%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LAT193-1 : TPTP v9.3.1. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n020.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 130.39s 16.98s
% Output : Proof 131.11s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.04 % Problem : LAT193-1 : TPTP v9.3.1. Released v3.1.0.
% 0.00/0.06 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.15/0.42 % Computer : n020.cluster.edu
% 0.15/0.42 % Model : x86_64 x86_64
% 0.15/0.42 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.42 % Memory : 8046.5625MB
% 0.15/0.42 % OS : Linux 6.8.0-71-generic
% 0.15/0.42 % CPULimit : 300
% 0.15/0.42 % WCLimit : 300
% 0.15/0.42 % DateTime : Sun Sep 27 14:06:34 UTC 2026
% 0.15/0.42 % CPUTime :
% 0.15/0.42 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 130.39/16.98 Command-line arguments: --no-flatten-goal
% 130.39/16.98
% 130.39/16.98 % SZS status Unsatisfiable
% 130.39/16.98
% 131.11/17.09 % SZS output start Proof
% 131.11/17.09 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 131.11/17.09 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 131.11/17.09 Axiom 3 (complement_join): join(X, complement(X)) = one.
% 131.11/17.09 Axiom 4 (idempotence_of_meet): meet(X, X) = X.
% 131.11/17.09 Axiom 5 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 131.11/17.09 Axiom 6 (complement_meet): meet(X, complement(X)) = zero.
% 131.11/17.09 Axiom 7 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 131.11/17.09 Axiom 8 (absorption2): join(X, meet(X, Y)) = X.
% 131.11/17.09 Axiom 9 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 131.11/17.09 Axiom 10 (absorption1): meet(X, join(X, Y)) = X.
% 131.11/17.09 Axiom 11 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 131.11/17.09 Axiom 12 (meet_join_complement): ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y) = Y.
% 131.11/17.09 Axiom 13 (equation_H24): join(meet(X, Y), meet(Y, Z)) = join(meet(X, Y), meet(Y, join(meet(X, Y), meet(Z, join(X, Y))))).
% 131.11/17.09
% 131.11/17.09 Lemma 14: meet(X, one) = X.
% 131.11/17.09 Proof:
% 131.11/17.09 meet(X, one)
% 131.11/17.09 = { by axiom 3 (complement_join) R->L }
% 131.11/17.09 meet(X, join(X, complement(X)))
% 131.11/17.09 = { by axiom 10 (absorption1) }
% 131.11/17.09 X
% 131.11/17.09
% 131.11/17.09 Lemma 15: complement(complement(X)) = X.
% 131.11/17.09 Proof:
% 131.11/17.09 complement(complement(X))
% 131.11/17.09 = { by axiom 7 (ifeq_axiom) R->L }
% 131.11/17.09 ifeq(zero, zero, complement(complement(X)), X)
% 131.11/17.09 = { by axiom 6 (complement_meet) R->L }
% 131.11/17.09 ifeq(meet(X, complement(X)), zero, complement(complement(X)), X)
% 131.11/17.09 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.09 ifeq(meet(complement(X), X), zero, complement(complement(X)), X)
% 131.11/17.09 = { by axiom 7 (ifeq_axiom) R->L }
% 131.11/17.09 ifeq(one, one, ifeq(meet(complement(X), X), zero, complement(complement(X)), X), X)
% 131.11/17.09 = { by axiom 3 (complement_join) R->L }
% 131.11/17.09 ifeq(join(X, complement(X)), one, ifeq(meet(complement(X), X), zero, complement(complement(X)), X), X)
% 131.11/17.09 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.09 ifeq(join(complement(X), X), one, ifeq(meet(complement(X), X), zero, complement(complement(X)), X), X)
% 131.11/17.09 = { by axiom 12 (meet_join_complement) }
% 131.11/17.09 X
% 131.11/17.09
% 131.11/17.09 Lemma 16: meet(X, meet(X, Y)) = meet(X, Y).
% 131.11/17.09 Proof:
% 131.11/17.09 meet(X, meet(X, Y))
% 131.11/17.09 = { by axiom 11 (associativity_of_meet) R->L }
% 131.11/17.09 meet(meet(X, X), Y)
% 131.11/17.09 = { by axiom 4 (idempotence_of_meet) }
% 131.11/17.09 meet(X, Y)
% 131.11/17.09
% 131.11/17.09 Lemma 17: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 131.11/17.09 Proof:
% 131.11/17.09 meet(X, meet(Y, Z))
% 131.11/17.09 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.09 meet(meet(Y, Z), X)
% 131.11/17.09 = { by axiom 11 (associativity_of_meet) }
% 131.11/17.09 meet(Y, meet(Z, X))
% 131.11/17.09 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.09 meet(Y, meet(X, Z))
% 131.11/17.09
% 131.11/17.09 Lemma 18: meet(X, join(Y, X)) = X.
% 131.11/17.09 Proof:
% 131.11/17.09 meet(X, join(Y, X))
% 131.11/17.09 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.09 meet(X, join(X, Y))
% 131.11/17.09 = { by axiom 10 (absorption1) }
% 131.11/17.09 X
% 131.11/17.09
% 131.11/17.09 Lemma 19: join(X, meet(Y, X)) = X.
% 131.11/17.09 Proof:
% 131.11/17.09 join(X, meet(Y, X))
% 131.11/17.09 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.09 join(X, meet(X, Y))
% 131.11/17.09 = { by axiom 8 (absorption2) }
% 131.11/17.09 X
% 131.11/17.09
% 131.11/17.09 Lemma 20: join(X, join(X, Y)) = join(X, Y).
% 131.11/17.09 Proof:
% 131.11/17.09 join(X, join(X, Y))
% 131.11/17.09 = { by axiom 9 (associativity_of_join) R->L }
% 131.11/17.09 join(join(X, X), Y)
% 131.11/17.09 = { by axiom 1 (idempotence_of_join) }
% 131.11/17.09 join(X, Y)
% 131.11/17.09
% 131.11/17.09 Lemma 21: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 131.11/17.09 Proof:
% 131.11/17.09 join(Y, join(X, Z))
% 131.11/17.09 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.09 join(join(X, Z), Y)
% 131.11/17.09 = { by axiom 9 (associativity_of_join) }
% 131.11/17.09 join(X, join(Z, Y))
% 131.11/17.09 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.09 join(X, join(Y, Z))
% 131.11/17.09
% 131.11/17.09 Lemma 22: join(X, join(Y, complement(X))) = one.
