0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.14/0.13	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.14/0.34	% Computer   : n006.cluster.edu
0.14/0.34	% Model      : x86_64 x86_64
0.14/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.14/0.34	% Memory     : 8042.1875MB
0.14/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.14/0.34	% CPULimit   : 1200
0.14/0.34	% WCLimit    : 120
0.14/0.34	% DateTime   : Tue Jul 13 15:05:10 EDT 2021
0.14/0.35	% CPUTime    : 
1.48/0.57	% SZS status Theorem
1.48/0.57	
1.48/0.58	% SZS output start Proof
1.48/0.58	Take the following subset of the input axioms:
1.48/0.58	  fof(goals, conjecture, ![X0]: (f(eh)=eg & (opp(f(X0))=f(inv(X0)) | ~g(X0)))).
1.48/0.58	  fof(sos02, axiom, ![A]: (g(inv(A)) <= g(A))).
1.48/0.58	  fof(sos03, axiom, g(eh)).
1.48/0.58	  fof(sos06, axiom, ![A]: (product(A, eh)=A <= g(A))).
1.48/0.58	  fof(sos07, axiom, ![A]: (product(A, inv(A))=eh <= g(A))).
1.48/0.58	  fof(sos10, axiom, ![A, B]: (h(opp(B)) <= h(A))).
1.48/0.58	  fof(sos11, axiom, h(eg)).
1.48/0.58	  fof(sos12, axiom, ![A, B, C]: ((h(C) & (h(B) & h(A))) => sum(A, sum(B, C))=sum(sum(A, B), C))).
1.48/0.58	  fof(sos13, axiom, ![A]: (A=sum(eg, A) <= h(A))).
1.48/0.58	  fof(sos14, axiom, ![A]: (h(A) => A=sum(A, eg))).
1.48/0.58	  fof(sos15, axiom, ![A]: (sum(A, opp(A))=eg <= h(A))).
1.48/0.58	  fof(sos16, axiom, ![A]: (sum(opp(A), A)=eg <= h(A))).
1.48/0.58	  fof(sos17, axiom, ![A]: (h(f(A)) <= g(A))).
1.48/0.58	  fof(sos18, axiom, ![A, B]: sum(f(A), f(B))=f(product(A, B))).
1.48/0.58	
1.48/0.58	Now clausify the problem and encode Horn clauses using encoding 3 of
1.48/0.58	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
1.48/0.58	We repeatedly replace C & s=t => u=v by the two clauses:
1.48/0.58	  fresh(y, y, x1...xn) = u
1.48/0.58	  C => fresh(s, t, x1...xn) = v
1.48/0.58	where fresh is a fresh function symbol and x1..xn are the free
1.48/0.58	variables of u and v.
1.48/0.58	A predicate p(X) is encoded as p(X)=true (this is sound, because the
1.48/0.58	input problem has no model of domain size 1).
1.48/0.58	
1.48/0.58	The encoding turns the above axioms into the following unit equations and goals:
1.48/0.58	
1.48/0.58	Axiom 1 (sos11): h(eg) = true.
1.48/0.58	Axiom 2 (sos03): g(eh) = true.
1.48/0.58	Axiom 3 (goals_1): fresh17(X, X) = true.
1.48/0.58	Axiom 4 (sos06): fresh(X, X, Y) = Y.
1.48/0.58	Axiom 5 (sos02): fresh15(X, X, Y) = true.
1.48/0.58	Axiom 6 (sos07): fresh13(X, X, Y) = eh.
1.48/0.58	Axiom 7 (sos10): fresh9(X, X, Y) = true.
1.48/0.58	Axiom 8 (sos15): fresh7(X, X, Y) = eg.
1.48/0.58	Axiom 9 (sos16): fresh6(X, X, Y) = eg.
1.48/0.58	Axiom 10 (sos17): fresh5(X, X, Y) = true.
1.48/0.58	Axiom 11 (sos14): fresh3(X, X, Y) = Y.
1.48/0.58	Axiom 12 (sos13): fresh2(X, X, Y) = Y.
1.48/0.58	Axiom 13 (goals_1): fresh17(f(eh), eg) = g(x0).
1.48/0.58	Axiom 14 (sos06): fresh(g(X), true, X) = product(X, eh).
1.48/0.58	Axiom 15 (sos02): fresh15(g(X), true, X) = g(inv(X)).
1.48/0.58	Axiom 16 (sos07): fresh13(g(X), true, X) = product(X, inv(X)).
