TSTP Solution File: GEO561+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GEO561+1 : TPTP v8.1.2. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n022.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Aug 30 23:29:19 EDT 2023
% Result : Theorem 78.37s 10.27s
% Output : Proof 80.07s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.13 % Problem : GEO561+1 : TPTP v8.1.2. Released v7.5.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n022.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Tue Aug 29 20:24:53 EDT 2023
% 0.13/0.35 % CPUTime :
% 78.37/10.27 Command-line arguments: --ground-connectedness --complete-subsets
% 78.37/10.27
% 78.37/10.27 % SZS status Theorem
% 78.37/10.27
% 80.00/10.41 % SZS output start Proof
% 80.00/10.41 Take the following subset of the input axioms:
% 80.07/10.42 fof(exemplo6GDDFULL214022, conjecture, ![A, B, C, D, E, M, O, P, Q, N, NWPNT1, NWPNT2]: ((circle(O, A, B, C) & (perp(P, O, A, C) & (coll(P, A, C) & (perp(Q, O, A, B) & (coll(Q, A, B) & (coll(N, O, P) & (circle(O, A, N, NWPNT1) & (coll(M, O, Q) & (circle(O, A, M, NWPNT2) & (coll(E, A, C) & (coll(E, N, M) & (coll(D, A, B) & coll(D, N, M))))))))))))) => eqangle(A, D, D, E, D, E, E, A))).
% 80.07/10.42 fof(ruleD1, axiom, ![A2, B2, C2]: (coll(A2, B2, C2) => coll(A2, C2, B2))).
% 80.07/10.42 fof(ruleD11, axiom, ![B2, A2_2, M2]: (midp(M2, B2, A2_2) => midp(M2, A2_2, B2))).
% 80.07/10.42 fof(ruleD17, axiom, ![B2, C2, D2, E2, A2_2]: ((cyclic(A2_2, B2, C2, D2) & cyclic(A2_2, B2, C2, E2)) => cyclic(B2, C2, D2, E2))).
% 80.07/10.42 fof(ruleD18, axiom, ![U, V, B2, C2, D2, A2_2, P2, Q2]: (eqangle(A2_2, B2, C2, D2, P2, Q2, U, V) => eqangle(B2, A2_2, C2, D2, P2, Q2, U, V))).
% 80.07/10.42 fof(ruleD19, axiom, ![B2, C2, D2, A2_2, P2, Q2, U2, V2]: (eqangle(A2_2, B2, C2, D2, P2, Q2, U2, V2) => eqangle(C2, D2, A2_2, B2, U2, V2, P2, Q2))).
% 80.07/10.42 fof(ruleD2, axiom, ![B2, C2, A2_2]: (coll(A2_2, B2, C2) => coll(B2, A2_2, C2))).
% 80.07/10.42 fof(ruleD20, axiom, ![B2, C2, D2, A2_2, P2, Q2, U2, V2]: (eqangle(A2_2, B2, C2, D2, P2, Q2, U2, V2) => eqangle(P2, Q2, U2, V2, A2_2, B2, C2, D2))).
% 80.07/10.42 fof(ruleD21, axiom, ![B2, C2, D2, A2_2, P2, Q2, U2, V2]: (eqangle(A2_2, B2, C2, D2, P2, Q2, U2, V2) => eqangle(A2_2, B2, P2, Q2, C2, D2, U2, V2))).
% 80.07/10.42 fof(ruleD3, axiom, ![B2, C2, D2, A2_2]: ((coll(A2_2, B2, C2) & coll(A2_2, B2, D2)) => coll(C2, D2, A2_2))).
% 80.07/10.42 fof(ruleD39, axiom, ![B2, C2, D2, A2_2, P2, Q2]: (eqangle(A2_2, B2, P2, Q2, C2, D2, P2, Q2) => para(A2_2, B2, C2, D2))).
% 80.07/10.42 fof(ruleD40, axiom, ![B2, C2, D2, A2_2, P2, Q2]: (para(A2_2, B2, C2, D2) => eqangle(A2_2, B2, P2, Q2, C2, D2, P2, Q2))).
% 80.07/10.42 fof(ruleD41, axiom, ![B2, A2_2, P2, Q2]: (cyclic(A2_2, B2, P2, Q2) => eqangle(P2, A2_2, P2, B2, Q2, A2_2, Q2, B2))).
% 80.07/10.42 fof(ruleD42b, axiom, ![B2, A2_2, P2, Q2]: ((eqangle(P2, A2_2, P2, B2, Q2, A2_2, Q2, B2) & coll(P2, Q2, B2)) => cyclic(A2_2, B2, P2, Q2))).
