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
0.13/0.34	% Computer   : n027.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 09:45:09 EDT 2019
0.13/0.34	% CPUTime    : 
0.54/0.71	% SZS status Unsatisfiable
0.54/0.71	
0.54/0.71	% SZS output start Proof
0.54/0.71	Take the following subset of the input axioms:
0.54/0.71	  fof(associativity, axiom, ![X, Y, Z]: multiply(multiply(X, Y), Z)=multiply(X, multiply(Y, Z))).
0.54/0.71	  fof(associativity_of_lub, axiom, ![X, Y, Z]: least_upper_bound(X, least_upper_bound(Y, Z))=least_upper_bound(least_upper_bound(X, Y), Z)).
0.54/0.71	  fof(idempotence_of_lub, axiom, ![X]: least_upper_bound(X, X)=X).
0.54/0.71	  fof(left_identity, axiom, ![X]: X=multiply(identity, X)).
0.54/0.71	  fof(left_inverse, axiom, ![X]: multiply(inverse(X), X)=identity).
0.54/0.71	  fof(lub_absorbtion, axiom, ![X, Y]: X=least_upper_bound(X, greatest_lower_bound(X, Y))).
0.54/0.71	  fof(monotony_lub1, axiom, ![X, Y, Z]: least_upper_bound(multiply(X, Y), multiply(X, Z))=multiply(X, least_upper_bound(Y, Z))).
0.54/0.71	  fof(monotony_lub2, axiom, ![X, Y, Z]: least_upper_bound(multiply(Y, X), multiply(Z, X))=multiply(least_upper_bound(Y, Z), X)).
0.54/0.71	  fof(p39d_1, hypothesis, b=greatest_lower_bound(a, b)).
0.54/0.71	  fof(prove_p39d, negated_conjecture, inverse(b)!=least_upper_bound(inverse(a), inverse(b))).
0.54/0.71	  fof(symmetry_of_glb, axiom, ![X, Y]: greatest_lower_bound(Y, X)=greatest_lower_bound(X, Y)).
0.54/0.71	  fof(symmetry_of_lub, axiom, ![X, Y]: least_upper_bound(X, Y)=least_upper_bound(Y, X)).
0.54/0.71	
0.54/0.71	Now clausify the problem and encode Horn clauses using encoding 3 of
0.54/0.71	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.54/0.71	We repeatedly replace C & s=t => u=v by the two clauses:
0.54/0.71	  fresh(y, y, x1...xn) = u
0.54/0.71	  C => fresh(s, t, x1...xn) = v
0.54/0.71	where fresh is a fresh function symbol and x1..xn are the free
0.54/0.71	variables of u and v.
0.54/0.71	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.54/0.71	input problem has no model of domain size 1).
0.54/0.71	
0.54/0.71	The encoding turns the above axioms into the following unit equations and goals:
0.54/0.71	
0.54/0.71	Axiom 1 (left_inverse): multiply(inverse(X), X) = identity.
0.54/0.71	Axiom 2 (associativity): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
0.54/0.71	Axiom 3 (left_identity): X = multiply(identity, X).
0.54/0.71	Axiom 4 (symmetry_of_lub): least_upper_bound(X, Y) = least_upper_bound(Y, X).
0.54/0.71	Axiom 5 (lub_absorbtion): X = least_upper_bound(X, greatest_lower_bound(X, Y)).
0.54/0.71	Axiom 6 (symmetry_of_glb): greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X).
0.54/0.71	Axiom 7 (idempotence_of_lub): least_upper_bound(X, X) = X.
0.54/0.71	Axiom 8 (monotony_lub2): least_upper_bound(multiply(X, Y), multiply(Z, Y)) = multiply(least_upper_bound(X, Z), Y).
0.54/0.71	Axiom 9 (monotony_lub1): least_upper_bound(multiply(X, Y), multiply(X, Z)) = multiply(X, least_upper_bound(Y, Z)).
0.54/0.71	Axiom 10 (associativity_of_lub): least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z).
0.54/0.71	Axiom 11 (p39d_1): b = greatest_lower_bound(a, b).
0.54/0.71	
0.54/0.71	Lemma 12: multiply(inverse(X), multiply(X, Y)) = Y.
0.54/0.71	Proof:
0.54/0.71	  multiply(inverse(X), multiply(X, Y))
0.54/0.71	= { by axiom 2 (associativity) }
0.54/0.71	  multiply(multiply(inverse(X), X), Y)
0.54/0.71	= { by axiom 1 (left_inverse) }
0.54/0.71	  multiply(identity, Y)
0.54/0.71	= { by axiom 3 (left_identity) }
0.54/0.71	  Y
0.54/0.71	
0.54/0.71	Lemma 13: multiply(inverse(inverse(X)), identity) = X.
0.54/0.71	Proof:
0.54/0.71	  multiply(inverse(inverse(X)), identity)
0.54/0.71	= { by axiom 1 (left_inverse) }
0.54/0.71	  multiply(inverse(inverse(X)), multiply(inverse(X), X))
0.54/0.71	= { by lemma 12 }
0.54/0.71	  X
0.54/0.71	
0.54/0.71	Lemma 14: multiply(X, identity) = X.
