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← Question derivation index · Original question batch
R0006-03873 · q-20261007-advance-20261009-r06-definite-integral-0073
On all R, f(x)=(23)x+(10). On a=0≤b=5, use N=23 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. Use reduced positive-denominator fractions, retaining /1 for integers.
G001 — On all R, f(x)=(23)x+(10). On a=0≤b=5, use N=23 uniform left cells. Find the signed error I−S_N, exact moment integral minus the finite left sum. Use reduced positive-denominator fractions, retaining /1 for integers.Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
explicit real polynomial moment or constant integrand model
After
oriented I(a,b)=H(b)−H(a);do not sort endpoints
Conditions: none
Real polynomials are continuous and integrable;use termwise primitive endpoint differences for exact oriented integrals,not absolute area or a finite sum.
Rule: polynomial_moment_scope · Uses: G001
Before
interval width := sub(5, 0)
After
interval width = 5
Conditions: none
Compute the nonnegative interval width,allowing zero width.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
partition step := div(5, 23)
After
partition step = 5/23
Conditions: none
Divide by positive integer N to get the exact step.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
left sample0 offset := mul(0, 5/23)
After
left sample0 offset = 0
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
left sample0 x := sum(0, 0)
After
left sample0 x = 0
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
left sample0 mx := mul(23, 0)
After
left sample0 mx = 0
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
left sample0 value := sum(0, 10)
After
left sample0 value = 10
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
left sample0 rectangle := mul(10, 5/23)
After
left sample0 rectangle = 50/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
[0, 10, 50/23]
After
finite left-sum term index=0
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S008
Before
left sample1 offset := mul(1, 5/23)
After
left sample1 offset = 5/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
left sample1 x := sum(0, 5/23)
After
left sample1 x = 5/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
left sample1 mx := mul(23, 5/23)
After
left sample1 mx = 5
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
left sample1 value := sum(5, 10)
After
left sample1 value = 15
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
left sample1 rectangle := mul(15, 5/23)
After
left sample1 rectangle = 75/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
[5/23, 15, 75/23]
After
finite left-sum term index=1
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S014
Before
left sample2 offset := mul(2, 5/23)
After
left sample2 offset = 10/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
left sample2 x := sum(0, 10/23)
After
left sample2 x = 10/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
left sample2 mx := mul(23, 10/23)
After
left sample2 mx = 10
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
left sample2 value := sum(10, 10)
After
left sample2 value = 20
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
left sample2 rectangle := mul(20, 5/23)
After
left sample2 rectangle = 100/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S019
Before
[10/23, 20, 100/23]
After
finite left-sum term index=2
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S020
Before
left sample3 offset := mul(3, 5/23)
After
left sample3 offset = 15/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S021
Before
left sample3 x := sum(0, 15/23)
After
left sample3 x = 15/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S022
Before
left sample3 mx := mul(23, 15/23)
After
left sample3 mx = 15
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S023
Before
left sample3 value := sum(15, 10)
After
left sample3 value = 25
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S024
Before
left sample3 rectangle := mul(25, 5/23)
After
left sample3 rectangle = 125/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S025
Before
[15/23, 25, 125/23]
After
finite left-sum term index=3
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S026
Before
left sample4 offset := mul(4, 5/23)
After
left sample4 offset = 20/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S027
Before
left sample4 x := sum(0, 20/23)
After
left sample4 x = 20/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S028
Before
left sample4 mx := mul(23, 20/23)
After
left sample4 mx = 20
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S029
Before
left sample4 value := sum(20, 10)
After
left sample4 value = 30
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S030
Before
left sample4 rectangle := mul(30, 5/23)
After
left sample4 rectangle = 150/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S031
Before
[20/23, 30, 150/23]
After
finite left-sum term index=4
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S032
Before
left sample5 offset := mul(5, 5/23)
After