1.48/0.58	Axiom 17 (sos10): fresh9(h(X), true, Y) = h(opp(Y)).
1.48/0.58	Axiom 18 (sos15): fresh7(h(X), true, X) = sum(X, opp(X)).
1.48/0.58	Axiom 19 (sos16): fresh6(h(X), true, X) = sum(opp(X), X).
1.48/0.58	Axiom 20 (sos17): fresh5(g(X), true, X) = h(f(X)).
1.48/0.58	Axiom 21 (sos14): fresh3(h(X), true, X) = sum(X, eg).
1.48/0.58	Axiom 22 (sos13): fresh2(h(X), true, X) = sum(eg, X).
1.48/0.58	Axiom 23 (sos18): sum(f(X), f(Y)) = f(product(X, Y)).
1.48/0.58	Axiom 24 (sos12): fresh22(X, X, Y, Z, W) = sum(sum(W, Z), Y).
1.48/0.58	Axiom 25 (sos12): fresh8(X, X, Y, Z, W) = sum(W, sum(Z, Y)).
1.48/0.58	Axiom 26 (sos12): fresh21(X, X, Y, Z, W) = fresh22(h(Y), true, Y, Z, W).
1.48/0.58	Axiom 27 (sos12): fresh21(h(X), true, Y, Z, X) = fresh8(h(Z), true, Y, Z, X).
1.48/0.58	
1.48/0.58	Lemma 28: g(eh) = h(eg).
1.48/0.58	Proof:
1.48/0.58	  g(eh)
1.48/0.58	= { by axiom 2 (sos03) }
1.48/0.58	  true
1.48/0.58	= { by axiom 1 (sos11) R->L }
1.48/0.58	  h(eg)
1.48/0.58	
1.48/0.58	Lemma 29: fresh5(g(X), g(eh), X) = h(f(X)).
1.48/0.58	Proof:
1.48/0.58	  fresh5(g(X), g(eh), X)
1.48/0.58	= { by lemma 28 }
1.48/0.58	  fresh5(g(X), h(eg), X)
1.48/0.58	= { by axiom 1 (sos11) }
1.48/0.58	  fresh5(g(X), true, X)
1.48/0.58	= { by axiom 20 (sos17) }
1.48/0.58	  h(f(X))
1.48/0.58	
1.48/0.58	Lemma 30: fresh5(X, X, Y) = g(eh).
1.48/0.58	Proof:
1.48/0.58	  fresh5(X, X, Y)
1.48/0.58	= { by axiom 10 (sos17) }
1.48/0.58	  true
1.48/0.58	= { by axiom 1 (sos11) R->L }
1.48/0.58	  h(eg)
1.48/0.58	= { by lemma 28 R->L }
1.48/0.58	  g(eh)
1.48/0.58	
1.48/0.58	Lemma 31: h(f(eh)) = g(eh).
1.48/0.58	Proof:
1.48/0.58	  h(f(eh))
1.48/0.58	= { by lemma 29 R->L }
1.48/0.58	  fresh5(g(eh), g(eh), eh)
1.48/0.58	= { by lemma 30 }
1.48/0.58	  g(eh)
1.48/0.58	
1.48/0.58	Lemma 32: fresh3(h(X), g(eh), X) = sum(X, eg).
1.48/0.58	Proof:
1.48/0.58	  fresh3(h(X), g(eh), X)
1.48/0.58	= { by lemma 28 }
1.48/0.58	  fresh3(h(X), h(eg), X)
1.48/0.58	= { by axiom 1 (sos11) }
1.48/0.58	  fresh3(h(X), true, X)
1.48/0.58	= { by axiom 21 (sos14) }
1.48/0.58	  sum(X, eg)
1.48/0.58	
1.48/0.58	Lemma 33: sum(f(eh), opp(f(eh))) = eg.
1.48/0.58	Proof:
1.48/0.58	  sum(f(eh), opp(f(eh)))
1.48/0.58	= { by axiom 18 (sos15) R->L }
1.48/0.58	  fresh7(h(f(eh)), true, f(eh))
1.48/0.58	= { by axiom 1 (sos11) R->L }
1.48/0.58	  fresh7(h(f(eh)), h(eg), f(eh))
1.48/0.58	= { by lemma 28 R->L }
1.48/0.58	  fresh7(h(f(eh)), g(eh), f(eh))
1.48/0.58	= { by lemma 31 }
1.48/0.58	  fresh7(g(eh), g(eh), f(eh))
1.48/0.58	= { by axiom 8 (sos15) }
1.48/0.58	  eg
1.48/0.58	
1.48/0.58	Lemma 34: fresh8(h(Z), g(eh), Y, Z, X) = fresh21(h(X), g(eh), Y, Z, X).
