0.00/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.11/0.12	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.14/0.33	% Computer   : n013.cluster.edu
0.14/0.33	% Model      : x86_64 x86_64
0.14/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.14/0.33	% Memory     : 8042.1875MB
0.14/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.14/0.33	% CPULimit   : 1200
0.14/0.33	% WCLimit    : 120
0.14/0.33	% DateTime   : Tue Jul 13 10:48:48 EDT 2021
0.14/0.34	% CPUTime    : 
0.20/0.39	% SZS status Theorem
0.20/0.39	
0.20/0.42	% SZS output start Proof
0.20/0.42	Take the following subset of the input axioms:
0.20/0.42	  fof(axiom_1, axiom, ![B, C, A]: times(B, times(C, A))=times(times(A, B), C)).
0.20/0.42	  fof(axiom_2, axiom, ![B]: (?[C]: (C=times(B, B) & times(B, C)=B) <=> element(B))).
0.20/0.42	  fof(conjecture_1, conjecture, ![B, C, A]: ((element(A) & (times(A, B)=C & element(B))) => element(C))).
0.20/0.42	
0.20/0.42	Now clausify the problem and encode Horn clauses using encoding 3 of
0.20/0.42	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.20/0.42	We repeatedly replace C & s=t => u=v by the two clauses:
0.20/0.42	  fresh(y, y, x1...xn) = u
0.20/0.42	  C => fresh(s, t, x1...xn) = v
0.20/0.42	where fresh is a fresh function symbol and x1..xn are the free
0.20/0.42	variables of u and v.
0.20/0.42	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.20/0.42	input problem has no model of domain size 1).
0.20/0.42	
0.20/0.42	The encoding turns the above axioms into the following unit equations and goals:
0.20/0.42	
0.20/0.42	Axiom 1 (conjecture_1_1): element(b) = true.
0.20/0.42	Axiom 2 (conjecture_1_2): element(a) = true.
0.20/0.42	Axiom 3 (conjecture_1): times(a, b) = c.
0.20/0.42	Axiom 4 (axiom_2_1): fresh(X, X, Y) = Y.
0.20/0.42	Axiom 5 (axiom_2): fresh4(X, X, Y) = true.
0.20/0.42	Axiom 6 (axiom_2_2): fresh2(X, X, Y) = c2(Y).
0.20/0.42	Axiom 7 (axiom_1): times(X, times(Y, Z)) = times(times(Z, X), Y).
0.20/0.42	Axiom 8 (axiom_2_1): fresh(element(X), true, X) = times(X, c2(X)).
0.20/0.42	Axiom 9 (axiom_2): fresh3(X, X, Y, Z) = element(Y).
0.20/0.42	Axiom 10 (axiom_2_2): fresh2(element(X), true, X) = times(X, X).
0.20/0.42	Axiom 11 (axiom_2): fresh3(X, times(Y, Y), Y, X) = fresh4(times(Y, X), Y, Y).
0.20/0.42	
0.20/0.42	Lemma 12: times(a, a) = c2(a).
0.20/0.42	Proof:
0.20/0.42	  times(a, a)
0.20/0.42	= { by axiom 10 (axiom_2_2) R->L }
0.20/0.42	  fresh2(element(a), true, a)
0.20/0.42	= { by axiom 2 (conjecture_1_2) }
0.20/0.42	  fresh2(true, true, a)
0.20/0.42	= { by axiom 6 (axiom_2_2) }
0.20/0.42	  c2(a)
0.20/0.42	
0.20/0.42	Lemma 13: times(b, b) = c2(b).
0.20/0.42	Proof:
0.20/0.42	  times(b, b)
0.20/0.42	= { by axiom 10 (axiom_2_2) R->L }
0.20/0.42	  fresh2(element(b), true, b)
0.20/0.42	= { by axiom 1 (conjecture_1_1) }
0.20/0.42	  fresh2(true, true, b)
0.20/0.42	= { by axiom 6 (axiom_2_2) }
0.20/0.42	  c2(b)
0.20/0.42	
0.20/0.42	Lemma 14: times(Z, times(W, times(X, Y))) = times(X, times(Y, times(Z, W))).
