%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LAT201-1 : TPTP v9.3.1. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n017.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 11:44:32 AM UTC 2026
% Result : Unsatisfiable 83.09s 10.99s
% Output : Proof 84.31s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : LAT201-1 : TPTP v9.3.1. Released v3.1.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.08/0.37 % Computer : n017.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 14:01:20 UTC 2026
% 0.08/0.37 % CPUTime :
% 0.08/0.37 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 83.09/10.99 Command-line arguments: --no-flatten-goal
% 83.09/10.99
% 83.09/10.99 % SZS status Unsatisfiable
% 83.09/10.99
% 84.31/11.10 % SZS output start Proof
% 84.31/11.10 Axiom 1 (commutativity_of_join): join(X, Y) = join(Y, X).
% 84.31/11.10 Axiom 2 (complement_join): join(X, complement(X)) = one.
% 84.31/11.10 Axiom 3 (idempotence_of_meet): meet(X, X) = X.
% 84.31/11.10 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 84.31/11.10 Axiom 5 (complement_meet): meet(X, complement(X)) = zero.
% 84.31/11.10 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 84.31/11.10 Axiom 7 (absorption2): join(X, meet(X, Y)) = X.
% 84.31/11.10 Axiom 8 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 84.31/11.10 Axiom 9 (absorption1): meet(X, join(X, Y)) = X.
% 84.31/11.10 Axiom 10 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 84.31/11.10 Axiom 11 (equation_H51): meet(X, join(Y, meet(Z, join(X, W)))) = meet(X, join(Y, join(meet(X, Z), meet(Z, W)))).
% 84.31/11.10 Axiom 12 (meet_join_complement): ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y) = Y.
% 84.31/11.10
% 84.31/11.10 Lemma 13: meet(X, one) = X.
% 84.31/11.10 Proof:
% 84.31/11.10 meet(X, one)
% 84.31/11.10 = { by axiom 2 (complement_join) R->L }
% 84.31/11.10 meet(X, join(X, complement(X)))
% 84.31/11.10 = { by axiom 9 (absorption1) }
% 84.31/11.10 X
% 84.31/11.10
% 84.31/11.10 Lemma 14: join(X, zero) = X.
% 84.31/11.10 Proof:
% 84.31/11.10 join(X, zero)
% 84.31/11.10 = { by axiom 5 (complement_meet) R->L }
% 84.31/11.10 join(X, meet(X, complement(X)))
% 84.31/11.10 = { by axiom 7 (absorption2) }
% 84.31/11.10 X
% 84.31/11.10
% 84.31/11.10 Lemma 15: join(zero, X) = X.
% 84.31/11.10 Proof:
% 84.31/11.10 join(zero, X)
% 84.31/11.10 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.10 join(X, zero)
% 84.31/11.10 = { by lemma 14 }
% 84.31/11.10 X
% 84.31/11.10
% 84.31/11.10 Lemma 16: meet(X, zero) = zero.
% 84.31/11.10 Proof:
% 84.31/11.10 meet(X, zero)
% 84.31/11.10 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.10 meet(zero, X)
% 84.31/11.10 = { by lemma 15 R->L }
% 84.31/11.10 join(zero, meet(zero, X))
% 84.31/11.10 = { by axiom 7 (absorption2) }
% 84.31/11.10 zero
% 84.31/11.10
% 84.31/11.10 Lemma 17: join(X, one) = one.
% 84.31/11.10 Proof:
% 84.31/11.10 join(X, one)
% 84.31/11.10 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.10 join(one, X)
% 84.31/11.10 = { by lemma 13 R->L }
% 84.31/11.10 join(one, meet(X, one))
% 84.31/11.10 = { by axiom 4 (commutativity_of_meet) }
% 84.31/11.10 join(one, meet(one, X))
% 84.31/11.10 = { by axiom 7 (absorption2) }
% 84.31/11.10 one
% 84.31/11.10
% 84.31/11.10 Lemma 18: meet(X, meet(Y, complement(X))) = zero.
