%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LAT217-1 : TPTP v9.3.1. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n006.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 11:44:33 AM UTC 2026
% Result : Unsatisfiable 233.78s 29.98s
% Output : Proof 234.08s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : LAT217-1 : TPTP v9.3.1. Released v3.1.0.
% 0.00/0.07 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.18/0.45 % Computer : n006.cluster.edu
% 0.18/0.45 % Model : x86_64 x86_64
% 0.18/0.45 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.45 % Memory : 8046.5625MB
% 0.18/0.45 % OS : Linux 6.8.0-71-generic
% 0.18/0.45 % CPULimit : 300
% 0.18/0.45 % WCLimit : 300
% 0.18/0.45 % DateTime : Sun Sep 27 14:06:10 UTC 2026
% 0.18/0.45 % CPUTime :
% 0.18/0.45 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 233.78/29.98 Command-line arguments: --no-flatten-goal
% 233.78/29.98
% 233.78/29.98 % SZS status Unsatisfiable
% 233.78/29.98
% 234.08/30.04 % SZS output start Proof
% 234.08/30.04 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 234.08/30.04 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 234.08/30.04 Axiom 3 (complement_join): join(X, complement(X)) = one.
% 234.08/30.04 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 234.08/30.04 Axiom 5 (complement_meet): meet(X, complement(X)) = zero.
% 234.08/30.04 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 234.08/30.04 Axiom 7 (absorption2): join(X, meet(X, Y)) = X.
% 234.08/30.04 Axiom 8 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 234.08/30.04 Axiom 9 (absorption1): meet(X, join(X, Y)) = X.
% 234.08/30.04 Axiom 10 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 234.08/30.04 Axiom 11 (meet_join_complement): ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y) = Y.
% 234.08/30.04 Axiom 12 (equation_H82): meet(X, join(meet(Y, join(X, Z)), meet(Z, join(X, Y)))) = join(meet(X, Y), meet(X, Z)).
% 234.08/30.04
% 234.08/30.04 Lemma 13: meet(X, one) = X.
% 234.08/30.04 Proof:
% 234.08/30.04 meet(X, one)
% 234.08/30.04 = { by axiom 3 (complement_join) R->L }
% 234.08/30.04 meet(X, join(X, complement(X)))
% 234.08/30.04 = { by axiom 9 (absorption1) }
% 234.08/30.04 X
% 234.08/30.04
% 234.08/30.04 Lemma 14: meet(one, X) = X.
% 234.08/30.04 Proof:
% 234.08/30.04 meet(one, X)
% 234.08/30.04 = { by axiom 4 (commutativity_of_meet) R->L }
% 234.08/30.04 meet(X, one)
% 234.08/30.04 = { by lemma 13 }
% 234.08/30.04 X
% 234.08/30.04
% 234.08/30.04 Lemma 15: join(X, one) = one.
% 234.08/30.04 Proof:
% 234.08/30.04 join(X, one)
% 234.08/30.04 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.04 join(one, X)
% 234.08/30.04 = { by lemma 14 R->L }
% 234.08/30.04 join(one, meet(one, X))
% 234.08/30.04 = { by axiom 7 (absorption2) }
% 234.08/30.04 one
% 234.08/30.04
% 234.08/30.04 Lemma 16: join(X, zero) = X.
% 234.08/30.04 Proof:
% 234.08/30.04 join(X, zero)
% 234.08/30.04 = { by axiom 5 (complement_meet) R->L }
% 234.08/30.04 join(X, meet(X, complement(X)))
% 234.08/30.04 = { by axiom 7 (absorption2) }
% 234.08/30.04 X
% 234.08/30.04
% 234.08/30.04 Lemma 17: join(X, join(Y, complement(X))) = one.
% 234.08/30.04 Proof:
% 234.08/30.04 join(X, join(Y, complement(X)))
% 234.08/30.04 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.04 join(X, join(complement(X), Y))
% 234.08/30.04 = { by axiom 8 (associativity_of_join) R->L }
% 234.08/30.04 join(join(X, complement(X)), Y)
% 234.08/30.04 = { by axiom 3 (complement_join) }
% 234.08/30.04 join(one, Y)
% 234.08/30.04 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.04 join(Y, one)
% 234.08/30.04 = { by lemma 15 }
% 234.08/30.04 one
% 234.08/30.04
% 234.08/30.04 Lemma 18: meet(X, join(Y, meet(complement(X), join(X, Y)))) = meet(X, Y).
