TSTP Solution File: REL026+4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL026+4 : 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 : n029.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 13:44:08 EDT 2023
% Result : Theorem 13.76s 2.67s
% Output : Proof 15.00s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL026+4 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.32 % Computer : n029.cluster.edu
% 0.12/0.32 % Model : x86_64 x86_64
% 0.12/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.32 % Memory : 8042.1875MB
% 0.12/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.32 % CPULimit : 300
% 0.12/0.32 % WCLimit : 300
% 0.12/0.32 % DateTime : Fri Aug 25 19:45:24 EDT 2023
% 0.12/0.32 % CPUTime :
% 13.76/2.67 Command-line arguments: --flatten
% 13.76/2.67
% 13.76/2.67 % SZS status Theorem
% 13.76/2.67
% 15.00/2.87 % SZS output start Proof
% 15.00/2.88 Take the following subset of the input axioms:
% 15.00/2.88 fof(composition_associativity, axiom, ![X0, X1, X2]: composition(X0, composition(X1, X2))=composition(composition(X0, X1), X2)).
% 15.00/2.88 fof(composition_distributivity, axiom, ![X0_2, X1_2, X2_2]: composition(join(X0_2, X1_2), X2_2)=join(composition(X0_2, X2_2), composition(X1_2, X2_2))).
% 15.00/2.88 fof(composition_identity, axiom, ![X0_2]: composition(X0_2, one)=X0_2).
% 15.00/2.89 fof(converse_additivity, axiom, ![X0_2, X1_2]: converse(join(X0_2, X1_2))=join(converse(X0_2), converse(X1_2))).
% 15.00/2.89 fof(converse_cancellativity, axiom, ![X0_2, X1_2]: join(composition(converse(X0_2), complement(composition(X0_2, X1_2))), complement(X1_2))=complement(X1_2)).
% 15.00/2.89 fof(converse_idempotence, axiom, ![X0_2]: converse(converse(X0_2))=X0_2).
% 15.00/2.89 fof(converse_multiplicativity, axiom, ![X0_2, X1_2]: converse(composition(X0_2, X1_2))=composition(converse(X1_2), converse(X0_2))).
% 15.00/2.89 fof(def_top, axiom, ![X0_2]: top=join(X0_2, complement(X0_2))).
% 15.00/2.89 fof(def_zero, axiom, ![X0_2]: zero=meet(X0_2, complement(X0_2))).
% 15.00/2.89 fof(goals, conjecture, ![X0_2, X1_2]: (join(X0_2, one)=one => (join(meet(composition(X0_2, top), X1_2), composition(X0_2, X1_2))=composition(X0_2, X1_2) & join(composition(X0_2, X1_2), meet(composition(X0_2, top), X1_2))=meet(composition(X0_2, top), X1_2)))).
% 15.00/2.90 fof(maddux1_join_commutativity, axiom, ![X0_2, X1_2]: join(X0_2, X1_2)=join(X1_2, X0_2)).
% 15.00/2.90 fof(maddux2_join_associativity, axiom, ![X0_2, X1_2, X2_2]: join(X0_2, join(X1_2, X2_2))=join(join(X0_2, X1_2), X2_2)).
% 15.00/2.90 fof(maddux3_a_kind_of_de_Morgan, axiom, ![X0_2, X1_2]: X0_2=join(complement(join(complement(X0_2), complement(X1_2))), complement(join(complement(X0_2), X1_2)))).
% 15.00/2.90 fof(maddux4_definiton_of_meet, axiom, ![X0_2, X1_2]: meet(X0_2, X1_2)=complement(join(complement(X0_2), complement(X1_2)))).
% 15.00/2.90 fof(modular_law_1, axiom, ![X0_2, X1_2, X2_2]: join(meet(composition(X0_2, X1_2), X2_2), meet(composition(X0_2, meet(X1_2, composition(converse(X0_2), X2_2))), X2_2))=meet(composition(X0_2, meet(X1_2, composition(converse(X0_2), X2_2))), X2_2)).
% 15.00/2.90 fof(modular_law_2, axiom, ![X0_2, X1_2, X2_2]: join(meet(composition(X0_2, X1_2), X2_2), meet(composition(meet(X0_2, composition(X2_2, converse(X1_2))), X1_2), X2_2))=meet(composition(meet(X0_2, composition(X2_2, converse(X1_2))), X1_2), X2_2)).
% 15.00/2.90
% 15.00/2.91 Now clausify the problem and encode Horn clauses using encoding 3 of
% 15.00/2.91 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 15.00/2.91 We repeatedly replace C & s=t => u=v by the two clauses:
% 15.00/2.91 fresh(y, y, x1...xn) = u
% 15.00/2.91 C => fresh(s, t, x1...xn) = v
% 15.00/2.91 where fresh is a fresh function symbol and x1..xn are the free
% 15.00/2.91 variables of u and v.
% 15.00/2.91 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 15.00/2.91 input problem has no model of domain size 1).
% 15.00/2.91
% 15.00/2.91 The encoding turns the above axioms into the following unit equations and goals:
% 15.00/2.91
% 15.00/2.91 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 15.00/2.91 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 15.00/2.91 Axiom 3 (goals): join(x0, one) = one.
% 15.00/2.91 Axiom 4 (composition_identity): composition(X, one) = X.
% 15.00/2.91 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 15.00/2.91 Axiom 6 (def_top): top = join(X, complement(X)).
% 15.00/2.91 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 15.00/2.91 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 15.00/2.91 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 15.00/2.91 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 15.00/2.91 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 15.00/2.91 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 15.00/2.91 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 15.00/2.91 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 15.00/2.91 Axiom 15 (modular_law_1): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
% 15.00/2.91 Axiom 16 (modular_law_2): join(meet(composition(X, Y), Z), meet(composition(meet(X, composition(Z, converse(Y))), Y), Z)) = meet(composition(meet(X, composition(Z, converse(Y))), Y), Z).
% 15.00/2.91
% 15.00/2.91 Lemma 17: complement(top) = zero.
% 15.00/2.91 Proof:
% 15.00/2.91 complement(top)
% 15.00/2.91 = { by axiom 6 (def_top) }
% 15.00/2.91 complement(join(complement(X), complement(complement(X))))
% 15.00/2.91 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.00/2.91 meet(X, complement(X))
% 15.00/2.91 = { by axiom 5 (def_zero) R->L }
% 15.00/2.91 zero
% 15.00/2.91
% 15.00/2.91 Lemma 18: join(X, join(Y, complement(X))) = join(Y, top).
% 15.00/2.91 Proof:
% 15.00/2.91 join(X, join(Y, complement(X)))
% 15.00/2.91 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.91 join(X, join(complement(X), Y))
% 15.00/2.91 = { by axiom 8 (maddux2_join_associativity) }
% 15.00/2.91 join(join(X, complement(X)), Y)
% 15.00/2.91 = { by axiom 6 (def_top) R->L }
% 15.00/2.91 join(top, Y)
% 15.00/2.91 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.91 join(Y, top)
% 15.00/2.91
% 15.00/2.91 Lemma 19: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 15.00/2.91 Proof:
% 15.00/2.91 converse(composition(converse(X), Y))
% 15.00/2.91 = { by axiom 9 (converse_multiplicativity) }
% 15.00/2.91 composition(converse(Y), converse(converse(X)))
% 15.00/2.91 = { by axiom 1 (converse_idempotence) }
% 15.00/2.91 composition(converse(Y), X)
% 15.00/2.91
% 15.00/2.91 Lemma 20: composition(X, join(x0, one)) = X.
% 15.00/2.91 Proof:
% 15.00/2.91 composition(X, join(x0, one))
% 15.00/2.91 = { by axiom 3 (goals) }
% 15.00/2.91 composition(X, one)
% 15.00/2.91 = { by axiom 4 (composition_identity) }
% 15.00/2.91 X
% 15.00/2.91
% 15.00/2.91 Lemma 21: composition(converse(join(x0, one)), X) = X.
% 15.00/2.91 Proof:
% 15.00/2.91 composition(converse(join(x0, one)), X)
% 15.00/2.91 = { by lemma 19 R->L }
% 15.00/2.91 converse(composition(converse(X), join(x0, one)))
% 15.00/2.91 = { by lemma 20 }
% 15.00/2.91 converse(converse(X))
% 15.00/2.91 = { by axiom 1 (converse_idempotence) }
% 15.00/2.91 X
% 15.00/2.91
% 15.00/2.91 Lemma 22: composition(join(x0, one), X) = X.
% 15.00/2.91 Proof:
% 15.00/2.91 composition(join(x0, one), X)
% 15.00/2.91 = { by lemma 21 R->L }
% 15.00/2.91 composition(converse(join(x0, one)), composition(join(x0, one), X))
% 15.00/2.91 = { by axiom 10 (composition_associativity) }
% 15.00/2.91 composition(composition(converse(join(x0, one)), join(x0, one)), X)
% 15.00/2.91 = { by lemma 20 }
% 15.00/2.91 composition(converse(join(x0, one)), X)
% 15.00/2.91 = { by lemma 21 }
% 15.00/2.91 X
% 15.00/2.91
% 15.00/2.91 Lemma 23: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 15.00/2.91 Proof:
% 15.00/2.91 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 15.00/2.91 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.91 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 15.00/2.91 = { by axiom 13 (converse_cancellativity) }
% 15.00/2.91 complement(X)
% 15.00/2.91
% 15.00/2.91 Lemma 24: join(complement(X), complement(X)) = complement(X).
% 15.00/2.91 Proof:
% 15.00/2.91 join(complement(X), complement(X))
% 15.00/2.91 = { by lemma 21 R->L }
% 15.00/2.91 join(complement(X), composition(converse(join(x0, one)), complement(X)))
% 15.00/2.91 = { by lemma 22 R->L }
% 15.00/2.91 join(complement(X), composition(converse(join(x0, one)), complement(composition(join(x0, one), X))))
% 15.00/2.91 = { by lemma 23 }
% 15.00/2.91 complement(X)
% 15.00/2.91
% 15.00/2.91 Lemma 25: join(top, complement(X)) = top.
% 15.00/2.91 Proof:
% 15.00/2.91 join(top, complement(X))
% 15.00/2.91 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.91 join(complement(X), top)
% 15.00/2.91 = { by lemma 18 R->L }
% 15.00/2.91 join(X, join(complement(X), complement(X)))
% 15.00/2.91 = { by lemma 24 }
% 15.00/2.91 join(X, complement(X))
% 15.00/2.91 = { by axiom 6 (def_top) R->L }
% 15.00/2.91 top
% 15.00/2.91
% 15.00/2.91 Lemma 26: join(Y, top) = join(X, top).
