TSTP Solution File: GEO564+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GEO564+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 : n025.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 23:29:20 EDT 2023
% Result : Theorem 174.63s 22.98s
% Output : Proof 174.63s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GEO564+1 : TPTP v8.1.2. Released v7.5.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n025.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Tue Aug 29 21:15:40 EDT 2023
% 0.14/0.34 % CPUTime :
% 174.63/22.98 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 174.63/22.98
% 174.63/22.98 % SZS status Theorem
% 174.63/22.98
% 174.63/23.01 % SZS output start Proof
% 174.63/23.01 Take the following subset of the input axioms:
% 174.63/23.03 fof(exemplo6GDDFULL214026, conjecture, ![A, B, C, O, H, OC, OA, OB, MIDPNT1, MIDPNT2, MIDPNT3, MIDPNT4, MIDPNT5, MIDPNT6, MIDPNT7, MIDPNT8, MIDPNT9, MIDPNT01, MIDPNT11, MIDPNT21]: ((perp(A, B, C, H) & (perp(A, C, B, H) & (perp(B, C, A, H) & (midp(MIDPNT1, A, B) & (perp(A, B, MIDPNT1, O) & (midp(MIDPNT2, A, C) & (perp(A, C, MIDPNT2, O) & (midp(MIDPNT3, B, C) & (perp(B, C, MIDPNT3, O) & (midp(MIDPNT4, A, H) & (perp(A, H, MIDPNT4, OC) & (midp(MIDPNT5, A, B) & (perp(A, B, MIDPNT5, OC) & (midp(MIDPNT6, H, B) & (perp(H, B, MIDPNT6, OC) & (midp(MIDPNT7, B, H) & (perp(B, H, MIDPNT7, OA) & (midp(MIDPNT8, B, C) & (perp(B, C, MIDPNT8, OA) & (midp(MIDPNT9, H, C) & (perp(H, C, MIDPNT9, OA) & (midp(MIDPNT01, C, H) & (perp(C, H, MIDPNT01, OB) & (midp(MIDPNT11, C, A) & (perp(C, A, MIDPNT11, OB) & (midp(MIDPNT21, H, A) & perp(H, A, MIDPNT21, OB))))))))))))))))))))))))))) => (eqangle(B, A, A, C, OB, OA, OA, OC) & ((eqangle(A, B, B, C, OA, OB, OB, OC) | eqangle(B, A, A, C, OB, OA, OA, OC)) & ((eqangle(A, B, B, C, OA, OC, OC, OB) | eqangle(B, A, A, C, OA, OB, OB, OC)) & ((eqangle(A, B, B, C, OB, OA, OA, OC) | eqangle(B, A, A, C, OA, OB, OB, OC)) & ((eqangle(A, B, B, C, OA, OC, OC, OB) | eqangle(B, A, A, C, OA, OC, OC, OB)) & ((eqangle(A, B, B, C, OB, OA, OA, OC) | eqangle(B, A, A, C, OA, OC, OC, OB)) & eqangle(A, B, B, C, OA, OB, OB, OC))))))))).
% 174.63/23.03 fof(ruleD1, axiom, ![A2, B2, C2]: (coll(A2, B2, C2) => coll(A2, C2, B2))).
% 174.63/23.03 fof(ruleD17, axiom, ![D, E, B2, C2, A2_2]: ((cyclic(A2_2, B2, C2, D) & cyclic(A2_2, B2, C2, E)) => cyclic(B2, C2, D, E))).
% 174.63/23.03 fof(ruleD18, axiom, ![P, Q, U, V, B2, C2, D2, A2_2]: (eqangle(A2_2, B2, C2, D2, P, Q, U, V) => eqangle(B2, A2_2, C2, D2, P, Q, U, V))).
% 174.63/23.03 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))).
% 174.63/23.03 fof(ruleD2, axiom, ![B2, C2, A2_2]: (coll(A2_2, B2, C2) => coll(B2, A2_2, C2))).
% 174.63/23.03 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))).
% 174.63/23.03 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))).
% 174.63/23.03 fof(ruleD22, axiom, ![F, G, B2, C2, D2, E2, A2_2, P2, Q2, U2, V2, H2]: ((eqangle(A2_2, B2, C2, D2, P2, Q2, U2, V2) & eqangle(P2, Q2, U2, V2, E2, F, G, H2)) => eqangle(A2_2, B2, C2, D2, E2, F, G, H2))).
% 174.63/23.03 fof(ruleD3, axiom, ![B2, C2, D2, A2_2]: ((coll(A2_2, B2, C2) & coll(A2_2, B2, D2)) => coll(C2, D2, A2_2))).
% 174.63/23.03 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))).
% 174.63/23.03 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))).