% 131.11/17.09 Proof:
% 131.11/17.09 join(X, join(Y, complement(X)))
% 131.11/17.09 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.09 join(X, join(complement(X), Y))
% 131.11/17.09 = { by axiom 9 (associativity_of_join) R->L }
% 131.11/17.09 join(join(X, complement(X)), Y)
% 131.11/17.09 = { by axiom 3 (complement_join) }
% 131.11/17.09 join(one, Y)
% 131.11/17.09 = { by lemma 14 R->L }
% 131.11/17.09 join(one, meet(Y, one))
% 131.11/17.09 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.09 join(one, meet(one, Y))
% 131.11/17.09 = { by axiom 8 (absorption2) }
% 131.11/17.09 one
% 131.11/17.09
% 131.11/17.09 Lemma 23: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 131.11/17.09 Proof:
% 131.11/17.09 join(X, join(Y, meet(X, Z)))
% 131.11/17.09 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.09 join(X, join(meet(X, Z), Y))
% 131.11/17.09 = { by axiom 9 (associativity_of_join) R->L }
% 131.11/17.09 join(join(X, meet(X, Z)), Y)
% 131.11/17.09 = { by axiom 8 (absorption2) }
% 131.11/17.09 join(X, Y)
% 131.11/17.09
% 131.11/17.09 Lemma 24: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
% 131.11/17.09 Proof:
% 131.11/17.09 meet(X, meet(Y, join(Z, meet(X, Y))))
% 131.11/17.09 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.09 meet(X, meet(Y, join(Z, meet(Y, X))))
% 131.11/17.09 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.09 meet(X, meet(Y, join(meet(Y, X), Z)))
% 131.11/17.09 = { by lemma 17 R->L }
% 131.11/17.09 meet(Y, meet(X, join(meet(Y, X), Z)))
% 131.11/17.09 = { by axiom 11 (associativity_of_meet) R->L }
% 131.11/17.09 meet(meet(Y, X), join(meet(Y, X), Z))
% 131.11/17.09 = { by axiom 10 (absorption1) }
% 131.11/17.09 meet(Y, X)
% 131.11/17.09 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.09 meet(X, Y)
% 131.11/17.09
% 131.11/17.09 Lemma 25: meet(X, join(meet(X, Y), meet(Y, Z))) = meet(X, Y).
% 131.11/17.09 Proof:
% 131.11/17.09 meet(X, join(meet(X, Y), meet(Y, Z)))
% 131.11/17.09 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.09 meet(X, join(meet(X, Y), meet(Z, Y)))
% 131.11/17.09 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.09 meet(X, join(meet(Y, X), meet(Z, Y)))
% 131.11/17.09 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.09 meet(X, join(meet(Z, Y), meet(Y, X)))
% 131.11/17.09 = { by lemma 18 R->L }
% 131.11/17.09 meet(X, meet(join(meet(Z, Y), meet(Y, X)), join(Y, join(meet(Z, Y), meet(Y, X)))))
% 131.11/17.09 = { by lemma 23 }
% 131.11/17.09 meet(X, meet(join(meet(Z, Y), meet(Y, X)), join(Y, meet(Z, Y))))
% 131.11/17.09 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.10 meet(X, meet(join(Y, meet(Z, Y)), join(meet(Z, Y), meet(Y, X))))
% 131.11/17.10 = { by lemma 19 }
% 131.11/17.10 meet(X, meet(Y, join(meet(Z, Y), meet(Y, X))))
% 131.11/17.10 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.10 meet(X, meet(Y, join(meet(Y, X), meet(Z, Y))))
% 131.11/17.10 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.10 meet(X, meet(Y, join(meet(X, Y), meet(Z, Y))))
% 131.11/17.10 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.10 meet(X, meet(Y, join(meet(X, Y), meet(Y, Z))))
% 131.11/17.10 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.10 meet(X, meet(Y, join(meet(Y, Z), meet(X, Y))))
% 131.11/17.10 = { by lemma 24 }
% 131.11/17.10 meet(X, Y)
% 131.11/17.10
% 131.11/17.10 Lemma 26: join(meet(X, Y), meet(X, join(Z, meet(X, Y)))) = meet(X, join(Z, meet(X, Y))).
% 131.11/17.10 Proof:
% 131.11/17.10 join(meet(X, Y), meet(X, join(Z, meet(X, Y))))
% 131.11/17.10 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.10 join(meet(Y, X), meet(X, join(Z, meet(X, Y))))
% 131.11/17.10 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.10 join(meet(Y, X), meet(X, join(Z, meet(Y, X))))
% 131.11/17.10 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.10 join(meet(X, join(Z, meet(Y, X))), meet(Y, X))
% 131.11/17.10 = { by lemma 24 R->L }
% 131.11/17.10 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(Z, meet(Y, X)))))
% 131.11/17.10 = { by lemma 19 }
% 131.11/17.10 meet(X, join(Z, meet(Y, X)))
% 131.11/17.10 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.10 meet(X, join(Z, meet(X, Y)))
% 131.11/17.10
% 131.11/17.10 Lemma 27: meet(X, join(meet(Y, X), meet(Z, join(Y, X)))) = join(meet(Y, X), meet(X, Z)).
% 131.11/17.10 Proof:
% 131.11/17.10 meet(X, join(meet(Y, X), meet(Z, join(Y, X))))
% 131.11/17.10 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.10 meet(X, join(meet(Z, join(Y, X)), meet(Y, X)))
% 131.11/17.10 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.10 meet(X, join(meet(Z, join(Y, X)), meet(X, Y)))
% 131.11/17.10 = { by lemma 26 R->L }
% 131.11/17.10 join(meet(X, Y), meet(X, join(meet(Z, join(Y, X)), meet(X, Y))))
% 131.11/17.10 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.10 join(meet(Y, X), meet(X, join(meet(Z, join(Y, X)), meet(X, Y))))
% 131.11/17.10 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.10 join(meet(Y, X), meet(X, join(meet(Z, join(Y, X)), meet(Y, X))))
% 131.11/17.10 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.10 join(meet(Y, X), meet(X, join(meet(Y, X), meet(Z, join(Y, X)))))
% 131.11/17.10 = { by axiom 13 (equation_H24) R->L }
% 131.11/17.10 join(meet(Y, X), meet(X, Z))
% 131.11/17.10
% 131.11/17.10 Lemma 28: meet(join(X, complement(Y)), join(Z, meet(Y, join(X, complement(Y))))) = join(meet(Y, join(X, complement(Y))), meet(Z, join(X, complement(Y)))).
% 131.11/17.10 Proof:
% 131.11/17.10 meet(join(X, complement(Y)), join(Z, meet(Y, join(X, complement(Y)))))
% 131.11/17.10 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.10 meet(join(X, complement(Y)), join(meet(Y, join(X, complement(Y))), Z))
% 131.11/17.10 = { by lemma 14 R->L }
% 131.11/17.10 meet(join(X, complement(Y)), join(meet(Y, join(X, complement(Y))), meet(Z, one)))
% 131.11/17.10 = { by lemma 22 R->L }
% 131.11/17.10 meet(join(X, complement(Y)), join(meet(Y, join(X, complement(Y))), meet(Z, join(Y, join(X, complement(Y))))))
% 131.11/17.10 = { by lemma 27 }
% 131.11/17.10 join(meet(Y, join(X, complement(Y))), meet(join(X, complement(Y)), Z))
% 131.11/17.10 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.10 join(meet(Y, join(X, complement(Y))), meet(Z, join(X, complement(Y))))
% 131.11/17.10
% 131.11/17.10 Lemma 29: join(meet(X, join(Y, complement(X))), meet(join(Y, complement(X)), complement(meet(X, join(Y, complement(X)))))) = join(Y, complement(X)).