% 234.08/30.04 Proof:
% 234.08/30.04 meet(X, join(Y, meet(complement(X), join(X, Y))))
% 234.08/30.04 = { by lemma 13 R->L }
% 234.08/30.04 meet(X, join(meet(Y, one), meet(complement(X), join(X, Y))))
% 234.08/30.04 = { by axiom 3 (complement_join) R->L }
% 234.08/30.04 meet(X, join(meet(Y, join(X, complement(X))), meet(complement(X), join(X, Y))))
% 234.08/30.04 = { by axiom 12 (equation_H82) }
% 234.08/30.04 join(meet(X, Y), meet(X, complement(X)))
% 234.08/30.04 = { by axiom 5 (complement_meet) }
% 234.08/30.04 join(meet(X, Y), zero)
% 234.08/30.04 = { by lemma 16 }
% 234.08/30.04 meet(X, Y)
% 234.08/30.04
% 234.08/30.04 Lemma 19: meet(X, join(Y, meet(complement(X), join(Y, X)))) = meet(X, Y).
% 234.08/30.04 Proof:
% 234.08/30.04 meet(X, join(Y, meet(complement(X), join(Y, X))))
% 234.08/30.04 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.04 meet(X, join(Y, meet(complement(X), join(X, Y))))
% 234.08/30.04 = { by lemma 18 }
% 234.08/30.04 meet(X, Y)
% 234.08/30.04
% 234.08/30.04 Lemma 20: meet(X, complement(complement(X))) = X.
% 234.08/30.04 Proof:
% 234.08/30.04 meet(X, complement(complement(X)))
% 234.08/30.04 = { by axiom 6 (ifeq_axiom) R->L }
% 234.08/30.04 meet(X, ifeq(zero, zero, complement(complement(X)), join(X, complement(complement(X)))))
% 234.08/30.04 = { by axiom 6 (ifeq_axiom) R->L }
% 234.08/30.04 meet(X, ifeq(one, one, ifeq(zero, zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X)))))
% 234.08/30.04 = { by lemma 17 R->L }
% 234.08/30.04 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)))))
% 234.08/30.04 = { by axiom 5 (complement_meet) R->L }
% 234.08/30.04 meet(X, ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(meet(X, complement(X)), zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X)))))
% 234.08/30.04 = { by axiom 4 (commutativity_of_meet) R->L }
% 234.08/30.04 meet(X, ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(meet(complement(X), X), zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X)))))
% 234.08/30.04 = { by lemma 19 R->L }
% 234.08/30.04 meet(X, ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(meet(complement(X), join(X, meet(complement(complement(X)), join(X, complement(X))))), zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X)))))
% 234.08/30.04 = { by axiom 3 (complement_join) }
% 234.08/30.04 meet(X, ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(meet(complement(X), join(X, meet(complement(complement(X)), one))), zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X)))))
% 234.08/30.04 = { by lemma 13 }
% 234.08/30.04 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)))))
% 234.08/30.04 = { by axiom 11 (meet_join_complement) }
% 234.08/30.04 meet(X, join(X, complement(complement(X))))
% 234.08/30.04 = { by axiom 9 (absorption1) }
% 234.08/30.04 X
% 234.08/30.04
% 234.08/30.04 Lemma 21: complement(complement(X)) = X.
% 234.08/30.04 Proof:
% 234.08/30.04 complement(complement(X))
% 234.08/30.04 = { by axiom 6 (ifeq_axiom) R->L }
% 234.08/30.04 ifeq(one, one, complement(complement(X)), X)
% 234.08/30.04 = { by axiom 3 (complement_join) R->L }
% 234.08/30.04 ifeq(join(X, complement(X)), one, complement(complement(X)), X)
% 234.08/30.04 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.04 ifeq(join(complement(X), X), one, complement(complement(X)), X)
% 234.08/30.04 = { by lemma 20 R->L }
% 234.08/30.04 ifeq(join(complement(X), X), one, complement(complement(X)), meet(X, complement(complement(X))))
% 234.08/30.04 = { by lemma 20 R->L }
% 234.08/30.04 ifeq(join(complement(X), meet(X, complement(complement(X)))), one, complement(complement(X)), meet(X, complement(complement(X))))
% 234.08/30.04 = { by axiom 6 (ifeq_axiom) R->L }
% 234.08/30.04 ifeq(join(complement(X), meet(X, complement(complement(X)))), one, ifeq(zero, zero, complement(complement(X)), meet(X, complement(complement(X)))), meet(X, complement(complement(X))))
% 234.08/30.04 = { by axiom 7 (absorption2) R->L }
