0.04/0.09	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.10	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.09/0.30	% Computer   : n008.cluster.edu
0.09/0.30	% Model      : x86_64 x86_64
0.09/0.30	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.09/0.30	% Memory     : 8042.1875MB
0.09/0.30	% OS         : Linux 3.10.0-693.el7.x86_64
0.09/0.30	% CPULimit   : 1200
0.09/0.30	% WCLimit    : 120
0.09/0.30	% DateTime   : Tue Jul 13 16:41:44 EDT 2021
0.09/0.30	% CPUTime    : 
0.15/0.35	% SZS status Theorem
0.15/0.35	
0.15/0.37	% SZS output start Proof
0.15/0.37	Take the following subset of the input axioms:
0.15/0.38	  fof(axiom_1, axiom, ?[Y24]: ![X19]: ((r1(X19) & Y24=X19) | (X19!=Y24 & ~r1(X19)))).
0.15/0.38	  fof(axiom_1a, axiom, ![X1, X8]: ?[Y4]: (?[Y7]: (r3(X1, X8, Y7) & r2(Y7, Y4)) & ?[Y5]: (Y4=Y5 & ?[Y15]: (r3(X1, Y15, Y5) & r2(X8, Y15))))).
0.15/0.38	  fof(axiom_2, axiom, ![X11]: ?[Y21]: ![X12]: ((Y21!=X12 & ~r2(X11, X12)) | (r2(X11, X12) & X12=Y21))).
0.15/0.38	  fof(axiom_3, axiom, ![X13, X14]: ?[Y22]: ![X15]: ((Y22=X15 & r3(X13, X14, X15)) | (X15!=Y22 & ~r3(X13, X14, X15)))).
0.15/0.38	  fof(axiom_4a, axiom, ![X4]: ?[Y9]: (X4=Y9 & ?[Y16]: (r3(X4, Y16, Y9) & r1(Y16)))).
0.15/0.38	  fof(axiom_7a, axiom, ![X7, Y10]: (~r2(X7, Y10) | ![Y20]: (~r1(Y20) | Y10!=Y20))).
0.15/0.38	  fof(zeroplusoneidzero, conjecture, ![Y1]: (![Y5]: (Y5!=Y1 | ~r1(Y5)) | ![Y2]: (![Y3]: (~r3(Y3, Y2, Y1) | ~r1(Y3)) | ![Y4]: (~r2(Y4, Y2) | ~r1(Y4))))).
0.15/0.38	
0.15/0.38	Now clausify the problem and encode Horn clauses using encoding 3 of
0.15/0.38	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.15/0.38	We repeatedly replace C & s=t => u=v by the two clauses:
0.15/0.38	  fresh(y, y, x1...xn) = u
0.15/0.38	  C => fresh(s, t, x1...xn) = v
0.15/0.38	where fresh is a fresh function symbol and x1..xn are the free
0.15/0.38	variables of u and v.
0.15/0.38	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.15/0.38	input problem has no model of domain size 1).
0.15/0.38	
0.15/0.38	The encoding turns the above axioms into the following unit equations and goals:
0.15/0.38	
0.15/0.38	Axiom 1 (zeroplusoneidzero): y5 = y1.
0.15/0.38	Axiom 2 (axiom_4a): X = y9(X).
0.15/0.38	Axiom 3 (zeroplusoneidzero_3): r1(y5) = true2.
0.15/0.38	Axiom 4 (zeroplusoneidzero_4): r1(y3) = true2.
0.15/0.38	Axiom 5 (zeroplusoneidzero_5): r1(y4) = true2.
0.15/0.38	Axiom 6 (axiom_1a): y4_2(X, Y) = y5_2(X, Y).
0.15/0.38	Axiom 7 (zeroplusoneidzero_1): r2(y4, y2) = true2.
0.15/0.38	Axiom 8 (axiom_4a_2): r1(y16(X)) = true2.
0.15/0.38	Axiom 9 (axiom_1): fresh9(X, X, Y) = true2.
0.15/0.38	Axiom 10 (axiom_1): fresh9(X, y24, X) = r1(X).
0.15/0.38	Axiom 11 (axiom_1_1): fresh2(X, X, Y) = Y.
0.15/0.38	Axiom 12 (zeroplusoneidzero_2): r3(y3, y2, y1) = true2.
0.15/0.38	Axiom 13 (axiom_2_1): fresh(X, X, Y, Z) = Z.
0.15/0.38	Axiom 14 (axiom_2): fresh10(X, X, Y, Z) = true2.
0.15/0.38	Axiom 15 (axiom_1_1): fresh2(r1(X), true2, X) = y24.
0.15/0.38	Axiom 16 (axiom_1a_1): r2(X, y15(Y, X)) = true2.