% 15.00/2.91 Proof:
% 15.00/2.91 join(Y, top)
% 15.00/2.91 = { by lemma 25 R->L }
% 15.00/2.91 join(Y, join(top, complement(Y)))
% 15.00/2.91 = { by lemma 18 }
% 15.00/2.91 join(top, top)
% 15.00/2.91 = { by lemma 18 R->L }
% 15.00/2.91 join(X, join(top, complement(X)))
% 15.00/2.92 = { by lemma 25 }
% 15.00/2.92 join(X, top)
% 15.00/2.92
% 15.00/2.92 Lemma 27: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 15.00/2.92 Proof:
% 15.00/2.92 join(meet(X, Y), complement(join(complement(X), Y)))
% 15.00/2.92 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.00/2.92 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 15.00/2.92 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 15.00/2.92 X
% 15.00/2.92
% 15.00/2.92 Lemma 28: join(zero, meet(X, X)) = X.
% 15.00/2.92 Proof:
% 15.00/2.92 join(zero, meet(X, X))
% 15.00/2.92 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.00/2.92 join(zero, complement(join(complement(X), complement(X))))
% 15.00/2.92 = { by axiom 5 (def_zero) }
% 15.00/2.92 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 15.00/2.92 = { by lemma 27 }
% 15.00/2.92 X
% 15.00/2.92
% 15.00/2.92 Lemma 29: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 15.00/2.92 Proof:
% 15.00/2.92 join(zero, join(X, complement(complement(Y))))
% 15.00/2.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.92 join(zero, join(complement(complement(Y)), X))
% 15.00/2.92 = { by lemma 24 R->L }
% 15.00/2.92 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 15.00/2.92 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.00/2.92 join(zero, join(meet(Y, Y), X))
% 15.00/2.92 = { by axiom 8 (maddux2_join_associativity) }
% 15.00/2.92 join(join(zero, meet(Y, Y)), X)
% 15.00/2.92 = { by lemma 28 }
% 15.00/2.92 join(Y, X)
% 15.00/2.92 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.92 join(X, Y)
% 15.00/2.92
% 15.00/2.92 Lemma 30: join(zero, complement(complement(X))) = X.
% 15.00/2.92 Proof:
% 15.00/2.92 join(zero, complement(complement(X)))
% 15.00/2.92 = { by axiom 5 (def_zero) }
% 15.00/2.92 join(meet(X, complement(X)), complement(complement(X)))
% 15.00/2.92 = { by lemma 24 R->L }
% 15.00/2.92 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 15.00/2.92 = { by lemma 27 }
% 15.00/2.92 X
% 15.00/2.92
% 15.00/2.92 Lemma 31: join(X, zero) = join(X, X).
% 15.00/2.92 Proof:
% 15.00/2.92 join(X, zero)
% 15.00/2.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.92 join(zero, X)
% 15.00/2.92 = { by lemma 30 R->L }
% 15.00/2.92 join(zero, join(zero, complement(complement(X))))
% 15.00/2.92 = { by lemma 24 R->L }
% 15.00/2.92 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 15.00/2.92 = { by lemma 29 }
% 15.00/2.92 join(zero, join(complement(complement(X)), X))
% 15.00/2.92 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.92 join(zero, join(X, complement(complement(X))))
% 15.00/2.92 = { by lemma 29 }
% 15.00/2.92 join(X, X)
% 15.00/2.92
% 15.00/2.92 Lemma 32: join(zero, complement(X)) = complement(X).
% 15.00/2.92 Proof:
% 15.00/2.92 join(zero, complement(X))
% 15.00/2.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.92 join(complement(X), zero)
% 15.00/2.92 = { by lemma 31 }
% 15.00/2.92 join(complement(X), complement(X))
% 15.00/2.92 = { by lemma 24 }
% 15.00/2.92 complement(X)
% 15.00/2.92
% 15.00/2.92 Lemma 33: join(X, zero) = X.
% 15.00/2.92 Proof:
% 15.00/2.92 join(X, zero)
% 15.00/2.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.92 join(zero, X)
% 15.00/2.92 = { by lemma 29 R->L }
% 15.00/2.92 join(zero, join(zero, complement(complement(X))))
% 15.00/2.92 = { by lemma 32 }
% 15.00/2.92 join(zero, complement(complement(X)))
% 15.00/2.92 = { by lemma 30 }
% 15.00/2.92 X
% 15.00/2.92
% 15.00/2.92 Lemma 34: join(zero, X) = X.
% 15.00/2.92 Proof:
% 15.00/2.92 join(zero, X)
% 15.00/2.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.92 join(X, zero)
% 15.00/2.92 = { by lemma 33 }
% 15.00/2.92 X
% 15.00/2.92
% 15.00/2.92 Lemma 35: join(X, top) = top.
% 15.00/2.92 Proof:
% 15.00/2.92 join(X, top)
% 15.00/2.92 = { by lemma 26 }
% 15.00/2.92 join(zero, top)
% 15.00/2.92 = { by lemma 34 }
% 15.00/2.92 top
% 15.00/2.92
% 15.00/2.92 Lemma 36: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 15.00/2.92 Proof:
% 15.00/2.92 converse(join(X, converse(Y)))
% 15.00/2.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.92 converse(join(converse(Y), X))
% 15.00/2.92 = { by axiom 7 (converse_additivity) }
% 15.00/2.92 join(converse(converse(Y)), converse(X))
% 15.00/2.92 = { by axiom 1 (converse_idempotence) }
% 15.00/2.92 join(Y, converse(X))
% 15.00/2.92
% 15.00/2.92 Lemma 37: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 15.00/2.92 Proof:
% 15.00/2.92 converse(join(converse(X), Y))
% 15.00/2.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.92 converse(join(Y, converse(X)))
% 15.00/2.92 = { by lemma 36 }
% 15.00/2.92 join(X, converse(Y))
% 15.00/2.92
% 15.00/2.92 Lemma 38: join(X, converse(complement(converse(X)))) = converse(top).
% 15.00/2.92 Proof:
% 15.00/2.92 join(X, converse(complement(converse(X))))
% 15.00/2.92 = { by lemma 37 R->L }
% 15.00/2.92 converse(join(converse(X), complement(converse(X))))
% 15.00/2.92 = { by axiom 6 (def_top) R->L }
% 15.00/2.92 converse(top)
% 15.00/2.92
% 15.00/2.92 Lemma 39: join(X, join(complement(X), Y)) = top.
% 15.00/2.92 Proof:
% 15.00/2.92 join(X, join(complement(X), Y))
% 15.00/2.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.92 join(X, join(Y, complement(X)))
% 15.00/2.92 = { by lemma 18 }
% 15.00/2.92 join(Y, top)
% 15.00/2.92 = { by lemma 26 R->L }
% 15.00/2.92 join(Z, top)
% 15.00/2.92 = { by lemma 35 }
% 15.00/2.92 top
% 15.00/2.92
% 15.00/2.92 Lemma 40: join(X, converse(top)) = top.
% 15.00/2.92 Proof:
% 15.00/2.92 join(X, converse(top))
% 15.00/2.92 = { by lemma 38 R->L }
% 15.00/2.92 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 15.00/2.92 = { by lemma 39 }
% 15.00/2.92 top
% 15.00/2.92
% 15.00/2.92 Lemma 41: converse(top) = top.
% 15.00/2.92 Proof:
% 15.00/2.92 converse(top)
% 15.00/2.92 = { by lemma 35 R->L }
% 15.00/2.92 converse(join(X, top))
% 15.00/2.92 = { by axiom 7 (converse_additivity) }
% 15.00/2.92 join(converse(X), converse(top))
% 15.00/2.92 = { by lemma 40 }
% 15.00/2.92 top
% 15.00/2.92
% 15.00/2.92 Lemma 42: meet(Y, X) = meet(X, Y).
% 15.00/2.92 Proof:
% 15.00/2.92 meet(Y, X)
% 15.00/2.92 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.00/2.92 complement(join(complement(Y), complement(X)))
% 15.00/2.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.92 complement(join(complement(X), complement(Y)))
% 15.00/2.92 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.00/2.92 meet(X, Y)
% 15.00/2.92
% 15.00/2.92 Lemma 43: complement(join(zero, complement(X))) = meet(X, top).
% 15.00/2.92 Proof:
% 15.00/2.92 complement(join(zero, complement(X)))
% 15.00/2.92 = { by lemma 17 R->L }
% 15.00/2.92 complement(join(complement(top), complement(X)))
% 15.00/2.92 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.00/2.92 meet(top, X)
% 15.00/2.92 = { by lemma 42 R->L }
% 15.00/2.92 meet(X, top)
% 15.00/2.92
% 15.00/2.92 Lemma 44: join(X, complement(zero)) = top.
% 15.00/2.92 Proof:
% 15.00/2.92 join(X, complement(zero))
% 15.00/2.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.92 join(complement(zero), X)
% 15.00/2.92 = { by lemma 29 R->L }
% 15.00/2.92 join(zero, join(complement(zero), complement(complement(X))))
% 15.00/2.92 = { by lemma 39 }
% 15.00/2.92 top
% 15.00/2.92
% 15.00/2.92 Lemma 45: meet(X, zero) = zero.
% 15.00/2.92 Proof:
% 15.00/2.92 meet(X, zero)
% 15.00/2.92 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.00/2.92 complement(join(complement(X), complement(zero)))
% 15.00/2.92 = { by lemma 44 }
% 15.00/2.92 complement(top)
% 15.00/2.92 = { by lemma 17 }
% 15.00/2.92 zero
% 15.00/2.92
% 15.00/2.92 Lemma 46: join(meet(X, Y), meet(X, complement(Y))) = X.
% 15.00/2.92 Proof:
% 15.00/2.92 join(meet(X, Y), meet(X, complement(Y)))
% 15.00/2.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.92 join(meet(X, complement(Y)), meet(X, Y))
% 15.00/2.92 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.00/2.92 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 15.00/2.92 = { by lemma 27 }
% 15.00/2.92 X
% 15.00/2.92
% 15.00/2.92 Lemma 47: meet(X, top) = X.
