0.06/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.06/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.33	% Computer   : n023.cluster.edu
0.13/0.33	% Model      : x86_64 x86_64
0.13/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.33	% Memory     : 8042.1875MB
0.13/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.33	% CPULimit   : 180
0.13/0.33	% DateTime   : Thu Aug 29 12:55:37 EDT 2019
0.13/0.33	% CPUTime    : 
1.82/1.99	% SZS status Unsatisfiable
1.82/1.99	
1.82/1.99	% SZS output start Proof
1.82/1.99	Take the following subset of the input axioms:
1.82/1.99	  fof(associativity, axiom, ![X, Y, Z]: multiply(multiply(X, Y), Z)=multiply(X, multiply(Y, Z))).
1.82/1.99	  fof(left_identity, axiom, ![X]: X=multiply(identity, X)).
1.82/1.99	  fof(left_inverse, axiom, ![X]: multiply(inverse(X), X)=identity).
1.82/1.99	  fof(monotony_glb1, axiom, ![X, Y, Z]: multiply(X, greatest_lower_bound(Y, Z))=greatest_lower_bound(multiply(X, Y), multiply(X, Z))).
1.82/1.99	  fof(monotony_glb2, axiom, ![X, Y, Z]: greatest_lower_bound(multiply(Y, X), multiply(Z, X))=multiply(greatest_lower_bound(Y, Z), X)).
1.82/1.99	  fof(p12x_1, hypothesis, greatest_lower_bound(b, c)=greatest_lower_bound(a, c)).
1.82/1.99	  fof(p12x_2, hypothesis, least_upper_bound(a, c)=least_upper_bound(b, c)).
1.82/1.99	  fof(p12x_4, hypothesis, ![X, Y]: greatest_lower_bound(inverse(X), inverse(Y))=inverse(least_upper_bound(X, Y))).
1.82/1.99	  fof(prove_p12x, negated_conjecture, a!=b).
1.82/1.99	  fof(symmetry_of_glb, axiom, ![X, Y]: greatest_lower_bound(Y, X)=greatest_lower_bound(X, Y)).
1.82/1.99	  fof(symmetry_of_lub, axiom, ![X, Y]: least_upper_bound(X, Y)=least_upper_bound(Y, X)).
1.82/1.99	
1.82/1.99	Now clausify the problem and encode Horn clauses using encoding 3 of
1.82/1.99	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
1.82/1.99	We repeatedly replace C & s=t => u=v by the two clauses:
1.82/1.99	  fresh(y, y, x1...xn) = u
1.82/1.99	  C => fresh(s, t, x1...xn) = v
1.82/1.99	where fresh is a fresh function symbol and x1..xn are the free
1.82/1.99	variables of u and v.
1.82/1.99	A predicate p(X) is encoded as p(X)=true (this is sound, because the
1.82/1.99	input problem has no model of domain size 1).
1.82/1.99	
1.82/1.99	The encoding turns the above axioms into the following unit equations and goals:
1.82/1.99	
1.82/2.00	Axiom 1 (left_inverse): multiply(inverse(X), X) = identity.
1.82/2.00	Axiom 2 (associativity): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
1.82/2.00	Axiom 3 (left_identity): X = multiply(identity, X).
1.82/2.00	Axiom 4 (symmetry_of_lub): least_upper_bound(X, Y) = least_upper_bound(Y, X).
1.82/2.00	Axiom 5 (monotony_glb1): multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)).
1.82/2.00	Axiom 6 (symmetry_of_glb): greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X).
1.82/2.00	Axiom 7 (monotony_glb2): greatest_lower_bound(multiply(X, Y), multiply(Z, Y)) = multiply(greatest_lower_bound(X, Z), Y).
1.82/2.00	Axiom 8 (p12x_4): greatest_lower_bound(inverse(X), inverse(Y)) = inverse(least_upper_bound(X, Y)).
1.82/2.00	Axiom 9 (p12x_1): greatest_lower_bound(b, c) = greatest_lower_bound(a, c).
1.82/2.00	Axiom 10 (p12x_2): least_upper_bound(a, c) = least_upper_bound(b, c).
1.82/2.00	
1.82/2.00	Lemma 11: multiply(inverse(X), multiply(X, Y)) = Y.
1.82/2.00	Proof:
1.82/2.00	  multiply(inverse(X), multiply(X, Y))
1.82/2.00	= { by axiom 2 (associativity) }
1.82/2.00	  multiply(multiply(inverse(X), X), Y)
1.82/2.00	= { by axiom 1 (left_inverse) }
1.82/2.00	  multiply(identity, Y)
1.82/2.00	= { by axiom 3 (left_identity) }
1.82/2.00	  Y
1.82/2.00	
1.82/2.00	Lemma 12: multiply(inverse(inverse(X)), Y) = multiply(X, Y).
1.82/2.00	Proof:
1.82/2.00	  multiply(inverse(inverse(X)), Y)
1.82/2.00	= { by lemma 11 }
1.82/2.00	  multiply(inverse(inverse(X)), multiply(inverse(X), multiply(X, Y)))
1.82/2.00	= { by lemma 11 }
1.82/2.00	  multiply(X, Y)
1.82/2.00	
1.82/2.00	Lemma 13: multiply(inverse(inverse(X)), identity) = X.
1.82/2.00	Proof:
1.82/2.00	  multiply(inverse(inverse(X)), identity)
1.82/2.00	= { by axiom 1 (left_inverse) }
1.82/2.00	  multiply(inverse(inverse(X)), multiply(inverse(X), X))
1.82/2.00	= { by lemma 11 }
1.82/2.00	  X
1.82/2.00	
1.82/2.00	Lemma 14: multiply(X, identity) = X.
