第 32622 題：仿射式的左端有限和

原題與已知條件

全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-22x+(9)，區間 [-1,1] 分為 N=22 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。

• G001 — 全實數論域，以下函數／點均以明示實數模型解讀。f(x)=-22x+(9)，區間 [-1,1] 分為 N=22 等份。求左端有限和 ΔΣ f(a+iΔ)，i=0至N−1、Δ=(b−a)/N。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-definite-integral

逐步推導

source roles = [qf-integral-affine-left-sum, affine_left_sum, reals, []]

derived = [[finite_left_sum_fraction_RETAIN1]]

Left sum ORIGINAL upper minus lower endpoint := sub(1, -1)

Left sum ORIGINAL upper minus lower endpoint = 2

Left sum ORIGINAL width over positive cell count := div(2, 22)

Left sum ORIGINAL width over positive cell count = 1/11

Left sum ORIGINAL sample0 index times width := mul(0, 1/11)

Left sum ORIGINAL sample0 index times width = 0

Left sum ORIGINAL sample0 lower plus offset := sum(-1, 0)

Left sum ORIGINAL sample0 lower plus offset = -1

Left sum ORIGINAL sample0 slope times x := mul(-22, -1)

Left sum ORIGINAL sample0 slope times x = 22

Left sum ORIGINAL sample0 affine value plus c := sum(22, 9)

Left sum ORIGINAL sample0 affine value plus c = 31

Left sum ORIGINAL sample0 value times cell width := mul(31, 1/11)

Left sum ORIGINAL sample0 value times cell width = 31/11

Left sum ORIGINAL accumulated term0 := sum(0, 31/11)

Left sum ORIGINAL accumulated term0 = 31/11

source roles = [0, 22, -22, 9, -1, 1, 1/11]

derived = [-1/1, 31/1, 31/11, 31/11]

Left sum ORIGINAL sample1 index times width := mul(1, 1/11)

Left sum ORIGINAL sample1 index times width = 1/11

Left sum ORIGINAL sample1 lower plus offset := sum(-1, 1/11)

Left sum ORIGINAL sample1 lower plus offset = -10/11

Left sum ORIGINAL sample1 slope times x := mul(-22, -10/11)

Left sum ORIGINAL sample1 slope times x = 20

Left sum ORIGINAL sample1 affine value plus c := sum(20, 9)

Left sum ORIGINAL sample1 affine value plus c = 29

Left sum ORIGINAL sample1 value times cell width := mul(29, 1/11)

Left sum ORIGINAL sample1 value times cell width = 29/11

Left sum ORIGINAL accumulated term1 := sum(31/11, 29/11)

Left sum ORIGINAL accumulated term1 = 60/11

source roles = [1, 22, -22, 9, -1, 1, 1/11]

derived = [-10/11, 29/1, 29/11, 60/11]

Left sum ORIGINAL sample2 index times width := mul(2, 1/11)

Left sum ORIGINAL sample2 index times width = 2/11

Left sum ORIGINAL sample2 lower plus offset := sum(-1, 2/11)

Left sum ORIGINAL sample2 lower plus offset = -9/11

Left sum ORIGINAL sample2 slope times x := mul(-22, -9/11)

Left sum ORIGINAL sample2 slope times x = 18

Left sum ORIGINAL sample2 affine value plus c := sum(18, 9)

Left sum ORIGINAL sample2 affine value plus c = 27

Left sum ORIGINAL sample2 value times cell width := mul(27, 1/11)

Left sum ORIGINAL sample2 value times cell width = 27/11

Left sum ORIGINAL accumulated term2 := sum(60/11, 27/11)

Left sum ORIGINAL accumulated term2 = 87/11

source roles = [2, 22, -22, 9, -1, 1, 1/11]

derived = [-9/11, 27/1, 27/11, 87/11]

Left sum ORIGINAL sample3 index times width := mul(3, 1/11)

Left sum ORIGINAL sample3 index times width = 3/11

Left sum ORIGINAL sample3 lower plus offset := sum(-1, 3/11)

Left sum ORIGINAL sample3 lower plus offset = -8/11

Left sum ORIGINAL sample3 slope times x := mul(-22, -8/11)

