0.06/0.11	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.06/0.11	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.11/0.31	% Computer   : n018.cluster.edu
0.11/0.31	% Model      : x86_64 x86_64
0.11/0.31	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.11/0.31	% Memory     : 8042.1875MB
0.11/0.31	% OS         : Linux 3.10.0-693.el7.x86_64
0.11/0.32	% CPULimit   : 1200
0.11/0.32	% WCLimit    : 120
0.11/0.32	% DateTime   : Tue Jul 13 14:36:23 EDT 2021
0.16/0.32	% CPUTime    : 
6.21/1.18	% SZS status Theorem
6.21/1.18	
6.21/1.20	% SZS output start Proof
6.21/1.20	Take the following subset of the input axioms:
6.21/1.21	  fof(exemplo6GDDFULL416047, conjecture, ![A, B, C, D, O, U, V, NWPNT1]: (cyclic(A, U, D, V) <= (perp(A, B, D, U) & (perp(A, D, B, U) & (perp(B, D, A, U) & (perp(A, C, D, V) & (perp(A, D, C, V) & (perp(C, D, A, V) & (circle(O, A, D, NWPNT1) & circle(O, A, B, C)))))))))).
6.21/1.21	  fof(ruleD1, axiom, ![A, B, C]: (coll(A, B, C) => coll(A, C, B))).
6.21/1.21	  fof(ruleD17, axiom, ![A, B, C, D, E]: ((cyclic(A, B, C, D) & cyclic(A, B, C, E)) => cyclic(B, C, D, E))).
6.21/1.21	  fof(ruleD19, axiom, ![A, B, C, P, Q, D, U, V]: (eqangle(C, D, A, B, U, V, P, Q) <= eqangle(A, B, C, D, P, Q, U, V))).
6.21/1.21	  fof(ruleD2, axiom, ![A, B, C]: (coll(B, A, C) <= coll(A, B, C))).
6.21/1.21	  fof(ruleD21, axiom, ![A, B, C, P, Q, D, U, V]: (eqangle(A, B, P, Q, C, D, U, V) <= eqangle(A, B, C, D, P, Q, U, V))).
6.21/1.21	  fof(ruleD3, axiom, ![A, B, C, D]: (coll(C, D, A) <= (coll(A, B, C) & coll(A, B, D)))).
6.21/1.21	  fof(ruleD39, axiom, ![A, B, C, P, Q, D]: (para(A, B, C, D) <= eqangle(A, B, P, Q, C, D, P, Q))).
6.21/1.21	  fof(ruleD40, axiom, ![A, B, C, P, Q, D]: (eqangle(A, B, P, Q, C, D, P, Q) <= para(A, B, C, D))).
6.21/1.21	  fof(ruleD42b, axiom, ![A, B, P, Q]: ((eqangle(P, A, P, B, Q, A, Q, B) & coll(P, Q, B)) => cyclic(A, B, P, Q))).
6.21/1.21	  fof(ruleD66, axiom, ![A, B, C]: (coll(A, B, C) <= para(A, B, A, C))).
6.21/1.21	  fof(ruleD8, axiom, ![A, B, C, D]: (perp(C, D, A, B) <= perp(A, B, C, D))).
6.21/1.21	  fof(ruleD9, axiom, ![A, B, C, D, E, F]: (para(A, B, E, F) <= (perp(C, D, E, F) & perp(A, B, C, D)))).
6.21/1.21	
6.21/1.21	Now clausify the problem and encode Horn clauses using encoding 3 of
6.21/1.21	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
6.21/1.21	We repeatedly replace C & s=t => u=v by the two clauses:
6.21/1.21	  fresh(y, y, x1...xn) = u
6.21/1.21	  C => fresh(s, t, x1...xn) = v
6.21/1.21	where fresh is a fresh function symbol and x1..xn are the free
6.21/1.21	variables of u and v.
6.21/1.21	A predicate p(X) is encoded as p(X)=true (this is sound, because the
6.21/1.21	input problem has no model of domain size 1).
6.21/1.21	
6.21/1.21	The encoding turns the above axioms into the following unit equations and goals:
6.21/1.21	
6.21/1.21	Axiom 1 (exemplo6GDDFULL416047_4): perp(c, d, a, v) = true.
