%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LAT066-1 : TPTP v9.3.1. Released v2.5.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n017.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 11:42:56 AM UTC 2026
% Result : Unsatisfiable 127.39s 16.54s
% Output : Proof 129.64s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : LAT066-1 : TPTP v9.3.1. Released v2.5.0.
% 0.00/0.05 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.11/0.37 % Computer : n017.cluster.edu
% 0.11/0.37 % Model : x86_64 x86_64
% 0.11/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.37 % Memory : 8046.5625MB
% 0.11/0.37 % OS : Linux 6.8.0-71-generic
% 0.11/0.37 % CPULimit : 300
% 0.11/0.37 % WCLimit : 300
% 0.11/0.37 % DateTime : Sun Sep 27 13:54:19 UTC 2026
% 0.11/0.38 % CPUTime :
% 0.11/0.38 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 127.39/16.54 Command-line arguments: --no-flatten-goal
% 127.39/16.54
% 127.39/16.54 % SZS status Unsatisfiable
% 127.39/16.54
% 128.89/16.75 % SZS output start Proof
% 128.89/16.75 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 128.89/16.75 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 128.89/16.75 Axiom 3 (top): join(X, complement(X)) = n1.
% 128.89/16.75 Axiom 4 (idempotence_of_meet): meet(X, X) = X.
% 128.89/16.75 Axiom 5 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 128.89/16.75 Axiom 6 (bottom): meet(X, complement(X)) = n0.
% 128.89/16.75 Axiom 7 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 128.89/16.75 Axiom 8 (absorption2): join(X, meet(X, Y)) = X.
% 128.89/16.75 Axiom 9 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 128.89/16.75 Axiom 10 (absorption1): meet(X, join(X, Y)) = X.
% 128.89/16.75 Axiom 11 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 128.89/16.75 Axiom 12 (complements_are_unique): ifeq(join(X, Y), n1, ifeq(meet(X, Y), n0, complement(X), Y), Y) = Y.
% 128.89/16.75 Axiom 13 (g61): meet(X, join(Y, join(Z, meet(W, join(X, meet(Y, Z)))))) = meet(X, join(Y, join(Z, meet(X, W)))).
% 128.89/16.75
% 128.89/16.75 Lemma 14: meet(X, n1) = X.
% 128.89/16.75 Proof:
% 128.89/16.75 meet(X, n1)
% 128.89/16.75 = { by axiom 3 (top) R->L }
% 128.89/16.75 meet(X, join(X, complement(X)))
% 128.89/16.75 = { by axiom 10 (absorption1) }
% 128.89/16.75 X
% 128.89/16.75
% 128.89/16.75 Lemma 15: join(X, n0) = X.
% 128.89/16.75 Proof:
% 128.89/16.75 join(X, n0)
% 128.89/16.75 = { by axiom 6 (bottom) R->L }
% 128.89/16.75 join(X, meet(X, complement(X)))
% 128.89/16.75 = { by axiom 8 (absorption2) }
% 128.89/16.75 X
% 128.89/16.75
% 128.89/16.75 Lemma 16: join(n0, X) = X.
% 128.89/16.75 Proof:
% 128.89/16.75 join(n0, X)
% 128.89/16.75 = { by axiom 2 (commutativity_of_join) R->L }
% 128.89/16.75 join(X, n0)
% 128.89/16.75 = { by lemma 15 }
% 128.89/16.75 X
% 128.89/16.75
% 128.89/16.75 Lemma 17: meet(X, n0) = n0.
% 128.89/16.75 Proof:
% 128.89/16.75 meet(X, n0)
% 128.89/16.75 = { by axiom 5 (commutativity_of_meet) R->L }
% 128.89/16.76 meet(n0, X)
% 128.89/16.76 = { by lemma 16 R->L }
% 128.89/16.76 join(n0, meet(n0, X))
% 128.89/16.76 = { by axiom 8 (absorption2) }
% 128.89/16.76 n0
% 128.89/16.76
% 128.89/16.76 Lemma 18: join(X, n1) = n1.
% 128.89/16.76 Proof:
% 128.89/16.76 join(X, n1)
% 128.89/16.76 = { by axiom 2 (commutativity_of_join) R->L }
% 128.89/16.76 join(n1, X)
% 128.89/16.76 = { by lemma 14 R->L }
% 128.89/16.76 join(n1, meet(X, n1))
% 128.89/16.76 = { by axiom 5 (commutativity_of_meet) }
% 128.89/16.76 join(n1, meet(n1, X))
% 128.89/16.76 = { by axiom 8 (absorption2) }
% 128.89/16.76 n1
% 128.89/16.76
% 128.89/16.76 Lemma 19: meet(X, meet(Y, complement(X))) = n0.
% 128.89/16.76 Proof:
% 128.89/16.76 meet(X, meet(Y, complement(X)))
% 128.89/16.76 = { by axiom 5 (commutativity_of_meet) R->L }
% 128.89/16.76 meet(X, meet(complement(X), Y))
% 128.89/16.76 = { by axiom 11 (associativity_of_meet) R->L }
% 128.89/16.76 meet(meet(X, complement(X)), Y)
% 128.89/16.76 = { by axiom 6 (bottom) }
% 128.89/16.76 meet(n0, Y)
% 128.89/16.76 = { by axiom 5 (commutativity_of_meet) R->L }
% 128.89/16.76 meet(Y, n0)
% 128.89/16.76 = { by lemma 17 }
% 128.89/16.76 n0
% 128.89/16.76
% 128.89/16.76 Lemma 20: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 128.89/16.76 Proof:
% 128.89/16.76 join(X, join(Y, meet(X, Z)))
% 128.89/16.76 = { by axiom 2 (commutativity_of_join) R->L }
% 128.89/16.76 join(X, join(meet(X, Z), Y))
% 128.89/16.76 = { by axiom 9 (associativity_of_join) R->L }
% 128.89/16.76 join(join(X, meet(X, Z)), Y)
% 128.89/16.76 = { by axiom 8 (absorption2) }
% 128.89/16.76 join(X, Y)
% 128.89/16.76
% 128.89/16.76 Lemma 21: join(X, join(X, Y)) = join(X, Y).
