0.06/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.06/0.12 % Command : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn 0.13/0.33 % Computer : n024.cluster.edu 0.13/0.33 % Model : x86_64 x86_64 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.33 % Memory : 8042.1875MB 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.34 % CPULimit : 180 0.13/0.34 % DateTime : Thu Aug 29 13:55:35 EDT 2019 0.13/0.34 % CPUTime : 5.36/5.52 % SZS status Unsatisfiable 5.36/5.52 5.36/5.52 % SZS output start Proof 5.36/5.52 Take the following subset of the input axioms: 5.53/5.70 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X). 5.53/5.70 fof(absorption2, axiom, ![X, Y]: join(X, meet(X, Y))=X). 5.53/5.70 fof(associativity_of_join, axiom, ![X, Y, Z]: join(join(X, Y), Z)=join(X, join(Y, Z))). 5.53/5.70 fof(associativity_of_meet, axiom, ![X, Y, Z]: meet(X, meet(Y, Z))=meet(meet(X, Y), Z)). 5.53/5.70 fof(commutativity_of_join, axiom, ![X, Y]: join(Y, X)=join(X, Y)). 5.53/5.70 fof(commutativity_of_meet, axiom, ![X, Y]: meet(Y, X)=meet(X, Y)). 5.53/5.70 fof(complement_join, axiom, ![X]: one=join(X, complement(X))). 5.53/5.70 fof(complement_meet, axiom, ![X]: meet(X, complement(X))=zero). 5.53/5.70 fof(equation_H8, axiom, ![X, Y, Z]: meet(X, join(Y, meet(X, Z)))=join(meet(X, Y), meet(X, join(Y, meet(Z, join(X, meet(Y, Z))))))). 5.53/5.70 fof(ifeq_axiom, axiom, ![B, A, C]: B=ifeq(A, A, B, C)). 5.53/5.70 fof(meet_join_complement, axiom, ![X, Y]: Y=ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y)). 5.53/5.70 fof(prove_distributivity, negated_conjecture, join(complement(b), complement(a))!=complement(a)). 5.53/5.70 fof(prove_distributivity_hypothesis, hypothesis, a=meet(b, a)). 5.53/5.70 5.53/5.70 Now clausify the problem and encode Horn clauses using encoding 3 of 5.53/5.70 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 5.53/5.70 We repeatedly replace C & s=t => u=v by the two clauses: 5.53/5.70 fresh(y, y, x1...xn) = u 5.53/5.70 C => fresh(s, t, x1...xn) = v 5.53/5.70 where fresh is a fresh function symbol and x1..xn are the free 5.53/5.70 variables of u and v. 5.53/5.70 A predicate p(X) is encoded as p(X)=true (this is sound, because the 5.53/5.70 input problem has no model of domain size 1). 5.53/5.70 5.53/5.70 The encoding turns the above axioms into the following unit equations and goals: 5.53/5.70 5.53/5.70 Axiom 1 (absorption2): join(X, meet(X, Y)) = X. 5.53/5.70 Axiom 2 (complement_meet): meet(X, complement(X)) = zero. 5.53/5.70 Axiom 3 (absorption1): meet(X, join(X, Y)) = X. 5.53/5.70 Axiom 4 (prove_distributivity_hypothesis): a = meet(b, a). 5.53/5.70 Axiom 5 (ifeq_axiom): X = ifeq(Y, Y, X, Z). 5.53/5.70 Axiom 6 (complement_join): one = join(X, complement(X)). 5.53/5.70 Axiom 7 (commutativity_of_meet): meet(X, Y) = meet(Y, X). 5.53/5.70 Axiom 8 (meet_join_complement): X = ifeq(join(Y, X), one, ifeq(meet(Y, X), zero, complement(Y), X), X). 5.53/5.70 Axiom 9 (commutativity_of_join): join(X, Y) = join(Y, X). 5.53/5.70 Axiom 10 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)). 5.53/5.70 Axiom 11 (equation_H8): meet(X, join(Y, meet(X, Z))) = join(meet(X, Y), meet(X, join(Y, meet(Z, join(X, meet(Y, Z)))))). 5.53/5.70 Axiom 12 (associativity_of_meet): meet(X, meet(Y, Z)) = meet(meet(X, Y), Z). 5.53/5.70 5.53/5.70 Lemma 13: join(X, meet(Y, X)) = X. 5.53/5.70 Proof: 5.53/5.70 join(X, meet(Y, X)) 5.53/5.70 = { by axiom 7 (commutativity_of_meet) } 5.53/5.70 join(X, meet(X, Y)) 5.53/5.70 = { by axiom 1 (absorption2) } 5.53/5.70 X 5.53/5.70 5.53/5.70 Lemma 14: meet(X, meet(Z, join(X, Y))) = meet(X, Z). 5.53/5.70 Proof: 5.53/5.70 meet(X, meet(Z, join(X, Y))) 5.53/5.70 = { by axiom 7 (commutativity_of_meet) } 5.53/5.70 meet(X, meet(join(X, Y), Z)) 5.53/5.70 = { by axiom 12 (associativity_of_meet) } 5.53/5.70 meet(meet(X, join(X, Y)), Z) 5.53/5.70 = { by axiom 3 (absorption1) } 5.53/5.70 meet(X, Z) 5.53/5.70 5.53/5.70 Lemma 15: meet(X, one) = X. 5.53/5.70 Proof: 5.53/5.70 meet(X, one) 5.53/5.70 = { by axiom 6 (complement_join) } 5.53/5.70 meet(X, join(X, complement(X))) 5.53/5.70 = { by axiom 3 (absorption1) } 5.53/5.70 X 5.53/5.70 5.53/5.70 Lemma 16: meet(one, X) = X. 5.53/5.70 Proof: 5.53/5.70 meet(one, X) 5.53/5.70 = { by axiom 7 (commutativity_of_meet) } 5.53/5.70 meet(X, one) 5.53/5.70 = { by lemma 15 } 5.53/5.70 X 5.53/5.70 5.53/5.70 Lemma 17: join(X, one) = one. 5.53/5.70 Proof: 5.53/5.70 join(X, one) 5.53/5.70 = { by axiom 9 (commutativity_of_join) } 5.53/5.70 join(one, X) 5.53/5.70 = { by lemma 16 } 5.53/5.70 join(one, meet(one, X)) 5.53/5.70 = { by axiom 1 (absorption2) } 5.53/5.70 one 5.53/5.70 5.53/5.70 Lemma 18: join(X, join(Y, Z)) = join(Z, join(X, Y)). 