0.00/0.04 % 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 0.02/0.23 % Computer : n119.star.cs.uiowa.edu 0.02/0.23 % Model : x86_64 x86_64 0.02/0.23 % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz 0.02/0.23 % Memory : 32218.625MB 0.02/0.23 % OS : Linux 3.10.0-693.2.2.el7.x86_64 0.02/0.24 % CPULimit : 300 0.02/0.24 % DateTime : Sat Jul 14 06:10:54 CDT 2018 0.02/0.24 % CPUTime : 114.91/115.14 % SZS status Theorem 114.91/115.14 114.91/115.14 % SZS output start Proof 114.91/115.14 Take the following subset of the input axioms: 115.57/115.77 fof(goals_14, conjecture, 115.57/115.77 ![X17, X18, X19]: 115.57/115.77 ('>='(X19, X17) 115.57/115.77 <= (X19='==>'(X19, X18) & '>='('==>'(X17, X18), X17)))). 115.57/115.77 fof(sos_01, axiom, 115.57/115.77 ![A, B, C]: '+'('+'(A, B), C)='+'(A, '+'(B, C))). 115.57/115.77 fof(sos_02, axiom, ![A, B]: '+'(B, A)='+'(A, B)). 115.57/115.77 fof(sos_03, axiom, ![A]: '+'(A, '0')=A). 115.57/115.77 fof(sos_04, axiom, ![A]: '>='(A, A)). 115.57/115.77 fof(sos_06, axiom, 115.57/115.77 ![X3, X4]: (X4=X3 <= ('>='(X3, X4) & '>='(X4, X3)))). 115.57/115.77 fof(sos_07, axiom, 115.57/115.77 ![X5, X6, X7]: 115.57/115.77 ('>='(X6, '==>'(X5, X7)) <=> '>='('+'(X5, X6), X7))). 115.57/115.77 fof(sos_08, axiom, ![A]: '>='(A, '0')). 115.57/115.77 fof(sos_09, axiom, 115.57/115.77 ![X8, X9, X10]: 115.57/115.77 ('>='(X8, X9) => '>='('+'(X8, X10), '+'(X9, X10)))). 115.57/115.77 fof(sos_10, axiom, 115.57/115.77 ![X11, X12, X13]: 115.57/115.77 ('>='('==>'(X12, X13), '==>'(X11, X13)) <= '>='(X11, X12))). 115.57/115.77 fof(sos_11, axiom, 115.57/115.77 ![X14, X15, X16]: 115.57/115.77 ('>='('==>'(X16, X14), '==>'(X16, X15)) <= '>='(X14, X15))). 115.57/115.77 fof(sos_12, axiom, 115.57/115.77 ![A, B]: '+'(A, '==>'(A, B))='+'(B, '==>'(B, A))). 115.57/115.77 115.57/115.77 Now clausify the problem and encode Horn clauses using encoding 3 of 115.57/115.77 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf. 115.57/115.77 We repeatedly replace C & s=t => u=v by the two clauses: 115.57/115.77 $$fresh(y, y, x1...xn) = u 115.57/115.77 C => $$fresh(s, t, x1...xn) = v 115.57/115.77 where $$fresh is a fresh function symbol and x1..xn are the free 115.57/115.77 variables of u and v. 115.57/115.77 A predicate p(X) is encoded as p(X)=$$true (this is sound, because the 115.57/115.77 input problem has no model of domain size 1). 115.57/115.77 115.57/115.77 The encoding turns the above axioms into the following unit equations and goals: 115.57/115.77 115.57/115.77 Axiom 3 (sos_06): $$fresh(X, X, Y, Z) = Y. 115.57/115.77 Axiom 4 (sos_06): $$fresh2(X, X, Y, Z) = Z. 115.57/115.77 Axiom 5 (sos_07): $$fresh5(X, X, Y, Z, W) = $$true. 115.57/115.77 Axiom 6 (sos_07_1): $$fresh9(X, X, Y, Z, W) = $$true. 115.57/115.77 Axiom 7 (sos_09): $$fresh4(X, X, Y, Z, W) = $$true. 115.57/115.77 Axiom 8 (sos_10): $$fresh3(X, X, Y, Z, W) = $$true. 115.57/115.77 Axiom 9 (sos_11): $$fresh8(X, X, Y, Z, W) = $$true. 115.57/115.77 Axiom 10 (sos_11): $$fresh8(X >= Y, $$true, X, Y, Z) = (Z ==> X) >= (Z ==> Y). 115.57/115.77 Axiom 11 (sos_07_1): $$fresh9((X + Y) >= Z, $$true, X, Y, Z) = Y >= (X ==> Z). 115.57/115.77 Axiom 12 (sos_07): $$fresh5(X >= (Y ==> Z), $$true, Y, X, Z) = (Y + X) >= Z. 115.57/115.77 Axiom 14 (sos_01): (X + Y) + Z = X + (Y + Z). 115.57/115.77 Axiom 15 (sos_12): X + (X ==> Y) = Y + (Y ==> X). 115.57/115.77 Axiom 16 (sos_04): X >= X = $$true. 115.57/115.77 Axiom 17 (sos_09): $$fresh4(X >= Y, $$true, X, Y, Z) = (X + Z) >= (Y + Z). 115.57/115.77 Axiom 18 (sos_03): X + 0 = X. 115.57/115.77 Axiom 20 (sos_02): X + Y = Y + X. 115.57/115.77 Axiom 21 (sos_10): $$fresh3(X >= Y, $$true, X, Y, Z) = (Y ==> Z) >= (X ==> Z). 115.57/115.77 Axiom 22 (sos_06): $$fresh2(X >= Y, $$true, Y, X) = $$fresh(Y >= X, $$true, Y, X). 115.57/115.77 Axiom 23 (sos_08): X >= 0 = $$true. 115.57/115.77 Axiom 24 (goals_14): sK2_goals_14_X19 = sK2_goals_14_X19 ==> sK3_goals_14_X18. 