%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LAT194-1 : TPTP v9.3.1. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n005.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 11:44:31 AM UTC 2026
% Result : Unsatisfiable 121.29s 15.76s
% Output : Proof 122.87s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : LAT194-1 : TPTP v9.3.1. Released v3.1.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.10/0.37 % Computer : n005.cluster.edu
% 0.10/0.37 % Model : x86_64 x86_64
% 0.10/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.37 % Memory : 8046.5625MB
% 0.10/0.37 % OS : Linux 6.8.0-71-generic
% 0.10/0.37 % CPULimit : 300
% 0.10/0.37 % WCLimit : 300
% 0.10/0.37 % DateTime : Sun Sep 27 14:05:31 UTC 2026
% 0.10/0.37 % CPUTime :
% 0.10/0.37 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 121.29/15.76 Command-line arguments: --no-flatten-goal
% 121.29/15.76
% 121.29/15.76 % SZS status Unsatisfiable
% 121.29/15.76
% 122.87/15.96 % SZS output start Proof
% 122.87/15.96 Axiom 1 (commutativity_of_join): join(X, Y) = join(Y, X).
% 122.87/15.96 Axiom 2 (complement_join): join(X, complement(X)) = one.
% 122.87/15.96 Axiom 3 (idempotence_of_meet): meet(X, X) = X.
% 122.87/15.96 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 122.87/15.96 Axiom 5 (complement_meet): meet(X, complement(X)) = zero.
% 122.87/15.96 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 122.87/15.96 Axiom 7 (absorption2): join(X, meet(X, Y)) = X.
% 122.87/15.96 Axiom 8 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 122.87/15.96 Axiom 9 (absorption1): meet(X, join(X, Y)) = X.
% 122.87/15.96 Axiom 10 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 122.87/15.96 Axiom 11 (meet_join_complement): ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y) = Y.
% 122.87/15.96 Axiom 12 (equation_H32): meet(X, join(Y, meet(X, meet(Z, W)))) = meet(X, join(Y, meet(Z, join(meet(X, W), meet(Y, W))))).
% 122.87/15.96
% 122.87/15.96 Lemma 13: meet(X, one) = X.
% 122.87/15.96 Proof:
% 122.87/15.96 meet(X, one)
% 122.87/15.96 = { by axiom 2 (complement_join) R->L }
% 122.87/15.96 meet(X, join(X, complement(X)))
% 122.87/15.96 = { by axiom 9 (absorption1) }
% 122.87/15.96 X
% 122.87/15.96
% 122.87/15.96 Lemma 14: join(X, zero) = X.
% 122.87/15.96 Proof:
% 122.87/15.96 join(X, zero)
% 122.87/15.96 = { by axiom 5 (complement_meet) R->L }
% 122.87/15.97 join(X, meet(X, complement(X)))
% 122.87/15.97 = { by axiom 7 (absorption2) }
% 122.87/15.97 X
% 122.87/15.97
% 122.87/15.97 Lemma 15: join(zero, X) = X.
% 122.87/15.97 Proof:
% 122.87/15.97 join(zero, X)
% 122.87/15.97 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/15.97 join(X, zero)
% 122.87/15.97 = { by lemma 14 }
% 122.87/15.97 X
% 122.87/15.97
% 122.87/15.97 Lemma 16: meet(X, zero) = zero.
% 122.87/15.97 Proof:
% 122.87/15.97 meet(X, zero)
% 122.87/15.97 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/15.97 meet(zero, X)
% 122.87/15.97 = { by lemma 15 R->L }
% 122.87/15.97 join(zero, meet(zero, X))
% 122.87/15.97 = { by axiom 7 (absorption2) }
% 122.87/15.97 zero
% 122.87/15.97
% 122.87/15.97 Lemma 17: join(X, one) = one.
% 122.87/15.97 Proof:
% 122.87/15.97 join(X, one)
% 122.87/15.97 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/15.97 join(one, X)
% 122.87/15.97 = { by lemma 13 R->L }
% 122.87/15.97 join(one, meet(X, one))
% 122.87/15.97 = { by axiom 4 (commutativity_of_meet) }
% 122.87/15.97 join(one, meet(one, X))
% 122.87/15.97 = { by axiom 7 (absorption2) }
% 122.87/15.97 one
% 122.87/15.97
% 122.87/15.97 Lemma 18: meet(X, meet(Y, complement(X))) = zero.
