0.07/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.07/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.35	% Computer   : n020.cluster.edu
0.13/0.35	% Model      : x86_64 x86_64
0.13/0.35	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.35	% Memory     : 8042.1875MB
0.13/0.35	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.35	% CPULimit   : 180
0.13/0.35	% DateTime   : Thu Aug 29 13:32:25 EDT 2019
0.13/0.35	% CPUTime    : 
31.84/32.01	% SZS status Unsatisfiable
31.84/32.01	
31.84/32.02	% SZS output start Proof
31.84/32.02	Take the following subset of the input axioms:
31.89/32.05	  fof(absorption1, axiom, ![X, Y]: X=meet(X, join(X, Y))).
31.89/32.05	  fof(absorption2, axiom, ![X, Y]: join(X, meet(X, Y))=X).
31.89/32.05	  fof(associativity_of_join, axiom, ![X, Y, Z]: join(join(X, Y), Z)=join(X, join(Y, Z))).
31.89/32.05	  fof(associativity_of_meet, axiom, ![X, Y, Z]: meet(X, meet(Y, Z))=meet(meet(X, Y), Z)).
31.89/32.05	  fof(commutativity_of_join, axiom, ![X, Y]: join(Y, X)=join(X, Y)).
31.89/32.05	  fof(commutativity_of_meet, axiom, ![X, Y]: meet(Y, X)=meet(X, Y)).
31.89/32.05	  fof(complement_join, axiom, ![X]: one=join(X, complement(X))).
31.89/32.05	  fof(complement_meet, axiom, ![X]: meet(X, complement(X))=zero).
31.89/32.05	  fof(equation_H6, axiom, ![X, Y, Z]: meet(X, join(Y, meet(X, Z)))=meet(X, join(meet(X, join(Y, meet(X, Z))), meet(Z, join(X, Y))))).
31.89/32.05	  fof(idempotence_of_join, axiom, ![X]: X=join(X, X)).
31.89/32.05	  fof(ifeq_axiom, axiom, ![A, B, C]: ifeq(A, A, B, C)=B).
31.89/32.05	  fof(meet_join_complement, axiom, ![X, Y]: Y=ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y)).
31.89/32.05	  fof(prove_distributivity, negated_conjecture, complement(a)!=join(complement(b), complement(a))).
31.89/32.05	  fof(prove_distributivity_hypothesis, hypothesis, a=meet(b, a)).
31.89/32.05	
31.89/32.05	Now clausify the problem and encode Horn clauses using encoding 3 of
31.89/32.05	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
31.89/32.05	We repeatedly replace C & s=t => u=v by the two clauses:
31.89/32.05	  fresh(y, y, x1...xn) = u
31.89/32.05	  C => fresh(s, t, x1...xn) = v
31.89/32.05	where fresh is a fresh function symbol and x1..xn are the free
31.89/32.05	variables of u and v.
31.89/32.05	A predicate p(X) is encoded as p(X)=true (this is sound, because the
31.89/32.05	input problem has no model of domain size 1).
31.89/32.05	
31.89/32.05	The encoding turns the above axioms into the following unit equations and goals:
31.89/32.05	
31.89/32.05	Axiom 1 (complement_join): one = join(X, complement(X)).
31.89/32.05	Axiom 2 (absorption2): join(X, meet(X, Y)) = X.
31.89/32.05	Axiom 3 (complement_meet): meet(X, complement(X)) = zero.
31.89/32.05	Axiom 4 (associativity_of_meet): meet(X, meet(Y, Z)) = meet(meet(X, Y), Z).
31.89/32.05	Axiom 5 (equation_H6): meet(X, join(Y, meet(X, Z))) = meet(X, join(meet(X, join(Y, meet(X, Z))), meet(Z, join(X, Y)))).
31.89/32.05	Axiom 6 (meet_join_complement): X = ifeq(join(Y, X), one, ifeq(meet(Y, X), zero, complement(Y), X), X).
31.89/32.05	Axiom 7 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
31.89/32.05	Axiom 8 (absorption1): X = meet(X, join(X, Y)).
31.89/32.05	Axiom 9 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
31.89/32.05	Axiom 10 (commutativity_of_join): join(X, Y) = join(Y, X).