% 128.89/16.76 Proof:
% 128.89/16.76 join(X, join(X, Y))
% 128.89/16.76 = { by axiom 9 (associativity_of_join) R->L }
% 128.89/16.76 join(join(X, X), Y)
% 128.89/16.76 = { by axiom 1 (idempotence_of_join) }
% 128.89/16.76 join(X, Y)
% 128.89/16.76
% 128.89/16.76 Lemma 22: meet(X, join(Y, meet(Z, join(X, Y)))) = meet(X, join(Y, meet(X, Z))).
% 128.89/16.76 Proof:
% 128.89/16.76 meet(X, join(Y, meet(Z, join(X, Y))))
% 128.89/16.76 = { by axiom 4 (idempotence_of_meet) R->L }
% 128.89/16.76 meet(X, join(Y, meet(Z, join(X, meet(Y, Y)))))
% 128.89/16.76 = { by lemma 21 R->L }
% 128.89/16.76 meet(X, join(Y, join(Y, meet(Z, join(X, meet(Y, Y))))))
% 128.89/16.76 = { by axiom 13 (g61) }
% 128.89/16.76 meet(X, join(Y, join(Y, meet(X, Z))))
% 128.89/16.76 = { by lemma 21 }
% 128.89/16.76 meet(X, join(Y, meet(X, Z)))
% 128.89/16.76
% 128.89/16.76 Lemma 23: join(X, meet(Y, X)) = X.
% 128.89/16.76 Proof:
% 128.89/16.76 join(X, meet(Y, X))
% 128.89/16.76 = { by axiom 5 (commutativity_of_meet) R->L }
% 128.89/16.76 join(X, meet(X, Y))
% 128.89/16.76 = { by axiom 8 (absorption2) }
% 128.89/16.76 X
% 128.89/16.76
% 128.89/16.76 Lemma 24: join(X, meet(Y, join(X, meet(Y, Z)))) = join(X, meet(Y, Z)).
% 128.89/16.76 Proof:
% 128.89/16.76 join(X, meet(Y, join(X, meet(Y, Z))))
% 128.89/16.76 = { by axiom 5 (commutativity_of_meet) R->L }
% 128.89/16.76 join(X, meet(Y, join(X, meet(Z, Y))))
% 128.89/16.76 = { by lemma 23 R->L }
% 128.89/16.76 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, meet(Y, join(X, meet(Z, Y))))))
% 128.89/16.76 = { by axiom 5 (commutativity_of_meet) R->L }
% 128.89/16.76 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, meet(Y, join(X, meet(Y, Z))))))
% 128.89/16.76 = { by axiom 2 (commutativity_of_join) R->L }
% 128.89/16.76 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, meet(Y, join(meet(Y, Z), X)))))
% 128.89/16.76 = { by axiom 5 (commutativity_of_meet) R->L }
% 128.89/16.76 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, meet(join(meet(Y, Z), X), Y))))
% 128.89/16.76 = { by axiom 11 (associativity_of_meet) R->L }
% 128.89/16.76 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(meet(Z, join(meet(Y, Z), X)), Y)))
% 128.89/16.76 = { by axiom 5 (commutativity_of_meet) }
% 128.89/16.76 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Y, meet(Z, join(meet(Y, Z), X)))))
% 128.89/16.76 = { by axiom 11 (associativity_of_meet) R->L }
% 128.89/16.76 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(meet(Y, Z), join(meet(Y, Z), X))))
% 128.89/16.76 = { by axiom 10 (absorption1) }
% 128.89/16.76 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Y, Z)))
% 128.89/16.76 = { by axiom 5 (commutativity_of_meet) }
% 128.89/16.76 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, Y)))
% 128.89/16.76 = { by axiom 2 (commutativity_of_join) }
% 128.89/16.76 join(X, join(meet(Z, Y), meet(Y, join(X, meet(Z, Y)))))
% 128.89/16.76 = { by axiom 5 (commutativity_of_meet) }
% 128.89/16.76 join(X, join(meet(Z, Y), meet(Y, join(X, meet(Y, Z)))))
% 128.89/16.76 = { by axiom 5 (commutativity_of_meet) }
% 128.89/16.76 join(X, join(meet(Y, Z), meet(Y, join(X, meet(Y, Z)))))
% 128.89/16.76 = { by axiom 2 (commutativity_of_join) R->L }
% 128.89/16.76 join(X, join(meet(Y, Z), meet(Y, join(meet(Y, Z), X))))
% 128.89/16.77 = { by axiom 5 (commutativity_of_meet) R->L }
% 128.89/16.77 join(X, join(meet(Y, Z), meet(join(meet(Y, Z), X), Y)))
% 128.89/16.77 = { by axiom 2 (commutativity_of_join) R->L }
% 128.89/16.77 join(join(meet(Y, Z), meet(join(meet(Y, Z), X), Y)), X)
% 128.89/16.77 = { by axiom 9 (associativity_of_join) }
% 128.89/16.77 join(meet(Y, Z), join(meet(join(meet(Y, Z), X), Y), X))
% 128.89/16.77 = { by axiom 2 (commutativity_of_join) }
% 128.89/16.77 join(meet(Y, Z), join(X, meet(join(meet(Y, Z), X), Y)))
% 128.89/16.77 = { by axiom 9 (associativity_of_join) R->L }
% 128.89/16.77 join(join(meet(Y, Z), X), meet(join(meet(Y, Z), X), Y))
% 128.89/16.77 = { by axiom 8 (absorption2) }
% 128.89/16.77 join(meet(Y, Z), X)
% 128.89/16.77 = { by axiom 2 (commutativity_of_join) }
% 128.89/16.77 join(X, meet(Y, Z))
% 128.89/16.77
% 128.89/16.77 Lemma 25: meet(X, join(complement(X), meet(X, Y))) = meet(X, join(Y, complement(X))).
