TSTP Solution File: GEO589+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GEO589+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 : n006.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 23:29:26 EDT 2023
% Result : Theorem 5.66s 1.14s
% Output : Proof 6.41s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : GEO589+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
% 0.13/0.34 % Computer : n006.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Tue Aug 29 22:33:06 EDT 2023
% 0.13/0.35 % CPUTime :
% 5.66/1.14 Command-line arguments: --kbo-weight0 --lhs-weight 5 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --goal-heuristic
% 5.66/1.14
% 5.66/1.14 % SZS status Theorem
% 5.66/1.14
% 6.41/1.18 % SZS output start Proof
% 6.41/1.18 Take the following subset of the input axioms:
% 6.41/1.18 fof(exemplo6GDDFULL416051, conjecture, ![A, B, C, D, E, O, NWPNT1, NWPNT2]: ((circle(O, A, B, C) & (perp(A, C, B, D) & (circle(O, A, D, NWPNT1) & (circle(O, D, E, NWPNT2) & coll(E, D, O))))) => para(B, E, A, C))).
% 6.41/1.18 fof(ruleD1, axiom, ![A2, B2, C2]: (coll(A2, B2, C2) => coll(A2, C2, B2))).
% 6.41/1.18 fof(ruleD14, axiom, ![B2, C2, D2, A2_2]: (cyclic(A2_2, B2, C2, D2) => cyclic(A2_2, B2, D2, C2))).
% 6.41/1.18 fof(ruleD15, axiom, ![B2, C2, D2, A2_2]: (cyclic(A2_2, B2, C2, D2) => cyclic(A2_2, C2, B2, D2))).
% 6.41/1.18 fof(ruleD16, axiom, ![B2, C2, D2, A2_2]: (cyclic(A2_2, B2, C2, D2) => cyclic(B2, A2_2, C2, D2))).
% 6.41/1.18 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))).
% 6.41/1.18 fof(ruleD19, axiom, ![P, Q, U, V, B2, C2, D2, A2_2]: (eqangle(A2_2, B2, C2, D2, P, Q, U, V) => eqangle(C2, D2, A2_2, B2, U, V, P, Q))).
% 6.41/1.18 fof(ruleD2, axiom, ![B2, C2, A2_2]: (coll(A2_2, B2, C2) => coll(B2, A2_2, C2))).
% 6.41/1.18 fof(ruleD23, axiom, ![B2, C2, D2, A2_2]: (cong(A2_2, B2, C2, D2) => cong(A2_2, B2, D2, C2))).
% 6.41/1.18 fof(ruleD24, axiom, ![B2, C2, D2, A2_2]: (cong(A2_2, B2, C2, D2) => cong(C2, D2, A2_2, B2))).
% 6.41/1.18 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))).
% 6.41/1.18 fof(ruleD4, axiom, ![B2, C2, D2, A2_2]: (para(A2_2, B2, C2, D2) => para(A2_2, B2, D2, C2))).
% 6.41/1.18 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))).
% 6.41/1.18 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))).
% 6.41/1.18 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))).
% 6.41/1.18 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))).
% 6.41/1.18 fof(ruleD66, axiom, ![B2, C2, A2_2]: (para(A2_2, B2, A2_2, C2) => coll(A2_2, B2, C2))).
% 6.41/1.18 fof(ruleD7, axiom, ![B2, C2, D2, A2_2]: (perp(A2_2, B2, C2, D2) => perp(A2_2, B2, D2, C2))).
% 6.41/1.18 fof(ruleD8, axiom, ![B2, C2, D2, A2_2]: (perp(A2_2, B2, C2, D2) => perp(C2, D2, A2_2, B2))).
% 6.41/1.18 fof(ruleD9, axiom, ![F, B2, C2, D2, E2, A2_2]: ((perp(A2_2, B2, C2, D2) & perp(C2, D2, E2, F)) => para(A2_2, B2, E2, F))).
% 6.41/1.18
% 6.41/1.18 Now clausify the problem and encode Horn clauses using encoding 3 of
% 6.41/1.18 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 6.41/1.18 We repeatedly replace C & s=t => u=v by the two clauses:
% 6.41/1.18 fresh(y, y, x1...xn) = u
% 6.41/1.18 C => fresh(s, t, x1...xn) = v
% 6.41/1.18 where fresh is a fresh function symbol and x1..xn are the free
% 6.41/1.18 variables of u and v.