6.21/1.21	Axiom 2 (ruleD1): fresh147(X, X, Y, Z, W) = true.
6.21/1.21	Axiom 3 (ruleD2): fresh134(X, X, Y, Z, W) = true.
6.21/1.21	Axiom 4 (ruleD3): fresh120(X, X, Y, Z, W) = true.
6.21/1.21	Axiom 5 (ruleD66): fresh66(X, X, Y, Z, W) = true.
6.21/1.21	Axiom 6 (ruleD17): fresh137(X, X, Y, Z, W, V) = true.
6.21/1.21	Axiom 7 (ruleD3): fresh121(X, X, Y, Z, W, V) = coll(W, V, Y).
6.21/1.21	Axiom 8 (ruleD39): fresh107(X, X, Y, Z, W, V) = true.
6.21/1.21	Axiom 9 (ruleD42b): fresh103(X, X, Y, Z, W, V) = cyclic(Y, Z, W, V).
6.21/1.21	Axiom 10 (ruleD42b): fresh102(X, X, Y, Z, W, V) = true.
6.21/1.21	Axiom 11 (ruleD8): fresh52(X, X, Y, Z, W, V) = true.
6.21/1.21	Axiom 12 (ruleD9): fresh50(X, X, Y, Z, W, V) = true.
6.21/1.21	Axiom 13 (ruleD17): fresh138(X, X, Y, Z, W, V, U) = cyclic(Z, W, V, U).
6.21/1.21	Axiom 14 (ruleD1): fresh147(coll(X, Y, Z), true, X, Y, Z) = coll(X, Z, Y).
6.21/1.21	Axiom 15 (ruleD2): fresh134(coll(X, Y, Z), true, X, Y, Z) = coll(Y, X, Z).
6.21/1.21	Axiom 16 (ruleD40): fresh105(X, X, Y, Z, W, V, U, T) = true.
6.21/1.21	Axiom 17 (ruleD9): fresh51(X, X, Y, Z, W, V, U, T) = para(Y, Z, U, T).
6.21/1.21	Axiom 18 (ruleD3): fresh121(coll(X, Y, Z), true, X, Y, W, Z) = fresh120(coll(X, Y, W), true, X, W, Z).
6.21/1.21	Axiom 19 (ruleD66): fresh66(para(X, Y, X, Z), true, X, Y, Z) = coll(X, Y, Z).
6.21/1.21	Axiom 20 (ruleD19): fresh135(X, X, Y, Z, W, V, U, T, S, X2) = true.
6.21/1.21	Axiom 21 (ruleD21): fresh132(X, X, Y, Z, W, V, U, T, S, X2) = true.
6.21/1.21	Axiom 22 (ruleD8): fresh52(perp(X, Y, Z, W), true, X, Y, Z, W) = perp(Z, W, X, Y).
6.21/1.21	Axiom 23 (ruleD17): fresh138(cyclic(X, Y, Z, W), true, X, Y, Z, V, W) = fresh137(cyclic(X, Y, Z, V), true, Y, Z, V, W).
6.21/1.21	Axiom 24 (ruleD40): fresh105(para(X, Y, Z, W), true, X, Y, Z, W, V, U) = eqangle(X, Y, V, U, Z, W, V, U).
6.21/1.21	Axiom 25 (ruleD9): fresh51(perp(X, Y, Z, W), true, V, U, X, Y, Z, W) = fresh50(perp(V, U, X, Y), true, V, U, Z, W).
6.21/1.21	Axiom 26 (ruleD39): fresh107(eqangle(X, Y, Z, W, V, U, Z, W), true, X, Y, V, U) = para(X, Y, V, U).
6.21/1.21	Axiom 27 (ruleD42b): fresh103(eqangle(X, Y, X, Z, W, Y, W, Z), true, Y, Z, X, W) = fresh102(coll(X, W, Z), true, Y, Z, X, W).
6.21/1.21	Axiom 28 (ruleD19): fresh135(eqangle(X, Y, Z, W, V, U, T, S), true, X, Y, Z, W, V, U, T, S) = eqangle(Z, W, X, Y, T, S, V, U).
6.21/1.21	Axiom 29 (ruleD21): fresh132(eqangle(X, Y, Z, W, V, U, T, S), true, X, Y, Z, W, V, U, T, S) = eqangle(X, Y, V, U, Z, W, T, S).
