TSTP Solution File: REL033+4 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL033+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 : n032.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 13:44:19 EDT 2023
% Result : Theorem 230.17s 29.93s
% Output : Proof 232.24s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.10 % Problem : REL033+4 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.10 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.10/0.30 % Computer : n032.cluster.edu
% 0.10/0.30 % Model : x86_64 x86_64
% 0.10/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.30 % Memory : 8042.1875MB
% 0.10/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.10/0.30 % CPULimit : 300
% 0.10/0.30 % WCLimit : 300
% 0.10/0.30 % DateTime : Fri Aug 25 19:20:13 EDT 2023
% 0.10/0.30 % CPUTime :
% 230.17/29.93 Command-line arguments: --flatten
% 230.17/29.93
% 230.17/29.93 % SZS status Theorem
% 230.17/29.93
% 231.22/30.14 % SZS output start Proof
% 231.22/30.14 Take the following subset of the input axioms:
% 231.22/30.14 fof(composition_associativity, axiom, ![X0, X1, X2]: composition(X0, composition(X1, X2))=composition(composition(X0, X1), X2)).
% 231.22/30.14 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))).
% 231.22/30.14 fof(composition_identity, axiom, ![X0_2]: composition(X0_2, one)=X0_2).
% 231.22/30.14 fof(converse_additivity, axiom, ![X0_2, X1_2]: converse(join(X0_2, X1_2))=join(converse(X0_2), converse(X1_2))).
% 231.22/30.14 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)).
% 231.22/30.14 fof(converse_idempotence, axiom, ![X0_2]: converse(converse(X0_2))=X0_2).
% 231.22/30.14 fof(converse_multiplicativity, axiom, ![X0_2, X1_2]: converse(composition(X0_2, X1_2))=composition(converse(X1_2), converse(X0_2))).
% 231.22/30.14 fof(dedekind_law, axiom, ![X0_2, X1_2, X2_2]: join(meet(composition(X0_2, X1_2), X2_2), composition(meet(X0_2, composition(X2_2, converse(X1_2))), meet(X1_2, composition(converse(X0_2), X2_2))))=composition(meet(X0_2, composition(X2_2, converse(X1_2))), meet(X1_2, composition(converse(X0_2), X2_2)))).
% 231.22/30.14 fof(def_top, axiom, ![X0_2]: top=join(X0_2, complement(X0_2))).
% 231.22/30.14 fof(def_zero, axiom, ![X0_2]: zero=meet(X0_2, complement(X0_2))).
% 231.22/30.14 fof(goals, conjecture, ![X0_2, X1_2, X2_2]: (composition(X0_2, top)=X0_2 => (join(composition(meet(X0_2, X1_2), X2_2), meet(X0_2, composition(X1_2, X2_2)))=meet(X0_2, composition(X1_2, X2_2)) & join(meet(X0_2, composition(X1_2, X2_2)), composition(meet(X0_2, X1_2), X2_2))=composition(meet(X0_2, X1_2), X2_2)))).
% 231.22/30.14 fof(maddux1_join_commutativity, axiom, ![X0_2, X1_2]: join(X0_2, X1_2)=join(X1_2, X0_2)).
% 231.22/30.14 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)).
% 231.22/30.14 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)))).
% 231.22/30.14 fof(maddux4_definiton_of_meet, axiom, ![X0_2, X1_2]: meet(X0_2, X1_2)=complement(join(complement(X0_2), complement(X1_2)))).
% 231.22/30.14 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)).
% 231.22/30.14
% 231.22/30.14 Now clausify the problem and encode Horn clauses using encoding 3 of
% 231.22/30.14 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 231.22/30.14 We repeatedly replace C & s=t => u=v by the two clauses:
% 231.22/30.14 fresh(y, y, x1...xn) = u
% 231.22/30.14 C => fresh(s, t, x1...xn) = v
% 231.22/30.14 where fresh is a fresh function symbol and x1..xn are the free
% 231.22/30.14 variables of u and v.
% 231.22/30.14 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 231.22/30.14 input problem has no model of domain size 1).
% 231.22/30.14
% 231.22/30.14 The encoding turns the above axioms into the following unit equations and goals:
% 231.22/30.14
% 231.22/30.14 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 231.22/30.14 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 231.22/30.14 Axiom 3 (composition_identity): composition(X, one) = X.
% 231.22/30.15 Axiom 4 (goals): composition(x0, top) = x0.
% 231.22/30.15 Axiom 5 (def_top): top = join(X, complement(X)).
% 231.22/30.15 Axiom 6 (def_zero): zero = meet(X, complement(X)).
% 231.22/30.15 Axiom 7 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 231.22/30.15 Axiom 8 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 231.22/30.15 Axiom 9 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 231.22/30.15 Axiom 10 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 231.22/30.15 Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 231.22/30.15 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 231.22/30.15 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 231.22/30.15 Axiom 14 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 231.22/30.15 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).
% 231.22/30.15 Axiom 16 (dedekind_law): join(meet(composition(X, Y), Z), composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z)))) = composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z))).
% 231.22/30.15
% 231.22/30.15 Lemma 17: complement(top) = zero.
% 231.22/30.15 Proof:
% 231.22/30.15 complement(top)
% 231.22/30.15 = { by axiom 5 (def_top) }
% 231.22/30.15 complement(join(complement(X), complement(complement(X))))
% 231.22/30.15 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 231.22/30.15 meet(X, complement(X))
% 231.22/30.15 = { by axiom 6 (def_zero) R->L }
% 231.22/30.15 zero
% 231.22/30.15
% 231.22/30.15 Lemma 18: join(X, join(Y, complement(X))) = join(Y, top).
% 231.22/30.15 Proof:
% 231.22/30.15 join(X, join(Y, complement(X)))
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 join(X, join(complement(X), Y))
% 231.22/30.15 = { by axiom 8 (maddux2_join_associativity) }
% 231.22/30.15 join(join(X, complement(X)), Y)
% 231.22/30.15 = { by axiom 5 (def_top) R->L }
% 231.22/30.15 join(top, Y)
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) }
% 231.22/30.15 join(Y, top)
% 231.22/30.15
% 231.22/30.15 Lemma 19: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 231.22/30.15 Proof:
% 231.22/30.15 converse(composition(converse(X), Y))
% 231.22/30.15 = { by axiom 9 (converse_multiplicativity) }
% 231.22/30.15 composition(converse(Y), converse(converse(X)))
% 231.22/30.15 = { by axiom 1 (converse_idempotence) }
% 231.22/30.15 composition(converse(Y), X)
% 231.22/30.15
% 231.22/30.15 Lemma 20: composition(converse(one), X) = X.
% 231.22/30.15 Proof:
% 231.22/30.15 composition(converse(one), X)
% 231.22/30.15 = { by lemma 19 R->L }
% 231.22/30.15 converse(composition(converse(X), one))
% 231.22/30.15 = { by axiom 3 (composition_identity) }
% 231.22/30.15 converse(converse(X))
% 231.22/30.15 = { by axiom 1 (converse_idempotence) }
% 231.22/30.15 X
% 231.22/30.15
% 231.22/30.15 Lemma 21: composition(one, X) = X.
% 231.22/30.15 Proof:
% 231.22/30.15 composition(one, X)
% 231.22/30.15 = { by lemma 20 R->L }
% 231.22/30.15 composition(converse(one), composition(one, X))
% 231.22/30.15 = { by axiom 10 (composition_associativity) }
% 231.22/30.15 composition(composition(converse(one), one), X)
% 231.22/30.15 = { by axiom 3 (composition_identity) }
% 231.22/30.15 composition(converse(one), X)
% 231.22/30.15 = { by lemma 20 }
% 231.22/30.15 X
% 231.22/30.15
% 231.22/30.15 Lemma 22: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 231.22/30.15 Proof:
% 231.22/30.15 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 231.22/30.15 = { by axiom 13 (converse_cancellativity) }
% 231.22/30.15 complement(X)
% 231.22/30.15
% 231.22/30.15 Lemma 23: join(complement(X), complement(X)) = complement(X).
% 231.22/30.15 Proof:
% 231.22/30.15 join(complement(X), complement(X))
% 231.22/30.15 = { by lemma 20 R->L }
% 231.22/30.15 join(complement(X), composition(converse(one), complement(X)))
% 231.22/30.15 = { by lemma 21 R->L }
% 231.22/30.15 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 231.22/30.15 = { by lemma 22 }
% 231.22/30.15 complement(X)
% 231.22/30.15
% 231.22/30.15 Lemma 24: join(top, complement(X)) = top.
% 231.22/30.15 Proof:
% 231.22/30.15 join(top, complement(X))
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 join(complement(X), top)
% 231.22/30.15 = { by lemma 18 R->L }
% 231.22/30.15 join(X, join(complement(X), complement(X)))
% 231.22/30.15 = { by lemma 23 }
% 231.22/30.15 join(X, complement(X))
% 231.22/30.15 = { by axiom 5 (def_top) R->L }
% 231.22/30.15 top
% 231.22/30.15
% 231.22/30.15 Lemma 25: join(Y, top) = join(X, top).
% 231.22/30.15 Proof:
% 231.22/30.15 join(Y, top)
% 231.22/30.15 = { by lemma 24 R->L }
% 231.22/30.15 join(Y, join(top, complement(Y)))
% 231.22/30.15 = { by lemma 18 }
% 231.22/30.15 join(top, top)
% 231.22/30.15 = { by lemma 18 R->L }
% 231.22/30.15 join(X, join(top, complement(X)))
% 231.22/30.15 = { by lemma 24 }
% 231.22/30.15 join(X, top)
% 231.22/30.15
% 231.22/30.15 Lemma 26: join(X, top) = top.
% 231.22/30.15 Proof:
% 231.22/30.15 join(X, top)
% 231.22/30.15 = { by lemma 25 }
% 231.22/30.15 join(zero, top)
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 join(top, zero)
% 231.22/30.15 = { by lemma 17 R->L }
% 231.22/30.15 join(top, complement(top))
% 231.22/30.15 = { by axiom 5 (def_top) R->L }
% 231.22/30.15 top
% 231.22/30.15
% 231.22/30.15 Lemma 27: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 231.22/30.15 Proof:
% 231.22/30.15 converse(join(X, converse(Y)))
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 converse(join(converse(Y), X))
% 231.22/30.15 = { by axiom 7 (converse_additivity) }
% 231.22/30.15 join(converse(converse(Y)), converse(X))
% 231.22/30.15 = { by axiom 1 (converse_idempotence) }
% 231.22/30.15 join(Y, converse(X))
% 231.22/30.15
% 231.22/30.15 Lemma 28: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 231.22/30.15 Proof:
% 231.22/30.15 converse(join(converse(X), Y))
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 converse(join(Y, converse(X)))
% 231.22/30.15 = { by lemma 27 }
% 231.22/30.15 join(X, converse(Y))
% 231.22/30.15
% 231.22/30.15 Lemma 29: join(X, join(complement(X), Y)) = top.
% 231.22/30.15 Proof:
% 231.22/30.15 join(X, join(complement(X), Y))
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 join(X, join(Y, complement(X)))
% 231.22/30.15 = { by lemma 18 }
% 231.22/30.15 join(Y, top)
% 231.22/30.15 = { by lemma 25 R->L }
% 231.22/30.15 join(Z, top)
% 231.22/30.15 = { by lemma 26 }
% 231.22/30.15 top
% 231.22/30.15
% 231.22/30.15 Lemma 30: join(X, converse(top)) = top.
% 231.22/30.15 Proof:
% 231.22/30.15 join(X, converse(top))
% 231.22/30.15 = { by axiom 5 (def_top) }
% 231.22/30.15 join(X, converse(join(converse(complement(X)), complement(converse(complement(X))))))
% 231.22/30.15 = { by lemma 28 }
% 231.22/30.15 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 231.22/30.15 = { by lemma 29 }
% 231.22/30.15 top
% 231.22/30.15
% 231.22/30.15 Lemma 31: converse(top) = top.
