TSTP Solution File: REL016+4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL016+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 : n012.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:43:57 EDT 2023
% Result : Theorem 45.26s 6.17s
% Output : Proof 46.10s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL016+4 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.17/0.34 % Computer : n012.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Fri Aug 25 21:37:56 EDT 2023
% 0.17/0.35 % CPUTime :
% 45.26/6.17 Command-line arguments: --ground-connectedness --complete-subsets
% 45.26/6.17
% 45.26/6.17 % SZS status Theorem
% 45.26/6.17
% 45.58/6.24 % SZS output start Proof
% 45.58/6.24 Take the following subset of the input axioms:
% 45.58/6.25 fof(composition_associativity, axiom, ![X0, X1, X2]: composition(X0, composition(X1, X2))=composition(composition(X0, X1), X2)).
% 45.58/6.25 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))).
% 45.58/6.25 fof(composition_identity, axiom, ![X0_2]: composition(X0_2, one)=X0_2).
% 45.58/6.25 fof(converse_additivity, axiom, ![X0_2, X1_2]: converse(join(X0_2, X1_2))=join(converse(X0_2), converse(X1_2))).
% 45.58/6.25 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)).
% 45.58/6.25 fof(converse_idempotence, axiom, ![X0_2]: converse(converse(X0_2))=X0_2).
% 45.58/6.25 fof(converse_multiplicativity, axiom, ![X0_2, X1_2]: converse(composition(X0_2, X1_2))=composition(converse(X1_2), converse(X0_2))).
% 45.58/6.25 fof(def_top, axiom, ![X0_2]: top=join(X0_2, complement(X0_2))).
% 45.58/6.25 fof(def_zero, axiom, ![X0_2]: zero=meet(X0_2, complement(X0_2))).
% 45.58/6.25 fof(goals, conjecture, ![X0_2, X1_2, X2_2]: (join(meet(composition(X0_2, X1_2), complement(composition(X0_2, X2_2))), meet(composition(X0_2, meet(X1_2, complement(X2_2))), complement(composition(X0_2, X2_2))))=meet(composition(X0_2, meet(X1_2, complement(X2_2))), complement(composition(X0_2, X2_2))) & join(meet(composition(X0_2, meet(X1_2, complement(X2_2))), complement(composition(X0_2, X2_2))), meet(composition(X0_2, X1_2), complement(composition(X0_2, X2_2))))=meet(composition(X0_2, X1_2), complement(composition(X0_2, X2_2))))).
% 45.58/6.25 fof(maddux1_join_commutativity, axiom, ![X0_2, X1_2]: join(X0_2, X1_2)=join(X1_2, X0_2)).
% 45.58/6.25 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)).
% 45.58/6.25 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)))).
% 45.58/6.25 fof(maddux4_definiton_of_meet, axiom, ![X0_2, X1_2]: meet(X0_2, X1_2)=complement(join(complement(X0_2), complement(X1_2)))).
% 45.58/6.25
% 45.58/6.25 Now clausify the problem and encode Horn clauses using encoding 3 of
% 45.58/6.25 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 45.58/6.25 We repeatedly replace C & s=t => u=v by the two clauses:
% 45.58/6.25 fresh(y, y, x1...xn) = u
% 45.58/6.25 C => fresh(s, t, x1...xn) = v
% 45.58/6.25 where fresh is a fresh function symbol and x1..xn are the free
% 45.58/6.25 variables of u and v.
% 45.58/6.25 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 45.58/6.25 input problem has no model of domain size 1).
% 45.58/6.25
% 45.58/6.25 The encoding turns the above axioms into the following unit equations and goals:
% 45.58/6.25
% 45.58/6.25 Axiom 1 (converse_idempotence): converse(converse(X)) = X.
% 45.58/6.25 Axiom 2 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 45.58/6.25 Axiom 3 (composition_identity): composition(X, one) = X.
% 45.58/6.25 Axiom 4 (def_top): top = join(X, complement(X)).
% 45.58/6.25 Axiom 5 (def_zero): zero = meet(X, complement(X)).
% 45.58/6.25 Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 45.58/6.25 Axiom 7 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 45.58/6.25 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 45.58/6.25 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 45.58/6.25 Axiom 10 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 45.58/6.25 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 45.58/6.25 Axiom 12 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 45.58/6.25 Axiom 13 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 45.58/6.25
% 45.58/6.25 Lemma 14: complement(top) = zero.
% 45.58/6.25 Proof:
% 45.58/6.25 complement(top)
% 45.58/6.25 = { by axiom 4 (def_top) }
% 45.58/6.25 complement(join(complement(X), complement(complement(X))))
% 45.58/6.25 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 45.58/6.25 meet(X, complement(X))
% 45.58/6.25 = { by axiom 5 (def_zero) R->L }
% 45.58/6.25 zero
% 45.58/6.25
% 45.58/6.25 Lemma 15: join(X, join(Y, complement(X))) = join(Y, top).
% 45.58/6.25 Proof:
% 45.58/6.25 join(X, join(Y, complement(X)))
% 45.58/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 45.58/6.25 join(X, join(complement(X), Y))
% 45.58/6.25 = { by axiom 7 (maddux2_join_associativity) }
% 45.58/6.25 join(join(X, complement(X)), Y)
% 45.58/6.25 = { by axiom 4 (def_top) R->L }
% 45.58/6.25 join(top, Y)
% 45.58/6.25 = { by axiom 2 (maddux1_join_commutativity) }
% 45.58/6.25 join(Y, top)
% 45.58/6.25
% 45.58/6.25 Lemma 16: composition(converse(one), X) = X.
