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   : n010.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
0.13/0.33	% CPULimit   : 180
0.13/0.33	% DateTime   : Thu Aug 29 13:10:51 EDT 2019
0.13/0.33	% CPUTime    : 
38.46/38.63	% SZS status Unsatisfiable
38.46/38.63	
38.46/38.63	% SZS output start Proof
38.46/38.63	Take the following subset of the input axioms:
38.46/38.65	  fof(absorption1, axiom, ![X, Y]: X=meet(X, join(X, Y))).
38.46/38.65	  fof(absorption2, axiom, ![X, Y]: X=join(X, meet(X, Y))).
38.46/38.65	  fof(associativity_of_join, axiom, ![X, Y, Z]: join(join(X, Y), Z)=join(X, join(Y, Z))).
38.46/38.65	  fof(associativity_of_meet, axiom, ![X, Y, Z]: meet(X, meet(Y, Z))=meet(meet(X, Y), Z)).
38.46/38.65	  fof(commutativity_of_join, axiom, ![X, Y]: join(X, Y)=join(Y, X)).
38.46/38.65	  fof(commutativity_of_meet, axiom, ![X, Y]: meet(X, Y)=meet(Y, X)).
38.46/38.65	  fof(complement_join, axiom, ![X]: one=join(X, complement(X))).
38.46/38.65	  fof(complement_meet, axiom, ![X]: meet(X, complement(X))=zero).
38.46/38.65	  fof(equation_H18, axiom, ![X, Y, Z]: join(meet(X, Y), meet(X, Z))=meet(X, join(meet(X, Y), join(meet(X, Z), meet(Y, join(X, Z)))))).
38.46/38.65	  fof(idempotence_of_join, axiom, ![X]: X=join(X, X)).
38.46/38.65	  fof(ifeq_axiom, axiom, ![A, B, C]: ifeq(A, A, B, C)=B).
38.46/38.65	  fof(meet_join_complement, axiom, ![X, Y]: ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y)=Y).
38.46/38.65	  fof(prove_distributivity, negated_conjecture, complement(a)!=join(complement(b), complement(a))).
38.46/38.65	  fof(prove_distributivity_hypothesis, hypothesis, meet(b, a)=a).
38.46/38.65	
38.46/38.65	Now clausify the problem and encode Horn clauses using encoding 3 of
38.46/38.65	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
38.46/38.65	We repeatedly replace C & s=t => u=v by the two clauses:
38.46/38.65	  fresh(y, y, x1...xn) = u
38.46/38.65	  C => fresh(s, t, x1...xn) = v
38.46/38.65	where fresh is a fresh function symbol and x1..xn are the free
38.46/38.65	variables of u and v.
38.46/38.65	A predicate p(X) is encoded as p(X)=true (this is sound, because the
38.46/38.65	input problem has no model of domain size 1).
38.46/38.65	
38.46/38.65	The encoding turns the above axioms into the following unit equations and goals:
38.46/38.65	
38.46/38.65	Axiom 1 (meet_join_complement): ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y) = Y.
38.46/38.65	Axiom 2 (prove_distributivity_hypothesis): meet(b, a) = a.
38.46/38.65	Axiom 3 (absorption1): X = meet(X, join(X, Y)).
38.46/38.65	Axiom 4 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
38.46/38.65	Axiom 5 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
38.46/38.65	Axiom 6 (commutativity_of_join): join(X, Y) = join(Y, X).
38.46/38.65	Axiom 7 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
38.46/38.65	Axiom 8 (equation_H18): join(meet(X, Y), meet(X, Z)) = meet(X, join(meet(X, Y), join(meet(X, Z), meet(Y, join(X, Z))))).
38.46/38.65	Axiom 9 (idempotence_of_join): X = join(X, X).
38.46/38.65	Axiom 10 (absorption2): X = join(X, meet(X, Y)).
38.46/38.65	Axiom 11 (complement_join): one = join(X, complement(X)).
38.46/38.65	Axiom 12 (associativity_of_meet): meet(X, meet(Y, Z)) = meet(meet(X, Y), Z).
38.54/38.76	Axiom 13 (complement_meet): meet(X, complement(X)) = zero.