% 6.41/1.18 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 6.41/1.18 input problem has no model of domain size 1).
% 6.41/1.18
% 6.41/1.18 The encoding turns the above axioms into the following unit equations and goals:
% 6.41/1.18
% 6.41/1.18 Axiom 1 (exemplo6GDDFULL416051_1): perp(a, c, b, d) = true.
% 6.41/1.18 Axiom 2 (ruleD1): fresh146(X, X, Y, Z, W) = true.
% 6.41/1.18 Axiom 3 (ruleD2): fresh133(X, X, Y, Z, W) = true.
% 6.41/1.18 Axiom 4 (ruleD66): fresh66(X, X, Y, Z, W) = true.
% 6.41/1.18 Axiom 5 (ruleD43): fresh185(X, X, Y, Z, W, V) = true.
% 6.41/1.18 Axiom 6 (ruleD14): fresh140(X, X, Y, Z, W, V) = true.
% 6.41/1.18 Axiom 7 (ruleD15): fresh139(X, X, Y, Z, W, V) = true.
% 6.41/1.18 Axiom 8 (ruleD16): fresh138(X, X, Y, Z, W, V) = true.
% 6.41/1.18 Axiom 9 (ruleD17): fresh136(X, X, Y, Z, W, V) = true.
% 6.41/1.18 Axiom 10 (ruleD23): fresh128(X, X, Y, Z, W, V) = true.
% 6.41/1.18 Axiom 11 (ruleD24): fresh127(X, X, Y, Z, W, V) = true.
% 6.41/1.18 Axiom 12 (ruleD39): fresh106(X, X, Y, Z, W, V) = true.
% 6.41/1.18 Axiom 13 (ruleD4): fresh105(X, X, Y, Z, W, V) = true.
% 6.41/1.18 Axiom 14 (ruleD42b): fresh102(X, X, Y, Z, W, V) = cyclic(Y, Z, W, V).
% 6.41/1.18 Axiom 15 (ruleD42b): fresh101(X, X, Y, Z, W, V) = true.
% 6.41/1.18 Axiom 16 (ruleD56): fresh80(X, X, Y, Z, W, V) = perp(Y, Z, W, V).
% 6.41/1.18 Axiom 17 (ruleD56): fresh79(X, X, Y, Z, W, V) = true.
% 6.41/1.18 Axiom 18 (ruleD7): fresh61(X, X, Y, Z, W, V) = true.
% 6.41/1.18 Axiom 19 (ruleD8): fresh52(X, X, Y, Z, W, V) = true.
% 6.41/1.18 Axiom 20 (ruleD9): fresh50(X, X, Y, Z, W, V) = true.
% 6.41/1.18 Axiom 21 (ruleD43): fresh183(X, X, Y, Z, W, V, U) = cong(Y, Z, V, U).
% 6.41/1.18 Axiom 22 (ruleD1): fresh146(coll(X, Y, Z), true, X, Y, Z) = coll(X, Z, Y).
% 6.41/1.18 Axiom 23 (ruleD17): fresh137(X, X, Y, Z, W, V, U) = cyclic(Z, W, V, U).
% 6.41/1.18 Axiom 24 (ruleD2): fresh133(coll(X, Y, Z), true, X, Y, Z) = coll(Y, X, Z).
% 6.41/1.18 Axiom 25 (ruleD40): fresh104(X, X, Y, Z, W, V, U, T) = true.
% 6.41/1.18 Axiom 26 (ruleD66): fresh66(para(X, Y, X, Z), true, X, Y, Z) = coll(X, Y, Z).
% 6.41/1.18 Axiom 27 (ruleD9): fresh51(X, X, Y, Z, W, V, U, T) = para(Y, Z, U, T).
% 6.41/1.18 Axiom 28 (ruleD43): fresh184(X, X, Y, Z, W, V, U) = fresh185(cyclic(Y, Z, W, V), true, Y, Z, V, U).
% 6.41/1.18 Axiom 29 (ruleD14): fresh140(cyclic(X, Y, Z, W), true, X, Y, Z, W) = cyclic(X, Y, W, Z).