% 45.58/6.25 Proof:
% 45.58/6.25 composition(converse(one), X)
% 45.58/6.25 = { by axiom 1 (converse_idempotence) R->L }
% 45.58/6.25 composition(converse(one), converse(converse(X)))
% 45.58/6.25 = { by axiom 8 (converse_multiplicativity) R->L }
% 45.58/6.25 converse(composition(converse(X), one))
% 45.58/6.25 = { by axiom 3 (composition_identity) }
% 45.58/6.25 converse(converse(X))
% 45.58/6.25 = { by axiom 1 (converse_idempotence) }
% 45.58/6.25 X
% 45.58/6.25
% 45.58/6.25 Lemma 17: join(complement(X), complement(X)) = complement(X).
% 45.58/6.25 Proof:
% 45.58/6.25 join(complement(X), complement(X))
% 45.58/6.25 = { by lemma 16 R->L }
% 45.58/6.25 join(complement(X), composition(converse(one), complement(X)))
% 45.58/6.25 = { by lemma 16 R->L }
% 45.58/6.25 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 45.58/6.25 = { by axiom 3 (composition_identity) R->L }
% 45.58/6.25 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 45.58/6.25 = { by axiom 9 (composition_associativity) R->L }
% 45.58/6.25 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 45.58/6.25 = { by lemma 16 }
% 45.58/6.25 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 45.58/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 45.58/6.25 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 45.58/6.25 = { by axiom 12 (converse_cancellativity) }
% 45.58/6.25 complement(X)
% 45.58/6.25
% 45.58/6.25 Lemma 18: join(top, complement(X)) = top.
% 45.58/6.25 Proof:
% 45.58/6.25 join(top, complement(X))
% 45.58/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 45.58/6.25 join(complement(X), top)
% 45.58/6.25 = { by lemma 15 R->L }
% 45.58/6.25 join(X, join(complement(X), complement(X)))
% 45.58/6.25 = { by lemma 17 }
% 45.58/6.25 join(X, complement(X))
% 45.58/6.25 = { by axiom 4 (def_top) R->L }
% 45.58/6.25 top
% 45.58/6.25
% 45.58/6.25 Lemma 19: join(Y, top) = join(X, top).
% 45.58/6.25 Proof:
% 45.58/6.25 join(Y, top)
% 45.58/6.25 = { by lemma 18 R->L }
% 45.58/6.25 join(Y, join(top, complement(Y)))
% 45.58/6.25 = { by lemma 15 }
% 45.58/6.25 join(top, top)
% 45.58/6.25 = { by lemma 15 R->L }
% 45.58/6.25 join(X, join(top, complement(X)))
% 45.58/6.25 = { by lemma 18 }
% 45.58/6.25 join(X, top)
% 45.58/6.25
% 45.58/6.25 Lemma 20: join(X, top) = top.
% 45.58/6.25 Proof:
% 45.58/6.25 join(X, top)
% 45.58/6.25 = { by lemma 19 }
% 45.58/6.25 join(complement(Y), top)
% 45.58/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 45.58/6.25 join(top, complement(Y))
% 45.58/6.25 = { by lemma 18 }
% 45.58/6.25 top
% 45.58/6.25
% 45.58/6.25 Lemma 21: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 45.58/6.25 Proof:
% 45.58/6.25 join(meet(X, Y), complement(join(complement(X), Y)))
% 45.58/6.25 = { by axiom 10 (maddux4_definiton_of_meet) }
% 45.58/6.25 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 45.58/6.25 = { by axiom 13 (maddux3_a_kind_of_de_Morgan) R->L }
% 46.06/6.25 X
% 46.06/6.25
% 46.06/6.25 Lemma 22: join(zero, meet(X, X)) = X.
% 46.06/6.25 Proof:
% 46.06/6.25 join(zero, meet(X, X))
% 46.06/6.25 = { by axiom 10 (maddux4_definiton_of_meet) }
% 46.06/6.25 join(zero, complement(join(complement(X), complement(X))))
% 46.06/6.25 = { by axiom 5 (def_zero) }
% 46.06/6.25 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 46.06/6.25 = { by lemma 21 }
% 46.06/6.25 X
% 46.06/6.25
% 46.06/6.25 Lemma 23: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 46.06/6.25 Proof:
% 46.06/6.25 join(zero, join(X, complement(complement(Y))))
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.06/6.25 join(zero, join(complement(complement(Y)), X))
% 46.06/6.25 = { by lemma 17 R->L }
% 46.06/6.25 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 46.06/6.25 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 46.06/6.25 join(zero, join(meet(Y, Y), X))
% 46.06/6.25 = { by axiom 7 (maddux2_join_associativity) }
% 46.06/6.25 join(join(zero, meet(Y, Y)), X)
% 46.06/6.25 = { by lemma 22 }
% 46.06/6.25 join(Y, X)
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) }
% 46.06/6.25 join(X, Y)
% 46.06/6.25
% 46.06/6.25 Lemma 24: join(zero, complement(complement(X))) = X.
% 46.06/6.25 Proof:
% 46.06/6.25 join(zero, complement(complement(X)))
% 46.06/6.25 = { by axiom 5 (def_zero) }
% 46.06/6.25 join(meet(X, complement(X)), complement(complement(X)))
% 46.06/6.25 = { by lemma 17 R->L }
% 46.06/6.25 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 46.06/6.25 = { by lemma 21 }
% 46.06/6.25 X
% 46.06/6.25
% 46.06/6.25 Lemma 25: join(zero, complement(X)) = complement(X).