% 15.00/2.92 Proof:
% 15.00/2.92 meet(X, top)
% 15.00/2.92 = { by lemma 43 R->L }
% 15.00/2.92 complement(join(zero, complement(X)))
% 15.00/2.92 = { by lemma 32 R->L }
% 15.00/2.92 join(zero, complement(join(zero, complement(X))))
% 15.00/2.92 = { by lemma 43 }
% 15.00/2.92 join(zero, meet(X, top))
% 15.00/2.92 = { by lemma 44 R->L }
% 15.00/2.92 join(zero, meet(X, join(complement(zero), complement(zero))))
% 15.00/2.92 = { by lemma 24 }
% 15.00/2.92 join(zero, meet(X, complement(zero)))
% 15.00/2.92 = { by lemma 45 R->L }
% 15.00/2.92 join(meet(X, zero), meet(X, complement(zero)))
% 15.00/2.92 = { by lemma 46 }
% 15.00/2.92 X
% 15.00/2.92
% 15.00/2.92 Lemma 48: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 15.00/2.92 Proof:
% 15.00/2.92 join(meet(X, Y), meet(X, Y))
% 15.00/2.92 = { by lemma 42 }
% 15.00/2.92 join(meet(Y, X), meet(X, Y))
% 15.00/2.92 = { by lemma 42 }
% 15.00/2.92 join(meet(Y, X), meet(Y, X))
% 15.00/2.92 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.00/2.92 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 15.00/2.92 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.00/2.92 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 15.00/2.92 = { by lemma 24 }
% 15.00/2.92 complement(join(complement(Y), complement(X)))
% 15.00/2.92 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.00/2.92 meet(Y, X)
% 15.00/2.92 = { by lemma 42 R->L }
% 15.00/2.92 meet(X, Y)
% 15.00/2.92
% 15.00/2.92 Lemma 49: converse(zero) = zero.
% 15.00/2.92 Proof:
% 15.00/2.92 converse(zero)
% 15.00/2.92 = { by lemma 34 R->L }
% 15.00/2.92 join(zero, converse(zero))
% 15.00/2.92 = { by lemma 37 R->L }
% 15.00/2.92 converse(join(converse(zero), zero))
% 15.00/2.92 = { by lemma 31 }
% 15.00/2.92 converse(join(converse(zero), converse(zero)))
% 15.00/2.92 = { by lemma 36 }
% 15.00/2.92 join(zero, converse(converse(zero)))
% 15.00/2.92 = { by axiom 1 (converse_idempotence) }
% 15.00/2.92 join(zero, zero)
% 15.00/2.92 = { by lemma 47 R->L }
% 15.00/2.92 join(zero, meet(zero, top))
% 15.00/2.92 = { by lemma 47 R->L }
% 15.00/2.92 join(meet(zero, top), meet(zero, top))
% 15.00/2.92 = { by lemma 48 }
% 15.00/2.92 meet(zero, top)
% 15.00/2.92 = { by lemma 47 }
% 15.00/2.92 zero
% 15.00/2.92
% 15.00/2.92 Lemma 50: join(top, X) = top.
% 15.00/2.92 Proof:
% 15.00/2.92 join(top, X)
% 15.00/2.92 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.92 join(X, top)
% 15.00/2.92 = { by lemma 26 R->L }
% 15.00/2.93 join(Y, top)
% 15.00/2.93 = { by lemma 35 }
% 15.00/2.93 top
% 15.00/2.93
% 15.00/2.93 Lemma 51: complement(complement(X)) = X.
% 15.00/2.93 Proof:
% 15.00/2.93 complement(complement(X))
% 15.00/2.93 = { by lemma 32 R->L }
% 15.00/2.93 join(zero, complement(complement(X)))
% 15.00/2.93 = { by lemma 30 }
% 15.00/2.93 X
% 15.00/2.93
% 15.00/2.93 Lemma 52: meet(zero, X) = zero.
% 15.00/2.93 Proof:
% 15.00/2.93 meet(zero, X)
% 15.00/2.93 = { by lemma 42 }
% 15.00/2.93 meet(X, zero)
% 15.00/2.93 = { by lemma 45 }
% 15.00/2.93 zero
% 15.00/2.93
% 15.00/2.93 Lemma 53: composition(join(join(x0, one), Y), X) = join(X, composition(Y, X)).
% 15.00/2.93 Proof:
% 15.00/2.93 composition(join(join(x0, one), Y), X)
% 15.00/2.93 = { by axiom 12 (composition_distributivity) }
% 15.00/2.93 join(composition(join(x0, one), X), composition(Y, X))
% 15.00/2.93 = { by lemma 22 }
% 15.00/2.93 join(X, composition(Y, X))
% 15.00/2.93
% 15.00/2.93 Lemma 54: composition(top, zero) = zero.
% 15.00/2.93 Proof:
% 15.00/2.93 composition(top, zero)
% 15.00/2.93 = { by lemma 41 R->L }
% 15.00/2.93 composition(converse(top), zero)
% 15.00/2.93 = { by lemma 34 R->L }
% 15.00/2.93 join(zero, composition(converse(top), zero))
% 15.00/2.93 = { by lemma 17 R->L }
% 15.00/2.93 join(complement(top), composition(converse(top), zero))
% 15.00/2.93 = { by lemma 17 R->L }
% 15.00/2.93 join(complement(top), composition(converse(top), complement(top)))
% 15.00/2.93 = { by lemma 50 R->L }
% 15.00/2.93 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 15.00/2.93 = { by lemma 41 R->L }
% 15.00/2.93 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 15.00/2.93 = { by lemma 53 R->L }
% 15.00/2.93 join(complement(top), composition(converse(top), complement(composition(join(join(x0, one), converse(top)), top))))
% 15.00/2.93 = { by lemma 40 }
% 15.00/2.93 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 15.00/2.93 = { by lemma 23 }
% 15.00/2.93 complement(top)
% 15.00/2.93 = { by lemma 17 }
% 15.00/2.93 zero
% 15.00/2.93
% 15.00/2.93 Lemma 55: composition(X, zero) = zero.
% 15.00/2.93 Proof:
% 15.00/2.93 composition(X, zero)
% 15.00/2.93 = { by lemma 34 R->L }
% 15.00/2.93 join(zero, composition(X, zero))
% 15.00/2.93 = { by lemma 54 R->L }
% 15.00/2.93 join(composition(top, zero), composition(X, zero))
% 15.00/2.93 = { by axiom 12 (composition_distributivity) R->L }
% 15.00/2.93 composition(join(top, X), zero)
% 15.00/2.93 = { by lemma 50 }
% 15.00/2.93 composition(top, zero)
% 15.00/2.93 = { by lemma 54 }
% 15.00/2.93 zero
% 15.00/2.93
% 15.00/2.93 Lemma 56: composition(converse(X), complement(composition(X, top))) = zero.
% 15.00/2.93 Proof:
% 15.00/2.93 composition(converse(X), complement(composition(X, top)))
% 15.00/2.93 = { by lemma 34 R->L }
% 15.00/2.93 join(zero, composition(converse(X), complement(composition(X, top))))
% 15.00/2.93 = { by lemma 17 R->L }
% 15.00/2.93 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 15.00/2.93 = { by lemma 23 }
% 15.00/2.93 complement(top)
% 15.00/2.93 = { by lemma 17 }
% 15.00/2.93 zero
% 15.00/2.93
% 15.00/2.93 Lemma 57: composition(converse(X), composition(complement(composition(X, top)), Y)) = zero.
% 15.00/2.93 Proof:
% 15.00/2.93 composition(converse(X), composition(complement(composition(X, top)), Y))
% 15.00/2.93 = { by lemma 19 R->L }
% 15.00/2.93 converse(composition(converse(composition(complement(composition(X, top)), Y)), X))
% 15.00/2.93 = { by axiom 9 (converse_multiplicativity) }
% 15.00/2.93 converse(composition(composition(converse(Y), converse(complement(composition(X, top)))), X))
% 15.00/2.93 = { by axiom 10 (composition_associativity) R->L }
% 15.00/2.93 converse(composition(converse(Y), composition(converse(complement(composition(X, top))), X)))
% 15.00/2.93 = { by lemma 19 R->L }
% 15.00/2.93 converse(composition(converse(Y), converse(composition(converse(X), complement(composition(X, top))))))
% 15.00/2.93 = { by lemma 56 }
% 15.00/2.93 converse(composition(converse(Y), converse(zero)))
% 15.00/2.93 = { by lemma 49 }
% 15.00/2.93 converse(composition(converse(Y), zero))
% 15.00/2.93 = { by lemma 55 }
% 15.00/2.93 converse(zero)
% 15.00/2.93 = { by lemma 49 }
% 15.00/2.93 zero
% 15.00/2.93
% 15.00/2.93 Lemma 58: composition(zero, X) = zero.
