0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.34	% Computer   : n009.cluster.edu
0.13/0.34	% Model      : x86_64 x86_64
0.13/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.34	% Memory     : 8042.1875MB
0.13/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.34	% CPULimit   : 180
0.13/0.34	% DateTime   : Thu Aug 29 13:15:57 EDT 2019
0.13/0.34	% CPUTime    : 
2.98/3.18	% SZS status Unsatisfiable
2.98/3.18	
2.98/3.19	% SZS output start Proof
2.98/3.19	Take the following subset of the input axioms:
2.98/3.19	  fof(composition_associativity_5, axiom, ![A, B, C]: composition(composition(A, B), C)=composition(A, composition(B, C))).
2.98/3.19	  fof(composition_distributivity_7, axiom, ![A, B, C]: composition(join(A, B), C)=join(composition(A, C), composition(B, C))).
2.98/3.19	  fof(composition_identity_6, axiom, ![A]: A=composition(A, one)).
2.98/3.19	  fof(converse_additivity_9, axiom, ![A, B]: join(converse(A), converse(B))=converse(join(A, B))).
2.98/3.19	  fof(converse_cancellativity_11, axiom, ![A, B]: join(composition(converse(A), complement(composition(A, B))), complement(B))=complement(B)).
2.98/3.19	  fof(converse_idempotence_8, axiom, ![A]: converse(converse(A))=A).
2.98/3.19	  fof(converse_multiplicativity_10, axiom, ![A, B]: converse(composition(A, B))=composition(converse(B), converse(A))).
2.98/3.19	  fof(def_top_12, axiom, ![A]: join(A, complement(A))=top).
2.98/3.19	  fof(def_zero_13, axiom, ![A]: zero=meet(A, complement(A))).
2.98/3.19	  fof(goals_14, negated_conjecture, join(sk1, one)=one).
2.98/3.19	  fof(goals_15, negated_conjecture, meet(composition(sk1, top), sk2)!=composition(sk1, sk2)).
2.98/3.19	  fof(maddux1_join_commutativity_1, axiom, ![A, B]: join(B, A)=join(A, B)).
2.98/3.19	  fof(maddux2_join_associativity_2, axiom, ![A, B, C]: join(join(A, B), C)=join(A, join(B, C))).
2.98/3.19	  fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A, B]: join(complement(join(complement(A), complement(B))), complement(join(complement(A), B)))=A).
2.98/3.19	  fof(maddux4_definiton_of_meet_4, axiom, ![A, B]: complement(join(complement(A), complement(B)))=meet(A, B)).
2.98/3.19	
2.98/3.19	Now clausify the problem and encode Horn clauses using encoding 3 of
2.98/3.19	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
2.98/3.19	We repeatedly replace C & s=t => u=v by the two clauses:
2.98/3.19	  fresh(y, y, x1...xn) = u
2.98/3.19	  C => fresh(s, t, x1...xn) = v
2.98/3.19	where fresh is a fresh function symbol and x1..xn are the free
2.98/3.19	variables of u and v.
2.98/3.19	A predicate p(X) is encoded as p(X)=true (this is sound, because the
2.98/3.19	input problem has no model of domain size 1).
2.98/3.19	
2.98/3.19	The encoding turns the above axioms into the following unit equations and goals:
2.98/3.19	
2.98/3.19	Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
2.98/3.19	Axiom 2 (maddux4_definiton_of_meet_4): complement(join(complement(X), complement(Y))) = meet(X, Y).
2.98/3.19	Axiom 3 (composition_identity_6): X = composition(X, one).
2.98/3.19	Axiom 4 (maddux3_a_kind_of_de_Morgan_3): join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))) = X.
2.98/3.19	Axiom 5 (converse_idempotence_8): converse(converse(X)) = X.
2.98/3.19	Axiom 6 (maddux2_join_associativity_2): join(join(X, Y), Z) = join(X, join(Y, Z)).