% 46.06/6.25 Proof:
% 46.06/6.25 join(zero, complement(X))
% 46.06/6.25 = { by lemma 24 R->L }
% 46.06/6.25 join(zero, join(zero, complement(complement(complement(X)))))
% 46.06/6.25 = { by lemma 17 R->L }
% 46.06/6.25 join(zero, join(zero, join(complement(complement(complement(X))), complement(complement(complement(X))))))
% 46.06/6.25 = { by lemma 23 }
% 46.06/6.25 join(zero, join(complement(complement(complement(X))), complement(X)))
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) }
% 46.06/6.25 join(zero, join(complement(X), complement(complement(complement(X)))))
% 46.06/6.25 = { by lemma 23 }
% 46.06/6.25 join(complement(X), complement(X))
% 46.06/6.25 = { by lemma 17 }
% 46.06/6.25 complement(X)
% 46.06/6.25
% 46.06/6.25 Lemma 26: join(X, zero) = X.
% 46.06/6.25 Proof:
% 46.06/6.25 join(X, zero)
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.06/6.25 join(zero, X)
% 46.06/6.25 = { by lemma 23 R->L }
% 46.06/6.25 join(zero, join(zero, complement(complement(X))))
% 46.06/6.25 = { by lemma 25 }
% 46.06/6.25 join(zero, complement(complement(X)))
% 46.06/6.25 = { by lemma 24 }
% 46.06/6.25 X
% 46.06/6.25
% 46.06/6.25 Lemma 27: join(top, X) = top.
% 46.06/6.25 Proof:
% 46.06/6.25 join(top, X)
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.06/6.25 join(X, top)
% 46.06/6.25 = { by lemma 19 R->L }
% 46.06/6.25 join(Y, top)
% 46.06/6.25 = { by lemma 20 }
% 46.06/6.25 top
% 46.06/6.25
% 46.06/6.25 Lemma 28: join(zero, X) = X.
% 46.06/6.25 Proof:
% 46.06/6.25 join(zero, X)
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.06/6.25 join(X, zero)
% 46.06/6.25 = { by lemma 26 }
% 46.06/6.25 X
% 46.06/6.25
% 46.06/6.25 Lemma 29: complement(complement(X)) = X.
% 46.06/6.25 Proof:
% 46.06/6.25 complement(complement(X))
% 46.06/6.25 = { by lemma 25 R->L }
% 46.06/6.25 join(zero, complement(complement(X)))
% 46.06/6.25 = { by lemma 24 }
% 46.06/6.25 X
% 46.06/6.25
% 46.06/6.25 Lemma 30: meet(Y, X) = meet(X, Y).
% 46.06/6.25 Proof:
% 46.06/6.25 meet(Y, X)
% 46.06/6.25 = { by axiom 10 (maddux4_definiton_of_meet) }
% 46.06/6.25 complement(join(complement(Y), complement(X)))
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.06/6.25 complement(join(complement(X), complement(Y)))
% 46.06/6.25 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 46.06/6.25 meet(X, Y)
% 46.06/6.25
% 46.06/6.25 Lemma 31: complement(join(zero, complement(X))) = meet(X, top).
% 46.06/6.25 Proof:
% 46.06/6.25 complement(join(zero, complement(X)))
% 46.06/6.25 = { by lemma 14 R->L }
% 46.06/6.25 complement(join(complement(top), complement(X)))
% 46.06/6.25 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 46.06/6.25 meet(top, X)
% 46.06/6.25 = { by lemma 30 R->L }
% 46.06/6.25 meet(X, top)
% 46.06/6.25
% 46.06/6.25 Lemma 32: join(X, complement(zero)) = top.
% 46.06/6.25 Proof:
% 46.06/6.25 join(X, complement(zero))
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.06/6.25 join(complement(zero), X)
% 46.06/6.25 = { by lemma 23 R->L }
% 46.06/6.25 join(zero, join(complement(zero), complement(complement(X))))
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.06/6.25 join(zero, join(complement(complement(X)), complement(zero)))
% 46.06/6.25 = { by lemma 15 }
% 46.06/6.25 join(complement(complement(X)), top)
% 46.06/6.25 = { by lemma 20 }
% 46.06/6.25 top
% 46.06/6.25
% 46.06/6.25 Lemma 33: join(meet(X, Y), meet(X, complement(Y))) = X.
% 46.06/6.25 Proof:
% 46.06/6.25 join(meet(X, Y), meet(X, complement(Y)))
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.06/6.25 join(meet(X, complement(Y)), meet(X, Y))
% 46.06/6.25 = { by axiom 10 (maddux4_definiton_of_meet) }
% 46.06/6.25 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 46.06/6.25 = { by lemma 21 }
% 46.06/6.25 X
% 46.06/6.25
% 46.06/6.25 Lemma 34: meet(X, top) = X.