31.89/32.05	Axiom 11 (prove_distributivity_hypothesis): a = meet(b, a).
31.89/32.05	Axiom 12 (idempotence_of_join): X = join(X, X).
31.99/32.21	Axiom 13 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
31.99/32.21	
31.99/32.21	Lemma 14: ifeq(join(Y, X), one, ifeq(meet(X, Y), zero, complement(Y), X), X) = X.
31.99/32.21	Proof:
31.99/32.21	  ifeq(join(Y, X), one, ifeq(meet(X, Y), zero, complement(Y), X), X)
31.99/32.21	= { by axiom 9 (commutativity_of_meet) }
31.99/32.21	  ifeq(join(Y, X), one, ifeq(meet(Y, X), zero, complement(Y), X), X)
31.99/32.21	= { by axiom 6 (meet_join_complement) }
31.99/32.21	  X
31.99/32.21	
31.99/32.21	Lemma 15: complement(complement(X)) = X.
31.99/32.21	Proof:
31.99/32.21	  complement(complement(X))
31.99/32.21	= { by axiom 7 (ifeq_axiom) }
31.99/32.21	  ifeq(one, one, complement(complement(X)), X)
31.99/32.21	= { by axiom 1 (complement_join) }
31.99/32.21	  ifeq(join(X, complement(X)), one, complement(complement(X)), X)
31.99/32.21	= { by axiom 10 (commutativity_of_join) }
31.99/32.21	  ifeq(join(complement(X), X), one, complement(complement(X)), X)
31.99/32.21	= { by axiom 7 (ifeq_axiom) }
31.99/32.21	  ifeq(join(complement(X), X), one, ifeq(zero, zero, complement(complement(X)), X), X)
31.99/32.21	= { by axiom 3 (complement_meet) }
31.99/32.21	  ifeq(join(complement(X), X), one, ifeq(meet(X, complement(X)), zero, complement(complement(X)), X), X)
31.99/32.21	= { by lemma 14 }
31.99/32.21	  X
31.99/32.21	
31.99/32.21	Lemma 16: meet(a, b) = a.
31.99/32.21	Proof:
31.99/32.21	  meet(a, b)
31.99/32.21	= { by axiom 9 (commutativity_of_meet) }
31.99/32.21	  meet(b, a)
31.99/32.21	= { by axiom 11 (prove_distributivity_hypothesis) }
31.99/32.21	  a
31.99/32.21	
31.99/32.21	Lemma 17: join(X, meet(Y, X)) = X.
31.99/32.21	Proof:
31.99/32.21	  join(X, meet(Y, X))
31.99/32.21	= { by axiom 9 (commutativity_of_meet) }
31.99/32.21	  join(X, meet(X, Y))
31.99/32.21	= { by axiom 2 (absorption2) }
31.99/32.21	  X
31.99/32.21	
31.99/32.21	Lemma 18: meet(X, join(Y, X)) = X.
31.99/32.21	Proof:
31.99/32.21	  meet(X, join(Y, X))
31.99/32.21	= { by axiom 10 (commutativity_of_join) }
31.99/32.21	  meet(X, join(X, Y))
31.99/32.21	= { by axiom 8 (absorption1) }
31.99/32.21	  X
31.99/32.21	
31.99/32.21	Lemma 19: meet(join(X, Y), join(X, meet(Y, Z))) = join(X, meet(Y, Z)).
31.99/32.21	Proof:
31.99/32.21	  meet(join(X, Y), join(X, meet(Y, Z)))
31.99/32.21	= { by axiom 10 (commutativity_of_join) }
31.99/32.21	  meet(join(Y, X), join(X, meet(Y, Z)))
31.99/32.21	= { by axiom 9 (commutativity_of_meet) }
31.99/32.21	  meet(join(X, meet(Y, Z)), join(Y, X))
31.99/32.21	= { by axiom 2 (absorption2) }
31.99/32.21	  meet(join(X, meet(Y, Z)), join(join(Y, meet(Y, Z)), X))
31.99/32.21	= { by axiom 13 (associativity_of_join) }
31.99/32.21	  meet(join(X, meet(Y, Z)), join(Y, join(meet(Y, Z), X)))
31.99/32.21	= { by axiom 10 (commutativity_of_join) }
31.99/32.21	  meet(join(X, meet(Y, Z)), join(Y, join(X, meet(Y, Z))))
31.99/32.21	= { by lemma 18 }
31.99/32.21	  join(X, meet(Y, Z))
31.99/32.21	
31.99/32.21	Lemma 20: join(X, join(Y, Z)) = join(Z, join(X, Y)).
