0.07/0.12	% 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
0.13/0.34	% Computer   : n013.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 12:47:16 EDT 2019
0.13/0.34	% CPUTime    : 
0.47/0.68	% SZS status Unsatisfiable
0.47/0.68	
0.47/0.68	% SZS output start Proof
0.47/0.68	Take the following subset of the input axioms:
0.63/0.80	  fof(associativity, axiom, ![X, Y, Z]: multiply(multiply(X, Y), Z)=multiply(X, multiply(Y, Z))).
0.63/0.80	  fof(associativity_of_glb, axiom, ![X, Y, Z]: greatest_lower_bound(X, greatest_lower_bound(Y, Z))=greatest_lower_bound(greatest_lower_bound(X, Y), Z)).
0.63/0.80	  fof(glb_absorbtion, axiom, ![X, Y]: greatest_lower_bound(X, least_upper_bound(X, Y))=X).
0.63/0.80	  fof(idempotence_of_gld, axiom, ![X]: X=greatest_lower_bound(X, X)).
0.63/0.80	  fof(lat4_1, axiom, ![X]: positive_part(X)=least_upper_bound(X, identity)).
0.63/0.80	  fof(lat4_2, axiom, ![X]: negative_part(X)=greatest_lower_bound(X, identity)).
0.63/0.80	  fof(lat4_3, axiom, ![X, Y, Z]: least_upper_bound(X, greatest_lower_bound(Y, Z))=greatest_lower_bound(least_upper_bound(X, Y), least_upper_bound(X, Z))).
0.63/0.80	  fof(lat4_4, axiom, ![X, Y, Z]: greatest_lower_bound(X, least_upper_bound(Y, Z))=least_upper_bound(greatest_lower_bound(X, Y), greatest_lower_bound(X, Z))).
0.63/0.80	  fof(left_identity, axiom, ![X]: X=multiply(identity, X)).
0.63/0.80	  fof(left_inverse, axiom, ![X]: multiply(inverse(X), X)=identity).
0.63/0.80	  fof(monotony_glb1, axiom, ![X, Y, Z]: multiply(X, greatest_lower_bound(Y, Z))=greatest_lower_bound(multiply(X, Y), multiply(X, Z))).
0.63/0.80	  fof(monotony_glb2, axiom, ![X, Y, Z]: greatest_lower_bound(multiply(Y, X), multiply(Z, X))=multiply(greatest_lower_bound(Y, Z), X)).
0.63/0.80	  fof(monotony_lub2, axiom, ![X, Y, Z]: least_upper_bound(multiply(Y, X), multiply(Z, X))=multiply(least_upper_bound(Y, Z), X)).
0.63/0.80	  fof(prove_lat4, negated_conjecture, multiply(positive_part(a), negative_part(a))!=a).
0.63/0.80	  fof(symmetry_of_glb, axiom, ![X, Y]: greatest_lower_bound(Y, X)=greatest_lower_bound(X, Y)).
0.63/0.80	  fof(symmetry_of_lub, axiom, ![X, Y]: least_upper_bound(X, Y)=least_upper_bound(Y, X)).
0.63/0.80	
0.63/0.80	Now clausify the problem and encode Horn clauses using encoding 3 of
0.63/0.80	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.63/0.80	We repeatedly replace C & s=t => u=v by the two clauses:
0.63/0.80	  fresh(y, y, x1...xn) = u
0.63/0.80	  C => fresh(s, t, x1...xn) = v
0.63/0.80	where fresh is a fresh function symbol and x1..xn are the free
0.63/0.80	variables of u and v.
0.63/0.80	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.63/0.80	input problem has no model of domain size 1).
0.63/0.80	
0.63/0.80	The encoding turns the above axioms into the following unit equations and goals:
0.63/0.80	
0.63/0.80	Axiom 1 (left_inverse): multiply(inverse(X), X) = identity.
0.63/0.80	Axiom 2 (associativity): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
0.63/0.80	Axiom 3 (left_identity): X = multiply(identity, X).
