0.00/0.03 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.04 % Command : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn 0.03/0.23 % Computer : n151.star.cs.uiowa.edu 0.03/0.23 % Model : x86_64 x86_64 0.03/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.03/0.23 % Memory : 32218.625MB 0.03/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.03/0.23 % CPULimit : 300 0.03/0.23 % DateTime : Sat Jul 14 06:01:26 CDT 2018 0.03/0.23 % CPUTime : 75.40/75.64 % SZS status Theorem 75.40/75.64 75.40/75.64 % SZS output start Proof 75.40/75.64 Take the following subset of the input axioms: 76.07/76.27 fof(goals_13, conjecture, 76.07/76.27 ![X17, X18, X19]: 76.07/76.27 ('>='(X19, X17) 76.07/76.27 <= ('>='('==>'(X17, X18), X17) & X19='==>'(X19, X18)))). 76.07/76.27 fof(sos_01, axiom, 76.07/76.27 ![A, B, C]: '+'('+'(A, B), C)='+'(A, '+'(B, C))). 76.07/76.27 fof(sos_02, axiom, ![A, B]: '+'(B, A)='+'(A, B)). 76.07/76.27 fof(sos_03, axiom, ![A]: '+'(A, '0')=A). 76.07/76.27 fof(sos_04, axiom, ![A]: '>='(A, A)). 76.07/76.27 fof(sos_06, axiom, 76.07/76.27 ![X3, X4]: (X4=X3 <= ('>='(X4, X3) & '>='(X3, X4)))). 76.07/76.27 fof(sos_07, axiom, 76.07/76.27 ![X5, X6, X7]: 76.07/76.27 ('>='(X6, '==>'(X5, X7)) <=> '>='('+'(X5, X6), X7))). 76.07/76.27 fof(sos_08, axiom, ![A]: '>='(A, '0')). 76.07/76.27 fof(sos_09, axiom, 76.07/76.27 ![X8, X9, X10]: 76.07/76.27 ('>='('+'(X8, X10), '+'(X9, X10)) <= '>='(X8, X9))). 76.07/76.27 fof(sos_10, axiom, 76.07/76.27 ![X11, X12, X13]: 76.07/76.27 ('>='('==>'(X12, X13), '==>'(X11, X13)) <= '>='(X11, X12))). 76.07/76.27 fof(sos_11, axiom, 76.07/76.27 ![X14, X15, X16]: 76.07/76.27 ('>='('==>'(X16, X14), '==>'(X16, X15)) <= '>='(X14, X15))). 76.07/76.27 fof(sos_12, axiom, 76.07/76.27 ![A, B]: '+'(B, '==>'(B, A))='+'(A, '==>'(A, B))). 76.07/76.27 76.07/76.27 Now clausify the problem and encode Horn clauses using encoding 3 of 76.07/76.27 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 76.07/76.27 We repeatedly replace C & s=t => u=v by the two clauses: 76.07/76.27 $$fresh(y, y, x1...xn) = u 76.07/76.27 C => $$fresh(s, t, x1...xn) = v 76.07/76.27 where $$fresh is a fresh function symbol and x1..xn are the free 76.07/76.27 variables of u and v. 76.07/76.27 A predicate p(X) is encoded as p(X)=$$true (this is sound, because the 76.07/76.27 input problem has no model of domain size 1). 76.07/76.27 76.07/76.27 The encoding turns the above axioms into the following unit equations and goals: 76.07/76.27 76.07/76.27 Axiom 3 (sos_06): $$fresh(X, X, Y, Z) = Y. 76.07/76.27 Axiom 4 (sos_06): $$fresh2(X, X, Y, Z) = Z. 76.07/76.27 Axiom 5 (sos_07): $$fresh6(X, X, Y, Z, W) = $$true. 76.07/76.27 Axiom 6 (sos_07_1): $$fresh5(X, X, Y, Z, W) = $$true. 76.07/76.27 Axiom 7 (sos_09): $$fresh4(X, X, Y, Z, W) = $$true. 76.07/76.27 Axiom 8 (sos_10): $$fresh3(X, X, Y, Z, W) = $$true. 76.07/76.27 Axiom 9 (sos_11): $$fresh9(X, X, Y, Z, W) = $$true. 76.07/76.27 Axiom 10 (sos_11): $$fresh9(X >= Y, $$true, X, Y, Z) = (Z ==> X) >= (Z ==> Y). 76.07/76.27 Axiom 11 (sos_01): (X + Y) + Z = X + (Y + Z). 76.07/76.27 Axiom 12 (sos_09): $$fresh4(X >= Y, $$true, X, Y, Z) = (X + Z) >= (Y + Z). 76.07/76.27 Axiom 13 (sos_07_1): $$fresh5((X + Y) >= Z, $$true, X, Y, Z) = Y >= (X ==> Z). 76.07/76.27 Axiom 14 (sos_07): $$fresh6(X >= (Y ==> Z), $$true, Y, X, Z) = (Y + X) >= Z. 76.07/76.27 Axiom 15 (sos_04): X >= X = $$true. 76.07/76.27 Axiom 16 (sos_12): X + (X ==> Y) = Y + (Y ==> X). 76.07/76.27 Axiom 17 (sos_03): X + 0 = X. 76.07/76.27 Axiom 18 (sos_02): X + Y = Y + X. 76.07/76.27 Axiom 19 (sos_10): $$fresh3(X >= Y, $$true, X, Y, Z) = (Y ==> Z) >= (X ==> Z). 76.07/76.27 Axiom 20 (sos_08): X >= 0 = $$true. 76.07/76.27 Axiom 21 (sos_06): $$fresh2(X >= Y, $$true, Y, X) = $$fresh(Y >= X, $$true, Y, X). 76.07/76.27 Axiom 23 (goals_13): sK2_goals_13_X19 = sK2_goals_13_X19 ==> sK3_goals_13_X18. 