% 15.00/2.93 Proof:
% 15.00/2.93 composition(zero, X)
% 15.00/2.93 = { by lemma 57 R->L }
% 15.00/2.93 composition(composition(converse(Y), composition(complement(composition(Y, top)), Z)), X)
% 15.00/2.93 = { by axiom 10 (composition_associativity) R->L }
% 15.00/2.93 composition(converse(Y), composition(composition(complement(composition(Y, top)), Z), X))
% 15.00/2.93 = { by axiom 10 (composition_associativity) R->L }
% 15.00/2.93 composition(converse(Y), composition(complement(composition(Y, top)), composition(Z, X)))
% 15.00/2.93 = { by lemma 57 }
% 15.00/2.93 zero
% 15.00/2.93
% 15.00/2.93 Lemma 59: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 15.00/2.93 Proof:
% 15.00/2.93 meet(X, join(complement(Y), complement(Z)))
% 15.00/2.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.93 meet(X, join(complement(Z), complement(Y)))
% 15.00/2.93 = { by lemma 42 }
% 15.00/2.93 meet(join(complement(Z), complement(Y)), X)
% 15.00/2.93 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.00/2.93 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 15.00/2.93 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.00/2.93 complement(join(meet(Z, Y), complement(X)))
% 15.00/2.93 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.93 complement(join(complement(X), meet(Z, Y)))
% 15.00/2.93 = { by lemma 42 R->L }
% 15.00/2.93 complement(join(complement(X), meet(Y, Z)))
% 15.00/2.93
% 15.00/2.93 Lemma 60: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 15.00/2.93 Proof:
% 15.00/2.93 complement(join(X, complement(Y)))
% 15.00/2.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.93 complement(join(complement(Y), X))
% 15.00/2.93 = { by lemma 47 R->L }
% 15.00/2.93 complement(join(complement(Y), meet(X, top)))
% 15.00/2.93 = { by lemma 42 R->L }
% 15.00/2.93 complement(join(complement(Y), meet(top, X)))
% 15.00/2.93 = { by lemma 59 R->L }
% 15.00/2.93 meet(Y, join(complement(top), complement(X)))
% 15.00/2.93 = { by lemma 17 }
% 15.00/2.93 meet(Y, join(zero, complement(X)))
% 15.00/2.93 = { by lemma 32 }
% 15.00/2.93 meet(Y, complement(X))
% 15.00/2.93
% 15.00/2.93 Lemma 61: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 15.00/2.93 Proof:
% 15.00/2.93 complement(meet(X, complement(Y)))
% 15.00/2.93 = { by lemma 34 R->L }
% 15.00/2.93 complement(join(zero, meet(X, complement(Y))))
% 15.00/2.93 = { by lemma 60 R->L }
% 15.00/2.93 complement(join(zero, complement(join(Y, complement(X)))))
% 15.00/2.93 = { by lemma 43 }
% 15.00/2.93 meet(join(Y, complement(X)), top)
% 15.00/2.93 = { by lemma 47 }
% 15.00/2.93 join(Y, complement(X))
% 15.00/2.93
% 15.00/2.93 Lemma 62: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 15.00/2.93 Proof:
% 15.00/2.93 complement(join(complement(X), Y))
% 15.00/2.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.93 complement(join(Y, complement(X)))
% 15.00/2.93 = { by lemma 60 }
% 15.00/2.93 meet(X, complement(Y))
% 15.00/2.93
% 15.00/2.93 Lemma 63: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 15.00/2.93 Proof:
% 15.00/2.93 join(complement(X), complement(Y))
% 15.00/2.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.93 join(complement(Y), complement(X))
% 15.00/2.93 = { by lemma 28 R->L }
% 15.00/2.93 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 15.00/2.93 = { by lemma 59 }
% 15.00/2.93 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 15.00/2.93 = { by lemma 32 }
% 15.00/2.93 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 15.00/2.93 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.00/2.93 complement(join(meet(Y, X), meet(Y, X)))
% 15.00/2.93 = { by lemma 48 }
% 15.00/2.93 complement(meet(Y, X))
% 15.00/2.93 = { by lemma 42 R->L }
% 15.00/2.93 complement(meet(X, Y))
% 15.00/2.93
% 15.00/2.93 Lemma 64: join(X, complement(meet(X, Y))) = top.
% 15.00/2.93 Proof:
% 15.00/2.93 join(X, complement(meet(X, Y)))
% 15.00/2.93 = { by lemma 42 }
% 15.00/2.93 join(X, complement(meet(Y, X)))
% 15.00/2.93 = { by lemma 63 R->L }
% 15.00/2.93 join(X, join(complement(Y), complement(X)))
% 15.00/2.93 = { by lemma 18 }
% 15.00/2.93 join(complement(Y), top)
% 15.00/2.93 = { by lemma 35 }
% 15.00/2.93 top
% 15.00/2.93
% 15.00/2.93 Lemma 65: meet(X, meet(Y, complement(X))) = zero.
% 15.00/2.93 Proof:
% 15.00/2.93 meet(X, meet(Y, complement(X)))
% 15.00/2.93 = { by lemma 42 }
% 15.00/2.93 meet(X, meet(complement(X), Y))
% 15.00/2.93 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.00/2.93 complement(join(complement(X), complement(meet(complement(X), Y))))
% 15.00/2.93 = { by lemma 64 }
% 15.00/2.93 complement(top)
% 15.00/2.93 = { by lemma 17 }
% 15.00/2.93 zero
% 15.00/2.93
% 15.00/2.93 Lemma 66: join(X, meet(X, Y)) = X.
% 15.00/2.93 Proof:
% 15.00/2.93 join(X, meet(X, Y))
% 15.00/2.93 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.00/2.93 join(X, complement(join(complement(X), complement(Y))))
% 15.00/2.93 = { by lemma 61 R->L }
% 15.00/2.93 complement(meet(join(complement(X), complement(Y)), complement(X)))
% 15.00/2.93 = { by lemma 42 R->L }
% 15.00/2.93 complement(meet(complement(X), join(complement(X), complement(Y))))
% 15.00/2.93 = { by lemma 61 R->L }
% 15.00/2.93 complement(meet(complement(X), complement(meet(Y, complement(complement(X))))))
% 15.00/2.93 = { by lemma 62 R->L }
% 15.00/2.93 complement(complement(join(complement(complement(X)), meet(Y, complement(complement(X))))))
% 15.00/2.93 = { by lemma 32 R->L }
% 15.00/2.93 complement(join(zero, complement(join(complement(complement(X)), meet(Y, complement(complement(X)))))))
% 15.00/2.93 = { by lemma 65 R->L }
% 15.00/2.93 complement(join(meet(complement(X), meet(Y, complement(complement(X)))), complement(join(complement(complement(X)), meet(Y, complement(complement(X)))))))
% 15.00/2.93 = { by lemma 27 }
% 15.00/2.93 complement(complement(X))
% 15.00/2.93 = { by lemma 51 }
% 15.00/2.93 X
% 15.00/2.93
% 15.00/2.93 Lemma 67: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 15.00/2.93 Proof:
% 15.00/2.93 join(Y, join(X, Z))
% 15.00/2.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.93 join(join(X, Z), Y)
% 15.00/2.93 = { by axiom 8 (maddux2_join_associativity) R->L }
% 15.00/2.93 join(X, join(Z, Y))
% 15.00/2.93 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.93 join(X, join(Y, Z))
% 15.00/2.93
% 15.00/2.93 Lemma 68: join(Z, join(X, Y)) = join(X, join(Y, Z)).
% 15.00/2.93 Proof:
% 15.00/2.93 join(Z, join(X, Y))
% 15.00/2.93 = { by lemma 67 }
% 15.00/2.93 join(X, join(Z, Y))
% 15.00/2.93 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.93 join(X, join(Y, Z))
% 15.00/2.93
% 15.00/2.93 Lemma 69: join(composition(x0, X), X) = X.
% 15.00/2.93 Proof:
% 15.00/2.93 join(composition(x0, X), X)
% 15.00/2.93 = { by lemma 22 R->L }
% 15.00/2.93 join(composition(x0, X), composition(join(x0, one), X))
% 15.00/2.93 = { by axiom 3 (goals) }
% 15.00/2.93 join(composition(x0, X), composition(one, X))
% 15.00/2.93 = { by axiom 12 (composition_distributivity) R->L }
% 15.00/2.93 composition(join(x0, one), X)
% 15.00/2.93 = { by lemma 22 }
% 15.00/2.93 X
% 15.00/2.93
% 15.00/2.93 Lemma 70: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 15.00/2.93 Proof:
% 15.00/2.93 meet(complement(X), complement(Y))
% 15.00/2.93 = { by lemma 42 }
% 15.00/2.93 meet(complement(Y), complement(X))
% 15.00/2.93 = { by lemma 32 R->L }
% 15.00/2.93 meet(join(zero, complement(Y)), complement(X))
% 15.00/2.93 = { by lemma 60 R->L }
% 15.00/2.93 complement(join(X, complement(join(zero, complement(Y)))))
% 15.00/2.93 = { by lemma 43 }
% 15.00/2.93 complement(join(X, meet(Y, top)))
% 15.00/2.93 = { by lemma 47 }
% 15.00/2.93 complement(join(X, Y))
% 15.00/2.93
% 15.00/2.93 Lemma 71: meet(complement(Z), meet(Y, X)) = meet(X, meet(Y, complement(Z))).
% 15.00/2.93 Proof:
% 15.00/2.93 meet(complement(Z), meet(Y, X))
% 15.00/2.93 = { by lemma 42 }
% 15.00/2.93 meet(complement(Z), meet(X, Y))
% 15.00/2.93 = { by lemma 42 }
% 15.00/2.93 meet(meet(X, Y), complement(Z))
% 15.00/2.93 = { by axiom 11 (maddux4_definiton_of_meet) }
% 15.00/2.93 meet(complement(join(complement(X), complement(Y))), complement(Z))
% 15.00/2.93 = { by lemma 70 }
% 15.00/2.93 complement(join(join(complement(X), complement(Y)), Z))
% 15.00/2.93 = { by axiom 8 (maddux2_join_associativity) R->L }
% 15.00/2.93 complement(join(complement(X), join(complement(Y), Z)))
% 15.00/2.93 = { by lemma 62 }
% 15.00/2.93 meet(X, complement(join(complement(Y), Z)))
% 15.00/2.93 = { by lemma 62 }
% 15.00/2.93 meet(X, meet(Y, complement(Z)))
% 15.00/2.93
% 15.00/2.93 Lemma 72: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 15.00/2.93 Proof:
% 15.00/2.93 meet(Y, meet(Z, X))
% 15.00/2.93 = { by lemma 47 R->L }
% 15.00/2.93 meet(meet(Y, top), meet(Z, X))
% 15.00/2.93 = { by lemma 43 R->L }
% 15.00/2.93 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 15.00/2.93 = { by lemma 71 }
% 15.00/2.93 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 15.00/2.93 = { by lemma 43 }
% 15.00/2.93 meet(X, meet(Z, meet(Y, top)))
% 15.00/2.93 = { by lemma 47 }
% 15.00/2.93 meet(X, meet(Z, Y))
% 15.00/2.93 = { by lemma 42 R->L }
% 15.00/2.93 meet(X, meet(Y, Z))
% 15.00/2.93
% 15.00/2.93 Lemma 73: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 15.00/2.93 Proof:
% 15.00/2.93 meet(Y, meet(X, Z))
% 15.00/2.93 = { by lemma 72 R->L }
% 15.00/2.93 meet(X, meet(Z, Y))
% 15.00/2.93 = { by lemma 42 R->L }
% 15.00/2.93 meet(X, meet(Y, Z))
% 15.00/2.93
% 15.00/2.93 Lemma 74: meet(meet(X, Y), complement(Y)) = zero.
