0.11/0.14	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.11/0.15	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.12/0.36	% Computer   : n008.cluster.edu
0.12/0.36	% Model      : x86_64 x86_64
0.12/0.36	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.36	% Memory     : 8042.1875MB
0.12/0.36	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.36	% CPULimit   : 1200
0.12/0.36	% WCLimit    : 120
0.12/0.36	% DateTime   : Tue Jul 13 14:02:59 EDT 2021
0.12/0.36	% CPUTime    : 
9.21/1.57	% SZS status Theorem
9.21/1.57	
9.53/1.58	% SZS output start Proof
9.53/1.58	Take the following subset of the input axioms:
9.53/1.58	  fof(aSatz7_10a, axiom, ![Xa, Xp]: (Xp!=s(Xa, Xp) | Xp=Xa)).
9.53/1.58	  fof(aSatz7_10b, axiom, ![Xa, Xp]: (Xp!=Xa | s(Xa, Xp)=Xp)).
9.53/1.58	  fof(aSatz7_13, axiom, ![Xa, Xp, Xq]: s_e(Xp, Xq, s(Xa, Xp), s(Xa, Xq))).
9.53/1.58	  fof(aSatz7_15a, axiom, ![Xa, Xp, Xq, Xr]: (s_t(s(Xa, Xp), s(Xa, Xq), s(Xa, Xr)) | ~s_t(Xp, Xq, Xr))).
9.53/1.58	  fof(aSatz7_16a, axiom, ![Xa, Xp, Xq, Xr, Xcs]: (s_e(s(Xa, Xp), s(Xa, Xq), s(Xa, Xr), s(Xa, Xcs)) | ~s_e(Xp, Xq, Xr, Xcs))).
9.53/1.58	  fof(aSatz7_17, axiom, ![Xa, Xp, Xb, Xq]: (~s_m(Xp, Xa, Xq) | (Xa=Xb | ~s_m(Xp, Xb, Xq)))).
9.53/1.58	  fof(aSatz7_19, conjecture, ![Xa, Xp, Xb]: (Xb=Xa | s(Xa, s(Xb, Xp))!=s(Xb, s(Xa, Xp)))).
9.53/1.58	  fof(aSatz7_4a, axiom, ![Xa, Xp]: s_m(Xp, Xa, s(Xa, Xp))).
9.53/1.58	  fof(aSatz7_7, axiom, ![Xa, Xp]: s(Xa, s(Xa, Xp))=Xp).
9.53/1.58	  fof(d_Defn7_1, axiom, ![Xa, Xb, Xm]: ((s_t(Xa, Xm, Xb) | ~s_m(Xa, Xm, Xb)) & ((~s_m(Xa, Xm, Xb) | s_e(Xm, Xa, Xm, Xb)) & (~s_e(Xm, Xa, Xm, Xb) | (s_m(Xa, Xm, Xb) | ~s_t(Xa, Xm, Xb)))))).
9.53/1.58	
9.53/1.58	Now clausify the problem and encode Horn clauses using encoding 3 of
9.53/1.58	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
9.53/1.58	We repeatedly replace C & s=t => u=v by the two clauses:
9.53/1.58	  fresh(y, y, x1...xn) = u
9.53/1.58	  C => fresh(s, t, x1...xn) = v
9.53/1.58	where fresh is a fresh function symbol and x1..xn are the free
9.53/1.58	variables of u and v.
9.53/1.58	A predicate p(X) is encoded as p(X)=true (this is sound, because the
9.53/1.58	input problem has no model of domain size 1).
9.53/1.58	
9.53/1.58	The encoding turns the above axioms into the following unit equations and goals:
9.53/1.58	
9.53/1.58	Axiom 1 (aSatz7_10b): s(X, X) = X.
9.53/1.58	Axiom 2 (aSatz7_7): s(X, s(X, Y)) = Y.
9.53/1.58	Axiom 3 (aSatz7_19): s(xa, s(xb, xp)) = s(xb, s(xa, xp)).