% 128.89/16.77 Proof:
% 128.89/16.77 meet(X, join(complement(X), meet(X, Y)))
% 128.89/16.77 = { by lemma 22 R->L }
% 128.89/16.77 meet(X, join(complement(X), meet(Y, join(X, complement(X)))))
% 128.89/16.77 = { by axiom 3 (top) }
% 128.89/16.77 meet(X, join(complement(X), meet(Y, n1)))
% 128.89/16.77 = { by lemma 14 }
% 128.89/16.77 meet(X, join(complement(X), Y))
% 128.89/16.77 = { by axiom 2 (commutativity_of_join) }
% 128.89/16.77 meet(X, join(Y, complement(X)))
% 128.89/16.77
% 128.89/16.77 Lemma 26: meet(X, join(complement(X), meet(Y, X))) = meet(X, join(Y, complement(X))).
% 128.89/16.77 Proof:
% 128.89/16.77 meet(X, join(complement(X), meet(Y, X)))
% 128.89/16.77 = { by axiom 5 (commutativity_of_meet) R->L }
% 128.89/16.77 meet(X, join(complement(X), meet(X, Y)))
% 128.89/16.77 = { by lemma 25 }
% 128.89/16.77 meet(X, join(Y, complement(X)))
% 128.89/16.77
% 128.89/16.77 Lemma 27: join(X, join(Y, complement(X))) = n1.
% 128.89/16.77 Proof:
% 128.89/16.77 join(X, join(Y, complement(X)))
% 128.89/16.77 = { by axiom 2 (commutativity_of_join) R->L }
% 128.89/16.77 join(X, join(complement(X), Y))
% 128.89/16.77 = { by axiom 9 (associativity_of_join) R->L }
% 128.89/16.77 join(join(X, complement(X)), Y)
% 128.89/16.77 = { by axiom 3 (top) }
% 128.89/16.77 join(n1, Y)
% 128.89/16.77 = { by axiom 2 (commutativity_of_join) R->L }
% 128.89/16.77 join(Y, n1)
% 128.89/16.77 = { by lemma 18 }
% 128.89/16.77 n1
% 128.89/16.77
% 128.89/16.77 Lemma 28: join(X, complement(complement(X))) = complement(complement(X)).
% 128.89/16.77 Proof:
% 128.89/16.77 join(X, complement(complement(X)))
% 128.89/16.77 = { by axiom 12 (complements_are_unique) R->L }
% 128.89/16.77 ifeq(join(complement(X), join(X, complement(complement(X)))), n1, ifeq(meet(complement(X), join(X, complement(complement(X)))), n0, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X))))
% 128.89/16.77 = { by lemma 26 R->L }
% 128.89/16.77 ifeq(join(complement(X), join(X, complement(complement(X)))), n1, ifeq(meet(complement(X), join(complement(complement(X)), meet(X, complement(X)))), n0, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X))))
% 128.89/16.77 = { by axiom 6 (bottom) }
% 128.89/16.77 ifeq(join(complement(X), join(X, complement(complement(X)))), n1, ifeq(meet(complement(X), join(complement(complement(X)), n0)), n0, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X))))
% 128.89/16.77 = { by lemma 15 }
% 128.89/16.77 ifeq(join(complement(X), join(X, complement(complement(X)))), n1, ifeq(meet(complement(X), complement(complement(X))), n0, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X))))
% 128.89/16.77 = { by axiom 6 (bottom) }
% 128.89/16.77 ifeq(join(complement(X), join(X, complement(complement(X)))), n1, ifeq(n0, n0, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X))))
% 128.89/16.77 = { by lemma 27 }
% 128.89/16.77 ifeq(n1, n1, ifeq(n0, n0, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X))))
% 128.89/16.77 = { by axiom 7 (ifeq_axiom) }
% 128.89/16.77 ifeq(n0, n0, complement(complement(X)), join(X, complement(complement(X))))
% 128.89/16.77 = { by axiom 7 (ifeq_axiom) }
% 128.89/16.77 complement(complement(X))
% 128.89/16.77
% 128.89/16.77 Lemma 29: complement(complement(X)) = X.