2.98/3.19	Axiom 7 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
2.98/3.19	Axiom 8 (def_zero_13): zero = meet(X, complement(X)).
2.98/3.19	Axiom 9 (def_top_12): join(X, complement(X)) = top.
2.98/3.19	Axiom 10 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
2.98/3.19	Axiom 11 (converse_additivity_9): join(converse(X), converse(Y)) = converse(join(X, Y)).
2.98/3.19	Axiom 12 (composition_associativity_5): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)).
2.98/3.19	Axiom 13 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
2.98/3.20	Axiom 14 (goals_14): join(sk1, one) = one.
2.98/3.20	
2.98/3.20	Lemma 15: meet(X, Y) = meet(Y, X).
2.98/3.20	Proof:
2.98/3.20	  meet(X, Y)
2.98/3.20	= { by axiom 2 (maddux4_definiton_of_meet_4) }
2.98/3.20	  complement(join(complement(X), complement(Y)))
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  complement(join(complement(Y), complement(X)))
2.98/3.20	= { by axiom 2 (maddux4_definiton_of_meet_4) }
2.98/3.20	  meet(Y, X)
2.98/3.20	
2.98/3.20	Lemma 16: complement(top) = zero.
2.98/3.20	Proof:
2.98/3.20	  complement(top)
2.98/3.20	= { by axiom 9 (def_top_12) }
2.98/3.20	  complement(join(complement(?), complement(complement(?))))
2.98/3.20	= { by axiom 2 (maddux4_definiton_of_meet_4) }
2.98/3.20	  meet(?, complement(?))
2.98/3.20	= { by axiom 8 (def_zero_13) }
2.98/3.20	  zero
2.98/3.20	
2.98/3.20	Lemma 17: complement(join(zero, complement(X))) = meet(X, top).
2.98/3.20	Proof:
2.98/3.20	  complement(join(zero, complement(X)))
2.98/3.20	= { by lemma 16 }
2.98/3.20	  complement(join(complement(top), complement(X)))
2.98/3.20	= { by axiom 2 (maddux4_definiton_of_meet_4) }
2.98/3.20	  meet(top, X)
2.98/3.20	= { by lemma 15 }
2.98/3.20	  meet(X, top)
2.98/3.20	
2.98/3.20	Lemma 18: composition(converse(one), X) = X.
2.98/3.20	Proof:
2.98/3.20	  composition(converse(one), X)
2.98/3.20	= { by axiom 5 (converse_idempotence_8) }
2.98/3.20	  composition(converse(one), converse(converse(X)))
2.98/3.20	= { by axiom 10 (converse_multiplicativity_10) }
2.98/3.20	  converse(composition(converse(X), one))
2.98/3.20	= { by axiom 3 (composition_identity_6) }
2.98/3.20	  converse(converse(X))
2.98/3.20	= { by axiom 5 (converse_idempotence_8) }
2.98/3.20	  X
2.98/3.20	
2.98/3.20	Lemma 19: composition(one, X) = X.
2.98/3.20	Proof:
2.98/3.20	  composition(one, X)
2.98/3.20	= { by lemma 18 }
2.98/3.20	  composition(converse(one), composition(one, X))
2.98/3.20	= { by axiom 12 (composition_associativity_5) }
2.98/3.20	  composition(composition(converse(one), one), X)
2.98/3.20	= { by axiom 3 (composition_identity_6) }
2.98/3.20	  composition(converse(one), X)
2.98/3.20	= { by lemma 18 }
2.98/3.20	  X
2.98/3.20	
2.98/3.20	Lemma 20: join(complement(X), complement(X)) = complement(X).
2.98/3.20	Proof:
2.98/3.20	  join(complement(X), complement(X))
2.98/3.20	= { by lemma 18 }
2.98/3.20	  join(complement(X), composition(converse(one), complement(X)))
2.98/3.20	= { by lemma 19 }
2.98/3.20	  join(complement(X), composition(converse(one), complement(composition(one, X))))
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  join(composition(converse(one), complement(composition(one, X))), complement(X))
2.98/3.20	= { by axiom 7 (converse_cancellativity_11) }
2.98/3.20	  complement(X)
2.98/3.20	
2.98/3.20	Lemma 21: meet(X, X) = complement(complement(X)).