9.53/1.58	Axiom 4 (aSatz7_17): fresh10(X, X, Y, Z) = Z.
9.53/1.58	Axiom 5 (aSatz7_10a): fresh4(X, X, Y, Z) = Y.
9.53/1.58	Axiom 6 (aSatz7_4a): s_m(X, Y, s(Y, X)) = true2.
9.53/1.58	Axiom 7 (d_Defn7_1): fresh31(X, X, Y, Z, W) = s_m(Y, Z, W).
9.53/1.58	Axiom 8 (d_Defn7_1): fresh30(X, X, Y, Z, W) = true2.
9.53/1.58	Axiom 9 (d_Defn7_1_1): fresh29(X, X, Y, Z, W) = true2.
9.53/1.58	Axiom 10 (aSatz7_15a): fresh64(X, X, Y, Z, W, V) = true2.
9.53/1.58	Axiom 11 (aSatz7_17): fresh11(X, X, Y, Z, W, V) = Z.
9.53/1.58	Axiom 12 (aSatz7_10a): fresh4(X, s(Y, X), Y, X) = X.
9.53/1.58	Axiom 13 (aSatz7_16a): fresh62(X, X, Y, Z, W, V, U) = true2.
9.53/1.58	Axiom 14 (aSatz7_13): s_e(X, Y, s(Z, X), s(Z, Y)) = true2.
9.53/1.58	Axiom 15 (d_Defn7_1_1): fresh29(s_m(X, Y, Z), true2, X, Y, Z) = s_t(X, Y, Z).
9.53/1.58	Axiom 16 (aSatz7_15a): fresh64(s_t(X, Y, Z), true2, X, Y, Z, W) = s_t(s(W, X), s(W, Y), s(W, Z)).
9.53/1.58	Axiom 17 (d_Defn7_1): fresh31(s_t(X, Y, Z), true2, X, Y, Z) = fresh30(s_e(Y, X, Y, Z), true2, X, Y, Z).
9.53/1.58	Axiom 18 (aSatz7_17): fresh11(s_m(X, Y, Z), true2, X, W, Z, Y) = fresh10(s_m(X, W, Z), true2, W, Y).
9.53/1.58	Axiom 19 (aSatz7_16a): fresh62(s_e(X, Y, Z, W), true2, X, Y, Z, W, V) = s_e(s(V, X), s(V, Y), s(V, Z), s(V, W)).
9.53/1.58	
9.53/1.58	Lemma 20: s(xa, s(xb, s(xa, xp))) = s(xb, xp).
9.53/1.58	Proof:
9.53/1.58	  s(xa, s(xb, s(xa, xp)))
9.53/1.58	= { by axiom 3 (aSatz7_19) R->L }
9.53/1.58	  s(xa, s(xa, s(xb, xp)))
9.53/1.58	= { by axiom 2 (aSatz7_7) }
9.53/1.58	  s(xb, xp)
9.53/1.58	
9.53/1.58	Goal 1 (aSatz7_19_1): xb = xa.
