# Question 3875: Exact Moment Integral Minus a Finite Left Sum

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0006-03875 · q-20261007-advance-20261009-r06-definite-integral-0075



## Original question and givens

On all R, f(x)=(25)x+(12). On a=-2≤b=-1, use N=50 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)=(25)x+(12). On a=-2≤b=-1, use N=50 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.





## Definition and scope

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.


- [mko-definite-integral](https://ocme.evemisslab.com/en/mko/mko-definite-integral/)





## Step-by-step derivation

S001 · impliesBefore

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.

Step identifiers and dependenciesRule: polynomial_moment_scope · Uses: G001





S002 · calculationBefore

interval width := sub(-1, -2)

After

interval width = 1

Conditions: none

Compute the nonnegative interval width,allowing zero width.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

partition step := div(1, 50)

After

partition step = 1/50

Conditions: none

Divide by positive integer N to get the exact step.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

left sample0 offset := mul(0, 1/50)

After

left sample0 offset = 0

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · calculationBefore

left sample0 x := sum(-2, 0)

After

left sample0 x = -2

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004





S006 · calculationBefore

left sample0 mx := mul(25, -2)

After

left sample0 mx = -50

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

left sample0 value := sum(-50, 12)

After

left sample0 value = -38

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · calculationBefore

left sample0 rectangle := mul(-38, 1/50)

After

left sample0 rectangle = -19/25

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007





S009 · impliesBefore

[-2, -38, -19/25]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S008





S010 · calculationBefore

left sample1 offset := mul(1, 1/50)

After

left sample1 offset = 1/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009





S011 · calculationBefore

left sample1 x := sum(-2, 1/50)

After

left sample1 x = -99/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010





S012 · calculationBefore

left sample1 mx := mul(25, -99/50)

After

left sample1 mx = -99/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011





S013 · calculationBefore

left sample1 value := sum(-99/2, 12)

After

left sample1 value = -75/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012





S014 · calculationBefore

left sample1 rectangle := mul(-75/2, 1/50)

After

left sample1 rectangle = -3/4

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013





S015 · impliesBefore

[-99/50, -75/2, -3/4]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S014





S016 · calculationBefore

left sample2 offset := mul(2, 1/50)

After

left sample2 offset = 1/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015





S017 · calculationBefore

left sample2 x := sum(-2, 1/25)

After

left sample2 x = -49/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016





S018 · calculationBefore

left sample2 mx := mul(25, -49/25)

After

left sample2 mx = -49

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017





S019 · calculationBefore

left sample2 value := sum(-49, 12)

After

left sample2 value = -37

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018





S020 · calculationBefore

left sample2 rectangle := mul(-37, 1/50)

After

left sample2 rectangle = -37/50

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019





S021 · impliesBefore

[-49/25, -37, -37/50]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S020





S022 · calculationBefore

left sample3 offset := mul(3, 1/50)

After

left sample3 offset = 3/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021





S023 · calculationBefore

left sample3 x := sum(-2, 3/50)

After

left sample3 x = -97/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022





S024 · calculationBefore

left sample3 mx := mul(25, -97/50)

After

left sample3 mx = -97/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023





S025 · calculationBefore

left sample3 value := sum(-97/2, 12)

After

left sample3 value = -73/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024





S026 · calculationBefore

left sample3 rectangle := mul(-73/2, 1/50)

After

left sample3 rectangle = -73/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025





S027 · impliesBefore

[-97/50, -73/2, -73/100]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S026





S028 · calculationBefore

left sample4 offset := mul(4, 1/50)

After

left sample4 offset = 2/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027





S029 · calculationBefore

left sample4 x := sum(-2, 2/25)

After

left sample4 x = -48/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028





S030 · calculationBefore

left sample4 mx := mul(25, -48/25)

After

left sample4 mx = -48

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029





S031 · calculationBefore

left sample4 value := sum(-48, 12)

After

left sample4 value = -36

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030





S032 · calculationBefore

left sample4 rectangle := mul(-36, 1/50)

After

left sample4 rectangle = -18/25

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031





S033 · impliesBefore

[-48/25, -36, -18/25]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S032





S034 · calculationBefore

left sample5 offset := mul(5, 1/50)

After

left sample5 offset = 1/10

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033





S035 · calculationBefore

left sample5 x := sum(-2, 1/10)

