TSTP Solution File: GEO549+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GEO549+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 : n020.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 23:29:16 EDT 2023
% Result : Theorem 27.25s 3.80s
% Output : Proof 27.25s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GEO549+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
% 0.12/0.34 % Computer : n020.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Tue Aug 29 22:20:58 EDT 2023
% 0.12/0.34 % CPUTime :
% 27.25/3.80 Command-line arguments: --no-flatten-goal
% 27.25/3.80
% 27.25/3.80 % SZS status Theorem
% 27.25/3.80
% 27.25/3.82 % SZS output start Proof
% 27.25/3.82 Take the following subset of the input axioms:
% 27.25/3.82 fof(exemplo6GDDFULL012009, conjecture, ![A, B, C, D, F, M, O, P, P1, F2, F3, F1, L, NWPNT1, NWPNT2, NWPNT3]: ((circle(O, A, B, D) & (circle(O, A, C, NWPNT1) & (circle(O, A, P, NWPNT2) & (perp(F, P, A, C) & (coll(F, A, C) & (circle(O, A, P1, NWPNT3) & (perp(F2, P1, A, C) & (coll(F2, A, C) & (perp(F3, P1, A, D) & (coll(F3, A, D) & (perp(F1, P, A, D) & (coll(F1, A, D) & (perp(L, P, A, B) & (coll(L, A, B) & (perp(M, P1, A, B) & coll(M, A, B)))))))))))))))) => eqangle(F, L, L, F1, F2, M, M, F3))).
% 27.25/3.82 fof(ruleD1, axiom, ![A2, B2, C2]: (coll(A2, B2, C2) => coll(A2, C2, B2))).
% 27.25/3.82 fof(ruleD17, axiom, ![E, B2, C2, D2, A2_2]: ((cyclic(A2_2, B2, C2, D2) & cyclic(A2_2, B2, C2, E)) => cyclic(B2, C2, D2, E))).
% 27.25/3.82 fof(ruleD19, axiom, ![Q, U, V, B2, C2, D2, A2_2, P2]: (eqangle(A2_2, B2, C2, D2, P2, Q, U, V) => eqangle(C2, D2, A2_2, B2, U, V, P2, Q))).
% 27.25/3.82 fof(ruleD2, axiom, ![B2, C2, A2_2]: (coll(A2_2, B2, C2) => coll(B2, A2_2, C2))).
% 27.25/3.82 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))).
% 27.25/3.82 fof(ruleD22, axiom, ![G, H, B2, C2, D2, E2, F4, A2_2, P2, Q2, U2, V2]: ((eqangle(A2_2, B2, C2, D2, P2, Q2, U2, V2) & eqangle(P2, Q2, U2, V2, E2, F4, G, H)) => eqangle(A2_2, B2, C2, D2, E2, F4, G, H))).
% 27.25/3.83 fof(ruleD3, axiom, ![B2, C2, D2, A2_2]: ((coll(A2_2, B2, C2) & coll(A2_2, B2, D2)) => coll(C2, D2, A2_2))).
% 27.25/3.83 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))).
% 27.25/3.83 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))).
% 27.25/3.83 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))).
% 27.25/3.83 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))).
% 27.25/3.83 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))).
% 27.25/3.83 fof(ruleD66, axiom, ![B2, C2, A2_2]: (para(A2_2, B2, A2_2, C2) => coll(A2_2, B2, C2))).
% 27.25/3.83 fof(ruleD8, axiom, ![B2, C2, D2, A2_2]: (perp(A2_2, B2, C2, D2) => perp(C2, D2, A2_2, B2))).
% 27.25/3.83 fof(ruleD9, axiom, ![B2, C2, D2, E2, F4, A2_2]: ((perp(A2_2, B2, C2, D2) & perp(C2, D2, E2, F4)) => para(A2_2, B2, E2, F4))).