1.48/0.58	Proof:
1.48/0.58	  fresh8(h(Z), g(eh), Y, Z, X)
1.48/0.58	= { by lemma 28 }
1.48/0.58	  fresh8(h(Z), h(eg), Y, Z, X)
1.48/0.58	= { by axiom 1 (sos11) }
1.48/0.58	  fresh8(h(Z), true, Y, Z, X)
1.48/0.58	= { by axiom 27 (sos12) R->L }
1.48/0.58	  fresh21(h(X), true, Y, Z, X)
1.48/0.58	= { by axiom 1 (sos11) R->L }
1.48/0.58	  fresh21(h(X), h(eg), Y, Z, X)
1.48/0.58	= { by lemma 28 R->L }
1.48/0.58	  fresh21(h(X), g(eh), Y, Z, X)
1.48/0.58	
1.48/0.58	Lemma 35: fresh22(h(X), g(eh), X, Y, Z) = fresh21(W, W, X, Y, Z).
1.48/0.58	Proof:
1.48/0.58	  fresh22(h(X), g(eh), X, Y, Z)
1.48/0.58	= { by lemma 28 }
1.48/0.58	  fresh22(h(X), h(eg), X, Y, Z)
1.48/0.58	= { by axiom 1 (sos11) }
1.48/0.58	  fresh22(h(X), true, X, Y, Z)
1.48/0.58	= { by axiom 26 (sos12) R->L }
1.48/0.59	  fresh21(W, W, X, Y, Z)
1.48/0.59	
1.48/0.59	Lemma 36: h(opp(X)) = g(eh).
1.48/0.59	Proof:
1.48/0.59	  h(opp(X))
1.48/0.59	= { by axiom 17 (sos10) R->L }
1.48/0.59	  fresh9(h(eg), true, X)
1.48/0.59	= { by axiom 1 (sos11) R->L }
1.48/0.59	  fresh9(h(eg), h(eg), X)
1.48/0.59	= { by lemma 28 R->L }
1.48/0.59	  fresh9(h(eg), g(eh), X)
1.48/0.59	= { by lemma 28 R->L }
1.48/0.59	  fresh9(g(eh), g(eh), X)
1.48/0.59	= { by axiom 7 (sos10) }
1.48/0.59	  true
1.48/0.59	= { by axiom 1 (sos11) R->L }
1.48/0.59	  h(eg)
1.48/0.59	= { by lemma 28 R->L }
1.48/0.59	  g(eh)
1.48/0.59	
1.48/0.59	Lemma 37: f(eh) = eg.
1.48/0.59	Proof:
1.48/0.59	  f(eh)
1.48/0.59	= { by axiom 11 (sos14) R->L }
1.48/0.59	  fresh3(g(eh), g(eh), f(eh))
1.48/0.59	= { by lemma 31 R->L }
1.48/0.59	  fresh3(h(f(eh)), g(eh), f(eh))
1.48/0.59	= { by lemma 32 }
1.48/0.59	  sum(f(eh), eg)
1.48/0.59	= { by lemma 33 R->L }
1.48/0.59	  sum(f(eh), sum(f(eh), opp(f(eh))))
1.48/0.59	= { by axiom 25 (sos12) R->L }
1.48/0.59	  fresh8(g(eh), g(eh), opp(f(eh)), f(eh), f(eh))
1.48/0.59	= { by lemma 31 R->L }
1.48/0.59	  fresh8(h(f(eh)), g(eh), opp(f(eh)), f(eh), f(eh))
1.48/0.59	= { by lemma 34 }
1.48/0.59	  fresh21(h(f(eh)), g(eh), opp(f(eh)), f(eh), f(eh))
1.48/0.59	= { by lemma 31 }
1.48/0.59	  fresh21(g(eh), g(eh), opp(f(eh)), f(eh), f(eh))
1.48/0.59	= { by lemma 35 R->L }
1.48/0.59	  fresh22(h(opp(f(eh))), g(eh), opp(f(eh)), f(eh), f(eh))
1.48/0.59	= { by lemma 36 }
1.48/0.59	  fresh22(g(eh), g(eh), opp(f(eh)), f(eh), f(eh))
1.48/0.59	= { by axiom 24 (sos12) }
1.48/0.59	  sum(sum(f(eh), f(eh)), opp(f(eh)))
1.48/0.59	= { by axiom 23 (sos18) }
1.48/0.59	  sum(f(product(eh, eh)), opp(f(eh)))
1.48/0.59	= { by axiom 14 (sos06) R->L }
1.48/0.59	  sum(f(fresh(g(eh), true, eh)), opp(f(eh)))
1.48/0.59	= { by axiom 1 (sos11) R->L }
1.48/0.59	  sum(f(fresh(g(eh), h(eg), eh)), opp(f(eh)))
1.48/0.59	= { by lemma 28 R->L }
1.48/0.59	  sum(f(fresh(g(eh), g(eh), eh)), opp(f(eh)))
1.48/0.59	= { by axiom 4 (sos06) }
1.48/0.59	  sum(f(eh), opp(f(eh)))
1.48/0.59	= { by lemma 33 }
1.48/0.59	  eg
1.48/0.59	
1.48/0.59	Lemma 38: g(eh) = g(x0).