% 231.22/30.15 Proof:
% 231.22/30.15 converse(top)
% 231.22/30.15 = { by lemma 26 R->L }
% 231.22/30.15 converse(join(X, top))
% 231.22/30.15 = { by axiom 7 (converse_additivity) }
% 231.22/30.15 join(converse(X), converse(top))
% 231.22/30.15 = { by lemma 30 }
% 231.22/30.15 top
% 231.22/30.15
% 231.22/30.15 Lemma 32: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 231.22/30.15 Proof:
% 231.22/30.15 join(meet(X, Y), complement(join(complement(X), Y)))
% 231.22/30.15 = { by axiom 11 (maddux4_definiton_of_meet) }
% 231.22/30.15 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 231.22/30.15 = { by axiom 14 (maddux3_a_kind_of_de_Morgan) R->L }
% 231.22/30.15 X
% 231.22/30.15
% 231.22/30.15 Lemma 33: join(zero, meet(X, X)) = X.
% 231.22/30.15 Proof:
% 231.22/30.15 join(zero, meet(X, X))
% 231.22/30.15 = { by axiom 11 (maddux4_definiton_of_meet) }
% 231.22/30.15 join(zero, complement(join(complement(X), complement(X))))
% 231.22/30.15 = { by axiom 6 (def_zero) }
% 231.22/30.15 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 231.22/30.15 = { by lemma 32 }
% 231.22/30.15 X
% 231.22/30.15
% 231.22/30.15 Lemma 34: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 231.22/30.15 Proof:
% 231.22/30.15 join(zero, join(X, complement(complement(Y))))
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 join(zero, join(complement(complement(Y)), X))
% 231.22/30.15 = { by lemma 23 R->L }
% 231.22/30.15 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 231.22/30.15 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 231.22/30.15 join(zero, join(meet(Y, Y), X))
% 231.22/30.15 = { by axiom 8 (maddux2_join_associativity) }
% 231.22/30.15 join(join(zero, meet(Y, Y)), X)
% 231.22/30.15 = { by lemma 33 }
% 231.22/30.15 join(Y, X)
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) }
% 231.22/30.15 join(X, Y)
% 231.22/30.15
% 231.22/30.15 Lemma 35: join(zero, complement(complement(X))) = X.
% 231.22/30.15 Proof:
% 231.22/30.15 join(zero, complement(complement(X)))
% 231.22/30.15 = { by axiom 6 (def_zero) }
% 231.22/30.15 join(meet(X, complement(X)), complement(complement(X)))
% 231.22/30.15 = { by lemma 23 R->L }
% 231.22/30.15 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 231.22/30.15 = { by lemma 32 }
% 231.22/30.15 X
% 231.22/30.15
% 231.22/30.15 Lemma 36: join(X, zero) = join(X, X).
% 231.22/30.15 Proof:
% 231.22/30.15 join(X, zero)
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 join(zero, X)
% 231.22/30.15 = { by lemma 35 R->L }
% 231.22/30.15 join(zero, join(zero, complement(complement(X))))
% 231.22/30.15 = { by lemma 23 R->L }
% 231.22/30.15 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 231.22/30.15 = { by lemma 34 }
% 231.22/30.15 join(zero, join(complement(complement(X)), X))
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) }
% 231.22/30.15 join(zero, join(X, complement(complement(X))))
% 231.22/30.15 = { by lemma 34 }
% 231.22/30.15 join(X, X)
% 231.22/30.15
% 231.22/30.15 Lemma 37: join(zero, complement(X)) = complement(X).
% 231.22/30.15 Proof:
% 231.22/30.15 join(zero, complement(X))
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 join(complement(X), zero)
% 231.22/30.15 = { by lemma 36 }
% 231.22/30.15 join(complement(X), complement(X))
% 231.22/30.15 = { by lemma 23 }
% 231.22/30.15 complement(X)
% 231.22/30.15
% 231.22/30.15 Lemma 38: join(X, zero) = X.
% 231.22/30.15 Proof:
% 231.22/30.15 join(X, zero)
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 join(zero, X)
% 231.22/30.15 = { by lemma 34 R->L }
% 231.22/30.15 join(zero, join(zero, complement(complement(X))))
% 231.22/30.15 = { by lemma 37 }
% 231.22/30.15 join(zero, complement(complement(X)))
% 231.22/30.15 = { by lemma 35 }
% 231.22/30.15 X
% 231.22/30.15
% 231.22/30.15 Lemma 39: join(zero, X) = X.
% 231.22/30.15 Proof:
% 231.22/30.15 join(zero, X)
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 join(X, zero)
% 231.22/30.15 = { by lemma 38 }
% 231.22/30.15 X
% 231.22/30.15
% 231.22/30.15 Lemma 40: meet(Y, X) = meet(X, Y).
% 231.22/30.15 Proof:
% 231.22/30.15 meet(Y, X)
% 231.22/30.15 = { by axiom 11 (maddux4_definiton_of_meet) }
% 231.22/30.15 complement(join(complement(Y), complement(X)))
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 complement(join(complement(X), complement(Y)))
% 231.22/30.15 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 231.22/30.15 meet(X, Y)
% 231.22/30.15
% 231.22/30.15 Lemma 41: complement(join(zero, complement(X))) = meet(X, top).
% 231.22/30.15 Proof:
% 231.22/30.15 complement(join(zero, complement(X)))
% 231.22/30.15 = { by lemma 17 R->L }
% 231.22/30.15 complement(join(complement(top), complement(X)))
% 231.22/30.15 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 231.22/30.15 meet(top, X)
% 231.22/30.15 = { by lemma 40 R->L }
% 231.22/30.15 meet(X, top)
% 231.22/30.15
% 231.22/30.15 Lemma 42: join(X, complement(zero)) = top.
% 231.22/30.15 Proof:
% 231.22/30.15 join(X, complement(zero))
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 join(complement(zero), X)
% 231.22/30.15 = { by lemma 34 R->L }
% 231.22/30.15 join(zero, join(complement(zero), complement(complement(X))))
% 231.22/30.15 = { by lemma 29 }
% 231.22/30.15 top
% 231.22/30.15
% 231.22/30.15 Lemma 43: meet(X, zero) = zero.
% 231.22/30.15 Proof:
% 231.22/30.15 meet(X, zero)
% 231.22/30.15 = { by axiom 11 (maddux4_definiton_of_meet) }
% 231.22/30.15 complement(join(complement(X), complement(zero)))
% 231.22/30.15 = { by lemma 42 }
% 231.22/30.15 complement(top)
% 231.22/30.15 = { by lemma 17 }
% 231.22/30.15 zero
% 231.22/30.15
% 231.22/30.15 Lemma 44: join(meet(X, Y), meet(X, complement(Y))) = X.
% 231.22/30.15 Proof:
% 231.22/30.15 join(meet(X, Y), meet(X, complement(Y)))
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 join(meet(X, complement(Y)), meet(X, Y))
% 231.22/30.15 = { by axiom 11 (maddux4_definiton_of_meet) }
% 231.22/30.15 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 231.22/30.15 = { by lemma 32 }
% 231.22/30.15 X
% 231.22/30.15
% 231.22/30.15 Lemma 45: meet(X, top) = X.
% 231.22/30.15 Proof:
% 231.22/30.15 meet(X, top)
% 231.22/30.15 = { by lemma 41 R->L }
% 231.22/30.15 complement(join(zero, complement(X)))
% 231.22/30.15 = { by lemma 37 R->L }
% 231.22/30.15 join(zero, complement(join(zero, complement(X))))
% 231.22/30.15 = { by lemma 41 }
% 231.22/30.15 join(zero, meet(X, top))
% 231.22/30.15 = { by lemma 42 R->L }
% 231.22/30.15 join(zero, meet(X, join(complement(zero), complement(zero))))
% 231.22/30.15 = { by lemma 23 }
% 231.22/30.15 join(zero, meet(X, complement(zero)))
% 231.22/30.15 = { by lemma 43 R->L }
% 231.22/30.15 join(meet(X, zero), meet(X, complement(zero)))
% 231.22/30.15 = { by lemma 44 }
% 231.22/30.15 X
% 231.22/30.15
% 231.22/30.15 Lemma 46: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 231.22/30.15 Proof:
% 231.22/30.15 join(meet(X, Y), meet(X, Y))
% 231.22/30.15 = { by lemma 40 }
% 231.22/30.15 join(meet(Y, X), meet(X, Y))
% 231.22/30.15 = { by lemma 40 }
% 231.22/30.15 join(meet(Y, X), meet(Y, X))
% 231.22/30.15 = { by axiom 11 (maddux4_definiton_of_meet) }
% 231.22/30.15 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 231.22/30.15 = { by axiom 11 (maddux4_definiton_of_meet) }
% 231.22/30.15 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 231.22/30.15 = { by lemma 23 }
% 231.22/30.15 complement(join(complement(Y), complement(X)))
% 231.22/30.15 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 231.22/30.15 meet(Y, X)
% 231.22/30.15 = { by lemma 40 R->L }
% 231.22/30.15 meet(X, Y)
% 231.22/30.15
% 231.22/30.15 Lemma 47: join(X, X) = X.
% 231.22/30.15 Proof:
% 231.22/30.15 join(X, X)
% 231.22/30.15 = { by lemma 45 R->L }
% 231.22/30.15 join(X, meet(X, top))
% 231.22/30.15 = { by lemma 45 R->L }
% 231.22/30.15 join(meet(X, top), meet(X, top))
% 231.22/30.15 = { by lemma 46 }
% 231.22/30.15 meet(X, top)
% 231.22/30.15 = { by lemma 45 }
% 231.22/30.15 X
% 231.22/30.15
% 231.22/30.15 Lemma 48: converse(zero) = zero.
% 231.22/30.15 Proof:
% 231.22/30.15 converse(zero)
% 231.22/30.15 = { by lemma 39 R->L }
% 231.22/30.15 join(zero, converse(zero))
% 231.22/30.15 = { by lemma 28 R->L }
% 231.22/30.15 converse(join(converse(zero), zero))
% 231.22/30.15 = { by lemma 36 }
% 231.22/30.15 converse(join(converse(zero), converse(zero)))
% 231.22/30.15 = { by lemma 27 }
% 231.22/30.15 join(zero, converse(converse(zero)))
% 231.22/30.15 = { by axiom 1 (converse_idempotence) }
% 231.22/30.15 join(zero, zero)
% 231.22/30.15 = { by lemma 47 }
% 231.22/30.15 zero
% 231.22/30.15
% 231.22/30.15 Lemma 49: join(top, X) = top.
% 231.22/30.15 Proof:
% 231.22/30.15 join(top, X)
% 231.22/30.15 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.15 join(X, top)
% 231.22/30.15 = { by lemma 25 R->L }
% 231.22/30.15 join(Y, top)
% 231.22/30.15 = { by lemma 26 }
% 231.22/30.15 top
% 231.22/30.15
% 231.22/30.15 Lemma 50: complement(complement(X)) = X.
% 231.22/30.15 Proof:
% 231.22/30.15 complement(complement(X))
% 231.22/30.15 = { by lemma 37 R->L }
% 231.22/30.15 join(zero, complement(complement(X)))
% 231.22/30.15 = { by lemma 35 }
% 231.22/30.15 X
% 231.22/30.15
% 231.22/30.15 Lemma 51: meet(zero, X) = zero.
% 231.22/30.15 Proof:
% 231.22/30.15 meet(zero, X)
% 231.22/30.15 = { by lemma 40 }
% 231.22/30.15 meet(X, zero)
% 231.22/30.15 = { by lemma 43 }
% 231.22/30.15 zero
% 231.22/30.15
% 231.22/30.15 Lemma 52: composition(top, zero) = zero.