76.07/76.27 Axiom 24 (goals_13_1): (sK1_goals_13_X17 ==> sK3_goals_13_X18) >= sK1_goals_13_X17 = $$true. 76.07/76.27 76.07/76.27 Lemma 25: 0 + X = X. 76.07/76.27 Proof: 76.07/76.27 0 + X 76.07/76.27 = { by axiom 18 (sos_02) } 76.07/76.27 X + 0 76.07/76.27 = { by axiom 17 (sos_03) } 76.07/76.27 X 76.07/76.27 76.07/76.27 Lemma 26: (X + Y) >= X = $$true. 76.07/76.27 Proof: 76.07/76.27 (X + Y) >= X 76.07/76.27 = { by axiom 18 (sos_02) } 76.07/76.27 (Y + X) >= X 76.07/76.27 = { by lemma 25 } 76.07/76.27 (Y + X) >= (0 + X) 76.07/76.27 = { by axiom 12 (sos_09) } 76.07/76.27 $$fresh4(Y >= 0, $$true, Y, 0, X) 76.07/76.27 = { by axiom 20 (sos_08) } 76.07/76.27 $$fresh4($$true, $$true, Y, 0, X) 76.07/76.27 = { by axiom 7 (sos_09) } 76.07/76.27 $$true 76.07/76.27 76.07/76.27 Lemma 27: (X + (X ==> Y)) >= Y = $$true. 76.07/76.27 Proof: 76.07/76.27 (X + (X ==> Y)) >= Y 76.07/76.27 = { by axiom 16 (sos_12) } 76.07/76.27 (Y + (Y ==> X)) >= Y 76.07/76.27 = { by lemma 26 } 76.67/76.90 $$true 76.67/76.90 76.67/76.90 Goal 1 (goals_13_2): sK2_goals_13_X19 >= sK1_goals_13_X17 = $$true. 76.67/76.90 Proof: 76.67/76.90 sK2_goals_13_X19 >= sK1_goals_13_X17 76.67/76.90 = { by axiom 17 (sos_03) } 76.67/76.90 (sK2_goals_13_X19 + 0) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 4 (sos_06) } 76.67/76.90 (sK2_goals_13_X19 + $$fresh2($$true, $$true, sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), 0)) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 6 (sos_07_1) } 76.67/76.90 (sK2_goals_13_X19 + $$fresh2($$fresh5($$true, $$true, sK2_goals_13_X19, 0, (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), $$true, sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), 0)) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 6 (sos_07_1) } 76.67/76.90 (sK2_goals_13_X19 + $$fresh2($$fresh5($$fresh5($$true, $$true, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19, sK1_goals_13_X17), $$true, sK2_goals_13_X19, 0, (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), $$true, sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), 0)) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 5 (sos_07) } 76.67/76.90 (sK2_goals_13_X19 + $$fresh2($$fresh5($$fresh5($$fresh6($$true, $$true, sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK1_goals_13_X17), $$true, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19, sK1_goals_13_X17), $$true, sK2_goals_13_X19, 0, (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), $$true, sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), 0)) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 9 (sos_11) } 76.67/76.90 (sK2_goals_13_X19 + $$fresh2($$fresh5($$fresh5($$fresh6($$fresh9($$true, $$true, sK1_goals_13_X17 ==> sK3_goals_13_X18, sK1_goals_13_X17, sK2_goals_13_X19), $$true, sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK1_goals_13_X17), $$true, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19, sK1_goals_13_X17), $$true, sK2_goals_13_X19, 0, (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), $$true, sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), 0)) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 24 (goals_13_1) } 76.67/76.90 (sK2_goals_13_X19 + $$fresh2($$fresh5($$fresh5($$fresh6($$fresh9((sK1_goals_13_X17 ==> sK3_goals_13_X18) >= sK1_goals_13_X17, $$true, sK1_goals_13_X17 ==> sK3_goals_13_X18, sK1_goals_13_X17, sK2_goals_13_X19), $$true, sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK1_goals_13_X17), $$true, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19, sK1_goals_13_X17), $$true, sK2_goals_13_X19, 0, (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), $$true, sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), 0)) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 10 (sos_11) } 76.67/76.90 (sK2_goals_13_X19 + $$fresh2($$fresh5($$fresh5($$fresh6((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) >= (sK2_goals_13_X19 ==> sK1_goals_13_X17), $$true, sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK1_goals_13_X17), $$true, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19, sK1_goals_13_X17), $$true, sK2_goals_13_X19, 0, (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), $$true, sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), 0)) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 