TSTP Solution File: GEO545+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GEO545+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 119.65s 15.64s
% Output : Proof 119.65s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.12 % Problem : GEO545+1 : TPTP v8.1.2. Released v7.5.0.
% 0.06/0.13 % 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 21:18:58 EDT 2023
% 0.12/0.34 % CPUTime :
% 119.65/15.64 Command-line arguments: --no-flatten-goal
% 119.65/15.64
% 119.65/15.64 % SZS status Theorem
% 119.65/15.64
% 119.65/15.68 % SZS output start Proof
% 119.65/15.68 Take the following subset of the input axioms:
% 119.65/15.68 fof(exemplo6GDDFULL012005, conjecture, ![A, B, C, O, H, C1, B1, A1, 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, B) & (perp(A, B, MIDPNT4, C1) & (midp(MIDPNT5, A, H) & (perp(A, H, MIDPNT5, C1) & (midp(MIDPNT6, B, H) & (perp(B, H, MIDPNT6, C1) & (midp(MIDPNT7, A, H) & (perp(A, H, MIDPNT7, B1) & (midp(MIDPNT8, A, C) & (perp(A, C, MIDPNT8, B1) & (midp(MIDPNT9, H, C) & (perp(H, C, MIDPNT9, B1) & (midp(MIDPNT01, B, H) & (perp(B, H, MIDPNT01, A1) & (midp(MIDPNT11, B, C) & (perp(B, C, MIDPNT11, A1) & (midp(MIDPNT21, H, C) & perp(H, C, MIDPNT21, A1))))))))))))))))))))))))))) => perp(A1, O, B1, C1))).
% 119.65/15.68 fof(ruleD1, axiom, ![A2, B2, C2]: (coll(A2, B2, C2) => coll(A2, C2, B2))).
% 119.65/15.68 fof(ruleD10, axiom, ![D, E, F, B2, C2, A2_2]: ((para(A2_2, B2, C2, D) & perp(C2, D, E, F)) => perp(A2_2, B2, E, F))).
% 119.65/15.68 fof(ruleD11, axiom, ![M, B2, A2_2]: (midp(M, B2, A2_2) => midp(M, A2_2, B2))).
% 119.65/15.68 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))).
% 119.65/15.68 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))).
% 119.65/15.68 fof(ruleD2, axiom, ![B2, C2, A2_2]: (coll(A2_2, B2, C2) => coll(B2, A2_2, C2))).
% 119.65/15.68 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))).
% 119.65/15.68 fof(ruleD3, axiom, ![B2, C2, D2, A2_2]: ((coll(A2_2, B2, C2) & coll(A2_2, B2, D2)) => coll(C2, D2, A2_2))).
% 119.65/15.68 fof(ruleD4, axiom, ![B2, C2, D2, A2_2]: (para(A2_2, B2, C2, D2) => para(A2_2, B2, D2, C2))).
% 119.65/15.68 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))).
% 119.65/15.68 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))).
% 119.65/15.68 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))).
% 119.65/15.68 fof(ruleD44, axiom, ![B2, C2, E2, F2, A2_2]: ((midp(E2, A2_2, B2) & midp(F2, A2_2, C2)) => para(E2, F2, B2, C2))).
% 119.65/15.68 fof(ruleD45, axiom, ![B2, C2, E2, F2, A2_2]: ((midp(E2, A2_2, B2) & (para(E2, F2, B2, C2) & coll(F2, A2_2, C2))) => midp(F2, A2_2, C2))).
% 119.65/15.68 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))).
% 119.65/15.68 fof(ruleD63, axiom, ![B2, C2, D2, A2_2, M2]: ((midp(M2, A2_2, B2) & midp(M2, C2, D2)) => para(A2_2, C2, B2, D2))).
% 119.65/15.68 fof(ruleD66, axiom, ![B2, C2, A2_2]: (para(A2_2, B2, A2_2, C2) => coll(A2_2, B2, C2))).
% 119.65/15.68 fof(ruleD68, axiom, ![B2, C2, A2_2]: (midp(A2_2, B2, C2) => cong(A2_2, B2, A2_2, C2))).
% 119.65/15.68 fof(ruleD73, axiom, ![B2, C2, D2, A2_2, P2, Q2, U2, V2]: ((eqangle(A2_2, B2, C2, D2, P2, Q2, U2, V2) & para(P2, Q2, U2, V2)) => para(A2_2, B2, C2, D2))).
% 119.65/15.68 fof(ruleD8, axiom, ![B2, C2, D2, A2_2]: (perp(A2_2, B2, C2, D2) => perp(C2, D2, A2_2, B2))).
