%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LAT200-1 : TPTP v9.3.1. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n013.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 11:44:32 AM UTC 2026
% Result : Unsatisfiable 137.09s 17.79s
% Output : Proof 139.47s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : LAT200-1 : TPTP v9.3.1. Released v3.1.0.
% 0.00/0.03 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.08/0.37 % Computer : n013.cluster.edu
% 0.08/0.37 % Model : x86_64 x86_64
% 0.08/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.37 % Memory : 8046.5625MB
% 0.08/0.37 % OS : Linux 6.8.0-71-generic
% 0.08/0.37 % CPULimit : 300
% 0.08/0.37 % WCLimit : 300
% 0.08/0.37 % DateTime : Sun Sep 27 14:05:21 UTC 2026
% 0.08/0.37 % CPUTime :
% 0.08/0.37 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 137.09/17.79 Command-line arguments: --no-flatten-goal
% 137.09/17.79
% 137.09/17.79 % SZS status Unsatisfiable
% 137.09/17.79
% 138.67/17.98 % SZS output start Proof
% 138.67/17.98 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 138.67/17.98 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 138.67/17.98 Axiom 3 (complement_join): join(X, complement(X)) = one.
% 138.67/17.98 Axiom 4 (idempotence_of_meet): meet(X, X) = X.
% 138.67/17.98 Axiom 5 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 138.67/17.98 Axiom 6 (complement_meet): meet(X, complement(X)) = zero.
% 138.67/17.98 Axiom 7 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 138.67/17.98 Axiom 8 (absorption2): join(X, meet(X, Y)) = X.
% 138.67/17.98 Axiom 9 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 138.67/17.98 Axiom 10 (absorption1): meet(X, join(X, Y)) = X.
% 138.67/17.98 Axiom 11 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 138.67/17.98 Axiom 12 (meet_join_complement): ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y) = Y.
% 138.67/17.98 Axiom 13 (equation_H50): meet(X, join(Y, meet(Z, join(X, W)))) = meet(X, join(Y, meet(Z, join(X, meet(Z, join(Y, W)))))).
% 138.67/17.98
% 138.67/17.98 Lemma 14: meet(X, one) = X.
% 138.67/17.98 Proof:
% 138.67/17.98 meet(X, one)
% 138.67/17.98 = { by axiom 3 (complement_join) R->L }
% 138.67/17.98 meet(X, join(X, complement(X)))
% 138.67/17.98 = { by axiom 10 (absorption1) }
% 138.67/17.98 X
% 138.67/17.98
% 138.67/17.98 Lemma 15: join(X, zero) = X.
% 138.67/17.98 Proof:
% 138.67/17.98 join(X, zero)
% 138.67/17.98 = { by axiom 6 (complement_meet) R->L }
% 138.67/17.98 join(X, meet(X, complement(X)))
% 138.67/17.98 = { by axiom 8 (absorption2) }
% 138.67/17.98 X
% 138.67/17.98
% 138.67/17.99 Lemma 16: meet(X, zero) = zero.
% 138.67/17.99 Proof:
% 138.67/17.99 meet(X, zero)
% 138.67/17.99 = { by axiom 5 (commutativity_of_meet) R->L }
% 138.67/17.99 meet(zero, X)
% 138.67/17.99 = { by lemma 15 R->L }
% 138.67/17.99 join(meet(zero, X), zero)
% 138.67/17.99 = { by axiom 2 (commutativity_of_join) }
% 138.67/17.99 join(zero, meet(zero, X))
% 138.67/17.99 = { by axiom 8 (absorption2) }
% 138.67/17.99 zero
% 138.67/17.99
% 138.67/17.99 Lemma 17: meet(one, X) = X.
% 138.67/17.99 Proof:
% 138.67/17.99 meet(one, X)
% 138.67/17.99 = { by axiom 5 (commutativity_of_meet) R->L }
% 138.67/17.99 meet(X, one)
% 138.67/17.99 = { by lemma 14 }
% 138.67/17.99 X
% 138.67/17.99
% 138.67/17.99 Lemma 18: join(X, one) = one.
% 138.67/17.99 Proof:
% 138.67/17.99 join(X, one)
% 138.67/17.99 = { by axiom 2 (commutativity_of_join) R->L }
% 138.67/17.99 join(one, X)
% 138.67/17.99 = { by lemma 17 R->L }
% 138.67/17.99 join(one, meet(one, X))
% 138.67/17.99 = { by axiom 8 (absorption2) }
% 138.67/17.99 one
% 138.67/17.99
% 138.67/17.99 Lemma 19: meet(X, meet(Y, complement(X))) = zero.
% 138.67/17.99 Proof:
% 138.67/17.99 meet(X, meet(Y, complement(X)))
% 138.67/17.99 = { by axiom 5 (commutativity_of_meet) R->L }
% 138.67/17.99 meet(X, meet(complement(X), Y))
% 138.67/17.99 = { by axiom 11 (associativity_of_meet) R->L }
% 138.67/17.99 meet(meet(X, complement(X)), Y)
% 138.67/17.99 = { by axiom 6 (complement_meet) }
% 138.67/17.99 meet(zero, Y)
% 138.67/17.99 = { by axiom 5 (commutativity_of_meet) R->L }
% 138.67/17.99 meet(Y, zero)
% 138.67/17.99 = { by lemma 16 }
% 138.67/17.99 zero
% 138.67/17.99
% 138.67/17.99 Lemma 20: join(X, join(Y, complement(X))) = one.
% 138.67/17.99 Proof:
% 138.67/17.99 join(X, join(Y, complement(X)))
% 138.67/17.99 = { by axiom 2 (commutativity_of_join) R->L }
% 138.67/17.99 join(X, join(complement(X), Y))
% 138.67/17.99 = { by axiom 9 (associativity_of_join) R->L }
% 138.67/17.99 join(join(X, complement(X)), Y)
% 138.67/17.99 = { by axiom 3 (complement_join) }
% 138.67/17.99 join(one, Y)
% 138.67/17.99 = { by axiom 2 (commutativity_of_join) R->L }
% 138.67/17.99 join(Y, one)
% 138.67/17.99 = { by lemma 18 }
% 138.67/17.99 one
% 138.67/17.99
% 138.67/17.99 Lemma 21: join(X, meet(Y, X)) = X.
