0.08/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.08/0.14	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.15/0.36	% Computer   : n026.cluster.edu
0.15/0.36	% Model      : x86_64 x86_64
0.15/0.36	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.15/0.36	% Memory     : 8042.1875MB
0.15/0.36	% OS         : Linux 3.10.0-693.el7.x86_64
0.15/0.36	% CPULimit   : 180
0.15/0.36	% DateTime   : Thu Aug 29 11:18:50 EDT 2019
0.15/0.36	% CPUTime    : 
0.58/0.74	% SZS status Unsatisfiable
0.58/0.74	
0.58/0.74	% SZS output start Proof
0.58/0.74	Take the following subset of the input axioms:
0.58/0.74	  fof(mckenzie1, axiom, ![X, Y, Z]: X=join(X, meet(Y, meet(X, Z)))).
0.58/0.74	  fof(mckenzie2, axiom, ![X, Y, Z]: X=meet(X, join(Y, join(X, Z)))).
0.58/0.74	  fof(mckenzie3, axiom, ![X, Y, Z]: Y=join(join(meet(X, Y), meet(Y, Z)), Y)).
0.58/0.74	  fof(mckenzie4, axiom, ![X, Y, Z]: Y=meet(meet(join(X, Y), join(Y, Z)), Y)).
0.58/0.74	  fof(prove_associativity_of_meet, negated_conjecture, meet(a, meet(b, c))!=meet(meet(a, b), c)).
0.58/0.74	
0.58/0.74	Now clausify the problem and encode Horn clauses using encoding 3 of
0.58/0.74	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.58/0.74	We repeatedly replace C & s=t => u=v by the two clauses:
0.58/0.74	  fresh(y, y, x1...xn) = u
0.58/0.74	  C => fresh(s, t, x1...xn) = v
0.58/0.74	where fresh is a fresh function symbol and x1..xn are the free
0.58/0.74	variables of u and v.
0.58/0.74	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.58/0.74	input problem has no model of domain size 1).
0.58/0.74	
0.58/0.74	The encoding turns the above axioms into the following unit equations and goals:
0.58/0.74	
0.58/0.74	Axiom 1 (mckenzie4): X = meet(meet(join(Y, X), join(X, Z)), X).
0.58/0.74	Axiom 2 (mckenzie1): X = join(X, meet(Y, meet(X, Z))).
0.58/0.74	Axiom 3 (mckenzie3): X = join(join(meet(Y, X), meet(X, Z)), X).
0.58/0.74	Axiom 4 (mckenzie2): X = meet(X, join(Y, join(X, Z))).
0.58/0.74	
0.58/0.74	Lemma 5: join(X, meet(X, Y)) = X.
0.58/0.74	Proof:
0.58/0.74	  join(X, meet(X, Y))
0.58/0.74	= { by axiom 1 (mckenzie4) }
0.58/0.74	  join(X, meet(meet(join(?, meet(X, Y)), join(meet(X, Y), ?)), meet(X, Y)))
0.58/0.74	= { by axiom 2 (mckenzie1) }
0.58/0.74	  X
0.58/0.74	
0.58/0.74	Lemma 6: join(X, meet(Y, X)) = X.
0.58/0.74	Proof:
0.58/0.74	  join(X, meet(Y, X))
0.58/0.74	= { by axiom 4 (mckenzie2) }
0.58/0.74	  join(X, meet(Y, meet(X, join(?, join(X, ?)))))
0.58/0.74	= { by axiom 2 (mckenzie1) }
0.58/0.74	  X
0.58/0.74	
0.58/0.74	Lemma 7: meet(X, join(X, Y)) = X.
0.58/0.74	Proof:
0.58/0.74	  meet(X, join(X, Y))
0.58/0.74	= { by axiom 3 (mckenzie3) }
0.58/0.74	  meet(X, join(join(meet(?, join(X, Y)), meet(join(X, Y), ?)), join(X, Y)))
0.58/0.74	= { by axiom 4 (mckenzie2) }
0.58/0.74	  X
0.58/0.74	
0.58/0.74	Lemma 8: meet(X, join(Y, X)) = X.
0.58/0.74	Proof:
0.58/0.74	  meet(X, join(Y, X))
0.58/0.74	= { by axiom 2 (mckenzie1) }
0.58/0.74	  meet(X, join(Y, join(X, meet(?, meet(X, ?)))))
0.58/0.74	= { by axiom 4 (mckenzie2) }
0.58/0.74	  X
0.58/0.74	
0.58/0.74	Lemma 9: join(join(Y, X), join(X, Y)) = join(X, Y).