% 174.63/23.03 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))).
% 174.63/23.03 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))).
% 174.63/23.03 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))).
% 174.63/23.03 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))).
% 174.63/23.03 fof(ruleD66, axiom, ![B2, C2, A2_2]: (para(A2_2, B2, A2_2, C2) => coll(A2_2, B2, C2))).
% 174.63/23.03 fof(ruleD68, axiom, ![B2, C2, A2_2]: (midp(A2_2, B2, C2) => cong(A2_2, B2, A2_2, C2))).
% 174.63/23.03 fof(ruleD8, axiom, ![B2, C2, D2, A2_2]: (perp(A2_2, B2, C2, D2) => perp(C2, D2, A2_2, B2))).
% 174.63/23.03 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))).
% 174.63/23.03
% 174.63/23.03 Now clausify the problem and encode Horn clauses using encoding 3 of
% 174.63/23.03 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 174.63/23.03 We repeatedly replace C & s=t => u=v by the two clauses:
% 174.63/23.03 fresh(y, y, x1...xn) = u
% 174.63/23.03 C => fresh(s, t, x1...xn) = v
% 174.63/23.03 where fresh is a fresh function symbol and x1..xn are the free
% 174.63/23.03 variables of u and v.
% 174.63/23.03 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 174.63/23.03 input problem has no model of domain size 1).
% 174.63/23.03
% 174.63/23.03 The encoding turns the above axioms into the following unit equations and goals:
% 174.63/23.03
% 174.63/23.03 Axiom 1 (exemplo6GDDFULL214026_30): fresh161(X, X) = true.
% 174.63/23.03 Axiom 2 (exemplo6GDDFULL214026_31): fresh159(X, X) = true.
% 174.63/23.03 Axiom 3 (exemplo6GDDFULL214026_30): fresh151(X, X) = or2.
% 174.63/23.03 Axiom 4 (exemplo6GDDFULL214026_31): fresh150(X, X) = or.
% 174.63/23.03 Axiom 5 (exemplo6GDDFULL214026_15): midp(midpnt1, a, b) = true.
% 174.63/23.03 Axiom 6 (ruleD1): fresh147(X, X, Y, Z, W) = true.
% 174.63/23.03 Axiom 7 (ruleD2): fresh133(X, X, Y, Z, W) = true.
% 174.63/23.03 Axiom 8 (ruleD3): fresh119(X, X, Y, Z, W) = true.
% 174.63/23.03 Axiom 9 (ruleD46): fresh97(X, X, Y, Z, W) = true.
% 174.63/23.03 Axiom 10 (ruleD66): fresh66(X, X, Y, Z, W) = true.
% 174.63/23.03 Axiom 11 (ruleD68): fresh63(X, X, Y, Z, W) = true.
% 174.63/23.03 Axiom 12 (ruleD43): fresh203(X, X, Y, Z, W, V) = true.
% 174.63/23.03 Axiom 13 (ruleD17): fresh136(X, X, Y, Z, W, V) = true.
% 174.63/23.03 Axiom 14 (ruleD3): fresh120(X, X, Y, Z, W, V) = coll(W, V, Y).
% 174.63/23.03 Axiom 15 (ruleD39): fresh106(X, X, Y, Z, W, V) = true.
% 174.63/23.03 Axiom 16 (ruleD42b): fresh102(X, X, Y, Z, W, V) = cyclic(Y, Z, W, V).
% 174.63/23.03 Axiom 17 (ruleD42b): fresh101(X, X, Y, Z, W, V) = true.
% 174.63/23.03 Axiom 18 (ruleD56): fresh80(X, X, Y, Z, W, V) = perp(Y, Z, W, V).
% 174.63/23.03 Axiom 19 (ruleD56): fresh79(X, X, Y, Z, W, V) = true.
% 174.63/23.03 Axiom 20 (ruleD8): fresh52(X, X, Y, Z, W, V) = true.
% 174.63/23.03 Axiom 21 (ruleD9): fresh50(X, X, Y, Z, W, V) = true.
% 174.63/23.03 Axiom 22 (ruleD43): fresh201(X, X, Y, Z, W, V, U) = cong(Y, Z, V, U).
% 174.63/23.03 Axiom 23 (ruleD17): fresh137(X, X, Y, Z, W, V, U) = cyclic(Z, W, V, U).
% 174.63/23.03 Axiom 24 (ruleD1): fresh147(coll(X, Y, Z), true, X, Y, Z) = coll(X, Z, Y).
% 174.63/23.03 Axiom 25 (ruleD2): fresh133(coll(X, Y, Z), true, X, Y, Z) = coll(Y, X, Z).
% 174.63/23.03 Axiom 26 (ruleD40): fresh104(X, X, Y, Z, W, V, U, T) = true.