After

left sample5 x = -19/10

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034





S036 · calculationBefore

left sample5 mx := mul(25, -19/10)

After

left sample5 mx = -95/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035





S037 · calculationBefore

left sample5 value := sum(-95/2, 12)

After

left sample5 value = -71/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036





S038 · calculationBefore

left sample5 rectangle := mul(-71/2, 1/50)

After

left sample5 rectangle = -71/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S037





S039 · impliesBefore

[-19/10, -71/2, -71/100]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S038





S040 · calculationBefore

left sample6 offset := mul(6, 1/50)

After

left sample6 offset = 3/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039





S041 · calculationBefore

left sample6 x := sum(-2, 3/25)

After

left sample6 x = -47/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040





S042 · calculationBefore

left sample6 mx := mul(25, -47/25)

After

left sample6 mx = -47

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041





S043 · calculationBefore

left sample6 value := sum(-47, 12)

After

left sample6 value = -35

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042





S044 · calculationBefore

left sample6 rectangle := mul(-35, 1/50)

After

left sample6 rectangle = -7/10

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043





S045 · impliesBefore

[-47/25, -35, -7/10]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S044





S046 · calculationBefore

left sample7 offset := mul(7, 1/50)

After

left sample7 offset = 7/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045





S047 · calculationBefore

left sample7 x := sum(-2, 7/50)

After

left sample7 x = -93/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S046





S048 · calculationBefore

left sample7 mx := mul(25, -93/50)

After

left sample7 mx = -93/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047





S049 · calculationBefore

left sample7 value := sum(-93/2, 12)

After

left sample7 value = -69/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048





S050 · calculationBefore

left sample7 rectangle := mul(-69/2, 1/50)

After

left sample7 rectangle = -69/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049





S051 · impliesBefore

[-93/50, -69/2, -69/100]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S050





S052 · calculationBefore

left sample8 offset := mul(8, 1/50)

After

left sample8 offset = 4/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051





S053 · calculationBefore

left sample8 x := sum(-2, 4/25)

After

left sample8 x = -46/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052





S054 · calculationBefore

left sample8 mx := mul(25, -46/25)

After

left sample8 mx = -46

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053





S055 · calculationBefore

left sample8 value := sum(-46, 12)

After

left sample8 value = -34

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054





S056 · calculationBefore

left sample8 rectangle := mul(-34, 1/50)

After

left sample8 rectangle = -17/25

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S055





S057 · impliesBefore

[-46/25, -34, -17/25]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S056





S058 · calculationBefore

left sample9 offset := mul(9, 1/50)

After

left sample9 offset = 9/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S057





S059 · calculationBefore

left sample9 x := sum(-2, 9/50)

After

left sample9 x = -91/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S058





S060 · calculationBefore

left sample9 mx := mul(25, -91/50)

After

left sample9 mx = -91/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S059





S061 · calculationBefore

left sample9 value := sum(-91/2, 12)

After

left sample9 value = -67/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S060





S062 · calculationBefore

left sample9 rectangle := mul(-67/2, 1/50)

After

left sample9 rectangle = -67/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S061





S063 · impliesBefore

[-91/50, -67/2, -67/100]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S062





S064 · calculationBefore

left sample10 offset := mul(10, 1/50)

After

left sample10 offset = 1/5

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S063





S065 · calculationBefore

left sample10 x := sum(-2, 1/5)

After

left sample10 x = -9/5

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S064





S066 · calculationBefore

left sample10 mx := mul(25, -9/5)

After

left sample10 mx = -45

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S065





S067 · calculationBefore

left sample10 value := sum(-45, 12)

After

left sample10 value = -33

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S066





S068 · calculationBefore

left sample10 rectangle := mul(-33, 1/50)

After

left sample10 rectangle = -33/50

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S067





S069 · impliesBefore

[-9/5, -33, -33/50]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S068





S070 · calculationBefore

left sample11 offset := mul(11, 1/50)

After

left sample11 offset = 11/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S069





S071 · calculationBefore

left sample11 x := sum(-2, 11/50)

After

left sample11 x = -89/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S070





S072 · calculationBefore

left sample11 mx := mul(25, -89/50)

After

left sample11 mx = -89/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S071





S073 · calculationBefore

left sample11 value := sum(-89/2, 12)

After

left sample11 value = -65/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S072





