0.07/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.07/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n014.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 09:50:09 EDT 2019
0.12/0.33	% CPUTime    : 
0.12/0.35	% SZS status Theorem
0.12/0.35	
0.12/0.35	% SZS output start Proof
0.12/0.35	Take the following subset of the input axioms:
0.12/0.35	  fof(f03, axiom, ![B, A]: A=mult(rd(A, B), B)).
0.12/0.35	  fof(f05, axiom, ![A]: A=mult(A, unit)).
0.12/0.35	  fof(f06, axiom, ![A]: mult(unit, A)=A).
0.12/0.35	  fof(f08, axiom, ![B, A]: mult(mult(op_c, A), B)=mult(op_c, mult(A, B))).
0.12/0.35	  fof(f09, axiom, ![B, A]: mult(A, mult(B, op_c))=mult(mult(A, B), op_c)).
0.12/0.35	  fof(f10, axiom, ![B, A]: mult(A, mult(op_c, B))=mult(mult(A, op_c), B)).
0.12/0.35	  fof(f12, axiom, ![B, A]: op_e=mult(mult(rd(op_c, mult(A, B)), B), A)).
0.12/0.35	  fof(goals, conjecture, ![X2, X3]: (mult(op_e, mult(X2, X3))=mult(mult(op_e, X2), X3) & (mult(mult(X2, op_e), X3)=mult(X2, mult(op_e, X3)) & mult(X2, mult(X3, op_e))=mult(mult(X2, X3), op_e)))).
0.12/0.35	
0.12/0.35	Now clausify the problem and encode Horn clauses using encoding 3 of
0.12/0.35	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.12/0.35	We repeatedly replace C & s=t => u=v by the two clauses:
0.12/0.35	  fresh(y, y, x1...xn) = u
0.12/0.35	  C => fresh(s, t, x1...xn) = v
0.12/0.35	where fresh is a fresh function symbol and x1..xn are the free
0.12/0.35	variables of u and v.
0.12/0.35	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.12/0.35	input problem has no model of domain size 1).
0.12/0.35	
0.12/0.35	The encoding turns the above axioms into the following unit equations and goals:
0.12/0.35	
0.12/0.35	Axiom 1 (f03): X = mult(rd(X, Y), Y).
0.12/0.35	Axiom 2 (f06): mult(unit, X) = X.
0.12/0.35	Axiom 3 (f10): mult(X, mult(op_c, Y)) = mult(mult(X, op_c), Y).
0.12/0.35	Axiom 4 (f08): mult(mult(op_c, X), Y) = mult(op_c, mult(X, Y)).
0.12/0.35	Axiom 5 (f09): mult(X, mult(Y, op_c)) = mult(mult(X, Y), op_c).
0.12/0.35	Axiom 6 (f05): X = mult(X, unit).
0.12/0.35	Axiom 7 (f12): op_e = mult(mult(rd(op_c, mult(X, Y)), Y), X).
0.12/0.35	
0.12/0.35	Lemma 8: op_e = op_c.
0.12/0.35	Proof:
0.12/0.35	  op_e
0.12/0.35	= { by axiom 7 (f12) }
0.12/0.35	  mult(mult(rd(op_c, mult(unit, ?)), ?), unit)
0.12/0.35	= { by axiom 2 (f06) }
0.12/0.35	  mult(mult(rd(op_c, ?), ?), unit)
0.12/0.35	= { by axiom 6 (f05) }
0.12/0.35	  mult(rd(op_c, ?), ?)
0.12/0.35	= { by axiom 1 (f03) }
0.12/0.35	  op_c
0.12/0.35	
0.12/0.35	Goal 1 (goals): tuple(mult(op_e, mult(sK6_goals_X2, sK5_goals_X3)), mult(mult(sK4_goals_X2, op_e), sK3_goals_X3), mult(sK2_goals_X2, mult(sK1_goals_X3, op_e))) = tuple(mult(mult(op_e, sK6_goals_X2), sK5_goals_X3), mult(sK4_goals_X2, mult(op_e, sK3_goals_X3)), mult(mult(sK2_goals_X2, sK1_goals_X3), op_e)).
0.12/0.35	Proof:
0.12/0.35	  tuple(mult(op_e, mult(sK6_goals_X2, sK5_goals_X3)), mult(mult(sK4_goals_X2, op_e), sK3_goals_X3), mult(sK2_goals_X2, mult(sK1_goals_X3, op_e)))
0.12/0.35	= { by lemma 8 }
0.12/0.35	  tuple(mult(op_c, mult(sK6_goals_X2, sK5_goals_X3)), mult(mult(sK4_goals_X2, op_e), sK3_goals_X3), mult(sK2_goals_X2, mult(sK1_goals_X3, op_e)))
0.12/0.35	= { by axiom 4 (f08) }
0.12/0.35	  tuple(mult(mult(op_c, sK6_goals_X2), sK5_goals_X3), mult(mult(sK4_goals_X2, op_e), sK3_goals_X3), mult(sK2_goals_X2, mult(sK1_goals_X3, op_e)))
0.12/0.35	= { by lemma 8 }
0.12/0.35	  tuple(mult(mult(op_e, sK6_goals_X2), sK5_goals_X3), mult(mult(sK4_goals_X2, op_e), sK3_goals_X3), mult(sK2_goals_X2, mult(sK1_goals_X3, op_e)))
0.12/0.35	= { by lemma 8 }
0.12/0.35	  tuple(mult(mult(op_e, sK6_goals_X2), sK5_goals_X3), mult(mult(sK4_goals_X2, op_c), sK3_goals_X3), mult(sK2_goals_X2, mult(sK1_goals_X3, op_e)))
0.12/0.35	= { by axiom 3 (f10) }
0.12/0.35	  tuple(mult(mult(op_e, sK6_goals_X2), sK5_goals_X3), mult(sK4_goals_X2, mult(op_c, sK3_goals_X3)), mult(sK2_goals_X2, mult(sK1_goals_X3, op_e)))
0.12/0.35	= { by lemma 8 }
0.12/0.35	  tuple(mult(mult(op_e, sK6_goals_X2), sK5_goals_X3), mult(sK4_goals_X2, mult(op_e, sK3_goals_X3)), mult(sK2_goals_X2, mult(sK1_goals_X3, op_e)))
0.12/0.35	= { by lemma 8 }
0.12/0.35	  tuple(mult(mult(op_e, sK6_goals_X2), sK5_goals_X3), mult(sK4_goals_X2, mult(op_e, sK3_goals_X3)), mult(sK2_goals_X2, mult(sK1_goals_X3, op_c)))
0.12/0.35	= { by axiom 5 (f09) }
0.12/0.35	  tuple(mult(mult(op_e, sK6_goals_X2), sK5_goals_X3), mult(sK4_goals_X2, mult(op_e, sK3_goals_X3)), mult(mult(sK2_goals_X2, sK1_goals_X3), op_c))
0.12/0.35	= { by lemma 8 }
0.12/0.35	  tuple(mult(mult(op_e, sK6_goals_X2), sK5_goals_X3), mult(sK4_goals_X2, mult(op_e, sK3_goals_X3)), mult(mult(sK2_goals_X2, sK1_goals_X3), op_e))
0.12/0.35	% SZS output end Proof
0.12/0.35	
0.12/0.35	RESULT: Theorem (the conjecture is true).
0.12/0.35	EOF