5.53/5.70 Proof: 5.53/5.70 join(X, join(Y, Z)) 5.53/5.70 = { by axiom 10 (associativity_of_join) } 5.53/5.70 join(join(X, Y), Z) 5.53/5.70 = { by axiom 9 (commutativity_of_join) } 5.53/5.70 join(Z, join(X, Y)) 5.53/5.70 5.53/5.70 Lemma 19: join(complement(meet(X, Y)), meet(X, join(Y, Z))) = one. 5.53/5.70 Proof: 5.53/5.70 join(complement(meet(X, Y)), meet(X, join(Y, Z))) 5.53/5.70 = { by axiom 7 (commutativity_of_meet) } 5.53/5.70 join(complement(meet(Y, X)), meet(X, join(Y, Z))) 5.53/5.70 = { by axiom 9 (commutativity_of_join) } 5.53/5.70 join(meet(X, join(Y, Z)), complement(meet(Y, X))) 5.53/5.70 = { by lemma 14 } 5.53/5.70 join(meet(X, join(Y, Z)), complement(meet(Y, meet(X, join(Y, Z))))) 5.53/5.70 = { by axiom 7 (commutativity_of_meet) } 5.53/5.70 join(meet(X, join(Y, Z)), complement(meet(meet(X, join(Y, Z)), Y))) 5.53/5.70 = { by axiom 9 (commutativity_of_join) } 5.53/5.70 join(complement(meet(meet(X, join(Y, Z)), Y)), meet(X, join(Y, Z))) 5.53/5.70 = { by axiom 1 (absorption2) } 5.53/5.70 join(complement(meet(meet(X, join(Y, Z)), Y)), join(meet(X, join(Y, Z)), meet(meet(X, join(Y, Z)), Y))) 5.53/5.70 = { by lemma 18 } 5.53/5.70 join(meet(meet(X, join(Y, Z)), Y), join(complement(meet(meet(X, join(Y, Z)), Y)), meet(X, join(Y, Z)))) 5.53/5.70 = { by lemma 18 } 5.53/5.70 join(meet(X, join(Y, Z)), join(meet(meet(X, join(Y, Z)), Y), complement(meet(meet(X, join(Y, Z)), Y)))) 5.53/5.70 = { by axiom 6 (complement_join) } 5.53/5.70 join(meet(X, join(Y, Z)), one) 5.53/5.70 = { by lemma 17 } 5.67/5.88 one 5.67/5.88 5.67/5.88 Goal 1 (prove_distributivity): join(complement(b), complement(a)) = complement(a). 5.67/5.88 Proof: 5.67/5.88 join(complement(b), complement(a)) 5.67/5.88 = { by axiom 9 (commutativity_of_join) } 5.67/5.88 join(complement(a), complement(b)) 5.67/5.88 = { by axiom 5 (ifeq_axiom) } 5.67/5.88 join(complement(a), ifeq(zero, zero, complement(b), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 2 (complement_meet) } 5.67/5.88 join(complement(a), ifeq(meet(meet(b, complement(a)), complement(meet(b, complement(a)))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 12 (associativity_of_meet) } 5.67/5.88 join(complement(a), ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 5 (ifeq_axiom) } 5.67/5.88 join(complement(a), ifeq(one, one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by lemma 19 } 5.67/5.88 join(complement(a), ifeq(join(complement(meet(b, complement(meet(a, join(a, ?))))), meet(b, join(complement(meet(a, join(a, ?))), meet(a, join(join(a, ?), ?))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by lemma 19 } 5.67/5.88 join(complement(a), ifeq(join(complement(meet(b, complement(meet(a, join(a, ?))))), meet(b, one)), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by lemma 15 } 5.67/5.88 join(complement(a), ifeq(join(complement(meet(b, complement(meet(a, join(a, ?))))), b), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 9 (commutativity_of_join) } 5.67/5.88 join(complement(a), ifeq(join(b, complement(meet(b, complement(meet(a, join(a, ?)))