0.08/0.10	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.08/0.11	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.11/0.31	% Computer   : n031.cluster.edu
0.11/0.31	% Model      : x86_64 x86_64
0.11/0.31	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.11/0.31	% Memory     : 8042.1875MB
0.11/0.31	% OS         : Linux 3.10.0-693.el7.x86_64
0.11/0.31	% CPULimit   : 960
0.11/0.31	% WCLimit    : 120
0.11/0.31	% DateTime   : Thu Jul  2 07:26:02 EDT 2020
0.11/0.31	% CPUTime    : 
0.16/0.36	% SZS status Theorem
0.16/0.36	
0.16/0.36	% SZS output start Proof
0.16/0.36	Take the following subset of the input axioms:
0.16/0.37	  fof(axiom_1, axiom, ![B, C, A]: times(times(A, B), C)=times(B, times(C, A))).
0.16/0.37	  fof(axiom_2, axiom, ![B]: (element(B) <=> ?[C]: (times(B, C)=B & times(B, B)=C))).
0.16/0.37	  fof(conjecture_1, conjecture, ![B, C, A]: (element(C) <= (element(A) & (times(A, B)=C & element(B))))).
0.16/0.37	
0.16/0.37	Now clausify the problem and encode Horn clauses using encoding 3 of
0.16/0.37	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.16/0.37	We repeatedly replace C & s=t => u=v by the two clauses:
0.16/0.37	  fresh(y, y, x1...xn) = u
0.16/0.37	  C => fresh(s, t, x1...xn) = v
0.16/0.37	where fresh is a fresh function symbol and x1..xn are the free
0.16/0.37	variables of u and v.
0.16/0.37	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.16/0.37	input problem has no model of domain size 1).
0.16/0.37	
0.16/0.37	The encoding turns the above axioms into the following unit equations and goals:
0.16/0.37	
0.16/0.37	Axiom 1 (axiom_2): fresh3(X, X, Y, Z) = element(Y).
0.16/0.37	Axiom 2 (axiom_2): fresh4(X, X, Y) = true.
0.16/0.37	Axiom 3 (axiom_2_1): fresh(X, X, Y) = Y.
0.16/0.37	Axiom 4 (axiom_2_2): fresh2(X, X, Y) = sK1_axiom_2_C(Y).
0.16/0.37	Axiom 5 (axiom_1): times(times(X, Y), Z) = times(Y, times(Z, X)).
0.16/0.37	Axiom 6 (axiom_2_2): fresh2(element(X), true, X) = times(X, X).
0.16/0.37	Axiom 7 (axiom_2_1): fresh(element(X), true, X) = times(X, sK1_axiom_2_C(X)).
0.16/0.37	Axiom 8 (axiom_2): fresh3(times(X, X), Y, X, Y) = fresh4(times(X, Y), X, X).
0.16/0.37	Axiom 9 (conjecture_1): times(sK2_conjecture_1_A, sK4_conjecture_1_B) = sK3_conjecture_1_C.
0.16/0.37	Axiom 10 (conjecture_1_1): element(sK4_conjecture_1_B) = true.
0.16/0.37	Axiom 11 (conjecture_1_2): element(sK2_conjecture_1_A) = true.
0.16/0.37	
0.16/0.37	Lemma 12: times(sK2_conjecture_1_A, times(sK4_conjecture_1_B, times(Y, X))) = times(Y, times(X, sK3_conjecture_1_C)).