left sample5 offset = 25/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S033
Before
left sample5 x := sum(0, 25/23)
After
left sample5 x = 25/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S034
Before
left sample5 mx := mul(23, 25/23)
After
left sample5 mx = 25
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S035
Before
left sample5 value := sum(25, 10)
After
left sample5 value = 35
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S036
Before
left sample5 rectangle := mul(35, 5/23)
After
left sample5 rectangle = 175/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S037
Before
[25/23, 35, 175/23]
After
finite left-sum term index=5
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S038
Before
left sample6 offset := mul(6, 5/23)
After
left sample6 offset = 30/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S039
Before
left sample6 x := sum(0, 30/23)
After
left sample6 x = 30/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S040
Before
left sample6 mx := mul(23, 30/23)
After
left sample6 mx = 30
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S041
Before
left sample6 value := sum(30, 10)
After
left sample6 value = 40
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S042
Before
left sample6 rectangle := mul(40, 5/23)
After
left sample6 rectangle = 200/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S043
Before
[30/23, 40, 200/23]
After
finite left-sum term index=6
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S044
Before
left sample7 offset := mul(7, 5/23)
After
left sample7 offset = 35/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S045
Before
left sample7 x := sum(0, 35/23)
After
left sample7 x = 35/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S046
Before
left sample7 mx := mul(23, 35/23)
After
left sample7 mx = 35
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S047
Before
left sample7 value := sum(35, 10)
After
left sample7 value = 45
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S048
Before
left sample7 rectangle := mul(45, 5/23)
After
left sample7 rectangle = 225/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S049
Before
[35/23, 45, 225/23]
After
finite left-sum term index=7
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S050
Before
left sample8 offset := mul(8, 5/23)
After
left sample8 offset = 40/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S051
Before
left sample8 x := sum(0, 40/23)
After
left sample8 x = 40/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S052
Before
left sample8 mx := mul(23, 40/23)
After
left sample8 mx = 40
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S053
Before
left sample8 value := sum(40, 10)
After
left sample8 value = 50
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S054
Before
left sample8 rectangle := mul(50, 5/23)
After
left sample8 rectangle = 250/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S055
Before
[40/23, 50, 250/23]
After
finite left-sum term index=8
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S056
Before
left sample9 offset := mul(9, 5/23)
After
left sample9 offset = 45/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S057
Before
left sample9 x := sum(0, 45/23)
After
left sample9 x = 45/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S058
Before
left sample9 mx := mul(23, 45/23)
After
left sample9 mx = 45
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S059
Before
left sample9 value := sum(45, 10)
After
left sample9 value = 55
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S060
Before
left sample9 rectangle := mul(55, 5/23)
After
left sample9 rectangle = 275/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S061
Before
[45/23, 55, 275/23]
After
finite left-sum term index=9
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S062
Before
left sample10 offset := mul(10, 5/23)
After
left sample10 offset = 50/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S063
Before
left sample10 x := sum(0, 50/23)
After
left sample10 x = 50/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S064
Before
left sample10 mx := mul(23, 50/23)
After
left sample10 mx = 50
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S065
Before
left sample10 value := sum(50, 10)
After
left sample10 value = 60
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S066
Before
left sample10 rectangle := mul(60, 5/23)
After
left sample10 rectangle = 300/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S067
Before
[50/23, 60, 300/23]
After
finite left-sum term index=10
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S068
Before
left sample11 offset := mul(11, 5/23)
After
left sample11 offset = 55/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S069
Before
left sample11 x := sum(0, 55/23)
After
left sample11 x = 55/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S070
Before
left sample11 mx := mul(23, 55/23)
After
left sample11 mx = 55
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S071
Before
left sample11 value := sum(55, 10)
After
left sample11 value = 65
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S072
Before