14 (sos_07) } 76.67/76.90 (sK2_goals_13_X19 + $$fresh2($$fresh5($$fresh5((sK2_goals_13_X19 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18))) >= sK1_goals_13_X17, $$true, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19, sK1_goals_13_X17), $$true, sK2_goals_13_X19, 0, (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), $$true, sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), 0)) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 18 (sos_02) } 76.67/76.90 (sK2_goals_13_X19 + $$fresh2($$fresh5($$fresh5(((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) + sK2_goals_13_X19) >= sK1_goals_13_X17, $$true, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19, sK1_goals_13_X17), $$true, sK2_goals_13_X19, 0, (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), $$true, sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), 0)) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 13 (sos_07_1) } 76.67/76.90 (sK2_goals_13_X19 + $$fresh2($$fresh5(sK2_goals_13_X19 >= ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), $$true, sK2_goals_13_X19, 0, (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), $$true, sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), 0)) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 17 (sos_03) } 76.67/76.90 (sK2_goals_13_X19 + $$fresh2($$fresh5((sK2_goals_13_X19 + 0) >= ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), $$true, sK2_goals_13_X19, 0, (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), $$true, sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), 0)) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 13 (sos_07_1) } 76.67/76.90 (sK2_goals_13_X19 + $$fresh2(0 >= (sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17)), $$true, sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), 0)) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 21 (sos_06) } 76.67/76.90 (sK2_goals_13_X19 + $$fresh((sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17)) >= 0, $$true, sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), 0)) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 20 (sos_08) } 76.67/76.90 (sK2_goals_13_X19 + $$fresh($$true, $$true, sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17), 0)) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 3 (sos_06) } 76.67/76.90 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 4 (sos_06) } 76.67/76.90 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh2($$true, $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 6 (sos_07_1) } 76.67/76.90 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh2($$fresh5($$true, $$true, sK1_goals_13_X17, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.90 = { by axiom 6 (sos_07_1) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh2($$fresh5($$fresh5($$true, $$true, sK2_goals_13_X19, sK1_goals_13_X17 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)), sK3_goals_13_X18), $$true, sK1_goals_13_X17, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 5 (sos_07) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh2($$fresh5($$fresh5($$fresh6($$true, $$true, sK1_goals_13_X17, sK2_goals_13_X19 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)), sK3_goals_13_X18), $$true, sK2_goals_13_X19, sK1_goals_13_X17 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)), sK3_goals_13_X18), $$true, sK1_goals_13_X17, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by lemma 27 } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh2($$fresh5($$fresh5($$fresh6((sK2_goals_13_X19 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18))) >= (sK1_goals_13_X17 ==> sK3_goals_13_X18), $$true, sK1_goals_13_X17, sK2_goals_13_X19 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)), sK3_goals_13_X18), $$true, sK2_goals_13_X19, sK1_goals_13_X17 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)), sK3_goals_13_X18), $$true, sK1_goals_13_X17, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 14 (sos_07) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh2($$fresh5($$fresh5((sK1_goals_13_X17 + (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)))) >= sK3_goals_13_X18, $$true, sK2_goals_13_X19, sK1_goals_13_X17 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)), sK3_goals_13_X18), $$true, sK1_goals_13_X17, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 18 (sos_02) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh2($$fresh5($$fresh5(((sK2_goals_13_X19 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18))) + sK1_goals_13_X17) >= sK3_goals_13_X18, $$true, sK2_goals_13_X19, sK1_goals_13_X17 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)), sK3_goals_13_X18), $$true, sK1_goals_13_X17, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 11 (sos_01) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh2($$fresh5($$fresh5((sK2_goals_13_X19 + ((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) + sK1_goals_13_X17)) >= sK3_goals_13_X18, $$true, sK2_goals_13_X19, sK1_goals_13_X17 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)), sK3_goals_13_X18), $$true, sK1_goals_13_X17, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 18 (sos_02) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh2($$fresh5($$fresh5((sK2_goals_13_X19 + (sK1_goals_13_X17 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)))) >= sK3_goals_13_X18, $$true, sK2_goals_13_X19, sK1_goals_13_X17 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)), sK3_goals_13_X18), $$true, sK1_goals_13_X17, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 13 (sos_07_1) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh2($$fresh5((sK1_goals_13_X17 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18))) >= (sK2_goals_13_X19 ==> sK3_goals_13_X18), $$true, sK1_goals_13_X17, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 23 (goals_13) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh2($$fresh5((sK1_goals_13_X17 + (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18))) >= sK2_goals_13_X19, $$true, sK1_goals_13_X17, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18), sK2_goals_13_X19), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 13 (sos_07_1) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh2((sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) >= (sK1_goals_13_X17 ==> sK2_goals_13_X19), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 21 (sos_06) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh((sK1_goals_13_X17 ==> sK2_goals_13_X19) >= (sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 13 (sos_07_1) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh($$fresh5((sK2_goals_13_X19 + (sK1_goals_13_X17 ==> sK2_goals_13_X19)) >= (sK1_goals_13_X17 ==> sK3_goals_13_X18), $$true, sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK3_goals_13_X18), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 13 (sos_07_1) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh($$fresh5($$fresh5((sK1_goals_13_X17 + (sK2_goals_13_X19 + (sK1_goals_13_X17 ==> sK2_goals_13_X19))) >= sK3_goals_13_X18, $$true, sK1_goals_13_X17, sK2_goals_13_X19 + (sK1_goals_13_X17 ==> sK2_goals_13_X19), sK3_goals_13_X18), $$true, sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK3_goals_13_X18), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 18 (sos_02) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh($$fresh5($$fresh5((sK1_goals_13_X17 + ((sK1_goals_13_X17 ==> sK2_goals_13_X19) + sK2_goals_13_X19)) >= sK3_goals_13_X18, $$true, sK1_goals_13_X17, sK2_goals_13_X19 + (sK1_goals_13_X17 ==> sK2_goals_13_X19), sK3_goals_13_X18), $$true, sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK3_goals_13_X18), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 11 (sos_01) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh($$fresh5($$fresh5(((sK1_goals_13_X17 + (sK1_goals_13_X17 ==> sK2_goals_13_X19)) + sK2_goals_13_X19) >= sK3_goals_13_X18, $$true, sK1_goals_13_X17, sK2_goals_13_X19 + (sK1_goals_13_X17 ==> sK2_goals_13_X19), sK3_goals_13_X18), $$true, sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK3_goals_13_X18), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 