% 119.65/15.68 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))).
% 119.65/15.68
% 119.65/15.68 Now clausify the problem and encode Horn clauses using encoding 3 of
% 119.65/15.68 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 119.65/15.68 We repeatedly replace C & s=t => u=v by the two clauses:
% 119.65/15.68 fresh(y, y, x1...xn) = u
% 119.65/15.68 C => fresh(s, t, x1...xn) = v
% 119.65/15.68 where fresh is a fresh function symbol and x1..xn are the free
% 119.65/15.68 variables of u and v.
% 119.65/15.68 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 119.65/15.68 input problem has no model of domain size 1).
% 119.65/15.68
% 119.65/15.68 The encoding turns the above axioms into the following unit equations and goals:
% 119.65/15.68
% 119.65/15.68 Axiom 1 (exemplo6GDDFULL012005_15): midp(midpnt1, a, b) = true.
% 119.65/15.68 Axiom 2 (ruleD45): fresh181(X, X, Y, Z, W) = true.
% 119.65/15.68 Axiom 3 (ruleD1): fresh146(X, X, Y, Z, W) = true.
% 119.65/15.68 Axiom 4 (ruleD11): fresh144(X, X, Y, Z, W) = true.
% 119.65/15.68 Axiom 5 (ruleD2): fresh133(X, X, Y, Z, W) = true.
% 119.65/15.68 Axiom 6 (ruleD3): fresh119(X, X, Y, Z, W) = true.
% 119.65/15.68 Axiom 7 (ruleD45): fresh98(X, X, Y, Z, W) = midp(W, Y, Z).
% 119.65/15.68 Axiom 8 (ruleD66): fresh66(X, X, Y, Z, W) = true.
% 119.65/15.68 Axiom 9 (ruleD68): fresh63(X, X, Y, Z, W) = true.
% 119.65/15.68 Axiom 10 (ruleD43): fresh185(X, X, Y, Z, W, V) = true.
% 119.65/15.68 Axiom 11 (ruleD10): fresh145(X, X, Y, Z, W, V) = true.
% 119.65/15.68 Axiom 12 (ruleD17): fresh136(X, X, Y, Z, W, V) = true.
% 119.65/15.68 Axiom 13 (ruleD3): fresh120(X, X, Y, Z, W, V) = coll(W, V, Y).
% 119.65/15.68 Axiom 14 (ruleD4): fresh105(X, X, Y, Z, W, V) = true.
% 119.65/15.68 Axiom 15 (ruleD42b): fresh102(X, X, Y, Z, W, V) = cyclic(Y, Z, W, V).
% 119.65/15.68 Axiom 16 (ruleD42b): fresh101(X, X, Y, Z, W, V) = true.
% 119.65/15.68 Axiom 17 (ruleD44): fresh99(X, X, Y, Z, W, V) = true.
% 119.65/15.68 Axiom 18 (ruleD56): fresh80(X, X, Y, Z, W, V) = perp(Y, Z, W, V).
% 119.65/15.68 Axiom 19 (ruleD56): fresh79(X, X, Y, Z, W, V) = true.
% 119.65/15.68 Axiom 20 (ruleD63): fresh69(X, X, Y, Z, W, V) = true.
% 119.65/15.68 Axiom 21 (ruleD73): fresh57(X, X, Y, Z, W, V) = true.
% 119.65/15.68 Axiom 22 (ruleD8): fresh52(X, X, Y, Z, W, V) = true.
% 119.65/15.68 Axiom 23 (ruleD9): fresh50(X, X, Y, Z, W, V) = true.
% 119.65/15.68 Axiom 24 (ruleD43): fresh183(X, X, Y, Z, W, V, U) = cong(Y, Z, V, U).
% 119.65/15.68 Axiom 25 (ruleD17): fresh137(X, X, Y, Z, W, V, U) = cyclic(Z, W, V, U).
% 119.65/15.68 Axiom 26 (ruleD44): fresh100(X, X, Y, Z, W, V, U) = para(V, U, Z, W).
% 119.65/15.68 Axiom 27 (ruleD63): fresh70(X, X, Y, Z, W, V, U) = para(Y, W, Z, V).
% 119.65/15.68 Axiom 28 (ruleD45): fresh180(X, X, Y, Z, W, V, U) = fresh181(coll(U, Y, W), true, Y, W, U).
% 119.65/15.68 Axiom 29 (ruleD10): fresh147(X, X, Y, Z, W, V, U, T) = perp(Y, Z, U, T).