% 84.31/11.10 Proof:
% 84.31/11.10 meet(X, meet(Y, complement(X)))
% 84.31/11.10 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.10 meet(X, meet(complement(X), Y))
% 84.31/11.10 = { by axiom 10 (associativity_of_meet) R->L }
% 84.31/11.10 meet(meet(X, complement(X)), Y)
% 84.31/11.10 = { by axiom 5 (complement_meet) }
% 84.31/11.10 meet(zero, Y)
% 84.31/11.10 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.10 meet(Y, zero)
% 84.31/11.10 = { by lemma 16 }
% 84.31/11.10 zero
% 84.31/11.10
% 84.31/11.10 Lemma 19: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 84.31/11.10 Proof:
% 84.31/11.10 join(X, join(Y, meet(X, Z)))
% 84.31/11.10 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.10 join(X, join(meet(X, Z), Y))
% 84.31/11.10 = { by axiom 8 (associativity_of_join) R->L }
% 84.31/11.10 join(join(X, meet(X, Z)), Y)
% 84.31/11.10 = { by axiom 7 (absorption2) }
% 84.31/11.10 join(X, Y)
% 84.31/11.10
% 84.31/11.10 Lemma 20: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 84.31/11.10 Proof:
% 84.31/11.10 join(X, join(Y, meet(Z, join(X, Y))))
% 84.31/11.10 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.10 join(X, join(Y, meet(Z, join(Y, X))))
% 84.31/11.10 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.10 join(X, join(Y, meet(join(Y, X), Z)))
% 84.31/11.10 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.10 join(join(Y, meet(join(Y, X), Z)), X)
% 84.31/11.10 = { by axiom 8 (associativity_of_join) }
% 84.31/11.10 join(Y, join(meet(join(Y, X), Z), X))
% 84.31/11.10 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.10 join(Y, join(X, meet(join(Y, X), Z)))
% 84.31/11.10 = { by axiom 8 (associativity_of_join) R->L }
% 84.31/11.10 join(join(Y, X), meet(join(Y, X), Z))
% 84.31/11.10 = { by axiom 7 (absorption2) }
% 84.31/11.10 join(Y, X)
% 84.31/11.10 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.10 join(X, Y)
% 84.31/11.10
% 84.31/11.10 Lemma 21: meet(X, join(Y, meet(Z, join(X, Y)))) = meet(X, join(Y, meet(X, Z))).
% 84.31/11.10 Proof:
% 84.31/11.10 meet(X, join(Y, meet(Z, join(X, Y))))
% 84.31/11.10 = { by lemma 19 R->L }
% 84.31/11.10 meet(X, join(Y, meet(Z, join(X, join(Y, meet(X, Z))))))
% 84.31/11.10 = { by axiom 11 (equation_H51) }
% 84.31/11.10 meet(X, join(Y, join(meet(X, Z), meet(Z, join(Y, meet(X, Z))))))
% 84.31/11.10 = { by lemma 20 }
% 84.31/11.10 meet(X, join(Y, meet(X, Z)))
% 84.31/11.10
% 84.31/11.10 Lemma 22: join(X, meet(Y, X)) = X.
% 84.31/11.10 Proof:
% 84.31/11.10 join(X, meet(Y, X))
% 84.31/11.10 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.10 join(X, meet(X, Y))
% 84.31/11.10 = { by axiom 7 (absorption2) }
% 84.31/11.10 X
% 84.31/11.10
% 84.31/11.10 Lemma 23: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 84.31/11.10 Proof:
% 84.31/11.10 meet(X, meet(Y, Z))
% 84.31/11.10 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.10 meet(meet(Y, Z), X)
% 84.31/11.10 = { by axiom 10 (associativity_of_meet) }
% 84.31/11.10 meet(Y, meet(Z, X))
% 84.31/11.10 = { by axiom 4 (commutativity_of_meet) }
% 84.31/11.10 meet(Y, meet(X, Z))
% 84.31/11.10
% 84.31/11.10 Lemma 24: join(X, meet(Y, join(X, meet(Y, Z)))) = join(X, meet(Y, Z)).