% 46.06/6.25 Proof:
% 46.06/6.25 meet(X, top)
% 46.06/6.25 = { by lemma 31 R->L }
% 46.06/6.25 complement(join(zero, complement(X)))
% 46.06/6.25 = { by lemma 25 R->L }
% 46.06/6.25 join(zero, complement(join(zero, complement(X))))
% 46.06/6.25 = { by lemma 31 }
% 46.06/6.25 join(zero, meet(X, top))
% 46.06/6.25 = { by lemma 32 R->L }
% 46.06/6.25 join(zero, meet(X, join(complement(zero), complement(zero))))
% 46.06/6.25 = { by lemma 17 }
% 46.06/6.25 join(zero, meet(X, complement(zero)))
% 46.06/6.25 = { by lemma 14 R->L }
% 46.06/6.25 join(complement(top), meet(X, complement(zero)))
% 46.06/6.25 = { by lemma 32 R->L }
% 46.06/6.25 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 46.06/6.25 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 46.06/6.25 join(meet(X, zero), meet(X, complement(zero)))
% 46.06/6.25 = { by lemma 33 }
% 46.06/6.25 X
% 46.06/6.25
% 46.06/6.25 Lemma 35: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 46.06/6.25 Proof:
% 46.06/6.25 join(Y, join(X, Z))
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.06/6.25 join(join(X, Z), Y)
% 46.06/6.25 = { by axiom 7 (maddux2_join_associativity) R->L }
% 46.06/6.25 join(X, join(Z, Y))
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) }
% 46.06/6.25 join(X, join(Y, Z))
% 46.06/6.25
% 46.06/6.25 Lemma 36: join(Z, join(X, Y)) = join(X, join(Y, Z)).
% 46.06/6.25 Proof:
% 46.06/6.25 join(Z, join(X, Y))
% 46.06/6.25 = { by lemma 35 }
% 46.06/6.25 join(X, join(Z, Y))
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) }
% 46.06/6.25 join(X, join(Y, Z))
% 46.06/6.25
% 46.06/6.25 Lemma 37: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 46.06/6.25 Proof:
% 46.06/6.25 meet(X, join(complement(Y), complement(Z)))
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.06/6.25 meet(X, join(complement(Z), complement(Y)))
% 46.06/6.25 = { by lemma 30 }
% 46.06/6.25 meet(join(complement(Z), complement(Y)), X)
% 46.06/6.25 = { by axiom 10 (maddux4_definiton_of_meet) }
% 46.06/6.25 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 46.06/6.25 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 46.06/6.25 complement(join(meet(Z, Y), complement(X)))
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) }
% 46.06/6.25 complement(join(complement(X), meet(Z, Y)))
% 46.06/6.25 = { by lemma 30 R->L }
% 46.06/6.25 complement(join(complement(X), meet(Y, Z)))
% 46.06/6.25
% 46.06/6.25 Lemma 38: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 46.06/6.25 Proof:
% 46.06/6.25 complement(join(X, complement(Y)))
% 46.06/6.25 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.06/6.25 complement(join(complement(Y), X))
% 46.06/6.25 = { by lemma 34 R->L }
% 46.06/6.26 complement(join(complement(Y), meet(X, top)))
% 46.06/6.26 = { by lemma 30 R->L }
% 46.06/6.26 complement(join(complement(Y), meet(top, X)))
% 46.06/6.26 = { by lemma 37 R->L }
% 46.06/6.26 meet(Y, join(complement(top), complement(X)))
% 46.06/6.26 = { by lemma 14 }
% 46.06/6.26 meet(Y, join(zero, complement(X)))
% 46.06/6.26 = { by lemma 25 }
% 46.06/6.26 meet(Y, complement(X))
% 46.06/6.26
% 46.06/6.26 Lemma 39: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 46.06/6.26 Proof:
% 46.06/6.26 complement(join(complement(X), Y))
% 46.06/6.26 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.06/6.26 complement(join(Y, complement(X)))
% 46.06/6.26 = { by lemma 38 }
% 46.06/6.26 meet(X, complement(Y))
% 46.06/6.26
% 46.06/6.26 Lemma 40: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 46.06/6.26 Proof:
% 46.06/6.26 complement(meet(X, complement(Y)))
% 46.06/6.26 = { by lemma 28 R->L }
% 46.06/6.26 complement(join(zero, meet(X, complement(Y))))
% 46.06/6.26 = { by lemma 38 R->L }
% 46.06/6.26 complement(join(zero, complement(join(Y, complement(X)))))
% 46.06/6.26 = { by lemma 31 }
% 46.06/6.26 meet(join(Y, complement(X)), top)
% 46.06/6.26 = { by lemma 34 }
% 46.06/6.26 join(Y, complement(X))
% 46.06/6.26
% 46.06/6.26 Lemma 41: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 46.06/6.26 Proof:
% 46.06/6.26 complement(meet(complement(X), Y))
% 46.06/6.26 = { by lemma 30 }
% 46.06/6.26 complement(meet(Y, complement(X)))
% 46.06/6.26 = { by lemma 40 }
% 46.06/6.26 join(X, complement(Y))
% 46.06/6.26
% 46.06/6.26 Lemma 42: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 46.06/6.26 Proof:
% 46.06/6.26 meet(complement(X), complement(Y))
% 46.06/6.26 = { by lemma 30 }
% 46.06/6.26 meet(complement(Y), complement(X))
% 46.06/6.26 = { by lemma 25 R->L }
% 46.06/6.26 meet(join(zero, complement(Y)), complement(X))
% 46.06/6.26 = { by lemma 38 R->L }
% 46.06/6.26 complement(join(X, complement(join(zero, complement(Y)))))
% 46.06/6.26 = { by lemma 31 }
% 46.06/6.26 complement(join(X, meet(Y, top)))