% 231.22/30.15 Proof:
% 231.22/30.15 composition(top, zero)
% 231.22/30.15 = { by lemma 31 R->L }
% 231.22/30.15 composition(converse(top), zero)
% 231.22/30.15 = { by lemma 39 R->L }
% 231.22/30.15 join(zero, composition(converse(top), zero))
% 231.22/30.15 = { by lemma 17 R->L }
% 231.22/30.15 join(complement(top), composition(converse(top), zero))
% 231.22/30.15 = { by lemma 17 R->L }
% 231.22/30.15 join(complement(top), composition(converse(top), complement(top)))
% 231.22/30.15 = { by lemma 49 R->L }
% 231.22/30.15 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 231.22/30.15 = { by lemma 31 R->L }
% 231.22/30.15 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 231.22/30.16 = { by lemma 21 R->L }
% 231.22/30.16 join(complement(top), composition(converse(top), complement(join(composition(one, top), composition(converse(top), top)))))
% 231.22/30.16 = { by axiom 12 (composition_distributivity) R->L }
% 231.22/30.16 join(complement(top), composition(converse(top), complement(composition(join(one, converse(top)), top))))
% 231.22/30.16 = { by lemma 30 }
% 231.22/30.16 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 231.22/30.16 = { by lemma 22 }
% 231.22/30.16 complement(top)
% 231.22/30.16 = { by lemma 17 }
% 231.22/30.16 zero
% 231.22/30.16
% 231.22/30.16 Lemma 53: composition(X, zero) = zero.
% 231.22/30.16 Proof:
% 231.22/30.16 composition(X, zero)
% 231.22/30.16 = { by lemma 39 R->L }
% 231.22/30.16 join(zero, composition(X, zero))
% 231.22/30.16 = { by lemma 52 R->L }
% 231.22/30.16 join(composition(top, zero), composition(X, zero))
% 231.22/30.16 = { by axiom 12 (composition_distributivity) R->L }
% 231.22/30.16 composition(join(top, X), zero)
% 231.22/30.16 = { by lemma 49 }
% 231.22/30.16 composition(top, zero)
% 231.22/30.16 = { by lemma 52 }
% 231.22/30.16 zero
% 231.22/30.16
% 231.22/30.16 Lemma 54: composition(zero, X) = zero.
% 231.22/30.16 Proof:
% 231.22/30.16 composition(zero, X)
% 231.22/30.16 = { by lemma 48 R->L }
% 231.22/30.16 composition(converse(zero), X)
% 231.22/30.16 = { by lemma 19 R->L }
% 231.22/30.16 converse(composition(converse(X), zero))
% 231.22/30.16 = { by lemma 53 }
% 231.22/30.16 converse(zero)
% 231.22/30.16 = { by lemma 48 }
% 231.22/30.16 zero
% 231.22/30.16
% 231.22/30.16 Lemma 55: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 231.22/30.16 Proof:
% 231.22/30.16 join(X, join(Y, Z))
% 231.22/30.16 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.16 join(join(Y, Z), X)
% 231.22/30.16 = { by axiom 8 (maddux2_join_associativity) R->L }
% 231.22/30.16 join(Y, join(Z, X))
% 231.22/30.16 = { by axiom 2 (maddux1_join_commutativity) }
% 231.22/30.16 join(Y, join(X, Z))
% 231.22/30.16
% 231.22/30.16 Lemma 56: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 231.22/30.16 Proof:
% 231.22/30.16 meet(X, join(complement(Y), complement(Z)))
% 231.22/30.16 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.16 meet(X, join(complement(Z), complement(Y)))
% 231.22/30.16 = { by lemma 40 }
% 231.22/30.16 meet(join(complement(Z), complement(Y)), X)
% 231.22/30.16 = { by axiom 11 (maddux4_definiton_of_meet) }
% 231.22/30.16 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 231.22/30.16 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 231.22/30.16 complement(join(meet(Z, Y), complement(X)))
% 231.22/30.16 = { by axiom 2 (maddux1_join_commutativity) }
% 231.22/30.16 complement(join(complement(X), meet(Z, Y)))
% 231.22/30.16 = { by lemma 40 R->L }
% 231.22/30.16 complement(join(complement(X), meet(Y, Z)))
% 231.22/30.16
% 231.22/30.16 Lemma 57: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 231.22/30.16 Proof:
% 231.22/30.16 complement(join(X, complement(Y)))
% 231.22/30.16 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.16 complement(join(complement(Y), X))
% 231.22/30.16 = { by lemma 45 R->L }
% 231.22/30.16 complement(join(complement(Y), meet(X, top)))
% 231.22/30.16 = { by lemma 40 R->L }
% 231.22/30.16 complement(join(complement(Y), meet(top, X)))
% 231.22/30.16 = { by lemma 56 R->L }
% 231.22/30.16 meet(Y, join(complement(top), complement(X)))
% 231.22/30.16 = { by lemma 17 }
% 231.22/30.16 meet(Y, join(zero, complement(X)))
% 231.22/30.16 = { by lemma 37 }
% 231.22/30.16 meet(Y, complement(X))
% 231.22/30.16
% 231.22/30.16 Lemma 58: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 231.22/30.16 Proof:
% 231.22/30.16 complement(meet(X, complement(Y)))
% 231.22/30.16 = { by lemma 39 R->L }
% 231.22/30.16 complement(join(zero, meet(X, complement(Y))))
% 231.22/30.16 = { by lemma 57 R->L }
% 231.22/30.16 complement(join(zero, complement(join(Y, complement(X)))))
% 231.22/30.16 = { by lemma 41 }
% 231.22/30.16 meet(join(Y, complement(X)), top)
% 231.22/30.16 = { by lemma 45 }
% 231.22/30.16 join(Y, complement(X))
% 231.22/30.16
% 231.22/30.16 Lemma 59: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 231.22/30.16 Proof:
% 231.22/30.16 complement(meet(complement(X), Y))
% 231.22/30.16 = { by lemma 40 }
% 231.22/30.16 complement(meet(Y, complement(X)))
% 231.22/30.16 = { by lemma 58 }
% 231.22/30.16 join(X, complement(Y))
% 231.22/30.16
% 231.22/30.16 Lemma 60: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 231.22/30.16 Proof:
% 231.22/30.16 complement(join(complement(X), Y))
% 231.22/30.16 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.16 complement(join(Y, complement(X)))
% 231.22/30.16 = { by lemma 57 }
% 231.22/30.16 meet(X, complement(Y))
% 231.22/30.16
% 231.22/30.16 Lemma 61: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 231.22/30.16 Proof:
% 231.22/30.16 join(complement(X), complement(Y))
% 231.22/30.16 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.16 join(complement(Y), complement(X))
% 231.22/30.16 = { by lemma 33 R->L }
% 231.22/30.16 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 231.22/30.16 = { by lemma 56 }
% 231.22/30.16 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 231.22/30.16 = { by lemma 37 }
% 231.22/30.16 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 231.22/30.16 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 231.22/30.16 complement(join(meet(Y, X), meet(Y, X)))
% 231.22/30.16 = { by lemma 46 }
% 231.22/30.16 complement(meet(Y, X))
% 231.22/30.16 = { by lemma 40 R->L }
% 231.22/30.16 complement(meet(X, Y))
% 231.22/30.16
% 231.22/30.16 Lemma 62: join(X, complement(meet(X, Y))) = top.
% 231.22/30.16 Proof:
% 231.22/30.16 join(X, complement(meet(X, Y)))
% 231.22/30.16 = { by lemma 40 }
% 231.22/30.16 join(X, complement(meet(Y, X)))
% 231.22/30.16 = { by lemma 61 R->L }
% 231.22/30.16 join(X, join(complement(Y), complement(X)))
% 231.22/30.16 = { by lemma 18 }
% 231.22/30.16 join(complement(Y), top)
% 231.22/30.16 = { by lemma 26 }
% 231.22/30.16 top
% 231.22/30.16
% 231.22/30.16 Lemma 63: meet(X, join(X, complement(Y))) = X.
% 231.22/30.16 Proof:
% 231.22/30.16 meet(X, join(X, complement(Y)))
% 231.22/30.16 = { by lemma 58 R->L }
% 231.22/30.16 meet(X, complement(meet(Y, complement(X))))
% 231.22/30.16 = { by lemma 60 R->L }
% 231.22/30.16 complement(join(complement(X), meet(Y, complement(X))))
% 231.22/30.16 = { by lemma 37 R->L }
% 231.22/30.16 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 231.22/30.16 = { by lemma 17 R->L }
% 231.22/30.16 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 231.22/30.16 = { by lemma 62 R->L }
% 231.22/30.16 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 231.22/30.16 = { by axiom 11 (maddux4_definiton_of_meet) R->L }
% 231.22/30.16 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 231.22/30.16 = { by lemma 40 R->L }
% 231.22/30.16 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 231.22/30.16 = { by lemma 32 }
% 231.22/30.16 X
% 231.22/30.16
% 231.22/30.16 Lemma 64: join(X, meet(X, Y)) = X.
% 231.22/30.16 Proof:
% 231.22/30.16 join(X, meet(X, Y))
% 231.22/30.16 = { by axiom 11 (maddux4_definiton_of_meet) }
% 231.22/30.16 join(X, complement(join(complement(X), complement(Y))))
% 231.22/30.16 = { by lemma 59 R->L }
% 231.22/30.16 complement(meet(complement(X), join(complement(X), complement(Y))))
% 231.22/30.16 = { by lemma 63 }
% 231.22/30.16 complement(complement(X))
% 231.22/30.16 = { by lemma 50 }
% 231.22/30.16 X
% 231.22/30.16
% 231.22/30.16 Lemma 65: join(Z, join(X, Y)) = join(X, join(Y, Z)).
% 231.22/30.16 Proof:
% 231.22/30.16 join(Z, join(X, Y))
% 231.22/30.16 = { by lemma 55 R->L }
% 231.22/30.16 join(X, join(Z, Y))
% 231.22/30.16 = { by axiom 2 (maddux1_join_commutativity) }
% 231.22/30.16 join(X, join(Y, Z))
% 231.22/30.16
% 231.22/30.16 Lemma 66: join(Z, join(Y, X)) = join(X, join(Y, Z)).
% 231.22/30.16 Proof:
% 231.22/30.16 join(Z, join(Y, X))
% 231.22/30.16 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.22/30.16 join(join(Y, X), Z)
% 231.22/30.16 = { by axiom 2 (maddux1_join_commutativity) }
% 231.22/30.16 join(join(X, Y), Z)
% 231.22/30.16 = { by axiom 8 (maddux2_join_associativity) R->L }
% 231.22/30.16 join(X, join(Y, Z))
% 231.22/30.16
% 231.22/30.16 Lemma 67: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 231.22/30.16 Proof:
% 231.22/30.16 meet(complement(X), complement(Y))
% 231.22/30.16 = { by lemma 40 }
% 231.22/30.16 meet(complement(Y), complement(X))
% 231.22/30.16 = { by lemma 37 R->L }
% 231.22/30.16 meet(join(zero, complement(Y)), complement(X))
% 231.22/30.16 = { by lemma 57 R->L }
% 231.22/30.16 complement(join(X, complement(join(zero, complement(Y)))))
% 231.22/30.16 = { by lemma 41 }
% 231.22/30.16 complement(join(X, meet(Y, top)))
% 231.22/30.16 = { by lemma 45 }
% 231.22/30.16 complement(join(X, Y))
% 231.22/30.16
% 231.22/30.16 Lemma 68: join(X, complement(meet(Y, X))) = top.
% 231.95/30.16 Proof:
% 231.95/30.16 join(X, complement(meet(Y, X)))
% 231.95/30.16 = { by lemma 40 }
% 231.95/30.16 join(X, complement(meet(X, Y)))
% 231.95/30.16 = { by lemma 62 }
% 231.95/30.16 top
% 231.95/30.16
% 231.95/30.16 Lemma 69: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 231.95/30.16 Proof:
% 231.95/30.16 join(meet(X, Y), meet(Y, complement(X)))
% 231.95/30.16 = { by lemma 40 }
% 231.95/30.16 join(meet(Y, X), meet(Y, complement(X)))
% 231.95/30.16 = { by lemma 44 }
% 231.95/30.16 Y
% 231.95/30.16
% 231.95/30.16 Lemma 70: join(meet(X, Y), Y) = Y.
