0.05/0.09	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.05/0.09	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.09/0.29	% Computer   : n029.cluster.edu
0.09/0.29	% Model      : x86_64 x86_64
0.09/0.29	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.09/0.29	% Memory     : 8042.1875MB
0.09/0.29	% OS         : Linux 3.10.0-693.el7.x86_64
0.09/0.29	% CPULimit   : 1200
0.09/0.29	% WCLimit    : 120
0.09/0.29	% DateTime   : Tue Jul 13 12:57:40 EDT 2021
0.09/0.29	% CPUTime    : 
10.93/1.71	% SZS status Theorem
10.93/1.71	
10.93/1.72	% SZS output start Proof
10.93/1.72	Take the following subset of the input axioms:
10.93/1.72	  fof(axiom_1, axiom, ?[Y24]: ![X19]: ((id(X19, Y24) & r1(X19)) | (~r1(X19) & ~id(X19, Y24)))).
10.93/1.72	  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)))).
10.93/1.72	  fof(axiom_2a, axiom, ![X2, X9]: ?[Y2]: (?[Y3]: (?[Y14]: (r4(X2, Y14, Y3) & r2(X9, Y14)) & id(Y3, Y2)) & ?[Y6]: (r3(Y6, X2, Y2) & r4(X2, X9, Y6)))).
10.93/1.72	  fof(axiom_4a, axiom, ![X4]: ?[Y9]: (?[Y16]: (r1(Y16) & r3(X4, Y16, Y9)) & id(Y9, X4))).
10.93/1.72	  fof(axiom_5, axiom, ![X20]: id(X20, X20)).
10.93/1.72	  fof(axiom_7a, axiom, ![X7, Y10]: (~r2(X7, Y10) | ![Y20]: (~id(Y20, Y10) | ~r1(Y20)))).
10.93/1.72	  fof(axiom_9, axiom, ![X28, X29, X30, X31]: (~id(X29, X31) | ((r2(X28, X29) & r2(X30, X31)) | ((~r2(X28, X29) & ~r2(X30, X31)) | ~id(X28, X30))))).
10.93/1.72	  fof(xplustwoidy, conjecture, ?[Y2, Y3, Y1]: (id(Y3, Y2) & ?[Y4]: (?[Y5]: (r2(Y5, Y4) & ?[Y6]: (r1(Y6) & r2(Y6, Y5))) & r3(Y1, Y4, Y3)))).
10.93/1.72	
10.93/1.72	Now clausify the problem and encode Horn clauses using encoding 3 of
10.93/1.72	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
10.93/1.72	We repeatedly replace C & s=t => u=v by the two clauses:
10.93/1.72	  fresh(y, y, x1...xn) = u
10.93/1.72	  C => fresh(s, t, x1...xn) = v
10.93/1.72	where fresh is a fresh function symbol and x1..xn are the free
10.93/1.72	variables of u and v.
10.93/1.72	A predicate p(X) is encoded as p(X)=true (this is sound, because the
10.93/1.72	input problem has no model of domain size 1).
10.93/1.72	
10.93/1.72	The encoding turns the above axioms into the following unit equations and goals:
10.93/1.72	
10.93/1.72	Axiom 1 (axiom_5): id(X, X) = true2.
10.93/1.72	Axiom 2 (axiom_4a_2): r1(y16(X)) = true2.
10.93/1.72	Axiom 3 (axiom_1_1): fresh17(X, X, Y) = true2.
10.93/1.72	Axiom 4 (axiom_2a_4): r2(X, y14(Y, X)) = true2.
10.93/1.72	Axiom 5 (axiom_9_1): fresh32(X, X, Y, Z) = true2.
10.93/1.72	Axiom 6 (axiom_1_1): fresh17(r1(X), true2, X) = id(X, y24).
10.93/1.72	Axiom 7 (axiom_1a_1): r3(X, Y, y7(X, Y)) = true2.
10.93/1.72	Axiom 8 (axiom_9_1): fresh(X, X, Y, Z, W) = r2(Y, Z).
10.93/1.72	Axiom 9 (axiom_9_1): fresh31(X, X, Y, Z, W, V) = fresh32(id(Y, W), true2, Y, Z).
10.93/1.72	Axiom 10 (axiom_9_1): fresh31(r2(X, Y), true2, Z, W, X, Y) = fresh(id(W, Y), true2, Z, W, X).