% 131.11/17.10 Proof:
% 131.11/17.10 join(meet(X, join(Y, complement(X))), meet(join(Y, complement(X)), complement(meet(X, join(Y, complement(X))))))
% 131.11/17.10 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.10 join(meet(X, join(Y, complement(X))), meet(complement(meet(X, join(Y, complement(X)))), join(Y, complement(X))))
% 131.11/17.10 = { by lemma 28 R->L }
% 131.11/17.10 meet(join(Y, complement(X)), join(complement(meet(X, join(Y, complement(X)))), meet(X, join(Y, complement(X)))))
% 131.11/17.10 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.10 meet(join(Y, complement(X)), join(meet(X, join(Y, complement(X))), complement(meet(X, join(Y, complement(X))))))
% 131.11/17.10 = { by axiom 3 (complement_join) }
% 131.11/17.10 meet(join(Y, complement(X)), one)
% 131.11/17.10 = { by lemma 14 }
% 131.11/17.10 join(Y, complement(X))
% 131.11/17.10
% 131.11/17.10 Lemma 30: meet(X, meet(Y, complement(meet(X, Y)))) = zero.
% 131.11/17.10 Proof:
% 131.11/17.10 meet(X, meet(Y, complement(meet(X, Y))))
% 131.11/17.10 = { by axiom 11 (associativity_of_meet) R->L }
% 131.11/17.10 meet(meet(X, Y), complement(meet(X, Y)))
% 131.11/17.10 = { by axiom 6 (complement_meet) }
% 131.11/17.10 zero
% 131.11/17.10
% 131.11/17.10 Lemma 31: meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))) = complement(Y).
% 131.11/17.10 Proof:
% 131.11/17.10 meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))
% 131.11/17.10 = { by axiom 12 (meet_join_complement) R->L }
% 131.11/17.10 ifeq(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 131.11/17.10 = { by lemma 23 R->L }
% 131.11/17.10 ifeq(join(Y, join(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))), meet(Y, join(X, complement(Y))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 131.11/17.10 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.10 ifeq(join(Y, join(meet(Y, join(X, complement(Y))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 131.11/17.10 = { by lemma 29 }
% 131.11/17.10 ifeq(join(Y, join(X, complement(Y))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 131.11/17.10 = { by lemma 22 }
% 131.11/17.10 ifeq(one, one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 131.11/17.10 = { by axiom 7 (ifeq_axiom) }
% 131.11/17.10 ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 131.11/17.10 = { by lemma 30 }
% 131.11/17.10 ifeq(zero, zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 131.11/17.10 = { by axiom 7 (ifeq_axiom) }
% 131.11/17.10 complement(Y)
% 131.11/17.10
% 131.11/17.10 Lemma 32: join(X, meet(complement(X), join(X, Y))) = join(X, Y).
% 131.11/17.10 Proof:
% 131.11/17.10 join(X, meet(complement(X), join(X, Y)))
% 131.11/17.10 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.10 join(X, meet(complement(X), join(Y, X)))
% 131.11/17.10 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.10 join(meet(complement(X), join(Y, X)), X)
% 131.11/17.10 = { by lemma 15 R->L }
% 131.11/17.10 join(meet(complement(X), join(Y, X)), complement(complement(X)))
% 131.11/17.10 = { by lemma 31 R->L }
% 131.11/17.10 join(meet(complement(X), join(Y, X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 131.11/17.10 = { by lemma 15 R->L }
% 131.11/17.10 join(meet(complement(X), join(Y, complement(complement(X)))), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 131.11/17.10 = { by lemma 29 }
% 131.11/17.10 join(Y, complement(complement(X)))
% 131.11/17.10 = { by lemma 15 }
% 131.11/17.10 join(Y, X)
% 131.11/17.10 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.10 join(X, Y)
% 131.11/17.10
% 131.11/17.10 Lemma 33: join(meet(X, Y), meet(X, complement(meet(X, Y)))) = X.
% 131.11/17.10 Proof:
% 131.11/17.10 join(meet(X, Y), meet(X, complement(meet(X, Y))))
% 131.11/17.10 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.10 join(meet(X, Y), meet(complement(meet(X, Y)), X))
% 131.11/17.10 = { by axiom 8 (absorption2) R->L }
% 131.11/17.10 join(meet(X, Y), meet(complement(meet(X, Y)), join(X, meet(X, Y))))
% 131.11/17.10 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.10 join(meet(X, Y), meet(complement(meet(X, Y)), join(meet(X, Y), X)))
% 131.11/17.10 = { by lemma 32 }
% 131.11/17.10 join(meet(X, Y), X)
% 131.11/17.10 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.10 join(X, meet(X, Y))
% 131.11/17.10 = { by axiom 8 (absorption2) }
% 131.11/17.11 X
% 131.11/17.11
% 131.11/17.11 Lemma 34: join(X, join(Y, meet(Z, X))) = join(X, Y).
% 131.11/17.11 Proof:
% 131.11/17.11 join(X, join(Y, meet(Z, X)))
% 131.11/17.11 = { by lemma 21 }
% 131.11/17.11 join(Y, join(X, meet(Z, X)))
% 131.11/17.11 = { by lemma 19 }
% 131.11/17.11 join(Y, X)
% 131.11/17.11 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.11 join(X, Y)
% 131.11/17.11
% 131.11/17.11 Lemma 35: ifeq(join(X, Y), one, complement(meet(X, complement(meet(X, Y)))), Y) = Y.
% 131.11/17.11 Proof:
% 131.11/17.11 ifeq(join(X, Y), one, complement(meet(X, complement(meet(X, Y)))), Y)
% 131.11/17.11 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.11 ifeq(join(X, Y), one, complement(meet(X, complement(meet(Y, X)))), Y)
% 131.11/17.11 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.11 ifeq(join(Y, X), one, complement(meet(X, complement(meet(Y, X)))), Y)
% 131.11/17.11 = { by lemma 33 R->L }
% 131.11/17.11 ifeq(join(Y, join(meet(X, Y), meet(X, complement(meet(X, Y))))), one, complement(meet(X, complement(meet(Y, X)))), Y)
% 131.11/17.11 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.11 ifeq(join(Y, join(meet(X, complement(meet(X, Y))), meet(X, Y))), one, complement(meet(X, complement(meet(Y, X)))), Y)
% 131.11/17.11 = { by lemma 34 }
% 131.11/17.11 ifeq(join(Y, meet(X, complement(meet(X, Y)))), one, complement(meet(X, complement(meet(Y, X)))), Y)
% 131.11/17.11 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.11 ifeq(join(Y, meet(X, complement(meet(Y, X)))), one, complement(meet(X, complement(meet(Y, X)))), Y)
% 131.11/17.11 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.11 ifeq(join(meet(X, complement(meet(Y, X))), Y), one, complement(meet(X, complement(meet(Y, X)))), Y)
% 131.11/17.11 = { by axiom 7 (ifeq_axiom) R->L }
% 131.11/17.11 ifeq(join(meet(X, complement(meet(Y, X))), Y), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 131.11/17.11 = { by lemma 30 R->L }
% 131.11/17.11 ifeq(join(meet(X, complement(meet(Y, X))), Y), one, ifeq(meet(Y, meet(X, complement(meet(Y, X)))), zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 131.11/17.11 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.11 ifeq(join(meet(X, complement(meet(Y, X))), Y), one, ifeq(meet(meet(X, complement(meet(Y, X))), Y), zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 131.11/17.11 = { by axiom 12 (meet_join_complement) }
% 131.11/17.11 Y
% 131.11/17.11
% 131.11/17.11 Lemma 36: meet(X, complement(meet(X, join(Y, complement(X))))) = complement(join(Y, complement(X))).