31.99/32.21	Proof:
31.99/32.21	  join(X, join(Y, Z))
31.99/32.21	= { by axiom 13 (associativity_of_join) }
31.99/32.21	  join(join(X, Y), Z)
31.99/32.21	= { by axiom 10 (commutativity_of_join) }
31.99/32.21	  join(Z, join(X, Y))
31.99/32.21	
31.99/32.21	Lemma 21: meet(X, one) = X.
31.99/32.21	Proof:
31.99/32.21	  meet(X, one)
31.99/32.21	= { by axiom 1 (complement_join) }
31.99/32.21	  meet(X, join(X, complement(X)))
31.99/32.21	= { by axiom 8 (absorption1) }
31.99/32.21	  X
31.99/32.21	
31.99/32.21	Lemma 22: join(X, one) = one.
31.99/32.21	Proof:
31.99/32.21	  join(X, one)
31.99/32.21	= { by axiom 10 (commutativity_of_join) }
31.99/32.21	  join(one, X)
31.99/32.21	= { by lemma 21 }
31.99/32.21	  join(one, meet(X, one))
31.99/32.21	= { by axiom 9 (commutativity_of_meet) }
31.99/32.21	  join(one, meet(one, X))
31.99/32.21	= { by axiom 2 (absorption2) }
31.99/32.21	  one
31.99/32.21	
31.99/32.21	Lemma 23: join(X, zero) = X.
31.99/32.21	Proof:
31.99/32.21	  join(X, zero)
31.99/32.21	= { by axiom 3 (complement_meet) }
31.99/32.21	  join(X, meet(X, complement(X)))
31.99/32.21	= { by axiom 2 (absorption2) }
31.99/32.21	  X
31.99/32.21	
31.99/32.21	Lemma 24: join(zero, X) = X.
31.99/32.21	Proof:
31.99/32.21	  join(zero, X)
31.99/32.21	= { by axiom 10 (commutativity_of_join) }
31.99/32.21	  join(X, zero)
31.99/32.21	= { by lemma 23 }
31.99/32.21	  X
31.99/32.21	
31.99/32.21	Lemma 25: complement(zero) = one.
31.99/32.21	Proof:
31.99/32.21	  complement(zero)
31.99/32.21	= { by lemma 24 }
31.99/32.21	  join(zero, complement(zero))
31.99/32.21	= { by axiom 1 (complement_join) }
31.99/32.21	  one
31.99/32.21	
31.99/32.21	Lemma 26: meet(a, meet(b, X)) = meet(X, a).
31.99/32.21	Proof:
31.99/32.21	  meet(a, meet(b, X))
31.99/32.21	= { by axiom 9 (commutativity_of_meet) }
31.99/32.21	  meet(a, meet(X, b))
31.99/32.21	= { by axiom 9 (commutativity_of_meet) }
31.99/32.21	  meet(a, meet(b, X))
31.99/32.21	= { by axiom 4 (associativity_of_meet) }
31.99/32.21	  meet(meet(a, b), X)
31.99/32.21	= { by lemma 16 }
31.99/32.21	  meet(a, X)
31.99/32.21	= { by axiom 9 (commutativity_of_meet) }
31.99/32.21	  meet(X, a)
31.99/32.21	
31.99/32.21	Lemma 27: join(b, meet(X, a)) = b.