% 231.95/30.16 Proof:
% 231.95/30.16 join(meet(X, Y), Y)
% 231.95/30.16 = { by lemma 45 R->L }
% 231.95/30.16 meet(join(meet(X, Y), Y), top)
% 231.95/30.16 = { by lemma 41 R->L }
% 231.95/30.16 complement(join(zero, complement(join(meet(X, Y), Y))))
% 231.95/30.16 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.95/30.16 complement(join(zero, complement(join(Y, meet(X, Y)))))
% 231.95/30.16 = { by lemma 67 R->L }
% 231.95/30.16 complement(join(zero, meet(complement(Y), complement(meet(X, Y)))))
% 231.95/30.16 = { by lemma 39 R->L }
% 231.95/30.16 complement(join(zero, join(zero, meet(complement(Y), complement(meet(X, Y))))))
% 231.95/30.16 = { by lemma 17 R->L }
% 231.95/30.16 complement(join(zero, join(complement(top), meet(complement(Y), complement(meet(X, Y))))))
% 231.95/30.16 = { by lemma 68 R->L }
% 231.95/30.16 complement(join(zero, join(complement(join(Y, complement(meet(X, Y)))), meet(complement(Y), complement(meet(X, Y))))))
% 231.95/30.16 = { by lemma 57 }
% 231.95/30.16 complement(join(zero, join(meet(meet(X, Y), complement(Y)), meet(complement(Y), complement(meet(X, Y))))))
% 231.95/30.16 = { by lemma 69 }
% 231.95/30.16 complement(join(zero, complement(Y)))
% 231.95/30.16 = { by lemma 41 }
% 231.95/30.16 meet(Y, top)
% 231.95/30.16 = { by lemma 45 }
% 231.95/30.16 Y
% 231.95/30.16
% 231.95/30.16 Lemma 71: meet(X, join(X, Y)) = X.
% 231.95/30.16 Proof:
% 231.95/30.16 meet(X, join(X, Y))
% 231.95/30.16 = { by lemma 45 R->L }
% 231.95/30.16 meet(X, join(X, meet(Y, top)))
% 231.95/30.16 = { by lemma 41 R->L }
% 231.95/30.16 meet(X, join(X, complement(join(zero, complement(Y)))))
% 231.95/30.16 = { by lemma 63 }
% 231.95/30.16 X
% 231.95/30.16
% 231.95/30.16 Lemma 72: meet(complement(Z), meet(Y, X)) = meet(X, meet(Y, complement(Z))).
% 231.95/30.16 Proof:
% 231.95/30.16 meet(complement(Z), meet(Y, X))
% 231.95/30.16 = { by lemma 40 }
% 231.95/30.16 meet(complement(Z), meet(X, Y))
% 231.95/30.16 = { by lemma 40 }
% 231.95/30.16 meet(meet(X, Y), complement(Z))
% 231.95/30.16 = { by axiom 11 (maddux4_definiton_of_meet) }
% 231.95/30.16 meet(complement(join(complement(X), complement(Y))), complement(Z))
% 231.95/30.16 = { by lemma 67 }
% 231.95/30.16 complement(join(join(complement(X), complement(Y)), Z))
% 231.95/30.16 = { by axiom 8 (maddux2_join_associativity) R->L }
% 231.95/30.16 complement(join(complement(X), join(complement(Y), Z)))
% 231.95/30.16 = { by lemma 60 }
% 231.95/30.16 meet(X, complement(join(complement(Y), Z)))
% 231.95/30.16 = { by lemma 60 }
% 231.95/30.16 meet(X, meet(Y, complement(Z)))
% 231.95/30.16
% 231.95/30.16 Lemma 73: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 231.95/30.16 Proof:
% 231.95/30.16 meet(Y, meet(Z, X))
% 231.95/30.16 = { by lemma 45 R->L }
% 231.95/30.16 meet(meet(Y, top), meet(Z, X))
% 231.95/30.16 = { by lemma 41 R->L }
% 231.95/30.16 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 231.95/30.16 = { by lemma 72 }
% 231.95/30.16 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 231.95/30.16 = { by lemma 41 }
% 231.95/30.16 meet(X, meet(Z, meet(Y, top)))
% 231.95/30.16 = { by lemma 45 }
% 231.95/30.16 meet(X, meet(Z, Y))
% 231.95/30.16 = { by lemma 40 R->L }
% 231.95/30.16 meet(X, meet(Y, Z))
% 231.95/30.16
% 231.95/30.16 Lemma 74: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 231.95/30.16 Proof:
% 231.95/30.16 meet(Y, meet(X, Z))
% 231.95/30.16 = { by lemma 40 }
% 231.95/30.16 meet(Y, meet(Z, X))
% 231.95/30.16 = { by lemma 73 }
% 231.95/30.16 meet(X, meet(Y, Z))
% 231.95/30.16
% 231.95/30.16 Lemma 75: join(complement(X), meet(Y, complement(Z))) = complement(meet(X, join(Z, complement(Y)))).
% 231.95/30.16 Proof:
% 231.95/30.16 join(complement(X), meet(Y, complement(Z)))
% 231.95/30.16 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.95/30.16 join(meet(Y, complement(Z)), complement(X))
% 231.95/30.16 = { by lemma 57 R->L }
% 231.95/30.16 join(complement(join(Z, complement(Y))), complement(X))
% 231.95/30.16 = { by lemma 61 }
% 231.95/30.16 complement(meet(join(Z, complement(Y)), X))
% 231.95/30.16 = { by lemma 40 R->L }
% 231.95/30.16 complement(meet(X, join(Z, complement(Y))))
% 231.95/30.16
% 231.95/30.16 Lemma 76: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 231.95/30.16 Proof:
% 231.95/30.16 join(meet(X, Y), meet(complement(X), Y))
% 231.95/30.16 = { by lemma 40 }
% 231.95/30.16 join(meet(X, Y), meet(Y, complement(X)))
% 231.95/30.16 = { by lemma 69 }
% 231.95/30.16 Y
% 231.95/30.16
% 231.95/30.16 Lemma 77: complement(meet(Y, join(X, complement(Y)))) = complement(meet(X, join(Y, complement(X)))).
% 231.95/30.16 Proof:
% 231.95/30.16 complement(meet(Y, join(X, complement(Y))))
% 231.95/30.16 = { by lemma 75 R->L }
% 231.95/30.16 join(complement(Y), meet(Y, complement(X)))
% 231.95/30.16 = { by axiom 2 (maddux1_join_commutativity) }
% 231.95/30.16 join(meet(Y, complement(X)), complement(Y))
% 231.95/30.16 = { by lemma 76 R->L }
% 231.95/30.16 join(meet(Y, complement(X)), join(meet(X, complement(Y)), meet(complement(X), complement(Y))))
% 231.95/30.16 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.95/30.16 join(meet(Y, complement(X)), join(meet(complement(X), complement(Y)), meet(X, complement(Y))))
% 231.95/30.16 = { by axiom 8 (maddux2_join_associativity) }
% 231.95/30.16 join(join(meet(Y, complement(X)), meet(complement(X), complement(Y))), meet(X, complement(Y)))
% 231.95/30.16 = { by lemma 69 }
% 231.95/30.16 join(complement(X), meet(X, complement(Y)))
% 231.95/30.16 = { by lemma 75 }
% 231.95/30.16 complement(meet(X, join(Y, complement(X))))
% 231.95/30.16
% 231.95/30.17 Lemma 78: join(complement(Y), meet(Y, X)) = join(X, complement(join(X, Y))).
% 231.95/30.17 Proof:
% 231.95/30.17 join(complement(Y), meet(Y, X))
% 231.95/30.17 = { by lemma 40 }
% 231.95/30.17 join(complement(Y), meet(X, Y))
% 231.95/30.17 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.95/30.17 join(meet(X, Y), complement(Y))
% 231.95/30.17 = { by lemma 58 R->L }
% 231.95/30.17 complement(meet(Y, complement(meet(X, Y))))
% 231.95/30.17 = { by lemma 61 R->L }
% 231.95/30.17 complement(meet(Y, join(complement(X), complement(Y))))
% 231.95/30.17 = { by lemma 77 R->L }
% 231.95/30.17 complement(meet(complement(X), join(Y, complement(complement(X)))))
% 231.95/30.17 = { by lemma 59 }
% 231.95/30.17 join(X, complement(join(Y, complement(complement(X)))))
% 231.95/30.17 = { by lemma 57 }
% 231.95/30.17 join(X, meet(complement(X), complement(Y)))
% 231.95/30.17 = { by lemma 67 }
% 231.95/30.17 join(X, complement(join(X, Y)))
% 231.95/30.17
% 231.95/30.17 Lemma 79: join(X, complement(join(X, Y))) = join(X, complement(Y)).
% 231.95/30.17 Proof:
% 231.95/30.17 join(X, complement(join(X, Y)))
% 231.95/30.17 = { by lemma 67 R->L }
% 231.95/30.17 join(X, meet(complement(X), complement(Y)))
% 231.95/30.17 = { by lemma 57 R->L }
% 231.95/30.17 join(X, complement(join(Y, complement(complement(X)))))
% 231.95/30.17 = { by lemma 59 R->L }
% 231.95/30.17 complement(meet(complement(X), join(Y, complement(complement(X)))))
% 231.95/30.17 = { by lemma 77 R->L }
% 231.95/30.17 complement(meet(Y, join(complement(X), complement(Y))))
% 231.95/30.17 = { by lemma 61 }
% 231.95/30.17 complement(meet(Y, complement(meet(X, Y))))
% 231.95/30.17 = { by lemma 58 }
% 231.95/30.17 join(meet(X, Y), complement(Y))
% 231.95/30.17 = { by lemma 76 R->L }
% 231.95/30.17 join(meet(X, Y), complement(join(meet(X, Y), meet(complement(X), Y))))
% 231.95/30.17 = { by lemma 78 R->L }
% 231.95/30.17 join(complement(meet(complement(X), Y)), meet(meet(complement(X), Y), meet(X, Y)))
% 231.95/30.17 = { by lemma 59 }
% 231.95/30.17 join(join(X, complement(Y)), meet(meet(complement(X), Y), meet(X, Y)))
% 231.95/30.17 = { by axiom 8 (maddux2_join_associativity) R->L }
% 231.95/30.17 join(X, join(complement(Y), meet(meet(complement(X), Y), meet(X, Y))))
% 231.95/30.17 = { by lemma 40 }
% 231.95/30.17 join(X, join(complement(Y), meet(meet(X, Y), meet(complement(X), Y))))
% 231.95/30.17 = { by lemma 73 R->L }
% 231.95/30.17 join(X, join(complement(Y), meet(complement(X), meet(Y, meet(X, Y)))))
% 231.95/30.17 = { by lemma 40 }
% 231.95/30.17 join(X, join(complement(Y), meet(meet(Y, meet(X, Y)), complement(X))))
% 231.95/30.17 = { by lemma 75 }
% 231.95/30.17 join(X, complement(meet(Y, join(X, complement(meet(Y, meet(X, Y)))))))
% 231.95/30.17 = { by lemma 40 }
% 231.95/30.17 join(X, complement(meet(Y, join(X, complement(meet(Y, meet(Y, X)))))))
% 231.95/30.17 = { by lemma 40 }
% 231.95/30.17 join(X, complement(meet(Y, join(X, complement(meet(meet(Y, X), Y))))))
% 231.95/30.17 = { by lemma 32 R->L }
% 231.95/30.17 join(X, complement(meet(Y, join(X, complement(meet(meet(Y, X), join(meet(Y, X), complement(join(complement(Y), X)))))))))
% 231.95/30.17 = { by lemma 63 }
% 231.95/30.17 join(X, complement(meet(Y, join(X, complement(meet(Y, X))))))
% 231.95/30.17 = { by lemma 68 }
% 231.95/30.17 join(X, complement(meet(Y, top)))
% 231.95/30.17 = { by lemma 45 }
% 231.95/30.17 join(X, complement(Y))
% 231.95/30.17
% 231.95/30.17 Lemma 80: meet(complement(X), join(Y, complement(Z))) = complement(join(X, meet(Z, complement(Y)))).
% 231.95/30.17 Proof:
% 231.95/30.17 meet(complement(X), join(Y, complement(Z)))
% 231.95/30.17 = { by lemma 40 }
% 231.95/30.17 meet(join(Y, complement(Z)), complement(X))
% 231.95/30.17 = { by lemma 57 R->L }
% 231.95/30.17 complement(join(X, complement(join(Y, complement(Z)))))
% 231.95/30.17 = { by lemma 57 }
% 231.95/30.17 complement(join(X, meet(Z, complement(Y))))
% 231.95/30.17
% 231.95/30.17 Lemma 81: meet(complement(X), join(Y, X)) = meet(Y, complement(X)).
