6.90 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.90 converse(composition(join(Y, converse(X)), Z))
% 50.09/6.90 = { by axiom 8 (converse_multiplicativity) }
% 50.09/6.90 composition(converse(Z), converse(join(Y, converse(X))))
% 50.09/6.90 = { by lemma 29 }
% 50.09/6.90 composition(converse(Z), join(X, converse(Y)))
% 50.09/6.90
% 50.09/6.90 Lemma 109: join(composition(X, converse(Y)), converse(composition(Y, Z))) = composition(join(X, converse(Z)), converse(Y)).
% 50.09/6.90 Proof:
% 50.09/6.90 join(composition(X, converse(Y)), converse(composition(Y, Z)))
% 50.09/6.90 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.90 join(converse(composition(Y, Z)), composition(X, converse(Y)))
% 50.09/6.90 = { by axiom 8 (converse_multiplicativity) }
% 50.09/6.90 join(composition(converse(Z), converse(Y)), composition(X, converse(Y)))
% 50.09/6.90 = { by axiom 13 (composition_distributivity) R->L }
% 50.09/6.90 composition(join(converse(Z), X), converse(Y))
% 50.09/6.90 = { by axiom 2 (maddux1_join_commutativity) }
% 50.09/6.90 composition(join(X, converse(Z)), converse(Y))
% 50.09/6.90
% 50.09/6.90 Goal 1 (goals_2): meet(composition(x0, x2), composition(x1, x2)) = composition(meet(x0, x1), x2).
% 50.09/6.90 Proof:
% 50.09/6.90 meet(composition(x0, x2), composition(x1, x2))
% 50.09/6.90 = { by axiom 5 (converse_idempotence) R->L }
% 50.09/6.90 converse(converse(meet(composition(x0, x2), composition(x1, x2))))
% 50.09/6.90 = { by lemma 64 R->L }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(meet(composition(x0, x2), composition(x1, x2)), composition(x0, top)))))
% 50.09/6.90 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x0, x2), composition(x1, x2))))))
% 50.09/6.90 = { by lemma 31 R->L }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x1, x2), composition(x0, x2))))))
% 50.09/6.90 = { by lemma 63 R->L }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x1, x2), meet(composition(x0, x2), join(composition(x0, x2), composition(x0, top))))))))
% 50.09/6.90 = { by lemma 88 R->L }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x1, x2), meet(composition(x0, x2), join(composition(x0, x2), composition(converse(x0), top))))))))
% 50.09/6.90 = { by axiom 5 (converse_idempotence) R->L }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x1, x2), meet(composition(x0, x2), join(converse(converse(composition(x0, x2))), composition(converse(x0), top))))))))
% 50.09/6.90 = { by lemma 57 R->L }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x1, x2), meet(composition(x0, x2), converse(join(converse(composition(x0, x2)), composition(converse(top), x0)))))))))
% 50.09/6.90 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x1, x2), meet(composition(x0, x2), converse(join(composition(converse(top), x0), converse(composition(x0, x2))))))))))
% 50.09/6.90 = { by lemma 88 R->L }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x1, x2), meet(composition(x0, x2), converse(join(composition(converse(top), converse(x0)), converse(composition(x0, x2))))))))))
% 50.09/6.90 = { by lemma 109 }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x1, x2), meet(composition(x0, x2), converse(composition(join(converse(top), converse(x2)), converse(x0)))))))))
% 50.09/6.90 = { by lemma 88 }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x1, x2), meet(composition(x0, x2), converse(composition(join(converse(top), converse(x2)), x0))))))))
% 50.09/6.90 = { by lemma 108 }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x1, x2), meet(composition(x0, x2), composition(converse(x0), join(top, converse(converse(x2))))))))))
% 50.09/6.90 = { by lemma 88 }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x1, x2), meet(composition(x0, x2), composition(x0, join(top, converse(converse(x2))))))))))
% 50.09/6.90 = { by axiom 5 (converse_idempotence) }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x1, x2), meet(composition(x0, x2), composition(x0, join(top, x2))))))))
% 50.09/6.90 = { by lemma 28 }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x1, x2), meet(composition(x0, x2), composition(x0, top)))))))
% 50.09/6.90 = { by lemma 103 }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(composition(x0, top), meet(composition(x0, x2), composition(x1, x2)))))))
% 50.09/6.90 = { by lemma 31 R->L }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x0, top), meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, top))))))