% 122.87/15.97 Proof:
% 122.87/15.97 meet(X, meet(Y, complement(X)))
% 122.87/15.97 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/15.97 meet(X, meet(complement(X), Y))
% 122.87/15.97 = { by axiom 10 (associativity_of_meet) R->L }
% 122.87/15.97 meet(meet(X, complement(X)), Y)
% 122.87/15.97 = { by axiom 5 (complement_meet) }
% 122.87/15.97 meet(zero, Y)
% 122.87/15.97 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/15.97 meet(Y, zero)
% 122.87/15.97 = { by lemma 16 }
% 122.87/15.97 zero
% 122.87/15.97
% 122.87/15.97 Lemma 19: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 122.87/15.97 Proof:
% 122.87/15.97 join(X, join(Y, meet(X, Z)))
% 122.87/15.97 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/15.97 join(X, join(meet(X, Z), Y))
% 122.87/15.97 = { by axiom 8 (associativity_of_join) R->L }
% 122.87/15.97 join(join(X, meet(X, Z)), Y)
% 122.87/15.97 = { by axiom 7 (absorption2) }
% 122.87/15.97 join(X, Y)
% 122.87/15.97
% 122.87/15.97 Lemma 20: meet(X, join(Y, meet(Z, join(X, Y)))) = meet(X, join(Y, meet(X, Z))).
% 122.87/15.97 Proof:
% 122.87/15.97 meet(X, join(Y, meet(Z, join(X, Y))))
% 122.87/15.97 = { by lemma 13 R->L }
% 122.87/15.97 meet(X, join(Y, meet(Z, join(X, meet(Y, one)))))
% 122.87/15.97 = { by lemma 13 R->L }
% 122.87/15.97 meet(X, join(Y, meet(Z, join(meet(X, one), meet(Y, one)))))
% 122.87/15.97 = { by axiom 12 (equation_H32) R->L }
% 122.87/15.97 meet(X, join(Y, meet(X, meet(Z, one))))
% 122.87/15.97 = { by lemma 13 }
% 122.87/15.97 meet(X, join(Y, meet(X, Z)))
% 122.87/15.97
% 122.87/15.97 Lemma 21: join(X, meet(Y, X)) = X.
% 122.87/15.97 Proof:
% 122.87/15.97 join(X, meet(Y, X))
% 122.87/15.97 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/15.97 join(X, meet(X, Y))
% 122.87/15.97 = { by axiom 7 (absorption2) }
% 122.87/15.97 X
% 122.87/15.97
% 122.87/15.97 Lemma 22: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 122.87/15.97 Proof:
% 122.87/15.97 meet(X, meet(Y, Z))
% 122.87/15.97 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/15.97 meet(meet(Y, Z), X)
% 122.87/15.97 = { by axiom 10 (associativity_of_meet) }
% 122.87/15.97 meet(Y, meet(Z, X))
% 122.87/15.97 = { by axiom 4 (commutativity_of_meet) }
% 122.87/15.97 meet(Y, meet(X, Z))
% 122.87/15.97
% 122.87/15.97 Lemma 23: join(X, meet(Y, join(X, meet(Y, Z)))) = join(X, meet(Y, Z)).
% 122.87/15.97 Proof:
% 122.87/15.97 join(X, meet(Y, join(X, meet(Y, Z))))
% 122.87/15.97 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/15.97 join(X, meet(Y, join(X, meet(Z, Y))))
% 122.87/15.97 = { by lemma 21 R->L }
% 122.87/15.97 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, meet(Y, join(X, meet(Z, Y))))))
% 122.87/15.97 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/15.97 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, meet(Y, join(X, meet(Y, Z))))))
% 122.87/15.97 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/15.97 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, meet(Y, join(meet(Y, Z), X)))))
% 122.87/15.97 = { by lemma 22 R->L }
% 122.87/15.97 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Y, meet(Z, join(meet(Y, Z), X)))))
% 122.87/15.97 = { by axiom 10 (associativity_of_meet) R->L }
% 122.87/15.97 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(meet(Y, Z), join(meet(Y, Z), X))))
% 122.87/15.97 = { by axiom 9 (absorption1) }
% 122.87/15.97 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Y, Z)))
% 122.87/15.97 = { by axiom 4 (commutativity_of_meet) }
% 122.87/15.97 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, Y)))
% 122.87/15.97 = { by axiom 1 (commutativity_of_join) }
% 122.87/15.97 join(X, join(meet(Z, Y), meet(Y, join(X, meet(Z, Y)))))
% 122.87/15.97 = { by axiom 4 (commutativity_of_meet) }
% 122.87/15.97 join(X, join(meet(Z, Y), meet(Y, join(X, meet(Y, Z)))))
% 122.87/15.97 = { by axiom 4 (commutativity_of_meet) }
% 122.87/15.98 join(X, join(meet(Y, Z), meet(Y, join(X, meet(Y, Z)))))
% 122.87/15.98 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/15.98 join(X, join(meet(Y, Z), meet(Y, join(meet(Y, Z), X))))
% 122.87/15.98 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/15.98 join(X, join(meet(Y, Z), meet(join(meet(Y, Z), X), Y)))
% 122.87/15.98 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/15.98 join(join(meet(Y, Z), meet(join(meet(Y, Z), X), Y)), X)
% 122.87/15.98 = { by axiom 8 (associativity_of_join) }
% 122.87/15.98 join(meet(Y, Z), join(meet(join(meet(Y, Z), X), Y), X))
% 122.87/15.98 = { by axiom 1 (commutativity_of_join) }
% 122.87/15.98 join(meet(Y, Z), join(X, meet(join(meet(Y, Z), X), Y)))
% 122.87/15.98 = { by axiom 8 (associativity_of_join) R->L }
% 122.87/15.98 join(join(meet(Y, Z), X), meet(join(meet(Y, Z), X), Y))
% 122.87/15.98 = { by axiom 7 (absorption2) }
% 122.87/15.98 join(meet(Y, Z), X)
% 122.87/15.98 = { by axiom 1 (commutativity_of_join) }
% 122.87/15.98 join(X, meet(Y, Z))
% 122.87/15.98
% 122.87/15.98 Lemma 24: meet(X, join(complement(X), meet(X, Y))) = meet(X, join(Y, complement(X))).
