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
0.12/0.34	% Computer   : n021.cluster.edu
0.12/0.34	% Model      : x86_64 x86_64
0.12/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.34	% Memory     : 8042.1875MB
0.12/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.34	% CPULimit   : 180
0.12/0.34	% DateTime   : Thu Aug 29 10:02:19 EDT 2019
0.12/0.34	% CPUTime    : 
0.12/0.36	% SZS status Unsatisfiable
0.12/0.36	
0.12/0.36	% SZS output start Proof
0.12/0.36	Take the following subset of the input axioms:
0.12/0.36	  fof(multiply, axiom, ![A, B]: divide(A, inverse(B))=multiply(A, B)).
0.12/0.36	  fof(prove_these_axioms_4, negated_conjecture, multiply(a, b)!=multiply(b, a)).
0.12/0.36	  fof(single_axiom, axiom, ![A, B, C]: divide(divide(A, inverse(divide(B, divide(A, C)))), C)=B).
0.12/0.36	
0.12/0.36	Now clausify the problem and encode Horn clauses using encoding 3 of
0.12/0.36	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.12/0.36	We repeatedly replace C & s=t => u=v by the two clauses:
0.12/0.36	  fresh(y, y, x1...xn) = u
0.12/0.36	  C => fresh(s, t, x1...xn) = v
0.12/0.36	where fresh is a fresh function symbol and x1..xn are the free
0.12/0.36	variables of u and v.
0.12/0.36	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.12/0.36	input problem has no model of domain size 1).
0.12/0.36	
0.12/0.36	The encoding turns the above axioms into the following unit equations and goals:
0.12/0.36	
0.12/0.36	Axiom 1 (single_axiom): divide(divide(X, inverse(divide(Y, divide(X, Z)))), Z) = Y.
0.12/0.36	Axiom 2 (multiply): divide(X, inverse(Y)) = multiply(X, Y).
0.12/0.36	
0.12/0.36	Lemma 3: multiply(multiply(X, divide(Y, multiply(X, Z))), Z) = Y.
0.12/0.36	Proof:
0.12/0.36	  multiply(multiply(X, divide(Y, multiply(X, Z))), Z)
0.12/0.36	= { by axiom 2 (multiply) }
0.12/0.36	  multiply(multiply(X, divide(Y, divide(X, inverse(Z)))), Z)
0.12/0.36	= { by axiom 2 (multiply) }
0.12/0.36	  multiply(divide(X, inverse(divide(Y, divide(X, inverse(Z))))), Z)
0.12/0.36	= { by axiom 2 (multiply) }
0.12/0.36	  divide(divide(X, inverse(divide(Y, divide(X, inverse(Z))))), inverse(Z))
0.12/0.36	= { by axiom 1 (single_axiom) }
0.12/0.36	  Y
0.12/0.36	
0.12/0.36	Lemma 4: multiply(X, divide(Y, divide(X, multiply(Z, W)))) = multiply(multiply(Z, Y), W).
0.12/0.36	Proof:
0.12/0.36	  multiply(X, divide(Y, divide(X, multiply(Z, W))))
0.12/0.36	= { by axiom 2 (multiply) }
0.12/0.36	  divide(X, inverse(divide(Y, divide(X, multiply(Z, W)))))
0.12/0.36	= { by lemma 3 }
0.12/0.36	  multiply(multiply(Z, divide(divide(X, inverse(divide(Y, divide(X, multiply(Z, W))))), multiply(Z, W))), W)
0.12/0.36	= { by axiom 1 (single_axiom) }
0.12/0.36	  multiply(multiply(Z, Y), W)
0.12/0.36	
0.12/0.36	Lemma 5: multiply(multiply(multiply(X, V), Z), divide(Y, V)) = multiply(multiply(X, Y), Z).
0.12/0.36	Proof:
0.12/0.36	  multiply(multiply(multiply(X, V), Z), divide(Y, V))
0.12/0.36	= { by lemma 4 }
0.12/0.36	  multiply(multiply(?, divide(V, divide(?, multiply(X, Z)))), divide(Y, V))
0.12/0.36	= { by axiom 2 (multiply) }
0.12/0.36	  multiply(divide(?, inverse(divide(V, divide(?, multiply(X, Z))))), divide(Y, V))
0.12/0.36	= { by axiom 1 (single_axiom) }
0.12/0.36	  multiply(divide(?, inverse(divide(V, divide(?, multiply(X, Z))))), divide(Y, divide(divide(?, inverse(divide(V, divide(?, multiply(X, Z))))), multiply(X, Z))))
0.12/0.36	= { by lemma 4 }
0.12/0.36	  multiply(multiply(X, Y), Z)
0.12/0.36	
0.12/0.36	Lemma 6: multiply(multiply(X, Y), Z) = multiply(multiply(X, Z), Y).