% 174.63/23.03 Axiom 27 (ruleD68): fresh63(midp(X, Y, Z), true, X, Y, Z) = cong(X, Y, X, Z).
% 174.63/23.03 Axiom 28 (ruleD9): fresh51(X, X, Y, Z, W, V, U, T) = para(Y, Z, U, T).
% 174.63/23.03 Axiom 29 (ruleD3): fresh120(coll(X, Y, Z), true, X, Y, W, Z) = fresh119(coll(X, Y, W), true, X, W, Z).
% 174.63/23.03 Axiom 30 (ruleD46): fresh97(cong(X, Y, X, Z), true, Y, Z, X) = eqangle(X, Y, Y, Z, Y, Z, X, Z).
% 174.63/23.03 Axiom 31 (ruleD66): fresh66(para(X, Y, X, Z), true, X, Y, Z) = coll(X, Y, Z).
% 174.63/23.03 Axiom 32 (ruleD43): fresh202(X, X, Y, Z, W, V, U) = fresh203(cyclic(Y, Z, W, V), true, Y, Z, V, U).
% 174.63/23.03 Axiom 33 (exemplo6GDDFULL214026_30): fresh160(X, X) = fresh161(eqangle(b, a, a, c, oa, ob, ob, oc), true).
% 174.63/23.03 Axiom 34 (exemplo6GDDFULL214026_31): fresh158(X, X) = fresh159(eqangle(b, a, a, c, oa, ob, ob, oc), true).
% 174.63/23.03 Axiom 35 (exemplo6GDDFULL214026_30): fresh160(eqangle(a, b, b, c, oa, ob, ob, oc), true) = fresh151(eqangle(b, a, a, c, ob, oa, oa, oc), true).
% 174.63/23.03 Axiom 36 (exemplo6GDDFULL214026_31): fresh158(or2, true) = fresh150(eqangle(a, b, b, c, oa, oc, oc, ob), true).
% 174.63/23.03 Axiom 37 (ruleD18): fresh135(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 174.63/23.03 Axiom 38 (ruleD19): fresh134(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 174.63/23.03 Axiom 39 (ruleD20): fresh132(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 174.63/23.03 Axiom 40 (ruleD21): fresh131(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 174.63/23.03 Axiom 41 (ruleD22): fresh129(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 174.63/23.03 Axiom 42 (ruleD56): fresh80(cong(X, Y, Z, Y), true, X, Z, W, Y) = fresh79(cong(X, W, Z, W), true, X, Z, W, Y).
% 174.63/23.03 Axiom 43 (ruleD8): fresh52(perp(X, Y, Z, W), true, X, Y, Z, W) = perp(Z, W, X, Y).
% 174.63/23.03 Axiom 44 (ruleD43): fresh200(X, X, Y, Z, W, V, U, T) = fresh201(cyclic(Y, Z, W, U), true, Y, Z, W, V, U).
% 174.63/23.03 Axiom 45 (ruleD17): fresh137(cyclic(X, Y, Z, W), true, X, Y, Z, V, W) = fresh136(cyclic(X, Y, Z, V), true, Y, Z, V, W).
% 174.63/23.03 Axiom 46 (ruleD40): fresh104(para(X, Y, Z, W), true, X, Y, Z, W, V, U) = eqangle(X, Y, V, U, Z, W, V, U).
% 174.63/23.03 Axiom 47 (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).
% 174.63/23.03 Axiom 48 (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).
% 174.63/23.03 Axiom 49 (ruleD39): fresh106(eqangle(X, Y, Z, W, V, U, Z, W), true, X, Y, V, U) = para(X, Y, V, U).
% 174.63/23.03 Axiom 50 (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).
% 174.63/23.03 Axiom 51 (ruleD43): fresh200(eqangle(X, Y, X, Z, W, V, W, U), true, Y, Z, X, V, U, W) = fresh202(cyclic(Y, Z, X, W), true, Y, Z, X, V, U).
% 174.63/23.03 Axiom 52 (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).
% 174.63/23.03 Axiom 53 (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).
% 174.63/23.03 Axiom 54 (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).
% 174.63/23.03 Axiom 55 (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).
% 174.63/23.03 Axiom 56 (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).
% 174.63/23.03
% 174.63/23.03 Lemma 57: eqangle(midpnt1, a, a, b, a, b, midpnt1, b) = true.