0.63/0.80	Axiom 4 (symmetry_of_lub): least_upper_bound(X, Y) = least_upper_bound(Y, X).
0.63/0.80	Axiom 5 (glb_absorbtion): greatest_lower_bound(X, least_upper_bound(X, Y)) = X.
0.63/0.80	Axiom 6 (idempotence_of_gld): X = greatest_lower_bound(X, X).
0.63/0.80	Axiom 7 (monotony_glb1): multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)).
0.63/0.80	Axiom 8 (symmetry_of_glb): greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X).
0.63/0.80	Axiom 9 (monotony_lub2): least_upper_bound(multiply(X, Y), multiply(Z, Y)) = multiply(least_upper_bound(X, Z), Y).
0.63/0.80	Axiom 10 (monotony_glb2): greatest_lower_bound(multiply(X, Y), multiply(Z, Y)) = multiply(greatest_lower_bound(X, Z), Y).
0.63/0.80	Axiom 11 (associativity_of_glb): greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z).
0.63/0.80	Axiom 12 (lat4_2): negative_part(X) = greatest_lower_bound(X, identity).
0.63/0.80	Axiom 13 (lat4_4): greatest_lower_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(greatest_lower_bound(X, Y), greatest_lower_bound(X, Z)).
0.63/0.80	Axiom 14 (lat4_1): positive_part(X) = least_upper_bound(X, identity).
0.63/0.80	Axiom 15 (lat4_3): least_upper_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(least_upper_bound(X, Y), least_upper_bound(X, Z)).
0.63/0.80	
0.63/0.80	Lemma 16: multiply(inverse(X), multiply(X, Y)) = Y.
0.63/0.80	Proof:
0.63/0.80	  multiply(inverse(X), multiply(X, Y))
0.63/0.80	= { by axiom 2 (associativity) }
0.63/0.80	  multiply(multiply(inverse(X), X), Y)
0.63/0.80	= { by axiom 1 (left_inverse) }
0.63/0.80	  multiply(identity, Y)
0.63/0.80	= { by axiom 3 (left_identity) }
0.63/0.80	  Y
0.63/0.80	
0.63/0.80	Lemma 17: multiply(inverse(inverse(X)), Y) = multiply(X, Y).
0.63/0.80	Proof:
0.63/0.80	  multiply(inverse(inverse(X)), Y)
0.63/0.80	= { by lemma 16 }
0.63/0.80	  multiply(inverse(inverse(X)), multiply(inverse(X), multiply(X, Y)))
0.63/0.80	= { by lemma 16 }
0.63/0.80	  multiply(X, Y)
0.63/0.80	
0.63/0.80	Lemma 18: multiply(inverse(inverse(X)), identity) = X.
0.63/0.80	Proof:
0.63/0.80	  multiply(inverse(inverse(X)), identity)
0.63/0.80	= { by axiom 1 (left_inverse) }
0.63/0.80	  multiply(inverse(inverse(X)), multiply(inverse(X), X))
0.63/0.80	= { by lemma 16 }
0.63/0.80	  X
0.63/0.80	
0.63/0.80	Lemma 19: multiply(X, identity) = X.
0.63/0.80	Proof:
0.63/0.80	  multiply(X, identity)
0.63/0.80	= { by lemma 17 }
0.63/0.80	  multiply(inverse(inverse(X)), identity)
0.63/0.80	= { by lemma 18 }
0.63/0.80	  X
0.63/0.80	
0.63/0.80	Lemma 20: least_upper_bound(identity, X) = positive_part(X).
0.63/0.80	Proof:
0.63/0.80	  least_upper_bound(identity, X)
0.63/0.80	= { by axiom 4 (symmetry_of_lub) }
0.63/0.80	  least_upper_bound(X, identity)
0.63/0.80	= { by axiom 14 (lat4_1) }
0.63/0.80	  positive_part(X)
0.63/0.80	
0.63/0.80	Lemma 21: least_upper_bound(Y, multiply(X, Y)) = multiply(positive_part(X), Y).