% 122.87/15.98 Proof:
% 122.87/15.98 meet(X, join(complement(X), meet(X, Y)))
% 122.87/15.98 = { by lemma 20 R->L }
% 122.87/15.98 meet(X, join(complement(X), meet(Y, join(X, complement(X)))))
% 122.87/15.98 = { by axiom 2 (complement_join) }
% 122.87/15.98 meet(X, join(complement(X), meet(Y, one)))
% 122.87/15.98 = { by lemma 13 }
% 122.87/15.98 meet(X, join(complement(X), Y))
% 122.87/15.98 = { by axiom 1 (commutativity_of_join) }
% 122.87/15.98 meet(X, join(Y, complement(X)))
% 122.87/15.98
% 122.87/15.98 Lemma 25: meet(X, join(complement(X), meet(Y, X))) = meet(X, join(Y, complement(X))).
% 122.87/15.98 Proof:
% 122.87/15.98 meet(X, join(complement(X), meet(Y, X)))
% 122.87/15.98 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/15.98 meet(X, join(complement(X), meet(X, Y)))
% 122.87/15.98 = { by lemma 24 }
% 122.87/15.98 meet(X, join(Y, complement(X)))
% 122.87/15.98
% 122.87/15.98 Lemma 26: join(X, join(Y, complement(X))) = one.
% 122.87/15.98 Proof:
% 122.87/15.98 join(X, join(Y, complement(X)))
% 122.87/15.98 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/15.98 join(X, join(complement(X), Y))
% 122.87/15.98 = { by axiom 8 (associativity_of_join) R->L }
% 122.87/15.98 join(join(X, complement(X)), Y)
% 122.87/15.98 = { by axiom 2 (complement_join) }
% 122.87/15.98 join(one, Y)
% 122.87/15.98 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/15.98 join(Y, one)
% 122.87/15.98 = { by lemma 17 }
% 122.87/15.98 one
% 122.87/15.98
% 122.87/15.98 Lemma 27: join(X, complement(complement(X))) = complement(complement(X)).
% 122.87/15.98 Proof:
% 122.87/15.98 join(X, complement(complement(X)))
% 122.87/15.98 = { by axiom 11 (meet_join_complement) R->L }
% 122.87/15.98 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))))
% 122.87/15.98 = { by lemma 25 R->L }
% 122.87/15.98 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))))
% 122.87/15.98 = { by axiom 5 (complement_meet) }
% 122.87/15.98 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))))
% 122.87/15.98 = { by lemma 14 }
% 122.87/15.98 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))))
% 122.87/15.98 = { by axiom 5 (complement_meet) }
% 122.87/15.98 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))))
% 122.87/15.98 = { by lemma 26 }
% 122.87/15.98 ifeq(one, one, ifeq(zero, zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X))))
% 122.87/15.98 = { by axiom 6 (ifeq_axiom) }
% 122.87/15.98 ifeq(zero, zero, complement(complement(X)), join(X, complement(complement(X))))
% 122.87/15.98 = { by axiom 6 (ifeq_axiom) }
% 122.87/15.98 complement(complement(X))
% 122.87/15.98
% 122.87/15.98 Lemma 28: complement(complement(X)) = X.