% 15.00/2.93 Proof:
% 15.00/2.93 meet(meet(X, Y), complement(Y))
% 15.00/2.93 = { by lemma 60 R->L }
% 15.00/2.93 complement(join(Y, complement(meet(X, Y))))
% 15.00/2.93 = { by lemma 42 }
% 15.00/2.93 complement(join(Y, complement(meet(Y, X))))
% 15.00/2.93 = { by lemma 64 }
% 15.00/2.93 complement(top)
% 15.00/2.93 = { by lemma 17 }
% 15.00/2.93 zero
% 15.00/2.93
% 15.00/2.93 Lemma 75: join(join(X, composition(x0, Y)), Y) = join(X, Y).
% 15.00/2.93 Proof:
% 15.00/2.93 join(join(X, composition(x0, Y)), Y)
% 15.00/2.93 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.93 join(Y, join(X, composition(x0, Y)))
% 15.00/2.93 = { by lemma 68 }
% 15.00/2.93 join(X, join(composition(x0, Y), Y))
% 15.00/2.93 = { by lemma 69 }
% 15.00/2.93 join(X, Y)
% 15.00/2.93
% 15.00/2.93 Lemma 76: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 15.00/2.93 Proof:
% 15.00/2.93 join(meet(X, Y), meet(Y, complement(X)))
% 15.00/2.93 = { by lemma 42 }
% 15.00/2.93 join(meet(Y, X), meet(Y, complement(X)))
% 15.00/2.93 = { by lemma 46 }
% 15.00/2.93 Y
% 15.00/2.93
% 15.00/2.93 Lemma 77: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 15.00/2.93 Proof:
% 15.00/2.93 join(meet(X, Y), meet(complement(X), Y))
% 15.00/2.93 = { by lemma 42 }
% 15.00/2.93 join(meet(X, Y), meet(Y, complement(X)))
% 15.00/2.93 = { by lemma 76 }
% 15.00/2.93 Y
% 15.00/2.93
% 15.00/2.93 Lemma 78: meet(composition(X, Y), complement(composition(X, top))) = zero.
% 15.00/2.93 Proof:
% 15.00/2.93 meet(composition(X, Y), complement(composition(X, top)))
% 15.00/2.93 = { by lemma 33 R->L }
% 15.00/2.93 join(meet(composition(X, Y), complement(composition(X, top))), zero)
% 15.00/2.93 = { by lemma 52 R->L }
% 15.00/2.93 join(meet(composition(X, Y), complement(composition(X, top))), meet(zero, complement(composition(X, top))))
% 15.00/2.93 = { by lemma 55 R->L }
% 15.00/2.93 join(meet(composition(X, Y), complement(composition(X, top))), meet(composition(X, zero), complement(composition(X, top))))
% 15.00/2.93 = { by lemma 45 R->L }
% 15.00/2.93 join(meet(composition(X, Y), complement(composition(X, top))), meet(composition(X, meet(Y, zero)), complement(composition(X, top))))
% 15.00/2.93 = { by lemma 56 R->L }
% 15.00/2.93 join(meet(composition(X, Y), complement(composition(X, top))), meet(composition(X, meet(Y, composition(converse(X), complement(composition(X, top))))), complement(composition(X, top))))
% 15.00/2.93 = { by axiom 15 (modular_law_1) }
% 15.00/2.93 meet(composition(X, meet(Y, composition(converse(X), complement(composition(X, top))))), complement(composition(X, top)))
% 15.00/2.93 = { by lemma 56 }
% 15.00/2.93 meet(composition(X, meet(Y, zero)), complement(composition(X, top)))
% 15.00/2.93 = { by lemma 45 }
% 15.00/2.93 meet(composition(X, zero), complement(composition(X, top)))
% 15.00/2.94 = { by lemma 55 }
% 15.00/2.94 meet(zero, complement(composition(X, top)))
% 15.00/2.94 = { by lemma 52 }
% 15.00/2.94 zero
% 15.00/2.94
% 15.00/2.94 Lemma 79: join(meet(composition(x0, top), X), composition(x0, X)) = meet(composition(x0, top), X).
% 15.00/2.94 Proof:
% 15.00/2.94 join(meet(composition(x0, top), X), composition(x0, X))
% 15.00/2.94 = { by lemma 27 R->L }
% 15.00/2.94 join(meet(composition(x0, top), X), join(meet(composition(x0, X), complement(composition(x0, top))), complement(join(complement(composition(x0, X)), complement(composition(x0, top))))))
% 15.00/2.94 = { by lemma 78 }
% 15.00/2.94 join(meet(composition(x0, top), X), join(zero, complement(join(complement(composition(x0, X)), complement(composition(x0, top))))))
% 15.00/2.94 = { by lemma 32 }
% 15.00/2.94 join(meet(composition(x0, top), X), complement(join(complement(composition(x0, X)), complement(composition(x0, top)))))
% 15.00/2.94 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.00/2.94 join(meet(composition(x0, top), X), meet(composition(x0, X), composition(x0, top)))
% 15.00/2.94 = { by lemma 42 R->L }
% 15.00/2.94 join(meet(composition(x0, top), X), meet(composition(x0, top), composition(x0, X)))
% 15.00/2.94 = { by lemma 27 R->L }
% 15.00/2.94 join(meet(composition(x0, top), X), meet(composition(x0, top), join(meet(composition(x0, X), complement(X)), complement(join(complement(composition(x0, X)), complement(X))))))
% 15.00/2.94 = { by lemma 60 R->L }
% 15.00/2.94 join(meet(composition(x0, top), X), meet(composition(x0, top), join(complement(join(X, complement(composition(x0, X)))), complement(join(complement(composition(x0, X)), complement(X))))))
% 15.00/2.94 = { by lemma 69 R->L }
% 15.00/2.94 join(meet(composition(x0, top), X), meet(composition(x0, top), join(complement(join(join(composition(x0, X), X), complement(composition(x0, X)))), complement(join(complement(composition(x0, X)), complement(X))))))
% 15.00/2.94 = { by axiom 8 (maddux2_join_associativity) R->L }
% 15.00/2.94 join(meet(composition(x0, top), X), meet(composition(x0, top), join(complement(join(composition(x0, X), join(X, complement(composition(x0, X))))), complement(join(complement(composition(x0, X)), complement(X))))))
% 15.00/2.94 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.94 join(meet(composition(x0, top), X), meet(composition(x0, top), join(complement(join(composition(x0, X), join(complement(composition(x0, X)), X))), complement(join(complement(composition(x0, X)), complement(X))))))
% 15.00/2.94 = { by lemma 39 }
% 15.00/2.94 join(meet(composition(x0, top), X), meet(composition(x0, top), join(complement(top), complement(join(complement(composition(x0, X)), complement(X))))))
% 15.00/2.94 = { by lemma 17 }
% 15.00/2.94 join(meet(composition(x0, top), X), meet(composition(x0, top), join(zero, complement(join(complement(composition(x0, X)), complement(X))))))
% 15.00/2.94 = { by lemma 32 }
% 15.00/2.94 join(meet(composition(x0, top), X), meet(composition(x0, top), complement(join(complement(composition(x0, X)), complement(X)))))
% 15.00/2.94 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.00/2.94 join(meet(composition(x0, top), X), meet(composition(x0, top), meet(composition(x0, X), X)))
% 15.00/2.94 = { by lemma 73 R->L }
% 15.00/2.94 join(meet(composition(x0, top), X), meet(composition(x0, X), meet(composition(x0, top), X)))
% 15.00/2.94 = { by lemma 42 R->L }
% 15.00/2.94 join(meet(composition(x0, top), X), meet(meet(composition(x0, top), X), composition(x0, X)))
% 15.00/2.94 = { by lemma 66 }
% 15.00/2.94 meet(composition(x0, top), X)
% 15.00/2.94
% 15.00/2.94 Lemma 80: meet(join(x0, one), composition(converse(complement(X)), X)) = zero.
% 15.00/2.94 Proof:
% 15.00/2.94 meet(join(x0, one), composition(converse(complement(X)), X))
% 15.00/2.94 = { by lemma 42 }
% 15.00/2.94 meet(composition(converse(complement(X)), X), join(x0, one))
% 15.00/2.94 = { by lemma 51 R->L }
% 15.00/2.94 meet(composition(converse(complement(X)), X), complement(complement(join(x0, one))))
% 15.00/2.94 = { by lemma 23 R->L }
% 15.00/2.94 meet(composition(converse(complement(X)), X), complement(join(complement(join(x0, one)), composition(converse(join(zero, complement(X))), complement(composition(join(zero, complement(X)), join(x0, one)))))))
% 15.00/2.94 = { by lemma 20 }
% 15.00/2.94 meet(composition(converse(complement(X)), X), complement(join(complement(join(x0, one)), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 15.00/2.94 = { by lemma 43 }
% 15.00/2.94 meet(composition(converse(complement(X)), X), complement(join(complement(join(x0, one)), composition(converse(join(zero, complement(X))), meet(X, top)))))
% 15.00/2.94 = { by lemma 32 }
% 15.00/2.94 meet(composition(converse(complement(X)), X), complement(join(complement(join(x0, one)), composition(converse(complement(X)), meet(X, top)))))
% 15.00/2.94 = { by lemma 47 }
% 15.00/2.94 meet(composition(converse(complement(X)), X), complement(join(complement(join(x0, one)), composition(converse(complement(X)), X))))
% 15.00/2.94 = { by lemma 62 }
% 15.00/2.94 meet(composition(converse(complement(X)), X), meet(join(x0, one), complement(composition(converse(complement(X)), X))))
% 15.00/2.94 = { by lemma 65 }
% 15.00/2.94 zero
% 15.00/2.94
% 15.00/2.94 Lemma 81: join(join(meet(composition(x0, top), X), composition(x0, X)), complement(X)) = join(composition(x0, top), complement(X)).