S074 · calculationBefore

left sample11 rectangle := mul(-65/2, 1/50)

After

left sample11 rectangle = -13/20

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S073





S075 · impliesBefore

[-89/50, -65/2, -13/20]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S074





S076 · calculationBefore

left sample12 offset := mul(12, 1/50)

After

left sample12 offset = 6/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S075





S077 · calculationBefore

left sample12 x := sum(-2, 6/25)

After

left sample12 x = -44/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S076





S078 · calculationBefore

left sample12 mx := mul(25, -44/25)

After

left sample12 mx = -44

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S077





S079 · calculationBefore

left sample12 value := sum(-44, 12)

After

left sample12 value = -32

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S078





S080 · calculationBefore

left sample12 rectangle := mul(-32, 1/50)

After

left sample12 rectangle = -16/25

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S079





S081 · impliesBefore

[-44/25, -32, -16/25]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S080





S082 · calculationBefore

left sample13 offset := mul(13, 1/50)

After

left sample13 offset = 13/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S081





S083 · calculationBefore

left sample13 x := sum(-2, 13/50)

After

left sample13 x = -87/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S082





S084 · calculationBefore

left sample13 mx := mul(25, -87/50)

After

left sample13 mx = -87/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S083





S085 · calculationBefore

left sample13 value := sum(-87/2, 12)

After

left sample13 value = -63/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S084





S086 · calculationBefore

left sample13 rectangle := mul(-63/2, 1/50)

After

left sample13 rectangle = -63/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S085





S087 · impliesBefore

[-87/50, -63/2, -63/100]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S086





S088 · calculationBefore

left sample14 offset := mul(14, 1/50)

After

left sample14 offset = 7/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S087





S089 · calculationBefore

left sample14 x := sum(-2, 7/25)

After

left sample14 x = -43/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S088





S090 · calculationBefore

left sample14 mx := mul(25, -43/25)

After

left sample14 mx = -43

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S089





S091 · calculationBefore

left sample14 value := sum(-43, 12)

After

left sample14 value = -31

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S090





S092 · calculationBefore

left sample14 rectangle := mul(-31, 1/50)

After

left sample14 rectangle = -31/50

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S091





S093 · impliesBefore

[-43/25, -31, -31/50]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S092





S094 · calculationBefore

left sample15 offset := mul(15, 1/50)

After

left sample15 offset = 3/10

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S093





S095 · calculationBefore

left sample15 x := sum(-2, 3/10)

After

left sample15 x = -17/10

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S094





S096 · calculationBefore

left sample15 mx := mul(25, -17/10)

After

left sample15 mx = -85/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S095





S097 · calculationBefore

left sample15 value := sum(-85/2, 12)

After

left sample15 value = -61/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S096





S098 · calculationBefore

left sample15 rectangle := mul(-61/2, 1/50)

After

left sample15 rectangle = -61/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S097





S099 · impliesBefore

[-17/10, -61/2, -61/100]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S098





S100 · calculationBefore

left sample16 offset := mul(16, 1/50)

After

left sample16 offset = 8/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S099





S101 · calculationBefore

left sample16 x := sum(-2, 8/25)

After

left sample16 x = -42/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S100





S102 · calculationBefore

left sample16 mx := mul(25, -42/25)

After

left sample16 mx = -42

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S101





S103 · calculationBefore

left sample16 value := sum(-42, 12)

After

left sample16 value = -30

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S102





S104 · calculationBefore

left sample16 rectangle := mul(-30, 1/50)

After

left sample16 rectangle = -3/5

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S103





S105 · impliesBefore

[-42/25, -30, -3/5]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S104





S106 · calculationBefore

left sample17 offset := mul(17, 1/50)

After

left sample17 offset = 17/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S105





S107 · calculationBefore

left sample17 x := sum(-2, 17/50)

After

left sample17 x = -83/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S106





S108 · calculationBefore

left sample17 mx := mul(25, -83/50)

After

left sample17 mx = -83/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S107





S109 · calculationBefore

left sample17 value := sum(-83/2, 12)

After

left sample17 value = -59/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S108





S110 · calculationBefore

left sample17 rectangle := mul(-59/2, 1/50)

After

left sample17 rectangle = -59/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S109





S111 · impliesBefore

[-83/50, -59/2, -59/100]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S110