% 122.87/15.98 Proof:
% 122.87/15.98 complement(complement(X))
% 122.87/15.98 = { by axiom 3 (idempotence_of_meet) R->L }
% 122.87/15.98 complement(complement(meet(X, X)))
% 122.87/15.98 = { by axiom 6 (ifeq_axiom) R->L }
% 122.87/15.98 ifeq(zero, zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X)))))
% 122.87/15.98 = { by lemma 18 R->L }
% 122.87/15.98 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)))))
% 122.87/15.98 = { by axiom 6 (ifeq_axiom) R->L }
% 122.87/15.98 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)))))
% 122.87/15.99 = { by axiom 2 (complement_join) R->L }
% 122.87/15.99 ifeq(join(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)))))
% 122.87/15.99 = { by lemma 13 R->L }
% 122.87/15.99 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), 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)))))
% 122.87/15.99 = { by lemma 17 R->L }
% 122.87/15.99 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), join(X, 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)))))
% 122.87/15.99 = { by axiom 2 (complement_join) R->L }
% 122.87/15.99 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), join(X, join(meet(X, 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)))))
% 122.87/15.99 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/15.99 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), join(X, join(complement(meet(X, X)), 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)))))
% 122.87/15.99 = { by lemma 19 }
% 122.87/15.99 ifeq(join(complement(meet(X, X)), meet(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)))))
% 122.87/15.99 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/15.99 ifeq(join(complement(meet(X, X)), meet(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)))))
% 122.87/15.99 = { by lemma 13 R->L }
% 122.87/15.99 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), join(complement(meet(X, X)), meet(X, 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)))))
% 122.87/15.99 = { by axiom 2 (complement_join) R->L }
% 122.87/15.99 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)))))))), 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)))))
% 122.87/15.99 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/15.99 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))))))), 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)))))
% 122.87/15.99 = { by lemma 20 }
% 122.87/15.99 ifeq(join(complement(meet(X, X)), meet(complement(complement(meet(X, X))), join(complement(meet(X, X)), meet(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)))))
% 122.87/15.99 = { by lemma 23 }
% 122.87/15.99 ifeq(join(complement(meet(X, X)), meet(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)))))
% 122.87/15.99 = { by axiom 4 (commutativity_of_meet) }
% 122.87/15.99 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)))))
% 122.87/15.99 = { by axiom 11 (meet_join_complement) }
% 122.87/15.99 meet(X, complement(complement(meet(X, X))))
% 122.87/15.99 = { by axiom 3 (idempotence_of_meet) }
% 122.87/15.99 meet(X, complement(complement(X)))
% 122.87/15.99 = { by lemma 27 R->L }
% 122.87/15.99 meet(X, join(X, complement(complement(X))))
% 122.87/15.99 = { by axiom 9 (absorption1) }
% 122.87/15.99 X
% 122.87/15.99
% 122.87/15.99 Lemma 29: meet(X, complement(join(X, Y))) = zero.
% 122.87/15.99 Proof:
% 122.87/15.99 meet(X, complement(join(X, Y)))
% 122.87/15.99 = { by axiom 9 (absorption1) R->L }
% 122.87/15.99 meet(meet(X, join(X, Y)), complement(join(X, Y)))
% 122.87/15.99 = { by axiom 10 (associativity_of_meet) }
% 122.87/15.99 meet(X, meet(join(X, Y), complement(join(X, Y))))
% 122.87/15.99 = { by axiom 5 (complement_meet) }
% 122.87/15.99 meet(X, zero)
% 122.87/15.99 = { by lemma 16 }
% 122.87/15.99 zero
% 122.87/15.99
% 122.87/15.99 Lemma 30: join(X, join(complement(X), Y)) = one.
% 122.87/15.99 Proof:
% 122.87/15.99 join(X, join(complement(X), Y))
% 122.87/15.99 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/15.99 join(X, join(Y, complement(X)))
% 122.87/15.99 = { by lemma 26 }
% 122.87/15.99 one
% 122.87/15.99
% 122.87/15.99 Lemma 31: join(complement(X), complement(join(X, Y))) = complement(X).
% 122.87/15.99 Proof:
% 122.87/15.99 join(complement(X), complement(join(X, Y)))
% 122.87/15.99 = { by axiom 11 (meet_join_complement) R->L }
% 122.87/15.99 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))))
% 122.87/15.99 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/15.99 ifeq(join(X, join(complement(X), complement(join(X, Y)))), one, ifeq(meet(X, join(complement(join(X, Y)), complement(X))), zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 122.87/15.99 = { by lemma 24 R->L }
% 122.87/15.99 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))))
% 122.87/15.99 = { by lemma 29 }
% 122.87/15.99 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))))
% 122.87/15.99 = { by lemma 14 }
% 122.87/15.99 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))))
% 122.87/15.99 = { by axiom 5 (complement_meet) }
% 122.87/15.99 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))))
% 122.87/15.99 = { by lemma 30 }
% 122.87/15.99 ifeq(one, one, ifeq(zero, zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 122.87/15.99 = { by axiom 6 (ifeq_axiom) }
% 122.87/16.00 ifeq(zero, zero, complement(X), join(complement(X), complement(join(X, Y))))
% 122.87/16.00 = { by axiom 6 (ifeq_axiom) }
% 122.87/16.00 complement(X)
% 122.87/16.00
% 122.87/16.00 Lemma 32: meet(X, join(Y, X)) = X.
% 122.87/16.00 Proof:
% 122.87/16.00 meet(X, join(Y, X))
% 122.87/16.00 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/16.00 meet(X, join(X, Y))
% 122.87/16.00 = { by axiom 9 (absorption1) }
% 122.87/16.00 X
% 122.87/16.00
% 122.87/16.00 Lemma 33: join(X, meet(Y, complement(X))) = join(X, Y).
