0.08/0.15 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.08/0.15 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof 0.16/0.38 % Computer : n029.cluster.edu 0.16/0.38 % Model : x86_64 x86_64 0.16/0.38 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.16/0.38 % Memory : 8042.1875MB 0.16/0.38 % OS : Linux 3.10.0-693.el7.x86_64 0.16/0.38 % CPULimit : 1200 0.16/0.38 % WCLimit : 120 0.16/0.38 % DateTime : Tue Jul 13 13:04:55 EDT 2021 0.16/0.38 % CPUTime : 299.89/38.55 % SZS status Theorem 299.89/38.55 299.89/38.55 % SZS output start Proof 299.89/38.55 Take the following subset of the input axioms: 299.89/38.55 fof('and(pred(comma_conjunct2(the(212)), 0), and(pred(comma_conjunct1(the(212)), 0), pred(the(212), 0)))', axiom, ron(vd1055, vd1063) & (rcenter(vd1057, vd1063) & rcircle(vd1063))). 299.89/38.55 fof('and(pred(conjunct2(210), 4), and(holds(conjunct2(210), 1059, 0), and(pred(conjunct2(210), 1), and(pred(conjunct1(210), 2), pred(conjunct1(210), 1)))))', axiom, vd1057!=vd1055 & (rpoint(vd1058) & (vd1056=vd1055 & (rpoint(vd1056) & vd1057=vd1058)))). 299.89/38.55 fof('qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))', axiom, ![Vd837, Vd838, Vd842, Vd843, Vd844]: (Vd844=Vd843 <= (vf(Vd842, Vd838)=vf(Vd842, Vd837) & (rcenter(Vd842, Vd844) & (rcenter(Vd842, Vd843) & (?[Vd841]: (rpoint(Vd841) & Vd838=Vd841) & (?[Vd840]: (Vd840=Vd837 & rpoint(Vd840)) & (ron(Vd837, Vd843) & ron(Vd838, Vd844))))))))). 299.89/38.55 fof('qu(theu(the(212), 1), imp(the(212)))', conjecture, ![Vdthex]: ((ron(vd1055, Vdthex) & (rcircle(Vdthex) & rcenter(vd1057, Vdthex))) => Vdthex=vd1063)). 299.89/38.55 299.89/38.55 Now clausify the problem and encode Horn clauses using encoding 3 of 299.89/38.55 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 299.89/38.55 We repeatedly replace C & s=t => u=v by the two clauses: 299.89/38.55 fresh(y, y, x1...xn) = u 299.89/38.55 C => fresh(s, t, x1...xn) = v 299.89/38.55 where fresh is a fresh function symbol and x1..xn are the free 299.89/38.55 variables of u and v. 299.89/38.55 A predicate p(X) is encoded as p(X)=true (this is sound, because the 299.89/38.55 input problem has no model of domain size 1). 299.89/38.55 299.89/38.55 The encoding turns the above axioms into the following unit equations and goals: 299.89/38.55 299.89/38.55 Axiom 1 (and(pred(conjunct2(210), 4), and(holds(conjunct2(210), 1059, 0), and(pred(conjunct2(210), 1), and(pred(conjunct1(210), 2), pred(conjunct1(210), 1)))))): vd1056 = vd1055. 299.89/38.55 Axiom 2 (and(pred(conjunct2(210), 4), and(holds(conjunct2(210), 1059, 0), and(pred(conjunct2(210), 1), and(pred(conjunct1(210), 2), pred(conjunct1(210), 1)))))_3): rpoint(vd1056) = true2. 299.89/38.55 Axiom 3 (and(pred(comma_conjunct2(the(212)), 0), and(pred(comma_conjunct1(the(212)), 0), pred(the(212), 0)))_1): ron(vd1055, vd1063) = true2. 299.89/38.55 Axiom 4 (qu(theu(the(212), 1), imp(the(212)))_1): ron(vd1055, vdthex) = true2. 299.89/38.55 Axiom 5 (and(pred(comma_conjunct2(the(212)), 0), and(pred(comma_conjunct1(the(212)), 0), pred(the(212), 0)))): rcenter(vd1057, vd1063) = true2. 299.89/38.55 Axiom 6 (qu(theu(the(212), 1), imp(the(212)))): rcenter(vd1057, vdthex) = true2. 299.89/38.55 Axiom 7 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))): fresh465(X, X, Y, Z) = Y. 299.89/38.55 Axiom 8 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))): fresh463(X, X, Y, Z, W, V, U) = V. 299.89/38.55 Axiom 9 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))): fresh464(X, X, Y, Z, W, V, U) = fresh465(vf(Z, U), vf(Z, Y), W, V). 299.89/38.55 Axiom 10 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))): fresh461(X, X, Y, Z, W, V, U) = fresh462(rpoint(Y), true2, Y, Z, W, V, U). 