left sample11 rectangle := mul(65, 5/23)
After
left sample11 rectangle = 325/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S073
Before
[55/23, 65, 325/23]
After
finite left-sum term index=11
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S074
Before
left sample12 offset := mul(12, 5/23)
After
left sample12 offset = 60/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S075
Before
left sample12 x := sum(0, 60/23)
After
left sample12 x = 60/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S076
Before
left sample12 mx := mul(23, 60/23)
After
left sample12 mx = 60
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S077
Before
left sample12 value := sum(60, 10)
After
left sample12 value = 70
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S078
Before
left sample12 rectangle := mul(70, 5/23)
After
left sample12 rectangle = 350/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S079
Before
[60/23, 70, 350/23]
After
finite left-sum term index=12
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S080
Before
left sample13 offset := mul(13, 5/23)
After
left sample13 offset = 65/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S081
Before
left sample13 x := sum(0, 65/23)
After
left sample13 x = 65/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S082
Before
left sample13 mx := mul(23, 65/23)
After
left sample13 mx = 65
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S083
Before
left sample13 value := sum(65, 10)
After
left sample13 value = 75
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S084
Before
left sample13 rectangle := mul(75, 5/23)
After
left sample13 rectangle = 375/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S085
Before
[65/23, 75, 375/23]
After
finite left-sum term index=13
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S086
Before
left sample14 offset := mul(14, 5/23)
After
left sample14 offset = 70/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S087
Before
left sample14 x := sum(0, 70/23)
After
left sample14 x = 70/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S088
Before
left sample14 mx := mul(23, 70/23)
After
left sample14 mx = 70
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S089
Before
left sample14 value := sum(70, 10)
After
left sample14 value = 80
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S090
Before
left sample14 rectangle := mul(80, 5/23)
After
left sample14 rectangle = 400/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S091
Before
[70/23, 80, 400/23]
After
finite left-sum term index=14
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S092
Before
left sample15 offset := mul(15, 5/23)
After
left sample15 offset = 75/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S093
Before
left sample15 x := sum(0, 75/23)
After
left sample15 x = 75/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S094
Before
left sample15 mx := mul(23, 75/23)
After
left sample15 mx = 75
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S095
Before
left sample15 value := sum(75, 10)
After
left sample15 value = 85
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S096
Before
left sample15 rectangle := mul(85, 5/23)
After
left sample15 rectangle = 425/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S097
Before
[75/23, 85, 425/23]
After
finite left-sum term index=15
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S098
Before
left sample16 offset := mul(16, 5/23)
After
left sample16 offset = 80/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S099
Before
left sample16 x := sum(0, 80/23)
After
left sample16 x = 80/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S100
Before
left sample16 mx := mul(23, 80/23)
After
left sample16 mx = 80
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S101
Before
left sample16 value := sum(80, 10)
After
left sample16 value = 90
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S102
Before
left sample16 rectangle := mul(90, 5/23)
After
left sample16 rectangle = 450/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S103
Before
[80/23, 90, 450/23]
After
finite left-sum term index=16
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S104
Before
left sample17 offset := mul(17, 5/23)
After
left sample17 offset = 85/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S105
Before
left sample17 x := sum(0, 85/23)
After
left sample17 x = 85/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S106
Before
left sample17 mx := mul(23, 85/23)
After
left sample17 mx = 85
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S107
Before
left sample17 value := sum(85, 10)
After
left sample17 value = 95
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S108
Before
left sample17 rectangle := mul(95, 5/23)
After
left sample17 rectangle = 475/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S109
Before
[85/23, 95, 475/23]
After
finite left-sum term index=17
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S110
Before
left sample18 offset := mul(18, 5/23)
After
left sample18 offset = 90/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S111