% 84.31/11.10 Proof:
% 84.31/11.10 join(X, meet(Y, join(X, meet(Y, Z))))
% 84.31/11.10 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.10 join(X, meet(Y, join(X, meet(Z, Y))))
% 84.31/11.10 = { by lemma 22 R->L }
% 84.31/11.10 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, meet(Y, join(X, meet(Z, Y))))))
% 84.31/11.10 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.10 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, meet(Y, join(X, meet(Y, Z))))))
% 84.31/11.10 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.10 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, meet(Y, join(meet(Y, Z), X)))))
% 84.31/11.10 = { by lemma 23 R->L }
% 84.31/11.10 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Y, meet(Z, join(meet(Y, Z), X)))))
% 84.31/11.10 = { by axiom 10 (associativity_of_meet) R->L }
% 84.31/11.11 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(meet(Y, Z), join(meet(Y, Z), X))))
% 84.31/11.11 = { by axiom 9 (absorption1) }
% 84.31/11.11 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Y, Z)))
% 84.31/11.11 = { by axiom 4 (commutativity_of_meet) }
% 84.31/11.11 join(X, join(meet(Y, join(X, meet(Z, Y))), meet(Z, Y)))
% 84.31/11.11 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.11 join(X, join(meet(Z, Y), meet(Y, join(X, meet(Z, Y)))))
% 84.31/11.11 = { by axiom 4 (commutativity_of_meet) }
% 84.31/11.11 join(X, join(meet(Z, Y), meet(Y, join(X, meet(Y, Z)))))
% 84.31/11.11 = { by axiom 4 (commutativity_of_meet) }
% 84.31/11.11 join(X, join(meet(Y, Z), meet(Y, join(X, meet(Y, Z)))))
% 84.31/11.11 = { by lemma 20 }
% 84.31/11.11 join(X, meet(Y, Z))
% 84.31/11.11
% 84.31/11.11 Lemma 25: meet(X, join(complement(X), meet(X, Y))) = meet(X, join(Y, complement(X))).
% 84.31/11.11 Proof:
% 84.31/11.11 meet(X, join(complement(X), meet(X, Y)))
% 84.31/11.11 = { by lemma 21 R->L }
% 84.31/11.11 meet(X, join(complement(X), meet(Y, join(X, complement(X)))))
% 84.31/11.11 = { by axiom 2 (complement_join) }
% 84.31/11.11 meet(X, join(complement(X), meet(Y, one)))
% 84.31/11.11 = { by lemma 13 }
% 84.31/11.11 meet(X, join(complement(X), Y))
% 84.31/11.11 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.11 meet(X, join(Y, complement(X)))
% 84.31/11.11
% 84.31/11.11 Lemma 26: meet(X, join(complement(X), meet(Y, X))) = meet(X, join(Y, complement(X))).
% 84.31/11.11 Proof:
% 84.31/11.11 meet(X, join(complement(X), meet(Y, X)))
% 84.31/11.11 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.11 meet(X, join(complement(X), meet(X, Y)))
% 84.31/11.11 = { by lemma 25 }
% 84.31/11.11 meet(X, join(Y, complement(X)))
% 84.31/11.11
% 84.31/11.11 Lemma 27: join(X, join(Y, complement(X))) = one.
% 84.31/11.11 Proof:
% 84.31/11.11 join(X, join(Y, complement(X)))
% 84.31/11.11 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.11 join(X, join(complement(X), Y))
% 84.31/11.11 = { by axiom 8 (associativity_of_join) R->L }
% 84.31/11.11 join(join(X, complement(X)), Y)
% 84.31/11.11 = { by axiom 2 (complement_join) }
% 84.31/11.11 join(one, Y)
% 84.31/11.11 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.11 join(Y, one)
% 84.31/11.11 = { by lemma 17 }
% 84.31/11.11 one
% 84.31/11.11
% 84.31/11.11 Lemma 28: join(X, complement(complement(X))) = complement(complement(X)).
% 84.31/11.11 Proof:
% 84.31/11.11 join(X, complement(complement(X)))
% 84.31/11.11 = { by axiom 12 (meet_join_complement) R->L }
% 84.31/11.11 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))))
% 84.31/11.11 = { by lemma 26 R->L }
% 84.31/11.11 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))))
% 84.31/11.11 = { by axiom 5 (complement_meet) }
% 84.31/11.11 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))))
% 84.31/11.11 = { by lemma 14 }
% 84.31/11.11 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))))
% 84.31/11.11 = { by axiom 5 (complement_meet) }
% 84.31/11.11 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))))
% 84.31/11.11 = { by lemma 27 }
% 84.31/11.11 ifeq(one, one, ifeq(zero, zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X))))
% 84.31/11.11 = { by axiom 6 (ifeq_axiom) }
% 84.31/11.11 ifeq(zero, zero, complement(complement(X)), join(X, complement(complement(X))))
% 84.31/11.11 = { by axiom 6 (ifeq_axiom) }
% 84.31/11.11 complement(complement(X))
% 84.31/11.11
% 84.31/11.11 Lemma 29: complement(complement(X)) = X.
