0.00/0.03	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.04	% Command    : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn
0.02/0.24	% Computer   : n013.star.cs.uiowa.edu
0.02/0.24	% Model      : x86_64 x86_64
0.02/0.24	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.02/0.24	% Memory     : 32218.625MB
0.02/0.24	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.02/0.24	% CPULimit   : 300
0.02/0.24	% DateTime   : Sat Jul 14 04:35:39 CDT 2018
0.02/0.24	% CPUTime    : 
0.02/0.25	% SZS status Theorem
0.02/0.25	
0.02/0.26	% SZS output start Proof
0.02/0.26	Take the following subset of the input axioms:
0.02/0.26	  fof(axiom_1, axiom,
0.02/0.26	      ![B, C, A]: times(times(A, B), C)=times(B, times(C, A))).
0.02/0.26	  fof(axiom_2, axiom,
0.02/0.26	      ![B]: (element(B) <=> ?[C]: (times(B, B)=C & B=times(B, C)))).
0.02/0.26	  fof(conjecture_1, conjecture,
0.02/0.26	      ![B, C, A]:
0.02/0.26	        ((element(A) & (element(B) & C=times(A, B))) => element(C))).
0.02/0.26	
0.02/0.26	Now clausify the problem and encode Horn clauses using encoding 3 of
0.02/0.26	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.02/0.26	We repeatedly replace C & s=t => u=v by the two clauses:
0.02/0.26	  $$fresh(y, y, x1...xn) = u
0.02/0.26	  C => $$fresh(s, t, x1...xn) = v
0.02/0.26	where $$fresh is a fresh function symbol and x1..xn are the free
0.02/0.26	variables of u and v.
0.02/0.26	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
0.02/0.26	input problem has no model of domain size 1).
0.02/0.26	
0.02/0.26	The encoding turns the above axioms into the following unit equations and goals:
0.02/0.26	
0.02/0.26	Axiom 1 (axiom_2): $$fresh4(X, X, Y, Z) = element(Y).
0.02/0.26	Axiom 2 (axiom_2): $$fresh3(X, X, Y) = $$true.
0.02/0.26	Axiom 3 (axiom_2_1): $$fresh(X, X, Y) = Y.
0.02/0.26	Axiom 4 (axiom_2_2): $$fresh2(X, X, Y) = sK4_axiom_2_C(Y).
0.02/0.26	Axiom 5 (axiom_1): times(times(X, Y), Z) = times(Y, times(Z, X)).
0.02/0.26	Axiom 6 (axiom_2_2): $$fresh2(element(X), $$true, X) = times(X, X).
0.02/0.26	Axiom 7 (axiom_2_1): $$fresh(element(X), $$true, X) = times(X, sK4_axiom_2_C(X)).
0.02/0.26	Axiom 8 (axiom_2): $$fresh4(times(X, X), Y, X, Y) = $$fresh3(X, times(X, Y), X).
0.02/0.26	Axiom 9 (conjecture_1): sK2_conjecture_1_C = times(sK1_conjecture_1_A, sK3_conjecture_1_B).
0.02/0.26	Axiom 10 (conjecture_1_1): element(sK3_conjecture_1_B) = $$true.
0.02/0.26	Axiom 11 (conjecture_1_2): element(sK1_conjecture_1_A) = $$true.
0.02/0.26	
0.02/0.26	Lemma 12: times(sK3_conjecture_1_B, sK3_conjecture_1_B) = sK4_axiom_2_C(sK3_conjecture_1_B).
0.02/0.26	Proof:
0.02/0.26	  times(sK3_conjecture_1_B, sK3_conjecture_1_B)
0.02/0.26	= { by axiom 6 (axiom_2_2) }
0.02/0.26	  $$fresh2(element(sK3_conjecture_1_B), $$true, sK3_conjecture_1_B)
0.02/0.26	= { by axiom 10 (conjecture_1_1) }
0.02/0.26	  $$fresh2($$true, $$true, sK3_conjecture_1_B)
0.02/0.26	= { by axiom 4 (axiom_2_2) }
0.02/0.26	  sK4_axiom_2_C(sK3_conjecture_1_B)
0.02/0.26	
0.02/0.26	Lemma 13: times(sK3_conjecture_1_B, sK4_axiom_2_C(sK3_conjecture_1_B)) = sK3_conjecture_1_B.