% 131.11/17.11 Proof:
% 131.11/17.11 meet(X, complement(meet(X, join(Y, complement(X)))))
% 131.11/17.11 = { by lemma 15 R->L }
% 131.11/17.11 complement(complement(meet(X, complement(meet(X, join(Y, complement(X)))))))
% 131.11/17.11 = { by axiom 7 (ifeq_axiom) R->L }
% 131.11/17.11 complement(ifeq(one, one, complement(meet(X, complement(meet(X, join(Y, complement(X)))))), join(Y, complement(X))))
% 131.11/17.11 = { by lemma 22 R->L }
% 131.11/17.11 complement(ifeq(join(X, join(Y, complement(X))), one, complement(meet(X, complement(meet(X, join(Y, complement(X)))))), join(Y, complement(X))))
% 131.11/17.11 = { by lemma 35 }
% 131.11/17.11 complement(join(Y, complement(X)))
% 131.11/17.11
% 131.11/17.11 Lemma 37: meet(complement(X), complement(meet(complement(X), join(Y, X)))) = complement(join(X, Y)).
% 131.11/17.11 Proof:
% 131.11/17.11 meet(complement(X), complement(meet(complement(X), join(Y, X))))
% 131.11/17.11 = { by lemma 15 R->L }
% 131.11/17.11 meet(complement(X), complement(meet(complement(X), join(Y, complement(complement(X))))))
% 131.11/17.11 = { by lemma 36 }
% 131.11/17.11 complement(join(Y, complement(complement(X))))
% 131.11/17.11 = { by lemma 15 }
% 131.11/17.11 complement(join(Y, X))
% 131.11/17.11 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.11 complement(join(X, Y))
% 131.11/17.11
% 131.11/17.11 Lemma 38: complement(meet(complement(X), join(X, Y))) = join(X, complement(join(X, Y))).
% 131.11/17.11 Proof:
% 131.11/17.11 complement(meet(complement(X), join(X, Y)))
% 131.11/17.11 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.11 complement(meet(complement(X), join(Y, X)))
% 131.11/17.11 = { by lemma 33 R->L }
% 131.11/17.11 join(meet(complement(meet(complement(X), join(Y, X))), join(complement(X), complement(complement(meet(complement(X), join(Y, X)))))), meet(complement(meet(complement(X), join(Y, X))), complement(meet(complement(meet(complement(X), join(Y, X))), join(complement(X), complement(complement(meet(complement(X), join(Y, X)))))))))
% 131.11/17.11 = { by lemma 36 }
% 131.11/17.11 join(meet(complement(meet(complement(X), join(Y, X))), join(complement(X), complement(complement(meet(complement(X), join(Y, X)))))), complement(join(complement(X), complement(complement(meet(complement(X), join(Y, X)))))))
% 131.11/17.11 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.11 join(complement(join(complement(X), complement(complement(meet(complement(X), join(Y, X)))))), meet(complement(meet(complement(X), join(Y, X))), join(complement(X), complement(complement(meet(complement(X), join(Y, X)))))))
% 131.11/17.11 = { by lemma 15 }
% 131.11/17.11 join(complement(join(complement(X), meet(complement(X), join(Y, X)))), meet(complement(meet(complement(X), join(Y, X))), join(complement(X), complement(complement(meet(complement(X), join(Y, X)))))))
% 131.11/17.11 = { by lemma 15 }
% 131.11/17.11 join(complement(join(complement(X), meet(complement(X), join(Y, X)))), meet(complement(meet(complement(X), join(Y, X))), join(complement(X), meet(complement(X), join(Y, X)))))
% 131.11/17.11 = { by axiom 8 (absorption2) }
% 131.11/17.11 join(complement(complement(X)), meet(complement(meet(complement(X), join(Y, X))), join(complement(X), meet(complement(X), join(Y, X)))))
% 131.11/17.11 = { by axiom 8 (absorption2) }
% 131.11/17.11 join(complement(complement(X)), meet(complement(meet(complement(X), join(Y, X))), complement(X)))
% 131.11/17.11 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.11 join(complement(complement(X)), meet(complement(X), complement(meet(complement(X), join(Y, X)))))
% 131.11/17.11 = { by lemma 37 }
% 131.11/17.11 join(complement(complement(X)), complement(join(X, Y)))
% 131.11/17.11 = { by lemma 15 }
% 131.11/17.11 join(X, complement(join(X, Y)))
% 131.11/17.11
% 131.11/17.11 Lemma 39: complement(meet(X, complement(meet(X, Y)))) = join(complement(X), meet(X, Y)).
% 131.11/17.11 Proof:
% 131.11/17.11 complement(meet(X, complement(meet(X, Y))))
% 131.11/17.11 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.11 complement(meet(complement(meet(X, Y)), X))
% 131.11/17.11 = { by lemma 25 R->L }
% 131.11/17.11 complement(meet(complement(meet(X, Y)), join(meet(complement(meet(X, Y)), X), meet(X, Y))))
% 131.11/17.11 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.11 complement(meet(complement(meet(X, Y)), join(meet(X, Y), meet(complement(meet(X, Y)), X))))
% 131.11/17.11 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.11 complement(meet(complement(meet(X, Y)), join(meet(X, Y), meet(X, complement(meet(X, Y))))))
% 131.11/17.11 = { by lemma 38 }
% 131.11/17.11 join(meet(X, Y), complement(join(meet(X, Y), meet(X, complement(meet(X, Y))))))
% 131.11/17.11 = { by lemma 33 }
% 131.11/17.11 join(meet(X, Y), complement(X))
% 131.11/17.11 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.11 join(complement(X), meet(X, Y))
% 131.11/17.11
% 131.11/17.11 Lemma 40: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 131.11/17.11 Proof:
% 131.11/17.11 join(X, join(Y, meet(Z, join(X, Y))))
% 131.11/17.11 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.11 join(X, join(Y, meet(Z, join(Y, X))))
% 131.11/17.11 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.11 join(X, join(Y, meet(join(Y, X), Z)))
% 131.11/17.11 = { by lemma 21 }
% 131.11/17.11 join(Y, join(X, meet(join(Y, X), Z)))
% 131.11/17.11 = { by axiom 9 (associativity_of_join) R->L }
% 131.11/17.11 join(join(Y, X), meet(join(Y, X), Z))
% 131.11/17.11 = { by axiom 8 (absorption2) }
% 131.11/17.11 join(Y, X)
% 131.11/17.11 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.11 join(X, Y)
% 131.11/17.11
% 131.11/17.11 Lemma 41: meet(join(X, Y), complement(meet(complement(Y), join(X, Y)))) = Y.
