TSTP Solution File: LAT198-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT198-1 : TPTP v8.1.2. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n007.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 06:27:44 EDT 2023
% Result : Unsatisfiable 28.14s 4.06s
% Output : Proof 29.15s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.13 % Problem : LAT198-1 : TPTP v8.1.2. Released v3.1.0.
% 0.11/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.15/0.35 % Computer : n007.cluster.edu
% 0.15/0.35 % Model : x86_64 x86_64
% 0.15/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.35 % Memory : 8042.1875MB
% 0.15/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.35 % CPULimit : 300
% 0.15/0.35 % WCLimit : 300
% 0.15/0.35 % DateTime : Thu Aug 24 05:34:39 EDT 2023
% 0.15/0.35 % CPUTime :
% 28.14/4.06 Command-line arguments: --no-flatten-goal
% 28.14/4.06
% 28.14/4.06 % SZS status Unsatisfiable
% 28.14/4.06
% 28.92/4.09 % SZS output start Proof
% 28.92/4.09 Take the following subset of the input axioms:
% 28.92/4.10 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 28.92/4.10 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 28.92/4.10 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 28.92/4.10 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 28.92/4.10 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 28.92/4.10 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 28.92/4.10 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 28.92/4.10 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 28.92/4.10 fof(equation_H42, axiom, ![U, X2, Y2, Z2]: meet(X2, join(Y2, meet(Z2, join(X2, U))))=meet(X2, join(Y2, meet(Z2, join(Y2, join(U, meet(X2, Z2))))))).
% 28.92/4.10 fof(idempotence_of_join, axiom, ![X2]: join(X2, X2)=X2).
% 28.92/4.10 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 28.92/4.10 fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
% 28.92/4.10
% 28.92/4.10 Now clausify the problem and encode Horn clauses using encoding 3 of
% 28.92/4.10 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 28.92/4.10 We repeatedly replace C & s=t => u=v by the two clauses:
% 28.92/4.10 fresh(y, y, x1...xn) = u
% 28.92/4.10 C => fresh(s, t, x1...xn) = v
% 28.92/4.10 where fresh is a fresh function symbol and x1..xn are the free
% 28.92/4.10 variables of u and v.
% 28.92/4.10 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 28.92/4.10 input problem has no model of domain size 1).
% 28.92/4.10
% 28.92/4.10 The encoding turns the above axioms into the following unit equations and goals:
% 28.92/4.10
% 28.92/4.10 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 28.92/4.10 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 28.92/4.10 Axiom 3 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 28.92/4.10 Axiom 4 (complement_join): join(X, complement(X)) = one.
% 28.92/4.10 Axiom 5 (complement_meet): meet(X, complement(X)) = zero.
% 28.92/4.10 Axiom 6 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 28.92/4.10 Axiom 7 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 28.92/4.10 Axiom 8 (absorption2): join(X, meet(X, Y)) = X.
% 28.92/4.10 Axiom 9 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 28.92/4.10 Axiom 10 (absorption1): meet(X, join(X, Y)) = X.
% 28.92/4.10 Axiom 11 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 28.92/4.10 Axiom 12 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 28.92/4.10 Axiom 13 (equation_H42): meet(X, join(Y, meet(Z, join(X, W)))) = meet(X, join(Y, meet(Z, join(Y, join(W, meet(X, Z)))))).
% 28.92/4.10
% 28.92/4.10 Lemma 14: complement(complement(X)) = X.
% 28.92/4.10 Proof:
% 28.92/4.10 complement(complement(X))
% 28.92/4.10 = { by axiom 7 (meet_join_complement) R->L }
% 28.92/4.10 fresh2(one, one, complement(X), X)
% 28.92/4.10 = { by axiom 4 (complement_join) R->L }
% 28.92/4.10 fresh2(join(X, complement(X)), one, complement(X), X)
% 28.92/4.10 = { by axiom 2 (commutativity_of_join) R->L }
% 28.92/4.10 fresh2(join(complement(X), X), one, complement(X), X)
% 28.92/4.10 = { by axiom 12 (meet_join_complement) }
% 28.92/4.10 fresh(meet(complement(X), X), zero, complement(X), X)
% 28.92/4.10 = { by axiom 3 (commutativity_of_meet) }
% 28.92/4.10 fresh(meet(X, complement(X)), zero, complement(X), X)
% 28.92/4.10 = { by axiom 5 (complement_meet) }
% 28.92/4.10 fresh(zero, zero, complement(X), X)
% 28.92/4.10 = { by axiom 6 (meet_join_complement) }
% 28.92/4.10 X
% 28.92/4.10
% 28.92/4.10 Lemma 15: meet(X, one) = X.