0.02/0.26	Proof:
0.02/0.26	  times(sK3_conjecture_1_B, sK4_axiom_2_C(sK3_conjecture_1_B))
0.02/0.26	= { by axiom 7 (axiom_2_1) }
0.02/0.26	  $$fresh(element(sK3_conjecture_1_B), $$true, sK3_conjecture_1_B)
0.02/0.26	= { by axiom 10 (conjecture_1_1) }
0.02/0.26	  $$fresh($$true, $$true, sK3_conjecture_1_B)
0.02/0.26	= { by axiom 3 (axiom_2_1) }
0.02/0.26	  sK3_conjecture_1_B
0.02/0.26	
0.02/0.26	Lemma 14: times(sK1_conjecture_1_A, sK4_axiom_2_C(sK1_conjecture_1_A)) = sK1_conjecture_1_A.
0.02/0.26	Proof:
0.02/0.26	  times(sK1_conjecture_1_A, sK4_axiom_2_C(sK1_conjecture_1_A))
0.02/0.26	= { by axiom 7 (axiom_2_1) }
0.02/0.26	  $$fresh(element(sK1_conjecture_1_A), $$true, sK1_conjecture_1_A)
0.02/0.26	= { by axiom 11 (conjecture_1_2) }
0.02/0.26	  $$fresh($$true, $$true, sK1_conjecture_1_A)
0.02/0.26	= { by axiom 3 (axiom_2_1) }
0.02/0.26	  sK1_conjecture_1_A
0.02/0.26	
0.02/0.26	Lemma 15: times(sK3_conjecture_1_B, times(X, sK1_conjecture_1_A)) = times(sK2_conjecture_1_C, X).
0.02/0.26	Proof:
0.02/0.26	  times(sK3_conjecture_1_B, times(X, sK1_conjecture_1_A))
0.02/0.26	= { by axiom 5 (axiom_1) }
0.02/0.26	  times(times(sK1_conjecture_1_A, sK3_conjecture_1_B), X)
0.02/0.26	= { by axiom 9 (conjecture_1) }
0.02/0.26	  times(sK2_conjecture_1_C, X)
0.02/0.26	
0.02/0.26	Lemma 16: times(sK3_conjecture_1_B, times(X, sK3_conjecture_1_B)) = times(sK4_axiom_2_C(sK3_conjecture_1_B), X).
0.02/0.26	Proof:
0.02/0.26	  times(sK3_conjecture_1_B, times(X, sK3_conjecture_1_B))
0.02/0.26	= { by axiom 5 (axiom_1) }
0.02/0.26	  times(times(sK3_conjecture_1_B, sK3_conjecture_1_B), X)
0.02/0.26	= { by lemma 12 }
0.02/0.26	  times(sK4_axiom_2_C(sK3_conjecture_1_B), X)
0.02/0.26	
0.02/0.26	Lemma 17: times(sK1_conjecture_1_A, times(X, sK1_conjecture_1_A)) = times(sK4_axiom_2_C(sK1_conjecture_1_A), X).
0.02/0.26	Proof:
0.02/0.26	  times(sK1_conjecture_1_A, times(X, sK1_conjecture_1_A))
0.02/0.26	= { by axiom 5 (axiom_1) }
0.02/0.26	  times(times(sK1_conjecture_1_A, sK1_conjecture_1_A), X)
0.02/0.26	= { by axiom 6 (axiom_2_2) }
0.02/0.26	  times($$fresh2(element(sK1_conjecture_1_A), $$true, sK1_conjecture_1_A), X)
0.02/0.26	= { by axiom 11 (conjecture_1_2) }
0.02/0.26	  times($$fresh2($$true, $$true, sK1_conjecture_1_A), X)
0.02/0.26	= { by axiom 4 (axiom_2_2) }
0.02/0.26	  times(sK4_axiom_2_C(sK1_conjecture_1_A), X)
0.02/0.26	
0.02/0.26	Lemma 18: times(sK4_axiom_2_C(sK3_conjecture_1_B), sK2_conjecture_1_C) = times(sK3_conjecture_1_B, sK1_conjecture_1_A).