% 131.11/17.11 Proof:
% 131.11/17.11 meet(join(X, Y), complement(meet(complement(Y), join(X, Y))))
% 131.11/17.11 = { by lemma 15 R->L }
% 131.11/17.11 meet(join(X, Y), complement(meet(complement(Y), join(X, complement(complement(Y))))))
% 131.11/17.11 = { by lemma 15 R->L }
% 131.11/17.11 meet(join(X, complement(complement(Y))), complement(meet(complement(Y), join(X, complement(complement(Y))))))
% 131.11/17.11 = { by lemma 31 }
% 131.11/17.11 complement(complement(Y))
% 131.11/17.11 = { by lemma 15 }
% 131.11/17.11 Y
% 131.11/17.11
% 131.11/17.11 Lemma 42: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 131.11/17.11 Proof:
% 131.11/17.11 meet(X, meet(Y, join(X, Z)))
% 131.11/17.11 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.11 meet(X, meet(join(X, Z), Y))
% 131.11/17.11 = { by axiom 11 (associativity_of_meet) R->L }
% 131.11/17.11 meet(meet(X, join(X, Z)), Y)
% 131.11/17.11 = { by axiom 10 (absorption1) }
% 131.11/17.11 meet(X, Y)
% 131.11/17.11
% 131.11/17.11 Lemma 43: meet(X, complement(meet(X, complement(meet(X, Y))))) = meet(X, Y).
% 131.11/17.11 Proof:
% 131.11/17.11 meet(X, complement(meet(X, complement(meet(X, Y)))))
% 131.11/17.11 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.11 meet(X, complement(meet(complement(meet(X, Y)), X)))
% 131.11/17.11 = { by axiom 8 (absorption2) R->L }
% 131.11/17.11 meet(X, complement(meet(complement(meet(X, Y)), join(X, meet(X, Y)))))
% 131.11/17.11 = { by lemma 15 R->L }
% 131.11/17.11 meet(X, complement(meet(complement(meet(X, Y)), join(X, complement(complement(meet(X, Y)))))))
% 131.11/17.11 = { by lemma 42 R->L }
% 131.11/17.11 meet(X, meet(complement(meet(complement(meet(X, Y)), join(X, complement(complement(meet(X, Y)))))), join(X, complement(complement(meet(X, Y))))))
% 131.11/17.11 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.11 meet(X, meet(join(X, complement(complement(meet(X, Y)))), complement(meet(complement(meet(X, Y)), join(X, complement(complement(meet(X, Y))))))))
% 131.11/17.11 = { by lemma 31 }
% 131.11/17.11 meet(X, complement(complement(meet(X, Y))))
% 131.11/17.11 = { by lemma 15 }
% 131.11/17.11 meet(X, meet(X, Y))
% 131.11/17.11 = { by lemma 16 }
% 131.11/17.11 meet(X, Y)
% 131.11/17.11
% 131.11/17.11 Lemma 44: meet(X, join(Y, meet(X, Z))) = join(meet(Y, X), meet(X, Z)).
% 131.11/17.11 Proof:
% 131.11/17.11 meet(X, join(Y, meet(X, Z)))
% 131.11/17.11 = { by lemma 43 R->L }
% 131.11/17.11 meet(X, join(Y, meet(X, complement(meet(X, complement(meet(X, Z)))))))
% 131.11/17.11 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.11 meet(X, join(Y, meet(complement(meet(X, complement(meet(X, Z)))), X)))
% 131.11/17.11 = { by axiom 8 (absorption2) R->L }
% 131.11/17.11 meet(X, join(Y, meet(complement(meet(X, complement(meet(X, Z)))), join(X, meet(X, complement(meet(X, Z)))))))
% 131.11/17.12 = { by axiom 8 (absorption2) R->L }
% 131.11/17.12 meet(join(X, meet(X, complement(meet(X, Z)))), join(Y, meet(complement(meet(X, complement(meet(X, Z)))), join(X, meet(X, complement(meet(X, Z)))))))
% 131.11/17.12 = { by lemma 15 R->L }
% 131.11/17.12 meet(join(X, meet(X, complement(meet(X, Z)))), join(Y, meet(complement(meet(X, complement(meet(X, Z)))), join(X, complement(complement(meet(X, complement(meet(X, Z)))))))))
% 131.11/17.12 = { by lemma 15 R->L }
% 131.11/17.12 meet(join(X, complement(complement(meet(X, complement(meet(X, Z)))))), join(Y, meet(complement(meet(X, complement(meet(X, Z)))), join(X, complement(complement(meet(X, complement(meet(X, Z)))))))))
% 131.11/17.12 = { by lemma 28 }
% 131.11/17.12 join(meet(complement(meet(X, complement(meet(X, Z)))), join(X, complement(complement(meet(X, complement(meet(X, Z))))))), meet(Y, join(X, complement(complement(meet(X, complement(meet(X, Z))))))))
% 131.11/17.12 = { by lemma 15 }
% 131.11/17.12 join(meet(complement(meet(X, complement(meet(X, Z)))), join(X, meet(X, complement(meet(X, Z))))), meet(Y, join(X, complement(complement(meet(X, complement(meet(X, Z))))))))
% 131.11/17.12 = { by lemma 15 }
% 131.11/17.12 join(meet(complement(meet(X, complement(meet(X, Z)))), join(X, meet(X, complement(meet(X, Z))))), meet(Y, join(X, meet(X, complement(meet(X, Z))))))
% 131.11/17.12 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.12 join(meet(Y, join(X, meet(X, complement(meet(X, Z))))), meet(complement(meet(X, complement(meet(X, Z)))), join(X, meet(X, complement(meet(X, Z))))))
% 131.11/17.12 = { by axiom 8 (absorption2) }
% 131.11/17.12 join(meet(Y, X), meet(complement(meet(X, complement(meet(X, Z)))), join(X, meet(X, complement(meet(X, Z))))))
% 131.11/17.12 = { by axiom 8 (absorption2) }
% 131.11/17.12 join(meet(Y, X), meet(complement(meet(X, complement(meet(X, Z)))), X))
% 131.11/17.12 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.12 join(meet(Y, X), meet(X, complement(meet(X, complement(meet(X, Z))))))
% 131.11/17.12 = { by lemma 43 }
% 131.11/17.12 join(meet(Y, X), meet(X, Z))
% 131.11/17.12
% 131.11/17.12 Lemma 45: meet(join(X, Y), join(X, Z)) = join(X, meet(Z, join(X, Y))).
% 131.11/17.12 Proof:
% 131.11/17.12 meet(join(X, Y), join(X, Z))
% 131.11/17.12 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.12 meet(join(X, Y), join(Z, X))
% 131.11/17.12 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.12 meet(join(Y, X), join(Z, X))
% 131.11/17.12 = { by lemma 41 R->L }
% 131.11/17.12 meet(join(Y, X), join(Z, meet(join(Y, X), complement(meet(complement(X), join(Y, X))))))
% 131.11/17.12 = { by lemma 44 }
% 131.11/17.12 join(meet(Z, join(Y, X)), meet(join(Y, X), complement(meet(complement(X), join(Y, X)))))
% 131.11/17.12 = { by lemma 41 }
% 131.11/17.12 join(meet(Z, join(Y, X)), X)
% 131.11/17.12 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.12 join(X, meet(Z, join(Y, X)))
% 131.11/17.12 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.12 join(X, meet(Z, join(X, Y)))
% 131.11/17.12
% 131.11/17.12 Lemma 46: meet(X, join(Y, complement(join(X, Y)))) = meet(X, Y).