% 84.31/11.11 Proof:
% 84.31/11.11 complement(complement(X))
% 84.31/11.11 = { by axiom 3 (idempotence_of_meet) R->L }
% 84.31/11.11 complement(complement(meet(X, X)))
% 84.31/11.11 = { by axiom 6 (ifeq_axiom) R->L }
% 84.31/11.11 ifeq(zero, zero, complement(complement(meet(X, X))), meet(X, complement(complement(meet(X, X)))))
% 84.31/11.11 = { by lemma 18 R->L }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by axiom 6 (ifeq_axiom) R->L }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by axiom 2 (complement_join) R->L }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by lemma 13 R->L }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by lemma 17 R->L }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by axiom 2 (complement_join) R->L }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by lemma 19 }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by lemma 13 R->L }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by axiom 2 (complement_join) R->L }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by lemma 21 }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by lemma 24 }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by axiom 4 (commutativity_of_meet) }
% 84.31/11.11 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)))))
% 84.31/11.11 = { by axiom 12 (meet_join_complement) }
% 84.31/11.11 meet(X, complement(complement(meet(X, X))))
% 84.31/11.11 = { by axiom 3 (idempotence_of_meet) }
% 84.31/11.11 meet(X, complement(complement(X)))
% 84.31/11.11 = { by lemma 28 R->L }
% 84.31/11.11 meet(X, join(X, complement(complement(X))))
% 84.31/11.11 = { by axiom 9 (absorption1) }
% 84.31/11.11 X
% 84.31/11.11
% 84.31/11.11 Lemma 30: meet(X, complement(join(X, Y))) = zero.
% 84.31/11.11 Proof:
% 84.31/11.11 meet(X, complement(join(X, Y)))
% 84.31/11.11 = { by axiom 9 (absorption1) R->L }
% 84.31/11.11 meet(meet(X, join(X, Y)), complement(join(X, Y)))
% 84.31/11.11 = { by axiom 10 (associativity_of_meet) }
% 84.31/11.11 meet(X, meet(join(X, Y), complement(join(X, Y))))
% 84.31/11.11 = { by axiom 5 (complement_meet) }
% 84.31/11.11 meet(X, zero)
% 84.31/11.11 = { by lemma 16 }
% 84.31/11.11 zero
% 84.31/11.11
% 84.31/11.11 Lemma 31: join(X, join(complement(X), Y)) = one.
% 84.31/11.11 Proof:
% 84.31/11.11 join(X, join(complement(X), Y))
% 84.31/11.11 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.11 join(X, join(Y, complement(X)))
% 84.31/11.11 = { by lemma 27 }
% 84.31/11.11 one
% 84.31/11.11
% 84.31/11.11 Lemma 32: join(complement(X), complement(join(X, Y))) = complement(X).
% 84.31/11.11 Proof:
% 84.31/11.11 join(complement(X), complement(join(X, Y)))
% 84.31/11.11 = { by axiom 12 (meet_join_complement) R->L }
% 84.31/11.11 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))))
% 84.31/11.11 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.11 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))))
% 84.31/11.11 = { by lemma 25 R->L }
% 84.31/11.11 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))))
% 84.31/11.11 = { by lemma 30 }
% 84.31/11.11 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))))
% 84.31/11.11 = { by lemma 14 }
% 84.31/11.11 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))))
% 84.31/11.11 = { by axiom 5 (complement_meet) }
% 84.31/11.11 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))))
% 84.31/11.11 = { by lemma 31 }
% 84.31/11.11 ifeq(one, one, ifeq(zero, zero, complement(X), join(complement(X), complement(join(X, Y)))), join(complement(X), complement(join(X, Y))))
% 84.31/11.11 = { by axiom 6 (ifeq_axiom) }
% 84.31/11.11 ifeq(zero, zero, complement(X), join(complement(X), complement(join(X, Y))))
% 84.31/11.11 = { by axiom 6 (ifeq_axiom) }
% 84.31/11.11 complement(X)
% 84.31/11.11
% 84.31/11.12 Lemma 33: meet(X, join(Y, X)) = X.
% 84.31/11.12 Proof:
% 84.31/11.12 meet(X, join(Y, X))
% 84.31/11.12 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.12 meet(X, join(X, Y))
% 84.31/11.12 = { by axiom 9 (absorption1) }
% 84.31/11.12 X
% 84.31/11.12
% 84.31/11.12 Lemma 34: join(X, meet(Y, complement(X))) = join(X, Y).