1.48/0.59	Proof:
1.48/0.59	  g(eh)
1.48/0.59	= { by lemma 28 }
1.48/0.59	  h(eg)
1.48/0.59	= { by axiom 1 (sos11) }
1.48/0.59	  true
1.48/0.59	= { by axiom 3 (goals_1) R->L }
1.48/0.59	  fresh17(f(eh), f(eh))
1.48/0.59	= { by lemma 37 }
1.48/0.59	  fresh17(f(eh), eg)
1.48/0.59	= { by axiom 13 (goals_1) }
1.48/0.59	  g(x0)
1.48/0.59	
1.48/0.59	Lemma 39: h(f(x0)) = g(eh).
1.48/0.59	Proof:
1.48/0.59	  h(f(x0))
1.48/0.59	= { by lemma 29 R->L }
1.48/0.59	  fresh5(g(x0), g(eh), x0)
1.48/0.59	= { by lemma 38 R->L }
1.48/0.59	  fresh5(g(eh), g(eh), x0)
1.48/0.59	= { by lemma 30 }
1.48/0.59	  g(eh)
1.48/0.59	
1.48/0.59	Lemma 40: h(f(inv(x0))) = g(eh).
1.48/0.59	Proof:
1.48/0.59	  h(f(inv(x0)))
1.48/0.59	= { by lemma 29 R->L }
1.48/0.59	  fresh5(g(inv(x0)), g(eh), inv(x0))
1.48/0.59	= { by axiom 15 (sos02) R->L }
1.48/0.59	  fresh5(fresh15(g(x0), true, x0), g(eh), inv(x0))
1.48/0.59	= { by axiom 1 (sos11) R->L }
1.48/0.59	  fresh5(fresh15(g(x0), h(eg), x0), g(eh), inv(x0))
1.48/0.59	= { by lemma 28 R->L }
1.48/0.59	  fresh5(fresh15(g(x0), g(eh), x0), g(eh), inv(x0))
1.48/0.59	= { by lemma 38 R->L }
1.48/0.59	  fresh5(fresh15(g(eh), g(eh), x0), g(eh), inv(x0))
1.48/0.59	= { by axiom 5 (sos02) }
1.48/0.59	  fresh5(true, g(eh), inv(x0))
1.48/0.59	= { by axiom 1 (sos11) R->L }
1.48/0.59	  fresh5(h(eg), g(eh), inv(x0))
1.48/0.59	= { by lemma 28 R->L }
1.48/0.59	  fresh5(g(eh), g(eh), inv(x0))
1.48/0.59	= { by lemma 30 }
1.48/0.59	  g(eh)
1.48/0.59	
1.48/0.59	Goal 1 (goals): tuple(opp(f(x0)), f(eh)) = tuple(f(inv(x0)), eg).