% 15.00/2.94 Proof:
% 15.00/2.94 join(join(meet(composition(x0, top), X), composition(x0, X)), complement(X))
% 15.00/2.94 = { by lemma 66 R->L }
% 15.00/2.94 join(join(join(meet(composition(x0, top), X), composition(x0, X)), complement(X)), meet(join(join(meet(composition(x0, top), X), composition(x0, X)), complement(X)), composition(x0, top)))
% 15.00/2.94 = { by lemma 42 R->L }
% 15.00/2.94 join(join(join(meet(composition(x0, top), X), composition(x0, X)), complement(X)), meet(composition(x0, top), join(join(meet(composition(x0, top), X), composition(x0, X)), complement(X))))
% 15.00/2.94 = { by lemma 79 }
% 15.00/2.94 join(join(join(meet(composition(x0, top), X), composition(x0, X)), complement(X)), meet(composition(x0, top), join(meet(composition(x0, top), X), complement(X))))
% 15.00/2.94 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.94 join(join(join(meet(composition(x0, top), X), composition(x0, X)), complement(X)), meet(composition(x0, top), join(complement(X), meet(composition(x0, top), X))))
% 15.00/2.94 = { by lemma 33 R->L }
% 15.00/2.94 join(join(join(meet(composition(x0, top), X), composition(x0, X)), complement(X)), join(meet(composition(x0, top), join(complement(X), meet(composition(x0, top), X))), zero))
% 15.00/2.94 = { by lemma 17 R->L }
% 15.00/2.94 join(join(join(meet(composition(x0, top), X), composition(x0, X)), complement(X)), join(meet(composition(x0, top), join(complement(X), meet(composition(x0, top), X))), complement(top)))
% 15.00/2.94 = { by axiom 6 (def_top) }
% 15.00/2.94 join(join(join(meet(composition(x0, top), X), composition(x0, X)), complement(X)), join(meet(composition(x0, top), join(complement(X), meet(composition(x0, top), X))), complement(join(join(complement(composition(x0, top)), complement(X)), complement(join(complement(composition(x0, top)), complement(X)))))))
% 15.00/2.94 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.00/2.94 join(join(join(meet(composition(x0, top), X), composition(x0, X)), complement(X)), join(meet(composition(x0, top), join(complement(X), meet(composition(x0, top), X))), complement(join(join(complement(composition(x0, top)), complement(X)), meet(composition(x0, top), X)))))
% 15.00/2.94 = { by axiom 8 (maddux2_join_associativity) R->L }
% 15.00/2.94 join(join(join(meet(composition(x0, top), X), composition(x0, X)), complement(X)), join(meet(composition(x0, top), join(complement(X), meet(composition(x0, top), X))), complement(join(complement(composition(x0, top)), join(complement(X), meet(composition(x0, top), X))))))
% 15.00/2.94 = { by lemma 27 }
% 15.00/2.94 join(join(join(meet(composition(x0, top), X), composition(x0, X)), complement(X)), composition(x0, top))
% 15.00/2.94 = { by axiom 8 (maddux2_join_associativity) R->L }
% 15.00/2.94 join(join(meet(composition(x0, top), X), composition(x0, X)), join(complement(X), composition(x0, top)))
% 15.00/2.94 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.94 join(join(meet(composition(x0, top), X), composition(x0, X)), join(composition(x0, top), complement(X)))
% 15.00/2.94 = { by axiom 8 (maddux2_join_associativity) }
% 15.00/2.94 join(join(join(meet(composition(x0, top), X), composition(x0, X)), composition(x0, top)), complement(X))
% 15.00/2.94 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.94 join(join(composition(x0, top), join(meet(composition(x0, top), X), composition(x0, X))), complement(X))
% 15.00/2.94 = { by lemma 67 }
% 15.00/2.94 join(join(meet(composition(x0, top), X), join(composition(x0, top), composition(x0, X))), complement(X))
% 15.00/2.94 = { by lemma 47 R->L }
% 15.00/2.94 join(join(meet(composition(x0, top), X), meet(join(composition(x0, top), composition(x0, X)), top)), complement(X))
% 15.00/2.94 = { by lemma 43 R->L }
% 15.00/2.94 join(join(meet(composition(x0, top), X), complement(join(zero, complement(join(composition(x0, top), composition(x0, X)))))), complement(X))
% 15.00/2.94 = { by lemma 70 R->L }
% 15.00/2.94 join(join(meet(composition(x0, top), X), complement(join(zero, meet(complement(composition(x0, top)), complement(composition(x0, X)))))), complement(X))
% 15.00/2.94 = { by lemma 34 R->L }
% 15.00/2.94 join(join(meet(composition(x0, top), X), complement(join(zero, join(zero, meet(complement(composition(x0, top)), complement(composition(x0, X))))))), complement(X))
% 15.00/2.94 = { by lemma 78 R->L }
% 15.00/2.94 join(join(meet(composition(x0, top), X), complement(join(zero, join(meet(composition(x0, X), complement(composition(x0, top))), meet(complement(composition(x0, top)), complement(composition(x0, X))))))), complement(X))
% 15.00/2.94 = { by lemma 76 }
% 15.00/2.94 join(join(meet(composition(x0, top), X), complement(join(zero, complement(composition(x0, top))))), complement(X))
% 15.00/2.94 = { by lemma 43 }
% 15.00/2.94 join(join(meet(composition(x0, top), X), meet(composition(x0, top), top)), complement(X))
% 15.00/2.94 = { by lemma 47 }
% 15.00/2.94 join(join(meet(composition(x0, top), X), composition(x0, top)), complement(X))
% 15.00/2.94 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.94 join(join(composition(x0, top), meet(composition(x0, top), X)), complement(X))
% 15.00/2.94 = { by lemma 66 }
% 15.00/2.94 join(composition(x0, top), complement(X))
% 15.00/2.94
% 15.00/2.94 Lemma 82: join(join(meet(composition(x0, top), X), composition(x0, X)), meet(X, complement(composition(x0, top)))) = X.
% 15.00/2.94 Proof:
% 15.00/2.94 join(join(meet(composition(x0, top), X), composition(x0, X)), meet(X, complement(composition(x0, top))))
% 15.00/2.94 = { by lemma 34 R->L }
% 15.00/2.94 join(join(meet(composition(x0, top), X), composition(x0, X)), join(zero, meet(X, complement(composition(x0, top)))))
% 15.00/2.94 = { by axiom 8 (maddux2_join_associativity) }
% 15.00/2.94 join(join(join(meet(composition(x0, top), X), composition(x0, X)), zero), meet(X, complement(composition(x0, top))))
% 15.00/2.94 = { by lemma 31 }
% 15.00/2.94 join(join(join(meet(composition(x0, top), X), composition(x0, X)), join(meet(composition(x0, top), X), composition(x0, X))), meet(X, complement(composition(x0, top))))
% 15.00/2.94 = { by axiom 8 (maddux2_join_associativity) R->L }
% 15.00/2.94 join(join(meet(composition(x0, top), X), composition(x0, X)), join(join(meet(composition(x0, top), X), composition(x0, X)), meet(X, complement(composition(x0, top)))))
% 15.00/2.94 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.94 join(join(meet(composition(x0, top), X), composition(x0, X)), join(meet(X, complement(composition(x0, top))), join(meet(composition(x0, top), X), composition(x0, X))))
% 15.00/2.94 = { by lemma 79 }
% 15.00/2.94 join(meet(composition(x0, top), X), join(meet(X, complement(composition(x0, top))), join(meet(composition(x0, top), X), composition(x0, X))))
% 15.00/2.94 = { by lemma 67 }
% 15.00/2.94 join(meet(X, complement(composition(x0, top))), join(meet(composition(x0, top), X), join(meet(composition(x0, top), X), composition(x0, X))))
% 15.00/2.94 = { by axiom 8 (maddux2_join_associativity) }
% 15.00/2.94 join(join(meet(X, complement(composition(x0, top))), meet(composition(x0, top), X)), join(meet(composition(x0, top), X), composition(x0, X)))
% 15.00/2.94 = { by lemma 68 }
% 15.00/2.94 join(meet(composition(x0, top), X), join(composition(x0, X), join(meet(X, complement(composition(x0, top))), meet(composition(x0, top), X))))
% 15.00/2.94 = { by lemma 67 R->L }
% 15.00/2.94 join(meet(composition(x0, top), X), join(meet(X, complement(composition(x0, top))), join(composition(x0, X), meet(composition(x0, top), X))))
% 15.00/2.94 = { by lemma 79 R->L }
% 15.00/2.94 join(join(meet(composition(x0, top), X), composition(x0, X)), join(meet(X, complement(composition(x0, top))), join(composition(x0, X), meet(composition(x0, top), X))))
% 15.00/2.94 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.94 join(join(meet(composition(x0, top), X), composition(x0, X)), join(join(composition(x0, X), meet(composition(x0, top), X)), meet(X, complement(composition(x0, top)))))
% 15.00/2.94 = { by axiom 8 (maddux2_join_associativity) R->L }
% 15.00/2.94 join(join(meet(composition(x0, top), X), composition(x0, X)), join(composition(x0, X), join(meet(composition(x0, top), X), meet(X, complement(composition(x0, top))))))
% 15.00/2.94 = { by lemma 76 }
% 15.00/2.94 join(join(meet(composition(x0, top), X), composition(x0, X)), join(composition(x0, X), X))
% 15.00/2.94 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.94 join(join(meet(composition(x0, top), X), composition(x0, X)), join(X, composition(x0, X)))
% 15.00/2.94 = { by axiom 8 (maddux2_join_associativity) }
% 15.00/2.94 join(join(join(meet(composition(x0, top), X), composition(x0, X)), X), composition(x0, X))
% 15.00/2.94 = { by lemma 75 }
% 15.00/2.94 join(join(meet(composition(x0, top), X), X), composition(x0, X))
% 15.00/2.94 = { by axiom 8 (maddux2_join_associativity) R->L }
% 15.00/2.94 join(meet(composition(x0, top), X), join(X, composition(x0, X)))
% 15.00/2.94 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.94 join(meet(composition(x0, top), X), join(composition(x0, X), X))
% 15.00/2.94 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.94 join(join(composition(x0, X), X), meet(composition(x0, top), X))
% 15.00/2.94 = { by axiom 8 (maddux2_join_associativity) R->L }
% 15.00/2.94 join(composition(x0, X), join(X, meet(composition(x0, top), X)))
% 15.00/2.94 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.94 join(composition(x0, X), join(meet(composition(x0, top), X), X))
% 15.00/2.94 = { by lemma 75 R->L }
% 15.00/2.94 join(composition(x0, X), join(join(meet(composition(x0, top), X), composition(x0, X)), X))
% 15.00/2.94 = { by lemma 47 R->L }
% 15.00/2.94 join(composition(x0, X), meet(join(join(meet(composition(x0, top), X), composition(x0, X)), X), top))
% 15.00/2.94 = { by lemma 43 R->L }
% 15.00/2.94 join(composition(x0, X), complement(join(zero, complement(join(join(meet(composition(x0, top), X), composition(x0, X)), X)))))
% 15.00/2.94 = { by lemma 75 }
% 15.00/2.94 join(composition(x0, X), complement(join(zero, complement(join(meet(composition(x0, top), X), X)))))
% 15.00/2.94 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.94 join(composition(x0, X), complement(join(zero, complement(join(X, meet(composition(x0, top), X))))))
% 15.00/2.94 = { by lemma 70 R->L }
% 15.00/2.94 join(composition(x0, X), complement(join(zero, meet(complement(X), complement(meet(composition(x0, top), X))))))
% 15.00/2.94 = { by lemma 34 R->L }
% 15.00/2.94 join(composition(x0, X), complement(join(zero, join(zero, meet(complement(X), complement(meet(composition(x0, top), X)))))))
% 15.00/2.94 = { by lemma 74 R->L }
% 15.00/2.94 join(composition(x0, X), complement(join(zero, join(meet(meet(composition(x0, top), X), complement(X)), meet(complement(X), complement(meet(composition(x0, top), X)))))))
% 15.00/2.94 = { by lemma 76 }
% 15.00/2.94 join(composition(x0, X), complement(join(zero, complement(X))))
% 15.00/2.94 = { by lemma 43 }
% 15.00/2.94 join(composition(x0, X), meet(X, top))
% 15.00/2.94 = { by lemma 47 }
% 15.00/2.94 join(composition(x0, X), X)
% 15.00/2.95 = { by lemma 69 }
% 15.00/2.95 X
% 15.00/2.95
% 15.00/2.95 Goal 1 (goals_1): tuple(join(meet(composition(x0, top), x1), composition(x0, x1)), join(composition(x0, x1), meet(composition(x0, top), x1))) = tuple(composition(x0, x1), meet(composition(x0, top), x1)).