% 119.65/15.68 Axiom 30 (ruleD1): fresh146(coll(X, Y, Z), true, X, Y, Z) = coll(X, Z, Y).
% 119.65/15.69 Axiom 31 (ruleD11): fresh144(midp(X, Y, Z), true, Z, Y, X) = midp(X, Z, Y).
% 119.65/15.69 Axiom 32 (ruleD2): fresh133(coll(X, Y, Z), true, X, Y, Z) = coll(Y, X, Z).
% 119.65/15.69 Axiom 33 (ruleD40): fresh104(X, X, Y, Z, W, V, U, T) = true.
% 119.65/15.69 Axiom 34 (ruleD68): fresh63(midp(X, Y, Z), true, X, Y, Z) = cong(X, Y, X, Z).
% 119.65/15.69 Axiom 35 (ruleD9): fresh51(X, X, Y, Z, W, V, U, T) = para(Y, Z, U, T).
% 119.65/15.69 Axiom 36 (ruleD3): fresh120(coll(X, Y, Z), true, X, Y, W, Z) = fresh119(coll(X, Y, W), true, X, W, Z).
% 119.65/15.69 Axiom 37 (ruleD66): fresh66(para(X, Y, X, Z), true, X, Y, Z) = coll(X, Y, Z).
% 119.65/15.69 Axiom 38 (ruleD43): fresh184(X, X, Y, Z, W, V, U) = fresh185(cyclic(Y, Z, W, V), true, Y, Z, V, U).
% 119.65/15.69 Axiom 39 (ruleD45): fresh180(midp(X, Y, Z), true, Y, Z, W, X, V) = fresh98(para(X, V, Z, W), true, Y, W, V).
% 119.65/15.69 Axiom 40 (ruleD19): fresh134(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 119.65/15.69 Axiom 41 (ruleD21): fresh131(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 119.65/15.69 Axiom 42 (ruleD4): fresh105(para(X, Y, Z, W), true, X, Y, Z, W) = para(X, Y, W, Z).
% 119.65/15.69 Axiom 43 (ruleD44): fresh100(midp(X, Y, Z), true, Y, W, Z, V, X) = fresh99(midp(V, Y, W), true, W, Z, V, X).
% 119.65/15.69 Axiom 44 (ruleD56): fresh80(cong(X, Y, Z, Y), true, X, Z, W, Y) = fresh79(cong(X, W, Z, W), true, X, Z, W, Y).
% 119.65/15.69 Axiom 45 (ruleD63): fresh70(midp(X, Y, Z), true, W, V, Y, Z, X) = fresh69(midp(X, W, V), true, W, V, Y, Z).
% 119.65/15.69 Axiom 46 (ruleD73): fresh58(X, X, Y, Z, W, V, U, T, S, X2) = para(Y, Z, W, V).
% 119.65/15.69 Axiom 47 (ruleD8): fresh52(perp(X, Y, Z, W), true, X, Y, Z, W) = perp(Z, W, X, Y).
% 119.65/15.69 Axiom 48 (ruleD43): fresh182(X, X, Y, Z, W, V, U, T) = fresh183(cyclic(Y, Z, W, U), true, Y, Z, W, V, U).
% 119.65/15.69 Axiom 49 (ruleD17): fresh137(cyclic(X, Y, Z, W), true, X, Y, Z, V, W) = fresh136(cyclic(X, Y, Z, V), true, Y, Z, V, W).
% 119.65/15.69 Axiom 50 (ruleD10): fresh147(perp(X, Y, Z, W), true, V, U, X, Y, Z, W) = fresh145(para(V, U, X, Y), true, V, U, Z, W).
% 119.65/15.69 Axiom 51 (ruleD40): fresh104(para(X, Y, Z, W), true, X, Y, Z, W, V, U) = eqangle(X, Y, V, U, Z, W, V, U).
% 119.65/15.69 Axiom 52 (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).
% 119.65/15.69 Axiom 53 (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).
% 119.65/15.69 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).
% 119.65/15.69 Axiom 55 (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).
% 119.65/15.69 Axiom 56 (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).
% 119.65/15.69 Axiom 57 (ruleD73): fresh58(eqangle(X, Y, Z, W, V, U, T, S), true, X, Y, Z, W, V, U, T, S) = fresh57(para(V, U, T, S), true, X, Y, Z, W).
% 119.65/15.69
% 119.65/15.69 Lemma 58: para(midpnt1, midpnt1, b, b) = true.