299.89/38.55 Axiom 11 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))): fresh460(X, X, Y, Z, W, V, U) = fresh461(rpoint(U), true2, Y, Z, W, V, U). 299.89/38.55 Axiom 12 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))): fresh459(X, X, Y, Z, W, V, U) = fresh464(rcenter(Z, V), true2, Y, Z, W, V, U). 299.89/38.55 Axiom 13 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))): fresh462(X, X, Y, Z, W, V, U) = fresh463(rcenter(Z, W), true2, Y, Z, W, V, U). 299.89/38.55 Axiom 14 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))): fresh459(ron(X, Y), true2, Z, W, V, Y, X) = fresh460(ron(Z, V), true2, Z, W, V, Y, X). 299.89/38.55 299.89/38.55 Lemma 15: rpoint(vd1055) = true2. 299.89/38.55 Proof: 299.89/38.55 rpoint(vd1055) 299.89/38.55 = { by axiom 1 (and(pred(conjunct2(210), 4), and(holds(conjunct2(210), 1059, 0), and(pred(conjunct2(210), 1), and(pred(conjunct1(210), 2), pred(conjunct1(210), 1)))))) R->L } 299.89/38.55 rpoint(vd1056) 299.89/38.55 = { by axiom 2 (and(pred(conjunct2(210), 4), and(holds(conjunct2(210), 1059, 0), and(pred(conjunct2(210), 1), and(pred(conjunct1(210), 2), pred(conjunct1(210), 1)))))_3) } 299.89/38.55 true2 299.89/38.55 299.89/38.55 Goal 1 (qu(theu(the(212), 1), imp(the(212)))_3): vdthex = vd1063. 299.89/38.55 Proof: 299.89/38.55 vdthex 299.89/38.55 = { by axiom 7 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))) R->L } 299.89/38.55 fresh465(vf(vd1057, vd1055), vf(vd1057, vd1055), vdthex, vd1063) 299.89/38.55 = { by axiom 9 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))) R->L } 299.89/38.55 fresh464(true2, true2, vd1055, vd1057, vdthex, vd1063, vd1055) 299.89/38.55 = { by axiom 5 (and(pred(comma_conjunct2(the(212)), 0), and(pred(comma_conjunct1(the(212)), 0), pred(the(212), 0)))) R->L } 299.89/38.55 fresh464(rcenter(vd1057, vd1063), true2, vd1055, vd1057, vdthex, vd1063, vd1055) 299.89/38.55 = { by axiom 12 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))) R->L } 299.89/38.55 fresh459(true2, true2, vd1055, vd1057, vdthex, vd1063, vd1055) 299.89/38.55 = { by axiom 3 (and(pred(comma_conjunct2(the(212)), 0), and(pred(comma_conjunct1(the(212)), 0), pred(the(212), 0)))_1) R->L } 299.89/38.55 fresh459(ron(vd1055, vd1063), true2, vd1055, vd1057, vdthex, vd1063, vd1055) 299.89/38.55 = { by axiom 14 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))) } 299.89/38.55 fresh460(ron(vd1055, vdthex), true2, vd1055, vd1057, vdthex, vd1063, vd1055) 299.89/38.55 = { by axiom 4 (qu(theu(the(212), 1), imp(the(212)))_1) } 299.89/38.55 fresh460(true2, true2, vd1055, vd1057, vdthex, vd1063, vd1055) 299.89/38.55 = { by axiom 11 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))) } 299.89/38.55 fresh461(rpoint(vd1055), true2, vd1055, vd1057, vdthex, vd1063, vd1055) 299.89/38.55 = { by lemma 15 } 299.89/38.55 fresh461(true2, true2, vd1055, vd1057, vdthex, vd1063, vd1055) 299.89/38.55 = { by axiom 10 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))) } 299.89/38.55 fresh462(rpoint(vd1055), true2, vd1055, vd1057, vdthex, vd1063, vd1055) 299.89/38.55 = { by lemma 15 } 299.89/38.55 fresh462(true2, true2, vd1055, vd1057, vdthex, vd1063, vd1055) 299.89/38.55 = { by axiom 13 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))) } 299.89/38.55 fresh463(rcenter(vd1057, vdthex), true2, vd1055, vd1057, vdthex, vd1063, vd1055) 299.89/38.55 = { by axiom 6 (qu(theu(the(212), 1), imp(the(212)))) } 299.89/38.55 fresh463(true2, true2, vd1055, vd1057, vdthex, vd1063, vd1055) 299.89/38.55 = { by axiom 8 (qu(cond(axiom(182), 0), imp(cond(axiom(182), 0)))) } 299.89/38.55 vd1063 299.89/38.55 % SZS output end Proof 299.89/38.55 299.89/38.55 RESULT: Theorem (the conjecture is true). 299.89/38.62 EOF