% 80.07/10.42 fof(ruleD43, axiom, ![R, B2, C2, A2_2, P2, Q2]: ((cyclic(A2_2, B2, C2, P2) & (cyclic(A2_2, B2, C2, Q2) & (cyclic(A2_2, B2, C2, R) & eqangle(C2, A2_2, C2, B2, R, P2, R, Q2)))) => cong(A2_2, B2, P2, Q2))).
% 80.07/10.42 fof(ruleD45, axiom, ![F, B2, C2, E2, A2_2]: ((midp(E2, A2_2, B2) & (para(E2, F, B2, C2) & coll(F, A2_2, C2))) => midp(F, A2_2, C2))).
% 80.07/10.42 fof(ruleD46, axiom, ![B2, A2_2, O2]: (cong(O2, A2_2, O2, B2) => eqangle(O2, A2_2, A2_2, B2, A2_2, B2, O2, B2))).
% 80.07/10.42 fof(ruleD51, axiom, ![B2, C2, A2_2, M2, O2]: ((circle(O2, A2_2, B2, C2) & (coll(M2, B2, C2) & eqangle(A2_2, B2, A2_2, C2, O2, B2, O2, M2))) => midp(M2, B2, C2))).
% 80.07/10.42 fof(ruleD56, axiom, ![B2, A2_2, P2, Q2]: ((cong(A2_2, P2, B2, P2) & cong(A2_2, Q2, B2, Q2)) => perp(A2_2, B2, P2, Q2))).
% 80.07/10.42 fof(ruleD66, axiom, ![B2, C2, A2_2]: (para(A2_2, B2, A2_2, C2) => coll(A2_2, B2, C2))).
% 80.07/10.42 fof(ruleD68, axiom, ![B2, C2, A2_2]: (midp(A2_2, B2, C2) => cong(A2_2, B2, A2_2, C2))).
% 80.07/10.42 fof(ruleD8, axiom, ![B2, C2, D2, A2_2]: (perp(A2_2, B2, C2, D2) => perp(C2, D2, A2_2, B2))).
% 80.07/10.42 fof(ruleD9, axiom, ![B2, C2, D2, E2, F2, A2_2]: ((perp(A2_2, B2, C2, D2) & perp(C2, D2, E2, F2)) => para(A2_2, B2, E2, F2))).
% 80.07/10.42
% 80.07/10.42 Now clausify the problem and encode Horn clauses using encoding 3 of
% 80.07/10.42 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 80.07/10.42 We repeatedly replace C & s=t => u=v by the two clauses:
% 80.07/10.42 fresh(y, y, x1...xn) = u
% 80.07/10.42 C => fresh(s, t, x1...xn) = v
% 80.07/10.42 where fresh is a fresh function symbol and x1..xn are the free
% 80.07/10.42 variables of u and v.
% 80.07/10.42 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 80.07/10.42 input problem has no model of domain size 1).
% 80.07/10.42
% 80.07/10.42 The encoding turns the above axioms into the following unit equations and goals:
% 80.07/10.42
% 80.07/10.42 Axiom 1 (exemplo6GDDFULL214022_10): circle(o, a, n, nwpnt1) = true.
% 80.07/10.42 Axiom 2 (exemplo6GDDFULL214022_8): perp(p, o, a, c) = true.
% 80.07/10.42 Axiom 3 (ruleD45): fresh181(X, X, Y, Z, W) = true.
% 80.07/10.42 Axiom 4 (ruleD51): fresh179(X, X, Y, Z, W) = true.
% 80.07/10.42 Axiom 5 (ruleD1): fresh146(X, X, Y, Z, W) = true.
% 80.07/10.42 Axiom 6 (ruleD11): fresh144(X, X, Y, Z, W) = true.
% 80.07/10.42 Axiom 7 (ruleD2): fresh133(X, X, Y, Z, W) = true.
% 80.07/10.42 Axiom 8 (ruleD3): fresh119(X, X, Y, Z, W) = true.
% 80.07/10.42 Axiom 9 (ruleD45): fresh98(X, X, Y, Z, W) = midp(W, Y, Z).
% 80.07/10.42 Axiom 10 (ruleD46): fresh97(X, X, Y, Z, W) = true.
% 80.07/10.42 Axiom 11 (ruleD51): fresh89(X, X, Y, Z, W) = midp(W, Y, Z).
% 80.07/10.42 Axiom 12 (ruleD66): fresh66(X, X, Y, Z, W) = true.
% 80.07/10.42 Axiom 13 (ruleD68): fresh63(X, X, Y, Z, W) = true.
% 80.07/10.42 Axiom 14 (ruleD43): fresh185(X, X, Y, Z, W, V) = true.
% 80.07/10.43 Axiom 15 (ruleD17): fresh136(X, X, Y, Z, W, V) = true.