% 27.25/3.83
% 27.25/3.83 Now clausify the problem and encode Horn clauses using encoding 3 of
% 27.25/3.83 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 27.25/3.83 We repeatedly replace C & s=t => u=v by the two clauses:
% 27.25/3.83 fresh(y, y, x1...xn) = u
% 27.25/3.83 C => fresh(s, t, x1...xn) = v
% 27.25/3.83 where fresh is a fresh function symbol and x1..xn are the free
% 27.25/3.83 variables of u and v.
% 27.25/3.83 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 27.25/3.83 input problem has no model of domain size 1).
% 27.25/3.83
% 27.25/3.83 The encoding turns the above axioms into the following unit equations and goals:
% 27.25/3.83
% 27.25/3.83 Axiom 1 (exemplo6GDDFULL012009_6): perp(f, p, a, c) = true.
% 27.25/3.83 Axiom 2 (ruleD1): fresh146(X, X, Y, Z, W) = true.
% 27.25/3.83 Axiom 3 (ruleD2): fresh133(X, X, Y, Z, W) = true.
% 27.25/3.83 Axiom 4 (ruleD3): fresh119(X, X, Y, Z, W) = true.
% 27.25/3.83 Axiom 5 (ruleD66): fresh66(X, X, Y, Z, W) = true.
% 27.25/3.83 Axiom 6 (ruleD43): fresh185(X, X, Y, Z, W, V) = true.
% 27.25/3.83 Axiom 7 (ruleD17): fresh136(X, X, Y, Z, W, V) = true.
% 27.25/3.83 Axiom 8 (ruleD3): fresh120(X, X, Y, Z, W, V) = coll(W, V, Y).
% 27.25/3.83 Axiom 9 (ruleD39): fresh106(X, X, Y, Z, W, V) = true.
% 27.25/3.83 Axiom 10 (ruleD42b): fresh102(X, X, Y, Z, W, V) = cyclic(Y, Z, W, V).
% 27.25/3.83 Axiom 11 (ruleD42b): fresh101(X, X, Y, Z, W, V) = true.
% 27.25/3.83 Axiom 12 (ruleD56): fresh80(X, X, Y, Z, W, V) = perp(Y, Z, W, V).
% 27.25/3.83 Axiom 13 (ruleD56): fresh79(X, X, Y, Z, W, V) = true.
% 27.25/3.83 Axiom 14 (ruleD8): fresh52(X, X, Y, Z, W, V) = true.
% 27.25/3.83 Axiom 15 (ruleD9): fresh50(X, X, Y, Z, W, V) = true.
% 27.25/3.83 Axiom 16 (ruleD43): fresh183(X, X, Y, Z, W, V, U) = cong(Y, Z, V, U).
% 27.25/3.83 Axiom 17 (ruleD17): fresh137(X, X, Y, Z, W, V, U) = cyclic(Z, W, V, U).
% 27.25/3.83 Axiom 18 (ruleD1): fresh146(coll(X, Y, Z), true, X, Y, Z) = coll(X, Z, Y).
% 27.25/3.83 Axiom 19 (ruleD2): fresh133(coll(X, Y, Z), true, X, Y, Z) = coll(Y, X, Z).
% 27.25/3.83 Axiom 20 (ruleD40): fresh104(X, X, Y, Z, W, V, U, T) = true.
% 27.25/3.83 Axiom 21 (ruleD9): fresh51(X, X, Y, Z, W, V, U, T) = para(Y, Z, U, T).
% 27.25/3.83 Axiom 22 (ruleD3): fresh120(coll(X, Y, Z), true, X, Y, W, Z) = fresh119(coll(X, Y, W), true, X, W, Z).
% 27.25/3.83 Axiom 23 (ruleD66): fresh66(para(X, Y, X, Z), true, X, Y, Z) = coll(X, Y, Z).
% 27.25/3.83 Axiom 24 (ruleD43): fresh184(X, X, Y, Z, W, V, U) = fresh185(cyclic(Y, Z, W, V), true, Y, Z, V, U).