0.20/0.42	Proof:
0.20/0.42	  times(Z, times(W, times(X, Y)))
0.20/0.42	= { by axiom 7 (axiom_1) }
0.20/0.42	  times(Z, times(times(Y, W), X))
0.20/0.42	= { by axiom 7 (axiom_1) }
0.20/0.42	  times(times(X, Z), times(Y, W))
0.20/0.42	= { by axiom 7 (axiom_1) }
0.20/0.42	  times(times(W, times(X, Z)), Y)
0.20/0.42	= { by axiom 7 (axiom_1) }
0.20/0.42	  times(times(times(Z, W), X), Y)
0.20/0.42	= { by axiom 7 (axiom_1) R->L }
0.20/0.42	  times(X, times(Y, times(Z, W)))
0.20/0.42	
0.20/0.42	Lemma 15: times(a, times(b, times(X, Y))) = times(X, times(Y, c)).
0.20/0.42	Proof:
0.20/0.42	  times(a, times(b, times(X, Y)))
0.20/0.42	= { by lemma 14 }
0.20/0.42	  times(X, times(Y, times(a, b)))
0.20/0.42	= { by axiom 3 (conjecture_1) }
0.20/0.42	  times(X, times(Y, c))
0.20/0.42	
0.20/0.42	Lemma 16: times(b, c2(b)) = b.
0.20/0.42	Proof:
0.20/0.42	  times(b, c2(b))
0.20/0.42	= { by axiom 8 (axiom_2_1) R->L }
0.20/0.42	  fresh(element(b), true, b)
0.20/0.42	= { by axiom 1 (conjecture_1_1) }
0.20/0.42	  fresh(true, true, b)
0.20/0.42	= { by axiom 4 (axiom_2_1) }
0.20/0.42	  b
0.20/0.42	
0.20/0.42	Lemma 17: times(b, times(b, c)) = c.
0.20/0.42	Proof:
0.20/0.42	  times(b, times(b, c))
0.20/0.42	= { by lemma 15 R->L }
0.20/0.42	  times(a, times(b, times(b, b)))
0.20/0.42	= { by lemma 13 }
0.20/0.42	  times(a, times(b, c2(b)))
0.20/0.42	= { by lemma 16 }
0.20/0.42	  times(a, b)
0.20/0.42	= { by axiom 3 (conjecture_1) }
0.20/0.42	  c
0.20/0.42	
0.20/0.42	Lemma 18: times(c2(b), times(X, b)) = times(b, X).
0.20/0.42	Proof:
0.20/0.42	  times(c2(b), times(X, b))
0.20/0.42	= { by axiom 7 (axiom_1) }
0.20/0.42	  times(times(b, c2(b)), X)
0.20/0.42	= { by lemma 16 }
0.20/0.42	  times(b, X)
0.20/0.42	
0.20/0.42	Lemma 19: times(b, times(c2(b), times(X, Y))) = times(X, times(Y, b)).
0.20/0.42	Proof:
0.20/0.42	  times(b, times(c2(b), times(X, Y)))
0.20/0.42	= { by lemma 14 }
0.20/0.42	  times(X, times(Y, times(b, c2(b))))
0.20/0.42	= { by lemma 16 }
0.20/0.42	  times(X, times(Y, b))
0.20/0.42	
0.20/0.42	Lemma 20: times(b, times(X, b)) = times(c2(b), X).
0.20/0.42	Proof:
0.20/0.42	  times(b, times(X, b))
0.20/0.42	= { by axiom 7 (axiom_1) }
0.20/0.42	  times(times(b, b), X)
0.20/0.42	= { by lemma 13 }
0.20/0.42	  times(c2(b), X)
0.20/0.42	
0.20/0.42	Lemma 21: times(b, c) = times(c, b).