% 84.31/11.12 Proof:
% 84.31/11.12 join(X, meet(Y, complement(X)))
% 84.31/11.12 = { by lemma 29 R->L }
% 84.31/11.12 complement(complement(join(X, meet(Y, complement(X)))))
% 84.31/11.12 = { by axiom 6 (ifeq_axiom) R->L }
% 84.31/11.12 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)))))
% 84.31/11.12 = { by axiom 6 (ifeq_axiom) R->L }
% 84.31/11.12 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)))))
% 84.31/11.12 = { by lemma 31 R->L }
% 84.31/11.12 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)))))
% 84.31/11.12 = { by axiom 5 (complement_meet) R->L }
% 84.31/11.12 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)))))
% 84.31/11.12 = { by lemma 14 R->L }
% 84.31/11.12 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)))))
% 84.31/11.12 = { by axiom 5 (complement_meet) R->L }
% 84.31/11.12 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)))))
% 84.31/11.12 = { by axiom 10 (associativity_of_meet) }
% 84.31/11.12 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)))))
% 84.31/11.12 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.12 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)))))
% 84.31/11.12 = { by axiom 10 (associativity_of_meet) R->L }
% 84.31/11.12 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)))))
% 84.31/11.13 = { by lemma 26 }
% 84.31/11.13 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)))))
% 84.31/11.13 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.13 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)))))
% 84.31/11.13 = { by axiom 4 (commutativity_of_meet) }
% 84.31/11.13 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)))))
% 84.31/11.13 = { by axiom 12 (meet_join_complement) }
% 84.31/11.13 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(join(X, meet(Y, complement(X)))), Y))))
% 84.31/11.13 = { by axiom 4 (commutativity_of_meet) }
% 84.31/11.13 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, complement(join(X, meet(Y, complement(X))))))))
% 84.31/11.13 = { by lemma 33 R->L }
% 84.31/11.13 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))))))))))
% 84.31/11.13 = { by lemma 32 }
% 84.31/11.13 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, meet(complement(join(X, meet(Y, complement(X)))), complement(X))))))
% 84.31/11.13 = { by axiom 4 (commutativity_of_meet) }
% 84.31/11.13 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, meet(complement(X), complement(join(X, meet(Y, complement(X)))))))))
% 84.31/11.13 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.13 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, meet(complement(X), complement(join(X, meet(complement(X), Y))))))))
% 84.31/11.13 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.13 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(Y, meet(complement(X), complement(join(meet(complement(X), Y), X)))))))
% 84.31/11.13 = { by lemma 23 R->L }
% 84.31/11.13 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(complement(X), meet(Y, complement(join(meet(complement(X), Y), X)))))))
% 84.31/11.13 = { by axiom 10 (associativity_of_meet) R->L }
% 84.31/11.13 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(meet(meet(complement(X), Y), complement(join(meet(complement(X), Y), X))))))
% 84.31/11.13 = { by lemma 30 }
% 84.31/11.13 join(complement(complement(join(X, meet(Y, complement(X))))), meet(Y, complement(zero)))
% 84.31/11.13 = { by lemma 29 }
% 84.31/11.13 join(join(X, meet(Y, complement(X))), meet(Y, complement(zero)))
% 84.31/11.13 = { by axiom 8 (associativity_of_join) }
% 84.31/11.13 join(X, join(meet(Y, complement(X)), meet(Y, complement(zero))))
% 84.31/11.13 = { by lemma 15 R->L }
% 84.31/11.13 join(X, join(meet(Y, complement(X)), meet(Y, join(zero, complement(zero)))))
% 84.31/11.13 = { by axiom 2 (complement_join) }
% 84.31/11.13 join(X, join(meet(Y, complement(X)), meet(Y, one)))
% 84.31/11.13 = { by lemma 13 }
% 84.31/11.13 join(X, join(meet(Y, complement(X)), Y))
% 84.31/11.13 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.13 join(X, join(Y, meet(Y, complement(X))))
% 84.31/11.13 = { by axiom 7 (absorption2) }
% 84.31/11.13 join(X, Y)
% 84.31/11.13
% 84.31/11.13 Lemma 35: meet(complement(X), join(Y, complement(join(X, Y)))) = complement(X).