% 138.67/17.99 Proof:
% 138.67/17.99 join(X, meet(Y, X))
% 138.67/17.99 = { by axiom 5 (commutativity_of_meet) R->L }
% 138.67/17.99 join(X, meet(X, Y))
% 138.67/17.99 = { by axiom 8 (absorption2) }
% 138.67/17.99 X
% 138.67/17.99
% 138.67/17.99 Lemma 22: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 138.67/17.99 Proof:
% 138.67/17.99 meet(X, meet(Y, Z))
% 138.67/17.99 = { by axiom 5 (commutativity_of_meet) R->L }
% 138.67/17.99 meet(meet(Y, Z), X)
% 138.67/17.99 = { by axiom 11 (associativity_of_meet) }
% 138.67/17.99 meet(Y, meet(Z, X))
% 138.67/17.99 = { by axiom 5 (commutativity_of_meet) }
% 138.67/17.99 meet(Y, meet(X, Z))
% 138.67/17.99
% 138.67/17.99 Lemma 23: meet(X, join(Y, X)) = X.
% 138.67/17.99 Proof:
% 138.67/17.99 meet(X, join(Y, X))
% 138.67/17.99 = { by axiom 2 (commutativity_of_join) R->L }
% 138.67/17.99 meet(X, join(X, Y))
% 138.67/17.99 = { by axiom 10 (absorption1) }
% 138.67/17.99 X
% 138.67/17.99
% 138.67/17.99 Lemma 24: meet(X, join(Y, meet(Z, join(X, complement(Y))))) = meet(X, join(Y, Z)).
% 138.67/17.99 Proof:
% 138.67/17.99 meet(X, join(Y, meet(Z, join(X, complement(Y)))))
% 138.67/17.99 = { by axiom 13 (equation_H50) }
% 138.67/17.99 meet(X, join(Y, meet(Z, join(X, meet(Z, join(Y, complement(Y)))))))
% 138.67/17.99 = { by axiom 3 (complement_join) }
% 138.67/17.99 meet(X, join(Y, meet(Z, join(X, meet(Z, one)))))
% 138.67/17.99 = { by lemma 14 }
% 138.67/17.99 meet(X, join(Y, meet(Z, join(X, Z))))
% 138.67/17.99 = { by lemma 23 }
% 138.67/17.99 meet(X, join(Y, Z))
% 138.67/17.99
% 138.67/17.99 Lemma 25: join(X, join(X, Y)) = join(X, Y).
% 138.67/17.99 Proof:
% 138.67/17.99 join(X, join(X, Y))
% 138.67/17.99 = { by axiom 9 (associativity_of_join) R->L }
% 138.67/17.99 join(join(X, X), Y)
% 138.67/17.99 = { by axiom 1 (idempotence_of_join) }
% 138.67/17.99 join(X, Y)
% 138.67/17.99
% 138.67/17.99 Lemma 26: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
% 138.67/17.99 Proof:
% 138.67/17.99 meet(X, meet(Y, join(Z, meet(X, Y))))
% 138.67/17.99 = { by axiom 5 (commutativity_of_meet) R->L }
% 138.67/17.99 meet(X, meet(Y, join(Z, meet(Y, X))))
% 138.67/17.99 = { by axiom 2 (commutativity_of_join) R->L }
% 138.67/17.99 meet(X, meet(Y, join(meet(Y, X), Z)))
% 138.67/17.99 = { by lemma 22 R->L }
% 138.67/17.99 meet(Y, meet(X, join(meet(Y, X), Z)))
% 138.67/17.99 = { by axiom 11 (associativity_of_meet) R->L }
% 138.67/17.99 meet(meet(Y, X), join(meet(Y, X), Z))
% 138.67/17.99 = { by axiom 10 (absorption1) }
% 138.67/17.99 meet(Y, X)
% 138.67/17.99 = { by axiom 5 (commutativity_of_meet) }
% 138.67/17.99 meet(X, Y)
% 138.67/17.99
% 138.67/17.99 Lemma 27: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 138.67/17.99 Proof:
% 138.67/17.99 join(X, join(Y, meet(Z, join(X, Y))))
% 138.67/17.99 = { by axiom 2 (commutativity_of_join) R->L }
% 138.67/17.99 join(X, join(Y, meet(Z, join(Y, X))))
% 138.67/17.99 = { by axiom 5 (commutativity_of_meet) R->L }
% 138.67/17.99 join(X, join(Y, meet(join(Y, X), Z)))
% 138.67/17.99 = { by axiom 2 (commutativity_of_join) R->L }
% 138.67/17.99 join(join(Y, meet(join(Y, X), Z)), X)
% 138.67/17.99 = { by axiom 9 (associativity_of_join) }
% 138.67/17.99 join(Y, join(meet(join(Y, X), Z), X))
% 138.67/17.99 = { by axiom 2 (commutativity_of_join) }
% 138.67/17.99 join(Y, join(X, meet(join(Y, X), Z)))
% 138.67/17.99 = { by axiom 9 (associativity_of_join) R->L }
% 138.67/17.99 join(join(Y, X), meet(join(Y, X), Z))
% 138.67/17.99 = { by axiom 8 (absorption2) }
% 138.67/17.99 join(Y, X)
% 138.67/17.99 = { by axiom 2 (commutativity_of_join) }
% 138.67/18.00 join(X, Y)
% 138.67/18.00
% 138.67/18.00 Lemma 28: meet(X, join(Y, meet(Z, join(X, Y)))) = meet(X, join(Y, meet(X, Z))).