0.58/0.74	Proof:
0.58/0.74	  join(join(Y, X), join(X, Y))
0.58/0.74	= { by lemma 8 }
0.58/0.74	  join(join(meet(Y, join(X, Y)), X), join(X, Y))
0.58/0.74	= { by lemma 7 }
0.58/0.74	  join(join(meet(Y, join(X, Y)), meet(X, join(X, Y))), join(X, Y))
0.58/0.74	= { by axiom 1 (mckenzie4) }
0.58/0.74	  join(join(meet(Y, join(X, Y)), meet(meet(join(join(X, Y), meet(X, join(X, Y))), join(meet(X, join(X, Y)), join(join(join(X, Y), meet(X, join(X, Y))), ?))), meet(X, join(X, Y)))), join(X, Y))
0.58/0.74	= { by axiom 4 (mckenzie2) }
0.58/0.74	  join(join(meet(Y, join(X, Y)), meet(join(join(X, Y), meet(X, join(X, Y))), meet(X, join(X, Y)))), join(X, Y))
0.58/0.74	= { by lemma 6 }
0.58/0.74	  join(join(meet(Y, join(X, Y)), meet(join(X, Y), meet(X, join(X, Y)))), join(X, Y))
0.58/0.74	= { by lemma 7 }
0.58/0.74	  join(join(meet(Y, join(X, Y)), meet(join(X, Y), X)), join(X, Y))
0.58/0.74	= { by axiom 3 (mckenzie3) }
0.58/0.74	  join(X, Y)
0.58/0.74	
0.58/0.74	Lemma 10: join(X, Y) = join(Y, X).
0.58/0.74	Proof:
0.58/0.74	  join(X, Y)
0.58/0.74	= { by lemma 9 }
0.58/0.74	  join(join(Y, X), join(X, Y))
0.58/0.74	= { by lemma 6 }
0.58/0.74	  join(join(join(Y, X), join(X, Y)), meet(join(Y, X), join(join(Y, X), join(X, Y))))
0.58/0.74	= { by lemma 9 }
0.58/0.74	  join(join(X, Y), meet(join(Y, X), join(join(Y, X), join(X, Y))))
0.58/0.74	= { by lemma 7 }
0.58/0.74	  join(join(X, Y), join(Y, X))
0.58/0.74	= { by lemma 9 }
0.58/0.74	  join(Y, X)
0.58/0.74	
0.58/0.74	Lemma 11: meet(meet(Y, X), meet(X, Y)) = meet(X, Y).
0.58/0.74	Proof:
0.58/0.74	  meet(meet(Y, X), meet(X, Y))
0.58/0.74	= { by lemma 6 }
0.58/0.74	  meet(meet(join(Y, meet(X, Y)), X), meet(X, Y))
0.58/0.74	= { by lemma 5 }
0.58/0.74	  meet(meet(join(Y, meet(X, Y)), join(X, meet(X, Y))), meet(X, Y))
0.58/0.74	= { by lemma 10 }
0.58/0.74	  meet(meet(join(Y, meet(X, Y)), join(meet(X, Y), X)), meet(X, Y))
0.58/0.74	= { by axiom 1 (mckenzie4) }
0.58/0.74	  meet(X, Y)
0.58/0.74	
0.58/0.74	Lemma 12: meet(meet(X, Y), X) = meet(X, Y).
0.58/0.74	Proof:
0.58/0.74	  meet(meet(X, Y), X)
0.58/0.74	= { by lemma 5 }
0.58/0.74	  meet(meet(X, Y), join(X, meet(X, Y)))
0.58/0.74	= { by lemma 8 }
0.58/0.74	  meet(X, Y)
0.58/0.74	
0.58/0.74	Lemma 13: meet(X, Y) = meet(Y, X).
0.58/0.74	Proof:
0.58/0.74	  meet(X, Y)
0.58/0.74	= { by lemma 11 }
0.58/0.74	  meet(meet(Y, X), meet(X, Y))
0.58/0.74	= { by lemma 12 }
0.58/0.74	  meet(meet(meet(Y, X), meet(X, Y)), meet(Y, X))
0.58/0.74	= { by lemma 11 }
0.58/0.74	  meet(meet(X, Y), meet(Y, X))
0.58/0.74	= { by lemma 11 }
0.58/0.74	  meet(Y, X)
0.58/0.74	
0.58/0.74	Lemma 14: meet(meet(X, Y), meet(X, meet(Y, Z))) = meet(X, meet(Y, Z)).