% 122.87/16.00 Proof:
% 122.87/16.00 join(X, meet(Y, complement(X)))
% 122.87/16.00 = { by lemma 28 R->L }
% 122.87/16.00 complement(complement(join(X, meet(Y, complement(X)))))
% 122.87/16.00 = { by axiom 6 (ifeq_axiom) R->L }
% 122.87/16.00 ifeq(zero, zero, 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)))))
% 122.87/16.00 = { by axiom 6 (ifeq_axiom) R->L }
% 122.87/16.00 ifeq(one, one, ifeq(zero, zero, 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)))))
% 122.87/16.00 = { by lemma 30 R->L }
% 122.87/16.00 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))))), one, ifeq(zero, zero, 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)))))
% 122.87/16.00 = { by axiom 5 (complement_meet) R->L }
% 122.87/16.00 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))))), one, ifeq(meet(complement(join(X, meet(Y, complement(X)))), complement(complement(join(X, meet(Y, complement(X)))))), zero, 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)))))
% 122.87/16.00 = { by lemma 14 R->L }
% 122.87/16.00 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))))), one, ifeq(meet(complement(join(X, meet(Y, complement(X)))), join(complement(complement(join(X, meet(Y, complement(X))))), zero)), zero, 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)))))
% 122.87/16.00 = { by axiom 5 (complement_meet) R->L }
% 122.87/16.00 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))))), one, 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))))))))), zero, 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)))))
% 122.87/16.01 = { by axiom 10 (associativity_of_meet) }
% 122.87/16.01 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))))), one, 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)))))))))), zero, 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)))))
% 122.87/16.01 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/16.01 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))))), one, 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)))))))), zero, 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)))))
% 122.87/16.01 = { by axiom 10 (associativity_of_meet) R->L }
% 122.87/16.01 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))))), one, 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))))))), zero, 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)))))
% 122.87/16.01 = { by lemma 25 }
% 122.87/16.01 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))))), one, 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))))))), zero, 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)))))
% 122.87/16.01 = { by axiom 1 (commutativity_of_join) }
% 122.87/16.01 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))))), one, 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))))))))), zero, 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)))))
% 122.87/16.01 = { by axiom 4 (commutativity_of_meet) }
% 122.87/16.01 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))))), one, 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))))), zero, 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)))))
% 122.87/16.01 = { by axiom 11 (meet_join_complement) }
% 122.87/16.01 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))
% 122.87/16.01 = { by axiom 4 (commutativity_of_meet) }
% 122.87/16.01 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, complement(join(X, meet(Y, complement(X))))))))
% 122.87/16.01 = { by lemma 32 R->L }
% 122.87/16.01 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))))))))))
% 122.87/16.01 = { by lemma 31 }
% 122.87/16.01 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, meet(complement(join(X, meet(Y, complement(X)))), complement(X))))))
% 122.87/16.01 = { by axiom 4 (commutativity_of_meet) }
% 122.87/16.01 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, meet(complement(X), complement(join(X, meet(Y, complement(X)))))))))
% 122.87/16.02 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/16.02 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, meet(complement(X), complement(join(X, meet(complement(X), Y))))))))
% 122.87/16.02 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/16.02 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, meet(complement(X), complement(join(meet(complement(X), Y), X)))))))
% 122.87/16.02 = { by lemma 22 R->L }
% 122.87/16.02 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(X), meet(Y, complement(join(meet(complement(X), Y), X)))))))
% 122.87/16.02 = { by axiom 10 (associativity_of_meet) R->L }
% 122.87/16.02 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(meet(complement(X), Y), complement(join(meet(complement(X), Y), X))))))
% 122.87/16.02 = { by lemma 29 }
% 122.87/16.02 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(zero)))
% 122.87/16.02 = { by lemma 28 }
% 122.87/16.02 join(join(X, meet(Y, complement(X))), meet(Y, complement(zero)))
% 122.87/16.02 = { by axiom 8 (associativity_of_join) }
% 122.87/16.02 join(X, join(meet(Y, complement(X)), meet(Y, complement(zero))))
% 122.87/16.02 = { by lemma 15 R->L }
% 122.87/16.02 join(X, join(meet(Y, complement(X)), meet(Y, join(zero, complement(zero)))))
% 122.87/16.02 = { by axiom 2 (complement_join) }
% 122.87/16.02 join(X, join(meet(Y, complement(X)), meet(Y, one)))
% 122.87/16.02 = { by lemma 13 }
% 122.87/16.02 join(X, join(meet(Y, complement(X)), Y))
% 122.87/16.02 = { by axiom 1 (commutativity_of_join) }
% 122.87/16.02 join(X, join(Y, meet(Y, complement(X))))
% 122.87/16.02 = { by axiom 7 (absorption2) }
% 122.87/16.02 join(X, Y)
% 122.87/16.02
% 122.87/16.02 Lemma 34: meet(complement(X), join(Y, complement(join(X, Y)))) = complement(X).