% 234.08/30.04 ifeq(join(complement(X), meet(X, complement(complement(X)))), one, ifeq(join(zero, meet(zero, X)), zero, complement(complement(X)), meet(X, complement(complement(X)))), meet(X, complement(complement(X))))
% 234.08/30.04 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.04 ifeq(join(complement(X), meet(X, complement(complement(X)))), one, ifeq(join(meet(zero, X), zero), zero, complement(complement(X)), meet(X, complement(complement(X)))), meet(X, complement(complement(X))))
% 234.08/30.04 = { by lemma 16 }
% 234.08/30.04 ifeq(join(complement(X), meet(X, complement(complement(X)))), one, ifeq(meet(zero, X), zero, complement(complement(X)), meet(X, complement(complement(X)))), meet(X, complement(complement(X))))
% 234.08/30.04 = { by axiom 5 (complement_meet) R->L }
% 234.08/30.04 ifeq(join(complement(X), meet(X, complement(complement(X)))), one, ifeq(meet(meet(complement(X), complement(complement(X))), X), zero, complement(complement(X)), meet(X, complement(complement(X)))), meet(X, complement(complement(X))))
% 234.08/30.04 = { by axiom 10 (associativity_of_meet) }
% 234.08/30.04 ifeq(join(complement(X), meet(X, complement(complement(X)))), one, ifeq(meet(complement(X), meet(complement(complement(X)), X)), zero, complement(complement(X)), meet(X, complement(complement(X)))), meet(X, complement(complement(X))))
% 234.08/30.04 = { by axiom 4 (commutativity_of_meet) }
% 234.08/30.04 ifeq(join(complement(X), meet(X, complement(complement(X)))), one, ifeq(meet(complement(X), meet(X, complement(complement(X)))), zero, complement(complement(X)), meet(X, complement(complement(X)))), meet(X, complement(complement(X))))
% 234.08/30.04 = { by axiom 11 (meet_join_complement) }
% 234.08/30.04 meet(X, complement(complement(X)))
% 234.08/30.04 = { by lemma 20 }
% 234.08/30.04 X
% 234.08/30.04
% 234.08/30.04 Lemma 22: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 234.08/30.04 Proof:
% 234.08/30.04 meet(X, meet(Y, Z))
% 234.08/30.04 = { by axiom 4 (commutativity_of_meet) R->L }
% 234.08/30.04 meet(meet(Y, Z), X)
% 234.08/30.04 = { by axiom 10 (associativity_of_meet) }
% 234.08/30.04 meet(Y, meet(Z, X))
% 234.08/30.04 = { by axiom 4 (commutativity_of_meet) }
% 234.08/30.04 meet(Y, meet(X, Z))
% 234.08/30.04
% 234.08/30.04 Lemma 23: meet(X, join(Y, X)) = X.
% 234.08/30.04 Proof:
% 234.08/30.04 meet(X, join(Y, X))
% 234.08/30.04 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.04 meet(X, join(X, Y))
% 234.08/30.04 = { by axiom 9 (absorption1) }
% 234.08/30.04 X
% 234.08/30.04
% 234.08/30.04 Lemma 24: join(X, meet(Y, X)) = X.
% 234.08/30.04 Proof:
% 234.08/30.04 join(X, meet(Y, X))
% 234.08/30.04 = { by axiom 4 (commutativity_of_meet) R->L }
% 234.08/30.04 join(X, meet(X, Y))
% 234.08/30.04 = { by axiom 7 (absorption2) }
% 234.08/30.04 X
% 234.08/30.04
% 234.08/30.04 Lemma 25: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 234.08/30.04 Proof:
% 234.08/30.04 join(Y, join(X, Z))
% 234.08/30.04 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.04 join(join(X, Z), Y)
% 234.08/30.04 = { by axiom 8 (associativity_of_join) }
% 234.08/30.04 join(X, join(Z, Y))
% 234.08/30.04 = { by axiom 2 (commutativity_of_join) }
% 234.08/30.04 join(X, join(Y, Z))
% 234.08/30.04
% 234.08/30.04 Lemma 26: join(X, join(Y, complement(join(X, Y)))) = one.
% 234.08/30.04 Proof:
% 234.08/30.04 join(X, join(Y, complement(join(X, Y))))
% 234.08/30.04 = { by axiom 8 (associativity_of_join) R->L }
% 234.08/30.04 join(join(X, Y), complement(join(X, Y)))
% 234.08/30.04 = { by axiom 3 (complement_join) }
% 234.08/30.04 one
% 234.08/30.04
% 234.08/30.04 Lemma 27: meet(X, meet(Y, join(Z, X))) = meet(X, Y).
% 234.08/30.04 Proof:
% 234.08/30.04 meet(X, meet(Y, join(Z, X)))
% 234.08/30.04 = { by lemma 22 R->L }
% 234.08/30.04 meet(Y, meet(X, join(Z, X)))
% 234.08/30.04 = { by lemma 23 }
% 234.08/30.04 meet(Y, X)
% 234.08/30.04 = { by axiom 4 (commutativity_of_meet) }
% 234.08/30.04 meet(X, Y)
% 234.08/30.04
% 234.08/30.04 Lemma 28: meet(X, meet(join(Y, X), Z)) = meet(X, Z).