% 27.25/3.83 Axiom 25 (ruleD19): fresh134(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 27.25/3.83 Axiom 26 (ruleD21): fresh131(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 27.25/3.83 Axiom 27 (ruleD22): fresh129(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 27.25/3.83 Axiom 28 (ruleD56): fresh80(cong(X, Y, Z, Y), true, X, Z, W, Y) = fresh79(cong(X, W, Z, W), true, X, Z, W, Y).
% 27.25/3.83 Axiom 29 (ruleD8): fresh52(perp(X, Y, Z, W), true, X, Y, Z, W) = perp(Z, W, X, Y).
% 27.25/3.83 Axiom 30 (ruleD43): fresh182(X, X, Y, Z, W, V, U, T) = fresh183(cyclic(Y, Z, W, U), true, Y, Z, W, V, U).
% 27.25/3.83 Axiom 31 (ruleD17): fresh137(cyclic(X, Y, Z, W), true, X, Y, Z, V, W) = fresh136(cyclic(X, Y, Z, V), true, Y, Z, V, W).
% 27.25/3.83 Axiom 32 (ruleD40): fresh104(para(X, Y, Z, W), true, X, Y, Z, W, V, U) = eqangle(X, Y, V, U, Z, W, V, U).
% 27.25/3.83 Axiom 33 (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).
% 27.25/3.83 Axiom 34 (ruleD22): fresh130(X, X, Y, Z, W, V, U, T, S, X2, Y2, Z2, W2, V2) = eqangle(Y, Z, W, V, Y2, Z2, W2, V2).
% 27.25/3.83 Axiom 35 (ruleD39): fresh106(eqangle(X, Y, Z, W, V, U, Z, W), true, X, Y, V, U) = para(X, Y, V, U).
% 27.25/3.83 Axiom 36 (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).
% 27.25/3.83 Axiom 37 (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).
% 27.25/3.83 Axiom 38 (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).
% 27.25/3.83 Axiom 39 (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).
% 27.25/3.83 Axiom 40 (ruleD22): fresh130(eqangle(X, Y, Z, W, V, U, T, S), true, X2, Y2, Z2, W2, X, Y, Z, W, V, U, T, S) = fresh129(eqangle(X2, Y2, Z2, W2, X, Y, Z, W), true, X2, Y2, Z2, W2, V, U, T, S).
% 27.25/3.83
% 27.25/3.83 Lemma 41: eqangle(a, c, X, Y, a, c, X, Y) = true.
% 27.25/3.83 Proof:
% 27.25/3.83 eqangle(a, c, X, Y, a, c, X, Y)
% 27.25/3.83 = { by axiom 32 (ruleD40) R->L }
% 27.25/3.83 fresh104(para(a, c, a, c), true, a, c, a, c, X, Y)
% 27.25/3.83 = { by axiom 21 (ruleD9) R->L }
% 27.25/3.83 fresh104(fresh51(true, true, a, c, f, p, a, c), true, a, c, a, c, X, Y)
% 27.25/3.83 = { by axiom 1 (exemplo6GDDFULL012009_6) R->L }
% 27.25/3.83 fresh104(fresh51(perp(f, p, a, c), true, a, c, f, p, a, c), true, a, c, a, c, X, Y)
% 27.25/3.83 = { by axiom 33 (ruleD9) }
% 27.25/3.83 fresh104(fresh50(perp(a, c, f, p), true, a, c, a, c), true, a, c, a, c, X, Y)
% 27.25/3.83 = { by axiom 29 (ruleD8) R->L }
% 27.25/3.83 fresh104(fresh50(fresh52(perp(f, p, a, c), true, f, p, a, c), true, a, c, a, c), true, a, c, a, c, X, Y)
% 27.25/3.83 = { by axiom 1 (exemplo6GDDFULL012009_6) }
% 27.25/3.83 fresh104(fresh50(fresh52(true, true, f, p, a, c), true, a, c, a, c), true, a, c, a, c, X, Y)
% 27.25/3.83 = { by axiom 14 (ruleD8) }
% 27.25/3.83 fresh104(fresh50(true, true, a, c, a, c), true, a, c, a, c, X, Y)
% 27.25/3.83 = { by axiom 15 (ruleD9) }
% 27.25/3.83 fresh104(true, true, a, c, a, c, X, Y)
% 27.25/3.83 = { by axiom 20 (ruleD40) }
% 27.25/3.83 true
% 27.25/3.83
% 27.25/3.83 Lemma 42: para(X, Y, X, Y) = true.