0.63/0.80	Proof:
0.63/0.80	  least_upper_bound(Y, multiply(X, Y))
0.63/0.80	= { by axiom 3 (left_identity) }
0.63/0.80	  least_upper_bound(multiply(identity, Y), multiply(X, Y))
0.63/0.80	= { by axiom 9 (monotony_lub2) }
0.63/0.80	  multiply(least_upper_bound(identity, X), Y)
0.63/0.80	= { by lemma 20 }
0.63/0.80	  multiply(positive_part(X), Y)
0.63/0.80	
0.63/0.80	Lemma 22: greatest_lower_bound(identity, X) = negative_part(X).
0.63/0.80	Proof:
0.63/0.80	  greatest_lower_bound(identity, X)
0.63/0.80	= { by axiom 8 (symmetry_of_glb) }
0.63/0.80	  greatest_lower_bound(X, identity)
0.63/0.80	= { by axiom 12 (lat4_2) }
0.63/0.80	  negative_part(X)
0.63/0.80	
0.63/0.80	Lemma 23: multiply(inverse(X), greatest_lower_bound(X, Y)) = negative_part(multiply(inverse(X), Y)).
0.63/0.80	Proof:
0.63/0.80	  multiply(inverse(X), greatest_lower_bound(X, Y))
0.63/0.80	= { by axiom 7 (monotony_glb1) }
0.63/0.80	  greatest_lower_bound(multiply(inverse(X), X), multiply(inverse(X), Y))
0.63/0.80	= { by axiom 1 (left_inverse) }
0.63/0.80	  greatest_lower_bound(identity, multiply(inverse(X), Y))
0.63/0.80	= { by lemma 22 }
0.63/0.80	  negative_part(multiply(inverse(X), Y))
0.63/0.80	
0.63/0.80	Lemma 24: greatest_lower_bound(Y, multiply(X, Y)) = multiply(negative_part(X), Y).
0.63/0.80	Proof:
0.63/0.80	  greatest_lower_bound(Y, multiply(X, Y))
0.63/0.80	= { by axiom 3 (left_identity) }
0.63/0.80	  greatest_lower_bound(multiply(identity, Y), multiply(X, Y))
0.63/0.80	= { by axiom 10 (monotony_glb2) }
0.63/0.80	  multiply(greatest_lower_bound(identity, X), Y)
0.63/0.80	= { by lemma 22 }
0.63/0.80	  multiply(negative_part(X), Y)
0.63/0.80	
0.63/0.80	Lemma 25: multiply(inverse(X), negative_part(X)) = negative_part(inverse(X)).
0.63/0.80	Proof:
0.63/0.80	  multiply(inverse(X), negative_part(X))
0.63/0.80	= { by axiom 12 (lat4_2) }
0.63/0.80	  multiply(inverse(X), greatest_lower_bound(X, identity))
0.63/0.80	= { by lemma 23 }
0.63/0.80	  negative_part(multiply(inverse(X), identity))
0.63/0.80	= { by lemma 22 }
0.63/0.80	  greatest_lower_bound(identity, multiply(inverse(X), identity))
0.63/0.80	= { by lemma 24 }
0.63/0.80	  multiply(negative_part(inverse(X)), identity)
0.63/0.80	= { by lemma 19 }
0.63/0.80	  negative_part(inverse(X))
0.63/0.80	
0.63/0.80	Lemma 26: least_upper_bound(negative_part(X), negative_part(Y)) = negative_part(least_upper_bound(X, Y)).
0.63/0.80	Proof:
0.63/0.80	  least_upper_bound(negative_part(X), negative_part(Y))
0.63/0.80	= { by lemma 22 }
0.63/0.80	  least_upper_bound(greatest_lower_bound(identity, X), negative_part(Y))
0.63/0.80	= { by lemma 22 }
0.63/0.80	  least_upper_bound(greatest_lower_bound(identity, X), greatest_lower_bound(identity, Y))
0.63/0.80	= { by axiom 13 (lat4_4) }
0.63/0.80	  greatest_lower_bound(identity, least_upper_bound(X, Y))
0.63/0.80	= { by lemma 22 }
0.71/0.90	  negative_part(least_upper_bound(X, Y))
0.71/0.90	
0.71/0.90	Goal 1 (prove_lat4): multiply(positive_part(a), negative_part(a)) = a.