38.54/38.76	
38.54/38.76	Lemma 14: ifeq(join(Y, X), one, ifeq(meet(X, Y), zero, complement(Y), X), X) = X.
38.54/38.76	Proof:
38.54/38.76	  ifeq(join(Y, X), one, ifeq(meet(X, Y), zero, complement(Y), X), X)
38.54/38.76	= { by axiom 5 (commutativity_of_meet) }
38.54/38.76	  ifeq(join(Y, X), one, ifeq(meet(Y, X), zero, complement(Y), X), X)
38.54/38.76	= { by axiom 1 (meet_join_complement) }
38.54/38.76	  X
38.54/38.76	
38.54/38.76	Lemma 15: complement(complement(X)) = X.
38.54/38.76	Proof:
38.54/38.76	  complement(complement(X))
38.54/38.76	= { by axiom 7 (ifeq_axiom) }
38.54/38.76	  ifeq(one, one, complement(complement(X)), X)
38.54/38.76	= { by axiom 11 (complement_join) }
38.54/38.76	  ifeq(join(X, complement(X)), one, complement(complement(X)), X)
38.54/38.76	= { by axiom 6 (commutativity_of_join) }
38.54/38.76	  ifeq(join(complement(X), X), one, complement(complement(X)), X)
38.54/38.76	= { by axiom 7 (ifeq_axiom) }
38.54/38.76	  ifeq(join(complement(X), X), one, ifeq(zero, zero, complement(complement(X)), X), X)
38.54/38.76	= { by axiom 13 (complement_meet) }
38.54/38.76	  ifeq(join(complement(X), X), one, ifeq(meet(X, complement(X)), zero, complement(complement(X)), X), X)
38.54/38.76	= { by lemma 14 }
38.54/38.76	  X
38.54/38.76	
38.54/38.76	Lemma 16: join(X, join(Z, meet(X, Y))) = join(X, Z).
38.54/38.76	Proof:
38.54/38.76	  join(X, join(Z, meet(X, Y)))
38.54/38.76	= { by axiom 6 (commutativity_of_join) }
38.54/38.76	  join(X, join(meet(X, Y), Z))
38.54/38.76	= { by axiom 4 (associativity_of_join) }
38.54/38.76	  join(join(X, meet(X, Y)), Z)
38.54/38.76	= { by axiom 10 (absorption2) }
38.54/38.76	  join(X, Z)
38.54/38.76	
38.54/38.76	Lemma 17: meet(X, join(Y, X)) = X.
38.54/38.76	Proof:
38.54/38.76	  meet(X, join(Y, X))
38.54/38.76	= { by axiom 6 (commutativity_of_join) }
38.54/38.76	  meet(X, join(X, Y))
38.54/38.76	= { by axiom 3 (absorption1) }
38.54/38.76	  X
38.54/38.76	
38.54/38.76	Lemma 18: meet(join(X, Y), join(X, meet(Y, Z))) = join(X, meet(Y, Z)).
38.54/38.76	Proof:
38.54/38.76	  meet(join(X, Y), join(X, meet(Y, Z)))
38.54/38.76	= { by axiom 6 (commutativity_of_join) }
38.54/38.76	  meet(join(Y, X), join(X, meet(Y, Z)))
38.54/38.76	= { by axiom 5 (commutativity_of_meet) }
38.54/38.76	  meet(join(X, meet(Y, Z)), join(Y, X))
38.54/38.76	= { by lemma 16 }
38.54/38.76	  meet(join(X, meet(Y, Z)), join(Y, join(X, meet(Y, Z))))
38.54/38.76	= { by lemma 17 }
38.54/38.76	  join(X, meet(Y, Z))
38.54/38.76	
38.54/38.76	Lemma 19: meet(b, join(a, meet(X, b))) = join(a, meet(X, b)).