% 80.07/10.43 Axiom 16 (ruleD3): fresh120(X, X, Y, Z, W, V) = coll(W, V, Y).
% 80.07/10.43 Axiom 17 (ruleD39): fresh106(X, X, Y, Z, W, V) = true.
% 80.07/10.43 Axiom 18 (ruleD41): fresh103(X, X, Y, Z, W, V) = true.
% 80.07/10.43 Axiom 19 (ruleD42b): fresh102(X, X, Y, Z, W, V) = cyclic(Y, Z, W, V).
% 80.07/10.43 Axiom 20 (ruleD42b): fresh101(X, X, Y, Z, W, V) = true.
% 80.07/10.43 Axiom 21 (ruleD56): fresh80(X, X, Y, Z, W, V) = perp(Y, Z, W, V).
% 80.07/10.43 Axiom 22 (ruleD56): fresh79(X, X, Y, Z, W, V) = true.
% 80.07/10.43 Axiom 23 (ruleD8): fresh52(X, X, Y, Z, W, V) = true.
% 80.07/10.43 Axiom 24 (ruleD9): fresh50(X, X, Y, Z, W, V) = true.
% 80.07/10.43 Axiom 25 (ruleD43): fresh183(X, X, Y, Z, W, V, U) = cong(Y, Z, V, U).
% 80.07/10.43 Axiom 26 (ruleD17): fresh137(X, X, Y, Z, W, V, U) = cyclic(Z, W, V, U).
% 80.07/10.43 Axiom 27 (ruleD45): fresh180(X, X, Y, Z, W, V, U) = fresh181(coll(U, Y, W), true, Y, W, U).
% 80.07/10.43 Axiom 28 (ruleD51): fresh178(X, X, Y, Z, W, V, U) = fresh179(coll(V, Z, W), true, Z, W, V).
% 80.07/10.43 Axiom 29 (ruleD1): fresh146(coll(X, Y, Z), true, X, Y, Z) = coll(X, Z, Y).
% 80.07/10.43 Axiom 30 (ruleD11): fresh144(midp(X, Y, Z), true, Z, Y, X) = midp(X, Z, Y).
% 80.07/10.43 Axiom 31 (ruleD2): fresh133(coll(X, Y, Z), true, X, Y, Z) = coll(Y, X, Z).
% 80.07/10.43 Axiom 32 (ruleD40): fresh104(X, X, Y, Z, W, V, U, T) = true.
% 80.07/10.43 Axiom 33 (ruleD68): fresh63(midp(X, Y, Z), true, X, Y, Z) = cong(X, Y, X, Z).
% 80.07/10.43 Axiom 34 (ruleD9): fresh51(X, X, Y, Z, W, V, U, T) = para(Y, Z, U, T).
% 80.07/10.43 Axiom 35 (ruleD3): fresh120(coll(X, Y, Z), true, X, Y, W, Z) = fresh119(coll(X, Y, W), true, X, W, Z).
% 80.07/10.43 Axiom 36 (ruleD46): fresh97(cong(X, Y, X, Z), true, Y, Z, X) = eqangle(X, Y, Y, Z, Y, Z, X, Z).
% 80.07/10.43 Axiom 37 (ruleD66): fresh66(para(X, Y, X, Z), true, X, Y, Z) = coll(X, Y, Z).
% 80.07/10.43 Axiom 38 (ruleD43): fresh184(X, X, Y, Z, W, V, U) = fresh185(cyclic(Y, Z, W, V), true, Y, Z, V, U).
% 80.07/10.43 Axiom 39 (ruleD45): fresh180(midp(X, Y, Z), true, Y, Z, W, X, V) = fresh98(para(X, V, Z, W), true, Y, W, V).
% 80.07/10.43 Axiom 40 (ruleD18): fresh135(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 80.07/10.43 Axiom 41 (ruleD19): fresh134(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 80.07/10.43 Axiom 42 (ruleD20): fresh132(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 80.07/10.43 Axiom 43 (ruleD21): fresh131(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 80.07/10.43 Axiom 44 (ruleD41): fresh103(cyclic(X, Y, Z, W), true, X, Y, Z, W) = eqangle(Z, X, Z, Y, W, X, W, Y).
% 80.07/10.43 Axiom 45 (ruleD56): fresh80(cong(X, Y, Z, Y), true, X, Z, W, Y) = fresh79(cong(X, W, Z, W), true, X, Z, W, Y).
% 80.07/10.43 Axiom 46 (ruleD8): fresh52(perp(X, Y, Z, W), true, X, Y, Z, W) = perp(Z, W, X, Y).
% 80.07/10.43 Axiom 47 (ruleD43): fresh182(X, X, Y, Z, W, V, U, T) = fresh183(cyclic(Y, Z, W, U), true, Y, Z, W, V, U).