% 27.25/3.83 Proof:
% 27.25/3.83 para(X, Y, X, Y)
% 27.25/3.83 = { by axiom 35 (ruleD39) R->L }
% 27.25/3.83 fresh106(eqangle(X, Y, a, c, X, Y, a, c), true, X, Y, X, Y)
% 27.25/3.83 = { by axiom 38 (ruleD19) R->L }
% 27.25/3.83 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)
% 27.25/3.83 = { by lemma 41 }
% 27.25/3.83 fresh106(fresh134(true, true, a, c, X, Y, a, c, X, Y), true, X, Y, X, Y)
% 27.25/3.83 = { by axiom 25 (ruleD19) }
% 27.25/3.83 fresh106(true, true, X, Y, X, Y)
% 27.25/3.83 = { by axiom 9 (ruleD39) }
% 27.25/3.83 true
% 27.25/3.83
% 27.25/3.83 Lemma 43: coll(X, X, Y) = true.
% 27.25/3.83 Proof:
% 27.25/3.83 coll(X, X, Y)
% 27.25/3.83 = { by axiom 18 (ruleD1) R->L }
% 27.25/3.83 fresh146(coll(X, Y, X), true, X, Y, X)
% 27.25/3.83 = { by axiom 19 (ruleD2) R->L }
% 27.25/3.83 fresh146(fresh133(coll(Y, X, X), true, Y, X, X), true, X, Y, X)
% 27.25/3.83 = { by axiom 23 (ruleD66) R->L }
% 27.25/3.83 fresh146(fresh133(fresh66(para(Y, X, Y, X), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 27.25/3.83 = { by lemma 42 }
% 27.25/3.83 fresh146(fresh133(fresh66(true, true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 27.25/3.83 = { by axiom 5 (ruleD66) }
% 27.25/3.83 fresh146(fresh133(true, true, Y, X, X), true, X, Y, X)
% 27.25/3.83 = { by axiom 3 (ruleD2) }
% 27.25/3.83 fresh146(true, true, X, Y, X)
% 27.25/3.83 = { by axiom 2 (ruleD1) }
% 27.25/3.83 true
% 27.25/3.83
% 27.25/3.83 Lemma 44: cyclic(c, c, a, X) = true.
% 27.25/3.83 Proof:
% 27.25/3.83 cyclic(c, c, a, X)
% 27.25/3.83 = { by axiom 10 (ruleD42b) R->L }
% 27.25/3.83 fresh102(true, true, c, c, a, X)
% 27.25/3.83 = { by axiom 26 (ruleD21) R->L }
% 27.25/3.83 fresh102(fresh131(true, true, a, c, X, c, a, c, X, c), true, c, c, a, X)
% 27.25/3.83 = { by lemma 41 R->L }
% 27.25/3.83 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)
% 27.25/3.83 = { by axiom 39 (ruleD21) }
% 27.25/3.83 fresh102(eqangle(a, c, a, c, X, c, X, c), true, c, c, a, X)
% 27.25/3.83 = { by axiom 36 (ruleD42b) }
% 27.25/3.83 fresh101(coll(a, X, c), true, c, c, a, X)
% 27.25/3.83 = { by axiom 8 (ruleD3) R->L }
% 27.25/3.83 fresh101(fresh120(true, true, c, c, a, X), true, c, c, a, X)
% 27.25/3.83 = { by lemma 43 R->L }
% 27.25/3.83 fresh101(fresh120(coll(c, c, X), true, c, c, a, X), true, c, c, a, X)
% 27.25/3.83 = { by axiom 22 (ruleD3) }
% 27.25/3.83 fresh101(fresh119(coll(c, c, a), true, c, a, X), true, c, c, a, X)
% 27.25/3.83 = { by lemma 43 }
% 27.25/3.83 fresh101(fresh119(true, true, c, a, X), true, c, c, a, X)
% 27.25/3.83 = { by axiom 4 (ruleD3) }
% 27.25/3.83 fresh101(true, true, c, c, a, X)
% 27.25/3.83 = { by axiom 11 (ruleD42b) }
% 27.25/3.83 true
% 27.25/3.83
% 27.25/3.83 Lemma 45: cyclic(c, a, X, Y) = true.