% 6.41/1.18 Axiom 30 (ruleD15): fresh139(cyclic(X, Y, Z, W), true, X, Y, Z, W) = cyclic(X, Z, Y, W).
% 6.41/1.18 Axiom 31 (ruleD16): fresh138(cyclic(X, Y, Z, W), true, X, Y, Z, W) = cyclic(Y, X, Z, W).
% 6.41/1.18 Axiom 32 (ruleD23): fresh128(cong(X, Y, Z, W), true, X, Y, Z, W) = cong(X, Y, W, Z).
% 6.41/1.18 Axiom 33 (ruleD24): fresh127(cong(X, Y, Z, W), true, X, Y, Z, W) = cong(Z, W, X, Y).
% 6.41/1.18 Axiom 34 (ruleD4): fresh105(para(X, Y, Z, W), true, X, Y, Z, W) = para(X, Y, W, Z).
% 6.41/1.18 Axiom 35 (ruleD56): fresh80(cong(X, Y, Z, Y), true, X, Z, W, Y) = fresh79(cong(X, W, Z, W), true, X, Z, W, Y).
% 6.41/1.18 Axiom 36 (ruleD7): fresh61(perp(X, Y, Z, W), true, X, Y, Z, W) = perp(X, Y, W, Z).
% 6.41/1.18 Axiom 37 (ruleD8): fresh52(perp(X, Y, Z, W), true, X, Y, Z, W) = perp(Z, W, X, Y).
% 6.41/1.18 Axiom 38 (ruleD43): fresh182(X, X, Y, Z, W, V, U, T) = fresh183(cyclic(Y, Z, W, U), true, Y, Z, W, V, U).
% 6.41/1.18 Axiom 39 (ruleD17): fresh137(cyclic(X, Y, Z, W), true, X, Y, Z, V, W) = fresh136(cyclic(X, Y, Z, V), true, Y, Z, V, W).
% 6.41/1.18 Axiom 40 (ruleD19): fresh134(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 6.41/1.18 Axiom 41 (ruleD40): fresh104(para(X, Y, Z, W), true, X, Y, Z, W, V, U) = eqangle(X, Y, V, U, Z, W, V, U).
% 6.41/1.18 Axiom 42 (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).
% 6.41/1.18 Axiom 43 (ruleD39): fresh106(eqangle(X, Y, Z, W, V, U, Z, W), true, X, Y, V, U) = para(X, Y, V, U).
% 6.41/1.18 Axiom 44 (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).
% 6.41/1.18 Axiom 45 (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).
% 6.41/1.18 Axiom 46 (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).
% 6.41/1.18
% 6.41/1.18 Lemma 47: fresh50(perp(X, Y, a, c), true, X, Y, b, d) = para(X, Y, b, d).
% 6.41/1.18 Proof:
% 6.41/1.18 fresh50(perp(X, Y, a, c), true, X, Y, b, d)
% 6.41/1.18 = { by axiom 42 (ruleD9) R->L }
% 6.41/1.18 fresh51(perp(a, c, b, d), true, X, Y, a, c, b, d)
% 6.41/1.18 = { by axiom 1 (exemplo6GDDFULL416051_1) }
% 6.41/1.18 fresh51(true, true, X, Y, a, c, b, d)
% 6.41/1.18 = { by axiom 27 (ruleD9) }
% 6.41/1.18 para(X, Y, b, d)
% 6.41/1.18
% 6.41/1.18 Lemma 48: eqangle(X, Y, b, d, X, Y, b, d) = true.