115.57/115.77 Axiom 25 (goals_14_1): (sK1_goals_14_X17 ==> sK3_goals_14_X18) >= sK1_goals_14_X17 = $$true. 115.57/115.77 115.57/115.77 Lemma 26: 0 + X = X. 115.57/115.77 Proof: 115.57/115.77 0 + X 115.57/115.77 = { by axiom 20 (sos_02) } 115.57/115.77 X + 0 115.57/115.77 = { by axiom 18 (sos_03) } 115.57/115.77 X 115.57/115.77 115.57/115.77 Lemma 27: (X + Y) >= X = $$true. 115.57/115.77 Proof: 115.57/115.77 (X + Y) >= X 115.57/115.77 = { by axiom 20 (sos_02) } 115.57/115.77 (Y + X) >= X 115.57/115.77 = { by lemma 26 } 115.57/115.77 (Y + X) >= (0 + X) 115.57/115.77 = { by axiom 17 (sos_09) } 115.57/115.77 $$fresh4(Y >= 0, $$true, Y, 0, X) 115.57/115.77 = { by axiom 23 (sos_08) } 115.57/115.77 $$fresh4($$true, $$true, Y, 0, X) 115.57/115.77 = { by axiom 7 (sos_09) } 115.57/115.77 $$true 115.57/115.77 115.57/115.77 Lemma 28: (X + (X ==> Y)) >= Y = $$true. 115.57/115.77 Proof: 115.57/115.77 (X + (X ==> Y)) >= Y 115.57/115.77 = { by axiom 15 (sos_12) } 115.57/115.77 (Y + (Y ==> X)) >= Y 115.57/115.77 = { by lemma 27 } 116.16/116.41 $$true 116.16/116.41 116.16/116.41 Goal 1 (goals_14_2): sK2_goals_14_X19 >= sK1_goals_14_X17 = $$true. 116.16/116.41 Proof: 116.16/116.41 sK2_goals_14_X19 >= sK1_goals_14_X17 116.16/116.41 = { by axiom 18 (sos_03) } 116.16/116.41 (sK2_goals_14_X19 + 0) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 4 (sos_06) } 116.16/116.41 (sK2_goals_14_X19 + $$fresh2($$true, $$true, sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), 0)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 6 (sos_07_1) } 116.16/116.41 (sK2_goals_14_X19 + $$fresh2($$fresh9($$true, $$true, sK2_goals_14_X19, 0, (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), $$true, sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), 0)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 6 (sos_07_1) } 116.16/116.41 (sK2_goals_14_X19 + $$fresh2($$fresh9($$fresh9($$true, $$true, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19, sK1_goals_14_X17), $$true, sK2_goals_14_X19, 0, (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), $$true, sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), 0)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 5 (sos_07) } 116.16/116.41 (sK2_goals_14_X19 + $$fresh2($$fresh9($$fresh9($$fresh5($$true, $$true, sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK1_goals_14_X17), $$true, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19, sK1_goals_14_X17), $$true, sK2_goals_14_X19, 0, (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), $$true, sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), 0)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 9 (sos_11) } 116.16/116.41 (sK2_goals_14_X19 + $$fresh2($$fresh9($$fresh9($$fresh5($$fresh8($$true, $$true, sK1_goals_14_X17 ==> sK3_goals_14_X18, sK1_goals_14_X17, sK2_goals_14_X19), $$true, sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK1_goals_14_X17), $$true, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19, sK1_goals_14_X17), $$true, sK2_goals_14_X19, 0, (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), $$true, sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), 0)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 25 (goals_14_1) } 116.16/116.41 (sK2_goals_14_X19 + $$fresh2($$fresh9($$fresh9($$fresh5($$fresh8((sK1_goals_14_X17 ==> sK3_goals_14_X18) >= sK1_goals_14_X17, $$true, sK1_goals_14_X17 ==> sK3_goals_14_X18, sK1_goals_14_X17, sK2_goals_14_X19), $$true, sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK1_goals_14_X17), $$true, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19, sK1_goals_14_X17), $$true, sK2_goals_14_X19, 0, (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), $$true, sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), 0)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 10 (sos_11) } 116.16/116.41 (sK2_goals_14_X19 + $$fresh2($$fresh9($$fresh9($$fresh5((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) >= (sK2_goals_14_X19 ==> sK1_goals_14_X17), $$true, sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK1_goals_14_X17), $$true, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19, sK1_goals_14_X17), $$true, sK2_goals_14_X19, 0, (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), $$true, sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), 0)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 