% 231.95/30.17 Proof:
% 231.95/30.17 meet(complement(X), join(Y, X))
% 231.95/30.17 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.95/30.17 meet(complement(X), join(X, Y))
% 231.95/30.17 = { by lemma 40 }
% 231.95/30.17 meet(join(X, Y), complement(X))
% 231.95/30.17 = { by lemma 57 R->L }
% 231.95/30.17 complement(join(X, complement(join(X, Y))))
% 231.95/30.17 = { by lemma 78 R->L }
% 231.95/30.17 complement(join(complement(Y), meet(Y, X)))
% 231.95/30.17 = { by lemma 60 }
% 231.95/30.17 meet(Y, complement(meet(Y, X)))
% 231.95/30.17 = { by lemma 61 R->L }
% 231.95/30.17 meet(Y, join(complement(Y), complement(X)))
% 231.95/30.17 = { by lemma 58 R->L }
% 231.95/30.17 meet(Y, complement(meet(X, complement(complement(Y)))))
% 231.95/30.17 = { by lemma 60 R->L }
% 231.95/30.17 complement(join(complement(Y), meet(X, complement(complement(Y)))))
% 231.95/30.17 = { by lemma 50 R->L }
% 231.95/30.17 complement(join(complement(Y), meet(X, complement(complement(complement(complement(Y)))))))
% 231.95/30.17 = { by lemma 60 R->L }
% 231.95/30.17 complement(join(complement(Y), complement(join(complement(X), complement(complement(complement(Y)))))))
% 231.95/30.17 = { by lemma 59 R->L }
% 231.95/30.17 complement(complement(meet(complement(complement(Y)), join(complement(X), complement(complement(complement(Y)))))))
% 231.95/30.17 = { by lemma 77 }
% 231.95/30.17 complement(complement(meet(complement(X), join(complement(complement(Y)), complement(complement(X))))))
% 231.95/30.17 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.95/30.17 complement(complement(meet(complement(X), join(complement(complement(X)), complement(complement(Y))))))
% 231.95/30.17 = { by lemma 80 }
% 231.95/30.17 complement(complement(complement(join(X, meet(complement(Y), complement(complement(complement(X))))))))
% 231.95/30.17 = { by lemma 50 }
% 231.95/30.17 complement(join(X, meet(complement(Y), complement(complement(complement(X))))))
% 231.95/30.17 = { by lemma 50 }
% 231.95/30.17 complement(join(X, meet(complement(Y), complement(X))))
% 231.95/30.17 = { by lemma 57 R->L }
% 231.95/30.17 complement(join(X, complement(join(X, complement(complement(Y))))))
% 231.95/30.17 = { by lemma 79 }
% 231.95/30.17 complement(join(X, complement(complement(complement(Y)))))
% 231.95/30.17 = { by lemma 50 }
% 231.95/30.17 complement(join(X, complement(Y)))
% 231.95/30.17 = { by lemma 57 }
% 231.95/30.17 meet(Y, complement(X))
% 231.95/30.17
% 231.95/30.17 Lemma 82: complement(join(X, join(Y, complement(Z)))) = meet(Z, complement(join(X, Y))).
% 231.95/30.17 Proof:
% 231.95/30.17 complement(join(X, join(Y, complement(Z))))
% 231.95/30.17 = { by lemma 55 R->L }
% 231.95/30.17 complement(join(Y, join(X, complement(Z))))
% 231.95/30.17 = { by axiom 8 (maddux2_join_associativity) }
% 231.95/30.17 complement(join(join(Y, X), complement(Z)))
% 231.95/30.17 = { by lemma 57 }
% 231.95/30.17 meet(Z, complement(join(Y, X)))
% 231.95/30.17 = { by axiom 2 (maddux1_join_commutativity) }
% 231.95/30.17 meet(Z, complement(join(X, Y)))
% 231.95/30.17
% 231.95/30.17 Lemma 83: meet(X, join(Y, join(X, Z))) = X.
% 231.95/30.17 Proof:
% 231.95/30.17 meet(X, join(Y, join(X, Z)))
% 231.95/30.17 = { by lemma 55 R->L }
% 231.95/30.17 meet(X, join(X, join(Y, Z)))
% 231.95/30.17 = { by lemma 71 }
% 231.95/30.17 X
% 231.95/30.17
% 231.95/30.17 Lemma 84: meet(meet(x0, X), complement(composition(x0, top))) = zero.
% 231.95/30.17 Proof:
% 231.95/30.17 meet(meet(x0, X), complement(composition(x0, top)))
% 231.95/30.17 = { by lemma 57 R->L }
% 231.95/30.17 complement(join(composition(x0, top), complement(meet(x0, X))))
% 231.95/30.17 = { by axiom 4 (goals) R->L }
% 231.95/30.17 complement(join(composition(x0, top), complement(meet(composition(x0, top), X))))
% 231.95/30.17 = { by lemma 62 }
% 231.95/30.17 complement(top)
% 231.95/30.17 = { by lemma 17 }
% 231.95/30.17 zero
% 231.95/30.17
% 231.95/30.17 Lemma 85: meet(meet(x0, X), composition(x0, top)) = meet(x0, X).
% 231.95/30.17 Proof:
% 231.95/30.17 meet(meet(x0, X), composition(x0, top))
% 231.95/30.17 = { by axiom 11 (maddux4_definiton_of_meet) }
% 231.95/30.17 complement(join(complement(meet(x0, X)), complement(composition(x0, top))))
% 231.95/30.17 = { by lemma 37 R->L }
% 231.95/30.17 join(zero, complement(join(complement(meet(x0, X)), complement(composition(x0, top)))))
% 231.95/30.17 = { by lemma 84 R->L }
% 231.95/30.17 join(meet(meet(x0, X), complement(composition(x0, top))), complement(join(complement(meet(x0, X)), complement(composition(x0, top)))))
% 231.95/30.17 = { by lemma 32 }
% 231.95/30.17 meet(x0, X)
% 231.95/30.17
% 231.95/30.17 Lemma 86: meet(composition(x0, top), meet(X, Y)) = meet(Y, meet(x0, X)).
% 231.95/30.17 Proof:
% 231.95/30.17 meet(composition(x0, top), meet(X, Y))
% 231.95/30.17 = { by lemma 73 }
% 231.95/30.17 meet(Y, meet(composition(x0, top), X))
% 231.95/30.17 = { by axiom 4 (goals) }
% 231.95/30.17 meet(Y, meet(x0, X))
% 231.95/30.17
% 231.95/30.17 Lemma 87: meet(composition(x0, top), meet(X, Y)) = meet(X, meet(x0, Y)).
% 231.95/30.17 Proof:
% 231.95/30.17 meet(composition(x0, top), meet(X, Y))
% 231.95/30.17 = { by lemma 40 }
% 231.95/30.17 meet(composition(x0, top), meet(Y, X))
% 231.95/30.17 = { by lemma 86 }
% 231.95/30.17 meet(X, meet(x0, Y))
% 231.95/30.17
% 231.95/30.17 Lemma 88: join(X, join(Y, complement(join(X, Y)))) = top.
% 231.95/30.17 Proof:
% 231.95/30.17 join(X, join(Y, complement(join(X, Y))))
% 231.95/30.17 = { by axiom 8 (maddux2_join_associativity) }
% 231.95/30.17 join(join(X, Y), complement(join(X, Y)))
% 231.95/30.17 = { by axiom 5 (def_top) R->L }
% 231.95/30.17 top
% 231.95/30.17
% 231.95/30.17 Lemma 89: join(meet(x0, X), meet(composition(x0, top), complement(X))) = composition(x0, top).
% 231.95/30.17 Proof:
% 231.95/30.17 join(meet(x0, X), meet(composition(x0, top), complement(X)))
% 231.95/30.17 = { by axiom 4 (goals) R->L }
% 231.95/30.17 join(meet(composition(x0, top), X), meet(composition(x0, top), complement(X)))
% 231.95/30.17 = { by lemma 44 }
% 231.95/30.17 composition(x0, top)
% 231.95/30.17
% 231.95/30.17 Lemma 90: join(meet(x0, X), complement(composition(x0, top))) = join(X, complement(composition(x0, top))).
% 231.95/30.17 Proof:
% 231.95/30.17 join(meet(x0, X), complement(composition(x0, top)))
% 231.95/30.17 = { by lemma 64 R->L }
% 231.95/30.17 join(meet(x0, X), complement(join(composition(x0, top), meet(composition(x0, top), complement(X)))))
% 231.95/30.17 = { by lemma 80 R->L }
% 231.95/30.17 join(meet(x0, X), meet(complement(composition(x0, top)), join(X, complement(composition(x0, top)))))
% 231.95/30.17 = { by lemma 40 R->L }
% 231.95/30.17 join(meet(x0, X), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.17 = { by axiom 4 (goals) R->L }
% 231.95/30.17 join(meet(composition(x0, top), X), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.17 = { by lemma 71 R->L }
% 231.95/30.17 join(meet(composition(x0, top), meet(X, join(X, complement(composition(x0, top))))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.17 = { by lemma 86 }
% 231.95/30.17 join(meet(join(X, complement(composition(x0, top))), meet(x0, X)), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.17 = { by lemma 58 R->L }
% 231.95/30.17 join(meet(complement(meet(composition(x0, top), complement(X))), meet(x0, X)), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.17 = { by lemma 85 R->L }
% 231.95/30.17 join(meet(complement(meet(composition(x0, top), complement(X))), meet(meet(x0, X), composition(x0, top))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.17 = { by lemma 74 R->L }
% 231.95/30.17 join(meet(meet(x0, X), meet(complement(meet(composition(x0, top), complement(X))), composition(x0, top))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.17 = { by lemma 89 R->L }
% 231.95/30.17 join(meet(meet(x0, X), meet(complement(meet(composition(x0, top), complement(X))), join(meet(x0, X), meet(composition(x0, top), complement(X))))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.17 = { by lemma 40 }
% 231.95/30.17 join(meet(meet(x0, X), meet(join(meet(x0, X), meet(composition(x0, top), complement(X))), complement(meet(composition(x0, top), complement(X))))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.17 = { by lemma 57 R->L }
% 231.95/30.17 join(meet(meet(x0, X), complement(join(meet(composition(x0, top), complement(X)), complement(join(meet(x0, X), meet(composition(x0, top), complement(X))))))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.17 = { by lemma 67 R->L }
% 231.95/30.17 join(meet(meet(x0, X), complement(join(meet(composition(x0, top), complement(X)), meet(complement(meet(x0, X)), complement(meet(composition(x0, top), complement(X))))))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.17 = { by lemma 38 R->L }
% 231.95/30.17 join(join(meet(meet(x0, X), complement(join(meet(composition(x0, top), complement(X)), meet(complement(meet(x0, X)), complement(meet(composition(x0, top), complement(X))))))), zero), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.17 = { by axiom 6 (def_zero) }
% 231.95/30.17 join(join(meet(meet(x0, X), complement(join(meet(composition(x0, top), complement(X)), meet(complement(meet(x0, X)), complement(meet(composition(x0, top), complement(X))))))), meet(join(meet(composition(x0, top), complement(X)), complement(complement(meet(x0, X)))), complement(join(meet(composition(x0, top), complement(X)), complement(complement(meet(x0, X))))))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 57 }
% 231.95/30.18 join(join(meet(meet(x0, X), complement(join(meet(composition(x0, top), complement(X)), meet(complement(meet(x0, X)), complement(meet(composition(x0, top), complement(X))))))), meet(join(meet(composition(x0, top), complement(X)), complement(complement(meet(x0, X)))), meet(complement(meet(x0, X)), complement(meet(composition(x0, top), complement(X)))))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 72 R->L }
% 231.95/30.18 join(join(meet(meet(x0, X), complement(join(meet(composition(x0, top), complement(X)), meet(complement(meet(x0, X)), complement(meet(composition(x0, top), complement(X))))))), meet(complement(meet(composition(x0, top), complement(X))), meet(complement(meet(x0, X)), join(meet(composition(x0, top), complement(X)), complement(complement(meet(x0, X))))))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 74 R->L }
% 231.95/30.18 join(join(meet(meet(x0, X), complement(join(meet(composition(x0, top), complement(X)), meet(complement(meet(x0, X)), complement(meet(composition(x0, top), complement(X))))))), meet(complement(meet(x0, X)), meet(complement(meet(composition(x0, top), complement(X))), join(meet(composition(x0, top), complement(X)), complement(complement(meet(x0, X))))))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 80 }
% 231.95/30.18 join(join(meet(meet(x0, X), complement(join(meet(composition(x0, top), complement(X)), meet(complement(meet(x0, X)), complement(meet(composition(x0, top), complement(X))))))), meet(complement(meet(x0, X)), complement(join(meet(composition(x0, top), complement(X)), meet(complement(meet(x0, X)), complement(meet(composition(x0, top), complement(X)))))))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 76 }
% 231.95/30.18 join(complement(join(meet(composition(x0, top), complement(X)), meet(complement(meet(x0, X)), complement(meet(composition(x0, top), complement(X)))))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 67 }
% 231.95/30.18 join(complement(join(meet(composition(x0, top), complement(X)), complement(join(meet(x0, X), meet(composition(x0, top), complement(X)))))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 57 }
% 231.95/30.18 join(meet(join(meet(x0, X), meet(composition(x0, top), complement(X))), complement(meet(composition(x0, top), complement(X)))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 40 R->L }
% 231.95/30.18 join(meet(complement(meet(composition(x0, top), complement(X))), join(meet(x0, X), meet(composition(x0, top), complement(X)))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 58 }
% 231.95/30.18 join(meet(join(X, complement(composition(x0, top))), join(meet(x0, X), meet(composition(x0, top), complement(X)))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 89 }
% 231.95/30.18 join(meet(join(X, complement(composition(x0, top))), composition(x0, top)), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 40 R->L }
% 231.95/30.18 join(meet(composition(x0, top), join(X, complement(composition(x0, top)))), meet(join(X, complement(composition(x0, top))), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 69 }
% 231.95/30.18 join(X, complement(composition(x0, top)))
% 231.95/30.18
% 231.95/30.18 Lemma 91: meet(composition(x0, top), complement(meet(x0, X))) = meet(composition(x0, top), complement(X)).