% 80.07/10.43 Axiom 48 (ruleD17): fresh137(cyclic(X, Y, Z, W), true, X, Y, Z, V, W) = fresh136(cyclic(X, Y, Z, V), true, Y, Z, V, W).
% 80.07/10.43 Axiom 49 (ruleD40): fresh104(para(X, Y, Z, W), true, X, Y, Z, W, V, U) = eqangle(X, Y, V, U, Z, W, V, U).
% 80.07/10.43 Axiom 50 (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).
% 80.07/10.43 Axiom 51 (ruleD39): fresh106(eqangle(X, Y, Z, W, V, U, Z, W), true, X, Y, V, U) = para(X, Y, V, U).
% 80.07/10.43 Axiom 52 (ruleD42b): fresh102(eqangle(X, Y, X, Z, W, Y, W, Z), true, Y, Z, X, W) = fresh101(coll(X, W, Z), true, Y, Z, X, W).
% 80.07/10.43 Axiom 53 (ruleD51): fresh178(eqangle(X, Y, X, Z, W, Y, W, V), true, X, Y, Z, V, W) = fresh89(circle(W, X, Y, Z), true, Y, Z, V).
% 80.07/10.43 Axiom 54 (ruleD43): fresh182(eqangle(X, Y, X, Z, W, V, W, U), true, Y, Z, X, V, U, W) = fresh184(cyclic(Y, Z, X, W), true, Y, Z, X, V, U).
% 80.07/10.43 Axiom 55 (ruleD18): fresh135(eqangle(X, Y, Z, W, V, U, T, S), true, X, Y, Z, W, V, U, T, S) = eqangle(Y, X, Z, W, V, U, T, S).
% 80.07/10.43 Axiom 56 (ruleD19): fresh134(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).
% 80.07/10.43 Axiom 57 (ruleD20): fresh132(eqangle(X, Y, Z, W, V, U, T, S), true, X, Y, Z, W, V, U, T, S) = eqangle(V, U, T, S, X, Y, Z, W).
% 80.07/10.43 Axiom 58 (ruleD21): fresh131(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).
% 80.07/10.43
% 80.07/10.43 Lemma 59: eqangle(a, c, X, Y, a, c, X, Y) = true.
% 80.07/10.43 Proof:
% 80.07/10.43 eqangle(a, c, X, Y, a, c, X, Y)
% 80.07/10.43 = { by axiom 49 (ruleD40) R->L }
% 80.07/10.43 fresh104(para(a, c, a, c), true, a, c, a, c, X, Y)
% 80.07/10.43 = { by axiom 34 (ruleD9) R->L }
% 80.07/10.43 fresh104(fresh51(true, true, a, c, p, o, a, c), true, a, c, a, c, X, Y)
% 80.07/10.43 = { by axiom 2 (exemplo6GDDFULL214022_8) R->L }
% 80.07/10.43 fresh104(fresh51(perp(p, o, a, c), true, a, c, p, o, a, c), true, a, c, a, c, X, Y)
% 80.07/10.43 = { by axiom 50 (ruleD9) }
% 80.07/10.43 fresh104(fresh50(perp(a, c, p, o), true, a, c, a, c), true, a, c, a, c, X, Y)
% 80.07/10.43 = { by axiom 46 (ruleD8) R->L }
% 80.07/10.43 fresh104(fresh50(fresh52(perp(p, o, a, c), true, p, o, a, c), true, a, c, a, c), true, a, c, a, c, X, Y)
% 80.07/10.43 = { by axiom 2 (exemplo6GDDFULL214022_8) }
% 80.07/10.43 fresh104(fresh50(fresh52(true, true, p, o, a, c), true, a, c, a, c), true, a, c, a, c, X, Y)
% 80.07/10.43 = { by axiom 23 (ruleD8) }
% 80.07/10.43 fresh104(fresh50(true, true, a, c, a, c), true, a, c, a, c, X, Y)
% 80.07/10.43 = { by axiom 24 (ruleD9) }
% 80.07/10.43 fresh104(true, true, a, c, a, c, X, Y)
% 80.07/10.43 = { by axiom 32 (ruleD40) }
% 80.07/10.43 true
% 80.07/10.43
% 80.07/10.43 Lemma 60: para(X, Y, X, Y) = true.