31.99/32.21	Proof:
31.99/32.21	  join(b, meet(X, a))
31.99/32.21	= { by lemma 26 }
31.99/32.21	  join(b, meet(a, meet(b, X)))
31.99/32.21	= { by axiom 9 (commutativity_of_meet) }
31.99/32.21	  join(b, meet(a, meet(X, b)))
31.99/32.21	= { by axiom 4 (associativity_of_meet) }
31.99/32.21	  join(b, meet(meet(a, X), b))
31.99/32.21	= { by lemma 17 }
31.99/32.21	  b
31.99/32.21	
31.99/32.21	Lemma 28: meet(X, meet(join(X, Z), Y)) = meet(X, Y).
31.99/32.21	Proof:
31.99/32.21	  meet(X, meet(join(X, Z), Y))
31.99/32.21	= { by axiom 9 (commutativity_of_meet) }
31.99/32.21	  meet(X, meet(Y, join(X, Z)))
31.99/32.21	= { by axiom 9 (commutativity_of_meet) }
31.99/32.21	  meet(X, meet(join(X, Z), Y))
31.99/32.21	= { by axiom 4 (associativity_of_meet) }
31.99/32.21	  meet(meet(X, join(X, Z)), Y)
31.99/32.21	= { by axiom 8 (absorption1) }
31.99/32.21	  meet(X, Y)
31.99/32.21	
31.99/32.21	Lemma 29: meet(X, meet(Y, join(Z, X))) = meet(X, Y).
31.99/32.21	Proof:
31.99/32.21	  meet(X, meet(Y, join(Z, X)))
31.99/32.21	= { by axiom 9 (commutativity_of_meet) }
31.99/32.21	  meet(X, meet(join(Z, X), Y))
31.99/32.21	= { by axiom 4 (associativity_of_meet) }
31.99/32.21	  meet(meet(X, join(Z, X)), Y)
31.99/32.21	= { by axiom 9 (commutativity_of_meet) }
31.99/32.21	  meet(Y, meet(X, join(Z, X)))
31.99/32.21	= { by lemma 18 }
31.99/32.21	  meet(Y, X)
31.99/32.21	= { by axiom 9 (commutativity_of_meet) }
32.20/32.38	  meet(X, Y)
32.20/32.38	
32.20/32.38	Lemma 30: join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))) = complement(a).
32.20/32.38	Proof:
32.20/32.38	  join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))
32.20/32.38	= { by axiom 6 (meet_join_complement) }
32.20/32.38	  ifeq(join(a, join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), one, ifeq(meet(a, join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 13 (associativity_of_join) }
32.20/32.38	  ifeq(join(join(a, meet(b, complement(a))), complement(join(a, meet(b, complement(a))))), one, ifeq(meet(a, join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 1 (complement_join) }
32.20/32.38	  ifeq(one, one, ifeq(meet(a, join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by lemma 15 }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 9 (commutativity_of_meet) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(meet(complement(a), b), complement(join(a, meet(b, complement(a)))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 9 (commutativity_of_meet) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(meet(complement(a), b), complement(join(a, meet(complement(a), b))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 10 (commutativity_of_join) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(complement(join(a, meet(complement(a), b))), meet(complement(a), b))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 10 (commutativity_of_join) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(complement(join(meet(complement(a), b), a)), meet(complement(a), b))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by lemma 15 }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), b))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by lemma 27 }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), join(b, meet(complement(a), a))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 5 (equation_H6) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), join(meet(complement(a), join(b, meet(complement(a), a))), meet(a, join(complement(a), b)))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by lemma 27 }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), join(meet(complement(a), b), meet(a, join(complement(a), b)))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 10 (commutativity_of_join) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), join(meet(complement(a), b), meet(a, join(b, complement(a))))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 9 (commutativity_of_meet) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), join(meet(complement(a), b), meet(join(b, complement(a)), a))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by lemma 26 }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), join(meet(complement(a), b), meet(a, meet(b, join(b, complement(a)))))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 8 (absorption1) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), join(meet(complement(a), b), meet(a, b))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by lemma 16 }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), join(meet(complement(a), b), a)))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by lemma 15 }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), join(meet(complement(a), b), complement(complement(a)))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by lemma 29 }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), join(meet(complement(a), b), complement(complement(a))))), join(meet(complement(a), b), complement(complement(a))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 9 (commutativity_of_meet) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), join(meet(complement(a), b), complement(complement(a))))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 9 (commutativity_of_meet) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(join(meet(complement(a), b), complement(complement(a))), complement(a))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 10 (commutativity_of_join) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(meet(join(meet(complement(a), b), complement(complement(a))), complement(a)), complement(join(meet(complement(a), b), complement(complement(a))))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by