Before
left sample18 x := sum(0, 90/23)
After
left sample18 x = 90/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S112
Before
left sample18 mx := mul(23, 90/23)
After
left sample18 mx = 90
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S113
Before
left sample18 value := sum(90, 10)
After
left sample18 value = 100
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S114
Before
left sample18 rectangle := mul(100, 5/23)
After
left sample18 rectangle = 500/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S115
Before
[90/23, 100, 500/23]
After
finite left-sum term index=18
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S116
Before
left sample19 offset := mul(19, 5/23)
After
left sample19 offset = 95/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S117
Before
left sample19 x := sum(0, 95/23)
After
left sample19 x = 95/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S118
Before
left sample19 mx := mul(23, 95/23)
After
left sample19 mx = 95
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S119
Before
left sample19 value := sum(95, 10)
After
left sample19 value = 105
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S120
Before
left sample19 rectangle := mul(105, 5/23)
After
left sample19 rectangle = 525/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S121
Before
[95/23, 105, 525/23]
After
finite left-sum term index=19
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S122
Before
left sample20 offset := mul(20, 5/23)
After
left sample20 offset = 100/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S123
Before
left sample20 x := sum(0, 100/23)
After
left sample20 x = 100/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S124
Before
left sample20 mx := mul(23, 100/23)
After
left sample20 mx = 100
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S125
Before
left sample20 value := sum(100, 10)
After
left sample20 value = 110
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S126
Before
left sample20 rectangle := mul(110, 5/23)
After
left sample20 rectangle = 550/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S127
Before
[100/23, 110, 550/23]
After
finite left-sum term index=20
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S128
Before
left sample21 offset := mul(21, 5/23)
After
left sample21 offset = 105/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S129
Before
left sample21 x := sum(0, 105/23)
After
left sample21 x = 105/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S130
Before
left sample21 mx := mul(23, 105/23)
After
left sample21 mx = 105
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S131
Before
left sample21 value := sum(105, 10)
After
left sample21 value = 115
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S132
Before
left sample21 rectangle := mul(115, 5/23)
After
left sample21 rectangle = 25
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S133
Before
[105/23, 115, 25]
After
finite left-sum term index=21
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S134
Before
left sample22 offset := mul(22, 5/23)
After
left sample22 offset = 110/23
Conditions: none
Left samples use i=0,...,N−1,not the final right endpoint.
Rule: exact_rational_arithmetic · Uses: G001, S135
Before
left sample22 x := sum(0, 110/23)
After
left sample22 x = 110/23
Conditions: none
Add the indexed offset to the start.
Rule: exact_rational_arithmetic · Uses: G001, S136
Before
left sample22 mx := mul(23, 110/23)
After
left sample22 mx = 110
Conditions: none
Evaluate the product term of the affine integrand.
Rule: exact_rational_arithmetic · Uses: G001, S137
Before
left sample22 value := sum(110, 10)
After
left sample22 value = 120
Conditions: none
Add c to obtain this sample value.
Rule: exact_rational_arithmetic · Uses: G001, S138
Before
left sample22 rectangle := mul(120, 5/23)
After
left sample22 rectangle = 600/23
Conditions: none
Multiply by the step to get the signed rectangle contribution,without absolute values.
Rule: exact_rational_arithmetic · Uses: G001, S139
Before
[110/23, 120, 600/23]
After
finite left-sum term index=22
Conditions: none
List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.
Rule: left_partition_term · Uses: G001, S140
Before
finite left sum := sum(50/23, 75/23, 100/23, 125/23, 150/23, 175/23, 200/23, 225/23, 250/23, 275/23, 300/23, 325/23, 350/23, 375/23, 400/23, 425/23, 450/23, 475/23, 500/23, 525/23, 550/23, 25, 600/23)
After
finite left sum = 325
Conditions: none
Add all N contributions.
Rule: exact_rational_arithmetic · Uses: G001, S141
Before
affine integral primitive coefficient1 := div(10, 1)
After
affine integral primitive coefficient1 = 10
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S142
Before
affine integral primitive coefficient2 := div(23, 2)
After
affine integral primitive coefficient2 = 23/2
Conditions: none
The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.
Rule: exact_rational_arithmetic · Uses: G001, S143
Before
[0, 10, 23/2]
After
H′=the supplied polynomial;integrate from 0 to 5
Conditions: none
Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.