% 6.41/1.19 Proof:
% 6.41/1.19 eqangle(X, Y, b, d, X, Y, b, d)
% 6.41/1.19 = { by axiom 46 (ruleD19) R->L }
% 6.41/1.19 fresh134(eqangle(b, d, X, Y, b, d, X, Y), true, b, d, X, Y, b, d, X, Y)
% 6.41/1.19 = { by axiom 41 (ruleD40) R->L }
% 6.41/1.19 fresh134(fresh104(para(b, d, b, d), true, b, d, b, d, X, Y), true, b, d, X, Y, b, d, X, Y)
% 6.41/1.19 = { by lemma 47 R->L }
% 6.41/1.19 fresh134(fresh104(fresh50(perp(b, d, a, c), true, b, d, b, d), true, b, d, b, d, X, Y), true, b, d, X, Y, b, d, X, Y)
% 6.41/1.19 = { by axiom 37 (ruleD8) R->L }
% 6.41/1.19 fresh134(fresh104(fresh50(fresh52(perp(a, c, b, d), true, a, c, b, d), true, b, d, b, d), true, b, d, b, d, X, Y), true, b, d, X, Y, b, d, X, Y)
% 6.41/1.19 = { by axiom 1 (exemplo6GDDFULL416051_1) }
% 6.41/1.19 fresh134(fresh104(fresh50(fresh52(true, true, a, c, b, d), true, b, d, b, d), true, b, d, b, d, X, Y), true, b, d, X, Y, b, d, X, Y)
% 6.41/1.19 = { by axiom 19 (ruleD8) }
% 6.41/1.19 fresh134(fresh104(fresh50(true, true, b, d, b, d), true, b, d, b, d, X, Y), true, b, d, X, Y, b, d, X, Y)
% 6.41/1.19 = { by axiom 20 (ruleD9) }
% 6.41/1.19 fresh134(fresh104(true, true, b, d, b, d, X, Y), true, b, d, X, Y, b, d, X, Y)
% 6.41/1.19 = { by axiom 25 (ruleD40) }
% 6.41/1.19 fresh134(true, true, b, d, X, Y, b, d, X, Y)
% 6.41/1.19 = { by axiom 40 (ruleD19) }
% 6.41/1.19 true
% 6.41/1.19
% 6.41/1.19 Lemma 49: cyclic(X, d, b, b) = true.
% 6.41/1.19 Proof:
% 6.41/1.19 cyclic(X, d, b, b)
% 6.41/1.19 = { by axiom 14 (ruleD42b) R->L }
% 6.41/1.19 fresh102(true, true, X, d, b, b)
% 6.41/1.19 = { by lemma 48 R->L }
% 6.41/1.19 fresh102(eqangle(b, X, b, d, b, X, b, d), true, X, d, b, b)
% 6.41/1.19 = { by axiom 44 (ruleD42b) }
% 6.41/1.19 fresh101(coll(b, b, d), true, X, d, b, b)
% 6.41/1.19 = { by axiom 22 (ruleD1) R->L }
% 6.41/1.19 fresh101(fresh146(coll(b, d, b), true, b, d, b), true, X, d, b, b)
% 6.41/1.19 = { by axiom 24 (ruleD2) R->L }
% 6.41/1.19 fresh101(fresh146(fresh133(coll(d, b, b), true, d, b, b), true, b, d, b), true, X, d, b, b)
% 6.41/1.19 = { by axiom 26 (ruleD66) R->L }
% 6.41/1.19 fresh101(fresh146(fresh133(fresh66(para(d, b, d, b), true, d, b, b), true, d, b, b), true, b, d, b), true, X, d, b, b)
% 6.41/1.19 = { by axiom 34 (ruleD4) R->L }
% 6.41/1.19 fresh101(fresh146(fresh133(fresh66(fresh105(para(d, b, b, d), true, d, b, b, d), true, d, b, b), true, d, b, b), true, b, d, b), true, X, d, b, b)
% 6.41/1.19 = { by lemma 47 R->L }
% 6.41/1.19 fresh101(fresh146(fresh133(fresh66(fresh105(fresh50(perp(d, b, a, c), true, d, b, b, d), true, d, b, b, d), true, d, b, b), true, d, b, b), true, b, d, b), true, X, d, b, b)
% 6.41/1.19 = { by axiom 37 (ruleD8) R->L }
% 6.41/1.19 fresh101(fresh146(fresh133(fresh66(fresh105(fresh50(fresh52(perp(a, c, d, b), true, a, c, d, b), true, d, b, b, d), true, d, b, b, d), true, d, b, b), true, d, b, b), true, b, d, b), true, X, d, b, b)