2.98/3.20	Proof:
2.98/3.20	  meet(X, X)
2.98/3.20	= { by axiom 2 (maddux4_definiton_of_meet_4) }
2.98/3.20	  complement(join(complement(X), complement(X)))
2.98/3.20	= { by lemma 20 }
2.98/3.20	  complement(complement(X))
2.98/3.20	
2.98/3.20	Lemma 22: join(meet(X, Y), complement(join(complement(X), Y))) = X.
2.98/3.20	Proof:
2.98/3.20	  join(meet(X, Y), complement(join(complement(X), Y)))
2.98/3.20	= { by axiom 2 (maddux4_definiton_of_meet_4) }
2.98/3.20	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
2.98/3.20	= { by axiom 4 (maddux3_a_kind_of_de_Morgan_3) }
2.98/3.20	  X
2.98/3.20	
2.98/3.20	Lemma 23: join(meet(X, Y), meet(X, complement(Y))) = X.
2.98/3.20	Proof:
2.98/3.20	  join(meet(X, Y), meet(X, complement(Y)))
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  join(meet(X, complement(Y)), meet(X, Y))
2.98/3.20	= { by axiom 2 (maddux4_definiton_of_meet_4) }
2.98/3.20	  join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
2.98/3.20	= { by lemma 22 }
2.98/3.20	  X
2.98/3.20	
2.98/3.20	Lemma 24: join(zero, complement(complement(X))) = X.
2.98/3.20	Proof:
2.98/3.20	  join(zero, complement(complement(X)))
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  join(complement(complement(X)), zero)
2.98/3.20	= { by lemma 21 }
2.98/3.20	  join(meet(X, X), zero)
2.98/3.20	= { by axiom 8 (def_zero_13) }
2.98/3.20	  join(meet(X, X), meet(X, complement(X)))
2.98/3.20	= { by lemma 23 }
2.98/3.20	  X
2.98/3.20	
2.98/3.20	Lemma 25: join(X, join(Y, Z)) = join(Z, join(X, Y)).
2.98/3.20	Proof:
2.98/3.20	  join(X, join(Y, Z))
2.98/3.20	= { by axiom 6 (maddux2_join_associativity_2) }
2.98/3.20	  join(join(X, Y), Z)
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  join(Z, join(X, Y))
2.98/3.20	
2.98/3.20	Lemma 26: join(X, top) = top.
2.98/3.20	Proof:
2.98/3.20	  join(X, top)
2.98/3.20	= { by axiom 9 (def_top_12) }
2.98/3.20	  join(X, join(complement(X), complement(complement(X))))
2.98/3.20	= { by lemma 25 }
2.98/3.20	  join(complement(X), join(complement(complement(X)), X))
2.98/3.20	= { by lemma 25 }
2.98/3.20	  join(complement(complement(X)), join(X, complement(X)))
2.98/3.20	= { by axiom 9 (def_top_12) }
2.98/3.20	  join(complement(complement(X)), top)
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  join(top, complement(complement(X)))
2.98/3.20	= { by axiom 9 (def_top_12) }
2.98/3.20	  join(join(complement(X), complement(complement(X))), complement(complement(X)))
2.98/3.20	= { by axiom 6 (maddux2_join_associativity_2) }
2.98/3.20	  join(complement(X), join(complement(complement(X)), complement(complement(X))))
2.98/3.20	= { by lemma 20 }
2.98/3.20	  join(complement(X), complement(complement(X)))
2.98/3.20	= { by axiom 9 (def_top_12) }
2.98/3.20	  top
2.98/3.20	
2.98/3.20	Lemma 27: join(zero, meet(X, top)) = X.