% 15.00/2.95 Proof:
% 15.00/2.95 tuple(join(meet(composition(x0, top), x1), composition(x0, x1)), join(composition(x0, x1), meet(composition(x0, top), x1)))
% 15.00/2.95 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.95 tuple(join(meet(composition(x0, top), x1), composition(x0, x1)), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.95 = { by lemma 51 R->L }
% 15.00/2.95 tuple(complement(complement(join(meet(composition(x0, top), x1), composition(x0, x1)))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.95 = { by lemma 79 }
% 15.00/2.95 tuple(complement(complement(meet(composition(x0, top), x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.95 = { by lemma 27 R->L }
% 15.00/2.95 tuple(complement(complement(join(meet(meet(composition(x0, top), x1), meet(x1, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), complement(join(complement(meet(composition(x0, top), x1)), meet(x1, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.95 = { by lemma 72 }
% 15.00/2.95 tuple(complement(complement(join(meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), meet(meet(composition(x0, top), x1), x1)), complement(join(complement(meet(composition(x0, top), x1)), meet(x1, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.95 = { by lemma 71 }
% 15.00/2.95 tuple(complement(complement(join(meet(x1, meet(meet(composition(x0, top), x1), complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), complement(join(complement(meet(composition(x0, top), x1)), meet(x1, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.95 = { by lemma 60 R->L }
% 15.00/2.95 tuple(complement(complement(join(meet(x1, complement(join(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(meet(composition(x0, top), x1))))), complement(join(complement(meet(composition(x0, top), x1)), meet(x1, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.96 tuple(complement(complement(join(meet(x1, complement(join(complement(meet(composition(x0, top), x1)), join(meet(composition(x0, top), x1), composition(x0, x1))))), complement(join(complement(meet(composition(x0, top), x1)), meet(x1, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 68 }
% 15.00/2.96 tuple(complement(complement(join(meet(x1, complement(join(meet(composition(x0, top), x1), join(composition(x0, x1), complement(meet(composition(x0, top), x1)))))), complement(join(complement(meet(composition(x0, top), x1)), meet(x1, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 18 }
% 15.00/2.96 tuple(complement(complement(join(meet(x1, complement(join(composition(x0, x1), top))), complement(join(complement(meet(composition(x0, top), x1)), meet(x1, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 35 }
% 15.00/2.96 tuple(complement(complement(join(meet(x1, complement(top)), complement(join(complement(meet(composition(x0, top), x1)), meet(x1, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 17 }
% 15.00/2.96 tuple(complement(complement(join(meet(x1, zero), complement(join(complement(meet(composition(x0, top), x1)), meet(x1, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 45 }
% 15.00/2.96 tuple(complement(complement(join(zero, complement(join(complement(meet(composition(x0, top), x1)), meet(x1, complement(join(meet(composition(x0, top), x1), composition(x0, x1))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 32 }
% 15.00/2.96 tuple(complement(complement(complement(join(complement(meet(composition(x0, top), x1)), meet(x1, complement(join(meet(composition(x0, top), x1), composition(x0, x1)))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 62 }
% 15.00/2.96 tuple(complement(complement(meet(meet(composition(x0, top), x1), complement(meet(x1, complement(join(meet(composition(x0, top), x1), composition(x0, x1)))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 61 }
% 15.00/2.96 tuple(complement(complement(meet(meet(composition(x0, top), x1), join(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 79 R->L }
% 15.00/2.96 tuple(complement(complement(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), join(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 42 }
% 15.00/2.96 tuple(complement(complement(meet(join(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(x1)), join(meet(composition(x0, top), x1), composition(x0, x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 81 }
% 15.00/2.96 tuple(complement(complement(meet(join(composition(x0, top), complement(x1)), join(meet(composition(x0, top), x1), composition(x0, x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 61 R->L }
% 15.00/2.96 tuple(complement(complement(meet(complement(meet(x1, complement(composition(x0, top)))), join(meet(composition(x0, top), x1), composition(x0, x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 42 }
% 15.00/2.96 tuple(complement(complement(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(meet(x1, complement(composition(x0, top))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 79 }
% 15.00/2.96 tuple(complement(complement(meet(meet(composition(x0, top), x1), complement(meet(x1, complement(composition(x0, top))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 27 R->L }
% 15.00/2.96 tuple(complement(complement(meet(join(meet(meet(composition(x0, top), x1), complement(x1)), complement(join(complement(meet(composition(x0, top), x1)), complement(x1)))), complement(meet(x1, complement(composition(x0, top))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 74 }
% 15.00/2.96 tuple(complement(complement(meet(join(zero, complement(join(complement(meet(composition(x0, top), x1)), complement(x1)))), complement(meet(x1, complement(composition(x0, top))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 32 }
% 15.00/2.96 tuple(complement(complement(meet(complement(join(complement(meet(composition(x0, top), x1)), complement(x1))), complement(meet(x1, complement(composition(x0, top))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.00/2.96 tuple(complement(complement(meet(meet(meet(composition(x0, top), x1), x1), complement(meet(x1, complement(composition(x0, top))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 42 }
% 15.00/2.96 tuple(complement(complement(meet(complement(meet(x1, complement(composition(x0, top)))), meet(meet(composition(x0, top), x1), x1)))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 72 R->L }
% 15.00/2.96 tuple(complement(complement(meet(meet(composition(x0, top), x1), meet(x1, complement(meet(x1, complement(composition(x0, top)))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 79 R->L }
% 15.00/2.96 tuple(complement(complement(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), meet(x1, complement(meet(x1, complement(composition(x0, top)))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.96 = { by lemma 42 R->L }
% 15.00/2.96 tuple(complement(complement(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), meet(complement(meet(x1, complement(composition(x0, top)))), x1)))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.97 = { by lemma 82 R->L }
% 15.00/2.97 tuple(complement(complement(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), meet(complement(meet(x1, complement(composition(x0, top)))), join(join(meet(composition(x0, top), x1), composition(x0, x1)), meet(x1, complement(composition(x0, top)))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.97 = { by lemma 42 }
% 15.00/2.97 tuple(complement(complement(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), meet(join(join(meet(composition(x0, top), x1), composition(x0, x1)), meet(x1, complement(composition(x0, top)))), complement(meet(x1, complement(composition(x0, top)))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.97 = { by lemma 60 R->L }
% 15.00/2.97 tuple(complement(complement(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(join(meet(x1, complement(composition(x0, top))), complement(join(join(meet(composition(x0, top), x1), composition(x0, x1)), meet(x1, complement(composition(x0, top)))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.97 = { by lemma 70 R->L }
% 15.00/2.97 tuple(complement(complement(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(join(meet(x1, complement(composition(x0, top))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), complement(meet(x1, complement(composition(x0, top)))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.97 = { by lemma 33 R->L }
% 15.00/2.97 tuple(complement(complement(join(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(join(meet(x1, complement(composition(x0, top))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), complement(meet(x1, complement(composition(x0, top)))))))), zero))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.97 = { by axiom 5 (def_zero) }
% 15.00/2.97 tuple(complement(complement(join(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(join(meet(x1, complement(composition(x0, top))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), complement(meet(x1, complement(composition(x0, top)))))))), meet(join(meet(x1, complement(composition(x0, top))), complement(complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), complement(join(meet(x1, complement(composition(x0, top))), complement(complement(join(meet(composition(x0, top), x1), composition(x0, x1)))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.97 = { by lemma 60 }
% 15.00/2.97 tuple(complement(complement(join(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(join(meet(x1, complement(composition(x0, top))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), complement(meet(x1, complement(composition(x0, top)))))))), meet(join(meet(x1, complement(composition(x0, top))), complement(complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), complement(meet(x1, complement(composition(x0, top))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.97 = { by lemma 71 R->L }
% 15.00/2.97 tuple(complement(complement(join(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(join(meet(x1, complement(composition(x0, top))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), complement(meet(x1, complement(composition(x0, top)))))))), meet(complement(meet(x1, complement(composition(x0, top)))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), join(meet(x1, complement(composition(x0, top))), complement(complement(join(meet(composition(x0, top), x1), composition(x0, x1)))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.97 = { by lemma 73 R->L }
% 15.00/2.97 tuple(complement(complement(join(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(join(meet(x1, complement(composition(x0, top))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), complement(meet(x1, complement(composition(x0, top)))))))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), meet(complement(meet(x1, complement(composition(x0, top)))), join(meet(x1, complement(composition(x0, top))), complement(complement(join(meet(composition(x0, top), x1), composition(x0, x1)))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.97 = { by lemma 42 }
% 15.00/2.97 tuple(complement(complement(join(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(join(meet(x1, complement(composition(x0, top))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), complement(meet(x1, complement(composition(x0, top)))))))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), meet(join(meet(x1, complement(composition(x0, top))), complement(complement(join(meet(composition(x0, top), x1), composition(x0, x1))))), complement(meet(x1, complement(composition(x0, top))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.97 = { by lemma 60 R->L }
% 15.00/2.98 tuple(complement(complement(join(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(join(meet(x1, complement(composition(x0, top))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), complement(meet(x1, complement(composition(x0, top)))))))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), complement(join(meet(x1, complement(composition(x0, top))), complement(join(meet(x1, complement(composition(x0, top))), complement(complement(join(meet(composition(x0, top), x1), composition(x0, x1)))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 60 }
% 15.00/2.98 tuple(complement(complement(join(meet(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(join(meet(x1, complement(composition(x0, top))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), complement(meet(x1, complement(composition(x0, top)))))))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), complement(join(meet(x1, complement(composition(x0, top))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), complement(meet(x1, complement(composition(x0, top))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 77 }
% 15.00/2.98 tuple(complement(complement(complement(join(meet(x1, complement(composition(x0, top))), meet(complement(join(meet(composition(x0, top), x1), composition(x0, x1))), complement(meet(x1, complement(composition(x0, top))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 70 }
% 15.00/2.98 tuple(complement(complement(complement(join(meet(x1, complement(composition(x0, top))), complement(join(join(meet(composition(x0, top), x1), composition(x0, x1)), meet(x1, complement(composition(x0, top))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 60 }
% 15.00/2.98 tuple(complement(complement(meet(join(join(meet(composition(x0, top), x1), composition(x0, x1)), meet(x1, complement(composition(x0, top)))), complement(meet(x1, complement(composition(x0, top))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 42 R->L }
% 15.00/2.98 tuple(complement(complement(meet(complement(meet(x1, complement(composition(x0, top)))), join(join(meet(composition(x0, top), x1), composition(x0, x1)), meet(x1, complement(composition(x0, top))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 61 }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, top), complement(x1)), join(join(meet(composition(x0, top), x1), composition(x0, x1)), meet(x1, complement(composition(x0, top))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 81 R->L }