% 174.63/23.03 Proof:
% 174.63/23.03 eqangle(midpnt1, a, a, b, a, b, midpnt1, b)
% 174.63/23.03 = { by axiom 30 (ruleD46) R->L }
% 174.63/23.03 fresh97(cong(midpnt1, a, midpnt1, b), true, a, b, midpnt1)
% 174.63/23.03 = { by axiom 27 (ruleD68) R->L }
% 174.63/23.03 fresh97(fresh63(midp(midpnt1, a, b), true, midpnt1, a, b), true, a, b, midpnt1)
% 174.63/23.03 = { by axiom 5 (exemplo6GDDFULL214026_15) }
% 174.63/23.03 fresh97(fresh63(true, true, midpnt1, a, b), true, a, b, midpnt1)
% 174.63/23.03 = { by axiom 11 (ruleD68) }
% 174.63/23.03 fresh97(true, true, a, b, midpnt1)
% 174.63/23.03 = { by axiom 9 (ruleD46) }
% 174.63/23.03 true
% 174.63/23.03
% 174.63/23.03 Lemma 58: para(X, Y, X, Y) = true.
% 174.63/23.03 Proof:
% 174.63/23.03 para(X, Y, X, Y)
% 174.63/23.03 = { by axiom 49 (ruleD39) R->L }
% 174.63/23.03 fresh106(eqangle(X, Y, a, b, X, Y, a, b), true, X, Y, X, Y)
% 174.63/23.03 = { by axiom 53 (ruleD19) R->L }
% 174.63/23.03 fresh106(fresh134(eqangle(a, b, X, Y, a, b, X, Y), true, a, b, X, Y, a, b, X, Y), true, X, Y, X, Y)
% 174.63/23.03 = { by axiom 46 (ruleD40) R->L }
% 174.63/23.03 fresh106(fresh134(fresh104(para(a, b, a, b), true, a, b, a, b, X, Y), true, a, b, X, Y, a, b, X, Y), true, X, Y, X, Y)
% 174.63/23.03 = { by axiom 49 (ruleD39) R->L }
% 174.63/23.03 fresh106(fresh134(fresh104(fresh106(eqangle(a, b, midpnt1, b, a, b, midpnt1, b), true, a, b, a, b), true, a, b, a, b, X, Y), true, a, b, X, Y, a, b, X, Y), true, X, Y, X, Y)
% 174.63/23.03 = { by axiom 48 (ruleD22) R->L }
% 174.63/23.03 fresh106(fresh134(fresh104(fresh106(fresh130(true, true, a, b, midpnt1, b, midpnt1, a, a, b, a, b, midpnt1, b), true, a, b, a, b), true, a, b, a, b, X, Y), true, a, b, X, Y, a, b, X, Y), true, X, Y, X, Y)
% 174.63/23.03 = { by lemma 57 R->L }
% 174.63/23.03 fresh106(fresh134(fresh104(fresh106(fresh130(eqangle(midpnt1, a, a, b, a, b, midpnt1, b), true, a, b, midpnt1, b, midpnt1, a, a, b, a, b, midpnt1, b), true, a, b, a, b), true, a, b, a, b, X, Y), true, a, b, X, Y, a, b, X, Y), true, X, Y, X, Y)
% 174.63/23.03 = { by axiom 56 (ruleD22) }
% 174.63/23.04 fresh106(fresh134(fresh104(fresh106(fresh129(eqangle(a, b, midpnt1, b, midpnt1, a, a, b), true, a, b, midpnt1, b, a, b, midpnt1, b), true, a, b, a, b), true, a, b, a, b, X, Y), true, a, b, X, Y, a, b, X, Y), true, X, Y, X, Y)
% 174.63/23.04 = { by axiom 54 (ruleD20) R->L }
% 174.63/23.04 fresh106(fresh134(fresh104(fresh106(fresh129(fresh132(eqangle(midpnt1, a, a, b, a, b, midpnt1, b), true, midpnt1, a, a, b, a, b, midpnt1, b), true, a, b, midpnt1, b, a, b, midpnt1, b), true, a, b, a, b), true, a, b, a, b, X, Y), true, a, b, X, Y, a, b, X, Y), true, X, Y, X, Y)
% 174.63/23.04 = { by lemma 57 }
% 174.63/23.04 fresh106(fresh134(fresh104(fresh106(fresh129(fresh132(true, true, midpnt1, a, a, b, a, b, midpnt1, b), true, a, b, midpnt1, b, a, b, midpnt1, b), true, a, b, a, b), true, a, b, a, b, X, Y), true, a, b, X, Y, a, b, X, Y), true, X, Y, X, Y)
% 174.63/23.04 = { by axiom 39 (ruleD20) }
% 174.63/23.04 fresh106(fresh134(fresh104(fresh106(fresh129(true, true, a, b, midpnt1, b, a, b, midpnt1, b), true, a, b, a, b), true, a, b, a, b, X, Y), true, a, b, X, Y, a, b, X, Y), true, X, Y, X, Y)
% 174.63/23.04 = { by axiom 41 (ruleD22) }
% 174.63/23.04 fresh106(fresh134(fresh104(fresh106(true, true, a, b, a, b), true, a, b, a, b, X, Y), true, a, b, X, Y, a, b, X, Y), true, X, Y, X, Y)
% 174.63/23.04 = { by axiom 15 (ruleD39) }
% 174.63/23.04 fresh106(fresh134(fresh104(true, true, a, b, a, b, X, Y), true, a, b, X, Y, a, b, X, Y), true, X, Y, X, Y)
% 174.63/23.04 = { by axiom 26 (ruleD40) }
% 174.63/23.04 fresh106(fresh134(true, true, a, b, X, Y, a, b, X, Y), true, X, Y, X, Y)
% 174.63/23.04 = { by axiom 38 (ruleD19) }
% 174.63/23.04 fresh106(true, true, X, Y, X, Y)
% 174.63/23.04 = { by axiom 15 (ruleD39) }
% 174.63/23.04 true
% 174.63/23.04
% 174.63/23.04 Lemma 59: eqangle(X, Y, Z, W, X, Y, Z, W) = true.