6.21/1.21	
6.21/1.21	Lemma 30: eqangle(a, v, X, Y, a, v, X, Y) = true.
6.21/1.21	Proof:
6.21/1.21	  eqangle(a, v, X, Y, a, v, X, Y)
6.21/1.21	= { by axiom 24 (ruleD40) R->L }
6.21/1.21	  fresh105(para(a, v, a, v), true, a, v, a, v, X, Y)
6.21/1.21	= { by axiom 17 (ruleD9) R->L }
6.21/1.21	  fresh105(fresh51(true, true, a, v, c, d, a, v), true, a, v, a, v, X, Y)
6.21/1.21	= { by axiom 1 (exemplo6GDDFULL416047_4) R->L }
6.21/1.21	  fresh105(fresh51(perp(c, d, a, v), true, a, v, c, d, a, v), true, a, v, a, v, X, Y)
6.21/1.21	= { by axiom 25 (ruleD9) }
6.21/1.21	  fresh105(fresh50(perp(a, v, c, d), true, a, v, a, v), true, a, v, a, v, X, Y)
6.21/1.21	= { by axiom 22 (ruleD8) R->L }
6.21/1.21	  fresh105(fresh50(fresh52(perp(c, d, a, v), true, c, d, a, v), true, a, v, a, v), true, a, v, a, v, X, Y)
6.21/1.21	= { by axiom 1 (exemplo6GDDFULL416047_4) }
6.21/1.21	  fresh105(fresh50(fresh52(true, true, c, d, a, v), true, a, v, a, v), true, a, v, a, v, X, Y)
6.21/1.21	= { by axiom 11 (ruleD8) }
6.21/1.21	  fresh105(fresh50(true, true, a, v, a, v), true, a, v, a, v, X, Y)
6.21/1.21	= { by axiom 12 (ruleD9) }
6.21/1.21	  fresh105(true, true, a, v, a, v, X, Y)
6.21/1.21	= { by axiom 16 (ruleD40) }
6.21/1.21	  true
6.21/1.21	
6.21/1.21	Lemma 31: coll(X, X, Y) = true.
6.21/1.21	Proof:
6.21/1.21	  coll(X, X, Y)
6.21/1.21	= { by axiom 14 (ruleD1) R->L }
6.21/1.21	  fresh147(coll(X, Y, X), true, X, Y, X)
6.21/1.21	= { by axiom 15 (ruleD2) R->L }
6.21/1.21	  fresh147(fresh134(coll(Y, X, X), true, Y, X, X), true, X, Y, X)
6.21/1.21	= { by axiom 19 (ruleD66) R->L }
6.21/1.21	  fresh147(fresh134(fresh66(para(Y, X, Y, X), true, Y, X, X), true, Y, X, X), true, X, Y, X)
6.21/1.21	= { by axiom 26 (ruleD39) R->L }
6.21/1.21	  fresh147(fresh134(fresh66(fresh107(eqangle(Y, X, a, v, Y, X, a, v), true, Y, X, Y, X), true, Y, X, X), true, Y, X, X), true, X, Y, X)
6.21/1.21	= { by axiom 28 (ruleD19) R->L }
6.21/1.21	  fresh147(fresh134(fresh66(fresh107(fresh135(eqangle(a, v, Y, X, a, v, Y, X), true, a, v, Y, X, a, v, Y, X), true, Y, X, Y, X), true, Y, X, X), true, Y, X, X), true, X, Y, X)
6.21/1.21	= { by lemma 30 }
6.21/1.21	  fresh147(fresh134(fresh66(fresh107(fresh135(true, true, a, v, Y, X, a, v, Y, X), true, Y, X, Y, X), true, Y, X, X), true, Y, X, X), true, X, Y, X)
6.21/1.21	= { by axiom 20 (ruleD19) }
6.21/1.21	  fresh147(fresh134(fresh66(fresh107(true, true, Y, X, Y, X), true, Y, X, X), true, Y, X, X), true, X, Y, X)
6.21/1.21	= { by axiom 8 (ruleD39) }
6.21/1.21	  fresh147(fresh134(fresh66(true, true, Y, X, X), true, Y, X, X), true, X, Y, X)
6.21/1.21	= { by axiom 5 (ruleD66) }
6.21/1.21	  fresh147(fresh134(true, true, Y, X, X), true, X, Y, X)
6.21/1.21	= { by axiom 3 (ruleD2) }
6.21/1.21	  fresh147(true, true, X, Y, X)
6.21/1.21	= { by axiom 2 (ruleD1) }
6.21/1.21	  true
6.21/1.21	
6.21/1.21	Lemma 32: cyclic(v, v, a, X) = true.