% 46.06/6.26 = { by lemma 34 }
% 46.06/6.26 complement(join(X, Y))
% 46.06/6.26
% 46.06/6.26 Lemma 43: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 46.06/6.26 Proof:
% 46.06/6.26 meet(Y, meet(Z, X))
% 46.06/6.26 = { by lemma 34 R->L }
% 46.06/6.26 meet(meet(Y, top), meet(Z, X))
% 46.06/6.26 = { by lemma 31 R->L }
% 46.06/6.26 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 46.06/6.26 = { by lemma 30 }
% 46.06/6.26 meet(complement(join(zero, complement(Y))), meet(X, Z))
% 46.06/6.26 = { by lemma 30 }
% 46.06/6.26 meet(meet(X, Z), complement(join(zero, complement(Y))))
% 46.06/6.26 = { by axiom 10 (maddux4_definiton_of_meet) }
% 46.06/6.26 meet(complement(join(complement(X), complement(Z))), complement(join(zero, complement(Y))))
% 46.06/6.26 = { by lemma 42 }
% 46.06/6.26 complement(join(join(complement(X), complement(Z)), join(zero, complement(Y))))
% 46.06/6.26 = { by axiom 7 (maddux2_join_associativity) R->L }
% 46.10/6.26 complement(join(complement(X), join(complement(Z), join(zero, complement(Y)))))
% 46.10/6.26 = { by lemma 39 }
% 46.10/6.26 meet(X, complement(join(complement(Z), join(zero, complement(Y)))))
% 46.10/6.26 = { by lemma 39 }
% 46.10/6.26 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 46.10/6.26 = { by lemma 31 }
% 46.10/6.26 meet(X, meet(Z, meet(Y, top)))
% 46.10/6.26 = { by lemma 34 }
% 46.10/6.26 meet(X, meet(Z, Y))
% 46.10/6.26 = { by lemma 30 R->L }
% 46.10/6.26 meet(X, meet(Y, Z))
% 46.10/6.26
% 46.10/6.26 Lemma 44: meet(X, join(X, complement(Y))) = X.
% 46.10/6.26 Proof:
% 46.10/6.26 meet(X, join(X, complement(Y)))
% 46.10/6.26 = { by lemma 40 R->L }
% 46.10/6.26 meet(X, complement(meet(Y, complement(X))))
% 46.10/6.26 = { by lemma 39 R->L }
% 46.10/6.26 complement(join(complement(X), meet(Y, complement(X))))
% 46.10/6.26 = { by lemma 25 R->L }
% 46.10/6.26 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by lemma 14 R->L }
% 46.10/6.26 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by lemma 20 R->L }
% 46.10/6.26 join(complement(join(complement(Y), top)), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by lemma 15 R->L }
% 46.10/6.26 join(complement(join(complement(X), join(complement(Y), complement(complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.10/6.26 join(complement(join(complement(X), join(complement(complement(X)), complement(Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by lemma 22 R->L }
% 46.10/6.26 join(complement(join(complement(X), join(zero, meet(join(complement(complement(X)), complement(Y)), join(complement(complement(X)), complement(Y)))))), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by lemma 37 }
% 46.10/6.26 join(complement(join(complement(X), join(zero, complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y)))))), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by lemma 25 }
% 46.10/6.26 join(complement(join(complement(X), complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y))))), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 46.10/6.26 join(complement(join(complement(X), complement(join(meet(complement(X), Y), meet(complement(X), Y))))), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by lemma 30 }
% 46.10/6.26 join(complement(join(complement(X), complement(join(meet(Y, complement(X)), meet(complement(X), Y))))), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by lemma 30 }
% 46.10/6.26 join(complement(join(complement(X), complement(join(meet(Y, complement(X)), meet(Y, complement(X)))))), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by axiom 10 (maddux4_definiton_of_meet) }
% 46.10/6.26 join(complement(join(complement(X), complement(join(meet(Y, complement(X)), complement(join(complement(Y), complement(complement(X)))))))), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by axiom 10 (maddux4_definiton_of_meet) }
% 46.10/6.26 join(complement(join(complement(X), complement(join(complement(join(complement(Y), complement(complement(X)))), complement(join(complement(Y), complement(complement(X)))))))), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by lemma 17 }
% 46.10/6.26 join(complement(join(complement(X), complement(complement(join(complement(Y), complement(complement(X))))))), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 46.10/6.26 join(complement(join(complement(X), complement(meet(Y, complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by lemma 30 R->L }
% 46.10/6.26 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 46.10/6.26 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by lemma 30 R->L }
% 46.10/6.26 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 46.10/6.26 = { by lemma 21 }
% 46.10/6.26 X
% 46.10/6.26
% 46.10/6.26 Lemma 45: join(meet(X, complement(Y)), Y) = join(X, Y).