0.58/0.74	Proof:
0.58/0.74	  meet(meet(X, Y), meet(X, meet(Y, Z)))
0.58/0.74	= { by lemma 13 }
0.58/0.74	  meet(meet(Y, X), meet(X, meet(Y, Z)))
0.58/0.74	= { by lemma 13 }
0.58/0.74	  meet(meet(X, meet(Y, Z)), meet(Y, X))
0.58/0.74	= { by lemma 5 }
0.58/0.74	  meet(meet(X, meet(Y, Z)), meet(Y, join(X, meet(X, meet(Y, Z)))))
0.58/0.74	= { by lemma 10 }
0.58/0.74	  meet(meet(X, meet(Y, Z)), meet(Y, join(meet(X, meet(Y, Z)), X)))
0.58/0.74	= { by lemma 13 }
0.58/0.74	  meet(meet(Y, join(meet(X, meet(Y, Z)), X)), meet(X, meet(Y, Z)))
0.58/0.74	= { by axiom 2 (mckenzie1) }
0.58/0.74	  meet(meet(join(Y, meet(X, meet(Y, Z))), join(meet(X, meet(Y, Z)), X)), meet(X, meet(Y, Z)))
0.58/0.74	= { by axiom 1 (mckenzie4) }
0.58/0.74	  meet(X, meet(Y, Z))
0.58/0.74	
0.58/0.74	Lemma 15: meet(meet(X, join(meet(X, Y), Z)), meet(X, Y)) = meet(X, Y).
0.58/0.74	Proof:
0.58/0.74	  meet(meet(X, join(meet(X, Y), Z)), meet(X, Y))
0.58/0.74	= { by lemma 5 }
0.58/0.74	  meet(meet(join(X, meet(X, Y)), join(meet(X, Y), Z)), meet(X, Y))
0.58/0.74	= { by axiom 1 (mckenzie4) }
0.58/0.74	  meet(X, Y)
0.58/0.74	
0.58/0.74	Lemma 16: meet(meet(X, Z), meet(Y, Z)) = meet(X, meet(Y, Z)).
0.58/0.74	Proof:
0.58/0.74	  meet(meet(X, Z), meet(Y, Z))
0.58/0.74	= { by lemma 13 }
0.58/0.74	  meet(meet(X, Z), meet(Z, Y))
0.58/0.74	= { by lemma 13 }
0.58/0.74	  meet(meet(Z, Y), meet(X, Z))
0.58/0.74	= { by lemma 14 }
0.58/0.74	  meet(meet(meet(Z, Y), X), meet(meet(Z, Y), meet(X, Z)))
0.58/0.74	= { by lemma 15 }
0.58/0.74	  meet(meet(meet(meet(Z, Y), join(meet(meet(Z, Y), X), meet(X, Z))), meet(meet(Z, Y), X)), meet(meet(Z, Y), meet(X, Z)))
0.58/0.74	= { by lemma 5 }
0.58/0.74	  meet(meet(meet(meet(Z, Y), join(meet(meet(Z, Y), X), meet(X, Z))), meet(meet(Z, Y), X)), meet(meet(Z, Y), join(meet(X, Z), meet(meet(X, Z), meet(X, meet(Z, Y))))))
0.58/0.74	= { by lemma 14 }
0.58/0.74	  meet(meet(meet(meet(Z, Y), join(meet(meet(Z, Y), X), meet(X, Z))), meet(meet(Z, Y), X)), meet(meet(Z, Y), join(meet(X, Z), meet(X, meet(Z, Y)))))
0.58/0.74	= { by lemma 13 }
0.58/0.74	  meet(meet(meet(meet(Z, Y), join(meet(meet(Z, Y), X), meet(X, Z))), meet(meet(Z, Y), X)), meet(meet(Z, Y), join(meet(X, Z), meet(meet(Z, Y), X))))
0.58/0.74	= { by lemma 10 }
0.58/0.74	  meet(meet(meet(meet(Z, Y), join(meet(meet(Z, Y), X), meet(X, Z))), meet(meet(Z, Y), X)), meet(meet(Z, Y), join(meet(meet(Z, Y), X), meet(X, Z))))
0.58/0.74	= { by lemma 12 }
0.58/0.74	  meet(meet(meet(Z, Y), join(meet(meet(Z, Y), X), meet(X, Z))), meet(meet(Z, Y), X))
0.58/0.74	= { by lemma 15 }
0.58/0.74	  meet(meet(Z, Y), X)
0.58/0.74	= { by lemma 13 }
0.58/0.74	  meet(X, meet(Z, Y))
0.58/0.74	= { by lemma 13 }
0.58/0.74	  meet(X, meet(Y, Z))
0.58/0.74	
0.58/0.74	Goal 1 (prove_associativity_of_meet): meet(a, meet(b, c)) = meet(meet(a, b), c).
0.58/0.74	Proof:
0.58/0.74	  meet(a, meet(b, c))
0.58/0.74	= { by lemma 13 }
0.58/0.74	  meet(a, meet(c, b))
0.58/0.74	= { by lemma 16 }
0.58/0.74	  meet(meet(a, b), meet(c, b))
0.58/0.74	= { by lemma 13 }
0.58/0.74	  meet(meet(c, b), meet(a, b))
0.58/0.74	= { by lemma 16 }
0.58/0.74	  meet(c, meet(a, b))
0.58/0.74	= { by lemma 13 }
0.58/0.74	  meet(meet(a, b), c)
0.58/0.74	% SZS output end Proof
0.58/0.74	
0.58/0.74	RESULT: Unsatisfiable (the axioms are contradictory).
0.58/0.75	EOF