18 (sos_02) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh($$fresh5($$fresh5((sK2_goals_13_X19 + (sK1_goals_13_X17 + (sK1_goals_13_X17 ==> sK2_goals_13_X19))) >= sK3_goals_13_X18, $$true, sK1_goals_13_X17, sK2_goals_13_X19 + (sK1_goals_13_X17 ==> sK2_goals_13_X19), sK3_goals_13_X18), $$true, sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK3_goals_13_X18), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 14 (sos_07) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh($$fresh5($$fresh5($$fresh6((sK1_goals_13_X17 + (sK1_goals_13_X17 ==> sK2_goals_13_X19)) >= (sK2_goals_13_X19 ==> sK3_goals_13_X18), $$true, sK2_goals_13_X19, sK1_goals_13_X17 + (sK1_goals_13_X17 ==> sK2_goals_13_X19), sK3_goals_13_X18), $$true, sK1_goals_13_X17, sK2_goals_13_X19 + (sK1_goals_13_X17 ==> sK2_goals_13_X19), sK3_goals_13_X18), $$true, sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK3_goals_13_X18), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 23 (goals_13) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh($$fresh5($$fresh5($$fresh6((sK1_goals_13_X17 + (sK1_goals_13_X17 ==> sK2_goals_13_X19)) >= sK2_goals_13_X19, $$true, sK2_goals_13_X19, sK1_goals_13_X17 + (sK1_goals_13_X17 ==> sK2_goals_13_X19), sK3_goals_13_X18), $$true, sK1_goals_13_X17, sK2_goals_13_X19 + (sK1_goals_13_X17 ==> sK2_goals_13_X19), sK3_goals_13_X18), $$true, sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK3_goals_13_X18), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by lemma 27 } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh($$fresh5($$fresh5($$fresh6($$true, $$true, sK2_goals_13_X19, sK1_goals_13_X17 + (sK1_goals_13_X17 ==> sK2_goals_13_X19), sK3_goals_13_X18), $$true, sK1_goals_13_X17, sK2_goals_13_X19 + (sK1_goals_13_X17 ==> sK2_goals_13_X19), sK3_goals_13_X18), $$true, sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK3_goals_13_X18), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 5 (sos_07) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh($$fresh5($$fresh5($$true, $$true, sK1_goals_13_X17, sK2_goals_13_X19 + (sK1_goals_13_X17 ==> sK2_goals_13_X19), sK3_goals_13_X18), $$true, sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK3_goals_13_X18), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 6 (sos_07_1) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh($$fresh5($$true, $$true, sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK3_goals_13_X18), $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 6 (sos_07_1) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ($$fresh($$true, $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK2_goals_13_X19 ==> (sK1_goals_13_X17 ==> sK3_goals_13_X18)) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 3 (sos_06) } 76.67/76.91 (sK2_goals_13_X19 + (sK2_goals_13_X19 ==> ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17))) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 16 (sos_12) } 76.67/76.91 (((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) + (((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19)) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 18 (sos_02) } 76.67/76.91 ((((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19) + ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17)) >= sK1_goals_13_X17 76.67/76.91 = { by axiom 14 (sos_07) } 76.67/76.91 $$fresh6(((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) >= ((((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19) ==> sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 19 (sos_10) } 76.67/76.91 $$fresh6($$fresh3((((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19) >= (sK1_goals_13_X17 ==> sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 19 (sos_10) } 76.67/76.91 $$fresh6($$fresh3($$fresh3(sK1_goals_13_X17 >= ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17), $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 4 (sos_06) } 76.67/76.91 $$fresh6($$fresh3($$fresh3($$fresh2($$true, $$true, 0 ==> sK1_goals_13_X17, sK1_goals_13_X17) >= ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17), $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 6 (sos_07_1) } 76.67/76.91 $$fresh6($$fresh3($$fresh3($$fresh2($$fresh5($$true, $$true, 0, sK1_goals_13_X17, 0 + sK1_goals_13_X17), $$true, 0 ==> sK1_goals_13_X17, sK1_goals_13_X17) >= ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17), $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 15 (sos_04) } 76.67/76.91 $$fresh6($$fresh3($$fresh3($$fresh2($$fresh5((0 + sK1_goals_13_X17) >= (0 + sK1_goals_13_X17), $$true, 0, sK1_goals_13_X17, 0 + sK1_goals_13_X17), $$true, 0 ==> sK1_goals_13_X17, sK1_goals_13_X17) >= ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17), $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 13 (sos_07_1) } 76.67/76.91 $$fresh6($$fresh3($$fresh3($$fresh2(sK1_goals_13_X17 >= (0 ==> (0 + sK1_goals_13_X17)), $$true, 0 ==> sK1_goals_13_X17, sK1_goals_13_X17) >= ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17), $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 18 (sos_02) } 76.67/76.91 $$fresh6($$fresh3($$fresh3($$fresh2(sK1_goals_13_X17 >= (0 ==> (sK1_goals_13_X17 + 0)), $$true, 0 ==> sK1_goals_13_X17, sK1_goals_13_X17) >= ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17), $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 17 (sos_03) } 76.67/76.91 $$fresh6($$fresh3($$fresh3($$fresh2(sK1_goals_13_X17 >= (0 ==> sK1_goals_13_X17), $$true, 0 ==> sK1_goals_13_X17, sK1_goals_13_X17) >= ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17), $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 21 (sos_06) } 76.67/76.91 $$fresh6($$fresh3($$fresh3($$fresh((0 ==> sK1_goals_13_X17) >= sK1_goals_13_X17, $$true, 0 ==> sK1_goals_13_X17, sK1_goals_13_X17) >= ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17), $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by lemma 25 } 76.67/76.91 $$fresh6($$fresh3($$fresh3($$fresh((0 + (0 ==> sK1_goals_13_X17)) >= sK1_goals_13_X17, $$true, 0 ==> sK1_goals_13_X17, sK1_goals_13_X17) >= ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17), $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 16 (sos_12) } 76.67/76.91 $$fresh6($$fresh3($$fresh3($$fresh((sK1_goals_13_X17 + (sK1_goals_13_X17 ==> 0)) >= sK1_goals_13_X17, $$true, 0 ==> sK1_goals_13_X17, sK1_goals_13_X17) >= ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17), $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by lemma 26 } 76.67/76.91 $$fresh6($$fresh3($$fresh3($$fresh($$true, $$true, 0 ==> sK1_goals_13_X17, sK1_goals_13_X17) >= ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17), $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 3 (sos_06) } 76.67/76.91 $$fresh6($$fresh3($$fresh3((0 ==> sK1_goals_13_X17) >= ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17), $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 19 (sos_10) } 76.67/76.91 $$fresh6($$fresh3($$fresh3($$fresh3((sK1_goals_13_X17 ==> sK2_goals_13_X19) >= 0, $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, 0, sK1_goals_13_X17), $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 20 (sos_08) } 76.67/76.91 $$fresh6($$fresh3($$fresh3($$fresh3($$true, $$true, sK1_goals_13_X17 ==> sK2_goals_13_X19, 0, sK1_goals_13_X17), $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 8 (sos_10) } 76.67/76.91 $$fresh6($$fresh3($$fresh3($$true, $$true, sK1_goals_13_X17, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK2_goals_13_X19), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 8 (sos_10) } 76.67/76.91 $$fresh6($$fresh3($$true, $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, sK1_goals_13_X17 ==> sK2_goals_13_X19, sK1_goals_13_X17), $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 8 (sos_10) } 76.67/76.91 $$fresh6($$true, $$true, ((sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17) ==> sK2_goals_13_X19, (sK1_goals_13_X17 ==> sK2_goals_13_X19) ==> sK1_goals_13_X17, sK1_goals_13_X17) 76.67/76.91 = { by axiom 5 (sos_07) } 76.67/76.91 $$true 76.67/76.91 % SZS output end Proof 76.67/76.91 76.67/76.91 RESULT: Theorem (the conjecture is true). 76.67/76.93 EOF