% 80.07/10.43 Proof:
% 80.07/10.43 para(X, Y, X, Y)
% 80.07/10.43 = { by axiom 51 (ruleD39) R->L }
% 80.07/10.43 fresh106(eqangle(X, Y, a, c, X, Y, a, c), true, X, Y, X, Y)
% 80.07/10.43 = { by axiom 56 (ruleD19) R->L }
% 80.07/10.43 fresh106(fresh134(eqangle(a, c, X, Y, a, c, X, Y), true, a, c, X, Y, a, c, X, Y), true, X, Y, X, Y)
% 80.07/10.43 = { by lemma 59 }
% 80.07/10.43 fresh106(fresh134(true, true, a, c, X, Y, a, c, X, Y), true, X, Y, X, Y)
% 80.07/10.43 = { by axiom 41 (ruleD19) }
% 80.07/10.43 fresh106(true, true, X, Y, X, Y)
% 80.07/10.43 = { by axiom 17 (ruleD39) }
% 80.07/10.43 true
% 80.07/10.43
% 80.07/10.43 Lemma 61: coll(X, X, Y) = true.
% 80.07/10.43 Proof:
% 80.07/10.43 coll(X, X, Y)
% 80.07/10.43 = { by axiom 29 (ruleD1) R->L }
% 80.07/10.43 fresh146(coll(X, Y, X), true, X, Y, X)
% 80.07/10.43 = { by axiom 31 (ruleD2) R->L }
% 80.07/10.43 fresh146(fresh133(coll(Y, X, X), true, Y, X, X), true, X, Y, X)
% 80.07/10.43 = { by axiom 37 (ruleD66) R->L }
% 80.07/10.43 fresh146(fresh133(fresh66(para(Y, X, Y, X), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 80.07/10.43 = { by lemma 60 }
% 80.07/10.43 fresh146(fresh133(fresh66(true, true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 80.07/10.43 = { by axiom 12 (ruleD66) }
% 80.07/10.43 fresh146(fresh133(true, true, Y, X, X), true, X, Y, X)
% 80.07/10.43 = { by axiom 7 (ruleD2) }
% 80.07/10.43 fresh146(true, true, X, Y, X)
% 80.07/10.43 = { by axiom 5 (ruleD1) }
% 80.07/10.43 true
% 80.07/10.43
% 80.07/10.43 Lemma 62: coll(X, Y, Z) = true.
% 80.07/10.43 Proof:
% 80.07/10.43 coll(X, Y, Z)
% 80.07/10.43 = { by axiom 16 (ruleD3) R->L }
% 80.07/10.43 fresh120(true, true, Z, Z, X, Y)
% 80.07/10.43 = { by lemma 61 R->L }
% 80.07/10.43 fresh120(coll(Z, Z, Y), true, Z, Z, X, Y)
% 80.07/10.43 = { by axiom 35 (ruleD3) }
% 80.07/10.43 fresh119(coll(Z, Z, X), true, Z, X, Y)
% 80.07/10.43 = { by lemma 61 }
% 80.07/10.43 fresh119(true, true, Z, X, Y)
% 80.07/10.43 = { by axiom 8 (ruleD3) }
% 80.07/10.43 true
% 80.07/10.43
% 80.07/10.43 Lemma 63: cyclic(c, c, a, X) = true.
% 80.07/10.43 Proof:
% 80.07/10.43 cyclic(c, c, a, X)
% 80.07/10.43 = { by axiom 19 (ruleD42b) R->L }
% 80.07/10.43 fresh102(true, true, c, c, a, X)
% 80.07/10.43 = { by axiom 43 (ruleD21) R->L }
% 80.07/10.43 fresh102(fresh131(true, true, a, c, X, c, a, c, X, c), true, c, c, a, X)
% 80.07/10.43 = { by lemma 59 R->L }
% 80.07/10.43 fresh102(fresh131(eqangle(a, c, X, c, a, c, X, c), true, a, c, X, c, a, c, X, c), true, c, c, a, X)
% 80.07/10.43 = { by axiom 58 (ruleD21) }
% 80.07/10.43 fresh102(eqangle(a, c, a, c, X, c, X, c), true, c, c, a, X)
% 80.07/10.43 = { by axiom 52 (ruleD42b) }
% 80.07/10.43 fresh101(coll(a, X, c), true, c, c, a, X)
% 80.07/10.43 = { by lemma 62 }
% 80.07/10.43 fresh101(true, true, c, c, a, X)
% 80.07/10.43 = { by axiom 20 (ruleD42b) }
% 80.07/10.43 true
% 80.07/10.43
% 80.07/10.43 Lemma 64: cyclic(c, a, X, Y) = true.