lemma 21 }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(meet(join(meet(complement(a), b), complement(complement(a))), complement(a)), meet(complement(join(meet(complement(a), b), complement(complement(a)))), one)))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by lemma 22 }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(meet(join(meet(complement(a), b), complement(complement(a))), complement(a)), meet(complement(join(meet(complement(a), b), complement(complement(a)))), join(meet(complement(a), b), one))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 10 (commutativity_of_join) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(meet(join(meet(complement(a), b), complement(complement(a))), complement(a)), meet(complement(join(meet(complement(a), b), complement(complement(a)))), join(one, meet(complement(a), b)))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 1 (complement_join) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(meet(join(meet(complement(a), b), complement(complement(a))), complement(a)), meet(complement(join(meet(complement(a), b), complement(complement(a)))), join(join(complement(a), complement(complement(a))), meet(complement(a), b)))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 13 (associativity_of_join) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(meet(join(meet(complement(a), b), complement(complement(a))), complement(a)), meet(complement(join(meet(complement(a), b), complement(complement(a)))), join(complement(a), join(complement(complement(a)), meet(complement(a), b))))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 10 (commutativity_of_join) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(meet(join(meet(complement(a), b), complement(complement(a))), complement(a)), meet(complement(join(meet(complement(a), b), complement(complement(a)))), join(complement(a), join(meet(complement(a), b), complement(complement(a)))))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 10 (commutativity_of_join) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(meet(join(meet(complement(a), b), complement(complement(a))), complement(a)), meet(complement(join(meet(complement(a), b), complement(complement(a)))), join(join(meet(complement(a), b), complement(complement(a))), complement(a)))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 10 (commutativity_of_join) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(meet(complement(join(meet(complement(a), b), complement(complement(a)))), join(join(meet(complement(a), b), complement(complement(a))), complement(a))), meet(join(meet(complement(a), b), complement(complement(a))), complement(a))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by lemma 23 }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(meet(complement(join(meet(complement(a), b), complement(complement(a)))), join(join(meet(complement(a), b), complement(complement(a))), complement(a))), meet(join(meet(complement(a), b), complement(complement(a))), join(complement(a), zero))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 3 (complement_meet) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(meet(complement(join(meet(complement(a), b), complement(complement(a)))), join(join(meet(complement(a), b), complement(complement(a))), complement(a))), meet(join(meet(complement(a), b), complement(complement(a))), join(complement(a), meet(join(meet(complement(a), b), complement(complement(a))), complement(join(meet(complement(a), b), complement(complement(a)))))))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 10 (commutativity_of_join) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(meet(join(meet(complement(a), b), complement(complement(a))), join(complement(a), meet(join(meet(complement(a), b), complement(complement(a))), complement(join(meet(complement(a), b), complement(complement(a))))))), meet(complement(join(meet(complement(a), b), complement(complement(a)))), join(join(meet(complement(a), b), complement(complement(a))), complement(a)))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 5 (equation_H6) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(complement(a), meet(join(meet(complement(a), b), complement(complement(a))), complement(join(meet(complement(a), b), complement(complement(a)))))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 3 (complement_meet) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(complement(a), zero))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by lemma 23 }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), complement(a))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 9 (commutativity_of_meet) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(complement(a), join(meet(complement(a), b), complement(complement(a))))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by lemma 29 }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(complement(a)), complement(a)), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 9 (commutativity_of_meet) }
32.20/32.38	  ifeq(one, one, ifeq(meet(complement(a), complement(complement(a))), zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 3 (complement_meet) }
32.20/32.38	  ifeq(one, one, ifeq(zero, zero, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 7 (ifeq_axiom) }
32.20/32.38	  ifeq(one, one, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.20/32.38	= { by axiom 7 (ifeq_axiom) }
32.20/32.39	  complement(a)
32.20/32.39	
32.20/32.39	Lemma 31: meet(b, complement(join(a, meet(b, complement(a))))) = zero.