% 131.11/17.12 Proof:
% 131.11/17.12 meet(X, join(Y, complement(join(X, Y))))
% 131.11/17.12 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.12 meet(X, join(Y, complement(join(Y, X))))
% 131.11/17.12 = { by lemma 24 R->L }
% 131.11/17.12 meet(X, meet(join(Y, complement(join(Y, X))), join(Y, meet(X, join(Y, complement(join(Y, X)))))))
% 131.11/17.12 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.12 meet(X, meet(join(complement(join(Y, X)), Y), join(Y, meet(X, join(Y, complement(join(Y, X)))))))
% 131.11/17.12 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.12 meet(X, meet(join(complement(join(Y, X)), Y), join(Y, meet(X, join(complement(join(Y, X)), Y)))))
% 131.11/17.12 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.12 meet(X, meet(join(Y, meet(X, join(complement(join(Y, X)), Y))), join(complement(join(Y, X)), Y)))
% 131.11/17.12 = { by lemma 40 R->L }
% 131.11/17.12 meet(X, meet(join(Y, meet(X, join(complement(join(Y, X)), Y))), join(complement(join(Y, X)), join(Y, meet(X, join(complement(join(Y, X)), Y))))))
% 131.11/17.12 = { by lemma 18 }
% 131.11/17.12 meet(X, join(Y, meet(X, join(complement(join(Y, X)), Y))))
% 131.11/17.12 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.12 meet(X, join(Y, meet(X, join(Y, complement(join(Y, X))))))
% 131.11/17.12 = { by lemma 32 R->L }
% 131.11/17.12 meet(X, join(Y, meet(complement(Y), join(Y, meet(X, join(Y, complement(join(Y, X))))))))
% 131.11/17.12 = { by lemma 45 R->L }
% 131.11/17.12 meet(X, join(Y, meet(complement(Y), meet(join(Y, complement(join(Y, X))), join(Y, X)))))
% 131.11/17.12 = { by axiom 11 (associativity_of_meet) R->L }
% 131.11/17.12 meet(X, join(Y, meet(meet(complement(Y), join(Y, complement(join(Y, X)))), join(Y, X))))
% 131.11/17.12 = { by lemma 45 R->L }
% 131.11/17.12 meet(X, meet(join(Y, X), join(Y, meet(complement(Y), join(Y, complement(join(Y, X)))))))
% 131.11/17.12 = { by lemma 15 R->L }
% 131.11/17.12 meet(X, meet(join(Y, X), join(complement(complement(Y)), meet(complement(Y), join(Y, complement(join(Y, X)))))))
% 131.11/17.12 = { by lemma 39 R->L }
% 131.11/17.12 meet(X, meet(join(Y, X), complement(meet(complement(Y), complement(meet(complement(Y), join(Y, complement(join(Y, X)))))))))
% 131.11/17.12 = { by lemma 15 R->L }
% 131.11/17.12 meet(X, meet(complement(complement(join(Y, X))), complement(meet(complement(Y), complement(meet(complement(Y), join(Y, complement(join(Y, X)))))))))
% 131.11/17.12 = { by lemma 37 R->L }
% 131.11/17.12 meet(X, meet(complement(meet(complement(Y), complement(meet(complement(Y), join(X, Y))))), complement(meet(complement(Y), complement(meet(complement(Y), join(Y, complement(join(Y, X)))))))))
% 131.11/17.12 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.12 meet(X, meet(complement(meet(complement(Y), complement(meet(complement(Y), join(Y, X))))), complement(meet(complement(Y), complement(meet(complement(Y), join(Y, complement(join(Y, X)))))))))
% 131.11/17.12 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.12 meet(X, meet(complement(meet(complement(meet(complement(Y), join(Y, X))), complement(Y))), complement(meet(complement(Y), complement(meet(complement(Y), join(Y, complement(join(Y, X)))))))))
% 131.11/17.12 = { by lemma 38 }
% 131.11/17.12 meet(X, meet(complement(meet(join(Y, complement(join(Y, X))), complement(Y))), complement(meet(complement(Y), complement(meet(complement(Y), join(Y, complement(join(Y, X)))))))))
% 131.11/17.12 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.12 meet(X, meet(complement(meet(complement(Y), join(Y, complement(join(Y, X))))), complement(meet(complement(Y), complement(meet(complement(Y), join(Y, complement(join(Y, X)))))))))
% 131.11/17.12 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.12 meet(X, meet(complement(meet(complement(Y), join(Y, complement(join(Y, X))))), complement(meet(complement(meet(complement(Y), join(Y, complement(join(Y, X))))), complement(Y)))))
% 131.11/17.12 = { by lemma 25 R->L }
% 131.11/17.12 meet(X, meet(complement(meet(complement(Y), join(Y, complement(join(Y, X))))), complement(meet(complement(meet(complement(Y), join(Y, complement(join(Y, X))))), join(meet(complement(meet(complement(Y), join(Y, complement(join(Y, X))))), complement(Y)), meet(complement(Y), join(Y, complement(join(Y, X)))))))))
% 131.11/17.12 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.12 meet(X, meet(complement(meet(complement(Y), join(Y, complement(join(Y, X))))), complement(meet(complement(meet(complement(Y), join(Y, complement(join(Y, X))))), join(meet(complement(Y), complement(meet(complement(Y), join(Y, complement(join(Y, X)))))), meet(complement(Y), join(Y, complement(join(Y, X)))))))))
% 131.11/17.12 = { by lemma 37 }
% 131.11/17.12 meet(X, complement(join(meet(complement(Y), join(Y, complement(join(Y, X)))), meet(complement(Y), complement(meet(complement(Y), join(Y, complement(join(Y, X)))))))))
% 131.11/17.12 = { by lemma 33 }
% 131.11/17.12 meet(X, complement(complement(Y)))
% 131.11/17.12 = { by lemma 15 }
% 131.11/17.12 meet(X, Y)
% 131.11/17.12
% 131.11/17.12 Lemma 47: join(complement(Y), meet(Y, X)) = join(X, complement(join(Y, X))).
% 131.11/17.12 Proof:
% 131.11/17.12 join(complement(Y), meet(Y, X))
% 131.11/17.12 = { by lemma 39 R->L }
% 131.11/17.12 complement(meet(Y, complement(meet(Y, X))))
% 131.11/17.12 = { by axiom 7 (ifeq_axiom) R->L }
% 131.11/17.12 ifeq(one, one, complement(meet(Y, complement(meet(Y, X)))), join(X, complement(join(Y, X))))
% 131.11/17.12 = { by axiom 3 (complement_join) R->L }
% 131.11/17.12 ifeq(join(join(Y, X), complement(join(Y, X))), one, complement(meet(Y, complement(meet(Y, X)))), join(X, complement(join(Y, X))))
% 131.11/17.12 = { by axiom 9 (associativity_of_join) }
% 131.11/17.12 ifeq(join(Y, join(X, complement(join(Y, X)))), one, complement(meet(Y, complement(meet(Y, X)))), join(X, complement(join(Y, X))))
% 131.11/17.12 = { by lemma 46 R->L }
% 131.11/17.12 ifeq(join(Y, join(X, complement(join(Y, X)))), one, complement(meet(Y, complement(meet(Y, join(X, complement(join(Y, X))))))), join(X, complement(join(Y, X))))
% 131.11/17.12 = { by lemma 35 }
% 131.11/17.12 join(X, complement(join(Y, X)))
% 131.11/17.12
% 131.11/17.13 Lemma 48: join(X, complement(join(X, Y))) = join(X, complement(Y)).