% 234.08/30.04 Proof:
% 234.08/30.04 meet(X, meet(join(Y, X), Z))
% 234.08/30.04 = { by axiom 4 (commutativity_of_meet) R->L }
% 234.08/30.04 meet(X, meet(Z, join(Y, X)))
% 234.08/30.04 = { by lemma 27 }
% 234.08/30.04 meet(X, Z)
% 234.08/30.04
% 234.08/30.04 Lemma 29: meet(X, join(Y, complement(join(X, Y)))) = meet(X, join(Y, complement(X))).
% 234.08/30.04 Proof:
% 234.08/30.04 meet(X, join(Y, complement(join(X, Y))))
% 234.08/30.04 = { by lemma 18 R->L }
% 234.08/30.04 meet(X, join(join(Y, complement(join(X, Y))), meet(complement(X), join(X, join(Y, complement(join(X, Y)))))))
% 234.08/30.04 = { by lemma 26 }
% 234.08/30.04 meet(X, join(join(Y, complement(join(X, Y))), meet(complement(X), one)))
% 234.08/30.04 = { by axiom 8 (associativity_of_join) }
% 234.08/30.04 meet(X, join(Y, join(complement(join(X, Y)), meet(complement(X), one))))
% 234.08/30.05 = { by lemma 13 }
% 234.08/30.05 meet(X, join(Y, join(complement(join(X, Y)), complement(X))))
% 234.08/30.05 = { by axiom 2 (commutativity_of_join) }
% 234.08/30.05 meet(X, join(Y, join(complement(X), complement(join(X, Y)))))
% 234.08/30.05 = { by axiom 11 (meet_join_complement) R->L }
% 234.08/30.05 meet(X, join(Y, 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))))))
% 234.08/30.05 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.05 meet(X, join(Y, 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))))))
% 234.08/30.05 = { by lemma 21 R->L }
% 234.08/30.05 meet(X, join(Y, ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(X, join(complement(X), complement(join(Y, complement(complement(X)))))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))))
% 234.08/30.05 = { by lemma 21 R->L }
% 234.08/30.05 meet(X, join(Y, ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(complement(complement(X)), join(complement(X), complement(join(Y, complement(complement(X)))))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))))
% 234.08/30.05 = { by lemma 28 R->L }
% 234.08/30.05 meet(X, join(Y, ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(complement(complement(X)), meet(join(Y, complement(complement(X))), join(complement(X), complement(join(Y, complement(complement(X))))))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))))
% 234.08/30.05 = { by lemma 13 R->L }
% 234.08/30.05 meet(X, join(Y, ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(complement(complement(X)), meet(join(Y, complement(complement(X))), join(complement(X), meet(complement(join(Y, complement(complement(X)))), one)))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))))
% 234.08/30.05 = { by lemma 17 R->L }
% 234.08/30.05 meet(X, join(Y, ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(complement(complement(X)), meet(join(Y, complement(complement(X))), join(complement(X), meet(complement(join(Y, complement(complement(X)))), join(complement(X), join(Y, complement(complement(X)))))))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))))
% 234.08/30.05 = { by lemma 19 }
% 234.08/30.05 meet(X, join(Y, ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(complement(complement(X)), meet(join(Y, complement(complement(X))), complement(X))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))))
% 234.08/30.05 = { by axiom 4 (commutativity_of_meet) }
% 234.08/30.05 meet(X, join(Y, ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(complement(complement(X)), meet(complement(X), join(Y, complement(complement(X))))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))))
% 234.08/30.05 = { by lemma 27 }
% 234.08/30.05 meet(X, join(Y, ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(complement(complement(X)), complement(X)), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))))
% 234.08/30.05 = { by axiom 4 (commutativity_of_meet) }
% 234.08/30.05 meet(X, join(Y, ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(complement(X), complement(complement(X))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))))
% 234.08/30.05 = { by axiom 5 (complement_meet) }
% 234.08/30.05 meet(X, join(Y, 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))))))
% 234.08/30.05 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.05 meet(X, join(Y, ifeq(join(X, join(complement(join(X, Y)), complement(X))), one, ifeq(zero, zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))))
% 234.08/30.05 = { by lemma 17 }
% 234.08/30.05 meet(X, join(Y, ifeq(one, one, ifeq(zero, zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))))
% 234.08/30.05 = { by axiom 6 (ifeq_axiom) }
% 234.08/30.05 meet(X, join(Y, ifeq(zero, zero, complement(X), join(complement(X), complement(join(X, Y))))))
% 234.08/30.05 = { by axiom 6 (ifeq_axiom) }
% 234.08/30.05 meet(X, join(Y, complement(X)))
% 234.08/30.05
% 234.08/30.05 Lemma 30: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
% 234.08/30.05 Proof:
% 234.08/30.05 meet(X, meet(join(X, Y), Z))
% 234.08/30.05 = { by axiom 10 (associativity_of_meet) R->L }
% 234.08/30.05 meet(meet(X, join(X, Y)), Z)
% 234.08/30.05 = { by axiom 9 (absorption1) }
% 234.08/30.05 meet(X, Z)
% 234.08/30.05
% 234.08/30.05 Lemma 31: join(complement(X), complement(join(complement(X), meet(X, Y)))) = complement(meet(X, Y)).