% 6.41/1.19 = { by axiom 36 (ruleD7) R->L }
% 6.41/1.19 fresh101(fresh146(fresh133(fresh66(fresh105(fresh50(fresh52(fresh61(perp(a, c, b, d), true, a, c, b, d), true, a, c, d, b), true, d, b, b, d), true, d, b, b, d), true, d, b, b), true, d, b, b), true, b, d, b), true, X, d, b, b)
% 6.41/1.19 = { by axiom 1 (exemplo6GDDFULL416051_1) }
% 6.41/1.19 fresh101(fresh146(fresh133(fresh66(fresh105(fresh50(fresh52(fresh61(true, true, a, c, b, d), true, a, c, d, b), true, d, b, b, d), true, d, b, b, d), true, d, b, b), true, d, b, b), true, b, d, b), true, X, d, b, b)
% 6.41/1.19 = { by axiom 18 (ruleD7) }
% 6.41/1.19 fresh101(fresh146(fresh133(fresh66(fresh105(fresh50(fresh52(true, true, a, c, d, b), true, d, b, b, d), true, d, b, b, d), true, d, b, b), true, d, b, b), true, b, d, b), true, X, d, b, b)
% 6.41/1.19 = { by axiom 19 (ruleD8) }
% 6.41/1.19 fresh101(fresh146(fresh133(fresh66(fresh105(fresh50(true, true, d, b, b, d), true, d, b, b, d), true, d, b, b), true, d, b, b), true, b, d, b), true, X, d, b, b)
% 6.41/1.19 = { by axiom 20 (ruleD9) }
% 6.41/1.19 fresh101(fresh146(fresh133(fresh66(fresh105(true, true, d, b, b, d), true, d, b, b), true, d, b, b), true, b, d, b), true, X, d, b, b)
% 6.41/1.19 = { by axiom 13 (ruleD4) }
% 6.41/1.19 fresh101(fresh146(fresh133(fresh66(true, true, d, b, b), true, d, b, b), true, b, d, b), true, X, d, b, b)
% 6.41/1.19 = { by axiom 4 (ruleD66) }
% 6.41/1.19 fresh101(fresh146(fresh133(true, true, d, b, b), true, b, d, b), true, X, d, b, b)
% 6.41/1.19 = { by axiom 3 (ruleD2) }
% 6.41/1.19 fresh101(fresh146(true, true, b, d, b), true, X, d, b, b)
% 6.41/1.19 = { by axiom 2 (ruleD1) }
% 6.41/1.19 fresh101(true, true, X, d, b, b)
% 6.41/1.19 = { by axiom 15 (ruleD42b) }
% 6.41/1.19 true
% 6.41/1.19
% 6.41/1.19 Lemma 50: cyclic(d, b, b, X) = true.
% 6.41/1.19 Proof:
% 6.41/1.19 cyclic(d, b, b, X)
% 6.41/1.19 = { by axiom 29 (ruleD14) R->L }
% 6.41/1.19 fresh140(cyclic(d, b, X, b), true, d, b, X, b)
% 6.41/1.19 = { by axiom 30 (ruleD15) R->L }
% 6.41/1.19 fresh140(fresh139(cyclic(d, X, b, b), true, d, X, b, b), true, d, b, X, b)
% 6.41/1.19 = { by axiom 31 (ruleD16) R->L }
% 6.41/1.19 fresh140(fresh139(fresh138(cyclic(X, d, b, b), true, X, d, b, b), true, d, X, b, b), true, d, b, X, b)
% 6.41/1.19 = { by lemma 49 }
% 6.41/1.19 fresh140(fresh139(fresh138(true, true, X, d, b, b), true, d, X, b, b), true, d, b, X, b)
% 6.41/1.19 = { by axiom 8 (ruleD16) }
% 6.41/1.19 fresh140(fresh139(true, true, d, X, b, b), true, d, b, X, b)
% 6.41/1.19 = { by axiom 7 (ruleD15) }
% 6.41/1.19 fresh140(true, true, d, b, X, b)
% 6.41/1.19 = { by axiom 6 (ruleD14) }
% 6.41/1.19 true
% 6.41/1.19
% 6.41/1.19 Lemma 51: cyclic(b, b, X, Y) = true.