38.54/38.76	Proof:
38.54/38.76	  meet(b, join(a, meet(X, b)))
38.54/38.76	= { by axiom 10 (absorption2) }
38.54/38.76	  meet(join(b, meet(b, a)), join(a, meet(X, b)))
38.54/38.76	= { by axiom 2 (prove_distributivity_hypothesis) }
38.54/38.76	  meet(join(b, a), join(a, meet(X, b)))
38.54/38.76	= { by axiom 6 (commutativity_of_join) }
38.54/38.76	  meet(join(a, b), join(a, meet(X, b)))
38.54/38.76	= { by axiom 5 (commutativity_of_meet) }
38.54/38.76	  meet(join(a, b), join(a, meet(b, X)))
38.54/38.76	= { by lemma 18 }
38.54/38.76	  join(a, meet(b, X))
38.54/38.76	= { by axiom 5 (commutativity_of_meet) }
38.54/38.76	  join(a, meet(X, b))
38.54/38.76	
38.54/38.76	Lemma 20: meet(X, one) = X.
38.54/38.76	Proof:
38.54/38.76	  meet(X, one)
38.54/38.76	= { by axiom 11 (complement_join) }
38.54/38.76	  meet(X, join(X, complement(X)))
38.54/38.76	= { by axiom 3 (absorption1) }
38.54/38.76	  X
38.54/38.76	
38.54/38.76	Lemma 21: join(X, one) = one.
38.54/38.76	Proof:
38.54/38.76	  join(X, one)
38.54/38.76	= { by axiom 6 (commutativity_of_join) }
38.54/38.76	  join(one, X)
38.54/38.76	= { by lemma 20 }
38.54/38.76	  meet(join(one, X), one)
38.54/38.76	= { by axiom 5 (commutativity_of_meet) }
38.54/38.76	  meet(one, join(one, X))
38.54/38.76	= { by axiom 3 (absorption1) }
38.54/38.76	  one
38.54/38.76	
38.54/38.76	Lemma 22: join(zero, X) = X.
38.54/38.76	Proof:
38.54/38.76	  join(zero, X)
38.54/38.76	= { by axiom 6 (commutativity_of_join) }
38.54/38.76	  join(X, zero)
38.54/38.76	= { by axiom 13 (complement_meet) }
38.54/38.76	  join(X, meet(X, complement(X)))
38.54/38.76	= { by axiom 10 (absorption2) }
38.54/38.76	  X
38.54/38.76	
38.54/38.76	Lemma 23: complement(zero) = one.
38.54/38.76	Proof:
38.54/38.76	  complement(zero)
38.54/38.76	= { by lemma 22 }
38.54/38.76	  join(zero, complement(zero))
38.54/38.76	= { by axiom 11 (complement_join) }
38.54/38.76	  one
38.54/38.76	
38.54/38.76	Lemma 24: meet(X, meet(Y, Z)) = meet(Z, meet(X, Y)).
38.54/38.76	Proof:
38.54/38.76	  meet(X, meet(Y, Z))
38.54/38.76	= { by axiom 12 (associativity_of_meet) }
38.54/38.76	  meet(meet(X, Y), Z)
38.54/38.76	= { by axiom 5 (commutativity_of_meet) }
38.54/38.76	  meet(Z, meet(X, Y))
38.54/38.76	
38.54/38.76	Lemma 25: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
38.54/38.76	Proof:
38.54/38.76	  meet(X, meet(join(X, Y), Z))
38.54/38.76	= { by lemma 24 }
38.54/38.76	  meet(join(X, Y), meet(Z, X))
38.54/38.76	= { by lemma 24 }
38.54/38.76	  meet(Z, meet(X, join(X, Y)))
38.54/38.76	= { by axiom 3 (absorption1) }
38.54/38.76	  meet(Z, X)
38.54/38.76	= { by axiom 5 (commutativity_of_meet) }
38.54/38.76	  meet(X, Z)
38.54/38.76	
38.54/38.76	Lemma 26: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
38.54/38.76	Proof:
38.54/38.76	  meet(X, meet(Y, join(Z, meet(X, Y))))
38.54/38.76	= { by axiom 6 (commutativity_of_join) }
38.54/38.76	  meet(X, meet(Y, join(meet(X, Y), Z)))
38.54/38.76	= { by axiom 12 (associativity_of_meet) }
38.54/38.76	  meet(meet(X, Y), join(meet(X, Y), Z))
38.54/38.76	= { by axiom 3 (absorption1) }
38.68/38.87	  meet(X, Y)
38.68/38.87	
38.68/38.87	Lemma 27: join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))) = complement(a).