12 (sos_07) } 116.16/116.41 (sK2_goals_14_X19 + $$fresh2($$fresh9($$fresh9((sK2_goals_14_X19 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18))) >= sK1_goals_14_X17, $$true, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19, sK1_goals_14_X17), $$true, sK2_goals_14_X19, 0, (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), $$true, sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), 0)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 20 (sos_02) } 116.16/116.41 (sK2_goals_14_X19 + $$fresh2($$fresh9($$fresh9(((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) + sK2_goals_14_X19) >= sK1_goals_14_X17, $$true, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19, sK1_goals_14_X17), $$true, sK2_goals_14_X19, 0, (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), $$true, sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), 0)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 11 (sos_07_1) } 116.16/116.41 (sK2_goals_14_X19 + $$fresh2($$fresh9(sK2_goals_14_X19 >= ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), $$true, sK2_goals_14_X19, 0, (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), $$true, sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), 0)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 18 (sos_03) } 116.16/116.41 (sK2_goals_14_X19 + $$fresh2($$fresh9((sK2_goals_14_X19 + 0) >= ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), $$true, sK2_goals_14_X19, 0, (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), $$true, sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), 0)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 11 (sos_07_1) } 116.16/116.41 (sK2_goals_14_X19 + $$fresh2(0 >= (sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17)), $$true, sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), 0)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 22 (sos_06) } 116.16/116.41 (sK2_goals_14_X19 + $$fresh((sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17)) >= 0, $$true, sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), 0)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 23 (sos_08) } 116.16/116.41 (sK2_goals_14_X19 + $$fresh($$true, $$true, sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17), 0)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 3 (sos_06) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 4 (sos_06) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh2($$true, $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 6 (sos_07_1) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh2($$fresh9($$true, $$true, sK1_goals_14_X17, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 6 (sos_07_1) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh2($$fresh9($$fresh9($$true, $$true, sK2_goals_14_X19, sK1_goals_14_X17 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)), sK3_goals_14_X18), $$true, sK1_goals_14_X17, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 5 (sos_07) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh2($$fresh9($$fresh9($$fresh5($$true, $$true, sK1_goals_14_X17, sK2_goals_14_X19 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)), sK3_goals_14_X18), $$true, sK2_goals_14_X19, sK1_goals_14_X17 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)), sK3_goals_14_X18), $$true, sK1_goals_14_X17, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by lemma 28 } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh2($$fresh9($$fresh9($$fresh5((sK2_goals_14_X19 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18))) >= (sK1_goals_14_X17 ==> sK3_goals_14_X18), $$true, sK1_goals_14_X17, sK2_goals_14_X19 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)), sK3_goals_14_X18), $$true, sK2_goals_14_X19, sK1_goals_14_X17 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)), sK3_goals_14_X18), $$true, sK1_goals_14_X17, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 12 (sos_07) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh2($$fresh9($$fresh9((sK1_goals_14_X17 + (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)))) >= sK3_goals_14_X18, $$true, sK2_goals_14_X19, sK1_goals_14_X17 