TSTP Solution File: GEO590+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GEO590+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 : n018.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 23:29:27 EDT 2023
% Result : Theorem 12.20s 1.93s
% Output : Proof 12.88s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.13 % Problem : GEO590+1 : TPTP v8.1.2. Released v7.5.0.
% 0.12/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n018.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Tue Aug 29 19:22:03 EDT 2023
% 0.14/0.35 % CPUTime :
% 12.20/1.93 Command-line arguments: --no-flatten-goal
% 12.20/1.93
% 12.20/1.93 % SZS status Theorem
% 12.20/1.93
% 12.20/1.97 % SZS output start Proof
% 12.20/1.97 Take the following subset of the input axioms:
% 12.20/1.98 fof(exemplo6GDDFULL416052, conjecture, ![A, C, D, E, F, O]: ((perp(E, C, E, D) & (midp(O, D, C) & (perp(C, D, C, A) & (perp(E, O, E, A) & (coll(F, C, A) & coll(F, D, E)))))) => cong(A, E, A, F))).
% 12.20/1.98 fof(ruleD1, axiom, ![B, A2, C2]: (coll(A2, B, C2) => coll(A2, C2, B))).
% 12.20/1.98 fof(ruleD11, axiom, ![M, B2, A2_2]: (midp(M, B2, A2_2) => midp(M, A2_2, B2))).
% 12.20/1.98 fof(ruleD17, axiom, ![B2, C2, D2, A2_2, E2]: ((cyclic(A2_2, B2, C2, D2) & cyclic(A2_2, B2, C2, E2)) => cyclic(B2, C2, D2, E2))).
% 12.20/1.98 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))).
% 12.20/1.98 fof(ruleD2, axiom, ![B2, C2, A2_2]: (coll(A2_2, B2, C2) => coll(B2, A2_2, C2))).
% 12.20/1.98 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))).
% 12.20/1.98 fof(ruleD3, axiom, ![B2, C2, D2, A2_2]: ((coll(A2_2, B2, C2) & coll(A2_2, B2, D2)) => coll(C2, D2, A2_2))).
% 12.20/1.98 fof(ruleD4, axiom, ![B2, C2, D2, A2_2]: (para(A2_2, B2, C2, D2) => para(A2_2, B2, D2, C2))).
% 12.20/1.98 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))).
% 12.20/1.98 fof(ruleD41, axiom, ![B2, A2_2, P2, Q2]: (cyclic(A2_2, B2, P2, Q2) => eqangle(P2, A2_2, P2, B2, Q2, A2_2, Q2, B2))).
% 12.20/1.98 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))).
% 12.20/1.98 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))).
% 12.20/1.98 fof(ruleD44, axiom, ![B2, C2, A2_2, E2, F2]: ((midp(E2, A2_2, B2) & midp(F2, A2_2, C2)) => para(E2, F2, B2, C2))).
% 12.20/1.98 fof(ruleD45, axiom, ![B2, C2, A2_2, E2, F2]: ((midp(E2, A2_2, B2) & (para(E2, F2, B2, C2) & coll(F2, A2_2, C2))) => midp(F2, A2_2, C2))).
% 12.20/1.98 fof(ruleD51, axiom, ![O2, B2, C2, A2_2, M2]: ((circle(O2, A2_2, B2, C2) & (coll(M2, B2, C2) & eqangle(A2_2, B2, A2_2, C2, O2, B2, O2, M2))) => midp(M2, B2, C2))).
% 12.20/1.98 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))).
% 12.20/1.98 fof(ruleD63, axiom, ![B2, C2, D2, A2_2, M2]: ((midp(M2, A2_2, B2) & midp(M2, C2, D2)) => para(A2_2, C2, B2, D2))).
% 12.20/1.98 fof(ruleD66, axiom, ![B2, C2, A2_2]: (para(A2_2, B2, A2_2, C2) => coll(A2_2, B2, C2))).
% 12.20/1.98 fof(ruleD68, axiom, ![B2, C2, A2_2]: (midp(A2_2, B2, C2) => cong(A2_2, B2, A2_2, C2))).
% 12.20/1.98 fof(ruleD7, axiom, ![B2, C2, D2, A2_2]: (perp(A2_2, B2, C2, D2) => perp(A2_2, B2, D2, C2))).
% 12.20/1.98 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))).
% 12.20/1.98 fof(ruleD8, axiom, ![B2, C2, D2, A2_2]: (perp(A2_2, B2, C2, D2) => perp(C2, D2, A2_2, B2))).
% 12.20/1.98 fof(ruleD9, axiom, ![B2, C2, D2, A2_2, E2, F2]: ((perp(A2_2, B2, C2, D2) & perp(C2, D2, E2, F2)) => para(A2_2, B2, E2, F2))).