2.98/3.20	Proof:
2.98/3.20	  join(zero, meet(X, top))
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  join(meet(X, top), zero)
2.98/3.20	= { by lemma 16 }
2.98/3.20	  join(meet(X, top), complement(top))
2.98/3.20	= { by lemma 26 }
2.98/3.20	  join(meet(X, top), complement(join(complement(X), top)))
2.98/3.20	= { by lemma 22 }
2.98/3.20	  X
2.98/3.20	
2.98/3.20	Lemma 28: join(zero, complement(X)) = complement(X).
2.98/3.20	Proof:
2.98/3.20	  join(zero, complement(X))
2.98/3.20	= { by lemma 24 }
2.98/3.20	  join(zero, complement(join(zero, complement(complement(X)))))
2.98/3.20	= { by lemma 17 }
2.98/3.20	  join(zero, meet(complement(X), top))
2.98/3.20	= { by lemma 27 }
2.98/3.20	  complement(X)
2.98/3.20	
2.98/3.20	Lemma 29: complement(complement(X)) = X.
2.98/3.20	Proof:
2.98/3.20	  complement(complement(X))
2.98/3.20	= { by lemma 28 }
2.98/3.20	  join(zero, complement(complement(X)))
2.98/3.20	= { by lemma 24 }
2.98/3.20	  X
2.98/3.20	
2.98/3.20	Lemma 30: join(X, zero) = X.
2.98/3.20	Proof:
2.98/3.20	  join(X, zero)
2.98/3.20	= { by lemma 29 }
2.98/3.20	  join(complement(complement(X)), zero)
2.98/3.20	= { by lemma 21 }
2.98/3.20	  join(meet(X, X), zero)
2.98/3.20	= { by axiom 8 (def_zero_13) }
2.98/3.20	  join(meet(X, X), meet(X, complement(X)))
2.98/3.20	= { by lemma 23 }
2.98/3.20	  X
2.98/3.20	
2.98/3.20	Lemma 31: meet(X, top) = X.
2.98/3.20	Proof:
2.98/3.20	  meet(X, top)
2.98/3.20	= { by lemma 17 }
2.98/3.20	  complement(join(zero, complement(X)))
2.98/3.20	= { by lemma 28 }
2.98/3.20	  join(zero, complement(join(zero, complement(X))))
2.98/3.20	= { by lemma 17 }
2.98/3.20	  join(zero, meet(X, top))
2.98/3.20	= { by lemma 27 }
2.98/3.20	  X
2.98/3.20	
2.98/3.20	Lemma 32: complement(join(X, complement(Y))) = meet(Y, complement(X)).
2.98/3.20	Proof:
2.98/3.20	  complement(join(X, complement(Y)))
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  complement(join(complement(Y), X))
2.98/3.20	= { by lemma 31 }
2.98/3.20	  complement(join(complement(Y), meet(X, top)))
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  complement(join(meet(X, top), complement(Y)))
2.98/3.20	= { by axiom 2 (maddux4_definiton_of_meet_4) }
2.98/3.20	  complement(join(complement(join(complement(X), complement(top))), complement(Y)))
2.98/3.20	= { by axiom 2 (maddux4_definiton_of_meet_4) }
2.98/3.20	  meet(join(complement(X), complement(top)), Y)
2.98/3.20	= { by lemma 15 }
2.98/3.20	  meet(Y, join(complement(X), complement(top)))
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  meet(Y, join(complement(top), complement(X)))
2.98/3.20	= { by lemma 16 }
2.98/3.20	  meet(Y, join(zero, complement(X)))
2.98/3.20	= { by lemma 28 }
2.98/3.20	  meet(Y, complement(X))
2.98/3.20	
2.98/3.20	Lemma 33: complement(join(complement(X), Y)) = meet(X, complement(Y)).