% 80.07/10.43 Proof:
% 80.07/10.43 cyclic(c, a, X, Y)
% 80.07/10.43 = { by axiom 26 (ruleD17) R->L }
% 80.07/10.43 fresh137(true, true, c, c, a, X, Y)
% 80.07/10.43 = { by lemma 63 R->L }
% 80.07/10.43 fresh137(cyclic(c, c, a, Y), true, c, c, a, X, Y)
% 80.07/10.43 = { by axiom 48 (ruleD17) }
% 80.07/10.43 fresh136(cyclic(c, c, a, X), true, c, a, X, Y)
% 80.07/10.43 = { by lemma 63 }
% 80.07/10.43 fresh136(true, true, c, a, X, Y)
% 80.07/10.43 = { by axiom 15 (ruleD17) }
% 80.07/10.43 true
% 80.07/10.43
% 80.07/10.43 Lemma 65: cyclic(a, X, Y, Z) = true.
% 80.07/10.43 Proof:
% 80.07/10.43 cyclic(a, X, Y, Z)
% 80.07/10.43 = { by axiom 26 (ruleD17) R->L }
% 80.07/10.43 fresh137(true, true, c, a, X, Y, Z)
% 80.07/10.43 = { by lemma 64 R->L }
% 80.07/10.43 fresh137(cyclic(c, a, X, Z), true, c, a, X, Y, Z)
% 80.07/10.43 = { by axiom 48 (ruleD17) }
% 80.07/10.43 fresh136(cyclic(c, a, X, Y), true, a, X, Y, Z)
% 80.07/10.43 = { by lemma 64 }
% 80.07/10.43 fresh136(true, true, a, X, Y, Z)
% 80.07/10.43 = { by axiom 15 (ruleD17) }
% 80.07/10.43 true
% 80.07/10.43
% 80.07/10.43 Lemma 66: cyclic(X, Y, Z, W) = true.
% 80.07/10.43 Proof:
% 80.07/10.43 cyclic(X, Y, Z, W)
% 80.07/10.43 = { by axiom 26 (ruleD17) R->L }
% 80.07/10.43 fresh137(true, true, a, X, Y, Z, W)
% 80.07/10.43 = { by lemma 65 R->L }
% 80.07/10.43 fresh137(cyclic(a, X, Y, W), true, a, X, Y, Z, W)
% 80.07/10.43 = { by axiom 48 (ruleD17) }
% 80.07/10.43 fresh136(cyclic(a, X, Y, Z), true, X, Y, Z, W)
% 80.07/10.43 = { by lemma 65 }
% 80.07/10.43 fresh136(true, true, X, Y, Z, W)
% 80.07/10.43 = { by axiom 15 (ruleD17) }
% 80.07/10.43 true
% 80.07/10.43
% 80.07/10.43 Lemma 67: cong(X, Y, X, Y) = true.
% 80.07/10.43 Proof:
% 80.07/10.43 cong(X, Y, X, Y)
% 80.07/10.43 = { by axiom 25 (ruleD43) R->L }
% 80.07/10.43 fresh183(true, true, X, Y, Z, X, Y)
% 80.07/10.43 = { by lemma 66 R->L }
% 80.07/10.43 fresh183(cyclic(X, Y, Z, Y), true, X, Y, Z, X, Y)
% 80.07/10.43 = { by axiom 47 (ruleD43) R->L }
% 80.07/10.43 fresh182(true, true, X, Y, Z, X, Y, Z)
% 80.07/10.43 = { by axiom 32 (ruleD40) R->L }
% 80.07/10.43 fresh182(fresh104(true, true, Z, X, Z, X, Z, Y), true, X, Y, Z, X, Y, Z)
% 80.07/10.43 = { by lemma 60 R->L }
% 80.07/10.43 fresh182(fresh104(para(Z, X, Z, X), true, Z, X, Z, X, Z, Y), true, X, Y, Z, X, Y, Z)
% 80.07/10.43 = { by axiom 49 (ruleD40) }
% 80.07/10.43 fresh182(eqangle(Z, X, Z, Y, Z, X, Z, Y), true, X, Y, Z, X, Y, Z)
% 80.07/10.44 = { by axiom 54 (ruleD43) }
% 80.07/10.44 fresh184(cyclic(X, Y, Z, Z), true, X, Y, Z, X, Y)
% 80.07/10.44 = { by lemma 66 }
% 80.07/10.44 fresh184(true, true, X, Y, Z, X, Y)
% 80.07/10.44 = { by axiom 38 (ruleD43) }
% 80.07/10.44 fresh185(cyclic(X, Y, Z, X), true, X, Y, X, Y)
% 80.07/10.44 = { by lemma 66 }
% 80.07/10.44 fresh185(true, true, X, Y, X, Y)
% 80.07/10.44 = { by axiom 14 (ruleD43) }
% 80.07/10.44 true
% 80.07/10.44
% 80.07/10.44 Lemma 68: perp(X, X, Y, Z) = true.