% 12.20/1.98 fof(ruleX14, axiom, ![B2, C2, D2, A2_2]: ?[O2]: ((perp(A2_2, C2, C2, B2) & cyclic(A2_2, B2, C2, D2)) => circle(O2, A2_2, B2, C2))).
% 12.20/1.98
% 12.20/1.98 Now clausify the problem and encode Horn clauses using encoding 3 of
% 12.20/1.98 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 12.20/1.98 We repeatedly replace C & s=t => u=v by the two clauses:
% 12.20/1.98 fresh(y, y, x1...xn) = u
% 12.20/1.98 C => fresh(s, t, x1...xn) = v
% 12.20/1.98 where fresh is a fresh function symbol and x1..xn are the free
% 12.20/1.98 variables of u and v.
% 12.20/1.98 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 12.20/1.98 input problem has no model of domain size 1).
% 12.20/1.98
% 12.20/1.98 The encoding turns the above axioms into the following unit equations and goals:
% 12.20/1.98
% 12.20/1.98 Axiom 1 (exemplo6GDDFULL416052_5): midp(o, d, c) = true.
% 12.20/1.98 Axiom 2 (exemplo6GDDFULL416052_2): perp(e, o, e, a) = true.
% 12.20/1.98 Axiom 3 (ruleD45): fresh181(X, X, Y, Z, W) = true.
% 12.20/1.98 Axiom 4 (ruleD51): fresh179(X, X, Y, Z, W) = true.
% 12.20/1.98 Axiom 5 (ruleD1): fresh146(X, X, Y, Z, W) = true.
% 12.20/1.98 Axiom 6 (ruleD11): fresh144(X, X, Y, Z, W) = true.
% 12.20/1.98 Axiom 7 (ruleD2): fresh133(X, X, Y, Z, W) = true.
% 12.20/1.98 Axiom 8 (ruleD3): fresh119(X, X, Y, Z, W) = true.
% 12.20/1.98 Axiom 9 (ruleD45): fresh98(X, X, Y, Z, W) = midp(W, Y, Z).
% 12.20/1.98 Axiom 10 (ruleD51): fresh89(X, X, Y, Z, W) = midp(W, Y, Z).
% 12.20/1.98 Axiom 11 (ruleD66): fresh66(X, X, Y, Z, W) = true.
% 12.20/1.98 Axiom 12 (ruleD68): fresh63(X, X, Y, Z, W) = true.
% 12.20/1.98 Axiom 13 (ruleX14): fresh36(X, X, Y, Z, W) = true.
% 12.20/1.98 Axiom 14 (ruleD43): fresh185(X, X, Y, Z, W, V) = true.
% 12.20/1.98 Axiom 15 (ruleD17): fresh136(X, X, Y, Z, W, V) = true.
% 12.20/1.98 Axiom 16 (ruleD3): fresh120(X, X, Y, Z, W, V) = coll(W, V, Y).
% 12.20/1.98 Axiom 17 (ruleD4): fresh105(X, X, Y, Z, W, V) = true.
% 12.20/1.98 Axiom 18 (ruleD41): fresh103(X, X, Y, Z, W, V) = true.
% 12.20/1.98 Axiom 19 (ruleD42b): fresh102(X, X, Y, Z, W, V) = cyclic(Y, Z, W, V).
% 12.20/1.98 Axiom 20 (ruleD42b): fresh101(X, X, Y, Z, W, V) = true.
% 12.20/1.98 Axiom 21 (ruleD44): fresh99(X, X, Y, Z, W, V) = true.
% 12.20/1.98 Axiom 22 (ruleD56): fresh80(X, X, Y, Z, W, V) = perp(Y, Z, W, V).
% 12.20/1.98 Axiom 23 (ruleD56): fresh79(X, X, Y, Z, W, V) = true.
% 12.20/1.98 Axiom 24 (ruleD63): fresh69(X, X, Y, Z, W, V) = true.
% 12.20/1.98 Axiom 25 (ruleD7): fresh61(X, X, Y, Z, W, V) = true.
% 12.20/1.98 Axiom 26 (ruleD73): fresh57(X, X, Y, Z, W, V) = true.
% 12.20/1.98 Axiom 27 (ruleD8): fresh52(X, X, Y, Z, W, V) = true.
% 12.20/1.98 Axiom 28 (ruleD9): fresh50(X, X, Y, Z, W, V) = true.
% 12.20/1.98 Axiom 29 (ruleD43): fresh183(X, X, Y, Z, W, V, U) = cong(Y, Z, V, U).
% 12.20/1.98 Axiom 30 (ruleD17): fresh137(X, X, Y, Z, W, V, U) = cyclic(Z, W, V, U).