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 3 (absorption1) } 5.67/5.88 join(complement(a), ifeq(join(b, complement(meet(b, complement(a)))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by lemma 16 } 5.67/5.88 join(complement(a), ifeq(join(b, meet(one, complement(meet(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by lemma 17 } 5.67/5.88 join(complement(a), ifeq(join(b, meet(join(b, one), complement(meet(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 9 (commutativity_of_join) } 5.67/5.88 join(complement(a), ifeq(join(b, meet(join(one, b), complement(meet(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 6 (complement_join) } 5.67/5.88 join(complement(a), ifeq(join(b, meet(join(join(a, complement(a)), b), complement(meet(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 10 (associativity_of_join) } 5.67/5.88 join(complement(a), ifeq(join(b, meet(join(a, join(complement(a), b)), complement(meet(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 9 (commutativity_of_join) } 5.67/5.88 join(complement(a), ifeq(join(b, meet(join(a, join(b, complement(a))), complement(meet(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 10 (associativity_of_join) } 5.67/5.88 join(complement(a), ifeq(join(b, meet(join(join(a, b), complement(a)), complement(meet(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 9 (commutativity_of_join) } 5.67/5.88 join(complement(a), ifeq(join(b, meet(join(join(b, a), complement(a)), complement(meet(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 4 (prove_distributivity_hypothesis) } 5.67/5.88 join(complement(a), ifeq(join(b, meet(join(join(b, meet(b, a)), complement(a)), complement(meet(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 1 (absorption2) } 5.67/5.88 join(complement(a), ifeq(join(b, meet(join(b, complement(a)), complement(meet(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 9 (commutativity_of_join) } 5.67/5.88 join(complement(a), ifeq(join(b, meet(join(complement(a), b), complement(meet(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 7 (commutativity_of_meet) } 5.67/5.88 join(complement(a), ifeq(join(b, meet(complement(meet(b, complement(a))), join(complement(a), b))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 9 (commutativity_of_join) } 5.67/5.88 join(complement(a), ifeq(join(b, meet(complement(meet(b, complement(a))), join(b, complement(a)))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by lemma 13 } 5.67/5.88 join(complement(a), ifeq(join(b, join(meet(complement(meet(b, complement(a))), join(b, complement(a))), meet(complement(a), meet(complement(meet(b, complement(a))), join(b, complement(a)))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 12 (associativity_of_meet) } 5.67/5.88 join(complement(a), ifeq(join(b, join(meet(complement(meet(b, complement(a))), join(b, complement(a))), meet(meet(complement(a), complement(meet(b, complement(a)))), join(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 7 (commutativity_of_meet) } 5.67/5.88 join(complement(a), ifeq(join(b, join(meet(complement(meet(b, complement(a))), join(b, complement(a))), meet(meet(complement(meet(b, complement(a))), complement(a)), join(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 12 (associativity_of_meet) } 5.67/5.88 join(complement(a), ifeq(join(b, join(meet(complement(meet(b, complement(a))), join(b, complement(a))), meet(complement(meet(b, complement(a))), meet(complement(a), join(b, complement(a)))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 9 (commutativity_of_join) } 5.67/5.88 join(complement(a), ifeq(join(b, join(meet(complement(meet(b, complement(a))), join(b, complement(a))), meet(complement(meet(b, complement(a))), meet(complement(a), join(complement(a), b))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 3 (absorption1) } 5.67/5.88 join(complement(a), ifeq(join(b, join(meet(complement(meet(b, complement(a))), join(b, complement(a))), meet(complement(meet(b, complement(a))), complement(a)))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 7 (commutativity_of_meet) } 5.67/5.88 join(complement(a), ifeq(join(b, join(meet(complement(meet(b, complement(a))), join(b, complement(a))), meet(complement(a), complement(meet(b, complement(a)))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 9 (commutativity_of_join) } 5.67/5.88 join(complement(a), ifeq(join(b, join(meet(complement(a), complement(meet(b, complement(a)))), meet(complement(meet(b, complement(a))), join(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 7 (commutativity_of_meet) } 5.67/5.88 join(complement(a), ifeq(join(b, join(meet(complement(a), complement(meet(b, complement(a)))), meet(join(b, complement(a)), complement(meet(b, complement(a)))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 