% 131.11/17.13 Proof:
% 131.11/17.13 join(X, complement(join(X, Y)))
% 131.11/17.13 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.13 join(X, complement(join(Y, X)))
% 131.11/17.13 = { by lemma 20 R->L }
% 131.11/17.13 join(X, join(X, complement(join(Y, X))))
% 131.11/17.13 = { by lemma 47 R->L }
% 131.11/17.13 join(X, join(complement(Y), meet(Y, X)))
% 131.11/17.13 = { by lemma 34 }
% 131.11/17.13 join(X, complement(Y))
% 131.11/17.13
% 131.11/17.13 Lemma 49: join(X, complement(join(Y, complement(X)))) = X.
% 131.11/17.13 Proof:
% 131.11/17.13 join(X, complement(join(Y, complement(X))))
% 131.11/17.13 = { by lemma 36 R->L }
% 131.11/17.13 join(X, meet(X, complement(meet(X, join(Y, complement(X))))))
% 131.11/17.13 = { by axiom 8 (absorption2) }
% 131.11/17.13 X
% 131.11/17.13
% 131.11/17.13 Lemma 50: join(meet(X, Y), meet(X, join(Y, Z))) = meet(X, join(Y, Z)).
% 131.11/17.13 Proof:
% 131.11/17.13 join(meet(X, Y), meet(X, join(Y, Z)))
% 131.11/17.13 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.13 join(meet(Y, X), meet(X, join(Y, Z)))
% 131.11/17.13 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.13 join(meet(X, join(Y, Z)), meet(Y, X))
% 131.11/17.13 = { by lemma 42 R->L }
% 131.11/17.13 join(meet(X, join(Y, Z)), meet(Y, meet(X, join(Y, Z))))
% 131.11/17.13 = { by lemma 19 }
% 131.11/17.13 meet(X, join(Y, Z))
% 131.11/17.13
% 131.11/17.13 Lemma 51: meet(X, join(Y, join(complement(X), Z))) = meet(X, join(Y, Z)).
% 131.11/17.13 Proof:
% 131.11/17.13 meet(X, join(Y, join(complement(X), Z)))
% 131.11/17.13 = { by lemma 16 R->L }
% 131.11/17.13 meet(X, meet(X, join(Y, join(complement(X), Z))))
% 131.11/17.13 = { by lemma 50 R->L }
% 131.11/17.13 meet(X, join(meet(X, Y), meet(X, join(Y, join(complement(X), Z)))))
% 131.11/17.13 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.13 meet(X, join(meet(Y, X), meet(X, join(Y, join(complement(X), Z)))))
% 131.11/17.13 = { by lemma 21 R->L }
% 131.11/17.13 meet(X, join(meet(Y, X), meet(X, join(complement(X), join(Y, Z)))))
% 131.11/17.13 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.13 meet(X, join(meet(Y, X), meet(X, join(join(Y, Z), complement(X)))))
% 131.11/17.13 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.13 meet(X, join(meet(X, join(join(Y, Z), complement(X))), meet(Y, X)))
% 131.11/17.13 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.13 meet(X, join(meet(join(join(Y, Z), complement(X)), X), meet(Y, X)))
% 131.11/17.13 = { by lemma 14 R->L }
% 131.11/17.13 meet(X, join(meet(join(join(Y, Z), complement(X)), X), meet(meet(Y, X), one)))
% 131.11/17.13 = { by lemma 22 R->L }
% 131.11/17.13 meet(X, join(meet(join(join(Y, Z), complement(X)), X), meet(meet(Y, X), join(X, join(join(Y, Z), complement(X))))))
% 131.11/17.13 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.13 meet(X, join(meet(join(join(Y, Z), complement(X)), X), meet(meet(Y, X), join(join(join(Y, Z), complement(X)), X))))
% 131.11/17.13 = { by lemma 27 }
% 131.11/17.13 join(meet(join(join(Y, Z), complement(X)), X), meet(X, meet(Y, X)))
% 131.11/17.13 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.13 join(meet(X, join(join(Y, Z), complement(X))), meet(X, meet(Y, X)))
% 131.11/17.13 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.13 join(meet(X, meet(Y, X)), meet(X, join(join(Y, Z), complement(X))))
% 131.11/17.13 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.13 join(meet(X, meet(X, Y)), meet(X, join(join(Y, Z), complement(X))))
% 131.11/17.13 = { by lemma 16 }
% 131.11/17.13 join(meet(X, Y), meet(X, join(join(Y, Z), complement(X))))
% 131.11/17.13 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.13 join(meet(X, Y), meet(X, join(complement(X), join(Y, Z))))
% 131.11/17.13 = { by axiom 10 (absorption1) R->L }
% 131.11/17.13 join(meet(X, Y), meet(X, join(complement(X), meet(join(Y, Z), join(join(Y, Z), complement(join(X, join(Y, Z))))))))
% 131.11/17.13 = { by lemma 47 R->L }
% 131.11/17.13 join(meet(X, Y), meet(X, join(complement(X), meet(join(Y, Z), join(complement(X), meet(X, join(Y, Z)))))))
% 131.11/17.13 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.13 join(meet(X, Y), meet(X, join(complement(X), meet(join(Y, Z), join(complement(X), meet(join(Y, Z), X))))))
% 131.11/17.13 = { by lemma 26 R->L }
% 131.11/17.13 join(meet(X, Y), meet(X, join(complement(X), join(meet(join(Y, Z), X), meet(join(Y, Z), join(complement(X), meet(join(Y, Z), X)))))))
% 131.11/17.13 = { by lemma 40 }
% 131.11/17.13 join(meet(X, Y), meet(X, join(complement(X), meet(join(Y, Z), X))))
% 131.11/17.13 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.13 join(meet(X, Y), meet(X, join(complement(X), meet(X, join(Y, Z)))))
% 131.11/17.13 = { by lemma 47 }
% 131.11/17.13 join(meet(X, Y), meet(X, join(join(Y, Z), complement(join(X, join(Y, Z))))))
% 131.11/17.13 = { by lemma 46 }
% 131.11/17.13 join(meet(X, Y), meet(X, join(Y, Z)))
% 131.11/17.13 = { by lemma 50 }
% 131.11/17.13 meet(X, join(Y, Z))
% 131.11/17.13
% 131.11/17.13 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 131.11/17.13 Proof:
% 131.11/17.13 meet(a, join(b, c))
% 131.11/17.13 = { by lemma 51 R->L }
% 131.11/17.13 meet(a, join(b, join(complement(a), c)))
% 131.11/17.13 = { by lemma 32 R->L }
% 131.11/17.13 meet(a, join(b, join(complement(a), meet(complement(complement(a)), join(complement(a), c)))))
% 131.11/17.13 = { by lemma 51 }
% 131.11/17.13 meet(a, join(b, meet(complement(complement(a)), join(complement(a), c))))
% 131.11/17.13 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.13 meet(a, join(b, meet(complement(complement(a)), join(c, complement(a)))))
% 131.11/17.13 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.13 meet(a, join(b, meet(join(c, complement(a)), complement(complement(a)))))
% 131.11/17.13 = { by lemma 15 R->L }
% 131.11/17.13 meet(a, join(b, complement(complement(meet(join(c, complement(a)), complement(complement(a)))))))
% 131.11/17.13 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.13 meet(a, join(b, complement(complement(meet(complement(complement(a)), join(c, complement(a)))))))
% 131.11/17.13 = { by lemma 49 R->L }
% 131.11/17.13 meet(a, join(b, complement(join(complement(meet(complement(complement(a)), join(c, complement(a)))), complement(join(complement(complement(a)), complement(complement(meet(complement(complement(a)), join(c, complement(a)))))))))))
