0.11/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.11/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.14/0.34	% Computer   : n024.cluster.edu
0.14/0.34	% Model      : x86_64 x86_64
0.14/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.14/0.34	% Memory     : 8042.1875MB
0.14/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.14/0.34	% CPULimit   : 180
0.14/0.34	% DateTime   : Thu Aug 29 09:15:20 EDT 2019
0.14/0.34	% CPUTime    : 
0.14/0.36	% SZS status Unsatisfiable
0.14/0.36	
0.14/0.36	% SZS output start Proof
0.14/0.36	Take the following subset of the input axioms:
0.14/0.36	  fof(absorption, axiom, ![X, Y]: X=meet(X, join(X, Y))).
0.14/0.36	  fof(distribution, axiom, ![X, Y, Z]: join(meet(Z, X), meet(Y, X))=meet(X, join(Y, Z))).
0.14/0.36	  fof(prove_absorbtion_dual, negated_conjecture, a!=join(a, meet(a, b))).
0.14/0.36	
0.14/0.36	Now clausify the problem and encode Horn clauses using encoding 3 of
0.14/0.36	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.14/0.36	We repeatedly replace C & s=t => u=v by the two clauses:
0.14/0.36	  fresh(y, y, x1...xn) = u
0.14/0.36	  C => fresh(s, t, x1...xn) = v
0.14/0.36	where fresh is a fresh function symbol and x1..xn are the free
0.14/0.36	variables of u and v.
0.14/0.36	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.14/0.36	input problem has no model of domain size 1).
0.14/0.36	
0.14/0.36	The encoding turns the above axioms into the following unit equations and goals:
0.14/0.36	
0.14/0.36	Axiom 1 (distribution): join(meet(X, Y), meet(Z, Y)) = meet(Y, join(Z, X)).
0.14/0.37	Axiom 2 (absorption): X = meet(X, join(X, Y)).
0.14/0.37	
0.14/0.37	Lemma 3: meet(meet(X, Y), Y) = meet(X, Y).
0.14/0.37	Proof:
0.14/0.37	  meet(meet(X, Y), Y)
0.14/0.37	= { by axiom 2 (absorption) }
0.14/0.37	  meet(meet(X, Y), meet(Y, join(Y, X)))
0.14/0.37	= { by axiom 1 (distribution) }
0.14/0.37	  meet(meet(X, Y), join(meet(X, Y), meet(Y, Y)))
0.14/0.37	= { by axiom 2 (absorption) }
0.14/0.37	  meet(X, Y)
0.14/0.37	
0.14/0.37	Lemma 4: meet(X, join(Y, X)) = meet(X, X).
0.14/0.37	Proof:
0.14/0.37	  meet(X, join(Y, X))
0.14/0.37	= { by axiom 1 (distribution) }
0.14/0.37	  join(meet(X, X), meet(Y, X))
0.14/0.37	= { by lemma 3 }
0.14/0.37	  join(meet(meet(X, X), X), meet(Y, X))
0.14/0.37	= { by lemma 3 }
0.14/0.37	  join(meet(meet(X, X), X), meet(meet(Y, X), X))
0.14/0.37	= { by axiom 1 (distribution) }
0.14/0.37	  meet(X, join(meet(Y, X), meet(X, X)))
0.14/0.37	= { by axiom 1 (distribution) }
0.14/0.37	  meet(X, meet(X, join(X, Y)))
0.14/0.37	= { by axiom 2 (absorption) }
0.14/0.37	  meet(X, X)
0.14/0.37	
0.14/0.37	Lemma 5: meet(X, X) = X.
0.14/0.37	Proof:
0.14/0.37	  meet(X, X)
0.14/0.37	= { by lemma 4 }
0.14/0.37	  meet(X, join(X, X))
0.14/0.37	= { by axiom 2 (absorption) }
0.14/0.37	  X
0.14/0.37	
0.14/0.37	Lemma 6: join(X, meet(Y, X)) = X.
0.14/0.37	Proof:
0.14/0.37	  join(X, meet(Y, X))
0.14/0.37	= { by lemma 5 }
0.14/0.37	  join(meet(X, X), meet(Y, X))
0.14/0.37	= { by axiom 1 (distribution) }
0.14/0.37	  meet(X, join(Y, X))
0.14/0.37	= { by lemma 4 }
0.14/0.37	  meet(X, X)
0.14/0.37	= { by lemma 5 }
0.14/0.37	  X
0.14/0.37	
0.14/0.37	Lemma 7: join(X, X) = X.
0.14/0.37	Proof:
0.14/0.37	  join(X, X)
0.14/0.37	= { by axiom 2 (absorption) }
0.14/0.37	  join(meet(X, join(X, meet(?, X))), X)
0.14/0.37	= { by axiom 2 (absorption) }
0.14/0.37	  join(meet(X, join(X, meet(?, X))), meet(X, join(X, meet(?, X))))
0.14/0.37	= { by axiom 1 (distribution) }
0.14/0.37	  meet(join(X, meet(?, X)), join(X, X))
0.14/0.37	= { by lemma 6 }
0.14/0.37	  meet(X, join(X, X))
0.14/0.37	= { by lemma 4 }
0.14/0.37	  meet(X, X)
0.14/0.37	= { by lemma 5 }
0.14/0.37	  X
0.14/0.37	
0.14/0.37	Goal 1 (prove_absorbtion_dual): a = join(a, meet(a, b)).
0.14/0.37	Proof:
0.14/0.37	  a
0.14/0.37	= { by lemma 6 }
0.14/0.37	  join(a, meet(b, a))
0.14/0.37	= { by lemma 7 }
0.14/0.37	  join(a, join(meet(b, a), meet(b, a)))
0.14/0.37	= { by axiom 1 (distribution) }
0.14/0.37	  join(a, meet(a, join(b, b)))
0.14/0.37	= { by lemma 7 }
0.14/0.37	  join(a, meet(a, b))
0.14/0.37	% SZS output end Proof
0.14/0.37	
0.14/0.37	RESULT: Unsatisfiable (the axioms are contradictory).
0.14/0.37	EOF