0.20/0.42	Proof:
0.20/0.42	  times(b, c)
0.20/0.42	= { by lemma 17 R->L }
0.20/0.42	  times(b, times(b, times(b, c)))
0.20/0.42	= { by lemma 18 R->L }
0.20/0.42	  times(b, times(c2(b), times(times(b, c), b)))
0.20/0.42	= { by lemma 19 }
0.20/0.42	  times(times(b, c), times(b, b))
0.20/0.42	= { by lemma 13 }
0.20/0.42	  times(times(b, c), c2(b))
0.20/0.42	= { by axiom 7 (axiom_1) R->L }
0.20/0.42	  times(c, times(c2(b), b))
0.20/0.42	= { by lemma 20 R->L }
0.20/0.42	  times(c, times(b, times(b, b)))
0.20/0.42	= { by lemma 13 }
0.20/0.42	  times(c, times(b, c2(b)))
0.20/0.42	= { by lemma 16 }
0.20/0.42	  times(c, b)
0.20/0.42	
0.20/0.42	Lemma 22: times(b, times(X, a)) = times(c, X).
0.20/0.42	Proof:
0.20/0.42	  times(b, times(X, a))
0.20/0.42	= { by axiom 7 (axiom_1) }
0.20/0.42	  times(times(a, b), X)
0.20/0.42	= { by axiom 3 (conjecture_1) }
0.20/0.42	  times(c, X)
0.20/0.42	
0.20/0.42	Lemma 23: times(b, c2(a)) = times(c, a).
0.20/0.42	Proof:
0.20/0.42	  times(b, c2(a))
0.20/0.42	= { by lemma 12 R->L }
0.20/0.42	  times(b, times(a, a))
0.20/0.42	= { by lemma 22 }
0.20/0.42	  times(c, a)
0.20/0.42	
0.20/0.42	Lemma 24: times(a, times(X, a)) = times(c2(a), X).
0.20/0.42	Proof:
0.20/0.42	  times(a, times(X, a))
0.20/0.42	= { by axiom 7 (axiom_1) }
0.20/0.42	  times(times(a, a), X)
0.20/0.42	= { by lemma 12 }
0.20/0.42	  times(c2(a), X)
0.20/0.42	
0.20/0.42	Lemma 25: times(b, times(X, c)) = times(c, times(b, X)).
0.20/0.42	Proof:
0.20/0.42	  times(b, times(X, c))
0.20/0.42	= { by axiom 3 (conjecture_1) R->L }
0.20/0.42	  times(b, times(X, times(a, b)))
0.20/0.42	= { by axiom 7 (axiom_1) }
0.20/0.42	  times(b, times(times(b, X), a))
0.20/0.42	= { by lemma 22 }
0.20/0.42	  times(c, times(b, X))
0.20/0.42	
0.20/0.42	Goal 1 (conjecture_1_3): element(c) = true.
0.20/0.42	Proof:
0.20/0.42	  element(c)
0.20/0.42	= { by axiom 9 (axiom_2) R->L }
0.20/0.42	  fresh3(times(c, c), times(c, c), c, times(c, c))
0.20/0.43	= { by axiom 11 (axiom_2) }
0.20/0.43	  fresh4(times(c, times(c, c)), c, c)
0.20/0.43	= { by lemma 17 R->L }
0.20/0.43	  fresh4(times(c, times(c, times(b, times(b, c)))), c, c)
0.20/0.43	= { by lemma 21 }
0.20/0.43	  fresh4(times(c, times(c, times(b, times(c, b)))), c, c)
0.20/0.43	= { by lemma 20 }
0.20/0.43	  fresh4(times(c, times(c, times(c2(b), c))), c, c)
0.20/0.43	= { by axiom 3 (conjecture_1) R->L }
0.20/0.43	  fresh4(times(c, times(c, times(c2(b), times(a, b)))), c, c)
0.20/0.43	= { by lemma 18 }
0.20/0.43	  fresh4(times(c, times(c, times(b, a))), c, c)
0.20/0.43	= { by lemma 25 R->L }
0.20/0.43	  fresh4(times(c, times(b, times(a, c))), c, c)
0.20/0.43	= { by lemma 25 R->L }
0.20/0.43	  fresh4(times(b, times(times(a, c), c)), c, c)
0.20/0.43	= { by axiom 7 (axiom_1) }