0.71/0.90	Proof:
0.71/0.90	  multiply(positive_part(a), negative_part(a))
0.71/0.90	= { by lemma 18 }
0.71/0.90	  multiply(multiply(inverse(inverse(positive_part(a))), identity), negative_part(a))
0.71/0.90	= { by lemma 19 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), identity)), identity), negative_part(a))
0.71/0.90	= { by axiom 5 (glb_absorbtion) }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), greatest_lower_bound(identity, least_upper_bound(identity, greatest_lower_bound(a, inverse(a)))))), identity), negative_part(a))
0.71/0.90	= { by lemma 20 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), greatest_lower_bound(identity, positive_part(greatest_lower_bound(a, inverse(a)))))), identity), negative_part(a))
0.71/0.90	= { by lemma 22 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(positive_part(greatest_lower_bound(a, inverse(a)))))), identity), negative_part(a))
0.71/0.90	= { by lemma 20 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(least_upper_bound(identity, greatest_lower_bound(a, inverse(a)))))), identity), negative_part(a))
0.71/0.90	= { by axiom 15 (lat4_3) }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(greatest_lower_bound(least_upper_bound(identity, a), least_upper_bound(identity, inverse(a)))))), identity), negative_part(a))
0.71/0.90	= { by lemma 20 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(greatest_lower_bound(positive_part(a), least_upper_bound(identity, inverse(a)))))), identity), negative_part(a))
0.71/0.90	= { by lemma 20 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(greatest_lower_bound(positive_part(a), positive_part(inverse(a)))))), identity), negative_part(a))
0.71/0.90	= { by lemma 18 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(greatest_lower_bound(positive_part(a), multiply(inverse(inverse(positive_part(inverse(a)))), identity))))), identity), negative_part(a))
0.71/0.90	= { by lemma 19 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(greatest_lower_bound(positive_part(a), inverse(inverse(positive_part(inverse(a)))))))), identity), negative_part(a))
0.71/0.90	= { by axiom 8 (symmetry_of_glb) }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(greatest_lower_bound(inverse(inverse(positive_part(inverse(a)))), positive_part(a))))), identity), negative_part(a))
0.71/0.90	= { by lemma 16 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(multiply(inverse(inverse(inverse(inverse(positive_part(inverse(a)))))), multiply(inverse(inverse(inverse(positive_part(inverse(a))))), greatest_lower_bound(inverse(inverse(positive_part(inverse(a)))), positive_part(a))))))), identity), negative_part(a))
0.71/0.90	= { by lemma 23 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(multiply(inverse(inverse(inverse(inverse(positive_part(inverse(a)))))), negative_part(multiply(inverse(inverse(inverse(positive_part(inverse(a))))), positive_part(a))))))), identity), negative_part(a))
0.71/0.90	= { by lemma 17 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(multiply(inverse(inverse(positive_part(inverse(a)))), negative_part(multiply(inverse(inverse(inverse(positive_part(inverse(a))))), positive_part(a))))))), identity), negative_part(a))
0.71/0.90	= { by lemma 17 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(multiply(inverse(inverse(positive_part(inverse(a)))), negative_part(multiply(inverse(positive_part(inverse(a))), positive_part(a))))))), identity), negative_part(a))
0.71/0.90	= { by axiom 14 (lat4_1) }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(multiply(inverse(inverse(positive_part(inverse(a)))), negative_part(multiply(inverse(positive_part(inverse(a))), least_upper_bound(a, identity))))))), identity), negative_part(a))
0.71/0.90	= { by axiom 1 (left_inverse) }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(multiply(inverse(inverse(positive_part(inverse(a)))), negative_part(multiply(inverse(positive_part(inverse(a))), least_upper_bound(a, multiply(inverse(a), a)))))))), identity), negative_part(a))