% 138.67/18.00 Proof:
% 138.67/18.00 meet(X, join(Y, meet(Z, join(X, Y))))
% 138.67/18.00 = { by lemma 26 R->L }
% 138.67/18.00 meet(X, join(Y, meet(Z, meet(join(X, Y), join(X, meet(Z, join(X, Y)))))))
% 138.67/18.00 = { by axiom 2 (commutativity_of_join) R->L }
% 138.67/18.00 meet(X, join(Y, meet(Z, meet(join(Y, X), join(X, meet(Z, join(X, Y)))))))
% 138.67/18.00 = { by axiom 2 (commutativity_of_join) R->L }
% 138.67/18.00 meet(X, join(Y, meet(Z, meet(join(Y, X), join(X, meet(Z, join(Y, X)))))))
% 138.67/18.00 = { by axiom 5 (commutativity_of_meet) R->L }
% 138.67/18.00 meet(X, join(Y, meet(Z, meet(join(X, meet(Z, join(Y, X))), join(Y, X)))))
% 138.67/18.00 = { by lemma 27 R->L }
% 138.67/18.00 meet(X, join(Y, meet(Z, meet(join(X, meet(Z, join(Y, X))), join(Y, join(X, meet(Z, join(Y, X))))))))
% 138.67/18.00 = { by lemma 23 }
% 138.67/18.00 meet(X, join(Y, meet(Z, join(X, meet(Z, join(Y, X))))))
% 138.67/18.00 = { by axiom 13 (equation_H50) R->L }
% 138.67/18.00 meet(X, join(Y, meet(Z, join(X, X))))
% 138.67/18.00 = { by axiom 1 (idempotence_of_join) }
% 138.67/18.00 meet(X, join(Y, meet(Z, X)))
% 138.67/18.00 = { by axiom 5 (commutativity_of_meet) }
% 138.67/18.00 meet(X, join(Y, meet(X, Z)))
% 138.67/18.00
% 138.67/18.00 Lemma 29: meet(X, join(complement(X), meet(X, Y))) = meet(X, join(Y, complement(X))).
% 138.67/18.00 Proof:
% 138.67/18.00 meet(X, join(complement(X), meet(X, Y)))
% 138.67/18.00 = { by lemma 28 R->L }
% 138.67/18.00 meet(X, join(complement(X), meet(Y, join(X, complement(X)))))
% 138.67/18.00 = { by axiom 3 (complement_join) }
% 138.67/18.00 meet(X, join(complement(X), meet(Y, one)))
% 138.67/18.00 = { by lemma 14 }
% 138.67/18.00 meet(X, join(complement(X), Y))
% 138.67/18.00 = { by axiom 2 (commutativity_of_join) }
% 138.67/18.00 meet(X, join(Y, complement(X)))
% 138.67/18.00
% 138.67/18.00 Lemma 30: meet(X, complement(complement(X))) = X.
% 138.67/18.00 Proof:
% 138.67/18.00 meet(X, complement(complement(X)))
% 138.67/18.00 = { by axiom 7 (ifeq_axiom) R->L }
% 138.67/18.00 meet(X, ifeq(zero, zero, complement(complement(X)), join(X, complement(complement(X)))))
% 138.67/18.00 = { by axiom 7 (ifeq_axiom) R->L }
% 138.67/18.00 meet(X, ifeq(one, one, ifeq(zero, zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X)))))
% 138.67/18.00 = { by lemma 20 R->L }
% 138.67/18.00 meet(X, ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(zero, zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X)))))
% 138.67/18.00 = { by axiom 6 (complement_meet) R->L }
% 138.67/18.00 meet(X, ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(meet(complement(X), complement(complement(X))), zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X)))))
% 139.47/18.00 = { by lemma 15 R->L }
% 139.47/18.00 meet(X, ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(meet(complement(X), join(complement(complement(X)), zero)), zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X)))))
% 139.47/18.00 = { by axiom 6 (complement_meet) R->L }
% 139.47/18.00 meet(X, ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(meet(complement(X), join(complement(complement(X)), meet(X, complement(X)))), zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X)))))
% 139.47/18.00 = { by axiom 5 (commutativity_of_meet) R->L }
% 139.47/18.00 meet(X, ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(meet(complement(X), join(complement(complement(X)), meet(complement(X), X))), zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X)))))
% 139.47/18.00 = { by lemma 29 }
% 139.47/18.00 meet(X, ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(meet(complement(X), join(X, complement(complement(X)))), zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X)))))
% 139.47/18.00 = { by axiom 12 (meet_join_complement) }
% 139.47/18.00 meet(X, join(X, complement(complement(X))))
% 139.47/18.00 = { by axiom 10 (absorption1) }
% 139.47/18.00 X
% 139.47/18.00
% 139.47/18.00 Lemma 31: complement(complement(X)) = X.