% 231.95/30.18 Proof:
% 231.95/30.18 meet(composition(x0, top), complement(meet(x0, X)))
% 231.95/30.18 = { by lemma 57 R->L }
% 231.95/30.18 complement(join(meet(x0, X), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 90 }
% 231.95/30.18 complement(join(X, complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 57 }
% 231.95/30.18 meet(composition(x0, top), complement(X))
% 231.95/30.18
% 231.95/30.18 Lemma 92: join(composition(x0, top), meet(meet(x0, X), Y)) = composition(x0, top).
% 231.95/30.18 Proof:
% 231.95/30.18 join(composition(x0, top), meet(meet(x0, X), Y))
% 231.95/30.18 = { by lemma 40 }
% 231.95/30.18 join(composition(x0, top), meet(Y, meet(x0, X)))
% 231.95/30.18 = { by lemma 86 R->L }
% 231.95/30.18 join(composition(x0, top), meet(composition(x0, top), meet(X, Y)))
% 231.95/30.18 = { by lemma 64 }
% 231.95/30.18 composition(x0, top)
% 231.95/30.18
% 231.95/30.18 Lemma 93: join(composition(meet(X, Y), Z), composition(Y, Z)) = composition(Y, Z).
% 231.95/30.18 Proof:
% 231.95/30.18 join(composition(meet(X, Y), Z), composition(Y, Z))
% 231.95/30.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.95/30.18 join(composition(Y, Z), composition(meet(X, Y), Z))
% 231.95/30.18 = { by axiom 12 (composition_distributivity) R->L }
% 231.95/30.18 composition(join(Y, meet(X, Y)), Z)
% 231.95/30.18 = { by axiom 2 (maddux1_join_commutativity) }
% 231.95/30.18 composition(join(meet(X, Y), Y), Z)
% 231.95/30.18 = { by lemma 70 }
% 231.95/30.18 composition(Y, Z)
% 231.95/30.18
% 231.95/30.18 Lemma 94: composition(converse(complement(composition(x0, top))), composition(x0, top)) = zero.
% 231.95/30.18 Proof:
% 231.95/30.18 composition(converse(complement(composition(x0, top))), composition(x0, top))
% 231.95/30.18 = { by lemma 19 R->L }
% 231.95/30.18 converse(composition(converse(composition(x0, top)), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 38 R->L }
% 231.95/30.18 converse(join(composition(converse(composition(x0, top)), complement(composition(x0, top))), zero))
% 231.95/30.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.95/30.18 converse(join(zero, composition(converse(composition(x0, top)), complement(composition(x0, top)))))
% 231.95/30.18 = { by lemma 17 R->L }
% 231.95/30.18 converse(join(complement(top), composition(converse(composition(x0, top)), complement(composition(x0, top)))))
% 231.95/30.18 = { by axiom 4 (goals) R->L }
% 231.95/30.18 converse(join(complement(top), composition(converse(composition(x0, top)), complement(composition(composition(x0, top), top)))))
% 231.95/30.18 = { by lemma 22 }
% 231.95/30.18 converse(complement(top))
% 231.95/30.18 = { by lemma 17 }
% 231.95/30.18 converse(zero)
% 231.95/30.18 = { by lemma 48 }
% 231.95/30.18 zero
% 231.95/30.18
% 231.95/30.18 Lemma 95: meet(meet(x0, X), meet(complement(composition(x0, top)), Y)) = zero.
% 231.95/30.18 Proof:
% 231.95/30.18 meet(meet(x0, X), meet(complement(composition(x0, top)), Y))
% 231.95/30.18 = { by lemma 74 }
% 231.95/30.18 meet(complement(composition(x0, top)), meet(meet(x0, X), Y))
% 231.95/30.18 = { by lemma 72 }
% 231.95/30.18 meet(Y, meet(meet(x0, X), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 84 }
% 231.95/30.18 meet(Y, zero)
% 231.95/30.18 = { by lemma 43 }
% 231.95/30.18 zero
% 231.95/30.18
% 231.95/30.18 Lemma 96: meet(composition(x0, top), composition(complement(composition(x0, top)), X)) = zero.
% 231.95/30.18 Proof:
% 231.95/30.18 meet(composition(x0, top), composition(complement(composition(x0, top)), X))
% 231.95/30.18 = { by lemma 40 }
% 231.95/30.18 meet(composition(complement(composition(x0, top)), X), composition(x0, top))
% 231.95/30.18 = { by lemma 38 R->L }
% 231.95/30.18 join(meet(composition(complement(composition(x0, top)), X), composition(x0, top)), zero)
% 231.95/30.18 = { by lemma 51 R->L }
% 231.95/30.18 join(meet(composition(complement(composition(x0, top)), X), composition(x0, top)), meet(zero, composition(x0, top)))
% 231.95/30.18 = { by lemma 53 R->L }
% 231.95/30.18 join(meet(composition(complement(composition(x0, top)), X), composition(x0, top)), meet(composition(complement(composition(x0, top)), zero), composition(x0, top)))
% 231.95/30.18 = { by lemma 43 R->L }
% 231.95/30.18 join(meet(composition(complement(composition(x0, top)), X), composition(x0, top)), meet(composition(complement(composition(x0, top)), meet(X, zero)), composition(x0, top)))
% 231.95/30.18 = { by lemma 94 R->L }
% 231.95/30.18 join(meet(composition(complement(composition(x0, top)), X), composition(x0, top)), meet(composition(complement(composition(x0, top)), meet(X, composition(converse(complement(composition(x0, top))), composition(x0, top)))), composition(x0, top)))
% 231.95/30.18 = { by axiom 15 (modular_law_1) }
% 231.95/30.18 meet(composition(complement(composition(x0, top)), meet(X, composition(converse(complement(composition(x0, top))), composition(x0, top)))), composition(x0, top))
% 231.95/30.18 = { by lemma 94 }
% 231.95/30.18 meet(composition(complement(composition(x0, top)), meet(X, zero)), composition(x0, top))
% 231.95/30.18 = { by lemma 43 }
% 231.95/30.18 meet(composition(complement(composition(x0, top)), zero), composition(x0, top))
% 231.95/30.18 = { by lemma 53 }
% 231.95/30.18 meet(zero, composition(x0, top))
% 231.95/30.18 = { by lemma 51 }
% 231.95/30.18 zero
% 231.95/30.18
% 231.95/30.18 Lemma 97: meet(complement(composition(x0, top)), composition(complement(composition(x0, top)), X)) = composition(complement(composition(x0, top)), X).
% 231.95/30.18 Proof:
% 231.95/30.18 meet(complement(composition(x0, top)), composition(complement(composition(x0, top)), X))
% 231.95/30.18 = { by lemma 40 }
% 231.95/30.18 meet(composition(complement(composition(x0, top)), X), complement(composition(x0, top)))
% 231.95/30.18 = { by lemma 39 R->L }
% 231.95/30.18 join(zero, meet(composition(complement(composition(x0, top)), X), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 96 R->L }
% 231.95/30.18 join(meet(composition(x0, top), composition(complement(composition(x0, top)), X)), meet(composition(complement(composition(x0, top)), X), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 69 }
% 231.95/30.18 composition(complement(composition(x0, top)), X)
% 231.95/30.18
% 231.95/30.18 Lemma 98: meet(meet(x0, X), composition(complement(composition(x0, top)), Y)) = zero.
% 231.95/30.18 Proof:
% 231.95/30.18 meet(meet(x0, X), composition(complement(composition(x0, top)), Y))
% 231.95/30.18 = { by lemma 97 R->L }
% 231.95/30.18 meet(meet(x0, X), meet(complement(composition(x0, top)), composition(complement(composition(x0, top)), Y)))
% 231.95/30.18 = { by lemma 95 }
% 231.95/30.18 zero
% 231.95/30.18
% 231.95/30.18 Lemma 99: meet(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)) = composition(meet(x0, X), Y).
% 231.95/30.18 Proof:
% 231.95/30.18 meet(meet(x0, composition(X, Y)), composition(meet(x0, X), Y))
% 231.95/30.18 = { by lemma 40 }
% 231.95/30.18 meet(composition(meet(x0, X), Y), meet(x0, composition(X, Y)))
% 231.95/30.18 = { by lemma 87 R->L }
% 231.95/30.18 meet(composition(x0, top), meet(composition(meet(x0, X), Y), composition(X, Y)))
% 231.95/30.18 = { by lemma 93 R->L }
% 231.95/30.18 meet(composition(x0, top), meet(composition(meet(x0, X), Y), join(composition(meet(x0, X), Y), composition(X, Y))))
% 231.95/30.18 = { by lemma 71 }
% 231.95/30.18 meet(composition(x0, top), composition(meet(x0, X), Y))
% 231.95/30.18 = { by lemma 38 R->L }
% 231.95/30.18 join(meet(composition(x0, top), composition(meet(x0, X), Y)), zero)
% 231.95/30.18 = { by lemma 54 R->L }
% 231.95/30.18 join(meet(composition(x0, top), composition(meet(x0, X), Y)), composition(zero, meet(Y, composition(converse(meet(x0, X)), complement(composition(x0, top))))))
% 231.95/30.18 = { by lemma 98 R->L }
% 231.95/30.18 join(meet(composition(x0, top), composition(meet(x0, X), Y)), composition(meet(meet(x0, X), composition(complement(composition(x0, top)), converse(Y))), meet(Y, composition(converse(meet(x0, X)), complement(composition(x0, top))))))
% 231.95/30.18 = { by axiom 16 (dedekind_law) R->L }
% 231.95/30.18 join(meet(composition(x0, top), composition(meet(x0, X), Y)), join(meet(composition(meet(x0, X), Y), complement(composition(x0, top))), composition(meet(meet(x0, X), composition(complement(composition(x0, top)), converse(Y))), meet(Y, composition(converse(meet(x0, X)), complement(composition(x0, top)))))))
% 231.95/30.18 = { by lemma 98 }
% 231.95/30.18 join(meet(composition(x0, top), composition(meet(x0, X), Y)), join(meet(composition(meet(x0, X), Y), complement(composition(x0, top))), composition(zero, meet(Y, composition(converse(meet(x0, X)), complement(composition(x0, top)))))))
% 231.95/30.18 = { by lemma 54 }
% 231.95/30.18 join(meet(composition(x0, top), composition(meet(x0, X), Y)), join(meet(composition(meet(x0, X), Y), complement(composition(x0, top))), zero))
% 231.95/30.18 = { by lemma 38 }
% 231.95/30.18 join(meet(composition(x0, top), composition(meet(x0, X), Y)), meet(composition(meet(x0, X), Y), complement(composition(x0, top))))
% 231.95/30.18 = { by lemma 69 }
% 231.95/30.18 composition(meet(x0, X), Y)
% 231.95/30.18
% 231.95/30.18 Lemma 100: join(meet(x0, X), join(Y, meet(composition(x0, top), complement(X)))) = join(Y, composition(x0, top)).