Left sum ORIGINAL sample3 slope times x = 16

Left sum ORIGINAL sample3 affine value plus c := sum(16, 9)

Left sum ORIGINAL sample3 affine value plus c = 25

Left sum ORIGINAL sample3 value times cell width := mul(25, 1/11)

Left sum ORIGINAL sample3 value times cell width = 25/11

Left sum ORIGINAL accumulated term3 := sum(87/11, 25/11)

Left sum ORIGINAL accumulated term3 = 112/11

source roles = [3, 22, -22, 9, -1, 1, 1/11]

derived = [-8/11, 25/1, 25/11, 112/11]

Left sum ORIGINAL sample4 index times width := mul(4, 1/11)

Left sum ORIGINAL sample4 index times width = 4/11

Left sum ORIGINAL sample4 lower plus offset := sum(-1, 4/11)

Left sum ORIGINAL sample4 lower plus offset = -7/11

Left sum ORIGINAL sample4 slope times x := mul(-22, -7/11)

Left sum ORIGINAL sample4 slope times x = 14

Left sum ORIGINAL sample4 affine value plus c := sum(14, 9)

Left sum ORIGINAL sample4 affine value plus c = 23

Left sum ORIGINAL sample4 value times cell width := mul(23, 1/11)

Left sum ORIGINAL sample4 value times cell width = 23/11

Left sum ORIGINAL accumulated term4 := sum(112/11, 23/11)

Left sum ORIGINAL accumulated term4 = 135/11

source roles = [4, 22, -22, 9, -1, 1, 1/11]

derived = [-7/11, 23/1, 23/11, 135/11]

Left sum ORIGINAL sample5 index times width := mul(5, 1/11)

Left sum ORIGINAL sample5 index times width = 5/11

Left sum ORIGINAL sample5 lower plus offset := sum(-1, 5/11)

Left sum ORIGINAL sample5 lower plus offset = -6/11

Left sum ORIGINAL sample5 slope times x := mul(-22, -6/11)

Left sum ORIGINAL sample5 slope times x = 12

Left sum ORIGINAL sample5 affine value plus c := sum(12, 9)

Left sum ORIGINAL sample5 affine value plus c = 21

Left sum ORIGINAL sample5 value times cell width := mul(21, 1/11)

Left sum ORIGINAL sample5 value times cell width = 21/11

Left sum ORIGINAL accumulated term5 := sum(135/11, 21/11)

Left sum ORIGINAL accumulated term5 = 156/11

source roles = [5, 22, -22, 9, -1, 1, 1/11]

derived = [-6/11, 21/1, 21/11, 156/11]

Left sum ORIGINAL sample6 index times width := mul(6, 1/11)

Left sum ORIGINAL sample6 index times width = 6/11

Left sum ORIGINAL sample6 lower plus offset := sum(-1, 6/11)

Left sum ORIGINAL sample6 lower plus offset = -5/11

Left sum ORIGINAL sample6 slope times x := mul(-22, -5/11)

Left sum ORIGINAL sample6 slope times x = 10

Left sum ORIGINAL sample6 affine value plus c := sum(10, 9)

Left sum ORIGINAL sample6 affine value plus c = 19

Left sum ORIGINAL sample6 value times cell width := mul(19, 1/11)

Left sum ORIGINAL sample6 value times cell width = 19/11

Left sum ORIGINAL accumulated term6 := sum(156/11, 19/11)

Left sum ORIGINAL accumulated term6 = 175/11

source roles = [6, 22, -22, 9, -1, 1, 1/11]

derived = [-5/11, 19/1, 19/11, 175/11]

Left sum ORIGINAL sample7 index times width := mul(7, 1/11)

Left sum ORIGINAL sample7 index times width = 7/11

Left sum ORIGINAL sample7 lower plus offset := sum(-1, 7/11)

Left sum ORIGINAL sample7 lower plus offset = -4/11

Left sum ORIGINAL sample7 slope times x := mul(-22, -4/11)

Left sum ORIGINAL sample7 slope times x = 8

Left sum ORIGINAL sample7 affine value plus c := sum(8, 9)

Left sum ORIGINAL sample7 affine value plus c = 17

Left sum ORIGINAL sample7 value times cell width := mul(17, 1/11)