% 139.47/18.00 Proof:
% 139.47/18.00 complement(complement(X))
% 139.47/18.00 = { by axiom 4 (idempotence_of_meet) R->L }
% 139.47/18.00 complement(complement(meet(X, X)))
% 139.47/18.01 = { by axiom 7 (ifeq_axiom) R->L }
% 139.47/18.01 ifeq(zero, zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by lemma 19 R->L }
% 139.47/18.01 ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by axiom 7 (ifeq_axiom) R->L }
% 139.47/18.01 ifeq(one, one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by lemma 20 R->L }
% 139.47/18.01 ifeq(join(complement(meet(X, X)), join(complement(complement(meet(X, X))), complement(complement(meet(X, X))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by lemma 17 R->L }
% 139.47/18.01 ifeq(join(complement(meet(X, X)), meet(one, join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by lemma 18 R->L }
% 139.47/18.01 ifeq(join(complement(meet(X, X)), meet(join(X, one), join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by axiom 3 (complement_join) R->L }
% 139.47/18.01 ifeq(join(complement(meet(X, X)), meet(join(X, join(meet(X, X), complement(meet(X, X)))), join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by axiom 9 (associativity_of_join) R->L }
% 139.47/18.01 ifeq(join(complement(meet(X, X)), meet(join(join(X, meet(X, X)), complement(meet(X, X))), join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by axiom 8 (absorption2) }
% 139.47/18.01 ifeq(join(complement(meet(X, X)), meet(join(X, complement(meet(X, X))), join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by axiom 5 (commutativity_of_meet) R->L }
% 139.47/18.01 ifeq(join(complement(meet(X, X)), meet(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), join(X, complement(meet(X, X))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by axiom 2 (commutativity_of_join) R->L }
% 139.47/18.01 ifeq(join(complement(meet(X, X)), meet(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), join(complement(meet(X, X)), X))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by lemma 21 R->L }
% 139.47/18.01 ifeq(join(complement(meet(X, X)), join(meet(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), join(complement(meet(X, X)), X)), meet(X, meet(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), join(complement(meet(X, X)), X))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by lemma 22 R->L }
% 139.47/18.01 ifeq(join(complement(meet(X, X)), join(meet(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), join(complement(meet(X, X)), X)), meet(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), meet(X, join(complement(meet(X, X)), X))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by lemma 23 }
% 139.47/18.01 ifeq(join(complement(meet(X, X)), join(meet(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), join(complement(meet(X, X)), X)), meet(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), X))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by axiom 5 (commutativity_of_meet) }
% 139.47/18.01 ifeq(join(complement(meet(X, X)), join(meet(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), join(complement(meet(X, X)), X)), meet(X, join(complement(complement(meet(X, X))), complement(complement(meet(X, X))))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.01 = { by axiom 2 (commutativity_of_join) }
% 139.47/18.02 ifeq(join(complement(meet(X, X)), join(meet(X, join(complement(complement(meet(X, X))), complement(complement(meet(X, X))))), meet(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), join(complement(meet(X, X)), X)))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.02 = { by axiom 5 (commutativity_of_meet) }
% 139.47/18.02 ifeq(join(complement(meet(X, X)), join(meet(X, join(complement(complement(meet(X, X))), complement(complement(meet(X, X))))), meet(join(complement(meet(X, X)), X), join(complement(complement(meet(X, X))), complement(complement(meet(X, X))))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.02 = { by axiom 9 (associativity_of_join) R->L }
% 139.47/18.02 ifeq(join(join(complement(meet(X, X)), meet(X, join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))))), meet(join(complement(meet(X, X)), X), join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.02 = { by axiom 5 (commutativity_of_meet) R->L }
% 139.47/18.02 ifeq(join(join(complement(meet(X, X)), meet(X, join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))))), meet(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), join(complement(meet(X, X)), X))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.02 = { by lemma 24 R->L }
% 139.47/18.02 ifeq(join(join(complement(meet(X, X)), meet(X, join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))))), meet(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), join(complement(meet(X, X)), meet(X, join(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), complement(complement(meet(X, X)))))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.02 = { by axiom 2 (commutativity_of_join) }
% 139.47/18.02 ifeq(join(join(complement(meet(X, X)), meet(X, join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))))), meet(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), join(complement(meet(X, X)), meet(X, join(complement(complement(meet(X, X))), join(complement(complement(meet(X, X))), complement(complement(meet(X, X))))))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.02 = { by lemma 25 }
% 139.47/18.02 ifeq(join(join(complement(meet(X, X)), meet(X, join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))))), meet(join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))), join(complement(meet(X, X)), meet(X, join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.02 = { by lemma 21 }
% 139.47/18.02 ifeq(join(complement(meet(X, X)), meet(X, join(complement(complement(meet(X, X))), complement(complement(meet(X, X)))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.02 = { by axiom 1 (idempotence_of_join) }
% 139.47/18.02 ifeq(join(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), one, ifeq(meet(complement(meet(X, X)), meet(X, complement(complement(meet(X, X))))), zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X))))), meet(X, complement(complement(meet(X, X)))))
% 139.47/18.02 = { by axiom 12 (meet_join_complement) }
% 139.47/18.02 meet(X, complement(complement(meet(X, X))))
% 139.47/18.02 = { by axiom 4 (idempotence_of_meet) }
% 139.47/18.02 meet(X, complement(complement(X)))
% 139.47/18.02 = { by lemma 30 }
% 139.47/18.02 X
% 139.47/18.02
% 139.47/18.02 Lemma 32: join(X, join(complement(X), Y)) = one.
% 139.47/18.02 Proof:
% 139.47/18.02 join(X, join(complement(X), Y))
% 139.47/18.02 = { by axiom 2 (commutativity_of_join) R->L }
% 139.47/18.02 join(X, join(Y, complement(X)))
% 139.47/18.02 = { by lemma 20 }
% 139.47/18.02 one
% 139.47/18.02
% 139.47/18.02 Lemma 33: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 139.47/18.02 Proof:
% 139.47/18.02 meet(X, meet(Y, join(X, Z)))
% 139.47/18.02 = { by axiom 5 (commutativity_of_meet) R->L }
% 139.47/18.02 meet(X, meet(join(X, Z), Y))
% 139.47/18.02 = { by axiom 11 (associativity_of_meet) R->L }
% 139.47/18.02 meet(meet(X, join(X, Z)), Y)
% 139.47/18.02 = { by axiom 10 (absorption1) }
% 139.47/18.02 meet(X, Y)
% 139.47/18.02
% 139.47/18.02 Lemma 34: join(complement(X), complement(join(X, Y))) = complement(X).