% 174.63/23.04 Proof:
% 174.63/23.04 eqangle(X, Y, Z, W, X, Y, Z, W)
% 174.63/23.04 = { by axiom 46 (ruleD40) R->L }
% 174.63/23.04 fresh104(para(X, Y, X, Y), true, X, Y, X, Y, Z, W)
% 174.63/23.04 = { by lemma 58 }
% 174.63/23.04 fresh104(true, true, X, Y, X, Y, Z, W)
% 174.63/23.04 = { by axiom 26 (ruleD40) }
% 174.63/23.04 true
% 174.63/23.04
% 174.63/23.04 Lemma 60: coll(X, X, Y) = true.
% 174.63/23.04 Proof:
% 174.63/23.04 coll(X, X, Y)
% 174.63/23.04 = { by axiom 24 (ruleD1) R->L }
% 174.63/23.04 fresh147(coll(X, Y, X), true, X, Y, X)
% 174.63/23.04 = { by axiom 25 (ruleD2) R->L }
% 174.63/23.04 fresh147(fresh133(coll(Y, X, X), true, Y, X, X), true, X, Y, X)
% 174.63/23.04 = { by axiom 31 (ruleD66) R->L }
% 174.63/23.04 fresh147(fresh133(fresh66(para(Y, X, Y, X), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 174.63/23.04 = { by lemma 58 }
% 174.63/23.04 fresh147(fresh133(fresh66(true, true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 174.63/23.04 = { by axiom 10 (ruleD66) }
% 174.63/23.04 fresh147(fresh133(true, true, Y, X, X), true, X, Y, X)
% 174.63/23.04 = { by axiom 7 (ruleD2) }
% 174.63/23.04 fresh147(true, true, X, Y, X)
% 174.63/23.04 = { by axiom 6 (ruleD1) }
% 174.63/23.04 true
% 174.63/23.04
% 174.63/23.04 Lemma 61: cyclic(X, X, Y, Z) = true.
% 174.63/23.04 Proof:
% 174.63/23.04 cyclic(X, X, Y, Z)
% 174.63/23.04 = { by axiom 16 (ruleD42b) R->L }
% 174.63/23.04 fresh102(true, true, X, X, Y, Z)
% 174.63/23.04 = { by axiom 40 (ruleD21) R->L }
% 174.63/23.04 fresh102(fresh131(true, true, Y, X, Z, X, Y, X, Z, X), true, X, X, Y, Z)
% 174.63/23.04 = { by lemma 59 R->L }
% 174.63/23.04 fresh102(fresh131(eqangle(Y, X, Z, X, Y, X, Z, X), true, Y, X, Z, X, Y, X, Z, X), true, X, X, Y, Z)
% 174.63/23.04 = { by axiom 55 (ruleD21) }
% 174.63/23.04 fresh102(eqangle(Y, X, Y, X, Z, X, Z, X), true, X, X, Y, Z)
% 174.63/23.04 = { by axiom 50 (ruleD42b) }
% 174.63/23.04 fresh101(coll(Y, Z, X), true, X, X, Y, Z)
% 174.63/23.04 = { by axiom 14 (ruleD3) R->L }
% 174.63/23.04 fresh101(fresh120(true, true, X, X, Y, Z), true, X, X, Y, Z)
% 174.63/23.04 = { by lemma 60 R->L }
% 174.63/23.04 fresh101(fresh120(coll(X, X, Z), true, X, X, Y, Z), true, X, X, Y, Z)
% 174.63/23.04 = { by axiom 29 (ruleD3) }
% 174.63/23.04 fresh101(fresh119(coll(X, X, Y), true, X, Y, Z), true, X, X, Y, Z)
% 174.63/23.04 = { by lemma 60 }
% 174.63/23.04 fresh101(fresh119(true, true, X, Y, Z), true, X, X, Y, Z)
% 174.63/23.04 = { by axiom 8 (ruleD3) }
% 174.63/23.04 fresh101(true, true, X, X, Y, Z)
% 174.63/23.04 = { by axiom 17 (ruleD42b) }
% 174.63/23.04 true
% 174.63/23.04
% 174.63/23.04 Lemma 62: cyclic(X, Y, Z, W) = true.