% 122.87/16.02 Proof:
% 122.87/16.02 meet(complement(X), join(Y, complement(join(X, Y))))
% 122.87/16.02 = { by axiom 11 (meet_join_complement) R->L }
% 122.87/16.02 ifeq(join(X, meet(complement(X), join(Y, complement(join(X, Y))))), one, ifeq(meet(X, meet(complement(X), join(Y, complement(join(X, Y))))), zero, complement(X), meet(complement(X), join(Y, complement(join(X, Y))))), meet(complement(X), join(Y, complement(join(X, Y)))))
% 122.87/16.02 = { by axiom 1 (commutativity_of_join) }
% 122.87/16.02 ifeq(join(meet(complement(X), join(Y, complement(join(X, Y)))), X), one, ifeq(meet(X, meet(complement(X), join(Y, complement(join(X, Y))))), zero, complement(X), meet(complement(X), join(Y, complement(join(X, Y))))), meet(complement(X), join(Y, complement(join(X, Y)))))
% 122.87/16.02 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/16.02 ifeq(join(meet(complement(X), join(Y, complement(join(X, Y)))), X), one, ifeq(meet(X, meet(join(Y, complement(join(X, Y))), complement(X))), zero, complement(X), meet(complement(X), join(Y, complement(join(X, Y))))), meet(complement(X), join(Y, complement(join(X, Y)))))
% 122.87/16.02 = { by lemma 18 }
% 122.87/16.02 ifeq(join(meet(complement(X), join(Y, complement(join(X, Y)))), X), one, ifeq(zero, zero, complement(X), meet(complement(X), join(Y, complement(join(X, Y))))), meet(complement(X), join(Y, complement(join(X, Y)))))
% 122.87/16.02 = { by axiom 6 (ifeq_axiom) }
% 122.87/16.02 ifeq(join(meet(complement(X), join(Y, complement(join(X, Y)))), X), one, complement(X), meet(complement(X), join(Y, complement(join(X, Y)))))
% 122.87/16.02 = { by axiom 1 (commutativity_of_join) }
% 122.87/16.02 ifeq(join(X, meet(complement(X), join(Y, complement(join(X, Y))))), one, complement(X), meet(complement(X), join(Y, complement(join(X, Y)))))
% 122.87/16.02 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/16.02 ifeq(join(X, meet(join(Y, complement(join(X, Y))), complement(X))), one, complement(X), meet(complement(X), join(Y, complement(join(X, Y)))))
% 122.87/16.02 = { by lemma 33 }
% 122.87/16.02 ifeq(join(X, join(Y, complement(join(X, Y)))), one, complement(X), meet(complement(X), join(Y, complement(join(X, Y)))))
% 122.87/16.02 = { by axiom 4 (commutativity_of_meet) }
% 122.87/16.02 ifeq(join(X, join(Y, complement(join(X, Y)))), one, complement(X), meet(join(Y, complement(join(X, Y))), complement(X)))
% 122.87/16.02 = { by axiom 8 (associativity_of_join) R->L }
% 122.87/16.02 ifeq(join(join(X, Y), complement(join(X, Y))), one, complement(X), meet(join(Y, complement(join(X, Y))), complement(X)))
% 122.87/16.02 = { by axiom 2 (complement_join) }
% 122.87/16.02 ifeq(one, one, complement(X), meet(join(Y, complement(join(X, Y))), complement(X)))
% 122.87/16.02 = { by axiom 6 (ifeq_axiom) }
% 122.87/16.02 complement(X)
% 122.87/16.02
% 122.87/16.02 Lemma 35: join(X, complement(join(X, Y))) = join(X, complement(Y)).
% 122.87/16.02 Proof:
% 122.87/16.02 join(X, complement(join(X, Y)))
% 122.87/16.02 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/16.02 join(X, complement(join(Y, X)))
% 122.87/16.02 = { by lemma 21 R->L }
% 122.87/16.02 join(join(X, complement(join(Y, X))), meet(complement(Y), join(X, complement(join(Y, X)))))
% 122.87/16.02 = { by lemma 34 }
% 122.87/16.02 join(join(X, complement(join(Y, X))), complement(Y))
% 122.87/16.02 = { by axiom 8 (associativity_of_join) }
% 122.87/16.02 join(X, join(complement(join(Y, X)), complement(Y)))
% 122.87/16.02 = { by axiom 1 (commutativity_of_join) }
% 122.87/16.02 join(X, join(complement(Y), complement(join(Y, X))))
% 122.87/16.02 = { by lemma 31 }
% 122.87/16.02 join(X, complement(Y))
% 122.87/16.02
% 122.87/16.02 Lemma 36: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 122.87/16.02 Proof:
% 122.87/16.02 complement(meet(X, complement(Y)))
% 122.87/16.02 = { by lemma 21 R->L }
% 122.87/16.02 join(complement(meet(X, complement(Y))), meet(Y, complement(meet(X, complement(Y)))))
% 122.87/16.02 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/16.02 join(complement(meet(X, complement(Y))), meet(Y, complement(meet(complement(Y), X))))
% 122.87/16.02 = { by lemma 31 R->L }
% 122.87/16.02 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), complement(join(meet(complement(Y), X), complement(Y))))))
% 122.87/16.03 = { by axiom 1 (commutativity_of_join) }
% 122.87/16.03 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), complement(join(complement(Y), meet(complement(Y), X))))))