% 84.31/11.13 Proof:
% 84.31/11.13 meet(complement(X), join(Y, complement(join(X, Y))))
% 84.31/11.13 = { by axiom 12 (meet_join_complement) R->L }
% 84.31/11.13 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)))))
% 84.31/11.13 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.13 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)))))
% 84.31/11.13 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.13 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)))))
% 84.31/11.13 = { by lemma 18 }
% 84.31/11.13 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)))))
% 84.31/11.13 = { by axiom 6 (ifeq_axiom) }
% 84.31/11.13 ifeq(join(meet(complement(X), join(Y, complement(join(X, Y)))), X), one, complement(X), meet(complement(X), join(Y, complement(join(X, Y)))))
% 84.31/11.13 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.13 ifeq(join(X, meet(complement(X), join(Y, complement(join(X, Y))))), one, complement(X), meet(complement(X), join(Y, complement(join(X, Y)))))
% 84.31/11.13 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.13 ifeq(join(X, meet(join(Y, complement(join(X, Y))), complement(X))), one, complement(X), meet(complement(X), join(Y, complement(join(X, Y)))))
% 84.31/11.13 = { by lemma 34 }
% 84.31/11.13 ifeq(join(X, join(Y, complement(join(X, Y)))), one, complement(X), meet(complement(X), join(Y, complement(join(X, Y)))))
% 84.31/11.13 = { by axiom 4 (commutativity_of_meet) }
% 84.31/11.13 ifeq(join(X, join(Y, complement(join(X, Y)))), one, complement(X), meet(join(Y, complement(join(X, Y))), complement(X)))
% 84.31/11.13 = { by axiom 8 (associativity_of_join) R->L }
% 84.31/11.13 ifeq(join(join(X, Y), complement(join(X, Y))), one, complement(X), meet(join(Y, complement(join(X, Y))), complement(X)))
% 84.31/11.13 = { by axiom 2 (complement_join) }
% 84.31/11.13 ifeq(one, one, complement(X), meet(join(Y, complement(join(X, Y))), complement(X)))
% 84.31/11.13 = { by axiom 6 (ifeq_axiom) }
% 84.31/11.13 complement(X)
% 84.31/11.13
% 84.31/11.13 Lemma 36: join(X, complement(join(X, Y))) = join(X, complement(Y)).
% 84.31/11.13 Proof:
% 84.31/11.13 join(X, complement(join(X, Y)))
% 84.31/11.13 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.13 join(X, complement(join(Y, X)))
% 84.31/11.13 = { by lemma 22 R->L }
% 84.31/11.13 join(join(X, complement(join(Y, X))), meet(complement(Y), join(X, complement(join(Y, X)))))
% 84.31/11.13 = { by lemma 35 }
% 84.31/11.13 join(join(X, complement(join(Y, X))), complement(Y))
% 84.31/11.13 = { by axiom 8 (associativity_of_join) }
% 84.31/11.13 join(X, join(complement(join(Y, X)), complement(Y)))
% 84.31/11.13 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.13 join(X, join(complement(Y), complement(join(Y, X))))
% 84.31/11.13 = { by lemma 32 }
% 84.31/11.13 join(X, complement(Y))
% 84.31/11.13
% 84.31/11.13 Lemma 37: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 84.31/11.13 Proof:
% 84.31/11.13 complement(meet(X, complement(Y)))
% 84.31/11.13 = { by lemma 22 R->L }
% 84.31/11.13 join(complement(meet(X, complement(Y))), meet(Y, complement(meet(X, complement(Y)))))
% 84.31/11.13 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.13 join(complement(meet(X, complement(Y))), meet(Y, complement(meet(complement(Y), X))))
% 84.31/11.13 = { by lemma 32 R->L }
% 84.31/11.13 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), complement(join(meet(complement(Y), X), complement(Y))))))
% 84.31/11.13 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.13 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), complement(join(complement(Y), meet(complement(Y), X))))))
% 84.31/11.13 = { by axiom 7 (absorption2) }
% 84.31/11.13 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), complement(complement(Y)))))
% 84.31/11.13 = { by lemma 28 R->L }
% 84.31/11.13 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), join(Y, complement(complement(Y))))))
% 84.31/11.13 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.13 join(complement(meet(X, complement(Y))), meet(Y, join(complement(meet(complement(Y), X)), join(complement(complement(Y)), Y))))
% 84.31/11.13 = { by axiom 8 (associativity_of_join) R->L }
% 84.31/11.13 join(complement(meet(X, complement(Y))), meet(Y, join(join(complement(meet(complement(Y), X)), complement(complement(Y))), Y)))
% 84.31/11.13 = { by lemma 33 }
% 84.31/11.13 join(complement(meet(X, complement(Y))), Y)
% 84.31/11.13 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.13 join(Y, complement(meet(X, complement(Y))))
% 84.31/11.13 = { by lemma 36 R->L }
% 84.31/11.13 join(Y, complement(join(Y, meet(X, complement(Y)))))
% 84.31/11.13 = { by lemma 34 }
% 84.31/11.13 join(Y, complement(join(Y, X)))
% 84.31/11.13 = { by lemma 36 }
% 84.31/11.13 join(Y, complement(X))
% 84.31/11.13
% 84.31/11.13 Lemma 38: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 84.31/11.13 Proof:
% 84.31/11.14 complement(join(X, complement(Y)))
% 84.31/11.14 = { by lemma 37 R->L }
% 84.31/11.14 complement(complement(meet(Y, complement(X))))
% 84.31/11.14 = { by lemma 29 }
% 84.31/11.14 meet(Y, complement(X))
% 84.31/11.14
% 84.31/11.14 Lemma 39: complement(join(X, Y)) = meet(complement(X), complement(Y)).