% 139.47/18.02 Proof:
% 139.47/18.02 join(complement(X), complement(join(X, Y)))
% 139.47/18.03 = { by axiom 12 (meet_join_complement) R->L }
% 139.47/18.03 ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(X, join(complement(X), complement(join(X, Y)))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 139.47/18.03 = { by axiom 2 (commutativity_of_join) R->L }
% 139.47/18.03 ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(X, join(complement(X), complement(join(Y, X)))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 139.47/18.03 = { by axiom 2 (commutativity_of_join) R->L }
% 139.47/18.03 ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(X, join(complement(join(Y, X)), complement(X))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 139.47/18.03 = { by lemma 29 R->L }
% 139.47/18.03 ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(X, join(complement(X), meet(X, complement(join(Y, X))))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 139.47/18.03 = { by axiom 2 (commutativity_of_join) R->L }
% 139.47/18.03 ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(X, join(complement(X), meet(X, complement(join(X, Y))))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 139.47/18.03 = { by lemma 33 R->L }
% 139.47/18.03 ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(X, join(complement(X), meet(X, meet(complement(join(X, Y)), join(X, Y))))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 139.47/18.03 = { by axiom 5 (commutativity_of_meet) }
% 139.47/18.03 ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(X, join(complement(X), meet(X, meet(join(X, Y), complement(join(X, Y)))))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 139.47/18.03 = { by axiom 6 (complement_meet) }
% 139.47/18.03 ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(X, join(complement(X), meet(X, zero))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 139.47/18.03 = { by lemma 16 }
% 139.47/18.03 ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(X, join(complement(X), zero)), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 139.47/18.03 = { by lemma 15 }
% 139.47/18.03 ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(X, complement(X)), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 139.47/18.03 = { by axiom 6 (complement_meet) }
% 139.47/18.03 ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(zero, zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 139.47/18.03 = { by lemma 32 }
% 139.47/18.03 ifeq(one, one, ifeq(zero, zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 139.47/18.03 = { by axiom 7 (ifeq_axiom) }
% 139.47/18.03 ifeq(zero, zero, complement(X), join(complement(X), complement(join(X, Y))))
% 139.47/18.03 = { by axiom 7 (ifeq_axiom) }
% 139.47/18.03 complement(X)
% 139.47/18.03
% 139.47/18.03 Lemma 35: meet(complement(X), complement(join(X, Y))) = complement(join(X, Y)).
% 139.47/18.03 Proof:
% 139.47/18.03 meet(complement(X), complement(join(X, Y)))
% 139.47/18.03 = { by axiom 5 (commutativity_of_meet) R->L }
% 139.47/18.03 meet(complement(join(X, Y)), complement(X))
% 139.47/18.03 = { by lemma 34 R->L }
% 139.47/18.03 meet(complement(join(X, Y)), join(complement(X), complement(join(X, Y))))
% 139.47/18.03 = { by lemma 23 }
% 139.47/18.03 complement(join(X, Y))
% 139.47/18.03
% 139.47/18.03 Lemma 36: join(meet(X, Y), meet(X, join(Y, Z))) = meet(X, join(Y, Z)).
% 139.47/18.03 Proof:
% 139.47/18.03 join(meet(X, Y), meet(X, join(Y, Z)))
% 139.47/18.03 = { by axiom 5 (commutativity_of_meet) R->L }
% 139.47/18.03 join(meet(Y, X), meet(X, join(Y, Z)))
% 139.47/18.03 = { by axiom 2 (commutativity_of_join) R->L }
% 139.47/18.03 join(meet(X, join(Y, Z)), meet(Y, X))
% 139.47/18.03 = { by lemma 33 R->L }
% 139.47/18.03 join(meet(X, join(Y, Z)), meet(Y, meet(X, join(Y, Z))))
% 139.47/18.03 = { by lemma 21 }
% 139.47/18.03 meet(X, join(Y, Z))
% 139.47/18.03
% 139.47/18.03 Lemma 37: meet(X, complement(meet(X, complement(Y)))) = meet(X, Y).
% 139.47/18.03 Proof:
% 139.47/18.03 meet(X, complement(meet(X, complement(Y))))
% 139.47/18.03 = { by lemma 23 R->L }
% 139.47/18.03 meet(meet(X, complement(meet(X, complement(Y)))), join(Y, meet(X, complement(meet(X, complement(Y))))))
% 139.47/18.04 = { by axiom 12 (meet_join_complement) R->L }
% 139.47/18.04 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(join(complement(Y), join(Y, meet(X, complement(meet(X, complement(Y)))))), one, ifeq(meet(complement(Y), join(Y, meet(X, complement(meet(X, complement(Y)))))), zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.04 = { by axiom 2 (commutativity_of_join) }
% 139.47/18.04 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(join(join(Y, meet(X, complement(meet(X, complement(Y))))), complement(Y)), one, ifeq(meet(complement(Y), join(Y, meet(X, complement(meet(X, complement(Y)))))), zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.04 = { by axiom 5 (commutativity_of_meet) R->L }
% 139.47/18.04 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(join(join(Y, meet(X, complement(meet(X, complement(Y))))), complement(Y)), one, ifeq(meet(complement(Y), join(Y, meet(complement(meet(X, complement(Y))), X))), zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.04 = { by lemma 24 R->L }
% 139.47/18.04 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(join(join(Y, meet(X, complement(meet(X, complement(Y))))), complement(Y)), one, ifeq(meet(complement(Y), join(Y, meet(meet(complement(meet(X, complement(Y))), X), join(complement(Y), complement(Y))))), zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.04 = { by axiom 11 (associativity_of_meet) }