% 122.87/16.03 = { by axiom 7 (absorption2) }
% 122.87/16.03 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), complement(complement(Y)))))
% 122.87/16.03 = { by lemma 27 R->L }
% 122.87/16.03 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), join(Y, complement(complement(Y))))))
% 122.87/16.03 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/16.03 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), join(complement(complement(Y)), Y))))
% 122.87/16.03 = { by axiom 8 (associativity_of_join) R->L }
% 122.87/16.03 join(complement(meet(X, complement(Y))), meet(Y, join(join(complement(meet(complement(Y), X)), complement(complement(Y))), Y)))
% 122.87/16.03 = { by lemma 32 }
% 122.87/16.03 join(complement(meet(X, complement(Y))), Y)
% 122.87/16.03 = { by axiom 1 (commutativity_of_join) }
% 122.87/16.03 join(Y, complement(meet(X, complement(Y))))
% 122.87/16.03 = { by lemma 35 R->L }
% 122.87/16.03 join(Y, complement(join(Y, meet(X, complement(Y)))))
% 122.87/16.03 = { by lemma 33 }
% 122.87/16.03 join(Y, complement(join(Y, X)))
% 122.87/16.03 = { by lemma 35 }
% 122.87/16.03 join(Y, complement(X))
% 122.87/16.03
% 122.87/16.03 Lemma 37: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 122.87/16.03 Proof:
% 122.87/16.03 complement(join(X, complement(Y)))
% 122.87/16.03 = { by lemma 36 R->L }
% 122.87/16.03 complement(complement(meet(Y, complement(X))))
% 122.87/16.03 = { by lemma 28 }
% 122.87/16.03 meet(Y, complement(X))
% 122.87/16.03
% 122.87/16.03 Lemma 38: complement(join(X, Y)) = meet(complement(X), complement(Y)).
% 122.87/16.03 Proof:
% 122.87/16.03 complement(join(X, Y))
% 122.87/16.03 = { by lemma 28 R->L }
% 122.87/16.03 complement(join(X, complement(complement(Y))))
% 122.87/16.03 = { by lemma 37 }
% 122.87/16.03 meet(complement(Y), complement(X))
% 122.87/16.03 = { by axiom 4 (commutativity_of_meet) }
% 122.87/16.03 meet(complement(X), complement(Y))
% 122.87/16.03
% 122.87/16.03 Lemma 39: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 122.87/16.03 Proof:
% 122.87/16.03 complement(meet(complement(X), Y))
% 122.87/16.03 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/16.03 complement(meet(Y, complement(X)))
% 122.87/16.03 = { by lemma 36 }
% 122.87/16.03 join(X, complement(Y))
% 122.87/16.03
% 122.87/16.03 Lemma 40: meet(complement(X), join(Y, meet(Z, complement(X)))) = meet(complement(X), join(Y, Z)).
% 122.87/16.03 Proof:
% 122.87/16.03 meet(complement(X), join(Y, meet(Z, complement(X))))
% 122.87/16.03 = { by lemma 32 R->L }
% 122.87/16.03 meet(complement(X), meet(join(Y, meet(Z, complement(X))), join(Z, join(Y, meet(Z, complement(X))))))
% 122.87/16.03 = { by lemma 19 }
% 122.87/16.03 meet(complement(X), meet(join(Y, meet(Z, complement(X))), join(Z, Y)))
% 122.87/16.03 = { by axiom 4 (commutativity_of_meet) }
% 122.87/16.03 meet(complement(X), meet(join(Z, Y), join(Y, meet(Z, complement(X)))))
% 122.87/16.03 = { by axiom 1 (commutativity_of_join) }
% 122.87/16.03 meet(complement(X), meet(join(Y, Z), join(Y, meet(Z, complement(X)))))
% 122.87/16.03 = { by lemma 37 R->L }
% 122.87/16.03 meet(complement(X), meet(join(Y, Z), join(Y, complement(join(X, complement(Z))))))
% 122.87/16.03 = { by lemma 35 R->L }
% 122.87/16.03 meet(complement(X), meet(join(Y, Z), join(Y, complement(join(Y, join(X, complement(Z)))))))
% 122.87/16.03 = { by lemma 28 R->L }
% 122.87/16.03 meet(complement(X), meet(join(Y, complement(complement(Z))), join(Y, complement(join(Y, join(X, complement(Z)))))))
% 122.87/16.03 = { by lemma 39 R->L }
% 122.87/16.03 meet(complement(X), meet(complement(meet(complement(Y), complement(Z))), join(Y, complement(join(Y, join(X, complement(Z)))))))
% 122.87/16.03 = { by axiom 10 (associativity_of_meet) R->L }
% 122.87/16.03 meet(meet(complement(X), complement(meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(X, complement(Z))))))
% 122.87/16.03 = { by lemma 38 R->L }
% 122.87/16.03 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(X, complement(Z))))))
% 122.87/16.03 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/16.03 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(complement(Z), X)))))
% 122.87/16.03 = { by axiom 8 (associativity_of_join) R->L }
% 122.87/16.03 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(join(Y, complement(Z)), X))))
% 122.87/16.03 = { by lemma 35 R->L }