1.48/0.59	Proof:
1.48/0.59	  tuple(opp(f(x0)), f(eh))
1.48/0.59	= { by lemma 37 }
1.48/0.59	  tuple(opp(f(x0)), eg)
1.48/0.59	= { by axiom 11 (sos14) R->L }
1.48/0.59	  tuple(fresh3(g(eh), g(eh), opp(f(x0))), eg)
1.48/0.59	= { by lemma 36 R->L }
1.48/0.59	  tuple(fresh3(h(opp(f(x0))), g(eh), opp(f(x0))), eg)
1.48/0.59	= { by lemma 32 }
1.48/0.59	  tuple(sum(opp(f(x0)), eg), eg)
1.48/0.59	= { by lemma 37 R->L }
1.48/0.59	  tuple(sum(opp(f(x0)), f(eh)), eg)
1.48/0.59	= { by axiom 6 (sos07) R->L }
1.48/0.59	  tuple(sum(opp(f(x0)), f(fresh13(g(eh), g(eh), x0))), eg)
1.48/0.59	= { by lemma 38 }
1.48/0.59	  tuple(sum(opp(f(x0)), f(fresh13(g(x0), g(eh), x0))), eg)
1.48/0.59	= { by lemma 28 }
1.48/0.59	  tuple(sum(opp(f(x0)), f(fresh13(g(x0), h(eg), x0))), eg)
1.48/0.59	= { by axiom 1 (sos11) }
1.48/0.59	  tuple(sum(opp(f(x0)), f(fresh13(g(x0), true, x0))), eg)
1.48/0.59	= { by axiom 16 (sos07) }
1.48/0.59	  tuple(sum(opp(f(x0)), f(product(x0, inv(x0)))), eg)
1.48/0.59	= { by axiom 23 (sos18) R->L }
1.48/0.59	  tuple(sum(opp(f(x0)), sum(f(x0), f(inv(x0)))), eg)
1.48/0.59	= { by axiom 25 (sos12) R->L }
1.48/0.59	  tuple(fresh8(g(eh), g(eh), f(inv(x0)), f(x0), opp(f(x0))), eg)
1.48/0.59	= { by lemma 39 R->L }
1.48/0.59	  tuple(fresh8(h(f(x0)), g(eh), f(inv(x0)), f(x0), opp(f(x0))), eg)
1.48/0.59	= { by lemma 34 }
1.48/0.59	  tuple(fresh21(h(opp(f(x0))), g(eh), f(inv(x0)), f(x0), opp(f(x0))), eg)
1.48/0.59	= { by lemma 36 }
1.48/0.59	  tuple(fresh21(g(eh), g(eh), f(inv(x0)), f(x0), opp(f(x0))), eg)
1.48/0.59	= { by lemma 35 R->L }
1.48/0.59	  tuple(fresh22(h(f(inv(x0))), g(eh), f(inv(x0)), f(x0), opp(f(x0))), eg)
1.48/0.59	= { by lemma 40 }
1.48/0.59	  tuple(fresh22(g(eh), g(eh), f(inv(x0)), f(x0), opp(f(x0))), eg)
1.48/0.59	= { by axiom 24 (sos12) }
1.48/0.59	  tuple(sum(sum(opp(f(x0)), f(x0)), f(inv(x0))), eg)
1.48/0.59	= { by axiom 19 (sos16) R->L }
1.48/0.59	  tuple(sum(fresh6(h(f(x0)), true, f(x0)), f(inv(x0))), eg)
1.48/0.59	= { by axiom 1 (sos11) R->L }
1.48/0.59	  tuple(sum(fresh6(h(f(x0)), h(eg), f(x0)), f(inv(x0))), eg)
1.48/0.59	= { by lemma 28 R->L }
1.48/0.59	  tuple(sum(fresh6(h(f(x0)), g(eh), f(x0)), f(inv(x0))), eg)
1.48/0.59	= { by lemma 39 }
1.48/0.59	  tuple(sum(fresh6(g(eh), g(eh), f(x0)), f(inv(x0))), eg)
1.48/0.59	= { by axiom 9 (sos16) }
1.48/0.59	  tuple(sum(eg, f(inv(x0))), eg)
1.48/0.59	= { by axiom 22 (sos13) R->L }
1.48/0.59	  tuple(fresh2(h(f(inv(x0))), true, f(inv(x0))), eg)
1.48/0.59	= { by axiom 1 (sos11) R->L }
1.48/0.59	  tuple(fresh2(h(f(inv(x0))), h(eg), f(inv(x0))), eg)
1.48/0.59	= { by lemma 28 R->L }
1.48/0.59	  tuple(fresh2(h(f(inv(x0))), g(eh), f(inv(x0))), eg)
1.48/0.59	= { by lemma 40 }
1.48/0.59	  tuple(fresh2(g(eh), g(eh), f(inv(x0))), eg)
1.48/0.59	= { by axiom 12 (sos13) }
1.48/0.59	  tuple(f(inv(x0)), eg)
1.48/0.59	% SZS output end Proof
1.48/0.59	
1.48/0.59	RESULT: Theorem (the conjecture is true).
1.48/0.59	EOF