% 231.95/30.18 Proof:
% 231.95/30.18 join(meet(x0, X), join(Y, meet(composition(x0, top), complement(X))))
% 231.95/30.18 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.95/30.18 join(meet(x0, X), join(meet(composition(x0, top), complement(X)), Y))
% 231.95/30.18 = { by axiom 8 (maddux2_join_associativity) }
% 231.95/30.18 join(join(meet(x0, X), meet(composition(x0, top), complement(X))), Y)
% 231.95/30.18 = { by lemma 89 }
% 231.95/30.18 join(composition(x0, top), Y)
% 231.95/30.18 = { by axiom 2 (maddux1_join_commutativity) }
% 231.95/30.18 join(Y, composition(x0, top))
% 231.95/30.18
% 231.95/30.18 Lemma 101: meet(join(meet(x0, X), composition(meet(x0, Y), Z)), composition(x0, top)) = join(meet(x0, X), composition(meet(x0, Y), Z)).
% 231.95/30.18 Proof:
% 231.95/30.18 meet(join(meet(x0, X), composition(meet(x0, Y), Z)), composition(x0, top))
% 231.95/30.18 = { by lemma 47 R->L }
% 231.95/30.18 meet(join(meet(x0, X), composition(meet(x0, Y), Z)), join(composition(x0, top), composition(x0, top)))
% 231.95/30.18 = { by lemma 92 R->L }
% 231.95/30.18 meet(join(meet(x0, X), composition(meet(x0, Y), Z)), join(composition(x0, top), join(composition(x0, top), meet(meet(x0, composition(Y, Z)), composition(meet(x0, Y), Z)))))
% 231.95/30.18 = { by lemma 99 }
% 231.95/30.18 meet(join(meet(x0, X), composition(meet(x0, Y), Z)), join(composition(x0, top), join(composition(x0, top), composition(meet(x0, Y), Z))))
% 231.95/30.18 = { by axiom 2 (maddux1_join_commutativity) }
% 231.95/30.18 meet(join(meet(x0, X), composition(meet(x0, Y), Z)), join(composition(x0, top), join(composition(meet(x0, Y), Z), composition(x0, top))))
% 231.95/30.18 = { by lemma 100 R->L }
% 231.95/30.18 meet(join(meet(x0, X), composition(meet(x0, Y), Z)), join(composition(x0, top), join(meet(x0, X), join(composition(meet(x0, Y), Z), meet(composition(x0, top), complement(X))))))
% 231.95/30.18 = { by lemma 65 R->L }
% 231.95/30.18 meet(join(meet(x0, X), composition(meet(x0, Y), Z)), join(composition(x0, top), join(meet(composition(x0, top), complement(X)), join(meet(x0, X), composition(meet(x0, Y), Z)))))
% 231.95/30.18 = { by axiom 2 (maddux1_join_commutativity) }
% 231.95/30.18 meet(join(meet(x0, X), composition(meet(x0, Y), Z)), join(composition(x0, top), join(join(meet(x0, X), composition(meet(x0, Y), Z)), meet(composition(x0, top), complement(X)))))
% 231.95/30.18 = { by lemma 91 R->L }
% 231.95/30.18 meet(join(meet(x0, X), composition(meet(x0, Y), Z)), join(composition(x0, top), join(join(meet(x0, X), composition(meet(x0, Y), Z)), meet(composition(x0, top), complement(meet(x0, X))))))
% 231.95/30.18 = { by lemma 83 }
% 231.95/30.18 join(meet(x0, X), composition(meet(x0, Y), Z))
% 231.95/30.18
% 231.95/30.18 Lemma 102: join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)) = meet(x0, composition(X, Y)).
% 231.95/30.18 Proof:
% 231.95/30.18 join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y))
% 231.95/30.19 = { by lemma 101 R->L }
% 231.95/30.19 meet(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)), composition(x0, top))
% 231.95/30.19 = { by lemma 40 }
% 231.95/30.19 meet(composition(x0, top), join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)))
% 231.95/30.19 = { by lemma 71 R->L }
% 231.95/30.19 meet(composition(x0, top), meet(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)), join(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)), composition(X, Y))))
% 231.95/30.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.95/30.19 meet(composition(x0, top), meet(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)), join(composition(X, Y), join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)))))
% 231.95/30.19 = { by lemma 65 }
% 231.95/30.19 meet(composition(x0, top), meet(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)), join(meet(x0, composition(X, Y)), join(composition(meet(x0, X), Y), composition(X, Y)))))
% 231.95/30.19 = { by lemma 93 }
% 231.95/30.19 meet(composition(x0, top), meet(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)), join(meet(x0, composition(X, Y)), composition(X, Y))))
% 231.95/30.19 = { by lemma 70 }
% 231.95/30.19 meet(composition(x0, top), meet(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)), composition(X, Y)))
% 231.95/30.19 = { by lemma 87 }
% 231.95/30.19 meet(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)), meet(x0, composition(X, Y)))
% 231.95/30.19 = { by lemma 40 }
% 231.95/30.19 meet(meet(x0, composition(X, Y)), join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)))
% 231.95/30.19 = { by axiom 11 (maddux4_definiton_of_meet) }
% 231.95/30.19 complement(join(complement(meet(x0, composition(X, Y))), complement(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)))))
% 231.95/30.19 = { by lemma 37 R->L }
% 231.95/30.19 join(zero, complement(join(complement(meet(x0, composition(X, Y))), complement(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y))))))
% 231.95/30.19 = { by lemma 17 R->L }
% 231.95/30.19 join(complement(top), complement(join(complement(meet(x0, composition(X, Y))), complement(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y))))))
% 231.95/30.19 = { by lemma 29 R->L }
% 231.95/30.19 join(complement(join(meet(x0, composition(X, Y)), join(complement(meet(x0, composition(X, Y))), composition(meet(x0, X), Y)))), complement(join(complement(meet(x0, composition(X, Y))), complement(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y))))))
% 231.95/30.19 = { by lemma 55 }
% 231.95/30.19 join(complement(join(complement(meet(x0, composition(X, Y))), join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)))), complement(join(complement(meet(x0, composition(X, Y))), complement(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y))))))
% 231.95/30.19 = { by axiom 2 (maddux1_join_commutativity) }
% 231.95/30.19 join(complement(join(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)), complement(meet(x0, composition(X, Y))))), complement(join(complement(meet(x0, composition(X, Y))), complement(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y))))))
% 231.95/30.19 = { by lemma 57 }
% 231.95/30.19 join(meet(meet(x0, composition(X, Y)), complement(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y)))), complement(join(complement(meet(x0, composition(X, Y))), complement(join(meet(x0, composition(X, Y)), composition(meet(x0, X), Y))))))
% 231.95/30.19 = { by lemma 32 }
% 231.95/30.19 meet(x0, composition(X, Y))
% 231.95/30.19
% 231.95/30.19 Goal 1 (goals_1): tuple(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2)))) = tuple(composition(meet(x0, x1), x2), meet(x0, composition(x1, x2_2))).
% 231.95/30.19 Proof:
% 231.95/30.19 tuple(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by lemma 101 R->L }
% 231.95/30.19 tuple(meet(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), composition(x0, top)), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by lemma 32 R->L }
% 231.95/30.19 tuple(meet(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), join(meet(composition(x0, top), composition(complement(composition(x0, top)), x2)), complement(join(complement(composition(x0, top)), composition(complement(composition(x0, top)), x2))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by lemma 96 }
% 231.95/30.19 tuple(meet(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), join(zero, complement(join(complement(composition(x0, top)), composition(complement(composition(x0, top)), x2))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by lemma 37 }
% 231.95/30.19 tuple(meet(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), complement(join(complement(composition(x0, top)), composition(complement(composition(x0, top)), x2)))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by lemma 60 }
% 231.95/30.19 tuple(meet(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), meet(composition(x0, top), complement(composition(complement(composition(x0, top)), x2)))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by lemma 102 }
% 231.95/30.19 tuple(meet(meet(x0, composition(x1, x2)), meet(composition(x0, top), complement(composition(complement(composition(x0, top)), x2)))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by lemma 73 }
% 231.95/30.19 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), meet(meet(x0, composition(x1, x2)), composition(x0, top))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by lemma 85 }
% 231.95/30.19 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), meet(x0, composition(x1, x2))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by lemma 102 R->L }
% 231.95/30.19 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by lemma 40 R->L }
% 231.95/30.19 tuple(meet(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), complement(composition(complement(composition(x0, top)), x2))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by lemma 81 R->L }
% 231.95/30.19 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), composition(complement(composition(x0, top)), x2))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 231.95/30.19 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by lemma 69 R->L }
% 231.95/30.19 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), complement(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by lemma 82 R->L }
% 231.95/30.19 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(composition(x1, x2), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by lemma 69 R->L }
% 231.95/30.19 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(join(meet(composition(x0, top), composition(x1, x2)), meet(composition(x1, x2), complement(composition(x0, top)))), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 231.95/30.19 = { by axiom 4 (goals) }
% 231.95/30.19 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(join(meet(x0, composition(x1, x2)), meet(composition(x1, x2), complement(composition(x0, top)))), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.19 = { by lemma 57 R->L }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(join(meet(x0, composition(x1, x2)), complement(join(composition(x0, top), complement(composition(x1, x2))))), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(join(meet(x0, composition(x1, x2)), complement(join(complement(composition(x1, x2)), composition(x0, top)))), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by lemma 100 R->L }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(join(meet(x0, composition(x1, x2)), complement(join(meet(x0, composition(x1, x2)), join(complement(composition(x1, x2)), meet(composition(x0, top), complement(composition(x1, x2))))))), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by lemma 40 }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(join(meet(x0, composition(x1, x2)), complement(join(meet(x0, composition(x1, x2)), join(complement(composition(x1, x2)), meet(complement(composition(x1, x2)), composition(x0, top)))))), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by lemma 64 }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(join(meet(x0, composition(x1, x2)), complement(join(meet(x0, composition(x1, x2)), complement(composition(x1, x2))))), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by lemma 57 }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(join(meet(x0, composition(x1, x2)), meet(composition(x1, x2), complement(meet(x0, composition(x1, x2))))), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by axiom 8 (maddux2_join_associativity) R->L }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), join(meet(composition(x1, x2), complement(meet(x0, composition(x1, x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by axiom 2 (maddux1_join_commutativity) }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), join(complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), meet(composition(x1, x2), complement(meet(x0, composition(x1, x2)))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by lemma 102 R->L }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), join(complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), meet(composition(x1, x2), complement(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by lemma 102 R->L }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), join(complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), meet(composition(x1, x2), complement(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by axiom 8 (maddux2_join_associativity) }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(composition(complement(composition(x0, top)), x2), join(join(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))), meet(composition(x1, x2), complement(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by axiom 8 (maddux2_join_associativity) }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(join(composition(complement(composition(x0, top)), x2), join(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))))), meet(composition(x1, x2), complement(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by lemma 88 }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(join(top, meet(composition(x1, x2), complement(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by lemma 49 }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), complement(top))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by lemma 17 }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), zero)), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.20 = { by lemma 38 }