% 12.20/1.98 Axiom 31 (ruleD44): fresh100(X, X, Y, Z, W, V, U) = para(V, U, Z, W).
% 12.20/1.98 Axiom 32 (ruleD63): fresh70(X, X, Y, Z, W, V, U) = para(Y, W, Z, V).
% 12.20/1.98 Axiom 33 (ruleX14): fresh37(X, X, Y, Z, W) = circle(o2(Y, Z, W), Y, Z, W).
% 12.20/1.98 Axiom 34 (ruleD45): fresh180(X, X, Y, Z, W, V, U) = fresh181(coll(U, Y, W), true, Y, W, U).
% 12.20/1.98 Axiom 35 (ruleD51): fresh178(X, X, Y, Z, W, V, U) = fresh179(coll(V, Z, W), true, Z, W, V).
% 12.20/1.98 Axiom 36 (ruleD1): fresh146(coll(X, Y, Z), true, X, Y, Z) = coll(X, Z, Y).
% 12.20/1.98 Axiom 37 (ruleD11): fresh144(midp(X, Y, Z), true, Z, Y, X) = midp(X, Z, Y).
% 12.20/1.98 Axiom 38 (ruleD2): fresh133(coll(X, Y, Z), true, X, Y, Z) = coll(Y, X, Z).
% 12.20/1.98 Axiom 39 (ruleD40): fresh104(X, X, Y, Z, W, V, U, T) = true.
% 12.20/1.98 Axiom 40 (ruleD68): fresh63(midp(X, Y, Z), true, X, Y, Z) = cong(X, Y, X, Z).
% 12.20/1.98 Axiom 41 (ruleD9): fresh51(X, X, Y, Z, W, V, U, T) = para(Y, Z, U, T).
% 12.20/1.98 Axiom 42 (ruleD3): fresh120(coll(X, Y, Z), true, X, Y, W, Z) = fresh119(coll(X, Y, W), true, X, W, Z).
% 12.20/1.98 Axiom 43 (ruleD66): fresh66(para(X, Y, X, Z), true, X, Y, Z) = coll(X, Y, Z).
% 12.20/1.98 Axiom 44 (ruleX14): fresh37(cyclic(X, Y, Z, W), true, X, Y, Z) = fresh36(perp(X, Z, Z, Y), true, X, Y, Z).
% 12.20/1.98 Axiom 45 (ruleD43): fresh184(X, X, Y, Z, W, V, U) = fresh185(cyclic(Y, Z, W, V), true, Y, Z, V, U).
% 12.20/1.98 Axiom 46 (ruleD45): fresh180(midp(X, Y, Z), true, Y, Z, W, X, V) = fresh98(para(X, V, Z, W), true, Y, W, V).
% 12.20/1.98 Axiom 47 (ruleD19): fresh134(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 12.20/1.98 Axiom 48 (ruleD21): fresh131(X, X, Y, Z, W, V, U, T, S, X2) = true.
% 12.20/1.98 Axiom 49 (ruleD4): fresh105(para(X, Y, Z, W), true, X, Y, Z, W) = para(X, Y, W, Z).
% 12.20/1.98 Axiom 50 (ruleD41): fresh103(cyclic(X, Y, Z, W), true, X, Y, Z, W) = eqangle(Z, X, Z, Y, W, X, W, Y).
% 12.20/1.98 Axiom 51 (ruleD44): fresh100(midp(X, Y, Z), true, Y, W, Z, V, X) = fresh99(midp(V, Y, W), true, W, Z, V, X).
% 12.20/1.98 Axiom 52 (ruleD56): fresh80(cong(X, Y, Z, Y), true, X, Z, W, Y) = fresh79(cong(X, W, Z, W), true, X, Z, W, Y).
% 12.20/1.98 Axiom 53 (ruleD63): fresh70(midp(X, Y, Z), true, W, V, Y, Z, X) = fresh69(midp(X, W, V), true, W, V, Y, Z).
% 12.20/1.98 Axiom 54 (ruleD7): fresh61(perp(X, Y, Z, W), true, X, Y, Z, W) = perp(X, Y, W, Z).
% 12.20/1.98 Axiom 55 (ruleD73): fresh58(X, X, Y, Z, W, V, U, T, S, X2) = para(Y, Z, W, V).
% 12.20/1.98 Axiom 56 (ruleD8): fresh52(perp(X, Y, Z, W), true, X, Y, Z, W) = perp(Z, W, X, Y).
% 12.20/1.98 Axiom 57 (ruleD43): fresh182(X, X, Y, Z, W, V, U, T) = fresh183(cyclic(Y, Z, W, U), true, Y, Z, W, V, U).