% 128.89/16.77 Proof:
% 128.89/16.77 complement(complement(X))
% 128.89/16.77 = { by axiom 4 (idempotence_of_meet) R->L }
% 128.89/16.77 complement(complement(meet(X, X)))
% 128.89/16.77 = { by axiom 7 (ifeq_axiom) R->L }
% 128.89/16.77 ifeq(n0, n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.77 = { by lemma 19 R->L }
% 128.89/16.77 ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.77 = { by axiom 7 (ifeq_axiom) R->L }
% 128.89/16.77 ifeq(n1, n1, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.77 = { by axiom 3 (top) R->L }
% 128.89/16.77 ifeq(join(complement(meet(X, X)), complement(complement(meet(X, X)))), n1, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.77 = { by lemma 14 R->L }
% 128.89/16.77 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), n1)), n1, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.77 = { by lemma 18 R->L }
% 128.89/16.77 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), join(X, n1))), n1, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.77 = { by axiom 3 (top) R->L }
% 128.89/16.77 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), join(X, join(meet(X, X), complement(meet(X, X)))))), n1, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.78 = { by axiom 2 (commutativity_of_join) R->L }
% 128.89/16.78 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), join(X, join(complement(meet(X, X)), meet(X, X))))), n1, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.78 = { by lemma 20 }
% 128.89/16.78 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), join(X, complement(meet(X, X))))), n1, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.78 = { by axiom 2 (commutativity_of_join) R->L }
% 128.89/16.78 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), join(complement(meet(X, X)), X))), n1, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.78 = { by lemma 14 R->L }
% 128.89/16.78 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), join(complement(meet(X, X)), meet(X, n1)))), n1, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.78 = { by axiom 3 (top) R->L }
% 128.89/16.78 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), join(complement(meet(X, X)), meet(X, join(complement(meet(X, X)), complement(complement(meet(X, X)))))))), n1, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.78 = { by axiom 2 (commutativity_of_join) R->L }
% 128.89/16.78 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), join(complement(meet(X, X)), meet(X, join(complement(complement(meet(X, X))), complement(meet(X, X))))))), n1, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.78 = { by lemma 22 }
% 128.89/16.78 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), X)))), n1, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.78 = { by lemma 24 }
% 128.89/16.78 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), X)), n1, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.78 = { by axiom 5 (commutativity_of_meet) }
% 128.89/16.78 ifeq(join(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n1, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), n0, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 128.89/16.78 = { by axiom 12 (complements_are_unique) }
% 128.89/16.78 meet(X, complement(complement(meet(X, X))))
% 128.89/16.78 = { by axiom 4 (idempotence_of_meet) }
% 128.89/16.78 meet(X, complement(complement(X)))
% 128.89/16.78 = { by lemma 28 R->L }
% 128.89/16.78 meet(X, join(X, complement(complement(X))))
% 128.89/16.78 = { by axiom 10 (absorption1) }
% 128.89/16.78 X
% 128.89/16.78
% 128.89/16.78 Lemma 30: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 128.89/16.78 Proof:
% 128.89/16.78 meet(X, meet(Y, join(X, Z)))
% 128.89/16.78 = { by axiom 5 (commutativity_of_meet) R->L }
% 128.89/16.78 meet(X, meet(join(X, Z), Y))
% 128.89/16.78 = { by axiom 11 (associativity_of_meet) R->L }
% 128.89/16.78 meet(meet(X, join(X, Z)), Y)
% 128.89/16.78 = { by axiom 10 (absorption1) }
% 128.89/16.78 meet(X, Y)
% 128.89/16.78
% 128.89/16.78 Lemma 31: meet(X, complement(join(X, Y))) = n0.
% 128.89/16.78 Proof:
% 128.89/16.78 meet(X, complement(join(X, Y)))
% 128.89/16.78 = { by lemma 30 R->L }
% 128.89/16.78 meet(X, meet(complement(join(X, Y)), join(X, Y)))
% 128.89/16.78 = { by axiom 5 (commutativity_of_meet) }
% 128.89/16.78 meet(X, meet(join(X, Y), complement(join(X, Y))))
% 128.89/16.78 = { by axiom 6 (bottom) }
% 128.89/16.78 meet(X, n0)
% 128.89/16.78 = { by lemma 17 }
% 128.89/16.78 n0
% 128.89/16.78
% 128.89/16.78 Lemma 32: join(X, join(complement(X), Y)) = n1.
% 128.89/16.78 Proof:
% 128.89/16.78 join(X, join(complement(X), Y))
% 128.89/16.78 = { by axiom 2 (commutativity_of_join) R->L }
% 128.89/16.78 join(X, join(Y, complement(X)))
% 128.89/16.78 = { by lemma 27 }
% 128.89/16.78 n1
% 128.89/16.78
% 128.89/16.78 Lemma 33: join(complement(X), complement(join(X, Y))) = complement(X).
% 128.89/16.78 Proof:
% 128.89/16.78 join(complement(X), complement(join(X, Y)))
% 128.89/16.78 = { by axiom 12 (complements_are_unique) R->L }
% 128.89/16.78 ifeq(join(X, join(complement(X), complement(join(X, Y)))), n1, ifeq(meet(X, join(complement(X), complement(join(X, Y)))), n0, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 128.89/16.78 = { by axiom 2 (commutativity_of_join) R->L }
% 128.89/16.78 ifeq(join(X, join(complement(X), complement(join(X, Y)))), n1, ifeq(meet(X, join(complement(join(X, Y)), complement(X))), n0, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 128.89/16.78 = { by lemma 25 R->L }
% 128.89/16.79 ifeq(join(X, join(complement(X), complement(join(X, Y)))), n1, ifeq(meet(X, join(complement(X), meet(X, complement(join(X, Y))))), n0, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 128.89/16.79 = { by lemma 31 }
% 128.89/16.79 ifeq(join(X, join(complement(X), complement(join(X, Y)))), n1, ifeq(meet(X, join(complement(X), n0)), n0, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 128.89/16.79 = { by lemma 15 }
% 128.89/16.79 ifeq(join(X, join(complement(X), complement(join(X, Y)))), n1, ifeq(meet(X, complement(X)), n0, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 128.89/16.79 = { by axiom 6 (bottom) }
% 128.89/16.79 ifeq(join(X, join(complement(X), complement(join(X, Y)))), n1, ifeq(n0, n0, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 128.89/16.79 = { by lemma 32 }
% 128.89/16.79 ifeq(n1, n1, ifeq(n0, n0, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 128.89/16.79 = { by axiom 7 (ifeq_axiom) }
% 128.89/16.79 ifeq(n0, n0, complement(X), join(complement(X), complement(join(X, Y))))
% 128.89/16.79 = { by axiom 7 (ifeq_axiom) }
% 128.89/16.79 complement(X)
% 128.89/16.79
% 128.89/16.79 Lemma 34: meet(X, join(Y, X)) = X.