% 234.08/30.05 Proof:
% 234.08/30.05 join(complement(X), complement(join(complement(X), meet(X, Y))))
% 234.08/30.05 = { by axiom 4 (commutativity_of_meet) R->L }
% 234.08/30.05 join(complement(X), complement(join(complement(X), meet(Y, X))))
% 234.08/30.05 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.05 join(complement(X), complement(join(meet(Y, X), complement(X))))
% 234.08/30.05 = { by lemma 24 R->L }
% 234.08/30.05 join(complement(join(X, meet(Y, X))), complement(join(meet(Y, X), complement(X))))
% 234.08/30.05 = { by lemma 24 R->L }
% 234.08/30.05 join(complement(join(X, meet(Y, X))), complement(join(meet(Y, X), complement(join(X, meet(Y, X))))))
% 234.08/30.05 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.05 join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(X, meet(Y, X))))))
% 234.08/30.05 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.05 join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X)))))
% 234.08/30.05 = { by axiom 11 (meet_join_complement) R->L }
% 234.08/30.05 ifeq(join(meet(Y, X), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X)))))), one, ifeq(meet(meet(Y, X), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X)))))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X)))))), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.05 = { by lemma 26 }
% 234.08/30.05 ifeq(one, one, ifeq(meet(meet(Y, X), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X)))))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X)))))), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.05 = { by axiom 6 (ifeq_axiom) }
% 234.08/30.05 ifeq(meet(meet(Y, X), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X)))))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.05 = { by lemma 29 }
% 234.08/30.05 ifeq(meet(meet(Y, X), join(complement(join(meet(Y, X), X)), complement(meet(Y, X)))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.05 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.05 ifeq(meet(meet(Y, X), join(complement(meet(Y, X)), complement(join(meet(Y, X), X)))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.05 = { by lemma 30 R->L }
% 234.08/30.05 ifeq(meet(meet(Y, X), meet(join(meet(Y, X), X), join(complement(meet(Y, X)), complement(join(meet(Y, X), X))))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.05 = { by lemma 29 R->L }
% 234.08/30.05 ifeq(meet(meet(Y, X), meet(join(meet(Y, X), X), join(complement(meet(Y, X)), complement(join(join(meet(Y, X), X), complement(meet(Y, X))))))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.05 = { by lemma 30 }
% 234.08/30.05 ifeq(meet(meet(Y, X), join(complement(meet(Y, X)), complement(join(join(meet(Y, X), X), complement(meet(Y, X)))))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.05 = { by axiom 8 (associativity_of_join) }
% 234.08/30.05 ifeq(meet(meet(Y, X), join(complement(meet(Y, X)), complement(join(meet(Y, X), join(X, complement(meet(Y, X))))))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.05 = { by axiom 4 (commutativity_of_meet) R->L }
% 234.08/30.05 ifeq(meet(meet(Y, X), join(complement(meet(Y, X)), complement(join(meet(Y, X), join(X, complement(meet(X, Y))))))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.05 = { by axiom 7 (absorption2) R->L }
% 234.08/30.05 ifeq(meet(meet(Y, X), join(complement(meet(Y, X)), complement(join(meet(Y, X), join(join(X, meet(X, Y)), complement(meet(X, Y))))))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.05 = { by axiom 8 (associativity_of_join) }
% 234.08/30.05 ifeq(meet(meet(Y, X), join(complement(meet(Y, X)), complement(join(meet(Y, X), join(X, join(meet(X, Y), complement(meet(X, Y)))))))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.05 = { by axiom 3 (complement_join) }
% 234.08/30.05 ifeq(meet(meet(Y, X), join(complement(meet(Y, X)), complement(join(meet(Y, X), join(X, one))))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.05 = { by lemma 15 }
% 234.08/30.05 ifeq(meet(meet(Y, X), join(complement(meet(Y, X)), complement(join(meet(Y, X), one)))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.05 = { by lemma 15 }
% 234.08/30.06 ifeq(meet(meet(Y, X), join(complement(meet(Y, X)), complement(one))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.06 = { by lemma 14 R->L }