2.98/3.20	Proof:
2.98/3.20	  complement(join(complement(X), Y))
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  complement(join(Y, complement(X)))
2.98/3.20	= { by lemma 32 }
2.98/3.20	  meet(X, complement(Y))
2.98/3.20	
2.98/3.20	Lemma 34: meet(complement(X), complement(Y)) = complement(join(X, Y)).
2.98/3.20	Proof:
2.98/3.20	  meet(complement(X), complement(Y))
2.98/3.20	= { by lemma 28 }
2.98/3.20	  meet(join(zero, complement(X)), complement(Y))
2.98/3.20	= { by lemma 32 }
2.98/3.20	  complement(join(Y, complement(join(zero, complement(X)))))
2.98/3.20	= { by lemma 17 }
2.98/3.20	  complement(join(Y, meet(X, top)))
2.98/3.20	= { by lemma 31 }
2.98/3.20	  complement(join(Y, X))
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  complement(join(X, Y))
2.98/3.20	
2.98/3.20	Lemma 35: meet(X, meet(Y, complement(Z))) = meet(complement(Z), meet(X, Y)).
2.98/3.20	Proof:
2.98/3.20	  meet(X, meet(Y, complement(Z)))
2.98/3.20	= { by lemma 33 }
2.98/3.20	  meet(X, complement(join(complement(Y), Z)))
2.98/3.20	= { by lemma 33 }
2.98/3.20	  complement(join(complement(X), join(complement(Y), Z)))
2.98/3.20	= { by axiom 6 (maddux2_join_associativity_2) }
2.98/3.20	  complement(join(join(complement(X), complement(Y)), Z))
2.98/3.20	= { by lemma 34 }
2.98/3.20	  meet(complement(join(complement(X), complement(Y))), complement(Z))
2.98/3.20	= { by axiom 2 (maddux4_definiton_of_meet_4) }
2.98/3.20	  meet(meet(X, Y), complement(Z))
2.98/3.20	= { by lemma 15 }
2.98/3.20	  meet(complement(Z), meet(X, Y))
2.98/3.20	
2.98/3.20	Lemma 36: meet(meet(X, Z), Y) = meet(X, meet(Y, Z)).
2.98/3.20	Proof:
2.98/3.20	  meet(meet(X, Z), Y)
2.98/3.20	= { by lemma 15 }
2.98/3.20	  meet(Y, meet(X, Z))
2.98/3.20	= { by lemma 31 }
2.98/3.20	  meet(meet(Y, top), meet(X, Z))
2.98/3.20	= { by lemma 17 }
2.98/3.20	  meet(complement(join(zero, complement(Y))), meet(X, Z))
2.98/3.20	= { by lemma 35 }
2.98/3.20	  meet(X, meet(Z, complement(join(zero, complement(Y)))))
2.98/3.20	= { by lemma 17 }
2.98/3.20	  meet(X, meet(Z, meet(Y, top)))
2.98/3.20	= { by lemma 31 }
2.98/3.20	  meet(X, meet(Z, Y))
2.98/3.20	= { by lemma 15 }
2.98/3.20	  meet(X, meet(Y, Z))
2.98/3.20	
2.98/3.20	Lemma 37: meet(X, join(Y, X)) = X.