% 6.41/1.19 Proof:
% 6.41/1.19 cyclic(b, b, X, Y)
% 6.41/1.19 = { by axiom 23 (ruleD17) R->L }
% 6.41/1.19 fresh137(true, true, d, b, b, X, Y)
% 6.41/1.19 = { by lemma 50 R->L }
% 6.41/1.19 fresh137(cyclic(d, b, b, Y), true, d, b, b, X, Y)
% 6.41/1.19 = { by axiom 39 (ruleD17) }
% 6.41/1.19 fresh136(cyclic(d, b, b, X), true, b, b, X, Y)
% 6.41/1.19 = { by lemma 50 }
% 6.41/1.19 fresh136(true, true, b, b, X, Y)
% 6.41/1.19 = { by axiom 9 (ruleD17) }
% 6.41/1.19 true
% 6.41/1.19
% 6.41/1.19 Lemma 52: cyclic(b, X, Y, Z) = true.
% 6.41/1.19 Proof:
% 6.41/1.19 cyclic(b, X, Y, Z)
% 6.41/1.19 = { by axiom 23 (ruleD17) R->L }
% 6.41/1.19 fresh137(true, true, b, b, X, Y, Z)
% 6.41/1.19 = { by lemma 51 R->L }
% 6.41/1.19 fresh137(cyclic(b, b, X, Z), true, b, b, X, Y, Z)
% 6.41/1.19 = { by axiom 39 (ruleD17) }
% 6.41/1.19 fresh136(cyclic(b, b, X, Y), true, b, X, Y, Z)
% 6.41/1.19 = { by lemma 51 }
% 6.41/1.19 fresh136(true, true, b, X, Y, Z)
% 6.41/1.19 = { by axiom 9 (ruleD17) }
% 6.41/1.19 true
% 6.41/1.19
% 6.41/1.19 Lemma 53: cyclic(X, Y, Z, W) = true.
% 6.41/1.19 Proof:
% 6.41/1.19 cyclic(X, Y, Z, W)
% 6.41/1.19 = { by axiom 23 (ruleD17) R->L }
% 6.41/1.19 fresh137(true, true, b, X, Y, Z, W)
% 6.41/1.19 = { by lemma 52 R->L }
% 6.41/1.19 fresh137(cyclic(b, X, Y, W), true, b, X, Y, Z, W)
% 6.41/1.19 = { by axiom 39 (ruleD17) }
% 6.41/1.19 fresh136(cyclic(b, X, Y, Z), true, X, Y, Z, W)
% 6.41/1.19 = { by lemma 52 }
% 6.41/1.19 fresh136(true, true, X, Y, Z, W)
% 6.41/1.19 = { by axiom 9 (ruleD17) }
% 6.41/1.19 true
% 6.41/1.19
% 6.41/1.19 Lemma 54: cong(b, X, b, X) = true.
% 6.41/1.19 Proof:
% 6.41/1.19 cong(b, X, b, X)
% 6.41/1.19 = { by axiom 32 (ruleD23) R->L }
% 6.41/1.19 fresh128(cong(b, X, X, b), true, b, X, X, b)
% 6.41/1.19 = { by axiom 33 (ruleD24) R->L }
% 6.41/1.19 fresh128(fresh127(cong(X, b, b, X), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by axiom 32 (ruleD23) R->L }
% 6.41/1.19 fresh128(fresh127(fresh128(cong(X, b, X, b), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by axiom 21 (ruleD43) R->L }
% 6.41/1.19 fresh128(fresh127(fresh128(fresh183(true, true, X, b, d, X, b), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by axiom 7 (ruleD15) R->L }
% 6.41/1.19 fresh128(fresh127(fresh128(fresh183(fresh139(true, true, X, d, b, b), true, X, b, d, X, b), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by lemma 49 R->L }
% 6.41/1.19 fresh128(fresh127(fresh128(fresh183(fresh139(cyclic(X, d, b, b), true, X, d, b, b), true, X, b, d, X, b), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by axiom 30 (ruleD15) }
% 6.41/1.19 fresh128(fresh127(fresh128(fresh183(cyclic(X, b, d, b), true, X, b, d, X, b), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by axiom 38 (ruleD43) R->L }
% 6.41/1.19 fresh128(fresh127(fresh128(fresh182(true, true, X, b, d, X, b, d), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by axiom 25 (ruleD40) R->L }
% 6.41/1.19 fresh128(fresh127(fresh128(fresh182(fresh104(true, true, d, X, d, X, d, b), true, X, b, d, X, b, d), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by axiom 12 (ruleD39) R->L }