% 139.47/18.04 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(join(join(Y, meet(X, complement(meet(X, complement(Y))))), complement(Y)), one, ifeq(meet(complement(Y), join(Y, meet(complement(meet(X, complement(Y))), meet(X, join(complement(Y), complement(Y)))))), zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.04 = { by lemma 22 }
% 139.47/18.04 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(join(join(Y, meet(X, complement(meet(X, complement(Y))))), complement(Y)), one, ifeq(meet(complement(Y), join(Y, meet(X, meet(complement(meet(X, complement(Y))), join(complement(Y), complement(Y)))))), zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.04 = { by axiom 1 (idempotence_of_join) }
% 139.47/18.04 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(join(join(Y, meet(X, complement(meet(X, complement(Y))))), complement(Y)), one, ifeq(meet(complement(Y), join(Y, meet(X, meet(complement(meet(X, complement(Y))), complement(Y))))), zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.04 = { by lemma 22 }
% 139.47/18.04 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(join(join(Y, meet(X, complement(meet(X, complement(Y))))), complement(Y)), one, ifeq(meet(complement(Y), join(Y, meet(complement(meet(X, complement(Y))), meet(X, complement(Y))))), zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.04 = { by axiom 5 (commutativity_of_meet) }
% 139.47/18.04 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(join(join(Y, meet(X, complement(meet(X, complement(Y))))), complement(Y)), one, ifeq(meet(complement(Y), join(Y, meet(meet(X, complement(Y)), complement(meet(X, complement(Y)))))), zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.04 = { by axiom 6 (complement_meet) }
% 139.47/18.04 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(join(join(Y, meet(X, complement(meet(X, complement(Y))))), complement(Y)), one, ifeq(meet(complement(Y), join(Y, zero)), zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.05 = { by lemma 15 }
% 139.47/18.05 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(join(join(Y, meet(X, complement(meet(X, complement(Y))))), complement(Y)), one, ifeq(meet(complement(Y), Y), zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.05 = { by axiom 5 (commutativity_of_meet) }
% 139.47/18.05 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(join(join(Y, meet(X, complement(meet(X, complement(Y))))), complement(Y)), one, ifeq(meet(Y, complement(Y)), zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.05 = { by axiom 6 (complement_meet) }
% 139.47/18.05 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(join(join(Y, meet(X, complement(meet(X, complement(Y))))), complement(Y)), one, ifeq(zero, zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.05 = { by axiom 9 (associativity_of_join) }
% 139.47/18.05 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(join(Y, join(meet(X, complement(meet(X, complement(Y)))), complement(Y))), one, ifeq(zero, zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.05 = { by lemma 20 }
% 139.47/18.05 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(one, one, ifeq(zero, zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.05 = { by axiom 7 (ifeq_axiom) }
% 139.47/18.05 meet(meet(X, complement(meet(X, complement(Y)))), ifeq(zero, zero, complement(complement(Y)), join(Y, meet(X, complement(meet(X, complement(Y)))))))
% 139.47/18.05 = { by axiom 7 (ifeq_axiom) }
% 139.47/18.05 meet(meet(X, complement(meet(X, complement(Y)))), complement(complement(Y)))
% 139.47/18.05 = { by lemma 31 }
% 139.47/18.05 meet(meet(X, complement(meet(X, complement(Y)))), Y)
% 139.47/18.05 = { by axiom 11 (associativity_of_meet) }
% 139.47/18.05 meet(X, meet(complement(meet(X, complement(Y))), Y))
% 139.47/18.05 = { by axiom 5 (commutativity_of_meet) }
% 139.47/18.05 meet(X, meet(Y, complement(meet(X, complement(Y)))))
% 139.47/18.05 = { by axiom 5 (commutativity_of_meet) R->L }
% 139.47/18.05 meet(X, meet(Y, complement(meet(complement(Y), X))))
% 139.47/18.05 = { by lemma 34 R->L }
% 139.47/18.05 meet(X, meet(Y, join(complement(meet(complement(Y), X)), complement(join(meet(complement(Y), X), complement(Y))))))
% 139.47/18.05 = { by axiom 2 (commutativity_of_join) }
% 139.47/18.05 meet(X, meet(Y, join(complement(meet(complement(Y), X)), complement(join(complement(Y), meet(complement(Y), X))))))
% 139.47/18.05 = { by axiom 8 (absorption2) }
% 139.47/18.05 meet(X, meet(Y, join(complement(meet(complement(Y), X)), complement(complement(Y)))))
% 139.47/18.05 = { by axiom 2 (commutativity_of_join) R->L }
% 139.47/18.05 meet(X, meet(Y, join(complement(complement(Y)), complement(meet(complement(Y), X)))))
% 139.47/18.05 = { by lemma 36 R->L }
% 139.47/18.05 meet(X, join(meet(Y, complement(complement(Y))), meet(Y, join(complement(complement(Y)), complement(meet(complement(Y), X))))))
% 139.47/18.05 = { by lemma 30 }
% 139.47/18.05 meet(X, join(Y, meet(Y, join(complement(complement(Y)), complement(meet(complement(Y), X))))))
% 139.47/18.05 = { by axiom 8 (absorption2) }
% 139.47/18.05 meet(X, Y)
% 139.47/18.05
% 139.47/18.05 Lemma 38: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 139.47/18.05 Proof:
% 139.47/18.05 meet(X, complement(meet(X, Y)))
% 139.47/18.05 = { by lemma 37 R->L }
% 139.47/18.05 meet(X, complement(meet(X, complement(meet(X, complement(Y))))))
% 139.47/18.05 = { by lemma 37 }
% 139.47/18.05 meet(X, meet(X, complement(Y)))
% 139.47/18.05 = { by axiom 11 (associativity_of_meet) R->L }
% 139.47/18.05 meet(meet(X, X), complement(Y))
% 139.47/18.05 = { by axiom 4 (idempotence_of_meet) }
% 139.47/18.05 meet(X, complement(Y))
% 139.47/18.05
% 139.47/18.05 Lemma 39: join(X, meet(complement(X), Y)) = join(X, Y).