1.82/2.00	Proof:
1.82/2.00	  multiply(X, identity)
1.82/2.00	= { by lemma 12 }
1.82/2.00	  multiply(inverse(inverse(X)), identity)
1.82/2.00	= { by lemma 13 }
1.82/2.00	  X
1.82/2.00	
1.82/2.00	Lemma 15: inverse(inverse(X)) = X.
1.82/2.00	Proof:
1.82/2.00	  inverse(inverse(X))
1.82/2.00	= { by lemma 14 }
1.82/2.00	  multiply(inverse(inverse(X)), identity)
1.82/2.00	= { by lemma 13 }
1.82/2.00	  X
1.82/2.00	
1.82/2.00	Lemma 16: multiply(X, inverse(multiply(Y, X))) = inverse(Y).
1.82/2.00	Proof:
1.82/2.00	  multiply(X, inverse(multiply(Y, X)))
1.82/2.00	= { by lemma 11 }
1.82/2.00	  multiply(inverse(Y), multiply(Y, multiply(X, inverse(multiply(Y, X)))))
1.82/2.00	= { by axiom 2 (associativity) }
1.82/2.00	  multiply(inverse(Y), multiply(multiply(Y, X), inverse(multiply(Y, X))))
1.82/2.00	= { by lemma 12 }
1.82/2.00	  multiply(inverse(Y), multiply(inverse(inverse(multiply(Y, X))), inverse(multiply(Y, X))))
1.82/2.00	= { by axiom 1 (left_inverse) }
1.82/2.00	  multiply(inverse(Y), identity)
1.82/2.00	= { by lemma 14 }
1.82/2.00	  inverse(Y)
1.82/2.00	
1.82/2.00	Lemma 17: multiply(inverse(least_upper_bound(Y, X)), Y) = multiply(inverse(X), greatest_lower_bound(Y, X)).
1.82/2.00	Proof:
1.82/2.00	  multiply(inverse(least_upper_bound(Y, X)), Y)
1.82/2.00	= { by axiom 4 (symmetry_of_lub) }
1.82/2.00	  multiply(inverse(least_upper_bound(X, Y)), Y)
1.82/2.00	= { by axiom 8 (p12x_4) }
1.82/2.00	  multiply(greatest_lower_bound(inverse(X), inverse(Y)), Y)
1.82/2.00	= { by axiom 6 (symmetry_of_glb) }
1.82/2.00	  multiply(greatest_lower_bound(inverse(Y), inverse(X)), Y)
1.82/2.00	= { by axiom 7 (monotony_glb2) }
1.82/2.00	  greatest_lower_bound(multiply(inverse(Y), Y), multiply(inverse(X), Y))
1.82/2.00	= { by axiom 1 (left_inverse) }
1.82/2.00	  greatest_lower_bound(identity, multiply(inverse(X), Y))
1.82/2.00	= { by axiom 1 (left_inverse) }
1.82/2.00	  greatest_lower_bound(multiply(inverse(X), X), multiply(inverse(X), Y))
1.82/2.00	= { by axiom 5 (monotony_glb1) }
1.82/2.00	  multiply(inverse(X), greatest_lower_bound(X, Y))
1.82/2.00	= { by axiom 6 (symmetry_of_glb) }
1.82/2.00	  multiply(inverse(X), greatest_lower_bound(Y, X))
1.82/2.00	
1.82/2.00	Goal 1 (prove_p12x): a = b.
1.82/2.00	Proof:
1.82/2.00	  a
1.82/2.00	= { by lemma 15 }
1.82/2.00	  inverse(inverse(a))
1.82/2.00	= { by lemma 16 }
1.82/2.00	  multiply(greatest_lower_bound(c, b), inverse(multiply(inverse(a), greatest_lower_bound(c, b))))
1.82/2.00	= { by axiom 6 (symmetry_of_glb) }
1.82/2.00	  multiply(greatest_lower_bound(c, b), inverse(multiply(inverse(a), greatest_lower_bound(b, c))))
1.82/2.00	= { by axiom 9 (p12x_1) }
1.82/2.00	  multiply(greatest_lower_bound(c, b), inverse(multiply(inverse(a), greatest_lower_bound(a, c))))
1.82/2.00	= { by axiom 6 (symmetry_of_glb) }
1.82/2.00	  multiply(greatest_lower_bound(c, b), inverse(multiply(inverse(a), greatest_lower_bound(c, a))))
1.82/2.00	= { by lemma 17 }
1.82/2.00	  multiply(greatest_lower_bound(c, b), inverse(multiply(inverse(least_upper_bound(c, a)), c)))
1.82/2.00	= { by axiom 4 (symmetry_of_lub) }
1.82/2.00	  multiply(greatest_lower_bound(c, b), inverse(multiply(inverse(least_upper_bound(a, c)), c)))
1.82/2.00	= { by axiom 10 (p12x_2) }
1.82/2.00	  multiply(greatest_lower_bound(c, b), inverse(multiply(inverse(least_upper_bound(b, c)), c)))
1.82/2.00	= { by axiom 4 (symmetry_of_lub) }
1.82/2.00	  multiply(greatest_lower_bound(c, b), inverse(multiply(inverse(least_upper_bound(c, b)), c)))
1.82/2.00	= { by lemma 17 }
1.82/2.00	  multiply(greatest_lower_bound(c, b), inverse(multiply(inverse(b), greatest_lower_bound(c, b))))
1.82/2.00	= { by lemma 16 }
1.82/2.00	  inverse(inverse(b))
1.82/2.00	= { by lemma 15 }
1.82/2.00	  b
1.82/2.00	% SZS output end Proof
1.82/2.00	
1.82/2.00	RESULT: Unsatisfiable (the axioms are contradictory).
1.82/2.00	EOF