0.54/0.71	Proof:
0.54/0.71	  multiply(X, identity)
0.54/0.71	= { by lemma 12 }
0.54/0.71	  multiply(inverse(inverse(X)), multiply(inverse(X), multiply(X, identity)))
0.54/0.71	= { by lemma 12 }
0.54/0.71	  multiply(inverse(inverse(X)), identity)
0.54/0.71	= { by lemma 13 }
0.54/0.71	  X
0.54/0.71	
0.54/0.71	Goal 1 (prove_p39d): inverse(b) = least_upper_bound(inverse(a), inverse(b)).
0.54/0.71	Proof:
0.54/0.71	  inverse(b)
0.54/0.71	= { by axiom 3 (left_identity) }
0.54/0.71	  multiply(identity, inverse(b))
0.54/0.71	= { by axiom 1 (left_inverse) }
0.54/0.71	  multiply(multiply(inverse(least_upper_bound(inverse(inverse(b)), a)), least_upper_bound(inverse(inverse(b)), a)), inverse(b))
0.54/0.71	= { by axiom 7 (idempotence_of_lub) }
0.54/0.71	  multiply(multiply(inverse(least_upper_bound(inverse(inverse(b)), a)), least_upper_bound(least_upper_bound(inverse(inverse(b)), inverse(inverse(b))), a)), inverse(b))
0.54/0.71	= { by axiom 10 (associativity_of_lub) }
0.54/0.71	  multiply(multiply(inverse(least_upper_bound(inverse(inverse(b)), a)), least_upper_bound(inverse(inverse(b)), least_upper_bound(inverse(inverse(b)), a))), inverse(b))
0.54/0.71	= { by axiom 4 (symmetry_of_lub) }
0.54/0.71	  multiply(multiply(inverse(least_upper_bound(inverse(inverse(b)), a)), least_upper_bound(least_upper_bound(inverse(inverse(b)), a), inverse(inverse(b)))), inverse(b))
0.54/0.71	= { by axiom 9 (monotony_lub1) }
0.54/0.71	  multiply(least_upper_bound(multiply(inverse(least_upper_bound(inverse(inverse(b)), a)), least_upper_bound(inverse(inverse(b)), a)), multiply(inverse(least_upper_bound(inverse(inverse(b)), a)), inverse(inverse(b)))), inverse(b))
0.54/0.71	= { by axiom 1 (left_inverse) }
0.54/0.71	  multiply(least_upper_bound(identity, multiply(inverse(least_upper_bound(inverse(inverse(b)), a)), inverse(inverse(b)))), inverse(b))
0.54/0.71	= { by axiom 4 (symmetry_of_lub) }
0.54/0.71	  multiply(least_upper_bound(multiply(inverse(least_upper_bound(inverse(inverse(b)), a)), inverse(inverse(b))), identity), inverse(b))
0.54/0.71	= { by axiom 8 (monotony_lub2) }
0.54/0.71	  least_upper_bound(multiply(multiply(inverse(least_upper_bound(inverse(inverse(b)), a)), inverse(inverse(b))), inverse(b)), multiply(identity, inverse(b)))
0.54/0.71	= { by axiom 2 (associativity) }
0.54/0.71	  least_upper_bound(multiply(inverse(least_upper_bound(inverse(inverse(b)), a)), multiply(inverse(inverse(b)), inverse(b))), multiply(identity, inverse(b)))
0.54/0.71	= { by axiom 1 (left_inverse) }
0.54/0.71	  least_upper_bound(multiply(inverse(least_upper_bound(inverse(inverse(b)), a)), identity), multiply(identity, inverse(b)))
0.54/0.71	= { by lemma 14 }
0.54/0.71	  least_upper_bound(inverse(least_upper_bound(inverse(inverse(b)), a)), multiply(identity, inverse(b)))
0.54/0.71	= { by axiom 3 (left_identity) }
0.54/0.71	  least_upper_bound(inverse(least_upper_bound(inverse(inverse(b)), a)), inverse(b))
0.54/0.71	= { by axiom 4 (symmetry_of_lub) }
0.54/0.71	  least_upper_bound(inverse(b), inverse(least_upper_bound(inverse(inverse(b)), a)))
0.54/0.71	= { by axiom 4 (symmetry_of_lub) }
0.54/0.71	  least_upper_bound(inverse(b), inverse(least_upper_bound(a, inverse(inverse(b)))))
0.54/0.71	= { by lemma 14 }
0.54/0.71	  least_upper_bound(inverse(b), inverse(least_upper_bound(a, multiply(inverse(inverse(b)), identity))))
0.54/0.71	= { by lemma 13 }
0.54/0.71	  least_upper_bound(inverse(b), inverse(least_upper_bound(a, b)))
0.54/0.71	= { by axiom 11 (p39d_1) }
0.54/0.71	  least_upper_bound(inverse(b), inverse(least_upper_bound(a, greatest_lower_bound(a, b))))
0.54/0.71	= { by axiom 5 (lub_absorbtion) }
0.54/0.71	  least_upper_bound(inverse(b), inverse(a))
0.54/0.71	= { by axiom 4 (symmetry_of_lub) }
0.54/0.71	  least_upper_bound(inverse(a), inverse(b))
0.54/0.71	% SZS output end Proof
0.54/0.71	
0.54/0.71	RESULT: Unsatisfiable (the axioms are contradictory).
0.54/0.72	EOF