0.16/0.37	Proof:
0.16/0.37	  times(sK2_conjecture_1_A, times(sK4_conjecture_1_B, times(Y, X)))
0.16/0.37	= { by axiom 5 (axiom_1) }
0.16/0.37	  times(sK2_conjecture_1_A, times(times(X, sK4_conjecture_1_B), Y))
0.16/0.37	= { by axiom 5 (axiom_1) }
0.16/0.37	  times(times(Y, sK2_conjecture_1_A), times(X, sK4_conjecture_1_B))
0.16/0.37	= { by axiom 5 (axiom_1) }
0.16/0.37	  times(times(sK4_conjecture_1_B, times(Y, sK2_conjecture_1_A)), X)
0.16/0.37	= { by axiom 5 (axiom_1) }
0.16/0.37	  times(times(times(sK2_conjecture_1_A, sK4_conjecture_1_B), Y), X)
0.16/0.37	= { by axiom 5 (axiom_1) }
0.16/0.37	  times(Y, times(X, times(sK2_conjecture_1_A, sK4_conjecture_1_B)))
0.16/0.37	= { by axiom 9 (conjecture_1) }
0.16/0.37	  times(Y, times(X, sK3_conjecture_1_C))
0.16/0.37	
0.16/0.37	Lemma 13: times(sK4_conjecture_1_B, sK4_conjecture_1_B) = sK1_axiom_2_C(sK4_conjecture_1_B).
0.16/0.37	Proof:
0.16/0.37	  times(sK4_conjecture_1_B, sK4_conjecture_1_B)
0.16/0.37	= { by axiom 6 (axiom_2_2) }
0.16/0.37	  fresh2(element(sK4_conjecture_1_B), true, sK4_conjecture_1_B)
0.16/0.37	= { by axiom 10 (conjecture_1_1) }
0.16/0.37	  fresh2(true, true, sK4_conjecture_1_B)
0.16/0.37	= { by axiom 4 (axiom_2_2) }
0.16/0.37	  sK1_axiom_2_C(sK4_conjecture_1_B)
0.16/0.37	
0.16/0.37	Lemma 14: times(sK4_conjecture_1_B, sK1_axiom_2_C(sK4_conjecture_1_B)) = sK4_conjecture_1_B.
0.16/0.37	Proof:
0.16/0.37	  times(sK4_conjecture_1_B, sK1_axiom_2_C(sK4_conjecture_1_B))
0.16/0.37	= { by axiom 7 (axiom_2_1) }
0.16/0.37	  fresh(element(sK4_conjecture_1_B), true, sK4_conjecture_1_B)
0.16/0.37	= { by axiom 10 (conjecture_1_1) }
0.16/0.37	  fresh(true, true, sK4_conjecture_1_B)
0.16/0.37	= { by axiom 3 (axiom_2_1) }
0.16/0.37	  sK4_conjecture_1_B
0.16/0.37	
0.16/0.37	Lemma 15: times(sK1_axiom_2_C(sK4_conjecture_1_B), sK3_conjecture_1_C) = times(sK4_conjecture_1_B, sK2_conjecture_1_A).
0.16/0.37	Proof:
0.16/0.37	  times(sK1_axiom_2_C(sK4_conjecture_1_B), sK3_conjecture_1_C)
0.16/0.37	= { by axiom 9 (conjecture_1) }
0.16/0.37	  times(sK1_axiom_2_C(sK4_conjecture_1_B), times(sK2_conjecture_1_A, sK4_conjecture_1_B))
0.16/0.37	= { by axiom 5 (axiom_1) }
0.16/0.37	  times(times(sK4_conjecture_1_B, sK1_axiom_2_C(sK4_conjecture_1_B)), sK2_conjecture_1_A)
0.16/0.37	= { by lemma 14 }
0.16/0.37	  times(sK4_conjecture_1_B, sK2_conjecture_1_A)
0.16/0.37	
0.16/0.37	Lemma 16: times(sK4_conjecture_1_B, times(X, sK2_conjecture_1_A)) = times(sK3_conjecture_1_C, X).
0.16/0.37	Proof:
0.16/0.37	  times(sK4_conjecture_1_B, times(X, sK2_conjecture_1_A))
0.16/0.37	= { by axiom 5 (axiom_1) }
0.16/0.37	  times(times(sK2_conjecture_1_A, sK4_conjecture_1_B), X)
0.16/0.37	= { by axiom 9 (conjecture_1) }
0.16/0.37	  times(sK3_conjecture_1_C, X)
0.16/0.37	
0.16/0.37	Lemma 17: times(sK4_conjecture_1_B, times(X, sK4_conjecture_1_B)) = times(sK1_axiom_2_C(sK4_conjecture_1_B), X).