% 234.08/30.06 ifeq(meet(meet(Y, X), join(complement(meet(Y, X)), meet(one, complement(one)))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.06 = { by axiom 5 (complement_meet) }
% 234.08/30.06 ifeq(meet(meet(Y, X), join(complement(meet(Y, X)), zero)), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.06 = { by lemma 16 }
% 234.08/30.06 ifeq(meet(meet(Y, X), complement(meet(Y, X))), zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.06 = { by axiom 5 (complement_meet) }
% 234.08/30.06 ifeq(zero, zero, complement(meet(Y, X)), join(complement(join(meet(Y, X), X)), complement(join(meet(Y, X), complement(join(meet(Y, X), X))))))
% 234.08/30.06 = { by axiom 6 (ifeq_axiom) }
% 234.08/30.06 complement(meet(Y, X))
% 234.08/30.06 = { by axiom 4 (commutativity_of_meet) }
% 234.08/30.06 complement(meet(X, Y))
% 234.08/30.06
% 234.08/30.06 Lemma 32: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
% 234.08/30.06 Proof:
% 234.08/30.06 meet(X, meet(Y, join(Z, meet(X, Y))))
% 234.08/30.06 = { by axiom 4 (commutativity_of_meet) R->L }
% 234.08/30.06 meet(X, meet(Y, join(Z, meet(Y, X))))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.06 meet(X, meet(Y, join(meet(Y, X), Z)))
% 234.08/30.06 = { by lemma 22 R->L }
% 234.08/30.06 meet(Y, meet(X, join(meet(Y, X), Z)))
% 234.08/30.06 = { by axiom 10 (associativity_of_meet) R->L }
% 234.08/30.06 meet(meet(Y, X), join(meet(Y, X), Z))
% 234.08/30.06 = { by axiom 9 (absorption1) }
% 234.08/30.06 meet(Y, X)
% 234.08/30.06 = { by axiom 4 (commutativity_of_meet) }
% 234.08/30.06 meet(X, Y)
% 234.08/30.06
% 234.08/30.06 Lemma 33: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 234.08/30.06 Proof:
% 234.08/30.06 join(X, join(Y, meet(Z, join(X, Y))))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.06 join(X, join(Y, meet(Z, join(Y, X))))
% 234.08/30.06 = { by axiom 4 (commutativity_of_meet) R->L }
% 234.08/30.06 join(X, join(Y, meet(join(Y, X), Z)))
% 234.08/30.06 = { by lemma 25 }
% 234.08/30.06 join(Y, join(X, meet(join(Y, X), Z)))
% 234.08/30.06 = { by axiom 8 (associativity_of_join) R->L }
% 234.08/30.06 join(join(Y, X), meet(join(Y, X), Z))
% 234.08/30.06 = { by axiom 7 (absorption2) }
% 234.08/30.06 join(Y, X)
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) }
% 234.08/30.06 join(X, Y)
% 234.08/30.06
% 234.08/30.06 Lemma 34: meet(X, join(complement(X), meet(X, Y))) = meet(X, Y).
% 234.08/30.06 Proof:
% 234.08/30.06 meet(X, join(complement(X), meet(X, Y)))
% 234.08/30.06 = { by lemma 21 R->L }
% 234.08/30.06 complement(complement(meet(X, join(complement(X), meet(X, Y)))))
% 234.08/30.06 = { by lemma 31 R->L }
% 234.08/30.06 complement(join(complement(X), complement(join(complement(X), meet(X, join(complement(X), meet(X, Y)))))))
% 234.08/30.06 = { by axiom 4 (commutativity_of_meet) R->L }
% 234.08/30.06 complement(join(complement(X), complement(join(complement(X), meet(X, join(complement(X), meet(Y, X)))))))
% 234.08/30.06 = { by lemma 24 R->L }
% 234.08/30.06 complement(join(complement(X), complement(join(complement(X), join(meet(X, join(complement(X), meet(Y, X))), meet(Y, meet(X, join(complement(X), meet(Y, X)))))))))
% 234.08/30.06 = { by lemma 32 }
% 234.08/30.06 complement(join(complement(X), complement(join(complement(X), join(meet(X, join(complement(X), meet(Y, X))), meet(Y, X))))))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) }
% 234.08/30.06 complement(join(complement(X), complement(join(complement(X), join(meet(Y, X), meet(X, join(complement(X), meet(Y, X))))))))
% 234.08/30.06 = { by axiom 4 (commutativity_of_meet) }
% 234.08/30.06 complement(join(complement(X), complement(join(complement(X), join(meet(Y, X), meet(X, join(complement(X), meet(X, Y))))))))
% 234.08/30.06 = { by axiom 4 (commutativity_of_meet) }
% 234.08/30.06 complement(join(complement(X), complement(join(complement(X), join(meet(X, Y), meet(X, join(complement(X), meet(X, Y))))))))
% 234.08/30.06 = { by lemma 33 }
% 234.08/30.06 complement(join(complement(X), complement(join(complement(X), meet(X, Y)))))
% 234.08/30.06 = { by lemma 31 }
% 234.08/30.06 complement(complement(meet(X, Y)))
% 234.08/30.06 = { by lemma 21 }
% 234.08/30.06 meet(X, Y)
% 234.08/30.06
% 234.08/30.06 Lemma 35: meet(X, join(Y, complement(X))) = meet(X, Y).