% 232.24/30.20 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), meet(join(composition(complement(composition(x0, top)), x2), composition(x1, x2)), join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 40 R->L }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), join(composition(complement(composition(x0, top)), x2), composition(x1, x2)))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by axiom 2 (maddux1_join_commutativity) }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), join(composition(x1, x2), composition(complement(composition(x0, top)), x2)))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by axiom 12 (composition_distributivity) R->L }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), composition(join(x1, complement(composition(x0, top))), x2))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 90 R->L }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), composition(join(meet(x0, x1), complement(composition(x0, top))), x2))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by axiom 12 (composition_distributivity) }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), join(composition(meet(x0, x1), x2), composition(complement(composition(x0, top)), x2)))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), join(composition(complement(composition(x0, top)), x2), composition(meet(x0, x1), x2)))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 38 R->L }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), join(composition(complement(composition(x0, top)), x2), composition(meet(x0, x1), x2))), zero)), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 17 R->L }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), join(composition(complement(composition(x0, top)), x2), composition(meet(x0, x1), x2))), complement(top))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 26 R->L }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), join(composition(complement(composition(x0, top)), x2), composition(meet(x0, x1), x2))), complement(join(meet(x0, composition(x1, x2)), top)))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 88 R->L }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), join(composition(complement(composition(x0, top)), x2), composition(meet(x0, x1), x2))), complement(join(meet(x0, composition(x1, x2)), join(composition(meet(x0, x1), x2), join(composition(complement(composition(x0, top)), x2), complement(join(composition(meet(x0, x1), x2), composition(complement(composition(x0, top)), x2))))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 65 R->L }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), join(composition(complement(composition(x0, top)), x2), composition(meet(x0, x1), x2))), complement(join(join(composition(complement(composition(x0, top)), x2), complement(join(composition(meet(x0, x1), x2), composition(complement(composition(x0, top)), x2)))), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by axiom 8 (maddux2_join_associativity) R->L }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), join(composition(complement(composition(x0, top)), x2), composition(meet(x0, x1), x2))), complement(join(composition(complement(composition(x0, top)), x2), join(complement(join(composition(meet(x0, x1), x2), composition(complement(composition(x0, top)), x2))), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by axiom 2 (maddux1_join_commutativity) }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), join(composition(complement(composition(x0, top)), x2), composition(meet(x0, x1), x2))), complement(join(composition(complement(composition(x0, top)), x2), join(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), complement(join(composition(meet(x0, x1), x2), composition(complement(composition(x0, top)), x2)))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by axiom 2 (maddux1_join_commutativity) }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), join(composition(complement(composition(x0, top)), x2), composition(meet(x0, x1), x2))), complement(join(composition(complement(composition(x0, top)), x2), join(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), complement(join(composition(complement(composition(x0, top)), x2), composition(meet(x0, x1), x2)))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 82 }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(meet(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))), join(composition(complement(composition(x0, top)), x2), composition(meet(x0, x1), x2))), meet(join(composition(complement(composition(x0, top)), x2), composition(meet(x0, x1), x2)), complement(join(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 69 }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(composition(complement(composition(x0, top)), x2), composition(meet(x0, x1), x2))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by axiom 2 (maddux1_join_commutativity) }
% 232.24/30.21 tuple(meet(complement(composition(complement(composition(x0, top)), x2)), join(composition(meet(x0, x1), x2), composition(complement(composition(x0, top)), x2))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 81 }
% 232.24/30.21 tuple(meet(composition(meet(x0, x1), x2), complement(composition(complement(composition(x0, top)), x2))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 38 R->L }
% 232.24/30.21 tuple(meet(composition(meet(x0, x1), x2), join(complement(composition(complement(composition(x0, top)), x2)), zero)), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 95 R->L }
% 232.24/30.21 tuple(meet(composition(meet(x0, x1), x2), join(complement(composition(complement(composition(x0, top)), x2)), meet(meet(x0, composition(x1, x2)), meet(complement(composition(x0, top)), composition(complement(composition(x0, top)), x2))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 97 }
% 232.24/30.21 tuple(meet(composition(meet(x0, x1), x2), join(complement(composition(complement(composition(x0, top)), x2)), meet(meet(x0, composition(x1, x2)), composition(complement(composition(x0, top)), x2)))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 102 R->L }
% 232.24/30.21 tuple(meet(composition(meet(x0, x1), x2), join(complement(composition(complement(composition(x0, top)), x2)), meet(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), composition(complement(composition(x0, top)), x2)))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 40 }
% 232.24/30.21 tuple(meet(composition(meet(x0, x1), x2), join(complement(composition(complement(composition(x0, top)), x2)), meet(composition(complement(composition(x0, top)), x2), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 78 }
% 232.24/30.21 tuple(meet(composition(meet(x0, x1), x2), join(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), complement(join(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), composition(complement(composition(x0, top)), x2))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 79 }
% 232.24/30.21 tuple(meet(composition(meet(x0, x1), x2), join(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), complement(composition(complement(composition(x0, top)), x2)))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 232.24/30.21 tuple(meet(composition(meet(x0, x1), x2), join(complement(composition(complement(composition(x0, top)), x2)), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by axiom 11 (maddux4_definiton_of_meet) }
% 232.24/30.21 tuple(complement(join(complement(composition(meet(x0, x1), x2)), complement(join(complement(composition(complement(composition(x0, top)), x2)), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 37 R->L }
% 232.24/30.21 tuple(join(zero, complement(join(complement(composition(meet(x0, x1), x2)), complement(join(complement(composition(complement(composition(x0, top)), x2)), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 17 R->L }
% 232.24/30.21 tuple(join(complement(top), complement(join(complement(composition(meet(x0, x1), x2)), complement(join(complement(composition(complement(composition(x0, top)), x2)), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 26 R->L }
% 232.24/30.21 tuple(join(complement(join(complement(composition(complement(composition(x0, top)), x2)), top)), complement(join(complement(composition(meet(x0, x1), x2)), complement(join(complement(composition(complement(composition(x0, top)), x2)), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 26 R->L }
% 232.24/30.21 tuple(join(complement(join(complement(composition(complement(composition(x0, top)), x2)), join(meet(x0, composition(x1, x2)), top))), complement(join(complement(composition(meet(x0, x1), x2)), complement(join(complement(composition(complement(composition(x0, top)), x2)), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by axiom 5 (def_top) }
% 232.24/30.21 tuple(join(complement(join(complement(composition(complement(composition(x0, top)), x2)), join(meet(x0, composition(x1, x2)), join(composition(meet(x0, x1), x2), complement(composition(meet(x0, x1), x2)))))), complement(join(complement(composition(meet(x0, x1), x2)), complement(join(complement(composition(complement(composition(x0, top)), x2)), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 65 R->L }
% 232.24/30.21 tuple(join(complement(join(complement(composition(complement(composition(x0, top)), x2)), join(complement(composition(meet(x0, x1), x2)), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))), complement(join(complement(composition(meet(x0, x1), x2)), complement(join(complement(composition(complement(composition(x0, top)), x2)), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by axiom 2 (maddux1_join_commutativity) }
% 232.24/30.21 tuple(join(complement(join(complement(composition(complement(composition(x0, top)), x2)), join(join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2)), complement(composition(meet(x0, x1), x2))))), complement(join(complement(composition(meet(x0, x1), x2)), complement(join(complement(composition(complement(composition(x0, top)), x2)), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 82 }
% 232.24/30.21 tuple(join(meet(composition(meet(x0, x1), x2), complement(join(complement(composition(complement(composition(x0, top)), x2)), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))), complement(join(complement(composition(meet(x0, x1), x2)), complement(join(complement(composition(complement(composition(x0, top)), x2)), join(meet(x0, composition(x1, x2)), composition(meet(x0, x1), x2))))))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.21 = { by lemma 32 }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))
% 232.24/30.22 = { by lemma 83 R->L }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), join(composition(x0, top), join(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), meet(composition(x0, top), complement(meet(x0, composition(x1, x2_2))))))))
% 232.24/30.22 = { by lemma 91 }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), join(composition(x0, top), join(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), meet(composition(x0, top), complement(composition(x1, x2_2)))))))
% 232.24/30.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), join(composition(x0, top), join(meet(composition(x0, top), complement(composition(x1, x2_2))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2)))))))
% 232.24/30.22 = { by lemma 66 }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), join(composition(x0, top), join(meet(x0, composition(x1, x2_2)), join(composition(meet(x0, x1), x2_2), meet(composition(x0, top), complement(composition(x1, x2_2))))))))
% 232.24/30.22 = { by lemma 100 }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), join(composition(x0, top), join(composition(meet(x0, x1), x2_2), composition(x0, top)))))
% 232.24/30.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), join(composition(x0, top), join(composition(x0, top), composition(meet(x0, x1), x2_2)))))
% 232.24/30.22 = { by lemma 99 R->L }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), join(composition(x0, top), join(composition(x0, top), meet(meet(x0, composition(x1, x2_2)), composition(meet(x0, x1), x2_2))))))
% 232.24/30.22 = { by lemma 92 }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), join(composition(x0, top), composition(x0, top))))
% 232.24/30.22 = { by lemma 47 }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), composition(x0, top)))
% 232.24/30.22 = { by lemma 40 }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(composition(x0, top), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2)))))
% 232.24/30.22 = { by lemma 71 R->L }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(composition(x0, top), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), join(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), composition(x1, x2_2)))))
% 232.24/30.22 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(composition(x0, top), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), join(composition(x1, x2_2), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2)))))))
% 232.24/30.22 = { by lemma 66 }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(composition(x0, top), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), join(meet(x0, composition(x1, x2_2)), join(composition(meet(x0, x1), x2_2), composition(x1, x2_2))))))
% 232.24/30.22 = { by lemma 93 }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(composition(x0, top), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), join(meet(x0, composition(x1, x2_2)), composition(x1, x2_2)))))
% 232.24/30.22 = { by lemma 70 }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(composition(x0, top), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), composition(x1, x2_2))))
% 232.24/30.22 = { by lemma 87 }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), meet(x0, composition(x1, x2_2))))
% 232.24/30.22 = { by lemma 40 }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(meet(x0, composition(x1, x2_2)), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2)))))
% 232.24/30.22 = { by axiom 11 (maddux4_definiton_of_meet) }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), complement(join(complement(meet(x0, composition(x1, x2_2))), complement(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2)))))))
% 232.24/30.22 = { by lemma 37 R->L }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), join(zero, complement(join(complement(meet(x0, composition(x1, x2_2))), complement(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))))))
% 232.24/30.22 = { by lemma 17 R->L }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), join(complement(top), complement(join(complement(meet(x0, composition(x1, x2_2))), complement(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))))))
% 232.24/30.22 = { by lemma 26 R->L }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), join(complement(join(composition(meet(x0, x1), x2_2), top)), complement(join(complement(meet(x0, composition(x1, x2_2))), complement(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))))))
% 232.24/30.22 = { by lemma 18 R->L }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), join(complement(join(meet(x0, composition(x1, x2_2)), join(composition(meet(x0, x1), x2_2), complement(meet(x0, composition(x1, x2_2)))))), complement(join(complement(meet(x0, composition(x1, x2_2))), complement(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))))))
% 232.24/30.22 = { by lemma 66 R->L }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), join(complement(join(complement(meet(x0, composition(x1, x2_2))), join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))), complement(join(complement(meet(x0, composition(x1, x2_2))), complement(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))))))
% 232.24/30.22 = { by axiom 2 (maddux1_join_commutativity) }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), join(complement(join(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))), complement(meet(x0, composition(x1, x2_2))))), complement(join(complement(meet(x0, composition(x1, x2_2))), complement(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))))))
% 232.24/30.22 = { by lemma 57 }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), join(meet(meet(x0, composition(x1, x2_2)), complement(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))), complement(join(complement(meet(x0, composition(x1, x2_2))), complement(join(composition(meet(x0, x1), x2_2), meet(x0, composition(x1, x2_2))))))))
% 232.24/30.22 = { by lemma 32 }
% 232.24/30.22 tuple(composition(meet(x0, x1), x2), meet(x0, composition(x1, x2_2)))
% 232.24/30.22 % SZS output end Proof
% 232.24/30.22
% 232.24/30.22 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------