% 128.89/16.79 Proof:
% 128.89/16.79 meet(X, join(Y, X))
% 128.89/16.79 = { by axiom 2 (commutativity_of_join) R->L }
% 128.89/16.79 meet(X, join(X, Y))
% 128.89/16.79 = { by axiom 10 (absorption1) }
% 128.89/16.79 X
% 128.89/16.79
% 128.89/16.79 Lemma 35: join(X, meet(Y, complement(X))) = join(X, Y).
% 128.89/16.79 Proof:
% 128.89/16.79 join(X, meet(Y, complement(X)))
% 128.89/16.79 = { by lemma 29 R->L }
% 128.89/16.79 complement(complement(join(X, meet(Y, complement(X)))))
% 128.89/16.79 = { by axiom 7 (ifeq_axiom) R->L }
% 128.89/16.79 ifeq(n0, n0, complement(complement(join(X, meet(Y, complement(X))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y)))))
% 128.89/16.79 = { by axiom 7 (ifeq_axiom) R->L }
% 128.89/16.79 ifeq(n1, n1, ifeq(n0, n0, complement(complement(join(X, meet(Y, complement(X))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y)))))
% 128.89/16.79 = { by lemma 32 R->L }
% 128.89/16.79 ifeq(join(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), n1, ifeq(n0, n0, complement(complement(join(X, meet(Y, complement(X))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y)))))
% 128.89/16.79 = { by axiom 6 (bottom) R->L }
% 128.89/16.79 ifeq(join(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), n1, ifeq(meet(complement(join(X, meet(Y, complement(X)))), complement(complement(join(X, meet(Y, complement(X)))))), n0, complement(complement(join(X, meet(Y, complement(X))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y)))))
% 128.89/16.79 = { by lemma 15 R->L }
% 128.89/16.79 ifeq(join(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), n1, ifeq(meet(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), n0)), n0, complement(complement(join(X, meet(Y, complement(X))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y)))))
% 128.89/16.80 = { by axiom 6 (bottom) R->L }
% 128.89/16.80 ifeq(join(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), n1, ifeq(meet(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(meet(Y, complement(join(X, meet(Y, complement(X))))), complement(meet(Y, complement(join(X, meet(Y, complement(X))))))))), n0, complement(complement(join(X, meet(Y, complement(X))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y)))))
% 128.89/16.80 = { by axiom 11 (associativity_of_meet) }
% 128.89/16.80 ifeq(join(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), n1, ifeq(meet(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, meet(complement(join(X, meet(Y, complement(X)))), complement(meet(Y, complement(join(X, meet(Y, complement(X)))))))))), n0, complement(complement(join(X, meet(Y, complement(X))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y)))))
% 128.89/16.80 = { by axiom 5 (commutativity_of_meet) R->L }
% 128.89/16.80 ifeq(join(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), n1, ifeq(meet(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, meet(complement(meet(Y, complement(join(X, meet(Y, complement(X)))))), complement(join(X, meet(Y, complement(X)))))))), n0, complement(complement(join(X, meet(Y, complement(X))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y)))))
% 128.89/16.80 = { by axiom 11 (associativity_of_meet) R->L }
% 128.89/16.80 ifeq(join(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), n1, ifeq(meet(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(meet(Y, complement(meet(Y, complement(join(X, meet(Y, complement(X))))))), complement(join(X, meet(Y, complement(X))))))), n0, complement(complement(join(X, meet(Y, complement(X))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y)))))
% 129.64/16.81 = { by lemma 26 }
% 129.64/16.81 ifeq(join(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), n1, ifeq(meet(complement(join(X, meet(Y, complement(X)))), join(meet(Y, complement(meet(Y, complement(join(X, meet(Y, complement(X))))))), complement(complement(join(X, meet(Y, complement(X))))))), n0, complement(complement(join(X, meet(Y, complement(X))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y)))))
% 129.64/16.81 = { by axiom 2 (commutativity_of_join) }
% 129.64/16.81 ifeq(join(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), n1, ifeq(meet(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, complement(join(X, meet(Y, complement(X))))))))), n0, complement(complement(join(X, meet(Y, complement(X))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y)))))
% 129.64/16.81 = { by axiom 5 (commutativity_of_meet) }
% 129.64/16.81 ifeq(join(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), n1, ifeq(meet(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), n0, complement(complement(join(X, meet(Y, complement(X))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))), join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y)))))
% 129.64/16.81 = { by axiom 12 (complements_are_unique) }
% 129.64/16.81 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))
% 129.64/16.81 = { by axiom 5 (commutativity_of_meet) }
% 129.64/16.81 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, complement(join(X, meet(Y, complement(X))))))))
% 129.64/16.81 = { by lemma 34 R->L }
% 129.64/16.81 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, meet(complement(join(X, meet(Y, complement(X)))), join(complement(X), complement(join(X, meet(Y, complement(X))))))))))
% 129.64/16.81 = { by lemma 33 }
% 129.64/16.81 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, meet(complement(join(X, meet(Y, complement(X)))), complement(X))))))
% 129.64/16.81 = { by axiom 5 (commutativity_of_meet) }
% 129.64/16.81 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, meet(complement(X), complement(join(X, meet(Y, complement(X)))))))))
% 129.64/16.81 = { by axiom 11 (associativity_of_meet) R->L }
% 129.64/16.81 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(meet(Y, complement(X)), complement(join(X, meet(Y, complement(X))))))))
% 129.64/16.81 = { by axiom 2 (commutativity_of_join) R->L }
% 129.64/16.81 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(meet(Y, complement(X)), complement(join(meet(Y, complement(X)), X))))))
% 129.64/16.81 = { by lemma 31 }
% 129.64/16.81 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(n0)))
% 129.64/16.81 = { by lemma 29 }
% 129.64/16.81 join(join(X, meet(Y, complement(X))), meet(Y, complement(n0)))
% 129.64/16.81 = { by axiom 9 (associativity_of_join) }
% 129.64/16.81 join(X, join(meet(Y, complement(X)), meet(Y, complement(n0))))
% 129.64/16.81 = { by lemma 16 R->L }
% 129.64/16.81 join(X, join(meet(Y, complement(X)), meet(Y, join(n0, complement(n0)))))
% 129.64/16.81 = { by axiom 3 (top) }
% 129.64/16.81 join(X, join(meet(Y, complement(X)), meet(Y, n1)))
% 129.64/16.81 = { by lemma 14 }
% 129.64/16.81 join(X, join(meet(Y, complement(X)), Y))
% 129.64/16.81 = { by axiom 2 (commutativity_of_join) }
% 129.64/16.81 join(X, join(Y, meet(Y, complement(X))))
% 129.64/16.81 = { by axiom 8 (absorption2) }
% 129.64/16.81 join(X, Y)
% 129.64/16.81
% 129.64/16.81 Lemma 36: meet(complement(X), join(Y, complement(join(X, Y)))) = complement(X).