% 139.47/18.05 Proof:
% 139.47/18.05 join(X, meet(complement(X), Y))
% 139.47/18.05 = { by axiom 5 (commutativity_of_meet) R->L }
% 139.47/18.05 join(X, meet(Y, complement(X)))
% 139.47/18.05 = { by lemma 31 R->L }
% 139.47/18.05 complement(complement(join(X, meet(Y, complement(X)))))
% 139.47/18.05 = { by lemma 35 R->L }
% 139.47/18.05 complement(meet(complement(X), complement(join(X, meet(Y, complement(X))))))
% 139.47/18.05 = { by lemma 38 R->L }
% 139.47/18.05 complement(meet(complement(X), complement(meet(complement(X), join(X, meet(Y, complement(X)))))))
% 139.47/18.05 = { by axiom 5 (commutativity_of_meet) R->L }
% 139.47/18.05 complement(meet(complement(X), complement(meet(complement(X), join(X, meet(complement(X), Y))))))
% 139.47/18.05 = { by lemma 28 R->L }
% 139.47/18.06 complement(meet(complement(X), complement(meet(complement(X), join(X, meet(Y, join(complement(X), X)))))))
% 139.47/18.06 = { by axiom 2 (commutativity_of_join) }
% 139.47/18.06 complement(meet(complement(X), complement(meet(complement(X), join(X, meet(Y, join(X, complement(X))))))))
% 139.47/18.06 = { by axiom 3 (complement_join) }
% 139.47/18.06 complement(meet(complement(X), complement(meet(complement(X), join(X, meet(Y, one))))))
% 139.47/18.06 = { by lemma 14 }
% 139.47/18.06 complement(meet(complement(X), complement(meet(complement(X), join(X, Y)))))
% 139.47/18.06 = { by lemma 38 }
% 139.47/18.06 complement(meet(complement(X), complement(join(X, Y))))
% 139.47/18.06 = { by lemma 35 }
% 139.47/18.06 complement(complement(join(X, Y)))
% 139.47/18.06 = { by lemma 31 }
% 139.47/18.06 join(X, Y)
% 139.47/18.06
% 139.47/18.06 Lemma 40: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 139.47/18.06 Proof:
% 139.47/18.06 complement(meet(X, complement(Y)))
% 139.47/18.06 = { by axiom 5 (commutativity_of_meet) R->L }
% 139.47/18.06 complement(meet(complement(Y), X))
% 139.47/18.06 = { by axiom 7 (ifeq_axiom) R->L }
% 139.47/18.06 ifeq(zero, zero, complement(meet(complement(Y), X)), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y))))))
% 139.47/18.06 = { by axiom 7 (ifeq_axiom) R->L }
% 139.47/18.06 ifeq(one, one, ifeq(zero, zero, complement(meet(complement(Y), X)), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y))))))
% 139.47/18.06 = { by lemma 32 R->L }
% 139.47/18.06 ifeq(join(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), one, ifeq(zero, zero, complement(meet(complement(Y), X)), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y))))))
% 139.47/18.06 = { by axiom 6 (complement_meet) R->L }
% 139.47/18.06 ifeq(join(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), one, ifeq(meet(meet(complement(Y), X), complement(meet(complement(Y), X))), zero, complement(meet(complement(Y), X)), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y))))))
% 139.47/18.06 = { by lemma 15 R->L }
% 139.47/18.06 ifeq(join(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), one, ifeq(meet(meet(complement(Y), X), join(complement(meet(complement(Y), X)), zero)), zero, complement(meet(complement(Y), X)), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y))))))
% 139.47/18.06 = { by lemma 19 R->L }
% 139.47/18.06 ifeq(join(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), one, ifeq(meet(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(join(meet(complement(Y), X), complement(Y)), meet(Y, complement(join(meet(complement(Y), X), complement(Y))))))), zero, complement(meet(complement(Y), X)), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y))))))
% 139.47/18.06 = { by axiom 5 (commutativity_of_meet) R->L }
% 139.47/18.07 ifeq(join(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), one, ifeq(meet(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(meet(Y, complement(join(meet(complement(Y), X), complement(Y)))), join(meet(complement(Y), X), complement(Y))))), zero, complement(meet(complement(Y), X)), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y))))))
% 139.47/18.07 = { by axiom 13 (equation_H50) }
% 139.47/18.07 ifeq(join(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), one, ifeq(meet(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(meet(Y, complement(join(meet(complement(Y), X), complement(Y)))), join(meet(complement(Y), X), meet(meet(Y, complement(join(meet(complement(Y), X), complement(Y)))), join(complement(meet(complement(Y), X)), complement(Y))))))), zero, complement(meet(complement(Y), X)), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y))))))
% 139.47/18.07 = { by lemma 25 R->L }
% 139.47/18.07 ifeq(join(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), one, ifeq(meet(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(meet(Y, complement(join(meet(complement(Y), X), complement(Y)))), join(meet(complement(Y), X), meet(meet(Y, complement(join(meet(complement(Y), X), complement(Y)))), join(complement(meet(complement(Y), X)), join(complement(meet(complement(Y), X)), complement(Y)))))))), zero, complement(meet(complement(Y), X)), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y))))))
% 139.47/18.07 = { by axiom 13 (equation_H50) R->L }
% 139.47/18.07 ifeq(join(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), one, ifeq(meet(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(meet(Y, complement(join(meet(complement(Y), X), complement(Y)))), join(meet(complement(Y), X), join(complement(meet(complement(Y), X)), complement(Y)))))), zero, complement(meet(complement(Y), X)), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y))))))
% 139.47/18.07 = { by lemma 32 }
% 139.47/18.07 ifeq(join(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), one, ifeq(meet(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(meet(Y, complement(join(meet(complement(Y), X), complement(Y)))), one))), zero, complement(meet(complement(Y), X)), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y))))))
% 139.47/18.07 = { by lemma 14 }
% 139.47/18.07 ifeq(join(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), one, ifeq(meet(meet(complement(Y), X), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), zero, complement(meet(complement(Y), X)), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))), join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y))))))
% 139.47/18.07 = { by axiom 12 (meet_join_complement) }
% 139.47/18.07 join(complement(meet(complement(Y), X)), meet(Y, complement(join(meet(complement(Y), X), complement(Y)))))
% 139.47/18.07 = { by axiom 2 (commutativity_of_join) }
% 139.47/18.07 join(complement(meet(complement(Y), X)), meet(Y, complement(join(complement(Y), meet(complement(Y), X)))))
% 139.47/18.07 = { by axiom 8 (absorption2) }
% 139.47/18.07 join(complement(meet(complement(Y), X)), meet(Y, complement(complement(Y))))
% 139.47/18.07 = { by axiom 2 (commutativity_of_join) }
% 139.47/18.07 join(meet(Y, complement(complement(Y))), complement(meet(complement(Y), X)))
% 139.47/18.07 = { by lemma 30 }
% 139.47/18.07 join(Y, complement(meet(complement(Y), X)))
% 139.47/18.07 = { by lemma 39 R->L }
% 139.47/18.07 join(Y, meet(complement(Y), complement(meet(complement(Y), X))))
% 139.47/18.07 = { by lemma 38 }
% 139.47/18.07 join(Y, meet(complement(Y), complement(X)))
% 139.47/18.07 = { by lemma 39 }
% 139.47/18.07 join(Y, complement(X))
% 139.47/18.07
% 139.47/18.07 Lemma 41: complement(meet(X, Y)) = join(complement(X), complement(Y)).