Left sum ORIGINAL sample7 value times cell width = 17/11

Left sum ORIGINAL accumulated term7 := sum(175/11, 17/11)

Left sum ORIGINAL accumulated term7 = 192/11

source roles = [7, 22, -22, 9, -1, 1, 1/11]

derived = [-4/11, 17/1, 17/11, 192/11]

Left sum ORIGINAL sample8 index times width := mul(8, 1/11)

Left sum ORIGINAL sample8 index times width = 8/11

Left sum ORIGINAL sample8 lower plus offset := sum(-1, 8/11)

Left sum ORIGINAL sample8 lower plus offset = -3/11

Left sum ORIGINAL sample8 slope times x := mul(-22, -3/11)

Left sum ORIGINAL sample8 slope times x = 6

Left sum ORIGINAL sample8 affine value plus c := sum(6, 9)

Left sum ORIGINAL sample8 affine value plus c = 15

Left sum ORIGINAL sample8 value times cell width := mul(15, 1/11)

Left sum ORIGINAL sample8 value times cell width = 15/11

Left sum ORIGINAL accumulated term8 := sum(192/11, 15/11)

Left sum ORIGINAL accumulated term8 = 207/11

source roles = [8, 22, -22, 9, -1, 1, 1/11]

derived = [-3/11, 15/1, 15/11, 207/11]

Left sum ORIGINAL sample9 index times width := mul(9, 1/11)

Left sum ORIGINAL sample9 index times width = 9/11

Left sum ORIGINAL sample9 lower plus offset := sum(-1, 9/11)

Left sum ORIGINAL sample9 lower plus offset = -2/11

Left sum ORIGINAL sample9 slope times x := mul(-22, -2/11)

Left sum ORIGINAL sample9 slope times x = 4

Left sum ORIGINAL sample9 affine value plus c := sum(4, 9)

Left sum ORIGINAL sample9 affine value plus c = 13

Left sum ORIGINAL sample9 value times cell width := mul(13, 1/11)

Left sum ORIGINAL sample9 value times cell width = 13/11

Left sum ORIGINAL accumulated term9 := sum(207/11, 13/11)

Left sum ORIGINAL accumulated term9 = 20

source roles = [9, 22, -22, 9, -1, 1, 1/11]

derived = [-2/11, 13/1, 13/11, 20/1]

Left sum ORIGINAL sample10 index times width := mul(10, 1/11)

Left sum ORIGINAL sample10 index times width = 10/11

Left sum ORIGINAL sample10 lower plus offset := sum(-1, 10/11)

Left sum ORIGINAL sample10 lower plus offset = -1/11

Left sum ORIGINAL sample10 slope times x := mul(-22, -1/11)

Left sum ORIGINAL sample10 slope times x = 2

Left sum ORIGINAL sample10 affine value plus c := sum(2, 9)

Left sum ORIGINAL sample10 affine value plus c = 11

Left sum ORIGINAL sample10 value times cell width := mul(11, 1/11)

Left sum ORIGINAL sample10 value times cell width = 1

Left sum ORIGINAL accumulated term10 := sum(20, 1)

Left sum ORIGINAL accumulated term10 = 21

source roles = [10, 22, -22, 9, -1, 1, 1/11]

derived = [-1/11, 11/1, 1/1, 21/1]

Left sum ORIGINAL sample11 index times width := mul(11, 1/11)

Left sum ORIGINAL sample11 index times width = 1

Left sum ORIGINAL sample11 lower plus offset := sum(-1, 1)

Left sum ORIGINAL sample11 lower plus offset = 0

Left sum ORIGINAL sample11 slope times x := mul(-22, 0)

Left sum ORIGINAL sample11 slope times x = 0

Left sum ORIGINAL sample11 affine value plus c := sum(0, 9)

Left sum ORIGINAL sample11 affine value plus c = 9

Left sum ORIGINAL sample11 value times cell width := mul(9, 1/11)

Left sum ORIGINAL sample11 value times cell width = 9/11

Left sum ORIGINAL accumulated term11 := sum(21, 9/11)

Left sum ORIGINAL accumulated term11 = 240/11

source roles = [11, 22, -22, 9, -1, 1, 1/11]

derived = [0/1, 9/1, 9/11, 240/11]