% 46.10/6.26 Proof:
% 46.10/6.26 join(meet(X, complement(Y)), Y)
% 46.10/6.26 = { by lemma 29 R->L }
% 46.10/6.26 join(meet(X, complement(Y)), complement(complement(Y)))
% 46.10/6.26 = { by lemma 44 R->L }
% 46.10/6.26 join(meet(X, complement(Y)), complement(meet(complement(Y), join(complement(Y), complement(X)))))
% 46.10/6.26 = { by lemma 41 }
% 46.10/6.26 join(meet(X, complement(Y)), join(Y, complement(join(complement(Y), complement(X)))))
% 46.10/6.26 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 46.10/6.26 join(meet(X, complement(Y)), join(Y, meet(Y, X)))
% 46.10/6.26 = { by lemma 30 R->L }
% 46.10/6.26 join(meet(X, complement(Y)), join(Y, meet(X, Y)))
% 46.10/6.26 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.10/6.26 join(meet(X, complement(Y)), join(meet(X, Y), Y))
% 46.10/6.26 = { by axiom 7 (maddux2_join_associativity) }
% 46.10/6.26 join(join(meet(X, complement(Y)), meet(X, Y)), Y)
% 46.10/6.26 = { by lemma 30 }
% 46.10/6.26 join(join(meet(X, complement(Y)), meet(Y, X)), Y)
% 46.10/6.26 = { by axiom 10 (maddux4_definiton_of_meet) }
% 46.10/6.26 join(join(meet(X, complement(Y)), complement(join(complement(Y), complement(X)))), Y)
% 46.10/6.26 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.10/6.26 join(join(meet(X, complement(Y)), complement(join(complement(X), complement(Y)))), Y)
% 46.10/6.26 = { by lemma 21 }
% 46.10/6.26 join(X, Y)
% 46.10/6.26
% 46.10/6.26 Lemma 46: join(composition(X, Z), composition(X, Y)) = composition(X, join(Y, Z)).
% 46.10/6.26 Proof:
% 46.10/6.26 join(composition(X, Z), composition(X, Y))
% 46.10/6.26 = { by axiom 1 (converse_idempotence) R->L }
% 46.10/6.26 join(composition(X, Z), composition(X, converse(converse(Y))))
% 46.10/6.26 = { by axiom 1 (converse_idempotence) R->L }
% 46.10/6.26 converse(converse(join(composition(X, Z), composition(X, converse(converse(Y))))))
% 46.10/6.26 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.10/6.26 converse(converse(join(composition(X, converse(converse(Y))), composition(X, Z))))
% 46.10/6.26 = { by axiom 6 (converse_additivity) }
% 46.10/6.26 converse(join(converse(composition(X, converse(converse(Y)))), converse(composition(X, Z))))
% 46.10/6.26 = { by axiom 8 (converse_multiplicativity) }
% 46.10/6.26 converse(join(composition(converse(converse(converse(Y))), converse(X)), converse(composition(X, Z))))
% 46.10/6.26 = { by axiom 1 (converse_idempotence) }
% 46.10/6.26 converse(join(composition(converse(Y), converse(X)), converse(composition(X, Z))))
% 46.10/6.26 = { by axiom 2 (maddux1_join_commutativity) }
% 46.10/6.26 converse(join(converse(composition(X, Z)), composition(converse(Y), converse(X))))
% 46.10/6.26 = { by axiom 8 (converse_multiplicativity) }
% 46.10/6.26 converse(join(composition(converse(Z), converse(X)), composition(converse(Y), converse(X))))
% 46.10/6.26 = { by axiom 11 (composition_distributivity) R->L }
% 46.10/6.26 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 46.10/6.26 = { by axiom 2 (maddux1_join_commutativity) }
% 46.10/6.26 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 46.10/6.26 = { by axiom 1 (converse_idempotence) R->L }
% 46.10/6.26 converse(composition(join(converse(converse(converse(Y))), converse(Z)), converse(X)))
% 46.10/6.26 = { by axiom 6 (converse_additivity) R->L }
% 46.10/6.26 converse(composition(converse(join(converse(converse(Y)), Z)), converse(X)))
% 46.10/6.26 = { by axiom 2 (maddux1_join_commutativity) }
% 46.10/6.26 converse(composition(converse(join(Z, converse(converse(Y)))), converse(X)))
% 46.10/6.26 = { by axiom 8 (converse_multiplicativity) R->L }
% 46.10/6.26 converse(converse(composition(X, join(Z, converse(converse(Y))))))
% 46.10/6.26 = { by axiom 1 (converse_idempotence) }
% 46.10/6.26 composition(X, join(Z, converse(converse(Y))))
% 46.10/6.26 = { by axiom 1 (converse_idempotence) }
% 46.10/6.26 composition(X, join(Z, Y))
% 46.10/6.26 = { by axiom 2 (maddux1_join_commutativity) }
% 46.10/6.26 composition(X, join(Y, Z))
% 46.10/6.26
% 46.10/6.26 Lemma 47: meet(meet(X, complement(Z)), complement(Y)) = meet(X, complement(join(Y, Z))).
% 46.10/6.26 Proof:
% 46.10/6.26 meet(meet(X, complement(Z)), complement(Y))
% 46.10/6.26 = { by lemma 38 R->L }
% 46.10/6.26 complement(join(Y, complement(meet(X, complement(Z)))))
% 46.10/6.26 = { by lemma 40 }
% 46.10/6.26 complement(join(Y, join(Z, complement(X))))
% 46.10/6.26 = { by lemma 35 }
% 46.10/6.26 complement(join(Z, join(Y, complement(X))))
% 46.10/6.26 = { by axiom 7 (maddux2_join_associativity) }
% 46.10/6.26 complement(join(join(Z, Y), complement(X)))
% 46.10/6.26 = { by lemma 38 }
% 46.10/6.26 meet(X, complement(join(Z, Y)))
% 46.10/6.26 = { by axiom 2 (maddux1_join_commutativity) }
% 46.10/6.26 meet(X, complement(join(Y, Z)))
% 46.10/6.26
% 46.10/6.26 Lemma 48: meet(X, join(Y, meet(X, complement(Y)))) = X.