% 139.47/18.07 Proof:
% 139.47/18.07 complement(meet(X, Y))
% 139.47/18.07 = { by lemma 31 R->L }
% 139.47/18.07 complement(meet(X, complement(complement(Y))))
% 139.47/18.07 = { by lemma 40 }
% 139.47/18.07 join(complement(Y), complement(X))
% 139.47/18.07 = { by axiom 2 (commutativity_of_join) }
% 139.47/18.07 join(complement(X), complement(Y))
% 139.47/18.07
% 139.47/18.07 Lemma 42: meet(X, join(Y, meet(X, Z))) = meet(X, join(Z, Y)).
% 139.47/18.07 Proof:
% 139.47/18.07 meet(X, join(Y, meet(X, Z)))
% 139.47/18.07 = { by lemma 31 R->L }
% 139.47/18.07 meet(X, join(complement(complement(Y)), meet(X, Z)))
% 139.47/18.07 = { by lemma 31 R->L }
% 139.47/18.07 meet(X, join(complement(complement(Y)), meet(X, complement(complement(Z)))))
% 139.47/18.07 = { by lemma 31 R->L }
% 139.47/18.07 meet(X, join(complement(complement(Y)), complement(complement(meet(X, complement(complement(Z)))))))
% 139.47/18.07 = { by lemma 40 }
% 139.47/18.07 meet(X, join(complement(complement(Y)), complement(join(complement(Z), complement(X)))))
% 139.47/18.07 = { by axiom 2 (commutativity_of_join) }
% 139.47/18.07 meet(X, join(complement(complement(Y)), complement(join(complement(X), complement(Z)))))
% 139.47/18.07 = { by lemma 41 R->L }
% 139.47/18.07 meet(X, complement(meet(complement(Y), join(complement(X), complement(Z)))))
% 139.47/18.08 = { by lemma 38 R->L }
% 139.47/18.08 meet(X, complement(meet(X, meet(complement(Y), join(complement(X), complement(Z))))))
% 139.47/18.08 = { by axiom 5 (commutativity_of_meet) R->L }
% 139.47/18.08 meet(X, complement(meet(X, meet(join(complement(X), complement(Z)), complement(Y)))))
% 139.47/18.08 = { by lemma 41 R->L }
% 139.47/18.08 meet(X, complement(meet(X, meet(complement(meet(X, Z)), complement(Y)))))
% 139.47/18.08 = { by axiom 11 (associativity_of_meet) R->L }
% 139.47/18.08 meet(X, complement(meet(meet(X, complement(meet(X, Z))), complement(Y))))
% 139.47/18.08 = { by lemma 38 }
% 139.47/18.08 meet(X, complement(meet(meet(X, complement(Z)), complement(Y))))
% 139.47/18.08 = { by axiom 11 (associativity_of_meet) }
% 139.47/18.08 meet(X, complement(meet(X, meet(complement(Z), complement(Y)))))
% 139.47/18.08 = { by axiom 5 (commutativity_of_meet) }
% 139.47/18.08 meet(X, complement(meet(X, meet(complement(Y), complement(Z)))))
% 139.47/18.08 = { by lemma 38 }
% 139.47/18.08 meet(X, complement(meet(complement(Y), complement(Z))))
% 139.47/18.08 = { by lemma 40 }
% 139.47/18.08 meet(X, join(Z, complement(complement(Y))))
% 139.47/18.08 = { by lemma 31 }
% 139.47/18.08 meet(X, join(Z, Y))
% 139.47/18.08
% 139.47/18.08 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 139.47/18.08 Proof:
% 139.47/18.08 meet(a, join(b, c))
% 139.47/18.08 = { by axiom 2 (commutativity_of_join) R->L }
% 139.47/18.08 meet(a, join(c, b))
% 139.47/18.08 = { by lemma 42 R->L }
% 139.47/18.08 meet(a, join(b, meet(a, c)))
% 139.47/18.08 = { by lemma 36 R->L }
% 139.47/18.08 join(meet(a, b), meet(a, join(b, meet(a, c))))
% 139.47/18.08 = { by lemma 42 R->L }
% 139.47/18.08 join(meet(a, b), meet(a, join(meet(a, c), meet(a, b))))
% 139.47/18.08 = { by axiom 2 (commutativity_of_join) R->L }
% 139.47/18.08 join(meet(a, b), meet(a, join(meet(a, b), meet(a, c))))
% 139.47/18.08 = { by axiom 5 (commutativity_of_meet) R->L }
% 139.47/18.08 join(meet(a, b), meet(a, join(meet(a, b), meet(c, a))))
% 139.47/18.08 = { by lemma 21 R->L }
% 139.47/18.08 join(meet(a, b), join(meet(a, join(meet(a, b), meet(c, a))), meet(c, meet(a, join(meet(a, b), meet(c, a))))))
% 139.47/18.08 = { by lemma 26 }
% 139.47/18.08 join(meet(a, b), join(meet(a, join(meet(a, b), meet(c, a))), meet(c, a)))
% 139.47/18.08 = { by axiom 2 (commutativity_of_join) }
% 139.47/18.08 join(meet(a, b), join(meet(c, a), meet(a, join(meet(a, b), meet(c, a)))))
% 139.47/18.08 = { by axiom 5 (commutativity_of_meet) }
% 139.47/18.08 join(meet(a, b), join(meet(c, a), meet(a, join(meet(a, b), meet(a, c)))))
% 139.47/18.08 = { by axiom 5 (commutativity_of_meet) }
% 139.47/18.08 join(meet(a, b), join(meet(a, c), meet(a, join(meet(a, b), meet(a, c)))))
% 139.47/18.08 = { by lemma 27 }
% 139.47/18.08 join(meet(a, b), meet(a, c))
% 139.47/18.08 % SZS output end Proof
% 139.47/18.08
% 139.47/18.08 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------