% 131.11/17.14 = { by lemma 15 }
% 131.11/17.14 meet(a, join(b, complement(join(complement(meet(complement(complement(a)), join(c, complement(a)))), complement(join(complement(complement(a)), meet(complement(complement(a)), join(c, complement(a)))))))))
% 131.11/17.14 = { by axiom 8 (absorption2) }
% 131.11/17.14 meet(a, join(b, complement(join(complement(meet(complement(complement(a)), join(c, complement(a)))), complement(complement(complement(a)))))))
% 131.11/17.14 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.14 meet(a, join(b, complement(join(complement(complement(complement(a))), complement(meet(complement(complement(a)), join(c, complement(a))))))))
% 131.11/17.14 = { by lemma 15 }
% 131.11/17.14 meet(a, join(b, complement(join(complement(a), complement(meet(complement(complement(a)), join(c, complement(a))))))))
% 131.11/17.14 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.14 meet(a, join(b, complement(join(complement(a), complement(meet(join(c, complement(a)), complement(complement(a))))))))
% 131.11/17.14 = { by lemma 48 R->L }
% 131.11/17.14 meet(a, join(b, complement(join(complement(a), complement(join(complement(a), meet(join(c, complement(a)), complement(complement(a)))))))))
% 131.11/17.14 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.14 meet(a, join(b, complement(join(complement(a), complement(join(complement(a), meet(complement(complement(a)), join(c, complement(a)))))))))
% 131.11/17.14 = { by lemma 20 R->L }
% 131.11/17.14 meet(a, join(b, complement(join(complement(a), complement(join(complement(a), join(complement(a), meet(complement(complement(a)), join(c, complement(a))))))))))
% 131.11/17.14 = { by lemma 15 R->L }
% 131.11/17.14 meet(a, join(b, complement(join(complement(a), complement(join(complement(a), join(complement(complement(complement(a))), meet(complement(complement(a)), join(c, complement(a))))))))))
% 131.11/17.14 = { by lemma 47 }
% 131.11/17.14 meet(a, join(b, complement(join(complement(a), complement(join(complement(a), join(join(c, complement(a)), complement(join(complement(complement(a)), join(c, complement(a)))))))))))
% 131.11/17.14 = { by lemma 21 }
% 131.11/17.14 meet(a, join(b, complement(join(complement(a), complement(join(join(c, complement(a)), join(complement(a), complement(join(complement(complement(a)), join(c, complement(a)))))))))))
% 131.11/17.14 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.14 meet(a, join(b, complement(join(complement(a), complement(join(join(c, complement(a)), join(complement(a), complement(join(join(c, complement(a)), complement(complement(a)))))))))))
% 131.11/17.14 = { by lemma 49 }
% 131.11/17.14 meet(a, join(b, complement(join(complement(a), complement(join(join(c, complement(a)), complement(a)))))))
% 131.11/17.14 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.14 meet(a, join(b, complement(join(complement(a), complement(join(complement(a), join(c, complement(a))))))))
% 131.11/17.14 = { by lemma 48 }
% 131.11/17.14 meet(a, join(b, complement(join(complement(a), complement(join(c, complement(a)))))))
% 131.11/17.14 = { by lemma 47 R->L }
% 131.11/17.14 meet(a, join(b, complement(join(complement(c), meet(c, complement(a))))))
% 131.11/17.14 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.14 meet(a, join(b, complement(join(meet(c, complement(a)), complement(c)))))
% 131.11/17.14 = { by lemma 15 R->L }
% 131.11/17.14 meet(a, join(b, complement(join(meet(c, complement(a)), complement(complement(complement(c)))))))
% 131.11/17.14 = { by lemma 36 R->L }
% 131.11/17.14 meet(a, join(b, meet(complement(complement(c)), complement(meet(complement(complement(c)), join(meet(c, complement(a)), complement(complement(complement(c)))))))))
% 131.11/17.14 = { by lemma 16 R->L }
% 131.11/17.14 meet(a, join(b, meet(complement(complement(c)), meet(complement(complement(c)), complement(meet(complement(complement(c)), join(meet(c, complement(a)), complement(complement(complement(c))))))))))
% 131.11/17.14 = { by lemma 36 }
% 131.11/17.14 meet(a, join(b, meet(complement(complement(c)), complement(join(meet(c, complement(a)), complement(complement(complement(c))))))))
% 131.11/17.14 = { by lemma 15 }
% 131.11/17.14 meet(a, join(b, meet(complement(complement(c)), complement(join(meet(c, complement(a)), complement(c))))))
% 131.11/17.14 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.14 meet(a, join(b, meet(complement(complement(c)), complement(join(complement(c), meet(c, complement(a)))))))
% 131.11/17.14 = { by lemma 47 }
% 131.11/17.14 meet(a, join(b, meet(complement(complement(c)), complement(join(complement(a), complement(join(c, complement(a))))))))
% 131.11/17.14 = { by lemma 15 }
% 131.11/17.15 meet(a, join(b, meet(c, complement(join(complement(a), complement(join(c, complement(a))))))))
% 131.11/17.15 = { by axiom 2 (commutativity_of_join) R->L }
% 131.11/17.15 meet(a, join(b, meet(c, complement(join(complement(a), complement(join(complement(a), c)))))))
% 131.11/17.15 = { by lemma 38 R->L }
% 131.11/17.15 meet(a, join(b, meet(c, complement(complement(meet(complement(complement(a)), join(complement(a), c)))))))
% 131.11/17.15 = { by lemma 15 }
% 131.11/17.15 meet(a, join(b, meet(c, meet(complement(complement(a)), join(complement(a), c)))))
% 131.11/17.15 = { by lemma 17 R->L }
% 131.11/17.15 meet(a, join(b, meet(complement(complement(a)), meet(c, join(complement(a), c)))))
% 131.11/17.15 = { by lemma 18 }
% 131.11/17.15 meet(a, join(b, meet(complement(complement(a)), c)))
% 131.11/17.15 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.15 meet(a, join(b, meet(c, complement(complement(a)))))
% 131.11/17.15 = { by lemma 15 }
% 131.11/17.15 meet(a, join(b, meet(c, a)))
% 131.11/17.15 = { by axiom 5 (commutativity_of_meet) R->L }
% 131.11/17.15 meet(a, join(b, meet(a, c)))
% 131.11/17.15 = { by lemma 44 }
% 131.11/17.15 join(meet(b, a), meet(a, c))
% 131.11/17.15 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.15 join(meet(a, c), meet(b, a))
% 131.11/17.15 = { by axiom 5 (commutativity_of_meet) }
% 131.11/17.15 join(meet(a, c), meet(a, b))
% 131.11/17.15 = { by axiom 2 (commutativity_of_join) }
% 131.11/17.15 join(meet(a, b), meet(a, c))
% 131.11/17.15 % SZS output end Proof
% 131.11/17.15
% 131.11/17.15 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------