% 174.63/23.04 Proof:
% 174.63/23.04 cyclic(X, Y, Z, W)
% 174.63/23.04 = { by axiom 23 (ruleD17) R->L }
% 174.63/23.04 fresh137(true, true, X, X, Y, Z, W)
% 174.63/23.04 = { by lemma 61 R->L }
% 174.63/23.04 fresh137(cyclic(X, X, Y, W), true, X, X, Y, Z, W)
% 174.63/23.04 = { by axiom 45 (ruleD17) }
% 174.63/23.04 fresh136(cyclic(X, X, Y, Z), true, X, Y, Z, W)
% 174.63/23.04 = { by lemma 61 }
% 174.63/23.04 fresh136(true, true, X, Y, Z, W)
% 174.63/23.04 = { by axiom 13 (ruleD17) }
% 174.63/23.04 true
% 174.63/23.04
% 174.63/23.04 Lemma 63: cong(X, Y, X, Y) = true.
% 174.63/23.04 Proof:
% 174.63/23.04 cong(X, Y, X, Y)
% 174.63/23.04 = { by axiom 22 (ruleD43) R->L }
% 174.63/23.04 fresh201(true, true, X, Y, X, X, Y)
% 174.63/23.04 = { by lemma 62 R->L }
% 174.63/23.04 fresh201(cyclic(X, Y, X, Y), true, X, Y, X, X, Y)
% 174.63/23.04 = { by axiom 44 (ruleD43) R->L }
% 174.63/23.04 fresh200(true, true, X, Y, X, X, Y, X)
% 174.63/23.04 = { by axiom 37 (ruleD18) R->L }
% 174.63/23.04 fresh200(fresh135(true, true, X, X, X, Y, X, X, X, Y), true, X, Y, X, X, Y, X)
% 174.63/23.04 = { by lemma 59 R->L }
% 174.63/23.04 fresh200(fresh135(eqangle(X, X, X, Y, X, X, X, Y), true, X, X, X, Y, X, X, X, Y), true, X, Y, X, X, Y, X)
% 174.63/23.04 = { by axiom 52 (ruleD18) }
% 174.63/23.04 fresh200(eqangle(X, X, X, Y, X, X, X, Y), true, X, Y, X, X, Y, X)
% 174.63/23.04 = { by axiom 51 (ruleD43) }
% 174.63/23.04 fresh202(cyclic(X, Y, X, X), true, X, Y, X, X, Y)
% 174.63/23.04 = { by lemma 62 }
% 174.63/23.04 fresh202(true, true, X, Y, X, X, Y)
% 174.63/23.04 = { by axiom 32 (ruleD43) }
% 174.63/23.04 fresh203(cyclic(X, Y, X, X), true, X, Y, X, Y)
% 174.63/23.04 = { by lemma 62 }
% 174.63/23.04 fresh203(true, true, X, Y, X, Y)
% 174.63/23.04 = { by axiom 12 (ruleD43) }
% 174.63/23.04 true
% 174.63/23.04
% 174.63/23.04 Lemma 64: perp(X, X, Y, Z) = true.
% 174.63/23.04 Proof:
% 174.63/23.04 perp(X, X, Y, Z)
% 174.63/23.04 = { by axiom 18 (ruleD56) R->L }
% 174.63/23.04 fresh80(true, true, X, X, Y, Z)
% 174.63/23.04 = { by lemma 63 R->L }
% 174.63/23.04 fresh80(cong(X, Z, X, Z), true, X, X, Y, Z)
% 174.63/23.04 = { by axiom 42 (ruleD56) }
% 174.63/23.04 fresh79(cong(X, Y, X, Y), true, X, X, Y, Z)
% 174.63/23.04 = { by lemma 63 }
% 174.63/23.04 fresh79(true, true, X, X, Y, Z)
% 174.63/23.04 = { by axiom 19 (ruleD56) }
% 174.63/23.04 true
% 174.63/23.04
% 174.63/23.04 Lemma 65: eqangle(X, Y, Z, W, V, U, Z, W) = true.
