0.03/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n018.cluster.edu
0.12/0.33	% Model      : x86_64 x86_64
0.12/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.33	% Memory     : 8042.1875MB
0.12/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.33	% CPULimit   : 180
0.12/0.33	% DateTime   : Thu Aug 29 11:21:51 EDT 2019
0.12/0.33	% CPUTime    : 
0.19/0.46	% SZS status Unsatisfiable
0.19/0.46	
0.19/0.46	% SZS output start Proof
0.19/0.46	Take the following subset of the input axioms:
0.19/0.46	  fof(c01, axiom, ![A, B]: B=mult(A, ld(A, B))).
0.19/0.46	  fof(c02, axiom, ![A, B]: B=ld(A, mult(A, B))).
0.19/0.46	  fof(c05, axiom, ![A]: A=mult(A, unit)).
0.19/0.46	  fof(c07, axiom, ![A, B, C]: mult(mult(A, mult(A, B)), C)=mult(mult(A, A), mult(B, C))).
0.19/0.46	  fof(goals, negated_conjecture, mult(mult(a, mult(b, b)), c)!=mult(a, mult(b, mult(b, c)))).
0.19/0.46	
0.19/0.46	Now clausify the problem and encode Horn clauses using encoding 3 of
0.19/0.46	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.19/0.46	We repeatedly replace C & s=t => u=v by the two clauses:
0.19/0.46	  fresh(y, y, x1...xn) = u
0.19/0.46	  C => fresh(s, t, x1...xn) = v
0.19/0.46	where fresh is a fresh function symbol and x1..xn are the free
0.19/0.46	variables of u and v.
0.19/0.46	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.19/0.46	input problem has no model of domain size 1).
0.19/0.46	
0.19/0.46	The encoding turns the above axioms into the following unit equations and goals:
0.19/0.46	
0.19/0.46	Axiom 1 (c02): X = ld(Y, mult(Y, X)).
0.19/0.46	Axiom 2 (c01): X = mult(Y, ld(Y, X)).
0.19/0.46	Axiom 3 (c07): mult(mult(X, mult(X, Y)), Z) = mult(mult(X, X), mult(Y, Z)).
0.19/0.46	Axiom 4 (c05): X = mult(X, unit).
0.19/0.46	
0.19/0.46	Lemma 5: mult(X, mult(X, mult(ld(X, Y), Z))) = mult(mult(X, Y), Z).
0.19/0.46	Proof:
0.19/0.46	  mult(X, mult(X, mult(ld(X, Y), Z)))
0.19/0.46	= { by axiom 4 (c05) }
0.19/0.46	  mult(mult(X, mult(X, mult(ld(X, Y), Z))), unit)
0.19/0.46	= { by axiom 3 (c07) }
0.19/0.46	  mult(mult(X, X), mult(mult(ld(X, Y), Z), unit))
0.19/0.46	= { by axiom 4 (c05) }
0.19/0.46	  mult(mult(X, X), mult(ld(X, Y), Z))
0.19/0.46	= { by axiom 3 (c07) }
0.19/0.46	  mult(mult(X, mult(X, ld(X, Y))), Z)
0.19/0.46	= { by axiom 2 (c01) }
0.19/0.46	  mult(mult(X, Y), Z)
0.19/0.46	
0.19/0.46	Lemma 6: mult(X, mult(ld(X, unit), Y)) = Y.
0.19/0.46	Proof:
0.19/0.46	  mult(X, mult(ld(X, unit), Y))
0.19/0.46	= { by axiom 1 (c02) }
0.19/0.46	  ld(X, mult(X, mult(X, mult(ld(X, unit), Y))))
0.19/0.46	= { by lemma 5 }
0.19/0.46	  ld(X, mult(mult(X, unit), Y))
0.19/0.46	= { by axiom 4 (c05) }
0.19/0.46	  ld(X, mult(X, Y))
0.19/0.46	= { by axiom 1 (c02) }
0.19/0.46	  Y
0.19/0.46	
0.19/0.46	Lemma 7: ld(ld(X, unit), Y) = mult(X, Y).
0.19/0.46	Proof:
0.19/0.46	  ld(ld(X, unit), Y)
0.19/0.46	= { by lemma 6 }
0.19/0.46	  mult(X, mult(ld(X, unit), ld(ld(X, unit), Y)))
0.19/0.46	= { by axiom 2 (c01) }
0.19/0.46	  mult(X, Y)
0.19/0.46	
0.19/0.46	Lemma 8: ld(ld(X, ld(X, Y)), Z) = ld(Y, mult(X, mult(X, Z))).
0.19/0.46	Proof:
0.19/0.46	  ld(ld(X, ld(X, Y)), Z)
0.19/0.46	= { by axiom 1 (c02) }
0.19/0.46	  ld(mult(X, ld(X, Y)), mult(mult(X, ld(X, Y)), ld(ld(X, ld(X, Y)), Z)))
0.19/0.46	= { by axiom 2 (c01) }
0.19/0.46	  ld(Y, mult(mult(X, ld(X, Y)), ld(ld(X, ld(X, Y)), Z)))
0.19/0.46	= { by lemma 5 }
0.19/0.46	  ld(Y, mult(X, mult(X, mult(ld(X, ld(X, Y)), ld(ld(X, ld(X, Y)), Z)))))
0.19/0.46	= { by axiom 2 (c01) }
0.19/0.46	  ld(Y, mult(X, mult(X, Z)))
0.19/0.46	
0.19/0.46	Goal 1 (goals): mult(mult(a, mult(b, b)), c) = mult(a, mult(b, mult(b, c))).
0.19/0.46	Proof:
0.19/0.46	  mult(mult(a, mult(b, b)), c)
0.19/0.46	= { by lemma 7 }
0.19/0.46	  mult(ld(ld(a, unit), mult(b, b)), c)
0.19/0.46	= { by axiom 4 (c05) }
0.19/0.46	  mult(ld(ld(a, unit), mult(b, mult(b, unit))), c)
0.19/0.46	= { by lemma 8 }
0.19/0.46	  mult(ld(ld(b, ld(b, ld(a, unit))), unit), c)
0.19/0.46	= { by axiom 1 (c02) }
0.19/0.46	  ld(ld(b, ld(b, ld(a, unit))), mult(ld(b, ld(b, ld(a, unit))), mult(ld(ld(b, ld(b, ld(a, unit))), unit), c)))
0.19/0.46	= { by lemma 6 }
0.19/0.46	  ld(ld(b, ld(b, ld(a, unit))), c)
0.19/0.46	= { by lemma 8 }
0.19/0.46	  ld(ld(a, unit), mult(b, mult(b, c)))
0.19/0.46	= { by lemma 7 }
0.19/0.46	  mult(a, mult(b, mult(b, c)))
0.19/0.46	% SZS output end Proof
0.19/0.46	
0.19/0.46	RESULT: Unsatisfiable (the axioms are contradictory).
0.19/0.46	EOF