Rule: polynomial_primitive_identity · Uses: G001, S144
Before
affine integral H(b) ascending coefficients=[0, 10, 23/2]
After
evaluate at x=5
Conditions: none
Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.
Rule: polynomial_evaluation_order · Uses: G001, S145
Before
affine integral H(b) Horner multiply before degree1 := mul(23/2, 5)
After
affine integral H(b) Horner multiply before degree1 = 115/2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S146
Before
affine integral H(b) Horner add degree1 := sum(115/2, 10)
After
affine integral H(b) Horner add degree1 = 135/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S147
Before
affine integral H(b) Horner multiply before degree0 := mul(135/2, 5)
After
affine integral H(b) Horner multiply before degree0 = 675/2
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S148
Before
affine integral H(b) Horner add degree0 := sum(675/2, 0)
After
affine integral H(b) Horner add degree0 = 675/2
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S149
Before
affine integral H(a) ascending coefficients=[0, 10, 23/2]
After
evaluate at x=0
Conditions: none
Evaluate ascending coefficients by Horner from the highest nonzero term down;ignoring redundant top zero terms changes neither the function nor the retained original list.
Rule: polynomial_evaluation_order · Uses: G001, S150
Before
affine integral H(a) Horner multiply before degree1 := mul(23/2, 0)
After
affine integral H(a) Horner multiply before degree1 = 0
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S151
Before
affine integral H(a) Horner add degree1 := sum(0, 10)
After
affine integral H(a) Horner add degree1 = 10
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S152
Before
affine integral H(a) Horner multiply before degree0 := mul(10, 0)
After
affine integral H(a) Horner multiply before degree0 = 0
Conditions: none
Multiply by the same evaluation point first.
Rule: exact_rational_arithmetic · Uses: G001, S153
Before
affine integral H(a) Horner add degree0 := sum(0, 0)
After
affine integral H(a) Horner add degree0 = 0
Conditions: none
Then add the full coefficient at this position.
Rule: exact_rational_arithmetic · Uses: G001, S154
Before
affine integral oriented moment := sub(675/2, 0)
After
affine integral oriented moment = 675/2
Conditions: none
Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.
Rule: exact_rational_arithmetic · Uses: G001, S155
Before
integral minus left sum := sub(675/2, 325)
After
integral minus left sum = 25/2
Conditions: none
The requested direction is I−S_N;a negative slope may give a negative difference,not absolute error.
Rule: exact_rational_arithmetic · Uses: G001, S156
Before
I=675/2;S_N=325;N=23
After
signed finite-sum difference=25/2
Conditions: none
This is the exact difference for finite N,not evidence of a general convergence rate or integrability of arbitrary functions.
Rule: finite_sum_vs_integral · Uses: G001, S157
S158 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S157 · premise_and_result_audit — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals.S156 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals.S155 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.S154 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S153 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.S152 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S150 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.S149 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.S148 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.S147 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.S146 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S145 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S144 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals.S143 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals.S142 · premise_and_result_audit — Recompute finite left sum with exact rational arithmetic, not rounded decimals.S141 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S140 · premise_and_result_audit — Recompute left sample22 rectangle with exact rational arithmetic, not rounded decimals.S139 · premise_and_result_audit — Recompute left sample22 value with exact rational arithmetic, not rounded decimals.S138 · premise_and_result_audit — Recompute left sample22 mx with exact rational arithmetic, not rounded decimals.S137 · premise_and_result_audit — Recompute left sample22 x with exact rational arithmetic, not rounded decimals.S136 · premise_and_result_audit — Recompute left sample22 offset with exact rational arithmetic, not rounded decimals.S135 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S134 · premise_and_result_audit — Recompute left sample21 rectangle with exact rational arithmetic, not rounded decimals.S133 · premise_and_result_audit — Recompute left sample21 value with exact rational arithmetic, not rounded decimals.S132 · premise_and_result_audit — Recompute left sample21 mx with exact rational arithmetic, not rounded decimals.S131 · premise_and_result_audit — Recompute left sample21 x with exact rational arithmetic, not rounded decimals.S130 · premise_and_result_audit — Recompute left sample21 offset with exact rational arithmetic, not rounded decimals.S129 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S128 · premise_and_result_audit — Recompute left sample20 rectangle with exact rational arithmetic, not rounded decimals.S127 · premise_and_result_audit — Recompute left sample20 value with exact rational arithmetic, not rounded decimals.S126 · premise_and_result_audit — Recompute left sample20 mx with exact rational arithmetic, not rounded decimals.S125 · premise_and_result_audit — Recompute left sample20 x with exact rational arithmetic, not rounded decimals.S124 · premise_and_result_audit — Recompute left sample20 offset with exact rational arithmetic, not rounded decimals.S123 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S122 · premise_and_result_audit — Recompute left sample19 rectangle with exact rational arithmetic, not rounded decimals.S121 · premise_and_result_audit — Recompute left sample19 value with exact rational arithmetic, not rounded decimals.S120 · premise_and_result_audit — Recompute left sample19 mx with exact rational arithmetic, not rounded decimals.S119 · premise_and_result_audit — Recompute left sample19 x with exact rational arithmetic, not rounded decimals.S118 · premise_and_result_audit — Recompute left sample19 offset with exact rational arithmetic, not rounded decimals.S117 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S116 · premise_and_result_audit — Recompute left sample18 rectangle with exact rational arithmetic, not rounded decimals.S115 · premise_and_result_audit — Recompute left sample18 value with exact rational arithmetic, not rounded decimals.S114 · premise_and_result_audit — Recompute left sample18 mx with exact rational arithmetic, not rounded decimals.S113 · premise_and_result_audit — Recompute left sample18 x with exact rational arithmetic, not rounded decimals.S112 · premise_and_result_audit — Recompute left sample18 offset with exact rational arithmetic, not rounded decimals.S111 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S110 · premise_and_result_audit — Recompute left sample17 rectangle with exact rational arithmetic, not rounded decimals.S109 · premise_and_result_audit — Recompute left sample17 value with exact rational arithmetic, not rounded decimals.S108 · premise_and_result_audit — Recompute left sample17 mx with exact rational arithmetic, not rounded decimals.S107 · premise_and_result_audit — Recompute left sample17 x with exact rational arithmetic, not rounded decimals.S106 · premise_and_result_audit — Recompute left sample17 offset with exact rational arithmetic, not rounded decimals.S105 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S104 · premise_and_result_audit — Recompute left sample16 rectangle with exact rational arithmetic, not rounded decimals.S103 · premise_and_result_audit — Recompute left sample16 value with exact rational arithmetic, not rounded decimals.S102 · premise_and_result_audit — Recompute left sample16 mx with exact rational arithmetic, not rounded decimals.S101 · premise_and_result_audit — Recompute left sample16 x with exact rational arithmetic, not rounded decimals.S100 · premise_and_result_audit — Recompute left sample16 offset with exact rational arithmetic, not rounded decimals.S099 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S098 · premise_and_result_audit — Recompute left sample15 rectangle with exact rational arithmetic, not rounded decimals.S097 · premise_and_result_audit — Recompute left sample15 value with exact rational arithmetic, not rounded decimals.S096 · premise_and_result_audit — Recompute left sample15 mx with exact rational arithmetic, not rounded decimals.S095 · premise_and_result_audit — Recompute left sample15 x with exact rational arithmetic, not rounded decimals.S094 · premise_and_result_audit — Recompute left sample15 offset with exact rational arithmetic, not rounded decimals.S093 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S092 · premise_and_result_audit — Recompute left sample14 rectangle with exact rational arithmetic, not rounded decimals.S091 · premise_and_result_audit — Recompute left sample14 value with exact rational