% 6.41/1.19 fresh128(fresh127(fresh128(fresh182(fresh104(fresh106(true, true, d, X, d, X), true, d, X, d, X, d, b), true, X, b, d, X, b, d), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by lemma 48 R->L }
% 6.41/1.19 fresh128(fresh127(fresh128(fresh182(fresh104(fresh106(eqangle(d, X, b, d, d, X, b, d), true, d, X, d, X), true, d, X, d, X, d, b), true, X, b, d, X, b, d), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by axiom 43 (ruleD39) }
% 6.41/1.19 fresh128(fresh127(fresh128(fresh182(fresh104(para(d, X, d, X), true, d, X, d, X, d, b), true, X, b, d, X, b, d), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by axiom 41 (ruleD40) }
% 6.41/1.19 fresh128(fresh127(fresh128(fresh182(eqangle(d, X, d, b, d, X, d, b), true, X, b, d, X, b, d), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by axiom 45 (ruleD43) }
% 6.41/1.19 fresh128(fresh127(fresh128(fresh184(cyclic(X, b, d, d), true, X, b, d, X, b), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by lemma 53 }
% 6.41/1.19 fresh128(fresh127(fresh128(fresh184(true, true, X, b, d, X, b), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by axiom 28 (ruleD43) }
% 6.41/1.19 fresh128(fresh127(fresh128(fresh185(cyclic(X, b, d, X), true, X, b, X, b), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by lemma 53 }
% 6.41/1.19 fresh128(fresh127(fresh128(fresh185(true, true, X, b, X, b), true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by axiom 5 (ruleD43) }
% 6.41/1.19 fresh128(fresh127(fresh128(true, true, X, b, X, b), true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by axiom 10 (ruleD23) }
% 6.41/1.19 fresh128(fresh127(true, true, X, b, b, X), true, b, X, X, b)
% 6.41/1.19 = { by axiom 11 (ruleD24) }
% 6.41/1.19 fresh128(true, true, b, X, X, b)
% 6.41/1.19 = { by axiom 10 (ruleD23) }
% 6.41/1.19 true
% 6.41/1.19
% 6.41/1.19 Lemma 55: perp(b, b, X, Y) = true.
% 6.41/1.19 Proof:
% 6.41/1.19 perp(b, b, X, Y)
% 6.41/1.19 = { by axiom 16 (ruleD56) R->L }
% 6.41/1.19 fresh80(true, true, b, b, X, Y)
% 6.41/1.19 = { by lemma 54 R->L }
% 6.41/1.19 fresh80(cong(b, Y, b, Y), true, b, b, X, Y)
% 6.41/1.19 = { by axiom 35 (ruleD56) }
% 6.41/1.19 fresh79(cong(b, X, b, X), true, b, b, X, Y)
% 6.41/1.19 = { by lemma 54 }
% 6.41/1.19 fresh79(true, true, b, b, X, Y)
% 6.41/1.19 = { by axiom 17 (ruleD56) }
% 6.41/1.19 true
% 6.41/1.19
% 6.41/1.19 Goal 1 (exemplo6GDDFULL416051_5): para(b, e, a, c) = true.
% 6.41/1.19 Proof:
% 6.41/1.19 para(b, e, a, c)
% 6.41/1.19 = { by axiom 27 (ruleD9) R->L }
% 6.41/1.19 fresh51(true, true, b, e, b, b, a, c)
% 6.41/1.19 = { by lemma 55 R->L }
% 6.41/1.19 fresh51(perp(b, b, a, c), true, b, e, b, b, a, c)
% 6.41/1.19 = { by axiom 42 (ruleD9) }
% 6.41/1.19 fresh50(perp(b, e, b, b), true, b, e, a, c)
% 6.41/1.19 = { by axiom 37 (ruleD8) R->L }
% 6.41/1.19 fresh50(fresh52(perp(b, b, b, e), true, b, b, b, e), true, b, e, a, c)
% 6.41/1.19 = { by lemma 55 }
% 6.41/1.19 fresh50(fresh52(true, true, b, b, b, e), true, b, e, a, c)
% 6.41/1.19 = { by axiom 19 (ruleD8) }
% 6.41/1.19 fresh50(true, true, b, e, a, c)
% 6.41/1.19 = { by axiom 20 (ruleD9) }
% 6.41/1.19 true
% 6.41/1.19 % SZS output end Proof
% 6.41/1.19
% 6.41/1.19 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------