32.20/32.39	Proof:
32.20/32.39	  meet(b, complement(join(a, meet(b, complement(a)))))
32.20/32.39	= { by axiom 9 (commutativity_of_meet) }
32.20/32.39	  meet(complement(join(a, meet(b, complement(a)))), b)
32.20/32.39	= { by lemma 28 }
32.20/32.39	  meet(complement(join(a, meet(b, complement(a)))), meet(join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))), b))
32.20/32.39	= { by axiom 8 (absorption1) }
32.20/32.39	  meet(complement(join(a, meet(b, complement(a)))), meet(meet(join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))), b), join(meet(join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))), b), meet(?, a))))
32.20/32.39	= { by axiom 4 (associativity_of_meet) }
32.20/32.39	  meet(complement(join(a, meet(b, complement(a)))), meet(join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))), meet(b, join(meet(join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))), b), meet(?, a)))))
32.20/32.39	= { by axiom 10 (commutativity_of_join) }
32.20/32.39	  meet(complement(join(a, meet(b, complement(a)))), meet(join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))), meet(b, join(meet(?, a), meet(join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))), b)))))
32.20/32.39	= { by lemma 28 }
32.20/32.39	  meet(complement(join(a, meet(b, complement(a)))), meet(b, join(meet(?, a), meet(join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))), b))))
32.20/32.39	= { by lemma 27 }
32.20/32.39	  meet(complement(join(a, meet(b, complement(a)))), meet(join(b, meet(?, a)), join(meet(?, a), meet(join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))), b))))
32.20/32.39	= { by axiom 10 (commutativity_of_join) }
32.20/32.39	  meet(complement(join(a, meet(b, complement(a)))), meet(join(meet(?, a), b), join(meet(?, a), meet(join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))), b))))
32.20/32.39	= { by axiom 9 (commutativity_of_meet) }
32.20/32.39	  meet(complement(join(a, meet(b, complement(a)))), meet(join(meet(?, a), b), join(meet(?, a), meet(b, join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a)))))))
32.20/32.39	= { by lemma 19 }
32.20/32.39	  meet(complement(join(a, meet(b, complement(a)))), join(meet(?, a), meet(b, join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))))))
32.20/32.39	= { by axiom 10 (commutativity_of_join) }
32.20/32.39	  meet(complement(join(a, meet(b, complement(a)))), join(meet(?, a), meet(b, join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))))
32.20/32.39	= { by lemma 30 }
32.20/32.39	  meet(complement(join(a, meet(b, complement(a)))), join(meet(?, a), meet(b, complement(a))))
32.20/32.39	= { by axiom 10 (commutativity_of_join) }
32.20/32.39	  meet(complement(join(a, meet(b, complement(a)))), join(meet(b, complement(a)), meet(?, a)))
32.20/32.39	= { by axiom 9 (commutativity_of_meet) }
32.20/32.39	  meet(join(meet(b, complement(a)), meet(?, a)), complement(join(a, meet(b, complement(a)))))
32.20/32.39	= { by axiom 10 (commutativity_of_join) }
32.20/32.39	  meet(join(meet(b, complement(a)), meet(?, a)), complement(join(meet(b, complement(a)), a)))
32.20/32.39	= { by lemma 17 }
32.20/32.39	  meet(join(meet(b, complement(a)), meet(?, a)), complement(join(meet(b, complement(a)), join(a, meet(?, a)))))
32.20/32.39	= { by axiom 13 (associativity_of_join) }
32.20/32.39	  meet(join(meet(b, complement(a)), meet(?, a)), complement(join(join(meet(b, complement(a)), a), meet(?, a))))
32.20/32.39	= { by axiom 10 (commutativity_of_join) }
32.20/32.39	  meet(join(meet(b, complement(a)), meet(?, a)), complement(join(join(a, meet(b, complement(a))), meet(?, a))))
32.20/32.39	= { by axiom 13 (associativity_of_join) }
32.20/32.39	  meet(join(meet(b, complement(a)), meet(?, a)), complement(join(a, join(meet(b, complement(a)), meet(?, a)))))
32.20/32.39	= { by axiom 10 (commutativity_of_join) }
32.20/32.39	  meet(join(meet(b, complement(a)), meet(?, a)), complement(join(join(meet(b, complement(a)), meet(?, a)), a)))
32.20/32.39	= { by lemma 28 }
32.20/32.39	  meet(join(meet(b, complement(a)), meet(?, a)), meet(join(join(meet(b, complement(a)), meet(?, a)), a), complement(join(join(meet(b, complement(a)), meet(?, a)), a))))
32.20/32.39	= { by axiom 3 (complement_meet) }
32.20/32.39	  meet(join(meet(b, complement(a)), meet(?, a)), zero)
32.20/32.39	= { by axiom 9 (commutativity_of_meet) }
32.20/32.39	  meet(zero, join(meet(b, complement(a)), meet(?, a)))
32.20/32.39	= { by lemma 24 }
32.20/32.39	  join(zero, meet(zero, join(meet(b, complement(a)), meet(?, a))))
32.20/32.39	= { by axiom 2 (absorption2) }
32.26/32.42	  zero
32.26/32.42	
32.26/32.42	Goal 1 (prove_distributivity): complement(a) = join(complement(b), complement(a)).