% 28.92/4.10 Proof:
% 28.92/4.10 meet(X, one)
% 28.92/4.10 = { by axiom 4 (complement_join) R->L }
% 28.92/4.10 meet(X, join(X, complement(X)))
% 28.92/4.10 = { by axiom 10 (absorption1) }
% 28.92/4.10 X
% 28.92/4.10
% 28.92/4.10 Lemma 16: join(X, one) = one.
% 28.92/4.10 Proof:
% 28.92/4.10 join(X, one)
% 28.92/4.10 = { by axiom 2 (commutativity_of_join) R->L }
% 28.92/4.10 join(one, X)
% 28.92/4.10 = { by lemma 15 R->L }
% 28.92/4.10 join(one, meet(X, one))
% 28.92/4.10 = { by axiom 3 (commutativity_of_meet) }
% 28.92/4.10 join(one, meet(one, X))
% 28.92/4.10 = { by axiom 8 (absorption2) }
% 28.92/4.10 one
% 28.92/4.10
% 28.92/4.10 Lemma 17: join(X, zero) = X.
% 28.92/4.10 Proof:
% 28.92/4.10 join(X, zero)
% 28.92/4.10 = { by axiom 5 (complement_meet) R->L }
% 28.92/4.10 join(X, meet(X, complement(X)))
% 28.92/4.10 = { by axiom 8 (absorption2) }
% 28.92/4.10 X
% 28.92/4.10
% 28.92/4.10 Lemma 18: meet(X, join(Y, X)) = X.
% 28.92/4.10 Proof:
% 28.92/4.10 meet(X, join(Y, X))
% 28.92/4.10 = { by axiom 2 (commutativity_of_join) R->L }
% 28.92/4.10 meet(X, join(X, Y))
% 28.92/4.10 = { by axiom 10 (absorption1) }
% 28.92/4.10 X
% 28.92/4.10
% 28.92/4.10 Lemma 19: join(X, meet(Y, X)) = X.
% 28.92/4.10 Proof:
% 28.92/4.10 join(X, meet(Y, X))
% 28.92/4.10 = { by axiom 3 (commutativity_of_meet) R->L }
% 28.92/4.10 join(X, meet(X, Y))
% 28.92/4.10 = { by axiom 8 (absorption2) }
% 28.92/4.10 X
% 28.92/4.10
% 28.92/4.10 Lemma 20: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 28.92/4.10 Proof:
% 28.92/4.10 join(Y, join(X, Z))
% 28.92/4.10 = { by axiom 2 (commutativity_of_join) R->L }
% 28.92/4.10 join(join(X, Z), Y)
% 28.92/4.10 = { by axiom 9 (associativity_of_join) }
% 28.92/4.10 join(X, join(Z, Y))
% 28.92/4.10 = { by axiom 2 (commutativity_of_join) }
% 28.92/4.10 join(X, join(Y, Z))
% 28.92/4.10
% 28.92/4.10 Lemma 21: meet(X, join(Y, meet(Z, join(X, Y)))) = meet(X, join(Y, meet(X, Z))).
% 28.92/4.10 Proof:
% 28.92/4.10 meet(X, join(Y, meet(Z, join(X, Y))))
% 28.92/4.10 = { by axiom 2 (commutativity_of_join) R->L }
% 28.92/4.10 meet(X, join(Y, meet(Z, join(Y, X))))
% 28.92/4.10 = { by axiom 8 (absorption2) R->L }
% 28.92/4.10 meet(X, join(Y, meet(Z, join(Y, join(X, meet(X, Z))))))
% 28.92/4.10 = { by axiom 13 (equation_H42) R->L }
% 28.92/4.10 meet(X, join(Y, meet(Z, join(X, X))))
% 28.92/4.10 = { by axiom 1 (idempotence_of_join) }
% 28.92/4.10 meet(X, join(Y, meet(Z, X)))
% 28.92/4.10 = { by axiom 3 (commutativity_of_meet) }
% 28.92/4.10 meet(X, join(Y, meet(X, Z)))
% 28.92/4.10
% 28.92/4.10 Lemma 22: meet(X, join(complement(X), meet(X, Y))) = meet(X, join(Y, complement(X))).