% 80.07/10.44 Proof:
% 80.07/10.44 perp(X, X, Y, Z)
% 80.07/10.44 = { by axiom 21 (ruleD56) R->L }
% 80.07/10.44 fresh80(true, true, X, X, Y, Z)
% 80.07/10.44 = { by lemma 67 R->L }
% 80.07/10.44 fresh80(cong(X, Z, X, Z), true, X, X, Y, Z)
% 80.07/10.44 = { by axiom 45 (ruleD56) }
% 80.07/10.44 fresh79(cong(X, Y, X, Y), true, X, X, Y, Z)
% 80.07/10.44 = { by lemma 67 }
% 80.07/10.44 fresh79(true, true, X, X, Y, Z)
% 80.07/10.44 = { by axiom 22 (ruleD56) }
% 80.07/10.44 true
% 80.07/10.44
% 80.07/10.44 Lemma 69: para(X, Y, Z, W) = true.
% 80.07/10.44 Proof:
% 80.07/10.44 para(X, Y, Z, W)
% 80.07/10.44 = { by axiom 34 (ruleD9) R->L }
% 80.07/10.44 fresh51(true, true, X, Y, V, V, Z, W)
% 80.07/10.44 = { by lemma 68 R->L }
% 80.07/10.44 fresh51(perp(V, V, Z, W), true, X, Y, V, V, Z, W)
% 80.07/10.44 = { by axiom 50 (ruleD9) }
% 80.07/10.44 fresh50(perp(X, Y, V, V), true, X, Y, Z, W)
% 80.07/10.44 = { by axiom 46 (ruleD8) R->L }
% 80.07/10.44 fresh50(fresh52(perp(V, V, X, Y), true, V, V, X, Y), true, X, Y, Z, W)
% 80.07/10.44 = { by lemma 68 }
% 80.07/10.44 fresh50(fresh52(true, true, V, V, X, Y), true, X, Y, Z, W)
% 80.07/10.44 = { by axiom 23 (ruleD8) }
% 80.07/10.44 fresh50(true, true, X, Y, Z, W)
% 80.07/10.44 = { by axiom 24 (ruleD9) }
% 80.07/10.44 true
% 80.07/10.44
% 80.07/10.44 Lemma 70: fresh180(X, X, Y, Z, W, V, U) = true.
% 80.07/10.44 Proof:
% 80.07/10.44 fresh180(X, X, Y, Z, W, V, U)
% 80.07/10.44 = { by axiom 27 (ruleD45) }
% 80.07/10.44 fresh181(coll(U, Y, W), true, Y, W, U)
% 80.07/10.44 = { by lemma 62 }
% 80.07/10.44 fresh181(true, true, Y, W, U)
% 80.07/10.44 = { by axiom 3 (ruleD45) }
% 80.07/10.44 true
% 80.07/10.44
% 80.07/10.44 Goal 1 (exemplo6GDDFULL214022_13): eqangle(a, d, d, e, d, e, e, a) = true.
% 80.07/10.44 Proof:
% 80.07/10.44 eqangle(a, d, d, e, d, e, e, a)
% 80.07/10.44 = { by axiom 57 (ruleD20) R->L }
% 80.07/10.44 fresh132(eqangle(d, e, e, a, a, d, d, e), true, d, e, e, a, a, d, d, e)
% 80.07/10.44 = { by axiom 56 (ruleD19) R->L }
% 80.07/10.44 fresh132(fresh134(eqangle(e, a, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.44 = { by axiom 55 (ruleD18) R->L }
% 80.07/10.44 fresh132(fresh134(fresh135(eqangle(a, e, d, e, d, e, a, d), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.44 = { by axiom 57 (ruleD20) R->L }
% 80.07/10.44 fresh132(fresh134(fresh135(fresh132(eqangle(d, e, a, d, a, e, d, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.44 = { by axiom 56 (ruleD19) R->L }
% 80.07/10.44 fresh132(fresh134(fresh135(fresh132(fresh134(eqangle(a, d, d, e, d, e, a, e), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.44 = { by axiom 36 (ruleD46) R->L }
% 80.07/10.44 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(cong(a, d, a, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.44 = { by axiom 33 (ruleD68) R->L }
% 80.07/10.44 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(midp(a, d, e), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.45 = { by axiom 9 (ruleD45) R->L }
% 80.07/10.45 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh98(true, true, d, e, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.45 = { by lemma 69 R->L }
% 80.07/10.45 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh98(para(X, a, n, e), true, d, e, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.45 = { by axiom 39 (ruleD45) R->L }
% 80.07/10.45 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(midp(X, d, n), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.45 = { by axiom 30 (ruleD11) R->L }
% 80.07/10.45 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(fresh144(midp(X, n, d), true, d, n, X), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.45 = { by axiom 9 (ruleD45) R->L }
% 80.07/10.45 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(fresh144(fresh98(true, true, n, d, X), true, d, n, X), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.45 = { by lemma 69 R->L }
% 80.07/10.45 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(fresh144(fresh98(para(nwpnt1, X, nwpnt1, d), true, n, d, X), true, d, n, X), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.45 = { by axiom 39 (ruleD45) R->L }
% 80.07/10.45 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(fresh144(fresh180(midp(nwpnt1, n, nwpnt1), true, n, nwpnt1, d, nwpnt1, X), true, d, n, X), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.45 = { by axiom 11 (ruleD51) R->L }