32.26/32.42	Proof:
32.26/32.42	  complement(a)
32.26/32.42	= { by axiom 12 (idempotence_of_join) }
32.26/32.42	  join(complement(a), complement(a))
32.26/32.42	= { by lemma 30 }
32.26/32.42	  join(complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
32.26/32.42	= { by lemma 20 }
32.26/32.42	  join(meet(b, complement(a)), join(complement(join(a, meet(b, complement(a)))), complement(a)))
32.26/32.42	= { by lemma 20 }
32.26/32.42	  join(complement(join(a, meet(b, complement(a)))), join(complement(a), meet(b, complement(a))))
32.26/32.42	= { by lemma 17 }
32.26/32.42	  join(complement(join(a, meet(b, complement(a)))), complement(a))
32.26/32.42	= { by axiom 10 (commutativity_of_join) }
32.26/32.42	  join(complement(a), complement(join(a, meet(b, complement(a)))))
32.26/32.42	= { by lemma 15 }
32.26/32.42	  join(complement(a), complement(complement(complement(join(a, meet(b, complement(a)))))))
32.26/32.42	= { by axiom 7 (ifeq_axiom) }
32.26/32.42	  join(complement(a), complement(ifeq(one, one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by lemma 22 }
32.26/32.42	  join(complement(a), complement(ifeq(join(b, one), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by axiom 1 (complement_join) }
32.26/32.42	  join(complement(a), complement(ifeq(join(b, join(meet(b, join(a, meet(complement(a), b))), complement(meet(b, join(a, meet(complement(a), b)))))), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by lemma 20 }
32.26/32.42	  join(complement(a), complement(ifeq(join(meet(b, join(a, meet(complement(a), b))), join(complement(meet(b, join(a, meet(complement(a), b)))), b)), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by lemma 20 }
32.26/32.42	  join(complement(a), complement(ifeq(join(complement(meet(b, join(a, meet(complement(a), b)))), join(b, meet(b, join(a, meet(complement(a), b))))), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by axiom 2 (absorption2) }
32.26/32.42	  join(complement(a), complement(ifeq(join(complement(meet(b, join(a, meet(complement(a), b)))), b), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by axiom 10 (commutativity_of_join) }
32.26/32.42	  join(complement(a), complement(ifeq(join(b, complement(meet(b, join(a, meet(complement(a), b))))), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by lemma 17 }
32.26/32.42	  join(complement(a), complement(ifeq(join(b, complement(meet(join(b, meet(a, b)), join(a, meet(complement(a), b))))), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by lemma 16 }
32.26/32.42	  join(complement(a), complement(ifeq(join(b, complement(meet(join(b, a), join(a, meet(complement(a), b))))), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by axiom 10 (commutativity_of_join) }
32.26/32.42	  join(complement(a), complement(ifeq(join(b, complement(meet(join(a, b), join(a, meet(complement(a), b))))), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by axiom 9 (commutativity_of_meet) }
32.26/32.42	  join(complement(a), complement(ifeq(join(b, complement(meet(join(a, b), join(a, meet(b, complement(a)))))), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by lemma 19 }
32.26/32.42	  join(complement(a), complement(ifeq(join(b, complement(join(a, meet(b, complement(a))))), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by lemma 21 }
32.26/32.42	  join(complement(a), complement(ifeq(join(b, meet(complement(join(a, meet(b, complement(a)))), one)), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by lemma 25 }