% 27.25/3.83 Proof:
% 27.25/3.83 cyclic(c, a, X, Y)
% 27.25/3.83 = { by axiom 17 (ruleD17) R->L }
% 27.25/3.83 fresh137(true, true, c, c, a, X, Y)
% 27.25/3.83 = { by lemma 44 R->L }
% 27.25/3.83 fresh137(cyclic(c, c, a, Y), true, c, c, a, X, Y)
% 27.25/3.83 = { by axiom 31 (ruleD17) }
% 27.25/3.83 fresh136(cyclic(c, c, a, X), true, c, a, X, Y)
% 27.25/3.83 = { by lemma 44 }
% 27.25/3.83 fresh136(true, true, c, a, X, Y)
% 27.25/3.83 = { by axiom 7 (ruleD17) }
% 27.25/3.83 true
% 27.25/3.83
% 27.25/3.83 Lemma 46: cyclic(a, X, Y, Z) = true.
% 27.25/3.83 Proof:
% 27.25/3.83 cyclic(a, X, Y, Z)
% 27.25/3.83 = { by axiom 17 (ruleD17) R->L }
% 27.25/3.83 fresh137(true, true, c, a, X, Y, Z)
% 27.25/3.83 = { by lemma 45 R->L }
% 27.25/3.83 fresh137(cyclic(c, a, X, Z), true, c, a, X, Y, Z)
% 27.25/3.83 = { by axiom 31 (ruleD17) }
% 27.25/3.83 fresh136(cyclic(c, a, X, Y), true, a, X, Y, Z)
% 27.25/3.83 = { by lemma 45 }
% 27.25/3.83 fresh136(true, true, a, X, Y, Z)
% 27.25/3.83 = { by axiom 7 (ruleD17) }
% 27.25/3.83 true
% 27.25/3.83
% 27.25/3.83 Lemma 47: cyclic(X, Y, Z, W) = true.
% 27.25/3.83 Proof:
% 27.25/3.83 cyclic(X, Y, Z, W)
% 27.25/3.83 = { by axiom 17 (ruleD17) R->L }
% 27.25/3.83 fresh137(true, true, a, X, Y, Z, W)
% 27.25/3.83 = { by lemma 46 R->L }
% 27.25/3.83 fresh137(cyclic(a, X, Y, W), true, a, X, Y, Z, W)
% 27.25/3.83 = { by axiom 31 (ruleD17) }
% 27.25/3.83 fresh136(cyclic(a, X, Y, Z), true, X, Y, Z, W)
% 27.25/3.83 = { by lemma 46 }
% 27.25/3.83 fresh136(true, true, X, Y, Z, W)
% 27.25/3.83 = { by axiom 7 (ruleD17) }
% 27.25/3.83 true
% 27.25/3.83
% 27.25/3.83 Lemma 48: cong(X, Y, X, Y) = true.