% 12.20/1.98 Axiom 58 (ruleD17): fresh137(cyclic(X, Y, Z, W), true, X, Y, Z, V, W) = fresh136(cyclic(X, Y, Z, V), true, Y, Z, V, W).
% 12.20/1.98 Axiom 59 (ruleD40): fresh104(para(X, Y, Z, W), true, X, Y, Z, W, V, U) = eqangle(X, Y, V, U, Z, W, V, U).
% 12.20/1.98 Axiom 60 (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).
% 12.20/1.98 Axiom 61 (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).
% 12.20/1.98 Axiom 62 (ruleD51): fresh178(eqangle(X, Y, X, Z, W, Y, W, V), true, X, Y, Z, V, W) = fresh89(circle(W, X, Y, Z), true, Y, Z, V).
% 12.20/1.98 Axiom 63 (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).
% 12.20/1.98 Axiom 64 (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).
% 12.20/1.98 Axiom 65 (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).
% 12.20/1.98 Axiom 66 (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).
% 12.20/1.98
% 12.20/1.98 Lemma 67: para(o, o, c, c) = true.
% 12.20/1.98 Proof:
% 12.20/1.98 para(o, o, c, c)
% 12.20/1.98 = { by axiom 31 (ruleD44) R->L }
% 12.20/1.98 fresh100(true, true, d, c, c, o, o)
% 12.20/1.98 = { by axiom 1 (exemplo6GDDFULL416052_5) R->L }
% 12.20/1.98 fresh100(midp(o, d, c), true, d, c, c, o, o)
% 12.20/1.98 = { by axiom 51 (ruleD44) }
% 12.20/1.98 fresh99(midp(o, d, c), true, c, c, o, o)
% 12.20/1.98 = { by axiom 1 (exemplo6GDDFULL416052_5) }
% 12.20/1.98 fresh99(true, true, c, c, o, o)
% 12.20/1.98 = { by axiom 21 (ruleD44) }
% 12.20/1.98 true
% 12.20/1.98
% 12.20/1.98 Lemma 68: para(X, Y, X, Y) = true.
% 12.20/1.98 Proof:
% 12.20/1.98 para(X, Y, X, Y)
% 12.20/1.98 = { by axiom 55 (ruleD73) R->L }
% 12.20/1.98 fresh58(true, true, X, Y, X, Y, o, o, c, c)
% 12.20/1.98 = { by axiom 48 (ruleD21) R->L }
% 12.20/1.98 fresh58(fresh131(true, true, X, Y, o, o, X, Y, c, c), true, X, Y, X, Y, o, o, c, c)
% 12.20/1.98 = { by axiom 47 (ruleD19) R->L }
% 12.20/1.98 fresh58(fresh131(fresh134(true, true, o, o, X, Y, c, c, X, Y), true, X, Y, o, o, X, Y, c, c), true, X, Y, X, Y, o, o, c, c)
% 12.20/1.98 = { by axiom 39 (ruleD40) R->L }
% 12.20/1.98 fresh58(fresh131(fresh134(fresh104(true, true, o, o, c, c, X, Y), true, o, o, X, Y, c, c, X, Y), true, X, Y, o, o, X, Y, c, c), true, X, Y, X, Y, o, o, c, c)
% 12.20/1.98 = { by lemma 67 R->L }
% 12.20/1.98 fresh58(fresh131(fresh134(fresh104(para(o, o, c, c), true, o, o, c, c, X, Y), true, o, o, X, Y, c, c, X, Y), true, X, Y, o, o, X, Y, c, c), true, X, Y, X, Y, o, o, c, c)
% 12.20/1.98 = { by axiom 59 (ruleD40) }
% 12.20/1.98 fresh58(fresh131(fresh134(eqangle(o, o, X, Y, c, c, X, Y), true, o, o, X, Y, c, c, X, Y), true, X, Y, o, o, X, Y, c, c), true, X, Y, X, Y, o, o, c, c)
% 12.20/1.98 = { by axiom 64 (ruleD19) }
% 12.20/1.98 fresh58(fresh131(eqangle(X, Y, o, o, X, Y, c, c), true, X, Y, o, o, X, Y, c, c), true, X, Y, X, Y, o, o, c, c)
% 12.20/1.98 = { by axiom 65 (ruleD21) }
% 12.20/1.98 fresh58(eqangle(X, Y, X, Y, o, o, c, c), true, X, Y, X, Y, o, o, c, c)
% 12.20/1.98 = { by axiom 66 (ruleD73) }
% 12.20/1.98 fresh57(para(o, o, c, c), true, X, Y, X, Y)
% 12.20/1.98 = { by lemma 67 }
% 12.20/1.98 fresh57(true, true, X, Y, X, Y)
% 12.20/1.98 = { by axiom 26 (ruleD73) }
% 12.20/1.98 true
% 12.20/1.98
% 12.20/1.98 Lemma 69: coll(X, X, Y) = true.