arithmetic, not rounded decimals.S090 · premise_and_result_audit — Recompute left sample14 mx with exact rational arithmetic, not rounded decimals.S089 · premise_and_result_audit — Recompute left sample14 x with exact rational arithmetic, not rounded decimals.S088 · premise_and_result_audit — Recompute left sample14 offset with exact rational arithmetic, not rounded decimals.S087 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S086 · premise_and_result_audit — Recompute left sample13 rectangle with exact rational arithmetic, not rounded decimals.S085 · premise_and_result_audit — Recompute left sample13 value with exact rational arithmetic, not rounded decimals.S084 · premise_and_result_audit — Recompute left sample13 mx with exact rational arithmetic, not rounded decimals.S083 · premise_and_result_audit — Recompute left sample13 x with exact rational arithmetic, not rounded decimals.S082 · premise_and_result_audit — Recompute left sample13 offset with exact rational arithmetic, not rounded decimals.S081 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S080 · premise_and_result_audit — Recompute left sample12 rectangle with exact rational arithmetic, not rounded decimals.S079 · premise_and_result_audit — Recompute left sample12 value with exact rational arithmetic, not rounded decimals.S078 · premise_and_result_audit — Recompute left sample12 mx with exact rational arithmetic, not rounded decimals.S077 · premise_and_result_audit — Recompute left sample12 x with exact rational arithmetic, not rounded decimals.S076 · premise_and_result_audit — Recompute left sample12 offset with exact rational arithmetic, not rounded decimals.S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S074 · premise_and_result_audit — Recompute left sample11 rectangle with exact rational arithmetic, not rounded decimals.S073 · premise_and_result_audit — Recompute left sample11 value with exact rational arithmetic, not rounded decimals.S072 · premise_and_result_audit — Recompute left sample11 mx with exact rational arithmetic, not rounded decimals.S071 · premise_and_result_audit — Recompute left sample11 x with exact rational arithmetic, not rounded decimals.S070 · premise_and_result_audit — Recompute left sample11 offset with exact rational arithmetic, not rounded decimals.S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S068 · premise_and_result_audit — Recompute left sample10 rectangle with exact rational arithmetic, not rounded decimals.S067 · premise_and_result_audit — Recompute left sample10 value with exact rational arithmetic, not rounded decimals.S066 · premise_and_result_audit — Recompute left sample10 mx with exact rational arithmetic, not rounded decimals.S065 · premise_and_result_audit — Recompute left sample10 x with exact rational arithmetic, not rounded decimals.S064 · premise_and_result_audit — Recompute left sample10 offset with exact rational arithmetic, not rounded decimals.S063 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S062 · premise_and_result_audit — Recompute left sample9 rectangle with exact rational arithmetic, not rounded decimals.S061 · premise_and_result_audit — Recompute left sample9 value with exact rational arithmetic, not rounded decimals.S060 · premise_and_result_audit — Recompute left sample9 mx with exact rational arithmetic, not rounded decimals.S059 · premise_and_result_audit — Recompute left sample9 x with exact rational arithmetic, not rounded decimals.S058 · premise_and_result_audit — Recompute left sample9 offset with exact rational arithmetic, not rounded decimals.S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S056 · premise_and_result_audit — Recompute left sample8 rectangle with exact rational arithmetic, not rounded decimals.S055 · premise_and_result_audit — Recompute left sample8 value with exact rational arithmetic, not rounded decimals.S054 · premise_and_result_audit — Recompute left sample8 mx with exact rational arithmetic, not rounded decimals.S053 · premise_and_result_audit — Recompute left sample8 x with exact rational arithmetic, not rounded decimals.S052 · premise_and_result_audit — Recompute left sample8 offset with exact rational arithmetic, not rounded decimals.S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S050 · premise_and_result_audit — Recompute left sample7 rectangle with exact rational arithmetic, not rounded decimals.S049 · premise_and_result_audit — Recompute left sample7 value with exact rational arithmetic, not rounded decimals.S048 · premise_and_result_audit — Recompute left sample7 mx with exact rational arithmetic, not rounded decimals.S047 · premise_and_result_audit — Recompute left sample7 x with exact