% 119.65/15.69 Proof:
% 119.65/15.69 para(midpnt1, midpnt1, b, b)
% 119.65/15.69 = { by axiom 26 (ruleD44) R->L }
% 119.65/15.69 fresh100(true, true, a, b, b, midpnt1, midpnt1)
% 119.65/15.69 = { by axiom 1 (exemplo6GDDFULL012005_15) R->L }
% 119.65/15.69 fresh100(midp(midpnt1, a, b), true, a, b, b, midpnt1, midpnt1)
% 119.65/15.69 = { by axiom 43 (ruleD44) }
% 119.65/15.69 fresh99(midp(midpnt1, a, b), true, b, b, midpnt1, midpnt1)
% 119.65/15.69 = { by axiom 1 (exemplo6GDDFULL012005_15) }
% 119.65/15.69 fresh99(true, true, b, b, midpnt1, midpnt1)
% 119.65/15.69 = { by axiom 17 (ruleD44) }
% 119.65/15.69 true
% 119.65/15.69
% 119.65/15.69 Lemma 59: coll(X, X, Y) = true.
% 119.65/15.69 Proof:
% 119.65/15.69 coll(X, X, Y)
% 119.65/15.69 = { by axiom 30 (ruleD1) R->L }
% 119.65/15.69 fresh146(coll(X, Y, X), true, X, Y, X)
% 119.65/15.69 = { by axiom 32 (ruleD2) R->L }
% 119.65/15.69 fresh146(fresh133(coll(Y, X, X), true, Y, X, X), true, X, Y, X)
% 119.65/15.69 = { by axiom 37 (ruleD66) R->L }
% 119.65/15.69 fresh146(fresh133(fresh66(para(Y, X, Y, X), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 119.65/15.69 = { by axiom 46 (ruleD73) R->L }
% 119.65/15.69 fresh146(fresh133(fresh66(fresh58(true, true, Y, X, Y, X, midpnt1, midpnt1, b, b), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 119.65/15.69 = { by axiom 41 (ruleD21) R->L }
% 119.65/15.69 fresh146(fresh133(fresh66(fresh58(fresh131(true, true, Y, X, midpnt1, midpnt1, Y, X, b, b), true, Y, X, Y, X, midpnt1, midpnt1, b, b), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 119.65/15.69 = { by axiom 40 (ruleD19) R->L }
% 119.65/15.69 fresh146(fresh133(fresh66(fresh58(fresh131(fresh134(true, true, midpnt1, midpnt1, Y, X, b, b, Y, X), true, Y, X, midpnt1, midpnt1, Y, X, b, b), true, Y, X, Y, X, midpnt1, midpnt1, b, b), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 119.65/15.69 = { by axiom 33 (ruleD40) R->L }
% 119.65/15.69 fresh146(fresh133(fresh66(fresh58(fresh131(fresh134(fresh104(true, true, midpnt1, midpnt1, b, b, Y, X), true, midpnt1, midpnt1, Y, X, b, b, Y, X), true, Y, X, midpnt1, midpnt1, Y, X, b, b), true, Y, X, Y, X, midpnt1, midpnt1, b, b), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 119.65/15.69 = { by lemma 58 R->L }
% 119.65/15.69 fresh146(fresh133(fresh66(fresh58(fresh131(fresh134(fresh104(para(midpnt1, midpnt1, b, b), true, midpnt1, midpnt1, b, b, Y, X), true, midpnt1, midpnt1, Y, X, b, b, Y, X), true, Y, X, midpnt1, midpnt1, Y, X, b, b), true, Y, X, Y, X, midpnt1, midpnt1, b, b), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 119.65/15.69 = { by axiom 51 (ruleD40) }
% 119.65/15.69 fresh146(fresh133(fresh66(fresh58(fresh131(fresh134(eqangle(midpnt1, midpnt1, Y, X, b, b, Y, X), true, midpnt1, midpnt1, Y, X, b, b, Y, X), true, Y, X, midpnt1, midpnt1, Y, X, b, b), true, Y, X, Y, X, midpnt1, midpnt1, b, b), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 119.65/15.69 = { by axiom 55 (ruleD19) }
% 119.65/15.69 fresh146(fresh133(fresh66(fresh58(fresh131(eqangle(Y, X, midpnt1, midpnt1, Y, X, b, b), true, Y, X, midpnt1, midpnt1, Y, X, b, b), true, Y, X, Y, X, midpnt1, midpnt1, b, b), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 119.65/15.69 = { by axiom 56 (ruleD21) }
% 119.65/15.69 fresh146(fresh133(fresh66(fresh58(eqangle(Y, X, Y, X, midpnt1, midpnt1, b, b), true, Y, X, Y, X, midpnt1, midpnt1, b, b), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 119.65/15.69 = { by axiom 57 (ruleD73) }
% 119.65/15.69 fresh146(fresh133(fresh66(fresh57(para(midpnt1, midpnt1, b, b), true, Y, X, Y, X), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 119.65/15.69 = { by lemma 58 }
% 119.65/15.69 fresh146(fresh133(fresh66(fresh57(true, true, Y, X, Y, X), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 119.65/15.69 = { by axiom 21 (ruleD73) }
% 119.65/15.69 fresh146(fresh133(fresh66(true, true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 119.65/15.69 = { by axiom 8 (ruleD66) }
% 119.65/15.69 fresh146(fresh133(true, true, Y, X, X), true, X, Y, X)
% 119.65/15.69 = { by axiom 5 (ruleD2) }
% 119.65/15.69 fresh146(true, true, X, Y, X)
% 119.65/15.69 = { by axiom 3 (ruleD1) }
% 119.65/15.69 true
% 119.65/15.69
% 119.65/15.69 Lemma 60: coll(X, Y, Z) = true.