0.12/0.36	Proof:
0.12/0.36	  multiply(multiply(X, Y), Z)
0.12/0.36	= { by lemma 5 }
0.12/0.36	  multiply(multiply(multiply(X, ?), Z), divide(Y, ?))
0.12/0.36	= { by lemma 5 }
0.12/0.36	  multiply(multiply(multiply(multiply(X, ?), Y), divide(Y, ?)), divide(Z, Y))
0.12/0.36	= { by lemma 5 }
0.12/0.36	  multiply(multiply(multiply(X, Y), Y), divide(Z, Y))
0.12/0.36	= { by lemma 5 }
0.12/0.36	  multiply(multiply(X, Z), Y)
0.12/0.36	
0.12/0.36	Lemma 7: divide(multiply(Y, divide(X, divide(Y, Z))), Z) = X.
0.12/0.36	Proof:
0.12/0.36	  divide(multiply(Y, divide(X, divide(Y, Z))), Z)
0.12/0.36	= { by axiom 2 (multiply) }
0.12/0.36	  divide(divide(Y, inverse(divide(X, divide(Y, Z)))), Z)
0.12/0.36	= { by axiom 1 (single_axiom) }
0.12/0.37	  X
0.12/0.37	
0.12/0.37	Lemma 8: multiply(divide(Y, Y), X) = X.
0.12/0.37	Proof:
0.12/0.37	  multiply(divide(Y, Y), X)
0.12/0.37	= { by lemma 3 }
0.12/0.37	  multiply(multiply(multiply(?, divide(divide(Y, Y), multiply(?, ?))), ?), X)
0.12/0.37	= { by lemma 7 }
0.12/0.37	  multiply(multiply(multiply(?, divide(divide(Y, Y), multiply(?, ?))), ?), divide(multiply(Y, divide(X, divide(Y, Y))), Y))
0.12/0.37	= { by lemma 3 }
0.12/0.37	  multiply(multiply(multiply(?, divide(divide(Y, Y), multiply(?, ?))), ?), divide(multiply(multiply(multiply(?, divide(Y, multiply(?, ?))), ?), divide(X, divide(Y, Y))), Y))
0.12/0.37	= { by lemma 6 }
0.12/0.37	  multiply(multiply(multiply(?, divide(divide(Y, Y), multiply(?, ?))), ?), divide(multiply(multiply(multiply(?, divide(Y, multiply(?, ?))), divide(X, divide(Y, Y))), ?), Y))
0.12/0.37	= { by lemma 4 }
0.12/0.37	  multiply(multiply(multiply(?, divide(divide(Y, Y), multiply(?, ?))), ?), divide(multiply(?, divide(divide(X, divide(Y, Y)), divide(?, multiply(multiply(?, divide(Y, multiply(?, ?))), ?)))), Y))
0.12/0.37	= { by lemma 3 }
0.12/0.37	  multiply(multiply(multiply(?, divide(divide(Y, Y), multiply(?, ?))), ?), divide(multiply(?, divide(divide(X, divide(Y, Y)), divide(?, multiply(multiply(?, divide(Y, multiply(?, ?))), ?)))), multiply(multiply(?, divide(Y, multiply(?, ?))), ?)))
0.12/0.37	= { by lemma 7 }
0.12/0.37	  multiply(multiply(multiply(?, divide(divide(Y, Y), multiply(?, ?))), ?), divide(X, divide(Y, Y)))
0.12/0.37	= { by lemma 3 }
0.12/0.37	  multiply(multiply(multiply(?, divide(divide(Y, Y), multiply(?, ?))), ?), divide(X, multiply(multiply(?, divide(divide(Y, Y), multiply(?, ?))), ?)))
0.12/0.37	= { by lemma 6 }
0.12/0.37	  multiply(multiply(multiply(?, divide(divide(Y, Y), multiply(?, ?))), divide(X, multiply(multiply(?, divide(divide(Y, Y), multiply(?, ?))), ?))), ?)
0.12/0.37	= { by lemma 3 }
0.12/0.37	  X
0.12/0.37	
0.12/0.37	Goal 1 (prove_these_axioms_4): multiply(a, b) = multiply(b, a).
0.12/0.37	Proof:
0.12/0.37	  multiply(a, b)
0.12/0.37	= { by lemma 8 }
0.12/0.37	  multiply(multiply(divide(?, ?), a), b)
0.12/0.37	= { by lemma 6 }
0.12/0.37	  multiply(multiply(divide(?, ?), b), a)
0.12/0.37	= { by lemma 8 }
0.12/0.37	  multiply(b, a)
0.12/0.37	% SZS output end Proof
0.12/0.37	
0.12/0.37	RESULT: Unsatisfiable (the axioms are contradictory).
0.12/0.37	EOF