9.53/1.58	Proof:
9.53/1.58	  xb
9.53/1.58	= { by axiom 12 (aSatz7_10a) R->L }
9.53/1.58	  fresh4(xb, s(xa, xb), xa, xb)
9.53/1.58	= { by axiom 11 (aSatz7_17) R->L }
9.53/1.58	  fresh4(xb, fresh11(true2, true2, xp, s(xa, xb), s(xb, xp), xb), xa, xb)
9.53/1.58	= { by axiom 6 (aSatz7_4a) R->L }
9.53/1.58	  fresh4(xb, fresh11(s_m(xp, xb, s(xb, xp)), true2, xp, s(xa, xb), s(xb, xp), xb), xa, xb)
9.53/1.58	= { by axiom 18 (aSatz7_17) }
9.53/1.58	  fresh4(xb, fresh10(s_m(xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 7 (d_Defn7_1) R->L }
9.53/1.58	  fresh4(xb, fresh10(fresh31(true2, true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 10 (aSatz7_15a) R->L }
9.53/1.58	  fresh4(xb, fresh10(fresh31(fresh64(true2, true2, s(xa, xp), xb, s(xb, s(xa, xp)), xa), true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 9 (d_Defn7_1_1) R->L }
9.53/1.58	  fresh4(xb, fresh10(fresh31(fresh64(fresh29(true2, true2, s(xa, xp), xb, s(xb, s(xa, xp))), true2, s(xa, xp), xb, s(xb, s(xa, xp)), xa), true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 6 (aSatz7_4a) R->L }
9.53/1.58	  fresh4(xb, fresh10(fresh31(fresh64(fresh29(s_m(s(xa, xp), xb, s(xb, s(xa, xp))), true2, s(xa, xp), xb, s(xb, s(xa, xp))), true2, s(xa, xp), xb, s(xb, s(xa, xp)), xa), true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 15 (d_Defn7_1_1) }
9.53/1.58	  fresh4(xb, fresh10(fresh31(fresh64(s_t(s(xa, xp), xb, s(xb, s(xa, xp))), true2, s(xa, xp), xb, s(xb, s(xa, xp)), xa), true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 16 (aSatz7_15a) }
9.53/1.58	  fresh4(xb, fresh10(fresh31(s_t(s(xa, s(xa, xp)), s(xa, xb), s(xa, s(xb, s(xa, xp)))), true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by lemma 20 }
9.53/1.58	  fresh4(xb, fresh10(fresh31(s_t(s(xa, s(xa, xp)), s(xa, xb), s(xb, xp)), true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 2 (aSatz7_7) }
9.53/1.58	  fresh4(xb, fresh10(fresh31(s_t(xp, s(xa, xb), s(xb, xp)), true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 17 (d_Defn7_1) }
9.53/1.58	  fresh4(xb, fresh10(fresh30(s_e(s(xa, xb), xp, s(xa, xb), s(xb, xp)), true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 2 (aSatz7_7) R->L }
9.53/1.58	  fresh4(xb, fresh10(fresh30(s_e(s(xa, xb), s(xa, s(xa, xp)), s(xa, xb), s(xb, xp)), true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by lemma 20 R->L }
9.53/1.58	  fresh4(xb, fresh10(fresh30(s_e(s(xa, xb), s(xa, s(xa, xp)), s(xa, xb), s(xa, s(xb, s(xa, xp)))), true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 19 (aSatz7_16a) R->L }
9.53/1.58	  fresh4(xb, fresh10(fresh30(fresh62(s_e(xb, s(xa, xp), xb, s(xb, s(xa, xp))), true2, xb, s(xa, xp), xb, s(xb, s(xa, xp)), xa), true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 1 (aSatz7_10b) R->L }
9.53/1.58	  fresh4(xb, fresh10(fresh30(fresh62(s_e(xb, s(xa, xp), s(xb, xb), s(xb, s(xa, xp))), true2, xb, s(xa, xp), xb, s(xb, s(xa, xp)), xa), true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 14 (aSatz7_13) }
9.53/1.58	  fresh4(xb, fresh10(fresh30(fresh62(true2, true2, xb, s(xa, xp), xb, s(xb, s(xa, xp)), xa), true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 13 (aSatz7_16a) }
9.53/1.58	  fresh4(xb, fresh10(fresh30(true2, true2, xp, s(xa, xb), s(xb, xp)), true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 8 (d_Defn7_1) }
9.53/1.58	  fresh4(xb, fresh10(true2, true2, s(xa, xb), xb), xa, xb)
9.53/1.58	= { by axiom 4 (aSatz7_17) }
9.53/1.58	  fresh4(xb, xb, xa, xb)
9.53/1.58	= { by axiom 5 (aSatz7_10a) }
9.53/1.58	  xa
9.53/1.58	% SZS output end Proof
9.53/1.58	
9.53/1.58	RESULT: Theorem (the conjecture is true).
9.53/1.59	EOF