% 119.65/15.69 Proof:
% 119.65/15.69 coll(X, Y, Z)
% 119.65/15.69 = { by axiom 13 (ruleD3) R->L }
% 119.65/15.69 fresh120(true, true, Z, Z, X, Y)
% 119.65/15.69 = { by lemma 59 R->L }
% 119.65/15.69 fresh120(coll(Z, Z, Y), true, Z, Z, X, Y)
% 119.65/15.69 = { by axiom 36 (ruleD3) }
% 119.65/15.69 fresh119(coll(Z, Z, X), true, Z, X, Y)
% 119.65/15.69 = { by lemma 59 }
% 119.65/15.69 fresh119(true, true, Z, X, Y)
% 119.65/15.69 = { by axiom 6 (ruleD3) }
% 119.65/15.69 true
% 119.65/15.69
% 119.65/15.69 Lemma 61: eqangle(b, a, X, Y, b, a, X, Y) = true.
% 119.65/15.69 Proof:
% 119.65/15.69 eqangle(b, a, X, Y, b, a, X, Y)
% 119.65/15.69 = { by axiom 51 (ruleD40) R->L }
% 119.65/15.69 fresh104(para(b, a, b, a), true, b, a, b, a, X, Y)
% 119.65/15.69 = { by axiom 42 (ruleD4) R->L }
% 119.65/15.69 fresh104(fresh105(para(b, a, a, b), true, b, a, a, b), true, b, a, b, a, X, Y)
% 119.65/15.69 = { by axiom 27 (ruleD63) R->L }
% 119.65/15.69 fresh104(fresh105(fresh70(true, true, b, a, a, b, midpnt1), true, b, a, a, b), true, b, a, b, a, X, Y)
% 119.65/15.69 = { by axiom 1 (exemplo6GDDFULL012005_15) R->L }
% 119.65/15.69 fresh104(fresh105(fresh70(midp(midpnt1, a, b), true, b, a, a, b, midpnt1), true, b, a, a, b), true, b, a, b, a, X, Y)
% 119.65/15.69 = { by axiom 45 (ruleD63) }
% 119.65/15.69 fresh104(fresh105(fresh69(midp(midpnt1, b, a), true, b, a, a, b), true, b, a, a, b), true, b, a, b, a, X, Y)
% 119.65/15.69 = { by axiom 31 (ruleD11) R->L }
% 119.65/15.69 fresh104(fresh105(fresh69(fresh144(midp(midpnt1, a, b), true, b, a, midpnt1), true, b, a, a, b), true, b, a, a, b), true, b, a, b, a, X, Y)
% 119.65/15.69 = { by axiom 1 (exemplo6GDDFULL012005_15) }
% 119.65/15.69 fresh104(fresh105(fresh69(fresh144(true, true, b, a, midpnt1), true, b, a, a, b), true, b, a, a, b), true, b, a, b, a, X, Y)
% 119.65/15.69 = { by axiom 4 (ruleD11) }
% 119.65/15.69 fresh104(fresh105(fresh69(true, true, b, a, a, b), true, b, a, a, b), true, b, a, b, a, X, Y)
% 119.65/15.69 = { by axiom 20 (ruleD63) }
% 119.65/15.69 fresh104(fresh105(true, true, b, a, a, b), true, b, a, b, a, X, Y)
% 119.65/15.69 = { by axiom 14 (ruleD4) }
% 119.65/15.69 fresh104(true, true, b, a, b, a, X, Y)
% 119.65/15.69 = { by axiom 33 (ruleD40) }
% 119.65/15.69 true
% 119.65/15.69
% 119.65/15.69 Lemma 62: cyclic(a, a, b, X) = true.