0.20/0.43	  fresh4(times(times(c, b), times(a, c)), c, c)
0.20/0.43	= { by lemma 15 R->L }
0.20/0.43	  fresh4(times(a, times(b, times(times(c, b), a))), c, c)
0.20/0.43	= { by lemma 22 }
0.20/0.43	  fresh4(times(a, times(c, times(c, b))), c, c)
0.20/0.43	= { by lemma 21 R->L }
0.20/0.43	  fresh4(times(a, times(c, times(b, c))), c, c)
0.20/0.43	= { by lemma 25 R->L }
0.20/0.43	  fresh4(times(a, times(b, times(c, c))), c, c)
0.20/0.43	= { by axiom 3 (conjecture_1) R->L }
0.20/0.43	  fresh4(times(a, times(b, times(c, times(a, b)))), c, c)
0.20/0.43	= { by lemma 19 R->L }
0.20/0.43	  fresh4(times(a, times(b, times(b, times(c2(b), times(c, a))))), c, c)
0.20/0.43	= { by lemma 23 R->L }
0.20/0.43	  fresh4(times(a, times(b, times(b, times(c2(b), times(b, c2(a)))))), c, c)
0.20/0.43	= { by lemma 19 }
0.20/0.43	  fresh4(times(a, times(b, times(b, times(c2(a), b)))), c, c)
0.20/0.43	= { by lemma 20 }
0.20/0.43	  fresh4(times(a, times(b, times(c2(b), c2(a)))), c, c)
0.20/0.43	= { by lemma 12 R->L }
0.20/0.43	  fresh4(times(a, times(b, times(c2(b), times(a, a)))), c, c)
0.20/0.43	= { by lemma 19 }
0.20/0.43	  fresh4(times(a, times(a, times(a, b))), c, c)
0.20/0.43	= { by axiom 3 (conjecture_1) }
0.20/0.43	  fresh4(times(a, times(a, c)), c, c)
0.20/0.43	= { by lemma 15 R->L }
0.20/0.43	  fresh4(times(a, times(b, times(a, a))), c, c)
0.20/0.43	= { by lemma 12 }
0.20/0.43	  fresh4(times(a, times(b, c2(a))), c, c)
0.20/0.43	= { by lemma 23 }
0.20/0.43	  fresh4(times(a, times(c, a)), c, c)
0.20/0.43	= { by lemma 24 }
0.20/0.43	  fresh4(times(c2(a), c), c, c)
0.20/0.43	= { by axiom 3 (conjecture_1) R->L }
0.20/0.43	  fresh4(times(c2(a), times(a, b)), c, c)
0.20/0.43	= { by lemma 19 R->L }
0.20/0.43	  fresh4(times(b, times(c2(b), times(c2(a), a))), c, c)
0.20/0.43	= { by lemma 24 R->L }
0.20/0.43	  fresh4(times(b, times(c2(b), times(a, times(a, a)))), c, c)
0.20/0.43	= { by lemma 12 }
0.20/0.43	  fresh4(times(b, times(c2(b), times(a, c2(a)))), c, c)
0.20/0.43	= { by axiom 8 (axiom_2_1) R->L }
0.20/0.43	  fresh4(times(b, times(c2(b), fresh(element(a), true, a))), c, c)
0.20/0.43	= { by axiom 2 (conjecture_1_2) }
0.20/0.43	  fresh4(times(b, times(c2(b), fresh(true, true, a))), c, c)
0.20/0.43	= { by axiom 4 (axiom_2_1) }
0.20/0.43	  fresh4(times(b, times(c2(b), a)), c, c)
0.20/0.43	= { by lemma 22 }
0.20/0.43	  fresh4(times(c, c2(b)), c, c)
0.20/0.43	= { by lemma 13 R->L }
0.20/0.43	  fresh4(times(c, times(b, b)), c, c)
0.20/0.43	= { by lemma 25 R->L }
0.20/0.43	  fresh4(times(b, times(b, c)), c, c)
0.20/0.43	= { by lemma 17 }
0.20/0.43	  fresh4(c, c, c)
0.20/0.43	= { by axiom 5 (axiom_2) }
0.20/0.43	  true
0.20/0.43	% SZS output end Proof
0.20/0.43	
0.20/0.43	RESULT: Theorem (the conjecture is true).
0.20/0.43	EOF