% 46.10/6.26 Proof:
% 46.10/6.26 meet(X, join(Y, meet(X, complement(Y))))
% 46.10/6.26 = { by lemma 25 R->L }
% 46.10/6.26 meet(X, join(Y, meet(X, join(zero, complement(Y)))))
% 46.10/6.26 = { by lemma 34 R->L }
% 46.10/6.26 meet(X, join(meet(Y, top), meet(X, join(zero, complement(Y)))))
% 46.10/6.26 = { by lemma 31 R->L }
% 46.10/6.26 meet(X, join(complement(join(zero, complement(Y))), meet(X, join(zero, complement(Y)))))
% 46.10/6.26 = { by lemma 26 R->L }
% 46.10/6.26 join(meet(X, join(complement(join(zero, complement(Y))), meet(X, join(zero, complement(Y))))), zero)
% 46.10/6.26 = { by lemma 14 R->L }
% 46.10/6.26 join(meet(X, join(complement(join(zero, complement(Y))), meet(X, join(zero, complement(Y))))), complement(top))
% 46.10/6.26 = { by axiom 4 (def_top) }
% 46.10/6.26 join(meet(X, join(complement(join(zero, complement(Y))), meet(X, join(zero, complement(Y))))), complement(join(join(complement(X), complement(join(zero, complement(Y)))), complement(join(complement(X), complement(join(zero, complement(Y))))))))
% 46.10/6.26 = { by axiom 10 (maddux4_definiton_of_meet) R->L }
% 46.10/6.26 join(meet(X, join(complement(join(zero, complement(Y))), meet(X, join(zero, complement(Y))))), complement(join(join(complement(X), complement(join(zero, complement(Y)))), meet(X, join(zero, complement(Y))))))
% 46.10/6.26 = { by axiom 7 (maddux2_join_associativity) R->L }
% 46.10/6.26 join(meet(X, join(complement(join(zero, complement(Y))), meet(X, join(zero, complement(Y))))), complement(join(complement(X), join(complement(join(zero, complement(Y))), meet(X, join(zero, complement(Y)))))))
% 46.10/6.26 = { by lemma 21 }
% 46.10/6.26 X
% 46.10/6.26
% 46.10/6.26 Lemma 49: join(meet(composition(X, meet(Y, complement(Z))), complement(composition(X, Z))), composition(X, Z)) = join(meet(composition(X, Y), complement(composition(X, Z))), composition(X, Z)).
% 46.10/6.26 Proof:
% 46.10/6.26 join(meet(composition(X, meet(Y, complement(Z))), complement(composition(X, Z))), composition(X, Z))
% 46.10/6.26 = { by lemma 45 }
% 46.10/6.26 join(composition(X, meet(Y, complement(Z))), composition(X, Z))
% 46.10/6.26 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.10/6.26 join(composition(X, Z), composition(X, meet(Y, complement(Z))))
% 46.10/6.26 = { by lemma 46 }
% 46.10/6.26 composition(X, join(meet(Y, complement(Z)), Z))
% 46.10/6.26 = { by lemma 45 }
% 46.10/6.26 composition(X, join(Y, Z))
% 46.10/6.26 = { by lemma 46 R->L }
% 46.10/6.26 join(composition(X, Z), composition(X, Y))
% 46.10/6.26 = { by axiom 2 (maddux1_join_commutativity) }
% 46.10/6.26 join(composition(X, Y), composition(X, Z))
% 46.10/6.26 = { by lemma 45 R->L }
% 46.10/6.27 join(meet(composition(X, Y), complement(composition(X, Z))), composition(X, Z))
% 46.10/6.27
% 46.10/6.27 Goal 1 (goals): tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))) = tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(composition(x0, x1), complement(composition(x0, x2)))).
% 46.10/6.27 Proof:
% 46.10/6.27 tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.10/6.27 = { by lemma 29 R->L }
% 46.10/6.27 tuple(complement(complement(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.10/6.27 = { by axiom 2 (maddux1_join_commutativity) }
% 46.10/6.27 tuple(complement(complement(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2)))))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.10/6.27 = { by lemma 42 R->L }
% 46.10/6.27 tuple(complement(meet(complement(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), complement(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2)))))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.10/6.27 = { by lemma 28 R->L }
% 46.10/6.27 tuple(complement(join(zero, meet(complement(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), complement(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))))))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.10/6.27 = { by axiom 5 (def_zero) }
% 46.10/6.27 tuple(complement(join(meet(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), complement(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))))), meet(complement(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), complement(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))))))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.10/6.27 = { by lemma 47 }
% 46.10/6.27 tuple(complement(join(meet(composition(x0_2, x1_2), complement(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), composition(x0_2, x2_2)))), meet(complement(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), complement(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))))))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.10/6.27 = { by lemma 49 R->L }
% 46.10/6.27 tuple(complement(join(meet(composition(x0_2, x1_2), complement(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), composition(x0_2, x2_2)))), meet(complement(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), complement(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))))))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.10/6.27 = { by lemma 47 R->L }
% 46.10/6.27 tuple(complement(join(meet(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), complement(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))))), meet(complement(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), complement(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))))))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.10/6.27 = { by lemma 30 }
% 46.10/6.27 tuple(complement(join(meet(complement(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2)))), meet(complement(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), complement(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))))))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.10/6.27 = { by lemma 33 }
% 46.10/6.27 tuple(complement(complement(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.10/6.27 = { by lemma 29 }
% 46.10/6.27 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.10/6.27 = { by lemma 48 R->L }
% 46.10/6.27 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))))))
% 46.10/6.27 = { by lemma 44 R->L }
% 46.10/6.27 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))), complement(join(zero, complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))))))))))
% 46.10/6.27 = { by lemma 31 }
% 46.10/6.27 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))), meet(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), top))))))
% 46.10/6.27 = { by lemma 34 }
% 46.10/6.27 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))), meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))))))))
% 46.10/6.27 = { by axiom 2 (maddux1_join_commutativity) }
% 46.10/6.27 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))))))))
% 46.10/6.27 = { by lemma 38 R->L }
% 46.10/6.27 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))))))))))
% 46.10/6.27 = { by lemma 42 R->L }
% 46.10/6.27 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))))))))))
% 46.10/6.27 = { by lemma 41 R->L }
% 46.10/6.27 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))), complement(meet(complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))))))))))