% 174.63/23.04 Proof:
% 174.63/23.04 eqangle(X, Y, Z, W, V, U, Z, W)
% 174.63/23.04 = { by axiom 46 (ruleD40) R->L }
% 174.63/23.04 fresh104(para(X, Y, V, U), true, X, Y, V, U, Z, W)
% 174.63/23.04 = { by axiom 28 (ruleD9) R->L }
% 174.63/23.04 fresh104(fresh51(true, true, X, Y, T, T, V, U), true, X, Y, V, U, Z, W)
% 174.63/23.04 = { by lemma 64 R->L }
% 174.63/23.04 fresh104(fresh51(perp(T, T, V, U), true, X, Y, T, T, V, U), true, X, Y, V, U, Z, W)
% 174.63/23.04 = { by axiom 47 (ruleD9) }
% 174.63/23.04 fresh104(fresh50(perp(X, Y, T, T), true, X, Y, V, U), true, X, Y, V, U, Z, W)
% 174.63/23.04 = { by axiom 43 (ruleD8) R->L }
% 174.63/23.04 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)
% 174.63/23.04 = { by lemma 64 }
% 174.63/23.04 fresh104(fresh50(fresh52(true, true, T, T, X, Y), true, X, Y, V, U), true, X, Y, V, U, Z, W)
% 174.63/23.04 = { by axiom 20 (ruleD8) }
% 174.63/23.04 fresh104(fresh50(true, true, X, Y, V, U), true, X, Y, V, U, Z, W)
% 174.63/23.04 = { by axiom 21 (ruleD9) }
% 174.63/23.04 fresh104(true, true, X, Y, V, U, Z, W)
% 174.63/23.04 = { by axiom 26 (ruleD40) }
% 174.63/23.04 true
% 174.63/23.04
% 174.63/23.04 Lemma 66: eqangle(X, Y, Z, W, V, U, T, S) = true.
% 174.63/23.04 Proof:
% 174.63/23.04 eqangle(X, Y, Z, W, V, U, T, S)
% 174.63/23.04 = { by axiom 48 (ruleD22) R->L }
% 174.63/23.04 fresh130(true, true, X, Y, Z, W, X2, Y2, X2, Y2, V, U, T, S)
% 174.63/23.04 = { by axiom 40 (ruleD21) R->L }
% 174.63/23.04 fresh130(fresh131(true, true, X2, Y2, V, U, X2, Y2, T, S), true, X, Y, Z, W, X2, Y2, X2, Y2, V, U, T, S)
% 174.63/23.04 = { by axiom 38 (ruleD19) R->L }
% 174.63/23.04 fresh130(fresh131(fresh134(true, true, V, U, X2, Y2, T, S, X2, Y2), true, X2, Y2, V, U, X2, Y2, T, S), true, X, Y, Z, W, X2, Y2, X2, Y2, V, U, T, S)
% 174.63/23.04 = { by lemma 65 R->L }
% 174.63/23.04 fresh130(fresh131(fresh134(eqangle(V, U, X2, Y2, T, S, X2, Y2), true, V, U, X2, Y2, T, S, X2, Y2), true, X2, Y2, V, U, X2, Y2, T, S), true, X, Y, Z, W, X2, Y2, X2, Y2, V, U, T, S)
% 174.63/23.04 = { by axiom 53 (ruleD19) }
% 174.63/23.04 fresh130(fresh131(eqangle(X2, Y2, V, U, X2, Y2, T, S), true, X2, Y2, V, U, X2, Y2, T, S), true, X, Y, Z, W, X2, Y2, X2, Y2, V, U, T, S)
% 174.63/23.04 = { by axiom 55 (ruleD21) }
% 174.63/23.04 fresh130(eqangle(X2, Y2, X2, Y2, V, U, T, S), true, X, Y, Z, W, X2, Y2, X2, Y2, V, U, T, S)
% 174.63/23.04 = { by axiom 56 (ruleD22) }
% 174.63/23.04 fresh129(eqangle(X, Y, Z, W, X2, Y2, X2, Y2), true, X, Y, Z, W, V, U, T, S)
% 174.63/23.04 = { by axiom 55 (ruleD21) R->L }
% 174.63/23.04 fresh129(fresh131(eqangle(X, Y, X2, Y2, Z, W, X2, Y2), true, X, Y, X2, Y2, Z, W, X2, Y2), true, X, Y, Z, W, V, U, T, S)
% 174.63/23.04 = { by lemma 65 }
% 174.63/23.04 fresh129(fresh131(true, true, X, Y, X2, Y2, Z, W, X2, Y2), true, X, Y, Z, W, V, U, T, S)
% 174.63/23.04 = { by axiom 40 (ruleD21) }
% 174.63/23.04 fresh129(true, true, X, Y, Z, W, V, U, T, S)
% 174.63/23.04 = { by axiom 41 (ruleD22) }
% 174.63/23.04 true
% 174.63/23.04
% 174.63/23.04 Lemma 67: true = or2.