2.98/3.20	Proof:
2.98/3.20	  meet(X, join(Y, X))
2.98/3.20	= { by axiom 5 (converse_idempotence_8) }
2.98/3.20	  meet(converse(converse(X)), join(Y, X))
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  meet(converse(converse(X)), join(X, Y))
2.98/3.20	= { by axiom 5 (converse_idempotence_8) }
2.98/3.20	  meet(converse(converse(X)), converse(converse(join(X, Y))))
2.98/3.20	= { by axiom 11 (converse_additivity_9) }
2.98/3.20	  meet(converse(converse(X)), converse(join(converse(X), converse(Y))))
2.98/3.20	= { by lemma 30 }
2.98/3.20	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), zero)
2.98/3.20	= { by lemma 16 }
2.98/3.20	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(top))
2.98/3.20	= { by lemma 26 }
2.98/3.20	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(converse(converse(Y)), top)))
2.98/3.20	= { by axiom 9 (def_top_12) }
2.98/3.20	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(converse(converse(Y)), join(converse(converse(X)), complement(converse(converse(X)))))))
2.98/3.20	= { by axiom 6 (maddux2_join_associativity_2) }
2.98/3.20	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(join(converse(converse(Y)), converse(converse(X))), complement(converse(converse(X))))))
2.98/3.20	= { by axiom 11 (converse_additivity_9) }
2.98/3.20	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(converse(join(converse(Y), converse(X))), complement(converse(converse(X))))))
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(complement(converse(converse(X))), converse(join(converse(Y), converse(X))))))
2.98/3.20	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.20	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(complement(converse(converse(X))), converse(join(converse(X), converse(Y))))))
2.98/3.20	= { by lemma 22 }
2.98/3.20	  converse(converse(X))
2.98/3.20	= { by axiom 5 (converse_idempotence_8) }
2.98/3.21	  X
2.98/3.21	
2.98/3.21	Lemma 38: meet(X, join(complement(X), Y)) = meet(X, Y).
2.98/3.21	Proof:
2.98/3.21	  meet(X, join(complement(X), Y))
2.98/3.21	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.21	  meet(X, join(Y, complement(X)))
2.98/3.21	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.21	  meet(X, join(complement(X), Y))
2.98/3.21	= { by lemma 30 }
2.98/3.21	  meet(X, join(join(complement(X), Y), zero))
2.98/3.21	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.21	  meet(X, join(zero, join(complement(X), Y)))
2.98/3.21	= { by axiom 6 (maddux2_join_associativity_2) }
2.98/3.21	  meet(X, join(join(zero, complement(X)), Y))
2.98/3.21	= { by lemma 31 }
2.98/3.21	  meet(X, meet(join(join(zero, complement(X)), Y), top))
2.98/3.21	= { by lemma 36 }
2.98/3.21	  meet(meet(X, top), join(join(zero, complement(X)), Y))
2.98/3.21	= { by lemma 17 }
2.98/3.21	  meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), Y))
2.98/3.21	= { by lemma 29 }
2.98/3.21	  meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y))))
2.98/3.21	= { by lemma 23 }
2.98/3.21	  join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), complement(Y)))
2.98/3.21	= { by lemma 15 }
2.98/3.21	  join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), meet(complement(Y), meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y))))))
2.98/3.21	= { by lemma 15 }
2.98/3.21	  join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), meet(complement(Y), meet(join(join(zero, complement(X)), complement(complement(Y))), complement(join(zero, complement(X))))))
2.98/3.21	= { by lemma 33 }
2.98/3.21	  join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), meet(complement(Y), complement(join(complement(join(join(zero, complement(X)), complement(complement(Y)))), join(zero, complement(X))))))
2.98/3.21	= { by lemma 33 }
2.98/3.21	  join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), complement(join(complement(complement(Y)), join(complement(join(join(zero, complement(X)), complement(complement(Y)))), join(zero, complement(X))))))