Left sum ORIGINAL sample12 index times width := mul(12, 1/11)

Left sum ORIGINAL sample12 index times width = 12/11

Left sum ORIGINAL sample12 lower plus offset := sum(-1, 12/11)

Left sum ORIGINAL sample12 lower plus offset = 1/11

Left sum ORIGINAL sample12 slope times x := mul(-22, 1/11)

Left sum ORIGINAL sample12 slope times x = -2

Left sum ORIGINAL sample12 affine value plus c := sum(-2, 9)

Left sum ORIGINAL sample12 affine value plus c = 7

Left sum ORIGINAL sample12 value times cell width := mul(7, 1/11)

Left sum ORIGINAL sample12 value times cell width = 7/11

Left sum ORIGINAL accumulated term12 := sum(240/11, 7/11)

Left sum ORIGINAL accumulated term12 = 247/11

source roles = [12, 22, -22, 9, -1, 1, 1/11]

derived = [1/11, 7/1, 7/11, 247/11]

Left sum ORIGINAL sample13 index times width := mul(13, 1/11)

Left sum ORIGINAL sample13 index times width = 13/11

Left sum ORIGINAL sample13 lower plus offset := sum(-1, 13/11)

Left sum ORIGINAL sample13 lower plus offset = 2/11

Left sum ORIGINAL sample13 slope times x := mul(-22, 2/11)

Left sum ORIGINAL sample13 slope times x = -4

Left sum ORIGINAL sample13 affine value plus c := sum(-4, 9)

Left sum ORIGINAL sample13 affine value plus c = 5

Left sum ORIGINAL sample13 value times cell width := mul(5, 1/11)

Left sum ORIGINAL sample13 value times cell width = 5/11

Left sum ORIGINAL accumulated term13 := sum(247/11, 5/11)

Left sum ORIGINAL accumulated term13 = 252/11

source roles = [13, 22, -22, 9, -1, 1, 1/11]

derived = [2/11, 5/1, 5/11, 252/11]

Left sum ORIGINAL sample14 index times width := mul(14, 1/11)

Left sum ORIGINAL sample14 index times width = 14/11

Left sum ORIGINAL sample14 lower plus offset := sum(-1, 14/11)

Left sum ORIGINAL sample14 lower plus offset = 3/11

Left sum ORIGINAL sample14 slope times x := mul(-22, 3/11)

Left sum ORIGINAL sample14 slope times x = -6

Left sum ORIGINAL sample14 affine value plus c := sum(-6, 9)

Left sum ORIGINAL sample14 affine value plus c = 3

Left sum ORIGINAL sample14 value times cell width := mul(3, 1/11)

Left sum ORIGINAL sample14 value times cell width = 3/11

Left sum ORIGINAL accumulated term14 := sum(252/11, 3/11)

Left sum ORIGINAL accumulated term14 = 255/11

source roles = [14, 22, -22, 9, -1, 1, 1/11]

derived = [3/11, 3/1, 3/11, 255/11]

Left sum ORIGINAL sample15 index times width := mul(15, 1/11)

Left sum ORIGINAL sample15 index times width = 15/11

Left sum ORIGINAL sample15 lower plus offset := sum(-1, 15/11)

Left sum ORIGINAL sample15 lower plus offset = 4/11

Left sum ORIGINAL sample15 slope times x := mul(-22, 4/11)

Left sum ORIGINAL sample15 slope times x = -8

Left sum ORIGINAL sample15 affine value plus c := sum(-8, 9)

Left sum ORIGINAL sample15 affine value plus c = 1

Left sum ORIGINAL sample15 value times cell width := mul(1, 1/11)

Left sum ORIGINAL sample15 value times cell width = 1/11

Left sum ORIGINAL accumulated term15 := sum(255/11, 1/11)

Left sum ORIGINAL accumulated term15 = 256/11

source roles = [15, 22, -22, 9, -1, 1, 1/11]

derived = [4/11, 1/1, 1/11, 256/11]

Left sum ORIGINAL sample16 index times width := mul(16, 1/11)

Left sum ORIGINAL sample16 index times width = 16/11

Left sum ORIGINAL sample16 lower plus offset := sum(-1, 16/11)