S112 · calculationBefore

left sample18 offset := mul(18, 1/50)

After

left sample18 offset = 9/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S111





S113 · calculationBefore

left sample18 x := sum(-2, 9/25)

After

left sample18 x = -41/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S112





S114 · calculationBefore

left sample18 mx := mul(25, -41/25)

After

left sample18 mx = -41

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S113





S115 · calculationBefore

left sample18 value := sum(-41, 12)

After

left sample18 value = -29

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S114





S116 · calculationBefore

left sample18 rectangle := mul(-29, 1/50)

After

left sample18 rectangle = -29/50

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S115





S117 · impliesBefore

[-41/25, -29, -29/50]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S116





S118 · calculationBefore

left sample19 offset := mul(19, 1/50)

After

left sample19 offset = 19/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S117





S119 · calculationBefore

left sample19 x := sum(-2, 19/50)

After

left sample19 x = -81/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S118





S120 · calculationBefore

left sample19 mx := mul(25, -81/50)

After

left sample19 mx = -81/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S119





S121 · calculationBefore

left sample19 value := sum(-81/2, 12)

After

left sample19 value = -57/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S120





S122 · calculationBefore

left sample19 rectangle := mul(-57/2, 1/50)

After

left sample19 rectangle = -57/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S121





S123 · impliesBefore

[-81/50, -57/2, -57/100]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S122





S124 · calculationBefore

left sample20 offset := mul(20, 1/50)

After

left sample20 offset = 2/5

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S123





S125 · calculationBefore

left sample20 x := sum(-2, 2/5)

After

left sample20 x = -8/5

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S124





S126 · calculationBefore

left sample20 mx := mul(25, -8/5)

After

left sample20 mx = -40

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S125





S127 · calculationBefore

left sample20 value := sum(-40, 12)

After

left sample20 value = -28

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S126





S128 · calculationBefore

left sample20 rectangle := mul(-28, 1/50)

After

left sample20 rectangle = -14/25

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S127





S129 · impliesBefore

[-8/5, -28, -14/25]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S128





S130 · calculationBefore

left sample21 offset := mul(21, 1/50)

After

left sample21 offset = 21/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S129





S131 · calculationBefore

left sample21 x := sum(-2, 21/50)

After

left sample21 x = -79/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S130





S132 · calculationBefore

left sample21 mx := mul(25, -79/50)

After

left sample21 mx = -79/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S131





S133 · calculationBefore

left sample21 value := sum(-79/2, 12)

After

left sample21 value = -55/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S132





S134 · calculationBefore

left sample21 rectangle := mul(-55/2, 1/50)

After

left sample21 rectangle = -11/20

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S133





S135 · impliesBefore

[-79/50, -55/2, -11/20]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S134





S136 · calculationBefore

left sample22 offset := mul(22, 1/50)

After

left sample22 offset = 11/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S135





S137 · calculationBefore

left sample22 x := sum(-2, 11/25)

After

left sample22 x = -39/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S136





S138 · calculationBefore

left sample22 mx := mul(25, -39/25)

After

left sample22 mx = -39

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S137





S139 · calculationBefore

left sample22 value := sum(-39, 12)

After

left sample22 value = -27

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S138





S140 · calculationBefore

left sample22 rectangle := mul(-27, 1/50)

After

left sample22 rectangle = -27/50

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S139





S141 · impliesBefore

[-39/25, -27, -27/50]

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.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S140





S142 · calculationBefore

left sample23 offset := mul(23, 1/50)

After

left sample23 offset = 23/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S141





S143 · calculationBefore

left sample23 x := sum(-2, 23/50)

After

left sample23 x = -77/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S142





S144 · calculationBefore

left sample23 mx := mul(25, -77/50)

After

left sample23 mx = -77/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S143





S145 · calculationBefore

left sample23 value := sum(-77/2, 12)

After

left sample23 value = -53/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S144





S146 · calculationBefore

left sample23 rectangle := mul(-53/2, 1/50)

After

left sample23 rectangle = -53/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S145





S147 · impliesBefore

[-77/50, -53/2, -53/100]

After

finite left-sum term index=23

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S146





S148 · calculationBefore

left sample24 offset := mul(24, 1/50)

After

left sample24 offset = 12/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S147





S149 · calculationBefore

left sample24 x := sum(-2, 12/25)