% 12.20/1.98 Proof:
% 12.20/1.98 coll(X, X, Y)
% 12.20/1.98 = { by axiom 36 (ruleD1) R->L }
% 12.20/1.98 fresh146(coll(X, Y, X), true, X, Y, X)
% 12.20/1.98 = { by axiom 38 (ruleD2) R->L }
% 12.20/1.98 fresh146(fresh133(coll(Y, X, X), true, Y, X, X), true, X, Y, X)
% 12.20/1.98 = { by axiom 43 (ruleD66) R->L }
% 12.20/1.98 fresh146(fresh133(fresh66(para(Y, X, Y, X), true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 12.20/1.98 = { by lemma 68 }
% 12.20/1.98 fresh146(fresh133(fresh66(true, true, Y, X, X), true, Y, X, X), true, X, Y, X)
% 12.20/1.98 = { by axiom 11 (ruleD66) }
% 12.20/1.98 fresh146(fresh133(true, true, Y, X, X), true, X, Y, X)
% 12.20/1.98 = { by axiom 7 (ruleD2) }
% 12.20/1.98 fresh146(true, true, X, Y, X)
% 12.20/1.98 = { by axiom 5 (ruleD1) }
% 12.20/1.98 true
% 12.20/1.98
% 12.20/1.98 Lemma 70: coll(X, Y, Z) = true.
% 12.20/1.98 Proof:
% 12.20/1.98 coll(X, Y, Z)
% 12.20/1.98 = { by axiom 16 (ruleD3) R->L }
% 12.20/1.98 fresh120(true, true, Z, Z, X, Y)
% 12.20/1.98 = { by lemma 69 R->L }
% 12.20/1.98 fresh120(coll(Z, Z, Y), true, Z, Z, X, Y)
% 12.20/1.98 = { by axiom 42 (ruleD3) }
% 12.20/1.98 fresh119(coll(Z, Z, X), true, Z, X, Y)
% 12.20/1.98 = { by lemma 69 }
% 12.20/1.98 fresh119(true, true, Z, X, Y)
% 12.20/1.98 = { by axiom 8 (ruleD3) }
% 12.20/1.99 true
% 12.20/1.99
% 12.20/1.99 Lemma 71: cyclic(d, d, c, X) = true.
% 12.20/1.99 Proof:
% 12.20/1.99 cyclic(d, d, c, X)
% 12.20/1.99 = { by axiom 19 (ruleD42b) R->L }
% 12.20/1.99 fresh102(true, true, d, d, c, X)
% 12.20/1.99 = { by axiom 48 (ruleD21) R->L }
% 12.20/1.99 fresh102(fresh131(true, true, c, d, X, d, c, d, X, d), true, d, d, c, X)
% 12.20/1.99 = { by axiom 39 (ruleD40) R->L }
% 12.20/1.99 fresh102(fresh131(fresh104(true, true, c, d, c, d, X, d), true, c, d, X, d, c, d, X, d), true, d, d, c, X)
% 12.20/1.99 = { by axiom 17 (ruleD4) R->L }
% 12.20/1.99 fresh102(fresh131(fresh104(fresh105(true, true, c, d, d, c), true, c, d, c, d, X, d), true, c, d, X, d, c, d, X, d), true, d, d, c, X)
% 12.20/1.99 = { by axiom 24 (ruleD63) R->L }
% 12.20/1.99 fresh102(fresh131(fresh104(fresh105(fresh69(true, true, c, d, d, c), true, c, d, d, c), true, c, d, c, d, X, d), true, c, d, X, d, c, d, X, d), true, d, d, c, X)
% 12.20/1.99 = { by axiom 6 (ruleD11) R->L }
% 12.20/1.99 fresh102(fresh131(fresh104(fresh105(fresh69(fresh144(true, true, c, d, o), true, c, d, d, c), true, c, d, d, c), true, c, d, c, d, X, d), true, c, d, X, d, c, d, X, d), true, d, d, c, X)
% 12.20/1.99 = { by axiom 1 (exemplo6GDDFULL416052_5) R->L }
% 12.20/1.99 fresh102(fresh131(fresh104(fresh105(fresh69(fresh144(midp(o, d, c), true, c, d, o), true, c, d, d, c), true, c, d, d, c), true, c, d, c, d, X, d), true, c, d, X, d, c, d, X, d), true, d, d, c, X)