% 27.25/3.83 Proof:
% 27.25/3.83 cong(X, Y, X, Y)
% 27.25/3.83 = { by axiom 16 (ruleD43) R->L }
% 27.25/3.83 fresh183(true, true, X, Y, Z, X, Y)
% 27.25/3.83 = { by lemma 47 R->L }
% 27.25/3.83 fresh183(cyclic(X, Y, Z, Y), true, X, Y, Z, X, Y)
% 27.25/3.83 = { by axiom 30 (ruleD43) R->L }
% 27.25/3.83 fresh182(true, true, X, Y, Z, X, Y, Z)
% 27.25/3.83 = { by axiom 20 (ruleD40) R->L }
% 27.25/3.83 fresh182(fresh104(true, true, Z, X, Z, X, Z, Y), true, X, Y, Z, X, Y, Z)
% 27.25/3.83 = { by lemma 42 R->L }
% 27.25/3.83 fresh182(fresh104(para(Z, X, Z, X), true, Z, X, Z, X, Z, Y), true, X, Y, Z, X, Y, Z)
% 27.25/3.83 = { by axiom 32 (ruleD40) }
% 27.25/3.83 fresh182(eqangle(Z, X, Z, Y, Z, X, Z, Y), true, X, Y, Z, X, Y, Z)
% 27.25/3.83 = { by axiom 37 (ruleD43) }
% 27.25/3.83 fresh184(cyclic(X, Y, Z, Z), true, X, Y, Z, X, Y)
% 27.25/3.83 = { by lemma 47 }
% 27.25/3.83 fresh184(true, true, X, Y, Z, X, Y)
% 27.25/3.83 = { by axiom 24 (ruleD43) }
% 27.25/3.83 fresh185(cyclic(X, Y, Z, X), true, X, Y, X, Y)
% 27.25/3.83 = { by lemma 47 }
% 27.25/3.83 fresh185(true, true, X, Y, X, Y)
% 27.25/3.83 = { by axiom 6 (ruleD43) }
% 27.25/3.83 true
% 27.25/3.83
% 27.25/3.83 Lemma 49: perp(X, X, Y, Z) = true.
% 27.25/3.83 Proof:
% 27.25/3.83 perp(X, X, Y, Z)
% 27.25/3.83 = { by axiom 12 (ruleD56) R->L }
% 27.25/3.83 fresh80(true, true, X, X, Y, Z)
% 27.25/3.83 = { by lemma 48 R->L }
% 27.25/3.83 fresh80(cong(X, Z, X, Z), true, X, X, Y, Z)
% 27.25/3.83 = { by axiom 28 (ruleD56) }
% 27.25/3.83 fresh79(cong(X, Y, X, Y), true, X, X, Y, Z)
% 27.25/3.83 = { by lemma 48 }
% 27.25/3.83 fresh79(true, true, X, X, Y, Z)
% 27.25/3.83 = { by axiom 13 (ruleD56) }
% 27.25/3.83 true
% 27.25/3.83
% 27.25/3.83 Lemma 50: eqangle(X, Y, Z, W, V, U, Z, W) = true.
% 27.25/3.83 Proof:
% 27.25/3.83 eqangle(X, Y, Z, W, V, U, Z, W)
% 27.25/3.83 = { by axiom 32 (ruleD40) R->L }
% 27.25/3.83 fresh104(para(X, Y, V, U), true, X, Y, V, U, Z, W)
% 27.25/3.83 = { by axiom 21 (ruleD9) R->L }
% 27.25/3.83 fresh104(fresh51(true, true, X, Y, T, T, V, U), true, X, Y, V, U, Z, W)
% 27.25/3.83 = { by lemma 49 R->L }
% 27.25/3.83 fresh104(fresh51(perp(T, T, V, U), true, X, Y, T, T, V, U), true, X, Y, V, U, Z, W)
% 27.25/3.83 = { by axiom 33 (ruleD9) }
% 27.25/3.83 fresh104(fresh50(perp(X, Y, T, T), true, X, Y, V, U), true, X, Y, V, U, Z, W)
% 27.25/3.83 = { by axiom 29 (ruleD8) R->L }
% 27.25/3.83 fresh104(fresh50(fresh52(perp(T, T, X, Y), true, T, T, X, Y), true, X, Y, V, U), true, X, Y, V, U, Z, W)
% 27.25/3.83 = { by lemma 49 }
% 27.25/3.83 fresh104(fresh50(fresh52(true, true, T, T, X, Y), true, X, Y, V, U), true, X, Y, V, U, Z, W)
% 27.25/3.83 = { by axiom 14 (ruleD8) }
% 27.25/3.83 fresh104(fresh50(true, true, X, Y, V, U), true, X, Y, V, U, Z, W)
% 27.25/3.83 = { by axiom 15 (ruleD9) }
% 27.25/3.83 fresh104(true, true, X, Y, V, U, Z, W)
% 27.25/3.83 = { by axiom 20 (ruleD40) }
% 27.25/3.83 true
% 27.25/3.83
% 27.25/3.83 Goal 1 (exemplo6GDDFULL012009_16): eqangle(f, l, l, f1, f2, m, m, f3) = true.