% 122.87/16.03 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(join(Y, complement(join(Y, Z))), X))))
% 122.87/16.03 = { by axiom 8 (associativity_of_join) }
% 122.87/16.03 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(complement(join(Y, Z)), X)))))
% 122.87/16.03 = { by lemma 38 }
% 122.87/16.03 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(meet(complement(Y), complement(Z)), X)))))
% 122.87/16.04 = { by axiom 1 (commutativity_of_join) }
% 122.87/16.04 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(X, meet(complement(Y), complement(Z)))))))
% 122.87/16.04 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/16.04 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(join(X, meet(complement(Y), complement(Z))), Y))))
% 122.87/16.04 = { by lemma 34 }
% 122.87/16.04 complement(join(X, meet(complement(Y), complement(Z))))
% 122.87/16.04 = { by lemma 38 }
% 122.87/16.04 meet(complement(X), complement(meet(complement(Y), complement(Z))))
% 122.87/16.04 = { by lemma 39 }
% 122.87/16.04 meet(complement(X), join(Y, complement(complement(Z))))
% 122.87/16.04 = { by lemma 28 }
% 122.87/16.04 meet(complement(X), join(Y, Z))
% 122.87/16.04
% 122.87/16.04 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 122.87/16.04 Proof:
% 122.87/16.04 meet(a, join(b, c))
% 122.87/16.04 = { by lemma 28 R->L }
% 122.87/16.04 meet(complement(complement(a)), join(b, c))
% 122.87/16.04 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/16.04 meet(complement(complement(a)), join(c, b))
% 122.87/16.04 = { by lemma 40 R->L }
% 122.87/16.04 meet(complement(complement(a)), join(c, meet(b, complement(complement(a)))))
% 122.87/16.04 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/16.04 meet(complement(complement(a)), join(meet(b, complement(complement(a))), c))
% 122.87/16.04 = { by lemma 21 R->L }
% 122.87/16.04 join(meet(complement(complement(a)), join(meet(b, complement(complement(a))), c)), meet(c, meet(complement(complement(a)), join(meet(b, complement(complement(a))), c))))
% 122.87/16.04 = { by lemma 22 R->L }
% 122.87/16.04 join(meet(complement(complement(a)), join(meet(b, complement(complement(a))), c)), meet(complement(complement(a)), meet(c, join(meet(b, complement(complement(a))), c))))
% 122.87/16.04 = { by lemma 32 }
% 122.87/16.04 join(meet(complement(complement(a)), join(meet(b, complement(complement(a))), c)), meet(complement(complement(a)), c))
% 122.87/16.04 = { by axiom 4 (commutativity_of_meet) }
% 122.87/16.04 join(meet(complement(complement(a)), join(meet(b, complement(complement(a))), c)), meet(c, complement(complement(a))))
% 122.87/16.04 = { by axiom 1 (commutativity_of_join) }
% 122.87/16.04 join(meet(c, complement(complement(a))), meet(complement(complement(a)), join(meet(b, complement(complement(a))), c)))
% 122.87/16.04 = { by lemma 40 R->L }
% 122.87/16.04 join(meet(c, complement(complement(a))), meet(complement(complement(a)), join(meet(b, complement(complement(a))), meet(c, complement(complement(a))))))
% 122.87/16.04 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/16.04 join(meet(c, complement(complement(a))), meet(complement(complement(a)), join(meet(complement(complement(a)), b), meet(c, complement(complement(a))))))
% 122.87/16.04 = { by axiom 1 (commutativity_of_join) R->L }
% 122.87/16.04 join(meet(c, complement(complement(a))), meet(complement(complement(a)), join(meet(c, complement(complement(a))), meet(complement(complement(a)), b))))
% 122.87/16.04 = { by lemma 23 }
% 122.87/16.04 join(meet(c, complement(complement(a))), meet(complement(complement(a)), b))
% 122.87/16.04 = { by axiom 4 (commutativity_of_meet) }
% 122.87/16.04 join(meet(c, complement(complement(a))), meet(b, complement(complement(a))))
% 122.87/16.04 = { by axiom 1 (commutativity_of_join) }
% 122.87/16.04 join(meet(b, complement(complement(a))), meet(c, complement(complement(a))))
% 122.87/16.04 = { by lemma 28 }
% 122.87/16.04 join(meet(b, a), meet(c, complement(complement(a))))
% 122.87/16.04 = { by lemma 28 }
% 122.87/16.04 join(meet(b, a), meet(c, a))
% 122.87/16.04 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/16.04 join(meet(b, a), meet(a, c))
% 122.87/16.04 = { by axiom 4 (commutativity_of_meet) R->L }
% 122.87/16.04 join(meet(a, b), meet(a, c))
% 122.87/16.04 % SZS output end Proof
% 122.87/16.04
% 122.87/16.04 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------