Left sum ORIGINAL sample16 lower plus offset = 5/11

Left sum ORIGINAL sample16 slope times x := mul(-22, 5/11)

Left sum ORIGINAL sample16 slope times x = -10

Left sum ORIGINAL sample16 affine value plus c := sum(-10, 9)

Left sum ORIGINAL sample16 affine value plus c = -1

Left sum ORIGINAL sample16 value times cell width := mul(-1, 1/11)

Left sum ORIGINAL sample16 value times cell width = -1/11

Left sum ORIGINAL accumulated term16 := sum(256/11, -1/11)

Left sum ORIGINAL accumulated term16 = 255/11

source roles = [16, 22, -22, 9, -1, 1, 1/11]

derived = [5/11, -1/1, -1/11, 255/11]

Left sum ORIGINAL sample17 index times width := mul(17, 1/11)

Left sum ORIGINAL sample17 index times width = 17/11

Left sum ORIGINAL sample17 lower plus offset := sum(-1, 17/11)

Left sum ORIGINAL sample17 lower plus offset = 6/11

Left sum ORIGINAL sample17 slope times x := mul(-22, 6/11)

Left sum ORIGINAL sample17 slope times x = -12

Left sum ORIGINAL sample17 affine value plus c := sum(-12, 9)

Left sum ORIGINAL sample17 affine value plus c = -3

Left sum ORIGINAL sample17 value times cell width := mul(-3, 1/11)

Left sum ORIGINAL sample17 value times cell width = -3/11

Left sum ORIGINAL accumulated term17 := sum(255/11, -3/11)

Left sum ORIGINAL accumulated term17 = 252/11

source roles = [17, 22, -22, 9, -1, 1, 1/11]

derived = [6/11, -3/1, -3/11, 252/11]

Left sum ORIGINAL sample18 index times width := mul(18, 1/11)

Left sum ORIGINAL sample18 index times width = 18/11

Left sum ORIGINAL sample18 lower plus offset := sum(-1, 18/11)

Left sum ORIGINAL sample18 lower plus offset = 7/11

Left sum ORIGINAL sample18 slope times x := mul(-22, 7/11)

Left sum ORIGINAL sample18 slope times x = -14

Left sum ORIGINAL sample18 affine value plus c := sum(-14, 9)

Left sum ORIGINAL sample18 affine value plus c = -5

Left sum ORIGINAL sample18 value times cell width := mul(-5, 1/11)

Left sum ORIGINAL sample18 value times cell width = -5/11

Left sum ORIGINAL accumulated term18 := sum(252/11, -5/11)

Left sum ORIGINAL accumulated term18 = 247/11

source roles = [18, 22, -22, 9, -1, 1, 1/11]

derived = [7/11, -5/1, -5/11, 247/11]

Left sum ORIGINAL sample19 index times width := mul(19, 1/11)

Left sum ORIGINAL sample19 index times width = 19/11

Left sum ORIGINAL sample19 lower plus offset := sum(-1, 19/11)

Left sum ORIGINAL sample19 lower plus offset = 8/11

Left sum ORIGINAL sample19 slope times x := mul(-22, 8/11)

Left sum ORIGINAL sample19 slope times x = -16

Left sum ORIGINAL sample19 affine value plus c := sum(-16, 9)

Left sum ORIGINAL sample19 affine value plus c = -7

Left sum ORIGINAL sample19 value times cell width := mul(-7, 1/11)

Left sum ORIGINAL sample19 value times cell width = -7/11

Left sum ORIGINAL accumulated term19 := sum(247/11, -7/11)

Left sum ORIGINAL accumulated term19 = 240/11

source roles = [19, 22, -22, 9, -1, 1, 1/11]

derived = [8/11, -7/1, -7/11, 240/11]

Left sum ORIGINAL sample20 index times width := mul(20, 1/11)

Left sum ORIGINAL sample20 index times width = 20/11

Left sum ORIGINAL sample20 lower plus offset := sum(-1, 20/11)

Left sum ORIGINAL sample20 lower plus offset = 9/11

Left sum ORIGINAL sample20 slope times x := mul(-22, 9/11)