% 119.65/15.69 Proof:
% 119.65/15.69 cyclic(a, a, b, X)
% 119.65/15.69 = { by axiom 15 (ruleD42b) R->L }
% 119.65/15.69 fresh102(true, true, a, a, b, X)
% 119.65/15.69 = { by axiom 41 (ruleD21) R->L }
% 119.65/15.69 fresh102(fresh131(true, true, b, a, X, a, b, a, X, a), true, a, a, b, X)
% 119.65/15.69 = { by lemma 61 R->L }
% 119.65/15.69 fresh102(fresh131(eqangle(b, a, X, a, b, a, X, a), true, b, a, X, a, b, a, X, a), true, a, a, b, X)
% 119.65/15.69 = { by axiom 56 (ruleD21) }
% 119.65/15.69 fresh102(eqangle(b, a, b, a, X, a, X, a), true, a, a, b, X)
% 119.65/15.69 = { by axiom 53 (ruleD42b) }
% 119.65/15.69 fresh101(coll(b, X, a), true, a, a, b, X)
% 119.65/15.69 = { by lemma 60 }
% 119.65/15.69 fresh101(true, true, a, a, b, X)
% 119.65/15.69 = { by axiom 16 (ruleD42b) }
% 119.65/15.69 true
% 119.65/15.69
% 119.65/15.69 Lemma 63: cyclic(a, b, X, Y) = true.
% 119.65/15.69 Proof:
% 119.65/15.69 cyclic(a, b, X, Y)
% 119.65/15.69 = { by axiom 25 (ruleD17) R->L }
% 119.65/15.69 fresh137(true, true, a, a, b, X, Y)
% 119.65/15.69 = { by lemma 62 R->L }
% 119.65/15.69 fresh137(cyclic(a, a, b, Y), true, a, a, b, X, Y)
% 119.65/15.69 = { by axiom 49 (ruleD17) }
% 119.65/15.69 fresh136(cyclic(a, a, b, X), true, a, b, X, Y)
% 119.65/15.69 = { by lemma 62 }
% 119.65/15.69 fresh136(true, true, a, b, X, Y)
% 119.65/15.69 = { by axiom 12 (ruleD17) }
% 119.65/15.69 true
% 119.65/15.69
% 119.65/15.69 Lemma 64: cyclic(b, X, Y, Z) = true.
% 119.65/15.69 Proof:
% 119.65/15.69 cyclic(b, X, Y, Z)
% 119.65/15.69 = { by axiom 25 (ruleD17) R->L }
% 119.65/15.69 fresh137(true, true, a, b, X, Y, Z)
% 119.65/15.69 = { by lemma 63 R->L }
% 119.65/15.69 fresh137(cyclic(a, b, X, Z), true, a, b, X, Y, Z)
% 119.65/15.69 = { by axiom 49 (ruleD17) }
% 119.65/15.69 fresh136(cyclic(a, b, X, Y), true, b, X, Y, Z)
% 119.65/15.69 = { by lemma 63 }
% 119.65/15.69 fresh136(true, true, b, X, Y, Z)
% 119.65/15.69 = { by axiom 12 (ruleD17) }
% 119.65/15.69 true
% 119.65/15.69
% 119.65/15.69 Lemma 65: cyclic(X, Y, Z, W) = true.
% 119.65/15.69 Proof:
% 119.65/15.69 cyclic(X, Y, Z, W)
% 119.65/15.69 = { by axiom 25 (ruleD17) R->L }
% 119.65/15.69 fresh137(true, true, b, X, Y, Z, W)
% 119.65/15.69 = { by lemma 64 R->L }
% 119.65/15.69 fresh137(cyclic(b, X, Y, W), true, b, X, Y, Z, W)
% 119.65/15.69 = { by axiom 49 (ruleD17) }
% 119.65/15.69 fresh136(cyclic(b, X, Y, Z), true, X, Y, Z, W)
% 119.65/15.69 = { by lemma 64 }
% 119.65/15.69 fresh136(true, true, X, Y, Z, W)
% 119.65/15.69 = { by axiom 12 (ruleD17) }
% 119.65/15.69 true
% 119.65/15.69
% 119.65/15.69 Lemma 66: cong(a, X, a, X) = true.