% 129.64/16.81 Proof:
% 129.64/16.81 meet(complement(X), join(Y, complement(join(X, Y))))
% 129.64/16.81 = { by axiom 12 (complements_are_unique) R->L }
% 129.64/16.81 ifeq(join(X, meet(complement(X), join(Y, complement(join(X, Y))))), n1, ifeq(meet(X, meet(complement(X), join(Y, complement(join(X, Y))))), n0, complement(X), meet(complement(X), join(Y, complement(join(X, Y))))), meet(complement(X), join(Y, complement(join(X, Y)))))
% 129.64/16.81 = { by axiom 2 (commutativity_of_join) }
% 129.64/16.81 ifeq(join(meet(complement(X), join(Y, complement(join(X, Y)))), X), n1, ifeq(meet(X, meet(complement(X), join(Y, complement(join(X, Y))))), n0, complement(X), meet(complement(X), join(Y, complement(join(X, Y))))), meet(complement(X), join(Y, complement(join(X, Y)))))
% 129.64/16.81 = { by axiom 5 (commutativity_of_meet) R->L }
% 129.64/16.81 ifeq(join(meet(complement(X), join(Y, complement(join(X, Y)))), X), n1, ifeq(meet(X, meet(join(Y, complement(join(X, Y))), complement(X))), n0, complement(X), meet(complement(X), join(Y, complement(join(X, Y))))), meet(complement(X), join(Y, complement(join(X, Y)))))
% 129.64/16.81 = { by lemma 19 }
% 129.64/16.81 ifeq(join(meet(complement(X), join(Y, complement(join(X, Y)))), X), n1, ifeq(n0, n0, complement(X), meet(complement(X), join(Y, complement(join(X, Y))))), meet(complement(X), join(Y, complement(join(X, Y)))))
% 129.64/16.81 = { by axiom 7 (ifeq_axiom) }
% 129.64/16.81 ifeq(join(meet(complement(X), join(Y, complement(join(X, Y)))), X), n1, complement(X), meet(complement(X), join(Y, complement(join(X, Y)))))
% 129.64/16.81 = { by axiom 2 (commutativity_of_join) }
% 129.64/16.81 ifeq(join(X, meet(complement(X), join(Y, complement(join(X, Y))))), n1, complement(X), meet(complement(X), join(Y, complement(join(X, Y)))))
% 129.64/16.81 = { by axiom 5 (commutativity_of_meet) R->L }
% 129.64/16.81 ifeq(join(X, meet(join(Y, complement(join(X, Y))), complement(X))), n1, complement(X), meet(complement(X), join(Y, complement(join(X, Y)))))
% 129.64/16.81 = { by lemma 35 }
% 129.64/16.81 ifeq(join(X, join(Y, complement(join(X, Y)))), n1, complement(X), meet(complement(X), join(Y, complement(join(X, Y)))))
% 129.64/16.81 = { by axiom 5 (commutativity_of_meet) }
% 129.64/16.81 ifeq(join(X, join(Y, complement(join(X, Y)))), n1, complement(X), meet(join(Y, complement(join(X, Y))), complement(X)))
% 129.64/16.81 = { by axiom 9 (associativity_of_join) R->L }
% 129.64/16.81 ifeq(join(join(X, Y), complement(join(X, Y))), n1, complement(X), meet(join(Y, complement(join(X, Y))), complement(X)))
% 129.64/16.81 = { by axiom 3 (top) }
% 129.64/16.81 ifeq(n1, n1, complement(X), meet(join(Y, complement(join(X, Y))), complement(X)))
% 129.64/16.81 = { by axiom 7 (ifeq_axiom) }
% 129.64/16.81 complement(X)
% 129.64/16.82
% 129.64/16.82 Lemma 37: join(X, complement(join(X, Y))) = join(X, complement(Y)).