Left sum ORIGINAL sample20 slope times x = -18

Left sum ORIGINAL sample20 affine value plus c := sum(-18, 9)

Left sum ORIGINAL sample20 affine value plus c = -9

Left sum ORIGINAL sample20 value times cell width := mul(-9, 1/11)

Left sum ORIGINAL sample20 value times cell width = -9/11

Left sum ORIGINAL accumulated term20 := sum(240/11, -9/11)

Left sum ORIGINAL accumulated term20 = 21

source roles = [20, 22, -22, 9, -1, 1, 1/11]

derived = [9/11, -9/1, -9/11, 21/1]

Left sum ORIGINAL sample21 index times width := mul(21, 1/11)

Left sum ORIGINAL sample21 index times width = 21/11

Left sum ORIGINAL sample21 lower plus offset := sum(-1, 21/11)

Left sum ORIGINAL sample21 lower plus offset = 10/11

Left sum ORIGINAL sample21 slope times x := mul(-22, 10/11)

Left sum ORIGINAL sample21 slope times x = -20

Left sum ORIGINAL sample21 affine value plus c := sum(-20, 9)

Left sum ORIGINAL sample21 affine value plus c = -11

Left sum ORIGINAL sample21 value times cell width := mul(-11, 1/11)

Left sum ORIGINAL sample21 value times cell width = -1

Left sum ORIGINAL accumulated term21 := sum(21, -1)

Left sum ORIGINAL accumulated term21 = 20

source roles = [21, 22, -22, 9, -1, 1, 1/11]

derived = [10/11, -11/1, -1/1, 20/1]

source roles = [22, -22, 9, -1, 1]

derived = [1/11, 20/1]