% 12.20/1.99 = { by axiom 37 (ruleD11) }
% 12.20/1.99 fresh102(fresh131(fresh104(fresh105(fresh69(midp(o, c, d), true, c, d, d, c), true, c, d, d, c), true, c, d, c, d, X, d), true, c, d, X, d, c, d, X, d), true, d, d, c, X)
% 12.20/1.99 = { by axiom 53 (ruleD63) R->L }
% 12.20/1.99 fresh102(fresh131(fresh104(fresh105(fresh70(midp(o, d, c), true, c, d, d, c, o), true, c, d, d, c), true, c, d, c, d, X, d), true, c, d, X, d, c, d, X, d), true, d, d, c, X)
% 12.20/1.99 = { by axiom 1 (exemplo6GDDFULL416052_5) }
% 12.20/1.99 fresh102(fresh131(fresh104(fresh105(fresh70(true, true, c, d, d, c, o), true, c, d, d, c), true, c, d, c, d, X, d), true, c, d, X, d, c, d, X, d), true, d, d, c, X)
% 12.20/1.99 = { by axiom 32 (ruleD63) }
% 12.20/1.99 fresh102(fresh131(fresh104(fresh105(para(c, d, d, c), true, c, d, d, c), true, c, d, c, d, X, d), true, c, d, X, d, c, d, X, d), true, d, d, c, X)
% 12.20/1.99 = { by axiom 49 (ruleD4) }
% 12.20/1.99 fresh102(fresh131(fresh104(para(c, d, c, d), true, c, d, c, d, X, d), true, c, d, X, d, c, d, X, d), true, d, d, c, X)
% 12.20/1.99 = { by axiom 59 (ruleD40) }
% 12.20/1.99 fresh102(fresh131(eqangle(c, d, X, d, c, d, X, d), true, c, d, X, d, c, d, X, d), true, d, d, c, X)
% 12.20/1.99 = { by axiom 65 (ruleD21) }
% 12.20/1.99 fresh102(eqangle(c, d, c, d, X, d, X, d), true, d, d, c, X)
% 12.20/1.99 = { by axiom 61 (ruleD42b) }
% 12.20/1.99 fresh101(coll(c, X, d), true, d, d, c, X)
% 12.20/1.99 = { by lemma 70 }
% 12.20/1.99 fresh101(true, true, d, d, c, X)
% 12.20/1.99 = { by axiom 20 (ruleD42b) }
% 12.20/1.99 true
% 12.20/1.99
% 12.20/1.99 Lemma 72: cyclic(d, c, X, Y) = true.
% 12.20/1.99 Proof:
% 12.20/1.99 cyclic(d, c, X, Y)
% 12.20/1.99 = { by axiom 30 (ruleD17) R->L }
% 12.20/1.99 fresh137(true, true, d, d, c, X, Y)
% 12.20/1.99 = { by lemma 71 R->L }
% 12.20/1.99 fresh137(cyclic(d, d, c, Y), true, d, d, c, X, Y)
% 12.20/1.99 = { by axiom 58 (ruleD17) }
% 12.20/1.99 fresh136(cyclic(d, d, c, X), true, d, c, X, Y)
% 12.20/1.99 = { by lemma 71 }
% 12.20/1.99 fresh136(true, true, d, c, X, Y)
% 12.20/1.99 = { by axiom 15 (ruleD17) }
% 12.20/1.99 true
% 12.20/1.99
% 12.20/1.99 Lemma 73: cyclic(c, X, Y, Z) = true.
% 12.20/1.99 Proof:
% 12.20/1.99 cyclic(c, X, Y, Z)
% 12.20/1.99 = { by axiom 30 (ruleD17) R->L }
% 12.20/1.99 fresh137(true, true, d, c, X, Y, Z)
% 12.20/1.99 = { by lemma 72 R->L }
% 12.20/1.99 fresh137(cyclic(d, c, X, Z), true, d, c, X, Y, Z)
% 12.20/1.99 = { by axiom 58 (ruleD17) }
% 12.20/1.99 fresh136(cyclic(d, c, X, Y), true, c, X, Y, Z)
% 12.20/1.99 = { by lemma 72 }
% 12.20/1.99 fresh136(true, true, c, X, Y, Z)
% 12.20/1.99 = { by axiom 15 (ruleD17) }
% 12.20/1.99 true
% 12.20/1.99
% 12.20/1.99 Lemma 74: cyclic(X, Y, Z, W) = true.