% 234.08/30.06 Proof:
% 234.08/30.06 meet(X, join(Y, complement(X)))
% 234.08/30.06 = { by lemma 29 R->L }
% 234.08/30.06 meet(X, join(Y, complement(join(X, Y))))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.06 meet(X, join(Y, complement(join(Y, X))))
% 234.08/30.06 = { by lemma 28 R->L }
% 234.08/30.06 meet(X, meet(join(Y, X), join(Y, complement(join(Y, X)))))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.06 meet(X, meet(join(Y, X), join(complement(join(Y, X)), Y)))
% 234.08/30.06 = { by axiom 9 (absorption1) R->L }
% 234.08/30.06 meet(X, meet(join(Y, X), join(complement(join(Y, X)), meet(Y, join(Y, X)))))
% 234.08/30.06 = { by axiom 4 (commutativity_of_meet) R->L }
% 234.08/30.06 meet(X, meet(join(Y, X), join(complement(join(Y, X)), meet(join(Y, X), Y))))
% 234.08/30.06 = { by lemma 34 }
% 234.08/30.06 meet(X, meet(join(Y, X), Y))
% 234.08/30.06 = { by axiom 4 (commutativity_of_meet) }
% 234.08/30.06 meet(X, meet(Y, join(Y, X)))
% 234.08/30.06 = { by axiom 9 (absorption1) }
% 234.08/30.06 meet(X, Y)
% 234.08/30.06
% 234.08/30.06 Lemma 36: meet(X, join(Y, meet(X, join(Y, Z)))) = meet(X, join(Y, Z)).
% 234.08/30.06 Proof:
% 234.08/30.06 meet(X, join(Y, meet(X, join(Y, Z))))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.06 meet(X, join(Y, meet(X, join(Z, Y))))
% 234.08/30.06 = { by lemma 23 R->L }
% 234.08/30.06 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, join(Y, meet(X, join(Z, Y))))))
% 234.08/30.06 = { by lemma 33 }
% 234.08/30.06 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, Y)))
% 234.08/30.06 = { by axiom 4 (commutativity_of_meet) }
% 234.08/30.06 meet(X, meet(join(Z, Y), join(Y, meet(X, join(Z, Y)))))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) }
% 234.08/30.06 meet(X, meet(join(Z, Y), join(Y, meet(X, join(Y, Z)))))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) }
% 234.08/30.06 meet(X, meet(join(Y, Z), join(Y, meet(X, join(Y, Z)))))
% 234.08/30.06 = { by lemma 32 }
% 234.08/30.06 meet(X, join(Y, Z))
% 234.08/30.06
% 234.08/30.06 Lemma 37: meet(X, join(Y, join(Z, complement(X)))) = meet(X, join(Y, Z)).
% 234.08/30.06 Proof:
% 234.08/30.06 meet(X, join(Y, join(Z, complement(X))))
% 234.08/30.06 = { by lemma 25 }
% 234.08/30.06 meet(X, join(Z, join(Y, complement(X))))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.06 meet(X, join(Z, join(complement(X), Y)))
% 234.08/30.06 = { by lemma 34 R->L }
% 234.08/30.06 meet(X, join(complement(X), meet(X, join(Z, join(complement(X), Y)))))
% 234.08/30.06 = { by lemma 25 }
% 234.08/30.06 meet(X, join(complement(X), meet(X, join(complement(X), join(Z, Y)))))
% 234.08/30.06 = { by lemma 36 }
% 234.08/30.06 meet(X, join(complement(X), join(Z, Y)))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.06 meet(X, join(join(Z, Y), complement(X)))
% 234.08/30.06 = { by lemma 35 }
% 234.08/30.06 meet(X, join(Z, Y))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) }
% 234.08/30.06 meet(X, join(Y, Z))
% 234.08/30.06
% 234.08/30.06 Lemma 38: meet(X, join(Y, meet(join(X, Y), join(Z, complement(X))))) = join(meet(X, Y), meet(X, join(Z, complement(X)))).