% 28.92/4.10 Proof:
% 28.92/4.10 meet(X, join(complement(X), meet(X, Y)))
% 28.92/4.10 = { by lemma 21 R->L }
% 28.92/4.10 meet(X, join(complement(X), meet(Y, join(X, complement(X)))))
% 28.92/4.10 = { by axiom 4 (complement_join) }
% 28.92/4.10 meet(X, join(complement(X), meet(Y, one)))
% 28.92/4.10 = { by lemma 15 }
% 28.92/4.10 meet(X, join(complement(X), Y))
% 28.92/4.10 = { by axiom 2 (commutativity_of_join) }
% 28.92/4.10 meet(X, join(Y, complement(X)))
% 28.92/4.10
% 28.92/4.10 Lemma 23: join(X, join(Y, complement(X))) = one.
% 28.92/4.10 Proof:
% 28.92/4.10 join(X, join(Y, complement(X)))
% 28.92/4.10 = { by axiom 2 (commutativity_of_join) R->L }
% 28.92/4.10 join(X, join(complement(X), Y))
% 28.92/4.10 = { by axiom 9 (associativity_of_join) R->L }
% 28.92/4.10 join(join(X, complement(X)), Y)
% 28.92/4.10 = { by axiom 4 (complement_join) }
% 28.92/4.10 join(one, Y)
% 28.92/4.10 = { by axiom 2 (commutativity_of_join) R->L }
% 28.92/4.10 join(Y, one)
% 28.92/4.10 = { by lemma 16 }
% 28.92/4.10 one
% 28.92/4.10
% 28.92/4.10 Lemma 24: meet(complement(X), join(X, meet(complement(X), Y))) = meet(complement(X), join(X, Y)).
% 28.92/4.10 Proof:
% 28.92/4.10 meet(complement(X), join(X, meet(complement(X), Y)))
% 28.92/4.10 = { by lemma 14 R->L }
% 28.92/4.10 meet(complement(X), join(complement(complement(X)), meet(complement(X), Y)))
% 28.92/4.10 = { by lemma 22 }
% 28.92/4.10 meet(complement(X), join(Y, complement(complement(X))))
% 28.92/4.10 = { by lemma 14 }
% 28.92/4.10 meet(complement(X), join(Y, X))
% 28.92/4.10 = { by axiom 2 (commutativity_of_join) }
% 28.92/4.10 meet(complement(X), join(X, Y))
% 28.92/4.10
% 28.92/4.10 Lemma 25: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 28.92/4.10 Proof:
% 28.92/4.10 join(X, join(Y, meet(X, Z)))
% 28.92/4.10 = { by axiom 2 (commutativity_of_join) R->L }
% 28.92/4.10 join(X, join(meet(X, Z), Y))
% 28.92/4.10 = { by axiom 9 (associativity_of_join) R->L }
% 28.92/4.10 join(join(X, meet(X, Z)), Y)
% 28.92/4.10 = { by axiom 8 (absorption2) }
% 28.92/4.10 join(X, Y)
% 28.92/4.10
% 28.92/4.10 Lemma 26: meet(X, complement(meet(Y, complement(X)))) = X.