% 12.20/1.99 Proof:
% 12.20/1.99 cyclic(X, Y, Z, W)
% 12.20/1.99 = { by axiom 30 (ruleD17) R->L }
% 12.20/1.99 fresh137(true, true, c, X, Y, Z, W)
% 12.20/1.99 = { by lemma 73 R->L }
% 12.20/1.99 fresh137(cyclic(c, X, Y, W), true, c, X, Y, Z, W)
% 12.20/1.99 = { by axiom 58 (ruleD17) }
% 12.20/1.99 fresh136(cyclic(c, X, Y, Z), true, X, Y, Z, W)
% 12.20/1.99 = { by lemma 73 }
% 12.20/1.99 fresh136(true, true, X, Y, Z, W)
% 12.20/1.99 = { by axiom 15 (ruleD17) }
% 12.20/1.99 true
% 12.20/1.99
% 12.20/1.99 Lemma 75: cong(X, Y, X, Y) = true.
% 12.20/1.99 Proof:
% 12.20/1.99 cong(X, Y, X, Y)
% 12.20/1.99 = { by axiom 29 (ruleD43) R->L }
% 12.20/1.99 fresh183(true, true, X, Y, Z, X, Y)
% 12.20/1.99 = { by lemma 74 R->L }
% 12.20/1.99 fresh183(cyclic(X, Y, Z, Y), true, X, Y, Z, X, Y)
% 12.20/1.99 = { by axiom 57 (ruleD43) R->L }
% 12.20/1.99 fresh182(true, true, X, Y, Z, X, Y, Z)
% 12.20/1.99 = { by axiom 39 (ruleD40) R->L }
% 12.20/1.99 fresh182(fresh104(true, true, Z, X, Z, X, Z, Y), true, X, Y, Z, X, Y, Z)
% 12.20/1.99 = { by lemma 68 R->L }
% 12.20/1.99 fresh182(fresh104(para(Z, X, Z, X), true, Z, X, Z, X, Z, Y), true, X, Y, Z, X, Y, Z)
% 12.20/1.99 = { by axiom 59 (ruleD40) }
% 12.20/1.99 fresh182(eqangle(Z, X, Z, Y, Z, X, Z, Y), true, X, Y, Z, X, Y, Z)
% 12.20/1.99 = { by axiom 63 (ruleD43) }
% 12.20/1.99 fresh184(cyclic(X, Y, Z, Z), true, X, Y, Z, X, Y)
% 12.20/1.99 = { by lemma 74 }
% 12.20/1.99 fresh184(true, true, X, Y, Z, X, Y)
% 12.20/1.99 = { by axiom 45 (ruleD43) }
% 12.20/1.99 fresh185(cyclic(X, Y, Z, X), true, X, Y, X, Y)
% 12.20/1.99 = { by lemma 74 }
% 12.20/1.99 fresh185(true, true, X, Y, X, Y)
% 12.20/1.99 = { by axiom 14 (ruleD43) }
% 12.20/1.99 true
% 12.20/1.99
% 12.20/1.99 Lemma 76: perp(X, X, Y, Z) = true.
% 12.20/1.99 Proof:
% 12.20/1.99 perp(X, X, Y, Z)
% 12.20/1.99 = { by axiom 22 (ruleD56) R->L }
% 12.20/1.99 fresh80(true, true, X, X, Y, Z)
% 12.20/1.99 = { by lemma 75 R->L }
% 12.20/1.99 fresh80(cong(X, Z, X, Z), true, X, X, Y, Z)
% 12.20/1.99 = { by axiom 52 (ruleD56) }
% 12.20/1.99 fresh79(cong(X, Y, X, Y), true, X, X, Y, Z)
% 12.20/1.99 = { by lemma 75 }
% 12.20/1.99 fresh79(true, true, X, X, Y, Z)
% 12.20/1.99 = { by axiom 23 (ruleD56) }
% 12.20/1.99 true
% 12.20/1.99
% 12.20/1.99 Goal 1 (exemplo6GDDFULL416052_6): cong(a, e, a, f) = true.