% 80.07/10.45 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(fresh144(fresh180(fresh89(true, true, n, nwpnt1, nwpnt1), true, n, nwpnt1, d, nwpnt1, X), true, d, n, X), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.45 = { by axiom 1 (exemplo6GDDFULL214022_10) R->L }
% 80.07/10.45 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(fresh144(fresh180(fresh89(circle(o, a, n, nwpnt1), true, n, nwpnt1, nwpnt1), true, n, nwpnt1, d, nwpnt1, X), true, d, n, X), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.45 = { by axiom 53 (ruleD51) R->L }
% 80.07/10.45 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(fresh144(fresh180(fresh178(eqangle(a, n, a, nwpnt1, o, n, o, nwpnt1), true, a, n, nwpnt1, nwpnt1, o), true, n, nwpnt1, d, nwpnt1, X), true, d, n, X), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.45 = { by axiom 44 (ruleD41) R->L }
% 80.07/10.45 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(fresh144(fresh180(fresh178(fresh103(cyclic(n, nwpnt1, a, o), true, n, nwpnt1, a, o), true, a, n, nwpnt1, nwpnt1, o), true, n, nwpnt1, d, nwpnt1, X), true, d, n, X), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.45 = { by lemma 66 }
% 80.07/10.45 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(fresh144(fresh180(fresh178(fresh103(true, true, n, nwpnt1, a, o), true, a, n, nwpnt1, nwpnt1, o), true, n, nwpnt1, d, nwpnt1, X), true, d, n, X), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.45 = { by axiom 18 (ruleD41) }
% 80.07/10.45 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(fresh144(fresh180(fresh178(true, true, a, n, nwpnt1, nwpnt1, o), true, n, nwpnt1, d, nwpnt1, X), true, d, n, X), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.46 = { by axiom 28 (ruleD51) }
% 80.07/10.46 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(fresh144(fresh180(fresh179(coll(nwpnt1, n, nwpnt1), true, n, nwpnt1, nwpnt1), true, n, nwpnt1, d, nwpnt1, X), true, d, n, X), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.46 = { by lemma 62 }
% 80.07/10.46 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(fresh144(fresh180(fresh179(true, true, n, nwpnt1, nwpnt1), true, n, nwpnt1, d, nwpnt1, X), true, d, n, X), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.46 = { by axiom 4 (ruleD51) }
% 80.07/10.46 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(fresh144(fresh180(true, true, n, nwpnt1, d, nwpnt1, X), true, d, n, X), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.46 = { by lemma 70 }
% 80.07/10.46 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(fresh144(true, true, d, n, X), true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.46 = { by axiom 6 (ruleD11) }
% 80.07/10.46 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(fresh180(true, true, d, n, e, X, a), true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.46 = { by lemma 70 }
% 80.07/10.46 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(fresh63(true, true, a, d, e), true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.46 = { by axiom 13 (ruleD68) }
% 80.07/10.46 fresh132(fresh134(fresh135(fresh132(fresh134(fresh97(true, true, d, e, a), true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.46 = { by axiom 10 (ruleD46) }
% 80.07/10.46 fresh132(fresh134(fresh135(fresh132(fresh134(true, true, a, d, d, e, d, e, a, e), true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.46 = { by axiom 41 (ruleD19) }
% 80.07/10.46 fresh132(fresh134(fresh135(fresh132(true, true, d, e, a, d, a, e, d, e), true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.46 = { by axiom 42 (ruleD20) }
% 80.07/10.46 fresh132(fresh134(fresh135(true, true, a, e, d, e, d, e, a, d), true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.46 = { by axiom 40 (ruleD18) }
% 80.07/10.46 fresh132(fresh134(true, true, e, a, d, e, d, e, a, d), true, d, e, e, a, a, d, d, e)
% 80.07/10.46 = { by axiom 41 (ruleD19) }
% 80.07/10.46 fresh132(true, true, d, e, e, a, a, d, d, e)
% 80.07/10.46 = { by axiom 42 (ruleD20) }
% 80.07/10.46 true
% 80.07/10.46 % SZS output end Proof
% 80.07/10.46
% 80.07/10.46 RESULT: Theorem (the conjecture is true).
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