0.15/0.38	Axiom 17 (axiom_2): fresh10(y21(X), Y, X, Y) = r2(X, Y).
0.15/0.38	Axiom 18 (axiom_3_1): fresh3(X, X, Y, Z, W) = W.
0.15/0.38	Axiom 19 (axiom_1a_3): r3(X, Y, y7(X, Y)) = true2.
0.15/0.38	Axiom 20 (axiom_4a_1): r3(X, y16(X), y9(X)) = true2.
0.15/0.38	Axiom 21 (axiom_2_1): fresh(r2(X, Y), true2, X, Y) = y21(X).
0.15/0.38	Axiom 22 (axiom_1a_2): r2(y7(X, Y), y4_2(X, Y)) = true2.
0.15/0.38	Axiom 23 (axiom_1a_4): r3(X, y15(X, Y), y5_2(X, Y)) = true2.
0.15/0.38	Axiom 24 (axiom_3_1): fresh3(r3(X, Y, Z), true2, X, Y, Z) = y22(X, Y).
0.15/0.38	
0.15/0.38	Goal 1 (axiom_7a): tuple(r2(X, Y), r1(Y)) = tuple(true2, true2).
0.15/0.38	The goal is true when:
0.15/0.38	  X = y24
0.15/0.38	  Y = y24
0.15/0.38	
0.15/0.38	Proof:
0.15/0.38	  tuple(r2(y24, y24), r1(y24))
0.15/0.38	= { by axiom 15 (axiom_1_1) R->L }
0.15/0.38	  tuple(r2(y24, fresh2(r1(y1), true2, y1)), r1(y24))
0.15/0.38	= { by axiom 1 (zeroplusoneidzero) R->L }
0.15/0.38	  tuple(r2(y24, fresh2(r1(y5), true2, y1)), r1(y24))
0.15/0.38	= { by axiom 3 (zeroplusoneidzero_3) }
0.15/0.38	  tuple(r2(y24, fresh2(true2, true2, y1)), r1(y24))
0.15/0.38	= { by axiom 11 (axiom_1_1) }
0.15/0.38	  tuple(r2(y24, y1), r1(y24))
0.15/0.38	= { by axiom 18 (axiom_3_1) R->L }
0.15/0.38	  tuple(r2(y24, fresh3(true2, true2, y3, y2, y1)), r1(y24))
0.15/0.38	= { by axiom 12 (zeroplusoneidzero_2) R->L }
0.15/0.38	  tuple(r2(y24, fresh3(r3(y3, y2, y1), true2, y3, y2, y1)), r1(y24))
0.15/0.38	= { by axiom 24 (axiom_3_1) }
0.15/0.38	  tuple(r2(y24, y22(y3, y2)), r1(y24))
0.15/0.38	= { by axiom 11 (axiom_1_1) R->L }
0.15/0.38	  tuple(r2(y24, y22(fresh2(true2, true2, y3), y2)), r1(y24))
0.15/0.38	= { by axiom 4 (zeroplusoneidzero_4) R->L }
0.15/0.38	  tuple(r2(y24, y22(fresh2(r1(y3), true2, y3), y2)), r1(y24))
0.15/0.38	= { by axiom 15 (axiom_1_1) }
0.15/0.38	  tuple(r2(y24, y22(y24, y2)), r1(y24))
0.15/0.38	= { by axiom 13 (axiom_2_1) R->L }
0.15/0.38	  tuple(r2(y24, y22(y24, fresh(true2, true2, y4, y2))), r1(y24))
0.15/0.38	= { by axiom 7 (zeroplusoneidzero_1) R->L }
0.15/0.38	  tuple(r2(y24, y22(y24, fresh(r2(y4, y2), true2, y4, y2))), r1(y24))
0.15/0.38	= { by axiom 21 (axiom_2_1) }
0.15/0.38	  tuple(r2(y24, y22(y24, y21(y4))), r1(y24))
0.15/0.38	= { by axiom 11 (axiom_1_1) R->L }
0.15/0.38	  tuple(r2(y24, y22(y24, y21(fresh2(true2, true2, y4)))), r1(y24))
0.15/0.38	= { by axiom 5 (zeroplusoneidzero_5) R->L }
0.15/0.38	  tuple(r2(y24, y22(y24, y21(fresh2(r1(y4), true2, y4)))), r1(y24))
0.15/0.38	= { by axiom 15 (axiom_1_1) }
0.15/0.38	  tuple(r2(y24, y22(y24, y21(y24))), r1(y24))
0.15/0.38	= { by axiom 21 (axiom_2_1) R->L }
0.15/0.38	  tuple(r2(y24, y22(y24, fresh(r2(y24, y15(y24, y24)), true2, y24, y15(y24, y24)))), r1(y24))
0.15/0.38	= { by axiom 16 (axiom_1a_1) }
0.15/0.38	  tuple(r2(y24, y22(y24, fresh(true2, true2, y24, y15(y24, y24)))), r1(y24))