0.71/0.90	= { by lemma 21 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(multiply(inverse(inverse(positive_part(inverse(a)))), negative_part(multiply(inverse(positive_part(inverse(a))), multiply(positive_part(inverse(a)), a))))))), identity), negative_part(a))
0.71/0.90	= { by lemma 16 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(multiply(inverse(inverse(positive_part(inverse(a)))), negative_part(a))))), identity), negative_part(a))
0.71/0.90	= { by lemma 17 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(multiply(positive_part(inverse(a)), negative_part(a))))), identity), negative_part(a))
0.71/0.90	= { by lemma 21 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(least_upper_bound(negative_part(a), multiply(inverse(a), negative_part(a)))))), identity), negative_part(a))
0.71/0.90	= { by lemma 25 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(least_upper_bound(negative_part(a), negative_part(inverse(a)))))), identity), negative_part(a))
0.71/0.90	= { by lemma 26 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(negative_part(least_upper_bound(a, inverse(a)))))), identity), negative_part(a))
0.71/0.90	= { by axiom 12 (lat4_2) }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(greatest_lower_bound(least_upper_bound(a, inverse(a)), identity)))), identity), negative_part(a))
0.71/0.90	= { by axiom 12 (lat4_2) }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), greatest_lower_bound(greatest_lower_bound(least_upper_bound(a, inverse(a)), identity), identity))), identity), negative_part(a))
0.71/0.90	= { by axiom 11 (associativity_of_glb) }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), greatest_lower_bound(least_upper_bound(a, inverse(a)), greatest_lower_bound(identity, identity)))), identity), negative_part(a))
0.71/0.90	= { by axiom 6 (idempotence_of_gld) }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), greatest_lower_bound(least_upper_bound(a, inverse(a)), identity))), identity), negative_part(a))
0.71/0.90	= { by axiom 12 (lat4_2) }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(least_upper_bound(a, inverse(a))))), identity), negative_part(a))
0.71/0.90	= { by axiom 4 (symmetry_of_lub) }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), negative_part(least_upper_bound(inverse(a), a)))), identity), negative_part(a))
0.71/0.90	= { by lemma 26 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), least_upper_bound(negative_part(inverse(a)), negative_part(a)))), identity), negative_part(a))
0.71/0.90	= { by lemma 16 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), least_upper_bound(negative_part(inverse(a)), multiply(inverse(inverse(a)), multiply(inverse(a), negative_part(a)))))), identity), negative_part(a))
0.71/0.90	= { by lemma 25 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), least_upper_bound(negative_part(inverse(a)), multiply(inverse(inverse(a)), negative_part(inverse(a)))))), identity), negative_part(a))
0.71/0.90	= { by lemma 17 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), least_upper_bound(negative_part(inverse(a)), multiply(a, negative_part(inverse(a)))))), identity), negative_part(a))
0.71/0.90	= { by lemma 21 }
0.71/0.90	  multiply(multiply(inverse(multiply(inverse(positive_part(a)), multiply(positive_part(a), negative_part(inverse(a))))), identity), negative_part(a))
0.71/0.90	= { by lemma 16 }
0.71/0.90	  multiply(multiply(inverse(negative_part(inverse(a))), identity), negative_part(a))
0.71/0.90	= { by lemma 19 }
0.71/0.90	  multiply(inverse(negative_part(inverse(a))), negative_part(a))
0.71/0.90	= { by axiom 12 (lat4_2) }
0.71/0.90	  multiply(inverse(negative_part(inverse(a))), greatest_lower_bound(a, identity))
0.71/0.90	= { by axiom 1 (left_inverse) }
0.71/0.90	  multiply(inverse(negative_part(inverse(a))), greatest_lower_bound(a, multiply(inverse(a), a)))
0.71/0.90	= { by lemma 24 }
0.71/0.90	  multiply(inverse(negative_part(inverse(a))), multiply(negative_part(inverse(a)), a))
0.71/0.90	= { by lemma 16 }
0.71/0.90	  a
0.71/0.90	% SZS output end Proof
0.71/0.90	
0.71/0.90	RESULT: Unsatisfiable (the axioms are contradictory).
0.71/0.90	EOF