0.02/0.26	Proof:
0.02/0.26	  times(sK4_axiom_2_C(sK3_conjecture_1_B), sK2_conjecture_1_C)
0.02/0.26	= { by axiom 9 (conjecture_1) }
0.02/0.26	  times(sK4_axiom_2_C(sK3_conjecture_1_B), times(sK1_conjecture_1_A, sK3_conjecture_1_B))
0.02/0.26	= { by axiom 5 (axiom_1) }
0.02/0.26	  times(times(sK3_conjecture_1_B, sK4_axiom_2_C(sK3_conjecture_1_B)), sK1_conjecture_1_A)
0.02/0.26	= { by lemma 13 }
0.02/0.26	  times(sK3_conjecture_1_B, sK1_conjecture_1_A)
0.02/0.26	
0.02/0.26	Lemma 19: times(sK3_conjecture_1_B, times(sK3_conjecture_1_B, sK2_conjecture_1_C)) = times(sK2_conjecture_1_C, sK4_axiom_2_C(sK3_conjecture_1_B)).
0.02/0.26	Proof:
0.02/0.26	  times(sK3_conjecture_1_B, times(sK3_conjecture_1_B, sK2_conjecture_1_C))
0.02/0.26	= { by axiom 9 (conjecture_1) }
0.02/0.26	  times(sK3_conjecture_1_B, times(sK3_conjecture_1_B, times(sK1_conjecture_1_A, sK3_conjecture_1_B)))
0.02/0.26	= { by lemma 16 }
0.02/0.26	  times(sK3_conjecture_1_B, times(sK4_axiom_2_C(sK3_conjecture_1_B), sK1_conjecture_1_A))
0.02/0.26	= { by lemma 15 }
0.02/0.26	  times(sK2_conjecture_1_C, sK4_axiom_2_C(sK3_conjecture_1_B))
0.02/0.26	
0.02/0.26	Lemma 20: times(sK2_conjecture_1_C, times(sK4_axiom_2_C(sK1_conjecture_1_A), X)) = times(sK2_conjecture_1_C, X).
0.02/0.26	Proof:
0.02/0.26	  times(sK2_conjecture_1_C, times(sK4_axiom_2_C(sK1_conjecture_1_A), X))
0.02/0.26	= { by lemma 15 }
0.02/0.26	  times(sK3_conjecture_1_B, times(times(sK4_axiom_2_C(sK1_conjecture_1_A), X), sK1_conjecture_1_A))
0.02/0.26	= { by axiom 5 (axiom_1) }
0.02/0.26	  times(sK3_conjecture_1_B, times(X, times(sK1_conjecture_1_A, sK4_axiom_2_C(sK1_conjecture_1_A))))
0.02/0.26	= { by lemma 14 }
0.02/0.26	  times(sK3_conjecture_1_B, times(X, sK1_conjecture_1_A))
0.02/0.26	= { by lemma 15 }
0.02/0.26	  times(sK2_conjecture_1_C, X)
0.02/0.26	
0.02/0.26	Lemma 21: times(sK1_conjecture_1_A, times(sK3_conjecture_1_B, times(X, Y))) = times(X, times(Y, sK2_conjecture_1_C)).
0.02/0.26	Proof:
0.02/0.26	  times(sK1_conjecture_1_A, times(sK3_conjecture_1_B, times(X, Y)))
0.02/0.26	= { by axiom 5 (axiom_1) }
0.02/0.26	  times(sK1_conjecture_1_A, times(times(Y, sK3_conjecture_1_B), X))
0.02/0.26	= { by axiom 5 (axiom_1) }
0.02/0.26	  times(times(X, sK1_conjecture_1_A), times(Y, sK3_conjecture_1_B))
0.02/0.26	= { by axiom 5 (axiom_1) }
0.02/0.26	  times(times(sK3_conjecture_1_B, times(X, sK1_conjecture_1_A)), Y)
0.02/0.27	= { by axiom 5 (axiom_1) }
0.02/0.27	  times(times(times(sK1_conjecture_1_A, sK3_conjecture_1_B), X), Y)
0.02/0.27	= { by axiom 5 (axiom_1) }
0.02/0.27	  times(X, times(Y, times(sK1_conjecture_1_A, sK3_conjecture_1_B)))
0.02/0.27	= { by axiom 9 (conjecture_1) }
0.02/0.27	  times(X, times(Y, sK2_conjecture_1_C))
0.02/0.27	
0.02/0.27	Lemma 22: times(sK3_conjecture_1_B, sK1_conjecture_1_A) = sK2_conjecture_1_C.