% 119.65/15.69 Proof:
% 119.65/15.69 cong(a, X, a, X)
% 119.65/15.69 = { by axiom 24 (ruleD43) R->L }
% 119.65/15.69 fresh183(true, true, a, X, b, a, X)
% 119.65/15.69 = { by lemma 65 R->L }
% 119.65/15.69 fresh183(cyclic(a, X, b, X), true, a, X, b, a, X)
% 119.65/15.69 = { by axiom 48 (ruleD43) R->L }
% 119.65/15.69 fresh182(true, true, a, X, b, a, X, b)
% 119.65/15.69 = { by lemma 61 R->L }
% 119.65/15.69 fresh182(eqangle(b, a, b, X, b, a, b, X), true, a, X, b, a, X, b)
% 119.65/15.69 = { by axiom 54 (ruleD43) }
% 119.65/15.69 fresh184(cyclic(a, X, b, b), true, a, X, b, a, X)
% 119.65/15.69 = { by lemma 65 }
% 119.65/15.69 fresh184(true, true, a, X, b, a, X)
% 119.65/15.69 = { by axiom 38 (ruleD43) }
% 119.65/15.69 fresh185(cyclic(a, X, b, a), true, a, X, a, X)
% 119.65/15.69 = { by lemma 65 }
% 119.65/15.69 fresh185(true, true, a, X, a, X)
% 119.65/15.69 = { by axiom 10 (ruleD43) }
% 119.65/15.69 true
% 119.65/15.69
% 119.65/15.69 Lemma 67: perp(a, a, X, Y) = true.
% 119.65/15.69 Proof:
% 119.65/15.69 perp(a, a, X, Y)
% 119.65/15.69 = { by axiom 18 (ruleD56) R->L }
% 119.65/15.69 fresh80(true, true, a, a, X, Y)
% 119.65/15.69 = { by lemma 66 R->L }
% 119.65/15.69 fresh80(cong(a, Y, a, Y), true, a, a, X, Y)
% 119.65/15.69 = { by axiom 44 (ruleD56) }
% 119.65/15.69 fresh79(cong(a, X, a, X), true, a, a, X, Y)
% 119.65/15.69 = { by lemma 66 }
% 119.65/15.69 fresh79(true, true, a, a, X, Y)
% 119.65/15.69 = { by axiom 19 (ruleD56) }
% 119.65/15.69 true
% 119.65/15.69
% 119.65/15.69 Lemma 68: para(X, Y, Z, W) = true.
% 119.65/15.69 Proof:
% 119.65/15.69 para(X, Y, Z, W)
% 119.65/15.69 = { by axiom 35 (ruleD9) R->L }
% 119.65/15.69 fresh51(true, true, X, Y, a, a, Z, W)
% 119.65/15.69 = { by lemma 67 R->L }
% 119.65/15.69 fresh51(perp(a, a, Z, W), true, X, Y, a, a, Z, W)
% 119.65/15.69 = { by axiom 52 (ruleD9) }
% 119.65/15.69 fresh50(perp(X, Y, a, a), true, X, Y, Z, W)
% 119.65/15.69 = { by axiom 47 (ruleD8) R->L }
% 119.65/15.69 fresh50(fresh52(perp(a, a, X, Y), true, a, a, X, Y), true, X, Y, Z, W)
% 119.65/15.69 = { by lemma 67 }
% 119.65/15.69 fresh50(fresh52(true, true, a, a, X, Y), true, X, Y, Z, W)
% 119.65/15.69 = { by axiom 22 (ruleD8) }
% 119.65/15.69 fresh50(true, true, X, Y, Z, W)
% 119.65/15.69 = { by axiom 23 (ruleD9) }
% 119.65/15.69 true
% 119.65/15.69
% 119.65/15.69 Lemma 69: fresh180(X, X, Y, Z, W, V, U) = true.
% 119.65/15.69 Proof:
% 119.65/15.69 fresh180(X, X, Y, Z, W, V, U)
% 119.65/15.69 = { by axiom 28 (ruleD45) }
% 119.65/15.69 fresh181(coll(U, Y, W), true, Y, W, U)
% 119.65/15.69 = { by lemma 60 }
% 119.65/15.69 fresh181(true, true, Y, W, U)
% 119.65/15.69 = { by axiom 2 (ruleD45) }
% 119.65/15.69 true
% 119.65/15.69
% 119.65/15.69 Lemma 70: cong(X, Y, X, Z) = true.