2.98/3.21	= { by axiom 6 (maddux2_join_associativity_2) }
2.98/3.21	  join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), complement(join(join(complement(complement(Y)), complement(join(join(zero, complement(X)), complement(complement(Y))))), join(zero, complement(X)))))
2.98/3.21	= { by lemma 34 }
2.98/3.21	  join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), meet(complement(join(complement(complement(Y)), complement(join(join(zero, complement(X)), complement(complement(Y)))))), complement(join(zero, complement(X)))))
2.98/3.21	= { by axiom 2 (maddux4_definiton_of_meet_4) }
2.98/3.21	  join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), meet(meet(complement(Y), join(join(zero, complement(X)), complement(complement(Y)))), complement(join(zero, complement(X)))))
2.98/3.21	= { by lemma 15 }
2.98/3.21	  join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), meet(complement(join(zero, complement(X))), meet(complement(Y), join(join(zero, complement(X)), complement(complement(Y))))))
2.98/3.21	= { by lemma 15 }
2.98/3.21	  join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), meet(complement(join(zero, complement(X))), meet(join(join(zero, complement(X)), complement(complement(Y))), complement(Y))))
2.98/3.21	= { by lemma 35 }
2.98/3.21	  join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), meet(join(join(zero, complement(X)), complement(complement(Y))), meet(complement(Y), complement(join(zero, complement(X))))))
2.98/3.21	= { by lemma 32 }
2.98/3.21	  join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), meet(join(join(zero, complement(X)), complement(complement(Y))), complement(join(join(zero, complement(X)), complement(complement(Y))))))
2.98/3.21	= { by axiom 8 (def_zero_13) }
2.98/3.21	  join(meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y), zero)
2.98/3.21	= { by lemma 30 }
2.98/3.21	  meet(meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), complement(complement(Y)))), Y)
2.98/3.21	= { by lemma 36 }
2.98/3.21	  meet(complement(join(zero, complement(X))), meet(Y, join(join(zero, complement(X)), complement(complement(Y)))))
2.98/3.21	= { by lemma 29 }
2.98/3.21	  meet(complement(join(zero, complement(X))), meet(Y, join(join(zero, complement(X)), Y)))
2.98/3.21	= { by lemma 37 }
2.98/3.21	  meet(complement(join(zero, complement(X))), Y)
2.98/3.21	= { by lemma 15 }
2.98/3.21	  meet(Y, complement(join(zero, complement(X))))
2.98/3.21	= { by lemma 17 }
2.98/3.21	  meet(Y, meet(X, top))
2.98/3.21	= { by lemma 31 }
2.98/3.21	  meet(Y, X)
2.98/3.21	= { by lemma 15 }
2.98/3.21	  meet(X, Y)
2.98/3.21	
2.98/3.21	Lemma 39: join(X, composition(sk1, X)) = X.
2.98/3.21	Proof:
2.98/3.21	  join(X, composition(sk1, X))
2.98/3.21	= { by lemma 19 }
2.98/3.21	  join(composition(one, X), composition(sk1, X))
2.98/3.21	= { by axiom 13 (composition_distributivity_7) }
2.98/3.21	  composition(join(one, sk1), X)
2.98/3.21	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.21	  composition(join(sk1, one), X)
2.98/3.21	= { by axiom 14 (goals_14) }
2.98/3.21	  composition(one, X)
2.98/3.21	= { by lemma 19 }
2.98/3.22	  X
2.98/3.22	
2.98/3.22	Goal 1 (goals_15): meet(composition(sk1, top), sk2) = composition(sk1, sk2).
2.98/3.22	Proof:
2.98/3.22	  meet(composition(sk1, top), sk2)
2.98/3.22	= { by lemma 15 }
2.98/3.22	  meet(sk2, composition(sk1, top))
2.98/3.22	= { by lemma 38 }
2.98/3.22	  meet(sk2, join(complement(sk2), composition(sk1, top)))
2.98/3.22	= { by axiom 9 (def_top_12) }
2.98/3.22	  meet(sk2, join(complement(sk2), composition(sk1, join(complement(sk2), complement(complement(sk2))))))
2.98/3.22	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.22	  meet(sk2, join(complement(sk2), composition(sk1, join(complement(complement(sk2)), complement(sk2)))))