% 50.09/6.90 = { by lemma 102 }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(composition(x0, top))))
% 50.09/6.90 = { by axiom 8 (converse_multiplicativity) }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), composition(converse(top), converse(x0))))
% 50.09/6.90 = { by lemma 88 }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), composition(converse(top), x0)))
% 50.09/6.90 = { by lemma 86 R->L }
% 50.09/6.90 converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), meet(converse(top), composition(top, x0))))
% 50.09/6.91 = { by lemma 75 R->L }
% 50.09/6.91 converse(meet(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(top)), composition(top, x0)))
% 50.09/6.91 = { by lemma 86 }
% 50.09/6.91 converse(composition(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(top)), x0))
% 50.09/6.91 = { by lemma 30 }
% 50.09/6.91 converse(composition(meet(converse(meet(composition(x0, x2), composition(x1, x2))), top), x0))
% 50.09/6.91 = { by lemma 40 }
% 50.09/6.91 converse(composition(converse(meet(composition(x0, x2), composition(x1, x2))), x0))
% 50.09/6.91 = { by lemma 17 }
% 50.09/6.91 composition(converse(x0), meet(composition(x0, x2), composition(x1, x2)))
% 50.09/6.91 = { by lemma 88 }
% 50.09/6.91 composition(x0, meet(composition(x0, x2), composition(x1, x2)))
% 50.09/6.91 = { by lemma 94 R->L }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, meet(composition(x0, x2), composition(x1, x2))))
% 50.09/6.91 = { by axiom 5 (converse_idempotence) R->L }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(converse(meet(composition(x0, x2), composition(x1, x2))))))
% 50.09/6.91 = { by lemma 64 R->L }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(meet(composition(x0, x2), composition(x1, x2)), composition(x1, top)))))))
% 50.09/6.91 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(composition(x0, x2), composition(x1, x2))))))))
% 50.09/6.91 = { by lemma 104 R->L }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), composition(x1, x2))))))))
% 50.09/6.91 = { by lemma 63 R->L }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), meet(composition(x1, x2), join(composition(x1, x2), composition(x1, top))))))))))
% 50.09/6.91 = { by lemma 81 R->L }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), meet(composition(x1, x2), join(composition(x1, x2), composition(converse(x1), top))))))))))
% 50.09/6.91 = { by axiom 5 (converse_idempotence) R->L }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), meet(composition(x1, x2), join(converse(converse(composition(x1, x2))), composition(converse(x1), top))))))))))
% 50.09/6.91 = { by lemma 57 R->L }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), meet(composition(x1, x2), converse(join(converse(composition(x1, x2)), composition(converse(top), x1)))))))))))
% 50.09/6.91 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), meet(composition(x1, x2), converse(join(composition(converse(top), x1), converse(composition(x1, x2))))))))))))
% 50.09/6.91 = { by lemma 81 R->L }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), meet(composition(x1, x2), converse(join(composition(converse(top), converse(x1)), converse(composition(x1, x2))))))))))))
% 50.09/6.91 = { by lemma 109 }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), meet(composition(x1, x2), converse(composition(join(converse(top), converse(x2)), converse(x1)))))))))))
% 50.09/6.91 = { by lemma 81 }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), meet(composition(x1, x2), converse(composition(join(converse(top), converse(x2)), x1))))))))))
% 50.09/6.91 = { by lemma 108 }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), meet(composition(x1, x2), composition(converse(x1), join(top, converse(converse(x2))))))))))))
% 50.09/6.91 = { by lemma 81 }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), meet(composition(x1, x2), composition(x1, join(top, converse(converse(x2))))))))))))
% 50.09/6.91 = { by axiom 5 (converse_idempotence) }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), meet(composition(x1, x2), composition(x1, join(top, x2))))))))))
% 50.09/6.91 = { by lemma 28 }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), meet(composition(x1, x2), composition(x1, top)))))))))
% 50.09/6.91 = { by lemma 59 }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), composition(x1, x2)))))))))
% 50.09/6.91 = { by lemma 104 }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(composition(x1, top), meet(composition(x0, x2), composition(x1, x2)))))))))