0.16/0.37	Proof:
0.16/0.37	  times(sK4_conjecture_1_B, times(X, sK4_conjecture_1_B))
0.16/0.38	= { by axiom 5 (axiom_1) }
0.16/0.38	  times(times(sK4_conjecture_1_B, sK4_conjecture_1_B), X)
0.16/0.38	= { by lemma 13 }
0.16/0.38	  times(sK1_axiom_2_C(sK4_conjecture_1_B), X)
0.16/0.38	
0.16/0.38	Lemma 18: times(sK4_conjecture_1_B, times(sK4_conjecture_1_B, sK3_conjecture_1_C)) = times(sK3_conjecture_1_C, sK1_axiom_2_C(sK4_conjecture_1_B)).
0.16/0.38	Proof:
0.16/0.38	  times(sK4_conjecture_1_B, times(sK4_conjecture_1_B, sK3_conjecture_1_C))
0.16/0.38	= { by axiom 9 (conjecture_1) }
0.16/0.38	  times(sK4_conjecture_1_B, times(sK4_conjecture_1_B, times(sK2_conjecture_1_A, sK4_conjecture_1_B)))
0.16/0.38	= { by lemma 17 }
0.16/0.38	  times(sK4_conjecture_1_B, times(sK1_axiom_2_C(sK4_conjecture_1_B), sK2_conjecture_1_A))
0.16/0.38	= { by lemma 16 }
0.16/0.38	  times(sK3_conjecture_1_C, sK1_axiom_2_C(sK4_conjecture_1_B))
0.16/0.38	
0.16/0.38	Lemma 19: times(sK2_conjecture_1_A, sK1_axiom_2_C(sK2_conjecture_1_A)) = sK2_conjecture_1_A.
0.16/0.38	Proof:
0.16/0.38	  times(sK2_conjecture_1_A, sK1_axiom_2_C(sK2_conjecture_1_A))
0.16/0.38	= { by axiom 7 (axiom_2_1) }
0.16/0.38	  fresh(element(sK2_conjecture_1_A), true, sK2_conjecture_1_A)
0.16/0.38	= { by axiom 11 (conjecture_1_2) }
0.16/0.38	  fresh(true, true, sK2_conjecture_1_A)
0.16/0.38	= { by axiom 3 (axiom_2_1) }
0.16/0.38	  sK2_conjecture_1_A
0.16/0.38	
0.16/0.38	Lemma 20: times(sK3_conjecture_1_C, times(sK1_axiom_2_C(sK2_conjecture_1_A), X)) = times(sK3_conjecture_1_C, X).
0.16/0.38	Proof:
0.16/0.38	  times(sK3_conjecture_1_C, times(sK1_axiom_2_C(sK2_conjecture_1_A), X))
0.16/0.38	= { by lemma 16 }
0.16/0.38	  times(sK4_conjecture_1_B, times(times(sK1_axiom_2_C(sK2_conjecture_1_A), X), sK2_conjecture_1_A))
0.16/0.38	= { by axiom 5 (axiom_1) }
0.16/0.38	  times(sK4_conjecture_1_B, times(X, times(sK2_conjecture_1_A, sK1_axiom_2_C(sK2_conjecture_1_A))))
0.16/0.38	= { by lemma 19 }
0.16/0.38	  times(sK4_conjecture_1_B, times(X, sK2_conjecture_1_A))
0.16/0.38	= { by lemma 16 }
0.16/0.38	  times(sK3_conjecture_1_C, X)
0.16/0.38	
0.16/0.38	Lemma 21: times(sK4_conjecture_1_B, sK2_conjecture_1_A) = sK3_conjecture_1_C.