% 234.08/30.06 Proof:
% 234.08/30.06 meet(X, join(Y, meet(join(X, Y), join(Z, complement(X)))))
% 234.08/30.06 = { by axiom 4 (commutativity_of_meet) R->L }
% 234.08/30.06 meet(X, join(Y, meet(join(Z, complement(X)), join(X, Y))))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.06 meet(X, join(meet(join(Z, complement(X)), join(X, Y)), Y))
% 234.08/30.06 = { by lemma 13 R->L }
% 234.08/30.06 meet(X, join(meet(join(Z, complement(X)), join(X, Y)), meet(Y, one)))
% 234.08/30.06 = { by lemma 17 R->L }
% 234.08/30.06 meet(X, join(meet(join(Z, complement(X)), join(X, Y)), meet(Y, join(X, join(Z, complement(X))))))
% 234.08/30.06 = { by axiom 12 (equation_H82) }
% 234.08/30.06 join(meet(X, join(Z, complement(X))), meet(X, Y))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) }
% 234.08/30.06 join(meet(X, Y), meet(X, join(Z, complement(X))))
% 234.08/30.06
% 234.08/30.06 Lemma 39: meet(X, join(meet(X, join(Y, complement(X))), Z)) = join(meet(X, Z), meet(X, join(Y, complement(X)))).
% 234.08/30.06 Proof:
% 234.08/30.06 meet(X, join(meet(X, join(Y, complement(X))), Z))
% 234.08/30.06 = { by lemma 24 R->L }
% 234.08/30.06 meet(X, join(meet(X, join(Y, complement(X))), join(Z, meet(X, Z))))
% 234.08/30.06 = { by lemma 25 R->L }
% 234.08/30.06 meet(X, join(Z, join(meet(X, join(Y, complement(X))), meet(X, Z))))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) }
% 234.08/30.06 meet(X, join(Z, join(meet(X, Z), meet(X, join(Y, complement(X))))))
% 234.08/30.06 = { by lemma 38 R->L }
% 234.08/30.06 meet(X, join(Z, meet(X, join(Z, meet(join(X, Z), join(Y, complement(X)))))))
% 234.08/30.06 = { by lemma 36 }
% 234.08/30.06 meet(X, join(Z, meet(join(X, Z), join(Y, complement(X)))))
% 234.08/30.06 = { by lemma 38 }
% 234.08/30.06 join(meet(X, Z), meet(X, join(Y, complement(X))))
% 234.08/30.06
% 234.08/30.06 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 234.08/30.06 Proof:
% 234.08/30.06 meet(a, join(b, c))
% 234.08/30.06 = { by lemma 37 R->L }
% 234.08/30.06 meet(a, join(b, join(c, complement(a))))
% 234.08/30.06 = { by lemma 25 }
% 234.08/30.06 meet(a, join(c, join(b, complement(a))))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) R->L }
% 234.08/30.06 meet(a, join(c, join(complement(a), b)))
% 234.08/30.06 = { by axiom 1 (idempotence_of_join) R->L }
% 234.08/30.06 meet(a, join(c, join(join(complement(a), complement(a)), b)))
% 234.08/30.06 = { by axiom 8 (associativity_of_join) }
% 234.08/30.06 meet(a, join(c, join(complement(a), join(complement(a), b))))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) }
% 234.08/30.06 meet(a, join(c, join(complement(a), join(b, complement(a)))))
% 234.08/30.06 = { by lemma 14 R->L }
% 234.08/30.06 meet(a, join(c, join(complement(a), meet(one, join(b, complement(a))))))
% 234.08/30.06 = { by axiom 8 (associativity_of_join) R->L }
% 234.08/30.06 meet(a, join(join(c, complement(a)), meet(one, join(b, complement(a)))))
% 234.08/30.06 = { by lemma 17 R->L }
% 234.08/30.06 meet(a, join(join(c, complement(a)), meet(join(a, join(c, complement(a))), join(b, complement(a)))))
% 234.08/30.06 = { by lemma 38 }
% 234.08/30.06 join(meet(a, join(c, complement(a))), meet(a, join(b, complement(a))))
% 234.08/30.06 = { by axiom 2 (commutativity_of_join) }
% 234.08/30.06 join(meet(a, join(b, complement(a))), meet(a, join(c, complement(a))))
% 234.08/30.06 = { by lemma 39 R->L }
% 234.08/30.06 meet(a, join(meet(a, join(c, complement(a))), join(b, complement(a))))
% 234.08/30.06 = { by lemma 37 }
% 234.08/30.06 meet(a, join(meet(a, join(c, complement(a))), b))
% 234.08/30.06 = { by lemma 39 }
% 234.08/30.06 join(meet(a, b), meet(a, join(c, complement(a))))
% 234.08/30.06 = { by lemma 35 }
% 234.08/30.06 join(meet(a, b), meet(a, c))
% 234.08/30.06 % SZS output end Proof
% 234.08/30.06
% 234.08/30.06 RESULT: Unsatisfiable (the axioms are contradictory).
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