% 129.64/16.82 Proof:
% 129.64/16.82 join(X, complement(join(X, Y)))
% 129.64/16.82 = { by axiom 2 (commutativity_of_join) R->L }
% 129.64/16.82 join(X, complement(join(Y, X)))
% 129.64/16.82 = { by lemma 23 R->L }
% 129.64/16.82 join(join(X, complement(join(Y, X))), meet(complement(Y), join(X, complement(join(Y, X)))))
% 129.64/16.82 = { by lemma 36 }
% 129.64/16.82 join(join(X, complement(join(Y, X))), complement(Y))
% 129.64/16.82 = { by axiom 9 (associativity_of_join) }
% 129.64/16.82 join(X, join(complement(join(Y, X)), complement(Y)))
% 129.64/16.82 = { by axiom 2 (commutativity_of_join) }
% 129.64/16.82 join(X, join(complement(Y), complement(join(Y, X))))
% 129.64/16.82 = { by lemma 33 }
% 129.64/16.82 join(X, complement(Y))
% 129.64/16.82
% 129.64/16.82 Lemma 38: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 129.64/16.82 Proof:
% 129.64/16.82 complement(meet(X, complement(Y)))
% 129.64/16.82 = { by lemma 23 R->L }
% 129.64/16.82 join(complement(meet(X, complement(Y))), meet(Y, complement(meet(X, complement(Y)))))
% 129.64/16.82 = { by axiom 5 (commutativity_of_meet) R->L }
% 129.64/16.82 join(complement(meet(X, complement(Y))), meet(Y, complement(meet(complement(Y), X))))
% 129.64/16.82 = { by lemma 33 R->L }
% 129.64/16.82 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), complement(join(meet(complement(Y), X), complement(Y))))))
% 129.64/16.82 = { by axiom 2 (commutativity_of_join) }
% 129.64/16.82 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), complement(join(complement(Y), meet(complement(Y), X))))))
% 129.64/16.82 = { by axiom 8 (absorption2) }
% 129.64/16.82 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), complement(complement(Y)))))
% 129.64/16.82 = { by lemma 28 R->L }
% 129.64/16.82 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), join(Y, complement(complement(Y))))))
% 129.64/16.82 = { by axiom 2 (commutativity_of_join) R->L }
% 129.64/16.82 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), join(complement(complement(Y)), Y))))
% 129.64/16.82 = { by axiom 9 (associativity_of_join) R->L }
% 129.64/16.82 join(complement(meet(X, complement(Y))), meet(Y, join(join(complement(meet(complement(Y), X)), complement(complement(Y))), Y)))
% 129.64/16.82 = { by lemma 34 }
% 129.64/16.82 join(complement(meet(X, complement(Y))), Y)
% 129.64/16.82 = { by axiom 2 (commutativity_of_join) }
% 129.64/16.82 join(Y, complement(meet(X, complement(Y))))
% 129.64/16.82 = { by lemma 37 R->L }
% 129.64/16.82 join(Y, complement(join(Y, meet(X, complement(Y)))))
% 129.64/16.82 = { by lemma 35 }
% 129.64/16.82 join(Y, complement(join(Y, X)))
% 129.64/16.82 = { by lemma 37 }
% 129.64/16.82 join(Y, complement(X))
% 129.64/16.82
% 129.64/16.82 Lemma 39: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 129.64/16.82 Proof:
% 129.64/16.82 complement(join(X, complement(Y)))
% 129.64/16.82 = { by lemma 38 R->L }
% 129.64/16.82 complement(complement(meet(Y, complement(X))))
% 129.64/16.82 = { by lemma 29 }
% 129.64/16.82 meet(Y, complement(X))
% 129.64/16.82
% 129.64/16.82 Lemma 40: complement(join(X, Y)) = meet(complement(X), complement(Y)).
% 129.64/16.82 Proof:
% 129.64/16.82 complement(join(X, Y))
% 129.64/16.82 = { by lemma 29 R->L }
% 129.64/16.82 complement(join(X, complement(complement(Y))))
% 129.64/16.82 = { by lemma 39 }
% 129.64/16.82 meet(complement(Y), complement(X))
% 129.64/16.82 = { by axiom 5 (commutativity_of_meet) }
% 129.64/16.82 meet(complement(X), complement(Y))
% 129.64/16.82
% 129.64/16.82 Lemma 41: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 129.64/16.82 Proof:
% 129.64/16.82 complement(meet(complement(X), Y))
% 129.64/16.82 = { by axiom 5 (commutativity_of_meet) R->L }
% 129.64/16.82 complement(meet(Y, complement(X)))
% 129.64/16.82 = { by lemma 38 }
% 129.64/16.82 join(X, complement(Y))
% 129.64/16.82
% 129.64/16.82 Lemma 42: meet(complement(X), join(Y, meet(Z, complement(X)))) = meet(complement(X), join(Y, Z)).