10 (associativity_of_join) } 5.67/5.88 join(complement(a), ifeq(join(join(b, meet(complement(a), complement(meet(b, complement(a))))), meet(join(b, complement(a)), complement(meet(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 7 (commutativity_of_meet) } 5.67/5.88 join(complement(a), ifeq(join(join(b, meet(complement(a), complement(meet(b, complement(a))))), meet(complement(meet(b, complement(a))), join(b, complement(a)))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by lemma 15 } 5.67/5.88 join(complement(a), ifeq(join(join(b, meet(complement(a), complement(meet(b, complement(a))))), meet(complement(meet(b, complement(a))), join(b, meet(complement(a), one)))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 6 (complement_join) } 5.67/5.88 join(complement(a), ifeq(join(join(b, meet(complement(a), complement(meet(b, complement(a))))), meet(complement(meet(b, complement(a))), join(b, meet(complement(a), join(meet(b, complement(a)), complement(meet(b, complement(a)))))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 9 (commutativity_of_join) } 5.67/5.88 join(complement(a), ifeq(join(join(b, meet(complement(a), complement(meet(b, complement(a))))), meet(complement(meet(b, complement(a))), join(b, meet(complement(a), join(complement(meet(b, complement(a))), meet(b, complement(a))))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by lemma 13 } 5.67/5.88 join(complement(a), ifeq(join(join(b, meet(complement(a), complement(meet(b, complement(a))))), join(meet(complement(meet(b, complement(a))), join(b, meet(complement(a), join(complement(meet(b, complement(a))), meet(b, complement(a)))))), meet(b, meet(complement(meet(b, complement(a))), join(b, meet(complement(a), join(complement(meet(b, complement(a))), meet(b, complement(a))))))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by lemma 14 } 5.67/5.88 join(complement(a), ifeq(join(join(b, meet(complement(a), complement(meet(b, complement(a))))), join(meet(complement(meet(b, complement(a))), join(b, meet(complement(a), join(complement(meet(b, complement(a))), meet(b, complement(a)))))), meet(b, complement(meet(b, complement(a)))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 9 (commutativity_of_join) } 5.67/5.88 join(complement(a), ifeq(join(join(b, meet(complement(a), complement(meet(b, complement(a))))), join(meet(b, complement(meet(b, complement(a)))), meet(complement(meet(b, complement(a))), join(b, meet(complement(a), join(complement(meet(b, complement(a))), meet(b, complement(a)))))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 7 (commutativity_of_meet) } 5.67/5.88 join(complement(a), ifeq(join(join(b, meet(complement(a), complement(meet(b, complement(a))))), join(meet(complement(meet(b, complement(a))), b), meet(complement(meet(b, complement(a))), join(b, meet(complement(a), join(complement(meet(b, complement(a))), meet(b, complement(a)))))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 11 (equation_H8) } 5.67/5.88 join(complement(a), ifeq(join(join(b, meet(complement(a), complement(meet(b, complement(a))))), meet(complement(meet(b, complement(a))), join(b, meet(complement(meet(b, complement(a))), complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 7 (commutativity_of_meet) } 5.67/5.88 join(complement(a), ifeq(join(join(b, meet(complement(a), complement(meet(b, complement(a))))), meet(complement(meet(b, complement(a))), join(b, meet(complement(a), complement(meet(b, complement(a))))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by lemma 13 } 5.67/5.88 join(complement(a), ifeq(join(b, meet(complement(a), complement(meet(b, complement(a))))), one, ifeq(meet(b, meet(complement(a), complement(meet(b, complement(a))))), zero, complement(b), meet(complement(a), complement(meet(b, complement(a))))), meet(complement(a), complement(meet(b, complement(a)))))) 5.67/5.88 = { by axiom 8 (meet_join_complement) } 5.67/5.88 join(complement(a), meet(complement(a), complement(meet(b, complement(a))))) 5.67/5.88 = { by axiom 1 (absorption2) } 5.67/5.88 complement(a) 5.67/5.88 % SZS output end Proof 5.67/5.88 5.67/5.88 RESULT: Unsatisfiable (the axioms are contradictory). 5.67/5.88 EOF