% 46.10/6.27 = { by lemma 48 }
% 46.10/6.27 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))), complement(complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))))))))
% 46.10/6.28 = { by lemma 29 }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))), meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))))))
% 46.10/6.28 = { by lemma 30 }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2)))))))))
% 46.10/6.28 = { by lemma 43 R->L }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))))))))
% 46.10/6.28 = { by lemma 30 }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), meet(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), complement(meet(composition(x0, x1), complement(composition(x0, x2)))))))))
% 46.10/6.28 = { by lemma 43 }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))))))))
% 46.10/6.28 = { by lemma 30 }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), meet(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))))))
% 46.10/6.28 = { by axiom 10 (maddux4_definiton_of_meet) }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), complement(join(complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))))))))
% 46.10/6.28 = { by lemma 25 R->L }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), join(zero, complement(join(complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))))))))))
% 46.10/6.28 = { by lemma 14 R->L }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), join(complement(top), complement(join(complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))))))))))
% 46.10/6.28 = { by lemma 27 R->L }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), join(complement(join(top, meet(composition(x0, x1), complement(composition(x0, x2))))), complement(join(complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))))))))))
% 46.10/6.28 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), join(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), top)), complement(join(complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))))))))))
% 46.10/6.28 = { by lemma 15 R->L }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), join(complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), join(meet(composition(x0, x1), complement(composition(x0, x2))), complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))))))), complement(join(complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))))))))))
% 46.10/6.28 = { by lemma 36 R->L }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), join(complement(join(complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))), complement(join(complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))))))))))
% 46.10/6.28 = { by axiom 2 (maddux1_join_commutativity) }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), join(complement(join(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))))), complement(join(complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))))))))))
% 46.10/6.28 = { by lemma 38 }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))), complement(join(complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))))))))))
% 46.10/6.28 = { by lemma 21 }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2)))))))
% 46.10/6.28 = { by lemma 30 R->L }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))))))
% 46.10/6.28 = { by lemma 47 }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, meet(x1, complement(x2))), complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), composition(x0, x2)))))))
% 46.10/6.28 = { by lemma 49 R->L }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(composition(x0, meet(x1, complement(x2))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), composition(x0, x2)))))))
% 46.10/6.28 = { by lemma 47 R->L }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), complement(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))))))))
% 46.10/6.28 = { by axiom 5 (def_zero) R->L }
% 46.10/6.28 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), zero)))
% 46.10/6.29 = { by lemma 26 }
% 46.10/6.29 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.10/6.29 = { by lemma 30 }
% 46.10/6.29 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(meet(composition(x0, x1), complement(composition(x0, x2))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))))
% 46.10/6.29 = { by axiom 10 (maddux4_definiton_of_meet) }
% 46.10/6.29 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), complement(join(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))))))
% 46.10/6.29 = { by lemma 25 R->L }
% 46.10/6.29 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), join(zero, complement(join(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))))))
% 46.10/6.29 = { by lemma 14 R->L }
% 46.10/6.29 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), join(complement(top), complement(join(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))))))
% 46.10/6.29 = { by lemma 27 R->L }
% 46.10/6.29 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), join(complement(join(top, meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))))), complement(join(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))))))
% 46.10/6.29 = { by axiom 2 (maddux1_join_commutativity) R->L }
% 46.10/6.29 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), join(complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), top)), complement(join(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))))))
% 46.10/6.29 = { by axiom 4 (def_top) }
% 46.10/6.29 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), join(complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), join(meet(composition(x0, x1), complement(composition(x0, x2))), complement(meet(composition(x0, x1), complement(composition(x0, x2))))))), complement(join(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))))))
% 46.10/6.29 = { by lemma 36 R->L }
% 46.10/6.29 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), join(complement(join(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))), complement(join(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))))))
% 46.10/6.29 = { by axiom 2 (maddux1_join_commutativity) }
% 46.10/6.29 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), join(complement(join(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), complement(meet(composition(x0, x1), complement(composition(x0, x2)))))), complement(join(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))))))
% 46.10/6.29 = { by lemma 38 }
% 46.10/6.29 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), join(meet(meet(composition(x0, x1), complement(composition(x0, x2))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))), complement(join(complement(meet(composition(x0, x1), complement(composition(x0, x2)))), complement(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))))))
% 46.10/6.29 = { by lemma 21 }
% 46.10/6.29 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(composition(x0, x1), complement(composition(x0, x2))))
% 46.10/6.29 % SZS output end Proof
% 46.10/6.29
% 46.10/6.29 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------