After

left sample24 x = -38/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S148





S150 · calculationBefore

left sample24 mx := mul(25, -38/25)

After

left sample24 mx = -38

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S149





S151 · calculationBefore

left sample24 value := sum(-38, 12)

After

left sample24 value = -26

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S150





S152 · calculationBefore

left sample24 rectangle := mul(-26, 1/50)

After

left sample24 rectangle = -13/25

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S151





S153 · impliesBefore

[-38/25, -26, -13/25]

After

finite left-sum term index=24

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S152





S154 · calculationBefore

left sample25 offset := mul(25, 1/50)

After

left sample25 offset = 1/2

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S153





S155 · calculationBefore

left sample25 x := sum(-2, 1/2)

After

left sample25 x = -3/2

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S154





S156 · calculationBefore

left sample25 mx := mul(25, -3/2)

After

left sample25 mx = -75/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S155





S157 · calculationBefore

left sample25 value := sum(-75/2, 12)

After

left sample25 value = -51/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S156





S158 · calculationBefore

left sample25 rectangle := mul(-51/2, 1/50)

After

left sample25 rectangle = -51/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S157





S159 · impliesBefore

[-3/2, -51/2, -51/100]

After

finite left-sum term index=25

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S158





S160 · calculationBefore

left sample26 offset := mul(26, 1/50)

After

left sample26 offset = 13/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S159





S161 · calculationBefore

left sample26 x := sum(-2, 13/25)

After

left sample26 x = -37/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S160





S162 · calculationBefore

left sample26 mx := mul(25, -37/25)

After

left sample26 mx = -37

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S161





S163 · calculationBefore

left sample26 value := sum(-37, 12)

After

left sample26 value = -25

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S162





S164 · calculationBefore

left sample26 rectangle := mul(-25, 1/50)

After

left sample26 rectangle = -1/2

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S163





S165 · impliesBefore

[-37/25, -25, -1/2]

After

finite left-sum term index=26

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S164





S166 · calculationBefore

left sample27 offset := mul(27, 1/50)

After

left sample27 offset = 27/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S165





S167 · calculationBefore

left sample27 x := sum(-2, 27/50)

After

left sample27 x = -73/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S166





S168 · calculationBefore

left sample27 mx := mul(25, -73/50)

After

left sample27 mx = -73/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S167





S169 · calculationBefore

left sample27 value := sum(-73/2, 12)

After

left sample27 value = -49/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S168





S170 · calculationBefore

left sample27 rectangle := mul(-49/2, 1/50)

After

left sample27 rectangle = -49/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S169





S171 · impliesBefore

[-73/50, -49/2, -49/100]

After

finite left-sum term index=27

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S170





S172 · calculationBefore

left sample28 offset := mul(28, 1/50)

After

left sample28 offset = 14/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S171





S173 · calculationBefore

left sample28 x := sum(-2, 14/25)

After

left sample28 x = -36/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S172





S174 · calculationBefore

left sample28 mx := mul(25, -36/25)

After

left sample28 mx = -36

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S173





S175 · calculationBefore

left sample28 value := sum(-36, 12)

After

left sample28 value = -24

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S174





S176 · calculationBefore

left sample28 rectangle := mul(-24, 1/50)

After

left sample28 rectangle = -12/25

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S175





S177 · impliesBefore

[-36/25, -24, -12/25]

After

finite left-sum term index=28

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S176





S178 · calculationBefore

left sample29 offset := mul(29, 1/50)

After

left sample29 offset = 29/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S177





S179 · calculationBefore

left sample29 x := sum(-2, 29/50)

After

left sample29 x = -71/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S178





S180 · calculationBefore

left sample29 mx := mul(25, -71/50)

After

left sample29 mx = -71/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S179





S181 · calculationBefore

left sample29 value := sum(-71/2, 12)

After

left sample29 value = -47/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S180





S182 · calculationBefore

left sample29 rectangle := mul(-47/2, 1/50)

After

left sample29 rectangle = -47/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S181





S183 · impliesBefore

[-71/50, -47/2, -47/100]

After

finite left-sum term index=29

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S182





S184 · calculationBefore

left sample30 offset := mul(30, 1/50)

After

left sample30 offset = 3/5

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S183





S185 · calculationBefore

left sample30 x := sum(-2, 3/5)