% 129.64/16.82 Proof:
% 129.64/16.82 meet(complement(X), join(Y, meet(Z, complement(X))))
% 129.64/16.82 = { by lemma 34 R->L }
% 129.64/16.82 meet(complement(X), meet(join(Y, meet(Z, complement(X))), join(Z, join(Y, meet(Z, complement(X))))))
% 129.64/16.82 = { by lemma 20 }
% 129.64/16.82 meet(complement(X), meet(join(Y, meet(Z, complement(X))), join(Z, Y)))
% 129.64/16.82 = { by axiom 5 (commutativity_of_meet) }
% 129.64/16.82 meet(complement(X), meet(join(Z, Y), join(Y, meet(Z, complement(X)))))
% 129.64/16.82 = { by axiom 2 (commutativity_of_join) }
% 129.64/16.82 meet(complement(X), meet(join(Y, Z), join(Y, meet(Z, complement(X)))))
% 129.64/16.82 = { by lemma 39 R->L }
% 129.64/16.82 meet(complement(X), meet(join(Y, Z), join(Y, complement(join(X, complement(Z))))))
% 129.64/16.82 = { by lemma 37 R->L }
% 129.64/16.82 meet(complement(X), meet(join(Y, Z), join(Y, complement(join(Y, join(X, complement(Z)))))))
% 129.64/16.82 = { by lemma 29 R->L }
% 129.64/16.82 meet(complement(X), meet(join(Y, complement(complement(Z))), join(Y, complement(join(Y, join(X, complement(Z)))))))
% 129.64/16.82 = { by lemma 41 R->L }
% 129.64/16.82 meet(complement(X), meet(complement(meet(complement(Y), complement(Z))), join(Y, complement(join(Y, join(X, complement(Z)))))))
% 129.64/16.83 = { by axiom 11 (associativity_of_meet) R->L }
% 129.64/16.83 meet(meet(complement(X), complement(meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(X, complement(Z))))))
% 129.64/16.83 = { by lemma 40 R->L }
% 129.64/16.83 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(X, complement(Z))))))
% 129.64/16.83 = { by axiom 2 (commutativity_of_join) R->L }
% 129.64/16.83 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(complement(Z), X)))))
% 129.64/16.83 = { by axiom 9 (associativity_of_join) R->L }
% 129.64/16.83 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(join(Y, complement(Z)), X))))
% 129.64/16.83 = { by lemma 37 R->L }
% 129.64/16.83 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(join(Y, complement(join(Y, Z))), X))))
% 129.64/16.83 = { by axiom 9 (associativity_of_join) }
% 129.64/16.83 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(complement(join(Y, Z)), X)))))
% 129.64/16.83 = { by lemma 40 }
% 129.64/16.83 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(meet(complement(Y), complement(Z)), X)))))
% 129.64/16.83 = { by axiom 2 (commutativity_of_join) }
% 129.64/16.83 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(X, meet(complement(Y), complement(Z)))))))
% 129.64/16.83 = { by axiom 2 (commutativity_of_join) R->L }
% 129.64/16.83 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(join(X, meet(complement(Y), complement(Z))), Y))))
% 129.64/16.83 = { by lemma 36 }
% 129.64/16.83 complement(join(X, meet(complement(Y), complement(Z))))
% 129.64/16.83 = { by lemma 40 }
% 129.64/16.83 meet(complement(X), complement(meet(complement(Y), complement(Z))))
% 129.64/16.83 = { by lemma 41 }
% 129.64/16.83 meet(complement(X), join(Y, complement(complement(Z))))
% 129.64/16.83 = { by lemma 29 }
% 129.64/16.83 meet(complement(X), join(Y, Z))
% 129.64/16.83
% 129.64/16.83 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 129.64/16.83 Proof:
% 129.64/16.83 meet(a, join(b, c))
% 129.64/16.83 = { by lemma 29 R->L }
% 129.64/16.83 meet(complement(complement(a)), join(b, c))
% 129.64/16.83 = { by axiom 2 (commutativity_of_join) R->L }
% 129.64/16.83 meet(complement(complement(a)), join(c, b))
% 129.64/16.83 = { by lemma 42 R->L }
% 129.64/16.83 meet(complement(complement(a)), join(c, meet(b, complement(complement(a)))))
% 129.64/16.83 = { by lemma 23 R->L }
% 129.64/16.83 join(meet(complement(complement(a)), join(c, meet(b, complement(complement(a))))), meet(c, meet(complement(complement(a)), join(c, meet(b, complement(complement(a)))))))
% 129.64/16.83 = { by lemma 30 }
% 129.64/16.83 join(meet(complement(complement(a)), join(c, meet(b, complement(complement(a))))), meet(c, complement(complement(a))))
% 129.64/16.83 = { by axiom 2 (commutativity_of_join) }
% 129.64/16.83 join(meet(c, complement(complement(a))), meet(complement(complement(a)), join(c, meet(b, complement(complement(a))))))
% 129.64/16.83 = { by axiom 2 (commutativity_of_join) }
% 129.64/16.83 join(meet(c, complement(complement(a))), meet(complement(complement(a)), join(meet(b, complement(complement(a))), c)))
% 129.64/16.83 = { by lemma 42 R->L }
% 129.64/16.83 join(meet(c, complement(complement(a))), meet(complement(complement(a)), join(meet(b, complement(complement(a))), meet(c, complement(complement(a))))))
% 129.64/16.83 = { by axiom 5 (commutativity_of_meet) R->L }
% 129.64/16.83 join(meet(c, complement(complement(a))), meet(complement(complement(a)), join(meet(complement(complement(a)), b), meet(c, complement(complement(a))))))
% 129.64/16.83 = { by axiom 2 (commutativity_of_join) R->L }
% 129.64/16.83 join(meet(c, complement(complement(a))), meet(complement(complement(a)), join(meet(c, complement(complement(a))), meet(complement(complement(a)), b))))
% 129.64/16.83 = { by lemma 24 }
% 129.64/16.83 join(meet(c, complement(complement(a))), meet(complement(complement(a)), b))
% 129.64/16.83 = { by axiom 5 (commutativity_of_meet) }
% 129.64/16.83 join(meet(c, complement(complement(a))), meet(b, complement(complement(a))))
% 129.64/16.83 = { by axiom 2 (commutativity_of_join) }
% 129.64/16.84 join(meet(b, complement(complement(a))), meet(c, complement(complement(a))))
% 129.64/16.84 = { by lemma 29 }
% 129.64/16.84 join(meet(b, a), meet(c, complement(complement(a))))
% 129.64/16.84 = { by lemma 29 }
% 129.64/16.84 join(meet(b, a), meet(c, a))
% 129.64/16.84 = { by axiom 5 (commutativity_of_meet) R->L }
% 129.64/16.84 join(meet(b, a), meet(a, c))
% 129.64/16.84 = { by axiom 5 (commutativity_of_meet) R->L }
% 129.64/16.84 join(meet(a, b), meet(a, c))
% 129.64/16.84 % SZS output end Proof
% 129.64/16.84
% 129.64/16.84 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------