% 28.92/4.10 Proof:
% 28.92/4.10 meet(X, complement(meet(Y, complement(X))))
% 28.92/4.10 = { by axiom 3 (commutativity_of_meet) R->L }
% 28.92/4.10 meet(X, complement(meet(complement(X), Y)))
% 28.92/4.10 = { by lemma 14 R->L }
% 28.92/4.10 meet(complement(complement(X)), complement(meet(complement(X), Y)))
% 28.92/4.10 = { by axiom 6 (meet_join_complement) R->L }
% 28.92/4.10 fresh(zero, zero, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 8 (absorption2) R->L }
% 28.92/4.10 fresh(join(zero, meet(zero, complement(meet(complement(X), Y)))), zero, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 2 (commutativity_of_join) R->L }
% 28.92/4.10 fresh(join(meet(zero, complement(meet(complement(X), Y))), zero), zero, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by lemma 17 }
% 28.92/4.10 fresh(meet(zero, complement(meet(complement(X), Y))), zero, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 5 (complement_meet) R->L }
% 28.92/4.10 fresh(meet(meet(complement(X), complement(complement(X))), complement(meet(complement(X), Y))), zero, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 11 (associativity_of_meet) }
% 28.92/4.10 fresh(meet(complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y)))), zero, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 12 (meet_join_complement) R->L }
% 28.92/4.10 fresh2(join(complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y)))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by lemma 19 R->L }
% 28.92/4.10 fresh2(join(join(complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y)))), meet(complement(complement(X)), join(complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y)))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by lemma 24 }
% 28.92/4.10 fresh2(join(join(complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y)))), meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 9 (associativity_of_join) }
% 28.92/4.10 fresh2(join(complement(X), join(meet(complement(complement(X)), complement(meet(complement(X), Y))), meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 3 (commutativity_of_meet) R->L }
% 28.92/4.10 fresh2(join(complement(X), join(meet(complement(meet(complement(X), Y)), complement(complement(X))), meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 2 (commutativity_of_join) R->L }
% 28.92/4.10 fresh2(join(complement(X), join(meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))), meet(complement(meet(complement(X), Y)), complement(complement(X))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 3 (commutativity_of_meet) R->L }
% 28.92/4.10 fresh2(join(complement(X), join(meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))), meet(complement(complement(X)), complement(meet(complement(X), Y))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by lemma 18 R->L }
% 28.92/4.10 fresh2(join(complement(X), join(meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))), meet(complement(complement(X)), meet(complement(meet(complement(X), Y)), join(complement(X), complement(meet(complement(X), Y))))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 3 (commutativity_of_meet) R->L }
% 28.92/4.10 fresh2(join(complement(X), join(meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))), meet(meet(complement(meet(complement(X), Y)), join(complement(X), complement(meet(complement(X), Y)))), complement(complement(X))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 11 (associativity_of_meet) }
% 28.92/4.10 fresh2(join(complement(X), join(meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))), meet(complement(meet(complement(X), Y)), meet(join(complement(X), complement(meet(complement(X), Y))), complement(complement(X)))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 3 (commutativity_of_meet) }
% 28.92/4.10 fresh2(join(complement(X), join(meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y)))), meet(complement(meet(complement(X), Y)), meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y))))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by lemma 19 }
% 28.92/4.10 fresh2(join(complement(X), meet(complement(complement(X)), join(complement(X), complement(meet(complement(X), Y))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by lemma 25 R->L }
% 28.92/4.10 fresh2(join(complement(X), meet(complement(complement(X)), join(complement(X), join(complement(meet(complement(X), Y)), meet(complement(X), Y))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 2 (commutativity_of_join) }
% 28.92/4.10 fresh2(join(complement(X), meet(complement(complement(X)), join(complement(X), join(meet(complement(X), Y), complement(meet(complement(X), Y)))))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 4 (complement_join) }
% 28.92/4.10 fresh2(join(complement(X), meet(complement(complement(X)), join(complement(X), one))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by lemma 16 }
% 28.92/4.10 fresh2(join(complement(X), meet(complement(complement(X)), one)), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by lemma 15 }
% 28.92/4.10 fresh2(join(complement(X), complement(complement(X))), one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 4 (complement_join) }
% 28.92/4.10 fresh2(one, one, complement(X), meet(complement(complement(X)), complement(meet(complement(X), Y))))
% 28.92/4.10 = { by axiom 7 (meet_join_complement) }
% 28.92/4.10 complement(complement(X))
% 28.92/4.10 = { by lemma 14 }
% 28.92/4.10 X
% 28.92/4.10
% 28.92/4.10 Lemma 27: meet(X, complement(meet(Y, X))) = meet(X, complement(Y)).
% 28.92/4.10 Proof:
% 28.92/4.10 meet(X, complement(meet(Y, X)))
% 28.92/4.11 = { by axiom 3 (commutativity_of_meet) R->L }
% 28.92/4.11 meet(X, complement(meet(X, Y)))
% 28.92/4.11 = { by lemma 14 R->L }
% 28.92/4.11 meet(X, complement(meet(X, complement(complement(Y)))))
% 28.92/4.11 = { by lemma 18 R->L }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y)))))))
% 28.92/4.11 = { by axiom 6 (meet_join_complement) R->L }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(zero, zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.92/4.11 = { by axiom 5 (complement_meet) R->L }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), complement(complement(complement(Y)))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.92/4.11 = { by lemma 17 R->L }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(complement(complement(complement(Y))), zero)), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.92/4.11 = { by axiom 5 (complement_meet) R->L }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(complement(complement(complement(Y))), meet(meet(complement(complement(Y)), X), complement(meet(complement(complement(Y)), X))))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.92/4.11 = { by axiom 11 (associativity_of_meet) }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(complement(complement(complement(Y))), meet(complement(complement(Y)), meet(X, complement(meet(complement(complement(Y)), X)))))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.92/4.11 = { by lemma 22 }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(meet(X, complement(meet(complement(complement(Y)), X))), complement(complement(complement(Y))))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.92/4.11 = { by axiom 2 (commutativity_of_join) }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(complement(complement(complement(Y))), meet(X, complement(meet(complement(complement(Y)), X))))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.92/4.11 = { by lemma 14 }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(complement(Y), meet(X, complement(meet(complement(complement(Y)), X))))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.92/4.11 = { by axiom 3 (commutativity_of_meet) }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.92/4.11 = { by axiom 2 (commutativity_of_join) R->L }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(meet(X, complement(meet(X, complement(complement(Y))))), complement(Y))), zero, complement(complement(Y)), join(complement(Y), meet(X, complement(meet(X, complement(complement(Y))))))))
% 28.92/4.11 = { by axiom 2 (commutativity_of_join) R->L }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(meet(X, complement(meet(X, complement(complement(Y))))), complement(Y))), zero, complement(complement(Y)), join(meet(X, complement(meet(X, complement(complement(Y))))), complement(Y))))
% 28.92/4.11 = { by lemma 14 R->L }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(meet(X, complement(meet(X, complement(complement(Y))))), complement(Y))), zero, complement(complement(Y)), join(meet(X, complement(meet(X, complement(complement(Y))))), complement(complement(complement(Y))))))
% 28.92/4.11 = { by lemma 14 R->L }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh(meet(complement(complement(Y)), join(meet(X, complement(meet(X, complement(complement(Y))))), complement(complement(complement(Y))))), zero, complement(complement(Y)), join(meet(X, complement(meet(X, complement(complement(Y))))), complement(complement(complement(Y))))))
% 28.92/4.11 = { by axiom 12 (meet_join_complement) R->L }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh2(join(complement(complement(Y)), join(meet(X, complement(meet(X, complement(complement(Y))))), complement(complement(complement(Y))))), one, complement(complement(Y)), join(meet(X, complement(meet(X, complement(complement(Y))))), complement(complement(complement(Y))))))
% 28.92/4.11 = { by lemma 23 }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), fresh2(one, one, complement(complement(Y)), join(meet(X, complement(meet(X, complement(complement(Y))))), complement(complement(complement(Y))))))
% 28.92/4.11 = { by axiom 7 (meet_join_complement) }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), complement(complement(complement(Y))))
% 28.92/4.11 = { by lemma 14 }
% 28.92/4.11 meet(meet(X, complement(meet(X, complement(complement(Y))))), complement(Y))
% 28.92/4.11 = { by axiom 11 (associativity_of_meet) }
% 28.92/4.11 meet(X, meet(complement(meet(X, complement(complement(Y)))), complement(Y)))
% 28.92/4.11 = { by axiom 3 (commutativity_of_meet) }
% 28.92/4.11 meet(X, meet(complement(Y), complement(meet(X, complement(complement(Y))))))
% 28.92/4.11 = { by lemma 26 }
% 28.92/4.11 meet(X, complement(Y))
% 28.92/4.11
% 28.92/4.11 Lemma 28: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 28.92/4.11 Proof:
% 28.92/4.11 meet(complement(X), join(X, Y))
% 28.92/4.11 = { by lemma 14 R->L }
% 28.92/4.11 meet(complement(X), join(X, complement(complement(Y))))
% 28.92/4.11 = { by lemma 24 R->L }
% 28.92/4.11 meet(complement(X), join(X, meet(complement(X), complement(complement(Y)))))
% 28.92/4.11 = { by lemma 27 R->L }
% 28.92/4.11 meet(complement(X), join(X, meet(complement(X), complement(meet(complement(Y), complement(X))))))
% 28.92/4.11 = { by lemma 24 }
% 28.92/4.11 meet(complement(X), join(X, complement(meet(complement(Y), complement(X)))))
% 28.92/4.11 = { by axiom 2 (commutativity_of_join) R->L }
% 28.92/4.11 meet(complement(X), join(complement(meet(complement(Y), complement(X))), X))
% 28.92/4.11 = { by lemma 26 R->L }
% 28.92/4.11 meet(complement(X), join(complement(meet(complement(Y), complement(X))), meet(X, complement(meet(complement(Y), complement(X))))))
% 28.92/4.11 = { by lemma 19 }
% 28.92/4.11 meet(complement(X), complement(meet(complement(Y), complement(X))))
% 28.92/4.11 = { by lemma 27 }
% 28.92/4.11 meet(complement(X), complement(complement(Y)))
% 28.92/4.11 = { by lemma 14 }
% 28.92/4.11 meet(complement(X), Y)
% 28.92/4.11 = { by axiom 3 (commutativity_of_meet) }
% 28.92/4.11 meet(Y, complement(X))
% 28.92/4.11
% 28.92/4.11 Lemma 29: meet(X, join(Y, join(complement(X), meet(X, Z)))) = meet(X, join(Y, join(Z, complement(X)))).
% 28.92/4.11 Proof:
% 28.92/4.11 meet(X, join(Y, join(complement(X), meet(X, Z))))
% 28.92/4.11 = { by axiom 9 (associativity_of_join) R->L }
% 28.92/4.11 meet(X, join(join(Y, complement(X)), meet(X, Z)))
% 28.92/4.11 = { by lemma 21 R->L }
% 28.92/4.11 meet(X, join(join(Y, complement(X)), meet(Z, join(X, join(Y, complement(X))))))
% 28.92/4.11 = { by lemma 23 }
% 28.92/4.11 meet(X, join(join(Y, complement(X)), meet(Z, one)))
% 28.92/4.11 = { by axiom 9 (associativity_of_join) }
% 28.92/4.11 meet(X, join(Y, join(complement(X), meet(Z, one))))
% 28.92/4.11 = { by lemma 15 }
% 28.92/4.11 meet(X, join(Y, join(complement(X), Z)))
% 28.92/4.11 = { by axiom 2 (commutativity_of_join) }
% 28.92/4.11 meet(X, join(Y, join(Z, complement(X))))
% 28.92/4.11
% 28.92/4.11 Lemma 30: meet(X, join(Y, meet(X, Z))) = meet(X, join(Y, Z)).
% 28.92/4.11 Proof:
% 28.92/4.11 meet(X, join(Y, meet(X, Z)))
% 28.92/4.11 = { by axiom 3 (commutativity_of_meet) R->L }
% 28.92/4.11 meet(X, join(Y, meet(Z, X)))
% 28.92/4.11 = { by lemma 14 R->L }
% 28.92/4.11 meet(X, join(Y, meet(Z, complement(complement(X)))))
% 28.92/4.11 = { by lemma 14 R->L }
% 29.15/4.11 meet(complement(complement(X)), join(Y, meet(Z, complement(complement(X)))))
% 29.15/4.11 = { by axiom 3 (commutativity_of_meet) R->L }
% 29.15/4.11 meet(join(Y, meet(Z, complement(complement(X)))), complement(complement(X)))
% 29.15/4.11 = { by lemma 28 R->L }
% 29.15/4.11 meet(complement(complement(X)), join(complement(X), join(Y, meet(Z, complement(complement(X))))))
% 29.15/4.11 = { by axiom 3 (commutativity_of_meet) R->L }
% 29.15/4.11 meet(complement(complement(X)), join(complement(X), join(Y, meet(complement(complement(X)), Z))))
% 29.15/4.11 = { by lemma 20 }
% 29.15/4.11 meet(complement(complement(X)), join(Y, join(complement(X), meet(complement(complement(X)), Z))))
% 29.15/4.11 = { by lemma 14 R->L }
% 29.15/4.11 meet(complement(complement(X)), join(Y, join(complement(complement(complement(X))), meet(complement(complement(X)), Z))))
% 29.15/4.11 = { by lemma 29 }
% 29.15/4.11 meet(complement(complement(X)), join(Y, join(Z, complement(complement(complement(X))))))
% 29.15/4.11 = { by lemma 14 }
% 29.15/4.11 meet(complement(complement(X)), join(Y, join(Z, complement(X))))
% 29.15/4.11 = { by axiom 2 (commutativity_of_join) }
% 29.15/4.11 meet(complement(complement(X)), join(Y, join(complement(X), Z)))
% 29.15/4.11 = { by lemma 20 R->L }
% 29.15/4.11 meet(complement(complement(X)), join(complement(X), join(Y, Z)))
% 29.15/4.11 = { by lemma 28 }
% 29.15/4.11 meet(join(Y, Z), complement(complement(X)))
% 29.15/4.11 = { by axiom 3 (commutativity_of_meet) }
% 29.15/4.11 meet(complement(complement(X)), join(Y, Z))
% 29.15/4.11 = { by lemma 14 }
% 29.15/4.11 meet(X, join(Y, Z))
% 29.15/4.11
% 29.15/4.11 Lemma 31: meet(X, join(Y, join(Z, complement(X)))) = meet(X, join(Y, meet(X, Z))).
% 29.15/4.11 Proof:
% 29.15/4.11 meet(X, join(Y, join(Z, complement(X))))
% 29.15/4.11 = { by lemma 29 R->L }
% 29.15/4.11 meet(X, join(Y, join(complement(X), meet(X, Z))))
% 29.15/4.11 = { by axiom 2 (commutativity_of_join) }
% 29.15/4.11 meet(X, join(join(complement(X), meet(X, Z)), Y))
% 29.15/4.11 = { by axiom 9 (associativity_of_join) }
% 29.15/4.11 meet(X, join(complement(X), join(meet(X, Z), Y)))
% 29.15/4.11 = { by lemma 14 R->L }
% 29.15/4.11 meet(complement(complement(X)), join(complement(X), join(meet(X, Z), Y)))
% 29.15/4.11 = { by lemma 28 }
% 29.15/4.11 meet(join(meet(X, Z), Y), complement(complement(X)))
% 29.15/4.11 = { by lemma 14 }
% 29.15/4.11 meet(join(meet(X, Z), Y), X)
% 29.15/4.11 = { by axiom 3 (commutativity_of_meet) }
% 29.15/4.11 meet(X, join(meet(X, Z), Y))
% 29.15/4.11 = { by axiom 2 (commutativity_of_join) }
% 29.15/4.11 meet(X, join(Y, meet(X, Z)))
% 29.15/4.11
% 29.15/4.11 Lemma 32: join(meet(X, Y), meet(X, join(meet(X, Y), Z))) = meet(X, join(Z, meet(X, Y))).