% 15.00/2.98 tuple(complement(complement(meet(join(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(x1)), join(join(meet(composition(x0, top), x1), composition(x0, x1)), meet(x1, complement(composition(x0, top))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 82 }
% 15.00/2.98 tuple(complement(complement(meet(join(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(x1)), x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 42 R->L }
% 15.00/2.98 tuple(complement(complement(meet(x1, join(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 69 R->L }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(join(meet(composition(x0, top), x1), composition(x0, x1)), complement(x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 81 }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(composition(x0, top), complement(x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), composition(x0, top))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 41 R->L }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), composition(x0, converse(top)))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 38 R->L }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), composition(x0, join(x1, converse(complement(converse(x1))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by axiom 1 (converse_idempotence) R->L }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), converse(converse(composition(x0, join(x1, converse(complement(converse(x1))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by axiom 9 (converse_multiplicativity) }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), converse(composition(converse(join(x1, converse(complement(converse(x1))))), converse(x0))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 36 }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), converse(composition(join(complement(converse(x1)), converse(x1)), converse(x0))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), converse(composition(join(converse(x1), complement(converse(x1))), converse(x0))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by axiom 12 (composition_distributivity) }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), converse(join(composition(converse(x1), converse(x0)), composition(complement(converse(x1)), converse(x0)))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by axiom 9 (converse_multiplicativity) R->L }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), converse(join(converse(composition(x0, x1)), composition(complement(converse(x1)), converse(x0)))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), converse(join(composition(complement(converse(x1)), converse(x0)), converse(composition(x0, x1)))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by axiom 1 (converse_idempotence) R->L }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), converse(join(composition(converse(converse(complement(converse(x1)))), converse(x0)), converse(composition(x0, x1)))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by axiom 9 (converse_multiplicativity) R->L }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), converse(join(converse(composition(x0, converse(complement(converse(x1))))), converse(composition(x0, x1)))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by axiom 7 (converse_additivity) R->L }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), converse(converse(join(composition(x0, converse(complement(converse(x1)))), composition(x0, x1)))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), converse(converse(join(composition(x0, x1), composition(x0, converse(complement(converse(x1))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by axiom 1 (converse_idempotence) }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(converse(x1))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 32 R->L }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(zero, complement(converse(x1)))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 27 R->L }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(meet(join(zero, complement(converse(x1))), complement(converse(complement(converse(complement(join(zero, complement(converse(x1))))))))), complement(join(complement(join(zero, complement(converse(x1)))), complement(converse(complement(converse(complement(join(zero, complement(converse(x1)))))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 62 R->L }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(complement(join(complement(join(zero, complement(converse(x1)))), converse(complement(converse(complement(join(zero, complement(converse(x1))))))))), complement(join(complement(join(zero, complement(converse(x1)))), complement(converse(complement(converse(complement(join(zero, complement(converse(x1)))))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 38 }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(complement(converse(top)), complement(join(complement(join(zero, complement(converse(x1)))), complement(converse(complement(converse(complement(join(zero, complement(converse(x1)))))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.98 = { by lemma 41 }
% 15.00/2.98 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(complement(top), complement(join(complement(join(zero, complement(converse(x1)))), complement(converse(complement(converse(complement(join(zero, complement(converse(x1)))))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 17 }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(zero, complement(join(complement(join(zero, complement(converse(x1)))), complement(converse(complement(converse(complement(join(zero, complement(converse(x1)))))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 32 }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(complement(join(complement(join(zero, complement(converse(x1)))), complement(converse(complement(converse(complement(join(zero, complement(converse(x1))))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(meet(join(zero, complement(converse(x1))), converse(complement(converse(complement(join(zero, complement(converse(x1))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 33 R->L }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(meet(join(zero, complement(converse(x1))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), zero)))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 52 R->L }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(meet(join(zero, complement(converse(x1))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(converse(x1)))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 58 R->L }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(meet(join(zero, complement(converse(x1))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), meet(composition(zero, complement(join(zero, complement(converse(x1))))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 80 R->L }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(meet(join(zero, complement(converse(x1))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), meet(composition(meet(join(x0, one), composition(converse(complement(converse(complement(join(zero, complement(converse(x1))))))), converse(complement(join(zero, complement(converse(x1))))))), complement(join(zero, complement(converse(x1))))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by axiom 16 (modular_law_2) R->L }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(meet(join(zero, complement(converse(x1))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), join(meet(composition(join(x0, one), complement(join(zero, complement(converse(x1))))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), meet(composition(meet(join(x0, one), composition(converse(complement(converse(complement(join(zero, complement(converse(x1))))))), converse(complement(join(zero, complement(converse(x1))))))), complement(join(zero, complement(converse(x1))))), converse(complement(converse(complement(join(zero, complement(converse(x1))))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 80 }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(meet(join(zero, complement(converse(x1))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), join(meet(composition(join(x0, one), complement(join(zero, complement(converse(x1))))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), meet(composition(zero, complement(join(zero, complement(converse(x1))))), converse(complement(converse(complement(join(zero, complement(converse(x1))))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 22 }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(meet(join(zero, complement(converse(x1))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), join(meet(complement(join(zero, complement(converse(x1)))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), meet(composition(zero, complement(join(zero, complement(converse(x1))))), converse(complement(converse(complement(join(zero, complement(converse(x1))))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 58 }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(meet(join(zero, complement(converse(x1))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), join(meet(complement(join(zero, complement(converse(x1)))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(converse(x1))))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 52 }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(meet(join(zero, complement(converse(x1))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), join(meet(complement(join(zero, complement(converse(x1)))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), zero))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 33 }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(join(meet(join(zero, complement(converse(x1))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))), meet(complement(join(zero, complement(converse(x1)))), converse(complement(converse(complement(join(zero, complement(converse(x1)))))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 77 }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, converse(converse(complement(converse(complement(join(zero, complement(converse(x1)))))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by axiom 1 (converse_idempotence) }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, complement(converse(complement(join(zero, complement(converse(x1)))))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 43 }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, complement(converse(meet(converse(x1), top))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 47 }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, complement(converse(converse(x1))))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by axiom 1 (converse_idempotence) }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, x1), composition(x0, complement(x1))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), join(composition(x0, complement(x1)), composition(x0, x1)))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by axiom 8 (maddux2_join_associativity) }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(join(complement(x1), composition(x0, complement(x1))), composition(x0, x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 53 R->L }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(composition(join(join(x0, one), x0), complement(x1)), composition(x0, x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(composition(join(x0, join(x0, one)), complement(x1)), composition(x0, x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by axiom 3 (goals) }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(composition(join(x0, one), complement(x1)), composition(x0, x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 22 }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(complement(x1), composition(x0, x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), x1), join(composition(x0, x1), complement(x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 15.00/2.99 tuple(complement(complement(meet(join(x1, composition(x0, x1)), join(composition(x0, x1), complement(x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 51 R->L }
% 15.00/2.99 tuple(complement(complement(meet(join(x1, composition(x0, x1)), join(composition(x0, x1), complement(complement(complement(x1))))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 42 }
% 15.00/2.99 tuple(complement(complement(meet(join(composition(x0, x1), complement(complement(complement(x1)))), join(x1, composition(x0, x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 63 R->L }
% 15.00/2.99 tuple(complement(join(complement(join(composition(x0, x1), complement(complement(complement(x1))))), complement(join(x1, composition(x0, x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 60 }
% 15.00/2.99 tuple(complement(join(meet(complement(complement(x1)), complement(composition(x0, x1))), complement(join(x1, composition(x0, x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by axiom 2 (maddux1_join_commutativity) }
% 15.00/2.99 tuple(complement(join(complement(join(x1, composition(x0, x1))), meet(complement(complement(x1)), complement(composition(x0, x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 42 R->L }
% 15.00/2.99 tuple(complement(join(complement(join(x1, composition(x0, x1))), meet(complement(composition(x0, x1)), complement(complement(x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 70 R->L }
% 15.00/2.99 tuple(complement(join(meet(complement(x1), complement(composition(x0, x1))), meet(complement(composition(x0, x1)), complement(complement(x1))))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 76 }
% 15.00/2.99 tuple(complement(complement(composition(x0, x1))), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 51 }
% 15.00/2.99 tuple(composition(x0, x1), join(meet(composition(x0, top), x1), composition(x0, x1)))
% 15.00/2.99 = { by lemma 79 }
% 15.00/2.99 tuple(composition(x0, x1), meet(composition(x0, top), x1))
% 15.00/2.99 % SZS output end Proof
% 15.00/3.01
% 15.00/3.01 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------