38.68/38.87	Proof:
38.68/38.87	  join(meet(b, complement(a)), complement(join(a, meet(b, complement(a)))))
38.68/38.87	= { by axiom 1 (meet_join_complement) }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 4 (associativity_of_join) }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 11 (complement_join) }
38.68/38.87	  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))))))
38.68/38.87	= { by lemma 15 }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 5 (commutativity_of_meet) }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 5 (commutativity_of_meet) }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 6 (commutativity_of_join) }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 6 (commutativity_of_join) }
38.68/38.87	  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))))))
38.68/38.87	= { by lemma 15 }
38.68/38.87	  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))))))
38.68/38.87	= { by lemma 26 }
38.68/38.87	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), meet(b, 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))))))
38.68/38.87	= { by lemma 19 }
38.68/38.87	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), 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))))))
38.68/38.87	= { by axiom 6 (commutativity_of_join) }
38.68/38.87	  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))))))
38.68/38.87	= { by lemma 15 }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 5 (commutativity_of_meet) }
38.68/38.87	  ifeq(one, one, ifeq(meet(join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), join(meet(complement(a), b), complement(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))))))
38.68/38.87	= { by lemma 17 }
38.68/38.87	  ifeq(one, one, ifeq(meet(join(complement(join(meet(complement(a), b), complement(complement(a)))), meet(complement(a), join(meet(complement(a), b), complement(complement(a))))), meet(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))))))
38.68/38.87	= { by lemma 24 }
38.68/38.87	  ifeq(one, one, ifeq(meet(join(meet(complement(a), b), 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))))), 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))))))
38.68/38.87	= { by lemma 24 }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 5 (commutativity_of_meet) }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 6 (commutativity_of_join) }
38.68/38.87	  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))))))
38.68/38.87	= { by lemma 20 }
38.68/38.87	  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))))))
38.68/38.87	= { by lemma 21 }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 6 (commutativity_of_join) }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 11 (complement_join) }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 4 (associativity_of_join) }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 6 (commutativity_of_join) }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 6 (commutativity_of_join) }
38.68/38.87	  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))))))
38.68/38.87	= { by lemma 22 }
38.68/38.87	  ifeq(one, one, ifeq(meet(complement(complement(a)), meet(join(meet(complement(a), b), complement(complement(a))), join(zero, 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))))))
38.68/38.87	= { by axiom 13 (complement_meet) }
38.68/38.87	  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(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))))))
38.68/38.87	= { by axiom 8 (equation_H18) }
38.68/38.87	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(meet(join(meet(complement(a), b), complement(complement(a))), 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))))))
38.68/38.87	= { by axiom 13 (complement_meet) }
38.68/38.87	  ifeq(one, one, ifeq(meet(complement(complement(a)), join(zero, 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))))))
38.68/38.87	= { by lemma 22 }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 5 (commutativity_of_meet) }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 12 (associativity_of_meet) }
38.68/38.87	  ifeq(one, one, ifeq(meet(meet(complement(complement(a)), 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))))))
38.68/38.87	= { by axiom 5 (commutativity_of_meet) }
38.68/38.87	  ifeq(one, one, ifeq(meet(meet(complement(a), 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))))))
38.68/38.87	= { by axiom 12 (associativity_of_meet) }
38.68/38.87	  ifeq(one, one, ifeq(meet(complement(a), meet(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))))))
38.68/38.87	= { by lemma 17 }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 13 (complement_meet) }
38.68/38.87	  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))))))
38.68/38.87	= { by axiom 7 (ifeq_axiom) }
38.68/38.87	  ifeq(one, one, complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
38.68/38.87	= { by axiom 7 (ifeq_axiom) }
38.68/38.87	  complement(a)
38.68/38.87	
38.68/38.87	Lemma 28: join(X, join(Y, meet(Z, X))) = join(X, Y).