% 27.25/3.83 Proof:
% 27.25/3.83 eqangle(f, l, l, f1, f2, m, m, f3)
% 27.25/3.83 = { by axiom 34 (ruleD22) R->L }
% 27.25/3.83 fresh130(true, true, f, l, l, f1, X, Y, X, Y, f2, m, m, f3)
% 27.25/3.83 = { by axiom 26 (ruleD21) R->L }
% 27.25/3.83 fresh130(fresh131(true, true, X, Y, f2, m, X, Y, m, f3), true, f, l, l, f1, X, Y, X, Y, f2, m, m, f3)
% 27.25/3.83 = { by axiom 25 (ruleD19) R->L }
% 27.25/3.83 fresh130(fresh131(fresh134(true, true, f2, m, X, Y, m, f3, X, Y), true, X, Y, f2, m, X, Y, m, f3), true, f, l, l, f1, X, Y, X, Y, f2, m, m, f3)
% 27.25/3.83 = { by lemma 50 R->L }
% 27.25/3.83 fresh130(fresh131(fresh134(eqangle(f2, m, X, Y, m, f3, X, Y), true, f2, m, X, Y, m, f3, X, Y), true, X, Y, f2, m, X, Y, m, f3), true, f, l, l, f1, X, Y, X, Y, f2, m, m, f3)
% 27.25/3.83 = { by axiom 38 (ruleD19) }
% 27.25/3.83 fresh130(fresh131(eqangle(X, Y, f2, m, X, Y, m, f3), true, X, Y, f2, m, X, Y, m, f3), true, f, l, l, f1, X, Y, X, Y, f2, m, m, f3)
% 27.25/3.83 = { by axiom 39 (ruleD21) }
% 27.25/3.83 fresh130(eqangle(X, Y, X, Y, f2, m, m, f3), true, f, l, l, f1, X, Y, X, Y, f2, m, m, f3)
% 27.25/3.83 = { by axiom 40 (ruleD22) }
% 27.25/3.83 fresh129(eqangle(f, l, l, f1, X, Y, X, Y), true, f, l, l, f1, f2, m, m, f3)
% 27.25/3.83 = { by axiom 39 (ruleD21) R->L }
% 27.25/3.83 fresh129(fresh131(eqangle(f, l, X, Y, l, f1, X, Y), true, f, l, X, Y, l, f1, X, Y), true, f, l, l, f1, f2, m, m, f3)
% 27.25/3.83 = { by lemma 50 }
% 27.25/3.83 fresh129(fresh131(true, true, f, l, X, Y, l, f1, X, Y), true, f, l, l, f1, f2, m, m, f3)
% 27.25/3.83 = { by axiom 26 (ruleD21) }
% 27.25/3.83 fresh129(true, true, f, l, l, f1, f2, m, m, f3)
% 27.25/3.83 = { by axiom 27 (ruleD22) }
% 27.25/3.83 true
% 27.25/3.83 % SZS output end Proof
% 27.25/3.83
% 27.25/3.83 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------