% 119.65/15.69 Proof:
% 119.65/15.69 cong(X, Y, X, Z)
% 119.65/15.69 = { by axiom 34 (ruleD68) R->L }
% 119.65/15.69 fresh63(midp(X, Y, Z), true, X, Y, Z)
% 119.65/15.69 = { by axiom 7 (ruleD45) R->L }
% 119.65/15.69 fresh63(fresh98(true, true, Y, Z, X), true, X, Y, Z)
% 119.65/15.69 = { by lemma 68 R->L }
% 119.65/15.69 fresh63(fresh98(para(W, X, a, Z), true, Y, Z, X), true, X, Y, Z)
% 119.65/15.69 = { by axiom 39 (ruleD45) R->L }
% 119.65/15.69 fresh63(fresh180(midp(W, Y, a), true, Y, a, Z, W, X), true, X, Y, Z)
% 119.65/15.69 = { by axiom 31 (ruleD11) R->L }
% 119.65/15.69 fresh63(fresh180(fresh144(midp(W, a, Y), true, Y, a, W), true, Y, a, Z, W, X), true, X, Y, Z)
% 119.65/15.69 = { by axiom 7 (ruleD45) R->L }
% 119.65/15.69 fresh63(fresh180(fresh144(fresh98(true, true, a, Y, W), true, Y, a, W), true, Y, a, Z, W, X), true, X, Y, Z)
% 119.65/15.70 = { by lemma 68 R->L }
% 119.65/15.70 fresh63(fresh180(fresh144(fresh98(para(midpnt1, W, b, Y), true, a, Y, W), true, Y, a, W), true, Y, a, Z, W, X), true, X, Y, Z)
% 119.65/15.70 = { by axiom 39 (ruleD45) R->L }
% 119.65/15.70 fresh63(fresh180(fresh144(fresh180(midp(midpnt1, a, b), true, a, b, Y, midpnt1, W), true, Y, a, W), true, Y, a, Z, W, X), true, X, Y, Z)
% 119.65/15.70 = { by axiom 1 (exemplo6GDDFULL012005_15) }
% 119.65/15.70 fresh63(fresh180(fresh144(fresh180(true, true, a, b, Y, midpnt1, W), true, Y, a, W), true, Y, a, Z, W, X), true, X, Y, Z)
% 119.65/15.70 = { by lemma 69 }
% 119.65/15.70 fresh63(fresh180(fresh144(true, true, Y, a, W), true, Y, a, Z, W, X), true, X, Y, Z)
% 119.65/15.70 = { by axiom 4 (ruleD11) }
% 119.65/15.70 fresh63(fresh180(true, true, Y, a, Z, W, X), true, X, Y, Z)
% 119.65/15.70 = { by lemma 69 }
% 119.65/15.70 fresh63(true, true, X, Y, Z)
% 119.65/15.70 = { by axiom 9 (ruleD68) }
% 119.65/15.70 true
% 119.65/15.70
% 119.65/15.70 Goal 1 (exemplo6GDDFULL012005_27): perp(a1, o, b1, c1) = true.
% 119.65/15.70 Proof:
% 119.65/15.70 perp(a1, o, b1, c1)
% 119.65/15.70 = { by axiom 29 (ruleD10) R->L }
% 119.65/15.70 fresh147(true, true, a1, o, X, X, b1, c1)
% 119.65/15.70 = { by axiom 19 (ruleD56) R->L }
% 119.65/15.70 fresh147(fresh79(true, true, X, X, b1, c1), true, a1, o, X, X, b1, c1)
% 119.65/15.70 = { by lemma 70 R->L }
% 119.65/15.70 fresh147(fresh79(cong(X, b1, X, b1), true, X, X, b1, c1), true, a1, o, X, X, b1, c1)
% 119.65/15.70 = { by axiom 44 (ruleD56) R->L }
% 119.65/15.70 fresh147(fresh80(cong(X, c1, X, c1), true, X, X, b1, c1), true, a1, o, X, X, b1, c1)
% 119.65/15.70 = { by lemma 70 }
% 119.65/15.70 fresh147(fresh80(true, true, X, X, b1, c1), true, a1, o, X, X, b1, c1)
% 119.65/15.70 = { by axiom 18 (ruleD56) }
% 119.65/15.70 fresh147(perp(X, X, b1, c1), true, a1, o, X, X, b1, c1)
% 119.65/15.70 = { by axiom 50 (ruleD10) }
% 119.65/15.70 fresh145(para(a1, o, X, X), true, a1, o, b1, c1)
% 119.65/15.70 = { by lemma 68 }
% 119.65/15.70 fresh145(true, true, a1, o, b1, c1)
% 119.65/15.70 = { by axiom 11 (ruleD10) }
% 119.65/15.70 true
% 119.65/15.70 % SZS output end Proof
% 119.65/15.70
% 119.65/15.70 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------