0.02/0.27	Proof:
0.02/0.27	  times(sK3_conjecture_1_B, sK1_conjecture_1_A)
0.02/0.27	= { by lemma 18 }
0.02/0.27	  times(sK4_axiom_2_C(sK3_conjecture_1_B), sK2_conjecture_1_C)
0.02/0.27	= { by lemma 16 }
0.02/0.27	  times(sK3_conjecture_1_B, times(sK2_conjecture_1_C, sK3_conjecture_1_B))
0.02/0.27	= { by lemma 20 }
0.02/0.27	  times(sK3_conjecture_1_B, times(sK2_conjecture_1_C, times(sK4_axiom_2_C(sK1_conjecture_1_A), sK3_conjecture_1_B)))
0.02/0.27	= { by axiom 5 (axiom_1) }
0.02/0.27	  times(sK3_conjecture_1_B, times(times(sK3_conjecture_1_B, sK2_conjecture_1_C), sK4_axiom_2_C(sK1_conjecture_1_A)))
0.02/0.27	= { by axiom 5 (axiom_1) }
0.02/0.27	  times(times(sK4_axiom_2_C(sK1_conjecture_1_A), sK3_conjecture_1_B), times(sK3_conjecture_1_B, sK2_conjecture_1_C))
0.02/0.27	= { by axiom 5 (axiom_1) }
0.02/0.27	  times(times(sK2_conjecture_1_C, times(sK4_axiom_2_C(sK1_conjecture_1_A), sK3_conjecture_1_B)), sK3_conjecture_1_B)
0.02/0.27	= { by lemma 20 }
0.02/0.27	  times(times(sK2_conjecture_1_C, sK3_conjecture_1_B), sK3_conjecture_1_B)
0.02/0.27	= { by axiom 5 (axiom_1) }
0.02/0.27	  times(sK3_conjecture_1_B, times(sK3_conjecture_1_B, sK2_conjecture_1_C))
0.02/0.27	= { by lemma 21 }
0.02/0.27	  times(sK1_conjecture_1_A, times(sK3_conjecture_1_B, times(sK3_conjecture_1_B, sK3_conjecture_1_B)))
0.02/0.27	= { by lemma 12 }
0.02/0.27	  times(sK1_conjecture_1_A, times(sK3_conjecture_1_B, sK4_axiom_2_C(sK3_conjecture_1_B)))
0.02/0.27	= { by lemma 13 }
0.02/0.27	  times(sK1_conjecture_1_A, sK3_conjecture_1_B)
0.02/0.27	= { by axiom 9 (conjecture_1) }
0.07/0.27	  sK2_conjecture_1_C
0.07/0.27	
0.07/0.27	Goal 1 (conjecture_1_3): element(sK2_conjecture_1_C) = $$true.