0.16/0.38	Proof:
0.16/0.38	  times(sK4_conjecture_1_B, sK2_conjecture_1_A)
0.16/0.38	= { by lemma 15 }
0.16/0.38	  times(sK1_axiom_2_C(sK4_conjecture_1_B), sK3_conjecture_1_C)
0.16/0.38	= { by lemma 17 }
0.16/0.38	  times(sK4_conjecture_1_B, times(sK3_conjecture_1_C, sK4_conjecture_1_B))
0.16/0.38	= { by lemma 20 }
0.16/0.38	  times(sK4_conjecture_1_B, times(sK3_conjecture_1_C, times(sK1_axiom_2_C(sK2_conjecture_1_A), sK4_conjecture_1_B)))
0.16/0.38	= { by axiom 5 (axiom_1) }
0.16/0.38	  times(sK4_conjecture_1_B, times(times(sK4_conjecture_1_B, sK3_conjecture_1_C), sK1_axiom_2_C(sK2_conjecture_1_A)))
0.16/0.38	= { by axiom 5 (axiom_1) }
0.16/0.38	  times(times(sK1_axiom_2_C(sK2_conjecture_1_A), sK4_conjecture_1_B), times(sK4_conjecture_1_B, sK3_conjecture_1_C))
0.16/0.38	= { by axiom 5 (axiom_1) }
0.16/0.38	  times(times(sK3_conjecture_1_C, times(sK1_axiom_2_C(sK2_conjecture_1_A), sK4_conjecture_1_B)), sK4_conjecture_1_B)
0.16/0.38	= { by lemma 20 }
0.16/0.38	  times(times(sK3_conjecture_1_C, sK4_conjecture_1_B), sK4_conjecture_1_B)
0.16/0.38	= { by axiom 5 (axiom_1) }
0.16/0.38	  times(sK4_conjecture_1_B, times(sK4_conjecture_1_B, sK3_conjecture_1_C))
0.16/0.38	= { by lemma 12 }
0.16/0.38	  times(sK2_conjecture_1_A, times(sK4_conjecture_1_B, times(sK4_conjecture_1_B, sK4_conjecture_1_B)))
0.16/0.38	= { by lemma 13 }
0.16/0.38	  times(sK2_conjecture_1_A, times(sK4_conjecture_1_B, sK1_axiom_2_C(sK4_conjecture_1_B)))
0.16/0.38	= { by lemma 14 }
0.16/0.38	  times(sK2_conjecture_1_A, sK4_conjecture_1_B)
0.16/0.38	= { by axiom 9 (conjecture_1) }
0.16/0.38	  sK3_conjecture_1_C
0.16/0.38	
0.16/0.38	Lemma 22: times(sK2_conjecture_1_A, times(X, sK2_conjecture_1_A)) = times(sK1_axiom_2_C(sK2_conjecture_1_A), X).
0.16/0.38	Proof:
0.16/0.38	  times(sK2_conjecture_1_A, times(X, sK2_conjecture_1_A))
0.16/0.38	= { by axiom 5 (axiom_1) }
0.16/0.38	  times(times(sK2_conjecture_1_A, sK2_conjecture_1_A), X)
0.16/0.38	= { by axiom 6 (axiom_2_2) }
0.16/0.38	  times(fresh2(element(sK2_conjecture_1_A), true, sK2_conjecture_1_A), X)
0.16/0.38	= { by axiom 11 (conjecture_1_2) }
0.16/0.38	  times(fresh2(true, true, sK2_conjecture_1_A), X)
0.16/0.38	= { by axiom 4 (axiom_2_2) }
0.16/0.39	  times(sK1_axiom_2_C(sK2_conjecture_1_A), X)
0.16/0.39	
0.16/0.39	Goal 1 (conjecture_1_3): element(sK3_conjecture_1_C) = true.
