0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.12	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.12/0.33	% Computer   : n016.cluster.edu
0.12/0.33	% Model      : x86_64 x86_64
0.12/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.33	% Memory     : 8042.1875MB
0.12/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.33	% CPULimit   : 1200
0.12/0.33	% WCLimit    : 120
0.12/0.33	% DateTime   : Tue Jul 13 10:51:53 EDT 2021
0.12/0.33	% CPUTime    : 
0.19/0.38	% SZS status Theorem
0.19/0.38	
0.19/0.38	% SZS output start Proof
0.19/0.38	Take the following subset of the input axioms:
0.19/0.39	  fof(x2138, conjecture, ?[Y]: ![E]: ?[M]: ![W]: (big_s(M, W) => ![Z]: (big_f(W, Z) => big_d(Z, Y, E))) <= (?[X]: ![E]: ?[N]: ![W]: (big_d(W, X, E) <= big_s(N, W)) & (![E]: ?[D]: ![A, B]: (![Y, Z]: ((big_f(B, Z) & big_f(A, Y)) => big_d(Y, Z, E)) <= big_d(A, B, D)) & ![X]: ?[Y]: big_f(X, Y)))).
0.19/0.39	
0.19/0.39	Now clausify the problem and encode Horn clauses using encoding 3 of
0.19/0.39	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.19/0.39	We repeatedly replace C & s=t => u=v by the two clauses:
0.19/0.39	  fresh(y, y, x1...xn) = u
0.19/0.39	  C => fresh(s, t, x1...xn) = v
0.19/0.39	where fresh is a fresh function symbol and x1..xn are the free
0.19/0.39	variables of u and v.
0.19/0.39	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.19/0.39	input problem has no model of domain size 1).
0.19/0.39	
0.19/0.39	The encoding turns the above axioms into the following unit equations and goals:
0.19/0.39	
0.19/0.39	Axiom 1 (x2138_1): big_f(X, y(X)) = true2.
0.19/0.39	Axiom 2 (x2138): big_s(X, w(Y, X)) = true2.
0.19/0.39	Axiom 3 (x2138_2): big_f(w(X, Y), z(X, Y)) = true2.
0.19/0.39	Axiom 4 (x2138_3): fresh(X, X, Y, Z) = true2.
0.19/0.39	Axiom 5 (x2138_4): fresh4(X, X, Y, Z, W) = true2.
0.19/0.39	Axiom 6 (x2138_3): fresh(big_s(n(X), Y), true2, X, Y) = big_d(Y, x, X).
0.19/0.39	Axiom 7 (x2138_4): fresh2(X, X, Y, Z, W, V) = big_d(W, V, Y).
0.19/0.39	Axiom 8 (x2138_4): fresh3(X, X, Y, Z, W, V, U) = fresh4(big_f(Z, V), true2, Y, V, U).
0.19/0.39	Axiom 9 (x2138_4): fresh3(big_d(X, Y, d(Z)), true2, Z, X, Y, W, V) = fresh2(big_f(Y, V), true2, Z, X, W, V).
0.19/0.39	
0.19/0.39	Goal 1 (x2138_5): big_d(z(X, Y), X, e(X)) = true2.
0.19/0.39	The goal is true when:
0.19/0.39	  X = y(x)
0.19/0.39	  Y = n(d(e(y(x))))
0.19/0.39	
0.19/0.39	Proof:
0.19/0.39	  big_d(z(y(x), n(d(e(y(x))))), y(x), e(y(x)))
0.19/0.39	= { by axiom 7 (x2138_4) R->L }
0.19/0.39	  fresh2(true2, true2, e(y(x)), w(y(x), n(d(e(y(x))))), z(y(x), n(d(e(y(x))))), y(x))
0.19/0.39	= { by axiom 1 (x2138_1) R->L }
0.19/0.39	  fresh2(big_f(x, y(x)), true2, e(y(x)), w(y(x), n(d(e(y(x))))), z(y(x), n(d(e(y(x))))), y(x))
0.19/0.39	= { by axiom 9 (x2138_4) R->L }
0.19/0.39	  fresh3(big_d(w(y(x), n(d(e(y(x))))), x, d(e(y(x)))), true2, e(y(x)), w(y(x), n(d(e(y(x))))), x, z(y(x), n(d(e(y(x))))), y(x))
0.19/0.39	= { by axiom 6 (x2138_3) R->L }
0.19/0.39	  fresh3(fresh(big_s(n(d(e(y(x)))), w(y(x), n(d(e(y(x)))))), true2, d(e(y(x))), w(y(x), n(d(e(y(x)))))), true2, e(y(x)), w(y(x), n(d(e(y(x))))), x, z(y(x), n(d(e(y(x))))), y(x))
0.19/0.39	= { by axiom 2 (x2138) }
0.19/0.39	  fresh3(fresh(true2, true2, d(e(y(x))), w(y(x), n(d(e(y(x)))))), true2, e(y(x)), w(y(x), n(d(e(y(x))))), x, z(y(x), n(d(e(y(x))))), y(x))
0.19/0.39	= { by axiom 4 (x2138_3) }
0.19/0.39	  fresh3(true2, true2, e(y(x)), w(y(x), n(d(e(y(x))))), x, z(y(x), n(d(e(y(x))))), y(x))
0.19/0.39	= { by axiom 8 (x2138_4) }
0.19/0.39	  fresh4(big_f(w(y(x), n(d(e(y(x))))), z(y(x), n(d(e(y(x)))))), true2, e(y(x)), z(y(x), n(d(e(y(x))))), y(x))
0.19/0.39	= { by axiom 3 (x2138_2) }
0.19/0.39	  fresh4(true2, true2, e(y(x)), z(y(x), n(d(e(y(x))))), y(x))
0.19/0.39	= { by axiom 5 (x2138_4) }
0.19/0.39	  true2
0.19/0.39	% SZS output end Proof
0.19/0.39	
0.19/0.39	RESULT: Theorem (the conjecture is true).
0.19/0.39	EOF