38.68/38.87	Proof:
38.68/38.87	  join(X, join(Y, meet(Z, X)))
38.68/38.87	= { by axiom 6 (commutativity_of_join) }
38.68/38.87	  join(X, join(meet(Z, X), Y))
38.68/38.87	= { by axiom 4 (associativity_of_join) }
38.68/38.87	  join(join(X, meet(Z, X)), Y)
38.68/38.87	= { by axiom 6 (commutativity_of_join) }
38.68/38.87	  join(Y, join(X, meet(Z, X)))
38.68/38.87	= { by axiom 5 (commutativity_of_meet) }
38.68/38.87	  join(Y, join(X, meet(X, Z)))
38.68/38.87	= { by axiom 10 (absorption2) }
38.68/38.87	  join(Y, X)
38.68/38.87	= { by axiom 6 (commutativity_of_join) }
38.68/38.88	  join(X, Y)
38.68/38.88	
38.68/38.88	Lemma 29: meet(b, complement(join(a, meet(b, complement(a))))) = zero.
38.68/38.88	Proof:
38.68/38.88	  meet(b, complement(join(a, meet(b, complement(a)))))
38.68/38.88	= { by axiom 5 (commutativity_of_meet) }
38.68/38.88	  meet(complement(join(a, meet(b, complement(a)))), b)
38.68/38.88	= { by lemma 25 }
38.68/38.88	  meet(complement(join(a, meet(b, complement(a)))), meet(join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))), b))
38.68/38.88	= { by lemma 26 }
38.68/38.88	  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)))))
38.68/38.88	= { by lemma 25 }
38.68/38.88	  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))))
38.68/38.88	= { by axiom 10 (absorption2) }
38.68/38.88	  meet(complement(join(a, meet(b, complement(a)))), meet(join(b, meet(b, meet(?, a))), join(meet(?, a), meet(join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))), b))))
38.68/38.88	= { by axiom 5 (commutativity_of_meet) }
38.68/38.88	  meet(complement(join(a, meet(b, complement(a)))), meet(join(b, meet(b, meet(a, ?))), join(meet(?, a), meet(join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))), b))))
38.68/38.88	= { by axiom 12 (associativity_of_meet) }
38.68/38.88	  meet(complement(join(a, meet(b, complement(a)))), meet(join(b, meet(meet(b, a), ?)), join(meet(?, a), meet(join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))), b))))
38.68/38.88	= { by axiom 2 (prove_distributivity_hypothesis) }
38.68/38.88	  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))))
38.68/38.88	= { by axiom 5 (commutativity_of_meet) }
38.68/38.88	  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))))
38.68/38.88	= { by axiom 6 (commutativity_of_join) }
38.68/38.88	  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))))
38.68/38.88	= { by axiom 5 (commutativity_of_meet) }
38.68/38.88	  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)))))))
38.68/38.88	= { by lemma 18 }
38.68/38.88	  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))))))
38.68/38.88	= { by axiom 6 (commutativity_of_join) }
38.68/38.88	  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))))))))
38.68/38.88	= { by lemma 27 }
38.68/38.88	  meet(complement(join(a, meet(b, complement(a)))), join(meet(?, a), meet(b, complement(a))))
38.68/38.88	= { by axiom 6 (commutativity_of_join) }
38.68/38.88	  meet(complement(join(a, meet(b, complement(a)))), join(meet(b, complement(a)), meet(?, a)))
38.68/38.88	= { by axiom 5 (commutativity_of_meet) }
38.68/38.88	  meet(join(meet(b, complement(a)), meet(?, a)), complement(join(a, meet(b, complement(a)))))
38.68/38.88	= { by lemma 28 }
38.68/38.88	  meet(join(meet(b, complement(a)), meet(?, a)), complement(join(a, join(meet(b, complement(a)), meet(?, a)))))
38.68/38.88	= { by axiom 6 (commutativity_of_join) }
38.68/38.88	  meet(join(meet(b, complement(a)), meet(?, a)), complement(join(join(meet(b, complement(a)), meet(?, a)), a)))
38.68/38.88	= { by lemma 25 }
38.68/38.88	  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))))
38.68/38.88	= { by axiom 13 (complement_meet) }
38.68/38.88	  meet(join(meet(b, complement(a)), meet(?, a)), zero)
38.68/38.88	= { by axiom 5 (commutativity_of_meet) }
38.68/38.88	  meet(zero, join(meet(b, complement(a)), meet(?, a)))
38.68/38.88	= { by lemma 22 }
38.68/38.88	  meet(zero, join(zero, join(meet(b, complement(a)), meet(?, a))))
38.68/38.88	= { by axiom 3 (absorption1) }
38.68/38.90	  zero
38.68/38.90	
38.68/38.90	Goal 1 (prove_distributivity): complement(a) = join(complement(b), complement(a)).