rational arithmetic, not rounded decimals.S046 · premise_and_result_audit — Recompute left sample7 offset with exact rational arithmetic, not rounded decimals.S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S044 · premise_and_result_audit — Recompute left sample6 rectangle with exact rational arithmetic, not rounded decimals.S043 · premise_and_result_audit — Recompute left sample6 value with exact rational arithmetic, not rounded decimals.S042 · premise_and_result_audit — Recompute left sample6 mx with exact rational arithmetic, not rounded decimals.S041 · premise_and_result_audit — Recompute left sample6 x with exact rational arithmetic, not rounded decimals.S040 · premise_and_result_audit — Recompute left sample6 offset with exact rational arithmetic, not rounded decimals.S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S038 · premise_and_result_audit — Recompute left sample5 rectangle with exact rational arithmetic, not rounded decimals.S037 · premise_and_result_audit — Recompute left sample5 value with exact rational arithmetic, not rounded decimals.S036 · premise_and_result_audit — Recompute left sample5 mx with exact rational arithmetic, not rounded decimals.S035 · premise_and_result_audit — Recompute left sample5 x with exact rational arithmetic, not rounded decimals.S034 · premise_and_result_audit — Recompute left sample5 offset with exact rational arithmetic, not rounded decimals.S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S032 · premise_and_result_audit — Recompute left sample4 rectangle with exact rational arithmetic, not rounded decimals.S031 · premise_and_result_audit — Recompute left sample4 value with exact rational arithmetic, not rounded decimals.S030 · premise_and_result_audit — Recompute left sample4 mx with exact rational arithmetic, not rounded decimals.S029 · premise_and_result_audit — Recompute left sample4 x with exact rational arithmetic, not rounded decimals.S028 · premise_and_result_audit — Recompute left sample4 offset with exact rational arithmetic, not rounded decimals.S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S026 · premise_and_result_audit — Recompute left sample3 rectangle with exact rational arithmetic, not rounded decimals.S025 · premise_and_result_audit — Recompute left sample3 value with exact rational arithmetic, not rounded decimals.S024 · premise_and_result_audit — Recompute left sample3 mx with exact rational arithmetic, not rounded decimals.S023 · premise_and_result_audit — Recompute left sample3 x with exact rational arithmetic, not rounded decimals.S022 · premise_and_result_audit — Recompute left sample3 offset with exact rational arithmetic, not rounded decimals.S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S020 · premise_and_result_audit — Recompute left sample2 rectangle with exact rational arithmetic, not rounded decimals.S019 · premise_and_result_audit — Recompute left sample2 value with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute left sample2 mx with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute left sample2 x with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute left sample2 offset with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S014 · premise_and_result_audit — Recompute left sample1 rectangle with exact rational arithmetic, not rounded decimals.S013 · premise_and_result_audit — Recompute left sample1 value with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Recompute left sample1 mx with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute left sample1 x with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Recompute left sample1 offset with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Recompute left sample0 rectangle with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Recompute left sample0 value with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Recompute left sample0 mx with exact rational arithmetic, not rounded decimals.S005 · premise_and_result_audit — Recompute left sample0 x with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Recompute left sample0 offset with exact rational arithmetic, not rounded decimals.S003 · premise_and_result_audit — Recompute partition step with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Recompute interval width with exact rational arithmetic, not rounded decimals.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.25/2
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 9aee17b3258cf5156318a51faafbc2bf7b687a2867ab49ea062dcd71ff9094b9candidate record SHA-256: 802f536d8b099bb1db5e1fa4ce9ca1c5088367d51e6ae7f784287e8a18c7fe0fsemantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED