TSTP Solution File: LAT200-1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT200-1 : 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 : n009.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 24.58s 3.51s
% Output : Proof 25.18s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LAT200-1 : TPTP v8.1.2. Released v3.1.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.34 % Computer : n009.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Aug 24 04:17:51 EDT 2023
% 0.12/0.34 % CPUTime :
% 24.58/3.51 Command-line arguments: --no-flatten-goal
% 24.58/3.51
% 24.58/3.51 % SZS status Unsatisfiable
% 24.58/3.51
% 25.18/3.57 % SZS output start Proof
% 25.18/3.57 Take the following subset of the input axioms:
% 25.18/3.57 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 25.18/3.57 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 25.18/3.57 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 25.18/3.57 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 25.18/3.57 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 25.18/3.57 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 25.18/3.57 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 25.18/3.57 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 25.18/3.57 fof(equation_H50, axiom, ![U, X2, Y2, Z2]: meet(X2, join(Y2, meet(Z2, join(X2, U))))=meet(X2, join(Y2, meet(Z2, join(X2, meet(Z2, join(Y2, U))))))).
% 25.18/3.57 fof(idempotence_of_join, axiom, ![X2]: join(X2, X2)=X2).
% 25.18/3.57 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 25.18/3.57 fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
% 25.18/3.57
% 25.18/3.57 Now clausify the problem and encode Horn clauses using encoding 3 of
% 25.18/3.57 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 25.18/3.57 We repeatedly replace C & s=t => u=v by the two clauses:
% 25.18/3.57 fresh(y, y, x1...xn) = u
% 25.18/3.57 C => fresh(s, t, x1...xn) = v
% 25.18/3.57 where fresh is a fresh function symbol and x1..xn are the free
% 25.18/3.57 variables of u and v.
% 25.18/3.57 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 25.18/3.57 input problem has no model of domain size 1).
% 25.18/3.57
% 25.18/3.57 The encoding turns the above axioms into the following unit equations and goals:
% 25.18/3.57
% 25.18/3.57 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 25.18/3.57 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 25.18/3.57 Axiom 3 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 25.18/3.57 Axiom 4 (complement_join): join(X, complement(X)) = one.
% 25.18/3.57 Axiom 5 (complement_meet): meet(X, complement(X)) = zero.
% 25.18/3.57 Axiom 6 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 25.18/3.57 Axiom 7 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 25.18/3.57 Axiom 8 (absorption2): join(X, meet(X, Y)) = X.
% 25.18/3.57 Axiom 9 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 25.18/3.57 Axiom 10 (absorption1): meet(X, join(X, Y)) = X.
% 25.18/3.57 Axiom 11 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 25.18/3.57 Axiom 12 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 25.18/3.57 Axiom 13 (equation_H50): meet(X, join(Y, meet(Z, join(X, W)))) = meet(X, join(Y, meet(Z, join(X, meet(Z, join(Y, W)))))).
% 25.18/3.57
% 25.18/3.57 Lemma 14: complement(complement(X)) = X.
% 25.18/3.57 Proof:
% 25.18/3.57 complement(complement(X))
% 25.18/3.57 = { by axiom 7 (meet_join_complement) R->L }
% 25.18/3.57 fresh2(one, one, complement(X), X)
% 25.18/3.57 = { by axiom 4 (complement_join) R->L }
% 25.18/3.57 fresh2(join(X, complement(X)), one, complement(X), X)
% 25.18/3.57 = { by axiom 2 (commutativity_of_join) R->L }
% 25.18/3.57 fresh2(join(complement(X), X), one, complement(X), X)
% 25.18/3.57 = { by axiom 12 (meet_join_complement) }
% 25.18/3.57 fresh(meet(complement(X), X), zero, complement(X), X)
% 25.18/3.57 = { by axiom 3 (commutativity_of_meet) }
% 25.18/3.57 fresh(meet(X, complement(X)), zero, complement(X), X)
% 25.18/3.57 = { by axiom 5 (complement_meet) }
% 25.18/3.57 fresh(zero, zero, complement(X), X)
% 25.18/3.57 = { by axiom 6 (meet_join_complement) }
% 25.18/3.57 X
% 25.18/3.57
% 25.18/3.57 Lemma 15: meet(X, one) = X.
% 25.18/3.57 Proof:
% 25.18/3.57 meet(X, one)
% 25.18/3.57 = { by axiom 4 (complement_join) R->L }
% 25.18/3.57 meet(X, join(X, complement(X)))
% 25.18/3.57 = { by axiom 10 (absorption1) }
% 25.18/3.57 X
% 25.18/3.57
% 25.18/3.57 Lemma 16: join(X, zero) = X.
% 25.18/3.57 Proof:
% 25.18/3.57 join(X, zero)
% 25.18/3.57 = { by axiom 5 (complement_meet) R->L }
% 25.18/3.57 join(X, meet(X, complement(X)))
% 25.18/3.57 = { by axiom 8 (absorption2) }
% 25.18/3.57 X
% 25.18/3.57
% 25.18/3.57 Lemma 17: meet(X, zero) = zero.
% 25.18/3.57 Proof:
% 25.18/3.57 meet(X, zero)
% 25.18/3.57 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.57 meet(zero, X)
% 25.18/3.57 = { by lemma 16 R->L }
% 25.18/3.57 join(meet(zero, X), zero)
% 25.18/3.57 = { by axiom 2 (commutativity_of_join) }
% 25.18/3.57 join(zero, meet(zero, X))
% 25.18/3.57 = { by axiom 8 (absorption2) }
% 25.18/3.57 zero
% 25.18/3.57
% 25.18/3.57 Lemma 18: join(X, one) = one.
% 25.18/3.57 Proof:
% 25.18/3.57 join(X, one)
% 25.18/3.57 = { by axiom 2 (commutativity_of_join) R->L }
% 25.18/3.57 join(one, X)
% 25.18/3.57 = { by lemma 15 R->L }
% 25.18/3.57 join(one, meet(X, one))
% 25.18/3.57 = { by axiom 3 (commutativity_of_meet) }
% 25.18/3.57 join(one, meet(one, X))
% 25.18/3.57 = { by axiom 8 (absorption2) }
% 25.18/3.57 one
% 25.18/3.57
% 25.18/3.57 Lemma 19: meet(X, join(Y, X)) = X.
% 25.18/3.57 Proof:
% 25.18/3.57 meet(X, join(Y, X))
% 25.18/3.57 = { by axiom 2 (commutativity_of_join) R->L }
% 25.18/3.57 meet(X, join(X, Y))
% 25.18/3.57 = { by axiom 10 (absorption1) }
% 25.18/3.57 X
% 25.18/3.57
% 25.18/3.57 Lemma 20: meet(complement(X), join(X, meet(Y, complement(X)))) = meet(complement(X), join(X, Y)).
% 25.18/3.57 Proof:
% 25.18/3.57 meet(complement(X), join(X, meet(Y, complement(X))))
% 25.18/3.57 = { by axiom 1 (idempotence_of_join) R->L }
% 25.18/3.57 meet(complement(X), join(X, meet(Y, join(complement(X), complement(X)))))
% 25.18/3.57 = { by axiom 13 (equation_H50) }
% 25.18/3.57 meet(complement(X), join(X, meet(Y, join(complement(X), meet(Y, join(X, complement(X)))))))
% 25.18/3.57 = { by axiom 4 (complement_join) }
% 25.18/3.57 meet(complement(X), join(X, meet(Y, join(complement(X), meet(Y, one)))))
% 25.18/3.57 = { by lemma 15 }
% 25.18/3.57 meet(complement(X), join(X, meet(Y, join(complement(X), Y))))
% 25.18/3.57 = { by lemma 19 }
% 25.18/3.57 meet(complement(X), join(X, Y))
% 25.18/3.57
% 25.18/3.57 Lemma 21: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 25.18/3.57 Proof:
% 25.18/3.57 meet(X, meet(Y, Z))
% 25.18/3.57 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.57 meet(meet(Y, Z), X)
% 25.18/3.57 = { by axiom 11 (associativity_of_meet) }
% 25.18/3.57 meet(Y, meet(Z, X))
% 25.18/3.57 = { by axiom 3 (commutativity_of_meet) }
% 25.18/3.57 meet(Y, meet(X, Z))
% 25.18/3.57
% 25.18/3.57 Lemma 22: meet(X, join(complement(X), meet(Y, complement(meet(X, Y))))) = zero.
% 25.18/3.57 Proof:
% 25.18/3.57 meet(X, join(complement(X), meet(Y, complement(meet(X, Y)))))
% 25.18/3.57 = { by lemma 14 R->L }
% 25.18/3.57 meet(complement(complement(X)), join(complement(X), meet(Y, complement(meet(X, Y)))))
% 25.18/3.57 = { by lemma 20 R->L }
% 25.18/3.57 meet(complement(complement(X)), join(complement(X), meet(meet(Y, complement(meet(X, Y))), complement(complement(X)))))
% 25.18/3.57 = { by axiom 11 (associativity_of_meet) }
% 25.18/3.57 meet(complement(complement(X)), join(complement(X), meet(Y, meet(complement(meet(X, Y)), complement(complement(X))))))
% 25.18/3.57 = { by lemma 14 }
% 25.18/3.57 meet(X, join(complement(X), meet(Y, meet(complement(meet(X, Y)), complement(complement(X))))))
% 25.18/3.57 = { by lemma 14 }
% 25.18/3.57 meet(X, join(complement(X), meet(Y, meet(complement(meet(X, Y)), X))))
% 25.18/3.57 = { by axiom 3 (commutativity_of_meet) }
% 25.18/3.57 meet(X, join(complement(X), meet(Y, meet(X, complement(meet(X, Y))))))
% 25.18/3.57 = { by lemma 21 }
% 25.18/3.57 meet(X, join(complement(X), meet(X, meet(Y, complement(meet(X, Y))))))
% 25.18/3.57 = { by axiom 11 (associativity_of_meet) R->L }
% 25.18/3.57 meet(X, join(complement(X), meet(meet(X, Y), complement(meet(X, Y)))))
% 25.18/3.57 = { by axiom 5 (complement_meet) }
% 25.18/3.57 meet(X, join(complement(X), zero))
% 25.18/3.57 = { by lemma 16 }
% 25.18/3.57 meet(X, complement(X))
% 25.18/3.57 = { by axiom 5 (complement_meet) }
% 25.18/3.57 zero
% 25.18/3.57
% 25.18/3.57 Lemma 23: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 25.18/3.57 Proof:
% 25.18/3.57 join(X, join(Y, meet(Z, join(X, Y))))
% 25.18/3.57 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.57 join(X, join(Y, meet(join(X, Y), Z)))
% 25.18/3.57 = { by axiom 9 (associativity_of_join) R->L }
% 25.18/3.57 join(join(X, Y), meet(join(X, Y), Z))
% 25.18/3.57 = { by axiom 8 (absorption2) }
% 25.18/3.57 join(X, Y)
% 25.18/3.57
% 25.18/3.57 Lemma 24: fresh(meet(X, join(Y, complement(X))), zero, X, join(Y, complement(X))) = complement(X).
% 25.18/3.57 Proof:
% 25.18/3.57 fresh(meet(X, join(Y, complement(X))), zero, X, join(Y, complement(X)))
% 25.18/3.57 = { by axiom 12 (meet_join_complement) R->L }
% 25.18/3.57 fresh2(join(X, join(Y, complement(X))), one, X, join(Y, complement(X)))
% 25.18/3.57 = { by axiom 2 (commutativity_of_join) R->L }
% 25.18/3.57 fresh2(join(X, join(complement(X), Y)), one, X, join(Y, complement(X)))
% 25.18/3.57 = { by axiom 9 (associativity_of_join) R->L }
% 25.18/3.57 fresh2(join(join(X, complement(X)), Y), one, X, join(Y, complement(X)))
% 25.18/3.57 = { by axiom 4 (complement_join) }
% 25.18/3.57 fresh2(join(one, Y), one, X, join(Y, complement(X)))
% 25.18/3.57 = { by axiom 2 (commutativity_of_join) R->L }
% 25.18/3.57 fresh2(join(Y, one), one, X, join(Y, complement(X)))
% 25.18/3.57 = { by lemma 18 }
% 25.18/3.57 fresh2(one, one, X, join(Y, complement(X)))
% 25.18/3.57 = { by axiom 7 (meet_join_complement) }
% 25.18/3.57 complement(X)
% 25.18/3.57
% 25.18/3.57 Lemma 25: join(X, meet(Y, X)) = X.
% 25.18/3.57 Proof:
% 25.18/3.57 join(X, meet(Y, X))
% 25.18/3.57 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.57 join(X, meet(X, Y))
% 25.18/3.57 = { by axiom 8 (absorption2) }
% 25.18/3.57 X
% 25.18/3.57
% 25.18/3.57 Lemma 26: join(X, join(meet(X, Y), Z)) = join(X, Z).
% 25.18/3.57 Proof:
% 25.18/3.57 join(X, join(meet(X, Y), Z))
% 25.18/3.57 = { by axiom 9 (associativity_of_join) R->L }
% 25.18/3.57 join(join(X, meet(X, Y)), Z)
% 25.18/3.57 = { by axiom 8 (absorption2) }
% 25.18/3.57 join(X, Z)
% 25.18/3.57
% 25.18/3.57 Lemma 27: meet(complement(X), complement(meet(X, Y))) = complement(X).
% 25.18/3.57 Proof:
% 25.18/3.57 meet(complement(X), complement(meet(X, Y)))
% 25.18/3.57 = { by axiom 6 (meet_join_complement) R->L }
% 25.18/3.57 fresh(zero, zero, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.57 = { by lemma 17 R->L }
% 25.18/3.57 fresh(meet(complement(meet(X, Y)), zero), zero, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.57 = { by axiom 3 (commutativity_of_meet) }
% 25.18/3.57 fresh(meet(zero, complement(meet(X, Y))), zero, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by axiom 5 (complement_meet) R->L }
% 25.18/3.58 fresh(meet(meet(X, complement(X)), complement(meet(X, Y))), zero, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by axiom 11 (associativity_of_meet) }
% 25.18/3.58 fresh(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by axiom 12 (meet_join_complement) R->L }
% 25.18/3.58 fresh2(join(X, meet(complement(X), complement(meet(X, Y)))), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.58 fresh2(join(X, meet(complement(X), complement(meet(Y, X)))), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.58 fresh2(join(X, meet(complement(meet(Y, X)), complement(X))), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by lemma 25 R->L }
% 25.18/3.58 fresh2(join(join(X, meet(complement(meet(Y, X)), complement(X))), meet(complement(X), join(X, meet(complement(meet(Y, X)), complement(X))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by lemma 20 }
% 25.18/3.58 fresh2(join(join(X, meet(complement(meet(Y, X)), complement(X))), meet(complement(X), join(X, complement(meet(Y, X))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by axiom 9 (associativity_of_join) }
% 25.18/3.58 fresh2(join(X, join(meet(complement(meet(Y, X)), complement(X)), meet(complement(X), join(X, complement(meet(Y, X)))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by axiom 2 (commutativity_of_join) R->L }
% 25.18/3.58 fresh2(join(X, join(meet(complement(X), join(X, complement(meet(Y, X)))), meet(complement(meet(Y, X)), complement(X)))), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.58 fresh2(join(X, join(meet(complement(X), join(X, complement(meet(Y, X)))), meet(complement(X), complement(meet(Y, X))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by lemma 19 R->L }
% 25.18/3.58 fresh2(join(X, join(meet(complement(X), join(X, complement(meet(Y, X)))), meet(complement(X), meet(complement(meet(Y, X)), join(X, complement(meet(Y, X))))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by lemma 21 }
% 25.18/3.58 fresh2(join(X, join(meet(complement(X), join(X, complement(meet(Y, X)))), meet(complement(meet(Y, X)), meet(complement(X), join(X, complement(meet(Y, X))))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by lemma 25 }
% 25.18/3.58 fresh2(join(X, meet(complement(X), join(X, complement(meet(Y, X))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.58 fresh2(join(X, meet(complement(X), join(X, complement(meet(X, Y))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by lemma 26 R->L }
% 25.18/3.58 fresh2(join(X, meet(complement(X), join(X, join(meet(X, Y), complement(meet(X, Y)))))), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by axiom 4 (complement_join) }
% 25.18/3.58 fresh2(join(X, meet(complement(X), join(X, one))), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by lemma 18 }
% 25.18/3.58 fresh2(join(X, meet(complement(X), one)), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by lemma 15 }
% 25.18/3.58 fresh2(join(X, complement(X)), one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by axiom 4 (complement_join) }
% 25.18/3.58 fresh2(one, one, X, meet(complement(X), complement(meet(X, Y))))
% 25.18/3.58 = { by axiom 7 (meet_join_complement) }
% 25.18/3.58 complement(X)
% 25.18/3.58
% 25.18/3.58 Lemma 28: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 25.18/3.58 Proof:
% 25.18/3.58 meet(X, complement(meet(X, Y)))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.58 meet(X, complement(meet(Y, X)))
% 25.18/3.58 = { by lemma 16 R->L }
% 25.18/3.58 join(meet(X, complement(meet(Y, X))), zero)
% 25.18/3.58 = { by lemma 22 R->L }
% 25.18/3.58 join(meet(X, complement(meet(Y, X))), meet(Y, join(complement(Y), meet(X, complement(meet(Y, X))))))
% 25.18/3.58 = { by lemma 19 R->L }
% 25.18/3.58 meet(join(meet(X, complement(meet(Y, X))), meet(Y, join(complement(Y), meet(X, complement(meet(Y, X)))))), join(complement(Y), join(meet(X, complement(meet(Y, X))), meet(Y, join(complement(Y), meet(X, complement(meet(Y, X))))))))
% 25.18/3.58 = { by lemma 23 }
% 25.18/3.58 meet(join(meet(X, complement(meet(Y, X))), meet(Y, join(complement(Y), meet(X, complement(meet(Y, X)))))), join(complement(Y), meet(X, complement(meet(Y, X)))))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) }
% 25.18/3.58 meet(join(complement(Y), meet(X, complement(meet(Y, X)))), join(meet(X, complement(meet(Y, X))), meet(Y, join(complement(Y), meet(X, complement(meet(Y, X)))))))
% 25.18/3.58 = { by lemma 22 }
% 25.18/3.58 meet(join(complement(Y), meet(X, complement(meet(Y, X)))), join(meet(X, complement(meet(Y, X))), zero))
% 25.18/3.58 = { by axiom 2 (commutativity_of_join) R->L }
% 25.18/3.58 meet(join(meet(X, complement(meet(Y, X))), complement(Y)), join(meet(X, complement(meet(Y, X))), zero))
% 25.18/3.58 = { by axiom 6 (meet_join_complement) R->L }
% 25.18/3.58 meet(fresh(zero, zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))), join(meet(X, complement(meet(Y, X))), zero))
% 25.18/3.58 = { by lemma 22 R->L }
% 25.18/3.58 meet(fresh(meet(Y, join(complement(Y), meet(X, complement(meet(Y, X))))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))), join(meet(X, complement(meet(Y, X))), zero))
% 25.18/3.58 = { by axiom 2 (commutativity_of_join) R->L }
% 25.18/3.58 meet(fresh(meet(Y, join(meet(X, complement(meet(Y, X))), complement(Y))), zero, Y, join(meet(X, complement(meet(Y, X))), complement(Y))), join(meet(X, complement(meet(Y, X))), zero))
% 25.18/3.58 = { by lemma 24 }
% 25.18/3.58 meet(complement(Y), join(meet(X, complement(meet(Y, X))), zero))
% 25.18/3.58 = { by lemma 16 }
% 25.18/3.58 meet(complement(Y), meet(X, complement(meet(Y, X))))
% 25.18/3.58 = { by lemma 21 }
% 25.18/3.58 meet(X, meet(complement(Y), complement(meet(Y, X))))
% 25.18/3.58 = { by lemma 27 }
% 25.18/3.58 meet(X, complement(Y))
% 25.18/3.58
% 25.18/3.58 Lemma 29: meet(X, complement(meet(Y, X))) = meet(X, complement(Y)).
% 25.18/3.58 Proof:
% 25.18/3.58 meet(X, complement(meet(Y, X)))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.58 meet(X, complement(meet(X, Y)))
% 25.18/3.58 = { by lemma 28 }
% 25.18/3.58 meet(X, complement(Y))
% 25.18/3.58
% 25.18/3.58 Lemma 30: meet(complement(X), join(X, meet(complement(X), Y))) = meet(complement(X), join(X, Y)).
% 25.18/3.58 Proof:
% 25.18/3.58 meet(complement(X), join(X, meet(complement(X), Y)))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.58 meet(complement(X), join(X, meet(Y, complement(X))))
% 25.18/3.58 = { by lemma 20 }
% 25.18/3.58 meet(complement(X), join(X, Y))
% 25.18/3.58
% 25.18/3.58 Lemma 31: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 25.18/3.58 Proof:
% 25.18/3.58 meet(X, meet(Y, join(X, Z)))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.58 meet(X, meet(join(X, Z), Y))
% 25.18/3.58 = { by axiom 11 (associativity_of_meet) R->L }
% 25.18/3.58 meet(meet(X, join(X, Z)), Y)
% 25.18/3.58 = { by axiom 10 (absorption1) }
% 25.18/3.58 meet(X, Y)
% 25.18/3.58
% 25.18/3.58 Lemma 32: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 25.18/3.58 Proof:
% 25.18/3.58 complement(meet(X, complement(Y)))
% 25.18/3.58 = { by lemma 28 R->L }
% 25.18/3.58 complement(meet(X, complement(meet(X, Y))))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.58 complement(meet(X, complement(meet(Y, X))))
% 25.18/3.58 = { by lemma 14 R->L }
% 25.18/3.58 complement(meet(X, complement(meet(Y, complement(complement(X))))))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.58 complement(meet(X, complement(meet(complement(complement(X)), Y))))
% 25.18/3.58 = { by lemma 14 R->L }
% 25.18/3.58 complement(meet(X, complement(meet(complement(complement(X)), complement(complement(Y))))))
% 25.18/3.58 = { by lemma 29 R->L }
% 25.18/3.58 complement(meet(X, complement(meet(complement(complement(X)), complement(meet(complement(Y), complement(complement(X))))))))
% 25.18/3.58 = { by lemma 25 R->L }
% 25.18/3.58 complement(meet(X, complement(meet(complement(complement(X)), join(complement(meet(complement(Y), complement(complement(X)))), meet(complement(X), complement(meet(complement(Y), complement(complement(X))))))))))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.58 complement(meet(X, complement(meet(complement(complement(X)), join(complement(meet(complement(Y), complement(complement(X)))), meet(complement(X), complement(meet(complement(complement(X)), complement(Y)))))))))
% 25.18/3.58 = { by lemma 14 R->L }
% 25.18/3.58 complement(meet(X, complement(meet(complement(complement(X)), join(complement(meet(complement(Y), complement(complement(X)))), meet(complement(complement(complement(X))), complement(meet(complement(complement(X)), complement(Y)))))))))
% 25.18/3.58 = { by lemma 27 }
% 25.18/3.58 complement(meet(X, complement(meet(complement(complement(X)), join(complement(meet(complement(Y), complement(complement(X)))), complement(complement(complement(X))))))))
% 25.18/3.58 = { by lemma 14 }
% 25.18/3.58 complement(meet(X, complement(meet(complement(complement(X)), join(complement(meet(complement(Y), complement(complement(X)))), complement(X))))))
% 25.18/3.58 = { by axiom 2 (commutativity_of_join) }
% 25.18/3.58 complement(meet(X, complement(meet(complement(complement(X)), join(complement(X), complement(meet(complement(Y), complement(complement(X)))))))))
% 25.18/3.58 = { by lemma 30 R->L }
% 25.18/3.58 complement(meet(X, complement(meet(complement(complement(X)), join(complement(X), meet(complement(complement(X)), complement(meet(complement(Y), complement(complement(X))))))))))
% 25.18/3.58 = { by lemma 29 }
% 25.18/3.58 complement(meet(X, complement(meet(complement(complement(X)), join(complement(X), meet(complement(complement(X)), complement(complement(Y))))))))
% 25.18/3.58 = { by lemma 30 }
% 25.18/3.58 complement(meet(X, complement(meet(complement(complement(X)), join(complement(X), complement(complement(Y)))))))
% 25.18/3.58 = { by lemma 14 }
% 25.18/3.58 complement(meet(X, complement(meet(complement(complement(X)), join(complement(X), Y)))))
% 25.18/3.58 = { by lemma 14 }
% 25.18/3.58 complement(meet(X, complement(meet(X, join(complement(X), Y)))))
% 25.18/3.58 = { by axiom 2 (commutativity_of_join) }
% 25.18/3.58 complement(meet(X, complement(meet(X, join(Y, complement(X))))))
% 25.18/3.58 = { by lemma 28 }
% 25.18/3.58 complement(meet(X, complement(join(Y, complement(X)))))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), X))
% 25.18/3.58 = { by lemma 14 R->L }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), complement(complement(X))))
% 25.18/3.58 = { by lemma 24 R->L }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(complement(join(Y, complement(X))), complement(complement(X)))), zero, complement(X), join(complement(join(Y, complement(X))), complement(complement(X))))))
% 25.18/3.58 = { by lemma 14 }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(complement(join(Y, complement(X))), X)), zero, complement(X), join(complement(join(Y, complement(X))), complement(complement(X))))))
% 25.18/3.58 = { by lemma 14 }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(complement(join(Y, complement(X))), X)), zero, complement(X), join(complement(join(Y, complement(X))), X))))
% 25.18/3.58 = { by axiom 2 (commutativity_of_join) }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(complement(join(Y, complement(X))), X)), zero, complement(X), join(X, complement(join(Y, complement(X)))))))
% 25.18/3.58 = { by axiom 2 (commutativity_of_join) }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(X, complement(join(Y, complement(X))))), zero, complement(X), join(X, complement(join(Y, complement(X)))))))
% 25.18/3.58 = { by lemma 30 R->L }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(X, meet(complement(X), complement(join(Y, complement(X)))))), zero, complement(X), join(X, complement(join(Y, complement(X)))))))
% 25.18/3.58 = { by axiom 2 (commutativity_of_join) R->L }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(X, meet(complement(X), complement(join(complement(X), Y))))), zero, complement(X), join(X, complement(join(Y, complement(X)))))))
% 25.18/3.58 = { by lemma 31 R->L }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(X, meet(complement(X), meet(complement(join(complement(X), Y)), join(complement(X), Y))))), zero, complement(X), join(X, complement(join(Y, complement(X)))))))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(X, meet(complement(X), meet(join(complement(X), Y), complement(join(complement(X), Y)))))), zero, complement(X), join(X, complement(join(Y, complement(X)))))))
% 25.18/3.58 = { by axiom 5 (complement_meet) }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(X, meet(complement(X), zero))), zero, complement(X), join(X, complement(join(Y, complement(X)))))))
% 25.18/3.58 = { by lemma 17 }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), fresh(meet(complement(X), join(X, zero)), zero, complement(X), join(X, complement(join(Y, complement(X)))))))
% 25.18/3.58 = { by lemma 16 }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), fresh(meet(complement(X), X), zero, complement(X), join(X, complement(join(Y, complement(X)))))))
% 25.18/3.58 = { by axiom 3 (commutativity_of_meet) }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), fresh(meet(X, complement(X)), zero, complement(X), join(X, complement(join(Y, complement(X)))))))
% 25.18/3.58 = { by axiom 5 (complement_meet) }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), fresh(zero, zero, complement(X), join(X, complement(join(Y, complement(X)))))))
% 25.18/3.58 = { by axiom 6 (meet_join_complement) }
% 25.18/3.58 complement(meet(complement(join(Y, complement(X))), join(X, complement(join(Y, complement(X))))))
% 25.18/3.58 = { by lemma 19 }
% 25.18/3.58 complement(complement(join(Y, complement(X))))
% 25.18/3.58 = { by lemma 14 }
% 25.18/3.58 join(Y, complement(X))
% 25.18/3.58
% 25.18/3.58 Lemma 33: meet(X, join(Y, meet(X, Z))) = meet(X, join(Z, Y)).
% 25.18/3.58 Proof:
% 25.18/3.59 meet(X, join(Y, meet(X, Z)))
% 25.18/3.59 = { by lemma 14 R->L }
% 25.18/3.59 meet(X, join(complement(complement(Y)), meet(X, Z)))
% 25.18/3.59 = { by axiom 2 (commutativity_of_join) R->L }
% 25.18/3.59 meet(X, join(meet(X, Z), complement(complement(Y))))
% 25.18/3.59 = { by lemma 32 R->L }
% 25.18/3.59 meet(X, complement(meet(complement(Y), complement(meet(X, Z)))))
% 25.18/3.59 = { by lemma 28 R->L }
% 25.18/3.59 meet(X, complement(meet(X, meet(complement(Y), complement(meet(X, Z))))))
% 25.18/3.59 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.59 meet(X, complement(meet(X, meet(complement(meet(X, Z)), complement(Y)))))
% 25.18/3.59 = { by axiom 11 (associativity_of_meet) R->L }
% 25.18/3.59 meet(X, complement(meet(meet(X, complement(meet(X, Z))), complement(Y))))
% 25.18/3.59 = { by lemma 28 }
% 25.18/3.59 meet(X, complement(meet(meet(X, complement(Z)), complement(Y))))
% 25.18/3.59 = { by axiom 11 (associativity_of_meet) }
% 25.18/3.59 meet(X, complement(meet(X, meet(complement(Z), complement(Y)))))
% 25.18/3.59 = { by axiom 3 (commutativity_of_meet) }
% 25.18/3.59 meet(X, complement(meet(X, meet(complement(Y), complement(Z)))))
% 25.18/3.59 = { by lemma 28 }
% 25.18/3.59 meet(X, complement(meet(complement(Y), complement(Z))))
% 25.18/3.59 = { by lemma 32 }
% 25.18/3.59 meet(X, join(Z, complement(complement(Y))))
% 25.18/3.59 = { by lemma 14 }
% 25.18/3.59 meet(X, join(Z, Y))
% 25.18/3.59
% 25.18/3.59 Lemma 34: meet(join(X, join(Y, Z)), join(X, meet(Y, W))) = join(X, meet(Y, W)).
% 25.18/3.59 Proof:
% 25.18/3.59 meet(join(X, join(Y, Z)), join(X, meet(Y, W)))
% 25.18/3.59 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.59 meet(join(X, meet(Y, W)), join(X, join(Y, Z)))
% 25.18/3.59 = { by lemma 26 R->L }
% 25.18/3.59 meet(join(X, meet(Y, W)), join(X, join(Y, join(meet(Y, W), Z))))
% 25.18/3.59 = { by axiom 2 (commutativity_of_join) R->L }
% 25.18/3.59 meet(join(X, meet(Y, W)), join(X, join(join(meet(Y, W), Z), Y)))
% 25.18/3.59 = { by axiom 9 (associativity_of_join) }
% 25.18/3.59 meet(join(X, meet(Y, W)), join(X, join(meet(Y, W), join(Z, Y))))
% 25.18/3.59 = { by axiom 2 (commutativity_of_join) }
% 25.18/3.59 meet(join(X, meet(Y, W)), join(X, join(meet(Y, W), join(Y, Z))))
% 25.18/3.59 = { by axiom 9 (associativity_of_join) R->L }
% 25.18/3.59 meet(join(X, meet(Y, W)), join(join(X, meet(Y, W)), join(Y, Z)))
% 25.18/3.59 = { by axiom 10 (absorption1) }
% 25.18/3.59 join(X, meet(Y, W))
% 25.18/3.59
% 25.18/3.59 Lemma 35: join(meet(X, Y), meet(X, join(Z, meet(X, Y)))) = meet(X, join(Z, meet(X, Y))).
% 25.18/3.59 Proof:
% 25.18/3.59 join(meet(X, Y), meet(X, join(Z, meet(X, Y))))
% 25.18/3.59 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.59 join(meet(Y, X), meet(X, join(Z, meet(X, Y))))
% 25.18/3.59 = { by axiom 3 (commutativity_of_meet) R->L }
% 25.18/3.59 join(meet(Y, X), meet(X, join(Z, meet(Y, X))))
% 25.18/3.59 = { by axiom 2 (commutativity_of_join) R->L }
% 25.18/3.59 join(meet(X, join(Z, meet(Y, X))), meet(Y, X))
% 25.18/3.59 = { by axiom 10 (absorption1) R->L }
% 25.18/3.59 join(meet(X, join(Z, meet(Y, X))), meet(meet(Y, X), join(meet(Y, X), Z)))
% 25.18/3.59 = { by axiom 11 (associativity_of_meet) }
% 25.18/3.59 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(meet(Y, X), Z))))
% 25.18/3.59 = { by axiom 2 (commutativity_of_join) }
% 25.18/3.59 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(Z, meet(Y, X)))))
% 25.18/3.59 = { by lemma 25 }
% 25.18/3.59 meet(X, join(Z, meet(Y, X)))
% 25.18/3.59 = { by axiom 3 (commutativity_of_meet) }
% 25.18/3.59 meet(X, join(Z, meet(X, Y)))
% 25.18/3.59
% 25.18/3.59 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 25.18/3.59 Proof:
% 25.18/3.59 meet(a, join(b, c))
% 25.18/3.59 = { by lemma 33 R->L }
% 25.18/3.59 meet(a, join(c, meet(a, b)))
% 25.18/3.59 = { by lemma 35 R->L }
% 25.18/3.59 join(meet(a, b), meet(a, join(c, meet(a, b))))
% 25.18/3.59 = { by lemma 35 R->L }
% 25.18/3.59 join(meet(a, b), join(meet(a, b), meet(a, join(c, meet(a, b)))))
% 25.18/3.59 = { by lemma 33 R->L }
% 25.18/3.59 join(meet(a, b), join(meet(a, b), meet(a, join(meet(a, b), meet(a, c)))))
% 25.18/3.59 = { by lemma 35 R->L }
% 25.18/3.59 join(meet(a, b), join(meet(a, b), join(meet(a, c), meet(a, join(meet(a, b), meet(a, c))))))
% 25.18/3.59 = { by lemma 23 }
% 25.18/3.59 join(meet(a, b), join(meet(a, b), meet(a, c)))
% 25.18/3.59 = { by axiom 10 (absorption1) R->L }
% 25.18/3.59 join(meet(meet(a, b), join(meet(a, b), join(a, X))), join(meet(a, b), meet(a, c)))
% 25.18/3.59 = { by lemma 34 R->L }
% 25.18/3.59 join(meet(meet(a, b), join(meet(a, b), join(a, X))), meet(join(meet(a, b), join(a, X)), join(meet(a, b), meet(a, c))))
% 25.18/3.59 = { by axiom 2 (commutativity_of_join) R->L }
% 25.18/3.59 join(meet(join(meet(a, b), join(a, X)), join(meet(a, b), meet(a, c))), meet(meet(a, b), join(meet(a, b), join(a, X))))
% 25.18/3.59 = { by lemma 31 R->L }
% 25.18/3.59 join(meet(join(meet(a, b), join(a, X)), join(meet(a, b), meet(a, c))), meet(meet(a, b), meet(join(meet(a, b), join(a, X)), join(meet(a, b), meet(a, c)))))
% 25.18/3.59 = { by lemma 25 }
% 25.18/3.59 meet(join(meet(a, b), join(a, X)), join(meet(a, b), meet(a, c)))
% 25.18/3.59 = { by lemma 34 }
% 25.18/3.59 join(meet(a, b), meet(a, c))
% 25.18/3.59 % SZS output end Proof
% 25.18/3.59
% 25.18/3.59 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------