6.21/1.21	Proof:
6.21/1.21	  cyclic(v, v, a, X)
6.21/1.21	= { by axiom 9 (ruleD42b) R->L }
6.21/1.21	  fresh103(true, true, v, v, a, X)
6.21/1.21	= { by axiom 21 (ruleD21) R->L }
6.21/1.21	  fresh103(fresh132(true, true, a, v, X, v, a, v, X, v), true, v, v, a, X)
6.21/1.21	= { by lemma 30 R->L }
6.21/1.21	  fresh103(fresh132(eqangle(a, v, X, v, a, v, X, v), true, a, v, X, v, a, v, X, v), true, v, v, a, X)
6.21/1.21	= { by axiom 29 (ruleD21) }
6.21/1.21	  fresh103(eqangle(a, v, a, v, X, v, X, v), true, v, v, a, X)
6.21/1.21	= { by axiom 27 (ruleD42b) }
6.21/1.21	  fresh102(coll(a, X, v), true, v, v, a, X)
6.21/1.21	= { by axiom 7 (ruleD3) R->L }
6.21/1.21	  fresh102(fresh121(true, true, v, v, a, X), true, v, v, a, X)
6.21/1.21	= { by lemma 31 R->L }
6.21/1.21	  fresh102(fresh121(coll(v, v, X), true, v, v, a, X), true, v, v, a, X)
6.21/1.21	= { by axiom 18 (ruleD3) }
6.21/1.21	  fresh102(fresh120(coll(v, v, a), true, v, a, X), true, v, v, a, X)
6.21/1.21	= { by lemma 31 }
6.21/1.21	  fresh102(fresh120(true, true, v, a, X), true, v, v, a, X)
6.21/1.21	= { by axiom 4 (ruleD3) }
6.21/1.21	  fresh102(true, true, v, v, a, X)
6.21/1.21	= { by axiom 10 (ruleD42b) }
6.21/1.21	  true
6.21/1.21	
6.21/1.21	Lemma 33: cyclic(v, a, X, Y) = true.
6.21/1.21	Proof:
6.21/1.21	  cyclic(v, a, X, Y)
6.21/1.21	= { by axiom 13 (ruleD17) R->L }
6.21/1.21	  fresh138(true, true, v, v, a, X, Y)
6.21/1.21	= { by lemma 32 R->L }
6.21/1.21	  fresh138(cyclic(v, v, a, Y), true, v, v, a, X, Y)
6.21/1.21	= { by axiom 23 (ruleD17) }
6.21/1.21	  fresh137(cyclic(v, v, a, X), true, v, a, X, Y)
6.21/1.21	= { by lemma 32 }
6.21/1.21	  fresh137(true, true, v, a, X, Y)
6.21/1.21	= { by axiom 6 (ruleD17) }
6.21/1.21	  true
6.21/1.21	
6.21/1.21	Goal 1 (exemplo6GDDFULL416047_8): cyclic(a, u, d, v) = true.
6.21/1.21	Proof:
6.21/1.21	  cyclic(a, u, d, v)
6.21/1.21	= { by axiom 13 (ruleD17) R->L }
6.21/1.21	  fresh138(true, true, v, a, u, d, v)
6.21/1.21	= { by lemma 33 R->L }
6.21/1.21	  fresh138(cyclic(v, a, u, v), true, v, a, u, d, v)
6.21/1.21	= { by axiom 23 (ruleD17) }
6.21/1.21	  fresh137(cyclic(v, a, u, d), true, a, u, d, v)
6.21/1.21	= { by lemma 33 }
6.21/1.21	  fresh137(true, true, a, u, d, v)
6.21/1.21	= { by axiom 6 (ruleD17) }
6.21/1.21	  true
6.21/1.21	% SZS output end Proof
6.21/1.21	
6.21/1.21	RESULT: Theorem (the conjecture is true).
6.21/1.22	EOF