0.07/0.27	Proof:
0.07/0.27	  element(sK2_conjecture_1_C)
0.07/0.27	= { by axiom 1 (axiom_2) }
0.07/0.27	  $$fresh4(times(sK2_conjecture_1_C, sK2_conjecture_1_C), times(sK2_conjecture_1_C, sK2_conjecture_1_C), sK2_conjecture_1_C, times(sK2_conjecture_1_C, sK2_conjecture_1_C))
0.07/0.27	= { by axiom 8 (axiom_2) }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK2_conjecture_1_C, times(sK2_conjecture_1_C, sK2_conjecture_1_C)), sK2_conjecture_1_C)
0.07/0.27	= { by lemma 21 }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK1_conjecture_1_A, times(sK3_conjecture_1_B, times(sK2_conjecture_1_C, sK2_conjecture_1_C))), sK2_conjecture_1_C)
0.07/0.27	= { by lemma 21 }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK1_conjecture_1_A, times(sK1_conjecture_1_A, times(sK3_conjecture_1_B, times(sK3_conjecture_1_B, sK2_conjecture_1_C)))), sK2_conjecture_1_C)
0.07/0.27	= { by lemma 22 }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK1_conjecture_1_A, times(sK1_conjecture_1_A, times(sK3_conjecture_1_B, times(sK3_conjecture_1_B, times(sK3_conjecture_1_B, sK1_conjecture_1_A))))), sK2_conjecture_1_C)
0.07/0.27	= { by lemma 18 }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK1_conjecture_1_A, times(sK1_conjecture_1_A, times(sK3_conjecture_1_B, times(sK3_conjecture_1_B, times(sK4_axiom_2_C(sK3_conjecture_1_B), sK2_conjecture_1_C))))), sK2_conjecture_1_C)
0.07/0.27	= { by lemma 21 }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK1_conjecture_1_A, times(sK1_conjecture_1_A, times(sK3_conjecture_1_B, times(sK1_conjecture_1_A, times(sK3_conjecture_1_B, times(sK3_conjecture_1_B, sK4_axiom_2_C(sK3_conjecture_1_B))))))), sK2_conjecture_1_C)
0.07/0.27	= { by lemma 13 }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK1_conjecture_1_A, times(sK1_conjecture_1_A, times(sK3_conjecture_1_B, times(sK1_conjecture_1_A, times(sK3_conjecture_1_B, sK3_conjecture_1_B))))), sK2_conjecture_1_C)
0.07/0.27	= { by lemma 12 }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK1_conjecture_1_A, times(sK1_conjecture_1_A, times(sK3_conjecture_1_B, times(sK1_conjecture_1_A, sK4_axiom_2_C(sK3_conjecture_1_B))))), sK2_conjecture_1_C)
0.07/0.27	= { by lemma 21 }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK1_conjecture_1_A, times(sK1_conjecture_1_A, times(sK4_axiom_2_C(sK3_conjecture_1_B), sK2_conjecture_1_C))), sK2_conjecture_1_C)
0.07/0.27	= { by lemma 18 }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK1_conjecture_1_A, times(sK1_conjecture_1_A, times(sK3_conjecture_1_B, sK1_conjecture_1_A))), sK2_conjecture_1_C)
0.07/0.27	= { by lemma 22 }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK1_conjecture_1_A, times(sK1_conjecture_1_A, sK2_conjecture_1_C)), sK2_conjecture_1_C)
0.07/0.27	= { by axiom 9 (conjecture_1) }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK1_conjecture_1_A, times(sK1_conjecture_1_A, times(sK1_conjecture_1_A, sK3_conjecture_1_B))), sK2_conjecture_1_C)
0.07/0.27	= { by axiom 5 (axiom_1) }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK1_conjecture_1_A, times(times(sK3_conjecture_1_B, sK1_conjecture_1_A), sK1_conjecture_1_A)), sK2_conjecture_1_C)
0.07/0.27	= { by lemma 22 }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK1_conjecture_1_A, times(sK2_conjecture_1_C, sK1_conjecture_1_A)), sK2_conjecture_1_C)
0.07/0.27	= { by lemma 17 }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK4_axiom_2_C(sK1_conjecture_1_A), sK2_conjecture_1_C), sK2_conjecture_1_C)
0.07/0.27	= { by lemma 22 }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK4_axiom_2_C(sK1_conjecture_1_A), times(sK3_conjecture_1_B, sK1_conjecture_1_A)), sK2_conjecture_1_C)
0.07/0.27	= { by axiom 5 (axiom_1) }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(times(sK1_conjecture_1_A, sK4_axiom_2_C(sK1_conjecture_1_A)), sK3_conjecture_1_B), sK2_conjecture_1_C)
0.07/0.27	= { by lemma 14 }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, times(sK1_conjecture_1_A, sK3_conjecture_1_B), sK2_conjecture_1_C)
0.07/0.27	= { by axiom 9 (conjecture_1) }
0.07/0.27	  $$fresh3(sK2_conjecture_1_C, sK2_conjecture_1_C, sK2_conjecture_1_C)
0.07/0.27	= { by axiom 2 (axiom_2) }
0.07/0.27	  $$true
0.07/0.27	% SZS output end Proof
0.07/0.27	
0.07/0.27	RESULT: Theorem (the conjecture is true).
0.07/0.28	EOF