% 12.20/1.99 Proof:
% 12.20/1.99 cong(a, e, a, f)
% 12.20/1.99 = { by axiom 40 (ruleD68) R->L }
% 12.20/1.99 fresh63(midp(a, e, f), true, a, e, f)
% 12.20/1.99 = { by axiom 9 (ruleD45) R->L }
% 12.20/1.99 fresh63(fresh98(true, true, e, f, a), true, a, e, f)
% 12.20/1.99 = { by axiom 28 (ruleD9) R->L }
% 12.20/1.99 fresh63(fresh98(fresh50(true, true, e, a, o, f), true, e, f, a), true, a, e, f)
% 12.20/1.99 = { by axiom 27 (ruleD8) R->L }
% 12.20/1.99 fresh63(fresh98(fresh50(fresh52(true, true, X, X, e, a), true, e, a, o, f), true, e, f, a), true, a, e, f)
% 12.20/1.99 = { by lemma 76 R->L }
% 12.20/1.99 fresh63(fresh98(fresh50(fresh52(perp(X, X, e, a), true, X, X, e, a), true, e, a, o, f), true, e, f, a), true, a, e, f)
% 12.20/1.99 = { by axiom 56 (ruleD8) }
% 12.20/1.99 fresh63(fresh98(fresh50(perp(e, a, X, X), true, e, a, o, f), true, e, f, a), true, a, e, f)
% 12.20/1.99 = { by axiom 60 (ruleD9) R->L }
% 12.20/1.99 fresh63(fresh98(fresh51(perp(X, X, o, f), true, e, a, X, X, o, f), true, e, f, a), true, a, e, f)
% 12.20/1.99 = { by lemma 76 }
% 12.20/1.99 fresh63(fresh98(fresh51(true, true, e, a, X, X, o, f), true, e, f, a), true, a, e, f)
% 12.20/1.99 = { by axiom 41 (ruleD9) }
% 12.20/1.99 fresh63(fresh98(para(e, a, o, f), true, e, f, a), true, a, e, f)
% 12.20/1.99 = { by axiom 46 (ruleD45) R->L }
% 12.20/1.99 fresh63(fresh180(midp(e, e, o), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by axiom 37 (ruleD11) R->L }
% 12.20/1.99 fresh63(fresh180(fresh144(midp(e, o, e), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by axiom 10 (ruleD51) R->L }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh89(true, true, o, e, e), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by axiom 13 (ruleX14) R->L }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh89(fresh36(true, true, a, o, e), true, o, e, e), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by axiom 27 (ruleD8) R->L }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh89(fresh36(fresh52(true, true, e, o, a, e), true, a, o, e), true, o, e, e), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by axiom 25 (ruleD7) R->L }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh89(fresh36(fresh52(fresh61(true, true, e, o, e, a), true, e, o, a, e), true, a, o, e), true, o, e, e), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by axiom 2 (exemplo6GDDFULL416052_2) R->L }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh89(fresh36(fresh52(fresh61(perp(e, o, e, a), true, e, o, e, a), true, e, o, a, e), true, a, o, e), true, o, e, e), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by axiom 54 (ruleD7) }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh89(fresh36(fresh52(perp(e, o, a, e), true, e, o, a, e), true, a, o, e), true, o, e, e), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by axiom 56 (ruleD8) }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh89(fresh36(perp(a, e, e, o), true, a, o, e), true, o, e, e), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by axiom 44 (ruleX14) R->L }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh89(fresh37(cyclic(a, o, e, Y), true, a, o, e), true, o, e, e), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by lemma 74 }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh89(fresh37(true, true, a, o, e), true, o, e, e), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by axiom 33 (ruleX14) }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh89(circle(o2(a, o, e), a, o, e), true, o, e, e), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by axiom 62 (ruleD51) R->L }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh178(eqangle(a, o, a, e, o2(a, o, e), o, o2(a, o, e), e), true, a, o, e, e, o2(a, o, e)), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by axiom 50 (ruleD41) R->L }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh178(fresh103(cyclic(o, e, a, o2(a, o, e)), true, o, e, a, o2(a, o, e)), true, a, o, e, e, o2(a, o, e)), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by lemma 74 }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh178(fresh103(true, true, o, e, a, o2(a, o, e)), true, a, o, e, e, o2(a, o, e)), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by axiom 18 (ruleD41) }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh178(true, true, a, o, e, e, o2(a, o, e)), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.20/1.99 = { by axiom 35 (ruleD51) }
% 12.20/1.99 fresh63(fresh180(fresh144(fresh179(coll(e, o, e), true, o, e, e), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.88/1.99 = { by lemma 70 }
% 12.88/1.99 fresh63(fresh180(fresh144(fresh179(true, true, o, e, e), true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.88/1.99 = { by axiom 4 (ruleD51) }
% 12.88/1.99 fresh63(fresh180(fresh144(true, true, e, o, e), true, e, o, f, e, a), true, a, e, f)
% 12.88/1.99 = { by axiom 6 (ruleD11) }
% 12.88/1.99 fresh63(fresh180(true, true, e, o, f, e, a), true, a, e, f)
% 12.88/1.99 = { by axiom 34 (ruleD45) }
% 12.88/1.99 fresh63(fresh181(coll(a, e, f), true, e, f, a), true, a, e, f)
% 12.88/1.99 = { by lemma 70 }
% 12.88/1.99 fresh63(fresh181(true, true, e, f, a), true, a, e, f)
% 12.88/1.99 = { by axiom 3 (ruleD45) }
% 12.88/1.99 fresh63(true, true, a, e, f)
% 12.88/1.99 = { by axiom 12 (ruleD68) }
% 12.88/1.99 true
% 12.88/1.99 % SZS output end Proof
% 12.88/1.99
% 12.88/1.99 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------