% 174.63/23.04 Proof:
% 174.63/23.04 true
% 174.63/23.04 = { by axiom 1 (exemplo6GDDFULL214026_30) R->L }
% 174.63/23.04 fresh161(true, true)
% 174.63/23.04 = { by lemma 66 R->L }
% 174.63/23.04 fresh161(eqangle(b, a, a, c, oa, ob, ob, oc), true)
% 174.63/23.04 = { by axiom 33 (exemplo6GDDFULL214026_30) R->L }
% 174.63/23.04 fresh160(true, true)
% 174.63/23.04 = { by lemma 66 R->L }
% 174.63/23.04 fresh160(eqangle(a, b, b, c, oa, ob, ob, oc), true)
% 174.63/23.04 = { by axiom 35 (exemplo6GDDFULL214026_30) }
% 174.63/23.04 fresh151(eqangle(b, a, a, c, ob, oa, oa, oc), true)
% 174.63/23.04 = { by lemma 66 }
% 174.63/23.04 fresh151(true, true)
% 174.63/23.04 = { by axiom 3 (exemplo6GDDFULL214026_30) }
% 174.63/23.04 or2
% 174.63/23.04
% 174.63/23.04 Lemma 68: or2 = or.
% 174.63/23.04 Proof:
% 174.63/23.04 or2
% 174.63/23.04 = { by lemma 67 R->L }
% 174.63/23.04 true
% 174.63/23.04 = { by axiom 2 (exemplo6GDDFULL214026_31) R->L }
% 174.63/23.04 fresh159(true, true)
% 174.63/23.04 = { by lemma 66 R->L }
% 174.63/23.04 fresh159(eqangle(b, a, a, c, oa, ob, ob, oc), true)
% 174.63/23.04 = { by axiom 34 (exemplo6GDDFULL214026_31) R->L }
% 174.63/23.04 fresh158(or2, or2)
% 174.63/23.04 = { by lemma 67 R->L }
% 174.63/23.04 fresh158(or2, true)
% 174.63/23.04 = { by axiom 36 (exemplo6GDDFULL214026_31) }
% 174.63/23.04 fresh150(eqangle(a, b, b, c, oa, oc, oc, ob), true)
% 174.63/23.04 = { by lemma 67 }
% 174.63/23.04 fresh150(eqangle(a, b, b, c, oa, oc, oc, ob), or2)
% 174.63/23.04 = { by lemma 66 }
% 174.63/23.04 fresh150(true, or2)
% 174.63/23.04 = { by lemma 67 }
% 174.63/23.04 fresh150(or2, or2)
% 174.63/23.04 = { by axiom 4 (exemplo6GDDFULL214026_31) }
% 174.63/23.04 or
% 174.63/23.04
% 174.63/23.04 Goal 1 (exemplo6GDDFULL214026_34): tuple(eqangle(a, b, b, c, ob, oa, oa, oc), or) = tuple(true, true).
% 174.63/23.04 Proof:
% 174.63/23.04 tuple(eqangle(a, b, b, c, ob, oa, oa, oc), or)
% 174.63/23.04 = { by lemma 66 }
% 174.63/23.04 tuple(true, or)
% 174.63/23.04 = { by lemma 67 }
% 174.63/23.04 tuple(or2, or)
% 174.63/23.04 = { by lemma 68 R->L }
% 174.63/23.04 tuple(or2, or2)
% 174.63/23.04 = { by lemma 67 R->L }
% 174.63/23.04 tuple(true, or2)
% 174.63/23.04 = { by lemma 67 R->L }
% 174.63/23.04 tuple(true, true)
% 174.63/23.04
% 174.63/23.04 Goal 2 (exemplo6GDDFULL214026_29): tuple(eqangle(b, a, a, c, oa, oc, oc, ob), or) = tuple(true, true).
% 174.63/23.04 Proof:
% 174.63/23.04 tuple(eqangle(b, a, a, c, oa, oc, oc, ob), or)
% 174.63/23.04 = { by lemma 66 }
% 174.63/23.04 tuple(true, or)
% 174.63/23.04 = { by lemma 67 }
% 174.63/23.04 tuple(or2, or)
% 174.63/23.04 = { by lemma 68 R->L }
% 174.63/23.04 tuple(or2, or2)
% 174.63/23.04 = { by lemma 67 R->L }
% 174.63/23.04 tuple(true, or2)
% 174.63/23.04 = { by lemma 67 R->L }
% 174.63/23.04 tuple(true, true)
% 174.63/23.04 % SZS output end Proof
% 174.63/23.04
% 174.63/23.04 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------