% 29.15/4.11 Proof:
% 29.15/4.11 join(meet(X, Y), meet(X, join(meet(X, Y), Z)))
% 29.15/4.11 = { by axiom 2 (commutativity_of_join) R->L }
% 29.15/4.11 join(meet(X, Y), meet(X, join(Z, meet(X, Y))))
% 29.15/4.11 = { by axiom 3 (commutativity_of_meet) R->L }
% 29.15/4.11 join(meet(Y, X), meet(X, join(Z, meet(X, Y))))
% 29.15/4.11 = { by axiom 3 (commutativity_of_meet) R->L }
% 29.15/4.11 join(meet(Y, X), meet(X, join(Z, meet(Y, X))))
% 29.15/4.11 = { by axiom 2 (commutativity_of_join) R->L }
% 29.15/4.11 join(meet(X, join(Z, meet(Y, X))), meet(Y, X))
% 29.15/4.11 = { by axiom 10 (absorption1) R->L }
% 29.15/4.11 join(meet(X, join(Z, meet(Y, X))), meet(meet(Y, X), join(meet(Y, X), Z)))
% 29.15/4.11 = { by axiom 11 (associativity_of_meet) }
% 29.15/4.11 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(meet(Y, X), Z))))
% 29.15/4.11 = { by axiom 2 (commutativity_of_join) }
% 29.15/4.11 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(Z, meet(Y, X)))))
% 29.15/4.11 = { by lemma 19 }
% 29.15/4.11 meet(X, join(Z, meet(Y, X)))
% 29.15/4.11 = { by axiom 3 (commutativity_of_meet) }
% 29.15/4.11 meet(X, join(Z, meet(X, Y)))
% 29.15/4.11
% 29.15/4.11 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 29.15/4.11 Proof:
% 29.15/4.11 meet(a, join(b, c))
% 29.15/4.11 = { by lemma 30 R->L }
% 29.15/4.11 meet(a, join(b, meet(a, c)))
% 29.15/4.11 = { by lemma 31 R->L }
% 29.15/4.11 meet(a, join(b, join(c, complement(a))))
% 29.15/4.11 = { by axiom 2 (commutativity_of_join) R->L }
% 29.15/4.11 meet(a, join(b, join(complement(a), c)))
% 29.15/4.11 = { by axiom 9 (associativity_of_join) R->L }
% 29.15/4.11 meet(a, join(join(b, complement(a)), c))
% 29.15/4.11 = { by lemma 30 R->L }
% 29.15/4.11 meet(a, join(join(b, complement(a)), meet(a, c)))
% 29.15/4.11 = { by lemma 32 R->L }
% 29.15/4.11 join(meet(a, c), meet(a, join(meet(a, c), join(b, complement(a)))))
% 29.15/4.11 = { by lemma 31 }
% 29.15/4.11 join(meet(a, c), meet(a, join(meet(a, c), meet(a, b))))
% 29.15/4.11 = { by lemma 32 }
% 29.15/4.11 meet(a, join(meet(a, b), meet(a, c)))
% 29.15/4.11 = { by axiom 3 (commutativity_of_meet) R->L }
% 29.15/4.11 meet(a, join(meet(a, b), meet(c, a)))
% 29.15/4.11 = { by axiom 2 (commutativity_of_join) R->L }
% 29.15/4.11 meet(a, join(meet(c, a), meet(a, b)))
% 29.15/4.11 = { by axiom 3 (commutativity_of_meet) R->L }
% 29.15/4.11 meet(a, join(meet(c, a), meet(b, a)))
% 29.15/4.11 = { by axiom 3 (commutativity_of_meet) R->L }
% 29.15/4.11 meet(a, join(meet(a, c), meet(b, a)))
% 29.15/4.11 = { by axiom 2 (commutativity_of_join) R->L }
% 29.15/4.11 meet(a, join(meet(b, a), meet(a, c)))
% 29.15/4.11 = { by lemma 19 R->L }
% 29.15/4.11 meet(join(a, meet(b, a)), join(meet(b, a), meet(a, c)))
% 29.15/4.11 = { by axiom 3 (commutativity_of_meet) R->L }
% 29.15/4.11 meet(join(meet(b, a), meet(a, c)), join(a, meet(b, a)))
% 29.15/4.11 = { by lemma 25 R->L }
% 29.15/4.11 meet(join(meet(b, a), meet(a, c)), join(a, join(meet(b, a), meet(a, c))))
% 29.15/4.11 = { by lemma 18 }
% 29.15/4.11 join(meet(b, a), meet(a, c))
% 29.15/4.11 = { by axiom 3 (commutativity_of_meet) R->L }
% 29.15/4.11 join(meet(a, b), meet(a, c))
% 29.15/4.11 % SZS output end Proof
% 29.15/4.11
% 29.15/4.11 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------