0.16/0.39	Proof:
0.16/0.39	  element(sK3_conjecture_1_C)
0.16/0.39	= { by axiom 1 (axiom_2) }
0.16/0.39	  fresh3(times(sK3_conjecture_1_C, sK3_conjecture_1_C), times(sK3_conjecture_1_C, sK3_conjecture_1_C), sK3_conjecture_1_C, times(sK3_conjecture_1_C, sK3_conjecture_1_C))
0.16/0.39	= { by axiom 8 (axiom_2) }
0.16/0.39	  fresh4(times(sK3_conjecture_1_C, times(sK3_conjecture_1_C, sK3_conjecture_1_C)), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by lemma 12 }
0.16/0.39	  fresh4(times(sK2_conjecture_1_A, times(sK4_conjecture_1_B, times(sK3_conjecture_1_C, sK3_conjecture_1_C))), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by lemma 12 }
0.16/0.39	  fresh4(times(sK2_conjecture_1_A, times(sK2_conjecture_1_A, times(sK4_conjecture_1_B, times(sK4_conjecture_1_B, sK3_conjecture_1_C)))), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by lemma 21 }
0.16/0.39	  fresh4(times(sK2_conjecture_1_A, times(sK2_conjecture_1_A, times(sK4_conjecture_1_B, times(sK4_conjecture_1_B, times(sK4_conjecture_1_B, sK2_conjecture_1_A))))), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by lemma 15 }
0.16/0.39	  fresh4(times(sK2_conjecture_1_A, times(sK2_conjecture_1_A, times(sK4_conjecture_1_B, times(sK4_conjecture_1_B, times(sK1_axiom_2_C(sK4_conjecture_1_B), sK3_conjecture_1_C))))), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by lemma 12 }
0.16/0.39	  fresh4(times(sK2_conjecture_1_A, times(sK2_conjecture_1_A, times(sK4_conjecture_1_B, times(sK2_conjecture_1_A, times(sK4_conjecture_1_B, times(sK4_conjecture_1_B, sK1_axiom_2_C(sK4_conjecture_1_B))))))), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by lemma 14 }
0.16/0.39	  fresh4(times(sK2_conjecture_1_A, times(sK2_conjecture_1_A, times(sK4_conjecture_1_B, times(sK2_conjecture_1_A, times(sK4_conjecture_1_B, sK4_conjecture_1_B))))), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by lemma 13 }
0.16/0.39	  fresh4(times(sK2_conjecture_1_A, times(sK2_conjecture_1_A, times(sK4_conjecture_1_B, times(sK2_conjecture_1_A, sK1_axiom_2_C(sK4_conjecture_1_B))))), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by lemma 12 }
0.16/0.39	  fresh4(times(sK2_conjecture_1_A, times(sK2_conjecture_1_A, times(sK1_axiom_2_C(sK4_conjecture_1_B), sK3_conjecture_1_C))), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by lemma 15 }
0.16/0.39	  fresh4(times(sK2_conjecture_1_A, times(sK2_conjecture_1_A, times(sK4_conjecture_1_B, sK2_conjecture_1_A))), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by lemma 21 }
0.16/0.39	  fresh4(times(sK2_conjecture_1_A, times(sK2_conjecture_1_A, sK3_conjecture_1_C)), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by axiom 9 (conjecture_1) }
0.16/0.39	  fresh4(times(sK2_conjecture_1_A, times(sK2_conjecture_1_A, times(sK2_conjecture_1_A, sK4_conjecture_1_B))), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by axiom 5 (axiom_1) }
0.16/0.39	  fresh4(times(sK2_conjecture_1_A, times(times(sK4_conjecture_1_B, sK2_conjecture_1_A), sK2_conjecture_1_A)), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by lemma 21 }
0.16/0.39	  fresh4(times(sK2_conjecture_1_A, times(sK3_conjecture_1_C, sK2_conjecture_1_A)), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by lemma 22 }
0.16/0.39	  fresh4(times(sK1_axiom_2_C(sK2_conjecture_1_A), sK3_conjecture_1_C), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by lemma 21 }
0.16/0.39	  fresh4(times(sK1_axiom_2_C(sK2_conjecture_1_A), times(sK4_conjecture_1_B, sK2_conjecture_1_A)), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by axiom 5 (axiom_1) }
0.16/0.39	  fresh4(times(times(sK2_conjecture_1_A, sK1_axiom_2_C(sK2_conjecture_1_A)), sK4_conjecture_1_B), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by lemma 19 }
0.16/0.39	  fresh4(times(sK2_conjecture_1_A, sK4_conjecture_1_B), sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by axiom 9 (conjecture_1) }
0.16/0.39	  fresh4(sK3_conjecture_1_C, sK3_conjecture_1_C, sK3_conjecture_1_C)
0.16/0.39	= { by axiom 2 (axiom_2) }
0.16/0.39	  true
0.16/0.39	% SZS output end Proof
0.16/0.39	
0.16/0.39	RESULT: Theorem (the conjecture is true).
0.20/0.40	EOF