% 50.09/6.91 = { by lemma 31 R->L }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(join(composition(x1, top), meet(meet(composition(x0, x2), composition(x1, x2)), composition(x1, top))))))))
% 50.09/6.91 = { by lemma 102 }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(composition(x1, top))))))
% 50.09/6.91 = { by axiom 8 (converse_multiplicativity) }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), composition(converse(top), converse(x1))))))
% 50.09/6.91 = { by lemma 81 }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(meet(converse(meet(composition(x0, x2), composition(x1, x2))), composition(converse(top), x1)))))
% 50.09/6.91 = { by lemma 98 }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(composition(meet(converse(meet(composition(x0, x2), composition(x1, x2))), converse(top)), x1))))
% 50.09/6.91 = { by lemma 30 }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(composition(meet(converse(meet(composition(x0, x2), composition(x1, x2))), top), x1))))
% 50.09/6.91 = { by lemma 40 }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, converse(composition(converse(meet(composition(x0, x2), composition(x1, x2))), x1))))
% 50.09/6.91 = { by lemma 17 }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, composition(converse(x1), meet(composition(x0, x2), composition(x1, x2)))))
% 50.09/6.91 = { by lemma 81 }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(x0, composition(x1, meet(composition(x0, x2), composition(x1, x2)))))
% 50.09/6.91 = { by axiom 9 (composition_associativity) }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(composition(x0, x1), meet(composition(x0, x2), composition(x1, x2))))
% 50.09/6.91 = { by lemma 95 R->L }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), meet(composition(x0, x2), composition(x1, x2))))
% 50.09/6.91 = { by lemma 107 R->L }
% 50.09/6.91 meet(meet(composition(x0, x2), composition(x1, x2)), composition(meet(x0, x1), top))
% 50.09/6.91 = { by lemma 31 }
% 50.09/6.91 meet(composition(meet(x0, x1), top), meet(composition(x0, x2), composition(x1, x2)))
% 50.09/6.91 = { by lemma 103 R->L }
% 50.09/6.91 meet(composition(x1, x2), meet(composition(x0, x2), composition(meet(x0, x1), top)))
% 50.09/6.91 = { by lemma 107 }
% 50.09/6.91 meet(composition(x1, x2), meet(composition(x0, x2), composition(meet(x0, x1), composition(x0, x2))))
% 50.09/6.91 = { by lemma 103 }
% 50.09/6.91 meet(composition(meet(x0, x1), composition(x0, x2)), meet(composition(x0, x2), composition(x1, x2)))
% 50.09/6.91 = { by lemma 105 }
% 50.09/6.91 meet(composition(x1, composition(x0, composition(x0, x2))), meet(composition(x0, x2), composition(x1, x2)))
% 50.09/6.91 = { by axiom 9 (composition_associativity) }
% 50.09/6.91 meet(composition(x1, composition(composition(x0, x0), x2)), meet(composition(x0, x2), composition(x1, x2)))
% 50.09/6.91 = { by lemma 86 R->L }
% 50.09/6.92 meet(composition(x1, composition(meet(x0, composition(top, x0)), x2)), meet(composition(x0, x2), composition(x1, x2)))
% 50.09/6.92 = { by lemma 76 }
% 50.09/6.92 meet(composition(x1, composition(x0, x2)), meet(composition(x0, x2), composition(x1, x2)))
% 50.09/6.92 = { by lemma 105 R->L }
% 50.09/6.92 meet(composition(meet(x0, x1), x2), meet(composition(x0, x2), composition(x1, x2)))
% 50.09/6.92 = { by lemma 59 }
% 50.09/6.92 meet(composition(x1, x2), meet(composition(meet(x0, x1), x2), composition(x0, x2)))
% 50.09/6.92 = { by lemma 51 R->L }
% 50.09/6.92 meet(composition(x1, x2), meet(composition(meet(x0, x1), x2), composition(join(x0, meet(x0, x1)), x2)))
% 50.09/6.92 = { by lemma 106 R->L }
% 50.09/6.92 meet(composition(x1, x2), meet(composition(meet(x0, x1), x2), join(composition(meet(x0, x1), x2), composition(x0, x2))))
% 50.09/6.92 = { by lemma 63 }
% 50.09/6.92 meet(composition(x1, x2), composition(meet(x0, x1), x2))
% 50.09/6.92 = { by lemma 31 R->L }
% 50.09/6.92 meet(composition(meet(x0, x1), x2), composition(x1, x2))
% 50.09/6.92 = { by lemma 102 R->L }
% 50.09/6.92 meet(composition(meet(x0, x1), x2), composition(join(x1, meet(x0, x1)), x2))
% 50.09/6.92 = { by lemma 106 R->L }
% 50.09/6.92 meet(composition(meet(x0, x1), x2), join(composition(meet(x0, x1), x2), composition(x1, x2)))
% 50.09/6.92 = { by lemma 63 }
% 50.09/6.92 composition(meet(x0, x1), x2)
% 50.09/6.92 % SZS output end Proof
% 50.09/6.92
% 50.09/6.92 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------