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)), sK3_goals_14_X18), $$true, sK1_goals_14_X17, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 20 (sos_02) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh2($$fresh9($$fresh9(((sK2_goals_14_X19 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18))) + sK1_goals_14_X17) >= sK3_goals_14_X18, $$true, sK2_goals_14_X19, sK1_goals_14_X17 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)), sK3_goals_14_X18), $$true, sK1_goals_14_X17, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 14 (sos_01) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh2($$fresh9($$fresh9((sK2_goals_14_X19 + ((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) + sK1_goals_14_X17)) >= sK3_goals_14_X18, $$true, sK2_goals_14_X19, sK1_goals_14_X17 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)), sK3_goals_14_X18), $$true, sK1_goals_14_X17, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 20 (sos_02) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh2($$fresh9($$fresh9((sK2_goals_14_X19 + (sK1_goals_14_X17 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)))) >= sK3_goals_14_X18, $$true, sK2_goals_14_X19, sK1_goals_14_X17 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)), sK3_goals_14_X18), $$true, sK1_goals_14_X17, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 11 (sos_07_1) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh2($$fresh9((sK1_goals_14_X17 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18))) >= (sK2_goals_14_X19 ==> sK3_goals_14_X18), $$true, sK1_goals_14_X17, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 24 (goals_14) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh2($$fresh9((sK1_goals_14_X17 + (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18))) >= sK2_goals_14_X19, $$true, sK1_goals_14_X17, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18), sK2_goals_14_X19), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 11 (sos_07_1) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh2((sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) >= (sK1_goals_14_X17 ==> sK2_goals_14_X19), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 22 (sos_06) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh((sK1_goals_14_X17 ==> sK2_goals_14_X19) >= (sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 11 (sos_07_1) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh($$fresh9((sK2_goals_14_X19 + (sK1_goals_14_X17 ==> sK2_goals_14_X19)) >= (sK1_goals_14_X17 ==> sK3_goals_14_X18), $$true, sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK3_goals_14_X18), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 11 (sos_07_1) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh($$fresh9($$fresh9((sK1_goals_14_X17 + (sK2_goals_14_X19 + (sK1_goals_14_X17 ==> sK2_goals_14_X19))) >= sK3_goals_14_X18, $$true, sK1_goals_14_X17, sK2_goals_14_X19 + (sK1_goals_14_X17 ==> sK2_goals_14_X19), sK3_goals_14_X18), $$true, sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK3_goals_14_X18), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 20 (sos_02) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh($$fresh9($$fresh9((sK1_goals_14_X17 + ((sK1_goals_14_X17 ==> sK2_goals_14_X19) + sK2_goals_14_X19)) >= sK3_goals_14_X18, $$true, sK1_goals_14_X17, sK2_goals_14_X19 + (sK1_goals_14_X17 ==> sK2_goals_14_X19), sK3_goals_14_X18), $$true, sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK3_goals_14_X18), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 14 (sos_01) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh($$fresh9($$fresh9(((sK1_goals_14_X17 + (sK1_goals_14_X17 ==> sK2_goals_14_X19)) + sK2_goals_14_X19) >= sK3_goals_14_X18, $$true, sK1_goals_14_X17, sK2_goals_14_X19 + (sK1_goals_14_X17 ==> sK2_goals_14_X19), sK3_goals_14_X18), $$true, sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK3_goals_14_X18), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 20 (sos_02) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh($$fresh9($$fresh9((sK2_goals_14_X19 + (sK1_goals_14_X17 + (sK1_goals_14_X17 ==> sK2_goals_14_X19))) >= sK3_goals_14_X18, $$true, sK1_goals_14_X17, sK2_goals_14_X19 + (sK1_goals_14_X17 ==> sK2_goals_14_X19), sK3_goals_14_X18), $$true, sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK3_goals_14_X18), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 12 (sos_07) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh($$fresh9($$fresh9($$fresh5((sK1_goals_14_X17 + (sK1_goals_14_X17 ==> sK2_goals_14_X19)) >= (sK2_goals_14_X19 ==> sK3_goals_14_X18), $$true, sK2_goals_14_X19, sK1_goals_14_X17 + (sK1_goals_14_X17 ==> sK2_goals_14_X19), sK3_goals_14_X18), $$true, sK1_goals_14_X17, sK2_goals_14_X19 + (sK1_goals_14_X17 ==> sK2_goals_14_X19), sK3_goals_14_X18), $$true, sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK3_goals_14_X18), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 24 (goals_14) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh($$fresh9($$fresh9($$fresh5((sK1_goals_14_X17 + (sK1_goals_14_X17 ==> sK2_goals_14_X19)) >= sK2_goals_14_X19, $$true, sK2_goals_14_X19, sK1_goals_14_X17 + (sK1_goals_14_X17 ==> sK2_goals_14_X19), sK3_goals_14_X18), $$true, sK1_goals_14_X17, sK2_goals_14_X19 + (sK1_goals_14_X17 ==> sK2_goals_14_X19), sK3_goals_14_X18), $$true, sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK3_goals_14_X18), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by lemma 28 } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh($$fresh9($$fresh9($$fresh5($$true, $$true, sK2_goals_14_X19, sK1_goals_14_X17 + (sK1_goals_14_X17 ==> sK2_goals_14_X19), sK3_goals_14_X18), $$true, sK1_goals_14_X17, sK2_goals_14_X19 + (sK1_goals_14_X17 ==> sK2_goals_14_X19), sK3_goals_14_X18), $$true, sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK3_goals_14_X18), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 5 (sos_07) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh($$fresh9($$fresh9($$true, $$true, sK1_goals_14_X17, sK2_goals_14_X19 + (sK1_goals_14_X17 ==> sK2_goals_14_X19), sK3_goals_14_X18), $$true, sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK3_goals_14_X18), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 6 (sos_07_1) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh($$fresh9($$true, $$true, sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK3_goals_14_X18), $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 6 (sos_07_1) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ($$fresh($$true, $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK2_goals_14_X19 ==> (sK1_goals_14_X17 ==> sK3_goals_14_X18)) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 3 (sos_06) } 116.16/116.41 (sK2_goals_14_X19 + (sK2_goals_14_X19 ==> ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17))) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 15 (sos_12) } 116.16/116.41 (((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) + (((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 20 (sos_02) } 116.16/116.41 ((((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19) + ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17)) >= sK1_goals_14_X17 116.16/116.41 = { by axiom 12 (sos_07) } 116.16/116.41 $$fresh5(((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) >= ((((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19) ==> sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.41 = { by axiom 21 (sos_10) } 116.16/116.41 $$fresh5($$fresh3((((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19) >= (sK1_goals_14_X17 ==> sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.41 = { by axiom 21 (sos_10) } 116.16/116.41 $$fresh5($$fresh3($$fresh3(sK1_goals_14_X17 >= ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17), $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.41 = { by axiom 4 (sos_06) } 116.16/116.41 $$fresh5($$fresh3($$fresh3($$fresh2($$true, $$true, 0 ==> sK1_goals_14_X17, sK1_goals_14_X17) >= ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17), $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.41 = { by axiom 6 (sos_07_1) } 116.16/116.41 $$fresh5($$fresh3($$fresh3($$fresh2($$fresh9($$true, $$true, 0, sK1_goals_14_X17, 0 + sK1_goals_14_X17), $$true, 0 ==> sK1_goals_14_X17, sK1_goals_14_X17) >= ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17), $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.41 = { by axiom 16 (sos_04) } 116.16/116.41 $$fresh5($$fresh3($$fresh3($$fresh2($$fresh9((0 + sK1_goals_14_X17) >= (0 + sK1_goals_14_X17), $$true, 0, sK1_goals_14_X17, 0 + sK1_goals_14_X17), $$true, 0 ==> sK1_goals_14_X17, sK1_goals_14_X17) >= ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17), $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.41 = { by axiom 11 (sos_07_1) } 116.16/116.41 $$fresh5($$fresh3($$fresh3($$fresh2(sK1_goals_14_X17 >= (0 ==> (0 + sK1_goals_14_X17)), $$true, 0 ==> sK1_goals_14_X17, sK1_goals_14_X17) >= ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17), $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.41 = { by axiom 20 (sos_02) } 116.16/116.41 $$fresh5($$fresh3($$fresh3($$fresh2(sK1_goals_14_X17 >= (0 ==> (sK1_goals_14_X17 + 0)), $$true, 0 ==> sK1_goals_14_X17, sK1_goals_14_X17) >= ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17), $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.41 = { by axiom 18 (sos_03) } 116.16/116.41 $$fresh5($$fresh3($$fresh3($$fresh2(sK1_goals_14_X17 >= (0 ==> sK1_goals_14_X17), $$true, 0 ==> sK1_goals_14_X17, sK1_goals_14_X17) >= ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17), $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.41 = { by axiom 22 (sos_06) } 116.16/116.41 $$fresh5($$fresh3($$fresh3($$fresh((0 ==> sK1_goals_14_X17) >= sK1_goals_14_X17, $$true, 0 ==> sK1_goals_14_X17, sK1_goals_14_X17) >= ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17), $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.41 = { by lemma 26 } 116.16/116.41 $$fresh5($$fresh3($$fresh3($$fresh((0 + (0 ==> sK1_goals_14_X17)) >= sK1_goals_14_X17, $$true, 0 ==> sK1_goals_14_X17, sK1_goals_14_X17) >= ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17), $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.41 = { by axiom 15 (sos_12) } 116.16/116.41 $$fresh5($$fresh3($$fresh3($$fresh((sK1_goals_14_X17 + (sK1_goals_14_X17 ==> 0)) >= sK1_goals_14_X17, $$true, 0 ==> sK1_goals_14_X17, sK1_goals_14_X17) >= ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17), $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.41 = { by lemma 27 } 116.16/116.41 $$fresh5($$fresh3($$fresh3($$fresh($$true, $$true, 0 ==> sK1_goals_14_X17, sK1_goals_14_X17) >= ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17), $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.41 = { by axiom 3 (sos_06) } 116.16/116.42 $$fresh5($$fresh3($$fresh3((0 ==> sK1_goals_14_X17) >= ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17), $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.42 = { by axiom 21 (sos_10) } 116.16/116.42 $$fresh5($$fresh3($$fresh3($$fresh3((sK1_goals_14_X17 ==> sK2_goals_14_X19) >= 0, $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, 0, sK1_goals_14_X17), $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.42 = { by axiom 23 (sos_08) } 116.16/116.42 $$fresh5($$fresh3($$fresh3($$fresh3($$true, $$true, sK1_goals_14_X17 ==> sK2_goals_14_X19, 0, sK1_goals_14_X17), $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.42 = { by axiom 8 (sos_10) } 116.16/116.42 $$fresh5($$fresh3($$fresh3($$true, $$true, sK1_goals_14_X17, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK2_goals_14_X19), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.42 = { by axiom 8 (sos_10) } 116.16/116.42 $$fresh5($$fresh3($$true, $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, sK1_goals_14_X17 ==> sK2_goals_14_X19, sK1_goals_14_X17), $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.42 = { by axiom 8 (sos_10) } 116.16/116.42 $$fresh5($$true, $$true, ((sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17) ==> sK2_goals_14_X19, (sK1_goals_14_X17 ==> sK2_goals_14_X19) ==> sK1_goals_14_X17, sK1_goals_14_X17) 116.16/116.42 = { by axiom 5 (sos_07) } 116.16/116.42 $$true 116.16/116.42 % SZS output end Proof 116.16/116.42 116.16/116.42 RESULT: Theorem (the conjecture is true). 116.16/116.44 EOF