After

left sample30 x = -7/5

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S184





S186 · calculationBefore

left sample30 mx := mul(25, -7/5)

After

left sample30 mx = -35

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S185





S187 · calculationBefore

left sample30 value := sum(-35, 12)

After

left sample30 value = -23

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S186





S188 · calculationBefore

left sample30 rectangle := mul(-23, 1/50)

After

left sample30 rectangle = -23/50

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S187





S189 · impliesBefore

[-7/5, -23, -23/50]

After

finite left-sum term index=30

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S188





S190 · calculationBefore

left sample31 offset := mul(31, 1/50)

After

left sample31 offset = 31/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S189





S191 · calculationBefore

left sample31 x := sum(-2, 31/50)

After

left sample31 x = -69/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S190





S192 · calculationBefore

left sample31 mx := mul(25, -69/50)

After

left sample31 mx = -69/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S191





S193 · calculationBefore

left sample31 value := sum(-69/2, 12)

After

left sample31 value = -45/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S192





S194 · calculationBefore

left sample31 rectangle := mul(-45/2, 1/50)

After

left sample31 rectangle = -9/20

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S193





S195 · impliesBefore

[-69/50, -45/2, -9/20]

After

finite left-sum term index=31

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S194





S196 · calculationBefore

left sample32 offset := mul(32, 1/50)

After

left sample32 offset = 16/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S195





S197 · calculationBefore

left sample32 x := sum(-2, 16/25)

After

left sample32 x = -34/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S196





S198 · calculationBefore

left sample32 mx := mul(25, -34/25)

After

left sample32 mx = -34

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S197





S199 · calculationBefore

left sample32 value := sum(-34, 12)

After

left sample32 value = -22

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S198





S200 · calculationBefore

left sample32 rectangle := mul(-22, 1/50)

After

left sample32 rectangle = -11/25

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S199





S201 · impliesBefore

[-34/25, -22, -11/25]

After

finite left-sum term index=32

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S200





S202 · calculationBefore

left sample33 offset := mul(33, 1/50)

After

left sample33 offset = 33/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S201





S203 · calculationBefore

left sample33 x := sum(-2, 33/50)

After

left sample33 x = -67/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S202





S204 · calculationBefore

left sample33 mx := mul(25, -67/50)

After

left sample33 mx = -67/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S203





S205 · calculationBefore

left sample33 value := sum(-67/2, 12)

After

left sample33 value = -43/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S204





S206 · calculationBefore

left sample33 rectangle := mul(-43/2, 1/50)

After

left sample33 rectangle = -43/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S205





S207 · impliesBefore

[-67/50, -43/2, -43/100]

After

finite left-sum term index=33

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S206





S208 · calculationBefore

left sample34 offset := mul(34, 1/50)

After

left sample34 offset = 17/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S207





S209 · calculationBefore

left sample34 x := sum(-2, 17/25)

After

left sample34 x = -33/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S208





S210 · calculationBefore

left sample34 mx := mul(25, -33/25)

After

left sample34 mx = -33

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S209





S211 · calculationBefore

left sample34 value := sum(-33, 12)

After

left sample34 value = -21

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S210





S212 · calculationBefore

left sample34 rectangle := mul(-21, 1/50)

After

left sample34 rectangle = -21/50

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S211





S213 · impliesBefore

[-33/25, -21, -21/50]

After

finite left-sum term index=34

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S212





S214 · calculationBefore

left sample35 offset := mul(35, 1/50)

After

left sample35 offset = 7/10

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S213





S215 · calculationBefore

left sample35 x := sum(-2, 7/10)

After

left sample35 x = -13/10

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S214





S216 · calculationBefore

left sample35 mx := mul(25, -13/10)

After

left sample35 mx = -65/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S215





S217 · calculationBefore

left sample35 value := sum(-65/2, 12)

After

left sample35 value = -41/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S216





S218 · calculationBefore

left sample35 rectangle := mul(-41/2, 1/50)

After

left sample35 rectangle = -41/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S217





S219 · impliesBefore

[-13/10, -41/2, -41/100]

After

finite left-sum term index=35

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S218





S220 · calculationBefore

left sample36 offset := mul(36, 1/50)

After

left sample36 offset = 18/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S219





S221 · calculationBefore

left sample36 x := sum(-2, 18/25)

After

left sample36 x = -32/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S220





S222 · calculationBefore

left sample36 mx := mul(25, -32/25)

