0.12/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.12/0.12	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.12/0.34	% Computer   : n011.cluster.edu
0.12/0.34	% Model      : x86_64 x86_64
0.12/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.34	% Memory     : 8042.1875MB
0.12/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.34	% CPULimit   : 1200
0.12/0.34	% WCLimit    : 120
0.12/0.34	% DateTime   : Tue Jul 13 11:44:40 EDT 2021
0.12/0.34	% CPUTime    : 
0.19/0.55	% SZS status Theorem
0.19/0.55	
0.19/0.55	% SZS output start Proof
0.19/0.55	Take the following subset of the input axioms:
0.19/0.55	  fof(axiom_1, axiom, ?[Y24]: ![X19]: ((id(X19, Y24) & r1(X19)) | (~r1(X19) & ~id(X19, Y24)))).
0.19/0.55	  fof(axiom_1a, axiom, ![X1, X8]: ?[Y4]: (?[Y5]: (?[Y15]: (r2(X8, Y15) & r3(X1, Y15, Y5)) & id(Y5, Y4)) & ?[Y7]: (r2(Y7, Y4) & r3(X1, X8, Y7)))).
0.19/0.55	  fof(axiom_2, axiom, ![X11]: ?[Y21]: ![X12]: ((id(X12, Y21) & r2(X11, X12)) | (~r2(X11, X12) & ~id(X12, Y21)))).
0.19/0.55	  fof(axiom_5, axiom, ![X20]: id(X20, X20)).
0.19/0.55	  fof(axiom_6, axiom, ![X21, X22]: (id(X22, X21) | ~id(X21, X22))).
0.19/0.55	  fof(axiom_7a, axiom, ![X7, Y10]: (~r2(X7, Y10) | ![Y20]: (~id(Y20, Y10) | ~r1(Y20)))).
0.19/0.55	  fof(axiom_8, axiom, ![X26, X27]: (~id(X26, X27) | ((~r1(X26) & ~r1(X27)) | (r1(X26) & r1(X27))))).
0.19/0.55	  fof(infiniteNumbersid, conjecture, ![X1]: ?[Y2, Y1]: (?[Y3]: (id(Y3, Y2) & r3(X1, Y1, Y3)) & ![Y4]: (~r1(Y4) | ~id(Y1, Y4)))).
0.19/0.55	
0.19/0.55	Now clausify the problem and encode Horn clauses using encoding 3 of
0.19/0.55	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.19/0.55	We repeatedly replace C & s=t => u=v by the two clauses:
0.19/0.55	  fresh(y, y, x1...xn) = u
0.19/0.55	  C => fresh(s, t, x1...xn) = v
0.19/0.55	where fresh is a fresh function symbol and x1..xn are the free
0.19/0.55	variables of u and v.
0.19/0.55	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.19/0.55	input problem has no model of domain size 1).
0.19/0.55	
0.19/0.55	The encoding turns the above axioms into the following unit equations and goals:
0.19/0.55	
0.19/0.55	Axiom 1 (axiom_5): id(X, X) = true2.
0.19/0.55	Axiom 2 (infiniteNumbersid_1): fresh(X, X, Y) = true2.
0.19/0.55	Axiom 3 (axiom_1_1): fresh21(X, X, Y) = true2.
0.19/0.55	Axiom 4 (axiom_8_1): fresh7(X, X, Y) = true2.
0.19/0.55	Axiom 5 (infiniteNumbersid): fresh3(X, X, Y) = true2.
0.19/0.55	Axiom 6 (axiom_1a_1): r3(X, Y, y7(X, Y)) = true2.
0.19/0.55	Axiom 7 (axiom_1_1): fresh21(r1(X), true2, X) = id(X, y24).
0.19/0.55	Axiom 8 (axiom_2): fresh20(X, X, Y, Z) = true2.
0.19/0.55	Axiom 9 (axiom_6): fresh13(X, X, Y, Z) = true2.
0.19/0.55	Axiom 10 (axiom_8_1): fresh8(X, X, Y, Z) = r1(Y).
0.19/0.55	Axiom 11 (axiom_6): fresh13(id(X, Y), true2, X, Y) = id(Y, X).
0.19/0.55	Axiom 12 (axiom_8_1): fresh8(r1(X), true2, Y, X) = fresh7(id(Y, X), true2, Y).
0.19/0.55	Axiom 13 (infiniteNumbersid): fresh4(X, X, Y, Z, W) = id(Z, y4(Z)).
0.19/0.55	Axiom 14 (infiniteNumbersid_1): fresh2(X, X, Y, Z, W) = r1(y4(Z)).
0.19/0.55	Axiom 15 (axiom_2): fresh20(id(X, y21(Y)), true2, Y, X) = r2(Y, X).
0.19/0.55	Axiom 16 (infiniteNumbersid): fresh4(r3(x1, X, Y), true2, Z, X, Y) = fresh3(id(Y, Z), true2, X).