% 84.31/11.14 Proof:
% 84.31/11.14 complement(join(X, Y))
% 84.31/11.14 = { by lemma 29 R->L }
% 84.31/11.14 complement(join(X, complement(complement(Y))))
% 84.31/11.14 = { by lemma 38 }
% 84.31/11.14 meet(complement(Y), complement(X))
% 84.31/11.14 = { by axiom 4 (commutativity_of_meet) }
% 84.31/11.14 meet(complement(X), complement(Y))
% 84.31/11.14
% 84.31/11.14 Lemma 40: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 84.31/11.14 Proof:
% 84.31/11.14 complement(meet(complement(X), Y))
% 84.31/11.14 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.14 complement(meet(Y, complement(X)))
% 84.31/11.14 = { by lemma 37 }
% 84.31/11.14 join(X, complement(Y))
% 84.31/11.14
% 84.31/11.14 Lemma 41: meet(complement(X), join(Y, meet(Z, complement(X)))) = meet(complement(X), join(Y, Z)).
% 84.31/11.14 Proof:
% 84.31/11.14 meet(complement(X), join(Y, meet(Z, complement(X))))
% 84.31/11.14 = { by lemma 33 R->L }
% 84.31/11.14 meet(complement(X), meet(join(Y, meet(Z, complement(X))), join(Z, join(Y, meet(Z, complement(X))))))
% 84.31/11.14 = { by lemma 19 }
% 84.31/11.14 meet(complement(X), meet(join(Y, meet(Z, complement(X))), join(Z, Y)))
% 84.31/11.14 = { by axiom 4 (commutativity_of_meet) }
% 84.31/11.14 meet(complement(X), meet(join(Z, Y), join(Y, meet(Z, complement(X)))))
% 84.31/11.14 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.14 meet(complement(X), meet(join(Y, Z), join(Y, meet(Z, complement(X)))))
% 84.31/11.14 = { by lemma 38 R->L }
% 84.31/11.14 meet(complement(X), meet(join(Y, Z), join(Y, complement(join(X, complement(Z))))))
% 84.31/11.14 = { by lemma 36 R->L }
% 84.31/11.14 meet(complement(X), meet(join(Y, Z), join(Y, complement(join(Y, join(X, complement(Z)))))))
% 84.31/11.14 = { by lemma 29 R->L }
% 84.31/11.14 meet(complement(X), meet(join(Y, complement(complement(Z))), join(Y, complement(join(Y, join(X, complement(Z)))))))
% 84.31/11.14 = { by lemma 40 R->L }
% 84.31/11.14 meet(complement(X), meet(complement(meet(complement(Y), complement(Z))), join(Y, complement(join(Y, join(X, complement(Z)))))))
% 84.31/11.14 = { by axiom 10 (associativity_of_meet) R->L }
% 84.31/11.14 meet(meet(complement(X), complement(meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(X, complement(Z))))))
% 84.31/11.14 = { by lemma 39 R->L }
% 84.31/11.14 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(X, complement(Z))))))
% 84.31/11.14 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.14 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(complement(Z), X)))))
% 84.31/11.14 = { by axiom 8 (associativity_of_join) R->L }
% 84.31/11.14 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(join(Y, complement(Z)), X))))
% 84.31/11.14 = { by lemma 36 R->L }
% 84.31/11.14 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(join(Y, complement(join(Y, Z))), X))))
% 84.31/11.14 = { by axiom 8 (associativity_of_join) }
% 84.31/11.14 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(complement(join(Y, Z)), X)))))
% 84.31/11.14 = { by lemma 39 }
% 84.31/11.14 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(meet(complement(Y), complement(Z)), X)))))
% 84.31/11.14 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.14 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(Y, join(X, meet(complement(Y), complement(Z)))))))
% 84.31/11.14 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.14 meet(complement(join(X, meet(complement(Y), complement(Z)))), join(Y, complement(join(join(X, meet(complement(Y), complement(Z))), Y))))