2.98/3.22	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.22	  meet(sk2, join(composition(sk1, join(complement(complement(sk2)), complement(sk2))), complement(sk2)))
2.98/3.22	= { by axiom 5 (converse_idempotence_8) }
2.98/3.22	  meet(sk2, join(converse(converse(composition(sk1, join(complement(complement(sk2)), complement(sk2))))), complement(sk2)))
2.98/3.22	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.22	  meet(sk2, join(converse(converse(composition(sk1, join(complement(sk2), complement(complement(sk2)))))), complement(sk2)))
2.98/3.22	= { by axiom 10 (converse_multiplicativity_10) }
2.98/3.22	  meet(sk2, join(converse(composition(converse(join(complement(sk2), complement(complement(sk2)))), converse(sk1))), complement(sk2)))
2.98/3.22	= { by axiom 11 (converse_additivity_9) }
2.98/3.22	  meet(sk2, join(converse(composition(join(converse(complement(sk2)), converse(complement(complement(sk2)))), converse(sk1))), complement(sk2)))
2.98/3.22	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.22	  meet(sk2, join(converse(composition(join(converse(complement(complement(sk2))), converse(complement(sk2))), converse(sk1))), complement(sk2)))
2.98/3.22	= { by axiom 13 (composition_distributivity_7) }
2.98/3.22	  meet(sk2, join(converse(join(composition(converse(complement(complement(sk2))), converse(sk1)), composition(converse(complement(sk2)), converse(sk1)))), complement(sk2)))
2.98/3.22	= { by axiom 10 (converse_multiplicativity_10) }
2.98/3.22	  meet(sk2, join(converse(join(converse(composition(sk1, complement(complement(sk2)))), composition(converse(complement(sk2)), converse(sk1)))), complement(sk2)))
2.98/3.22	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.22	  meet(sk2, join(converse(join(composition(converse(complement(sk2)), converse(sk1)), converse(composition(sk1, complement(complement(sk2)))))), complement(sk2)))
2.98/3.22	= { by axiom 10 (converse_multiplicativity_10) }
2.98/3.22	  meet(sk2, join(converse(join(converse(composition(sk1, complement(sk2))), converse(composition(sk1, complement(complement(sk2)))))), complement(sk2)))
2.98/3.22	= { by axiom 11 (converse_additivity_9) }
2.98/3.22	  meet(sk2, join(converse(converse(join(composition(sk1, complement(sk2)), composition(sk1, complement(complement(sk2)))))), complement(sk2)))
2.98/3.22	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.22	  meet(sk2, join(converse(converse(join(composition(sk1, complement(complement(sk2))), composition(sk1, complement(sk2))))), complement(sk2)))
2.98/3.22	= { by axiom 5 (converse_idempotence_8) }
2.98/3.22	  meet(sk2, join(join(composition(sk1, complement(complement(sk2))), composition(sk1, complement(sk2))), complement(sk2)))
2.98/3.22	= { by axiom 6 (maddux2_join_associativity_2) }
2.98/3.22	  meet(sk2, join(composition(sk1, complement(complement(sk2))), join(composition(sk1, complement(sk2)), complement(sk2))))
2.98/3.22	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.22	  meet(sk2, join(composition(sk1, complement(complement(sk2))), join(complement(sk2), composition(sk1, complement(sk2)))))
2.98/3.22	= { by lemma 39 }
2.98/3.22	  meet(sk2, join(composition(sk1, complement(complement(sk2))), complement(sk2)))
2.98/3.22	= { by axiom 1 (maddux1_join_commutativity_1) }
2.98/3.22	  meet(sk2, join(complement(sk2), composition(sk1, complement(complement(sk2)))))
2.98/3.22	= { by lemma 38 }
2.98/3.22	  meet(sk2, composition(sk1, complement(complement(sk2))))
2.98/3.22	= { by lemma 29 }
2.98/3.22	  meet(sk2, composition(sk1, sk2))
2.98/3.22	= { by lemma 15 }
2.98/3.22	  meet(composition(sk1, sk2), sk2)
2.98/3.22	= { by lemma 39 }
2.98/3.22	  meet(composition(sk1, sk2), join(sk2, composition(sk1, sk2)))
2.98/3.22	= { by lemma 37 }
2.98/3.22	  composition(sk1, sk2)
2.98/3.22	% SZS output end Proof
2.98/3.22	
2.98/3.22	RESULT: Unsatisfiable (the axioms are contradictory).
2.98/3.22	EOF