38.68/38.90	Proof:
38.68/38.90	  complement(a)
38.68/38.90	= { by axiom 9 (idempotence_of_join) }
38.68/38.90	  join(complement(a), complement(a))
38.68/38.90	= { by lemma 27 }
38.68/38.90	  join(complement(a), join(meet(b, complement(a)), complement(join(a, meet(b, complement(a))))))
38.68/38.90	= { by axiom 6 (commutativity_of_join) }
38.68/38.90	  join(complement(a), join(complement(join(a, meet(b, complement(a)))), meet(b, complement(a))))
38.68/38.90	= { by lemma 28 }
38.68/38.90	  join(complement(a), complement(join(a, meet(b, complement(a)))))
38.68/38.90	= { by lemma 15 }
38.68/38.90	  join(complement(a), complement(complement(complement(join(a, meet(b, complement(a)))))))
38.68/38.90	= { by axiom 7 (ifeq_axiom) }
38.68/38.90	  join(complement(a), complement(ifeq(one, one, complement(complement(join(a, meet(b, complement(a))))), b)))
38.68/38.90	= { by lemma 21 }
38.68/38.90	  join(complement(a), complement(ifeq(join(b, one), one, complement(complement(join(a, meet(b, complement(a))))), b)))
38.68/38.90	= { by axiom 11 (complement_join) }
38.68/38.90	  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)))
38.68/38.90	= { by axiom 6 (commutativity_of_join) }
38.68/38.90	  join(complement(a), complement(ifeq(join(b, join(complement(meet(b, join(a, meet(complement(a), b)))), meet(b, join(a, meet(complement(a), b))))), one, complement(complement(join(a, meet(b, complement(a))))), b)))
38.68/38.90	= { by lemma 16 }
38.68/38.90	  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)))
38.68/38.90	= { by lemma 19 }
38.68/38.90	  join(complement(a), complement(ifeq(join(b, complement(join(a, meet(complement(a), b)))), one, complement(complement(join(a, meet(b, complement(a))))), b)))
38.68/38.90	= { by axiom 5 (commutativity_of_meet) }
38.68/38.90	  join(complement(a), complement(ifeq(join(b, complement(join(a, meet(b, complement(a))))), one, complement(complement(join(a, meet(b, complement(a))))), b)))
38.68/38.90	= { by lemma 20 }
38.68/38.90	  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)))
38.68/38.90	= { by lemma 23 }
38.68/38.90	  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)))
38.68/38.90	= { by lemma 29 }
38.68/38.90	  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)))
38.68/38.90	= { by axiom 6 (commutativity_of_join) }
38.68/38.90	  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)))
38.68/38.90	= { by lemma 20 }
38.68/38.90	  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)))
38.68/38.90	= { by lemma 23 }
38.68/38.90	  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)))
38.68/38.90	= { by lemma 29 }
38.68/38.90	  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)))
38.68/38.90	= { by axiom 7 (ifeq_axiom) }
38.68/38.90	  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)))
38.68/38.90	= { by axiom 13 (complement_meet) }
38.68/38.90	  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)))
38.68/38.90	= { by axiom 12 (associativity_of_meet) }
38.68/38.90	  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)))
38.68/38.90	= { by lemma 14 }
38.68/38.90	  join(complement(a), complement(b))
38.68/38.90	= { by axiom 6 (commutativity_of_join) }
38.68/38.90	  join(complement(b), complement(a))
38.68/38.90	% SZS output end Proof
38.68/38.90	
38.68/38.90	RESULT: Unsatisfiable (the axioms are contradictory).
38.68/38.90	EOF