% 84.31/11.14 = { by lemma 35 }
% 84.31/11.14 complement(join(X, meet(complement(Y), complement(Z))))
% 84.31/11.14 = { by lemma 39 }
% 84.31/11.14 meet(complement(X), complement(meet(complement(Y), complement(Z))))
% 84.31/11.14 = { by lemma 40 }
% 84.31/11.14 meet(complement(X), join(Y, complement(complement(Z))))
% 84.31/11.14 = { by lemma 29 }
% 84.31/11.14 meet(complement(X), join(Y, Z))
% 84.31/11.14
% 84.31/11.14 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 84.31/11.14 Proof:
% 84.31/11.14 meet(a, join(b, c))
% 84.31/11.14 = { by lemma 29 R->L }
% 84.31/11.14 meet(complement(complement(a)), join(b, c))
% 84.31/11.14 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.14 meet(complement(complement(a)), join(c, b))
% 84.31/11.14 = { by lemma 41 R->L }
% 84.31/11.14 meet(complement(complement(a)), join(c, meet(b, complement(complement(a)))))
% 84.31/11.14 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.14 meet(complement(complement(a)), join(meet(b, complement(complement(a))), c))
% 84.31/11.14 = { by lemma 22 R->L }
% 84.31/11.14 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))))
% 84.31/11.14 = { by lemma 23 R->L }
% 84.31/11.14 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))))
% 84.31/11.14 = { by lemma 33 }
% 84.31/11.14 join(meet(complement(complement(a)), join(meet(b, complement(complement(a))), c)), meet(complement(complement(a)), c))
% 84.31/11.14 = { by axiom 4 (commutativity_of_meet) }
% 84.31/11.14 join(meet(complement(complement(a)), join(meet(b, complement(complement(a))), c)), meet(c, complement(complement(a))))
% 84.31/11.14 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.14 join(meet(c, complement(complement(a))), meet(complement(complement(a)), join(meet(b, complement(complement(a))), c)))
% 84.31/11.14 = { by lemma 41 R->L }
% 84.31/11.14 join(meet(c, complement(complement(a))), meet(complement(complement(a)), join(meet(b, complement(complement(a))), meet(c, complement(complement(a))))))
% 84.31/11.14 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.14 join(meet(c, complement(complement(a))), meet(complement(complement(a)), join(meet(complement(complement(a)), b), meet(c, complement(complement(a))))))
% 84.31/11.14 = { by axiom 1 (commutativity_of_join) R->L }
% 84.31/11.14 join(meet(c, complement(complement(a))), meet(complement(complement(a)), join(meet(c, complement(complement(a))), meet(complement(complement(a)), b))))
% 84.31/11.14 = { by lemma 24 }
% 84.31/11.14 join(meet(c, complement(complement(a))), meet(complement(complement(a)), b))
% 84.31/11.14 = { by axiom 4 (commutativity_of_meet) }
% 84.31/11.14 join(meet(c, complement(complement(a))), meet(b, complement(complement(a))))
% 84.31/11.14 = { by axiom 1 (commutativity_of_join) }
% 84.31/11.14 join(meet(b, complement(complement(a))), meet(c, complement(complement(a))))
% 84.31/11.14 = { by lemma 29 }
% 84.31/11.14 join(meet(b, a), meet(c, complement(complement(a))))
% 84.31/11.14 = { by lemma 29 }
% 84.31/11.14 join(meet(b, a), meet(c, a))
% 84.31/11.14 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.14 join(meet(b, a), meet(a, c))
% 84.31/11.14 = { by axiom 4 (commutativity_of_meet) R->L }
% 84.31/11.14 join(meet(a, b), meet(a, c))
% 84.31/11.14 % SZS output end Proof
% 84.31/11.14
% 84.31/11.14 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------