0.19/0.55	Axiom 17 (infiniteNumbersid_1): fresh2(r3(x1, X, Y), true2, Z, X, Y) = fresh(id(Y, Z), true2, X).
0.19/0.55	
0.19/0.56	Lemma 18: r1(X) = true2.
0.19/0.56	Proof:
0.19/0.56	  r1(X)
0.19/0.56	= { by axiom 10 (axiom_8_1) R->L }
0.19/0.56	  fresh8(true2, true2, X, y4(X))
0.19/0.56	= { by axiom 2 (infiniteNumbersid_1) R->L }
0.19/0.56	  fresh8(fresh(true2, true2, X), true2, X, y4(X))
0.19/0.56	= { by axiom 1 (axiom_5) R->L }
0.19/0.56	  fresh8(fresh(id(y7(x1, X), y7(x1, X)), true2, X), true2, X, y4(X))
0.19/0.56	= { by axiom 17 (infiniteNumbersid_1) R->L }
0.19/0.56	  fresh8(fresh2(r3(x1, X, y7(x1, X)), true2, y7(x1, X), X, y7(x1, X)), true2, X, y4(X))
0.19/0.56	= { by axiom 6 (axiom_1a_1) }
0.19/0.56	  fresh8(fresh2(true2, true2, y7(x1, X), X, y7(x1, X)), true2, X, y4(X))
0.19/0.56	= { by axiom 14 (infiniteNumbersid_1) }
0.19/0.56	  fresh8(r1(y4(X)), true2, X, y4(X))
0.19/0.56	= { by axiom 12 (axiom_8_1) }
0.19/0.56	  fresh7(id(X, y4(X)), true2, X)
0.19/0.56	= { by axiom 13 (infiniteNumbersid) R->L }
0.19/0.56	  fresh7(fresh4(true2, true2, y7(x1, X), X, y7(x1, X)), true2, X)
0.19/0.56	= { by axiom 6 (axiom_1a_1) R->L }
0.19/0.56	  fresh7(fresh4(r3(x1, X, y7(x1, X)), true2, y7(x1, X), X, y7(x1, X)), true2, X)
0.19/0.56	= { by axiom 16 (infiniteNumbersid) }
0.19/0.56	  fresh7(fresh3(id(y7(x1, X), y7(x1, X)), true2, X), true2, X)
0.19/0.56	= { by axiom 1 (axiom_5) }
0.19/0.56	  fresh7(fresh3(true2, true2, X), true2, X)
0.19/0.56	= { by axiom 5 (infiniteNumbersid) }
0.19/0.56	  fresh7(true2, true2, X)
0.19/0.56	= { by axiom 4 (axiom_8_1) }
0.19/0.56	  true2
0.19/0.56	
0.19/0.56	Lemma 19: id(X, y24) = true2.
0.19/0.56	Proof:
0.19/0.56	  id(X, y24)
0.19/0.56	= { by axiom 7 (axiom_1_1) R->L }
0.19/0.56	  fresh21(r1(X), true2, X)
0.19/0.56	= { by lemma 18 }
0.19/0.56	  fresh21(true2, true2, X)
0.19/0.56	= { by axiom 3 (axiom_1_1) }
0.19/0.56	  true2
0.19/0.56	
0.19/0.56	Goal 1 (axiom_7a): tuple(id(X, Y), r2(Z, Y), r1(X)) = tuple(true2, true2, true2).
0.19/0.56	The goal is true when:
0.19/0.56	  X = X
0.19/0.56	  Y = y24
0.19/0.56	  Z = Y
0.19/0.56	
0.19/0.56	Proof:
0.19/0.56	  tuple(id(X, y24), r2(Y, y24), r1(X))
0.19/0.56	= { by axiom 15 (axiom_2) R->L }
0.19/0.56	  tuple(id(X, y24), fresh20(id(y24, y21(Y)), true2, Y, y24), r1(X))
0.19/0.56	= { by axiom 11 (axiom_6) R->L }
0.19/0.56	  tuple(id(X, y24), fresh20(fresh13(id(y21(Y), y24), true2, y21(Y), y24), true2, Y, y24), r1(X))
0.19/0.56	= { by lemma 19 }
0.19/0.56	  tuple(id(X, y24), fresh20(fresh13(true2, true2, y21(Y), y24), true2, Y, y24), r1(X))
0.19/0.56	= { by axiom 9 (axiom_6) }
0.19/0.56	  tuple(id(X, y24), fresh20(true2, true2, Y, y24), r1(X))
0.19/0.56	= { by axiom 8 (axiom_2) }
0.19/0.56	  tuple(id(X, y24), true2, r1(X))
0.19/0.56	= { by lemma 19 }
0.19/0.56	  tuple(true2, true2, r1(X))
0.19/0.56	= { by lemma 18 }
0.19/0.56	  tuple(true2, true2, true2)
0.19/0.56	% SZS output end Proof
0.19/0.56	
0.19/0.56	RESULT: Theorem (the conjecture is true).
0.19/0.56	EOF