32.26/32.42	  join(complement(a), complement(ifeq(join(b, meet(complement(join(a, meet(b, complement(a)))), complement(zero))), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by lemma 31 }
32.26/32.42	  join(complement(a), complement(ifeq(join(b, meet(complement(join(a, meet(b, complement(a)))), complement(meet(b, complement(join(a, meet(b, complement(a)))))))), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by axiom 10 (commutativity_of_join) }
32.26/32.42	  join(complement(a), complement(ifeq(join(meet(complement(join(a, meet(b, complement(a)))), complement(meet(b, complement(join(a, meet(b, complement(a))))))), b), one, complement(complement(join(a, meet(b, complement(a))))), b)))
32.26/32.42	= { by lemma 21 }
32.26/32.42	  join(complement(a), complement(ifeq(join(meet(complement(join(a, meet(b, complement(a)))), complement(meet(b, complement(join(a, meet(b, complement(a))))))), b), one, complement(meet(complement(join(a, meet(b, complement(a)))), one)), b)))
32.26/32.42	= { by lemma 25 }
32.26/32.42	  join(complement(a), complement(ifeq(join(meet(complement(join(a, meet(b, complement(a)))), complement(meet(b, complement(join(a, meet(b, complement(a))))))), b), one, complement(meet(complement(join(a, meet(b, complement(a)))), complement(zero))), b)))
32.26/32.42	= { by lemma 31 }
32.26/32.42	  join(complement(a), complement(ifeq(join(meet(complement(join(a, meet(b, complement(a)))), complement(meet(b, complement(join(a, meet(b, complement(a))))))), b), one, complement(meet(complement(join(a, meet(b, complement(a)))), complement(meet(b, complement(join(a, meet(b, complement(a)))))))), b)))
32.26/32.42	= { by axiom 7 (ifeq_axiom) }
32.26/32.42	  join(complement(a), complement(ifeq(join(meet(complement(join(a, meet(b, complement(a)))), complement(meet(b, complement(join(a, meet(b, complement(a))))))), b), one, ifeq(zero, zero, complement(meet(complement(join(a, meet(b, complement(a)))), complement(meet(b, complement(join(a, meet(b, complement(a)))))))), b), b)))
32.26/32.42	= { by axiom 3 (complement_meet) }
32.26/32.42	  join(complement(a), complement(ifeq(join(meet(complement(join(a, meet(b, complement(a)))), complement(meet(b, complement(join(a, meet(b, complement(a))))))), b), one, ifeq(meet(meet(b, complement(join(a, meet(b, complement(a))))), complement(meet(b, complement(join(a, meet(b, complement(a))))))), zero, complement(meet(complement(join(a, meet(b, complement(a)))), complement(meet(b, complement(join(a, meet(b, complement(a)))))))), b), b)))
32.26/32.42	= { by axiom 4 (associativity_of_meet) }
32.26/32.42	  join(complement(a), complement(ifeq(join(meet(complement(join(a, meet(b, complement(a)))), complement(meet(b, complement(join(a, meet(b, complement(a))))))), b), one, ifeq(meet(b, meet(complement(join(a, meet(b, complement(a)))), complement(meet(b, complement(join(a, meet(b, complement(a)))))))), zero, complement(meet(complement(join(a, meet(b, complement(a)))), complement(meet(b, complement(join(a, meet(b, complement(a)))))))), b), b)))
32.26/32.42	= { by lemma 14 }
32.26/32.42	  join(complement(a), complement(b))
32.26/32.42	= { by axiom 10 (commutativity_of_join) }
32.26/32.42	  join(complement(b), complement(a))
32.26/32.42	% SZS output end Proof
32.26/32.42	
32.26/32.42	RESULT: Unsatisfiable (the axioms are contradictory).
32.26/32.43	EOF