0.15/0.38	= { by axiom 13 (axiom_2_1) }
0.15/0.38	  tuple(r2(y24, y22(y24, y15(y24, y24))), r1(y24))
0.15/0.38	= { by axiom 24 (axiom_3_1) R->L }
0.15/0.38	  tuple(r2(y24, fresh3(r3(y24, y15(y24, y24), y4_2(y24, y24)), true2, y24, y15(y24, y24), y4_2(y24, y24))), r1(y24))
0.15/0.38	= { by axiom 6 (axiom_1a) }
0.15/0.38	  tuple(r2(y24, fresh3(r3(y24, y15(y24, y24), y5_2(y24, y24)), true2, y24, y15(y24, y24), y4_2(y24, y24))), r1(y24))
0.15/0.38	= { by axiom 23 (axiom_1a_4) }
0.15/0.39	  tuple(r2(y24, fresh3(true2, true2, y24, y15(y24, y24), y4_2(y24, y24))), r1(y24))
0.15/0.39	= { by axiom 18 (axiom_3_1) }
0.15/0.39	  tuple(r2(y24, y4_2(y24, y24)), r1(y24))
0.15/0.39	= { by axiom 13 (axiom_2_1) R->L }
0.15/0.39	  tuple(r2(y24, fresh(true2, true2, y7(y24, y24), y4_2(y24, y24))), r1(y24))
0.15/0.39	= { by axiom 22 (axiom_1a_2) R->L }
0.15/0.39	  tuple(r2(y24, fresh(r2(y7(y24, y24), y4_2(y24, y24)), true2, y7(y24, y24), y4_2(y24, y24))), r1(y24))
0.15/0.39	= { by axiom 21 (axiom_2_1) }
0.15/0.39	  tuple(r2(y24, y21(y7(y24, y24))), r1(y24))
0.15/0.39	= { by axiom 18 (axiom_3_1) R->L }
0.15/0.39	  tuple(r2(y24, y21(fresh3(true2, true2, y24, y24, y7(y24, y24)))), r1(y24))
0.15/0.39	= { by axiom 19 (axiom_1a_3) R->L }
0.15/0.39	  tuple(r2(y24, y21(fresh3(r3(y24, y24, y7(y24, y24)), true2, y24, y24, y7(y24, y24)))), r1(y24))
0.15/0.39	= { by axiom 24 (axiom_3_1) }
0.15/0.39	  tuple(r2(y24, y21(y22(y24, y24))), r1(y24))
0.15/0.39	= { by axiom 15 (axiom_1_1) R->L }
0.15/0.39	  tuple(r2(y24, y21(y22(y24, fresh2(r1(y16(y24)), true2, y16(y24))))), r1(y24))
0.15/0.39	= { by axiom 8 (axiom_4a_2) }
0.15/0.39	  tuple(r2(y24, y21(y22(y24, fresh2(true2, true2, y16(y24))))), r1(y24))
0.15/0.39	= { by axiom 11 (axiom_1_1) }
0.15/0.39	  tuple(r2(y24, y21(y22(y24, y16(y24)))), r1(y24))
0.15/0.39	= { by axiom 24 (axiom_3_1) R->L }
0.15/0.39	  tuple(r2(y24, y21(fresh3(r3(y24, y16(y24), y24), true2, y24, y16(y24), y24))), r1(y24))
0.15/0.39	= { by axiom 2 (axiom_4a) }
0.15/0.39	  tuple(r2(y24, y21(fresh3(r3(y24, y16(y24), y9(y24)), true2, y24, y16(y24), y24))), r1(y24))
0.15/0.39	= { by axiom 20 (axiom_4a_1) }
0.15/0.39	  tuple(r2(y24, y21(fresh3(true2, true2, y24, y16(y24), y24))), r1(y24))
0.15/0.39	= { by axiom 18 (axiom_3_1) }
0.15/0.39	  tuple(r2(y24, y21(y24)), r1(y24))
0.15/0.39	= { by axiom 17 (axiom_2) R->L }
0.15/0.39	  tuple(fresh10(y21(y24), y21(y24), y24, y21(y24)), r1(y24))
0.15/0.39	= { by axiom 14 (axiom_2) }
0.15/0.39	  tuple(true2, r1(y24))
0.15/0.39	= { by axiom 10 (axiom_1) R->L }
0.15/0.39	  tuple(true2, fresh9(y24, y24, y24))
0.15/0.39	= { by axiom 9 (axiom_1) }
0.15/0.39	  tuple(true2, true2)
0.15/0.39	% SZS output end Proof
0.15/0.39	
0.15/0.39	RESULT: Theorem (the conjecture is true).
0.15/0.39	EOF