10.93/1.72	
10.93/1.72	Goal 1 (xplustwoidy): tuple(id(X, Y), r3(Z, W, X), r2(V, W), r2(U, V), r1(U)) = tuple(true2, true2, true2, true2, true2).
10.93/1.72	The goal is true when:
10.93/1.72	  X = y7(W, y14(Z, y14(Y, y24)))
10.93/1.72	  Y = y7(W, y14(Z, y14(Y, y24)))
10.93/1.72	  Z = W
10.93/1.72	  W = y14(Z, y14(Y, y24))
10.93/1.72	  V = y14(Y, y24)
10.93/1.72	  U = y16(X)
10.93/1.72	
10.93/1.72	Proof:
10.93/1.72	  tuple(id(y7(W, y14(Z, y14(Y, y24))), y7(W, y14(Z, y14(Y, y24)))), r3(W, y14(Z, y14(Y, y24)), y7(W, y14(Z, y14(Y, y24)))), r2(y14(Y, y24), y14(Z, y14(Y, y24))), r2(y16(X), y14(Y, y24)), r1(y16(X)))
10.93/1.72	= { by axiom 1 (axiom_5) }
10.93/1.72	  tuple(true2, r3(W, y14(Z, y14(Y, y24)), y7(W, y14(Z, y14(Y, y24)))), r2(y14(Y, y24), y14(Z, y14(Y, y24))), r2(y16(X), y14(Y, y24)), r1(y16(X)))
10.93/1.72	= { by axiom 7 (axiom_1a_1) }
10.93/1.72	  tuple(true2, true2, r2(y14(Y, y24), y14(Z, y14(Y, y24))), r2(y16(X), y14(Y, y24)), r1(y16(X)))
10.93/1.72	= { by axiom 4 (axiom_2a_4) }
10.93/1.72	  tuple(true2, true2, true2, r2(y16(X), y14(Y, y24)), r1(y16(X)))
10.93/1.72	= { by axiom 8 (axiom_9_1) R->L }
10.93/1.72	  tuple(true2, true2, true2, fresh(true2, true2, y16(X), y14(Y, y24), y24), r1(y16(X)))
10.93/1.72	= { by axiom 1 (axiom_5) R->L }
10.93/1.72	  tuple(true2, true2, true2, fresh(id(y14(Y, y24), y14(Y, y24)), true2, y16(X), y14(Y, y24), y24), r1(y16(X)))
10.93/1.72	= { by axiom 10 (axiom_9_1) R->L }
10.93/1.72	  tuple(true2, true2, true2, fresh31(r2(y24, y14(Y, y24)), true2, y16(X), y14(Y, y24), y24, y14(Y, y24)), r1(y16(X)))
10.93/1.72	= { by axiom 4 (axiom_2a_4) }
10.93/1.72	  tuple(true2, true2, true2, fresh31(true2, true2, y16(X), y14(Y, y24), y24, y14(Y, y24)), r1(y16(X)))
10.93/1.72	= { by axiom 9 (axiom_9_1) }
10.93/1.72	  tuple(true2, true2, true2, fresh32(id(y16(X), y24), true2, y16(X), y14(Y, y24)), r1(y16(X)))
10.93/1.72	= { by axiom 6 (axiom_1_1) R->L }
10.93/1.72	  tuple(true2, true2, true2, fresh32(fresh17(r1(y16(X)), true2, y16(X)), true2, y16(X), y14(Y, y24)), r1(y16(X)))
10.93/1.72	= { by axiom 2 (axiom_4a_2) }
10.93/1.72	  tuple(true2, true2, true2, fresh32(fresh17(true2, true2, y16(X)), true2, y16(X), y14(Y, y24)), r1(y16(X)))
10.93/1.72	= { by axiom 3 (axiom_1_1) }
10.93/1.72	  tuple(true2, true2, true2, fresh32(true2, true2, y16(X), y14(Y, y24)), r1(y16(X)))
10.93/1.72	= { by axiom 5 (axiom_9_1) }
10.93/1.72	  tuple(true2, true2, true2, true2, r1(y16(X)))
10.93/1.72	= { by axiom 2 (axiom_4a_2) }
10.93/1.72	  tuple(true2, true2, true2, true2, true2)
10.93/1.72	% SZS output end Proof
10.93/1.72	
10.93/1.72	RESULT: Theorem (the conjecture is true).
10.93/1.73	EOF