逐步反向核對

• S158 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S157 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S156 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term21，不以小數近似替代。
• S155 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample21 value times cell width，不以小數近似替代。
• S154 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample21 affine value plus c，不以小數近似替代。
• S153 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample21 slope times x，不以小數近似替代。
• S152 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample21 lower plus offset，不以小數近似替代。
• S151 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample21 index times width，不以小數近似替代。
• S150 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S149 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term20，不以小數近似替代。
• S148 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample20 value times cell width，不以小數近似替代。
• S147 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample20 affine value plus c，不以小數近似替代。
• S146 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample20 slope times x，不以小數近似替代。
• S145 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample20 lower plus offset，不以小數近似替代。
• S144 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample20 index times width，不以小數近似替代。
• S143 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S142 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term19，不以小數近似替代。
• S141 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample19 value times cell width，不以小數近似替代。
• S140 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample19 affine value plus c，不以小數近似替代。
• S139 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample19 slope times x，不以小數近似替代。
• S138 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample19 lower plus offset，不以小數近似替代。
• S137 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample19 index times width，不以小數近似替代。
• S136 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S135 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term18，不以小數近似替代。
• S134 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample18 value times cell width，不以小數近似替代。
• S133 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample18 affine value plus c，不以小數近似替代。
• S132 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample18 slope times x，不以小數近似替代。
• S131 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample18 lower plus offset，不以小數近似替代。
• S130 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample18 index times width，不以小數近似替代。
• S129 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S128 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term17，不以小數近似替代。
• S127 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample17 value times cell width，不以小數近似替代。
• S126 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample17 affine value plus c，不以小數近似替代。
• S125 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample17 slope times x，不以小數近似替代。
• S124 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample17 lower plus offset，不以小數近似替代。
• S123 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample17 index times width，不以小數近似替代。
• S122 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S121 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term16，不以小數近似替代。
• S120 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample16 value times cell width，不以小數近似替代。
• S119 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample16 affine value plus c，不以小數近似替代。
• S118 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample16 slope times x，不以小數近似替代。
• S117 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample16 lower plus offset，不以小數近似替代。
• S116 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample16 index times width，不以小數近似替代。
• S115 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S114 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term15，不以小數近似替代。
• S113 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample15 value times cell width，不以小數近似替代。
• S112 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample15 affine value plus c，不以小數近似替代。
• S111 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample15 slope times x，不以小數近似替代。
• S110 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample15 lower plus offset，不以小數近似替代。
• S109 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample15 index times width，不以小數近似替代。
• S108 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S107 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term14，不以小數近似替代。
• S106 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample14 value times cell width，不以小數近似替代。
• S105 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample14 affine value plus c，不以小數近似替代。
• S104 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample14 slope times x，不以小數近似替代。
• S103 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample14 lower plus offset，不以小數近似替代。
• S102 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample14 index times width，不以小數近似替代。
• S101 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S100 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term13，不以小數近似替代。
• S099 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample13 value times cell width，不以小數近似替代。
• S098 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample13 affine value plus c，不以小數近似替代。
• S097 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample13 slope times x，不以小數近似替代。
• S096 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample13 lower plus offset，不以小數近似替代。
• S095 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample13 index times width，不以小數近似替代。
• S094 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S093 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term12，不以小數近似替代。
• S092 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample12 value times cell width，不以小數近似替代。
• S091 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample12 affine value plus c，不以小數近似替代。
• S090 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample12 slope times x，不以小數近似替代。
• S089 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample12 lower plus offset，不以小數近似替代。
• S088 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample12 index times width，不以小數近似替代。
• S087 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S086 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term11，不以小數近似替代。
• S085 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample11 value times cell width，不以小數近似替代。
• S084 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample11 affine value plus c，不以小數近似替代。
• S083 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample11 slope times x，不以小數近似替代。
• S082 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample11 lower plus offset，不以小數近似替代。
• S081 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample11 index times width，不以小數近似替代。
• S080 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S079 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term10，不以小數近似替代。
• S078 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample10 value times cell width，不以小數近似替代。
• S077 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample10 affine value plus c，不以小數近似替代。
• S076 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample10 slope times x，不以小數近似替代。
• S075 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample10 lower plus offset，不以小數近似替代。
• S074 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample10 index times width，不以小數近似替代。
• S073 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S072 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term9，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample9 value times cell width，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample9 affine value plus c，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample9 slope times x，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample9 lower plus offset，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample9 index times width，不以小數近似替代。
• S066 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S065 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term8，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample8 value times cell width，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample8 affine value plus c，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample8 slope times x，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample8 lower plus offset，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample8 index times width，不以小數近似替代。
• S059 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S058 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term7，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample7 value times cell width，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample7 affine value plus c，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample7 slope times x，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample7 lower plus offset，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample7 index times width，不以小數近似替代。
• S052 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S051 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term6，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample6 value times cell width，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample6 affine value plus c，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample6 slope times x，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample6 lower plus offset，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample6 index times width，不以小數近似替代。
• S045 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S044 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term5，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample5 value times cell width，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample5 affine value plus c，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample5 slope times x，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample5 lower plus offset，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample5 index times width，不以小數近似替代。
• S038 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S037 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term4，不以小數近似替代。
• S036 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample4 value times cell width，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample4 affine value plus c，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample4 slope times x，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample4 lower plus offset，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample4 index times width，不以小數近似替代。
• S031 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S030 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term3，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample3 value times cell width，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample3 affine value plus c，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample3 slope times x，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample3 lower plus offset，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample3 index times width，不以小數近似替代。
• S024 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S023 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term2，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample2 value times cell width，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample2 affine value plus c，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample2 slope times x，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample2 lower plus offset，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample2 index times width，不以小數近似替代。
• S017 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S016 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term1，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample1 value times cell width，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample1 affine value plus c，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample1 slope times x，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample1 lower plus offset，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample1 index times width，不以小數近似替代。
• S010 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。
• S009 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL accumulated term0，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample0 value times cell width，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample0 affine value plus c，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample0 slope times x，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample0 lower plus offset，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL sample0 index times width，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL width over positive cell count，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 Left sum ORIGINAL upper minus lower endpoint，不以小數近似替代。
• S001 · premise_and_result_audit — 以原始資料、欄位角色、型別、論域及本步前提反向代回核對；不表示結論能唯一還原前提。來源綁定候選、獨立審查、形式證明與發布分開。

結論

20/1

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。