After

left sample36 mx = -32

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S221





S223 · calculationBefore

left sample36 value := sum(-32, 12)

After

left sample36 value = -20

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S222





S224 · calculationBefore

left sample36 rectangle := mul(-20, 1/50)

After

left sample36 rectangle = -2/5

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S223





S225 · impliesBefore

[-32/25, -20, -2/5]

After

finite left-sum term index=36

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S224





S226 · calculationBefore

left sample37 offset := mul(37, 1/50)

After

left sample37 offset = 37/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S225





S227 · calculationBefore

left sample37 x := sum(-2, 37/50)

After

left sample37 x = -63/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S226





S228 · calculationBefore

left sample37 mx := mul(25, -63/50)

After

left sample37 mx = -63/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S227





S229 · calculationBefore

left sample37 value := sum(-63/2, 12)

After

left sample37 value = -39/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S228





S230 · calculationBefore

left sample37 rectangle := mul(-39/2, 1/50)

After

left sample37 rectangle = -39/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S229





S231 · impliesBefore

[-63/50, -39/2, -39/100]

After

finite left-sum term index=37

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S230





S232 · calculationBefore

left sample38 offset := mul(38, 1/50)

After

left sample38 offset = 19/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S231





S233 · calculationBefore

left sample38 x := sum(-2, 19/25)

After

left sample38 x = -31/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S232





S234 · calculationBefore

left sample38 mx := mul(25, -31/25)

After

left sample38 mx = -31

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S233





S235 · calculationBefore

left sample38 value := sum(-31, 12)

After

left sample38 value = -19

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S234





S236 · calculationBefore

left sample38 rectangle := mul(-19, 1/50)

After

left sample38 rectangle = -19/50

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S235





S237 · impliesBefore

[-31/25, -19, -19/50]

After

finite left-sum term index=38

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S236





S238 · calculationBefore

left sample39 offset := mul(39, 1/50)

After

left sample39 offset = 39/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S237





S239 · calculationBefore

left sample39 x := sum(-2, 39/50)

After

left sample39 x = -61/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S238





S240 · calculationBefore

left sample39 mx := mul(25, -61/50)

After

left sample39 mx = -61/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S239





S241 · calculationBefore

left sample39 value := sum(-61/2, 12)

After

left sample39 value = -37/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S240





S242 · calculationBefore

left sample39 rectangle := mul(-37/2, 1/50)

After

left sample39 rectangle = -37/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S241





S243 · impliesBefore

[-61/50, -37/2, -37/100]

After

finite left-sum term index=39

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S242





S244 · calculationBefore

left sample40 offset := mul(40, 1/50)

After

left sample40 offset = 4/5

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S243





S245 · calculationBefore

left sample40 x := sum(-2, 4/5)

After

left sample40 x = -6/5

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S244





S246 · calculationBefore

left sample40 mx := mul(25, -6/5)

After

left sample40 mx = -30

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S245





S247 · calculationBefore

left sample40 value := sum(-30, 12)

After

left sample40 value = -18

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S246





S248 · calculationBefore

left sample40 rectangle := mul(-18, 1/50)

After

left sample40 rectangle = -9/25

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S247





S249 · impliesBefore

[-6/5, -18, -9/25]

After

finite left-sum term index=40

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S248





S250 · calculationBefore

left sample41 offset := mul(41, 1/50)

After

left sample41 offset = 41/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S249





S251 · calculationBefore

left sample41 x := sum(-2, 41/50)

After

left sample41 x = -59/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S250





S252 · calculationBefore

left sample41 mx := mul(25, -59/50)

After

left sample41 mx = -59/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S251





S253 · calculationBefore

left sample41 value := sum(-59/2, 12)

After

left sample41 value = -35/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S252





S254 · calculationBefore

left sample41 rectangle := mul(-35/2, 1/50)

After

left sample41 rectangle = -7/20

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S253





S255 · impliesBefore

[-59/50, -35/2, -7/20]

After

finite left-sum term index=41

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S254





S256 · calculationBefore

left sample42 offset := mul(42, 1/50)

After

left sample42 offset = 21/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S255





S257 · calculationBefore

left sample42 x := sum(-2, 21/25)

After

left sample42 x = -29/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S256





S258 · calculationBefore

left sample42 mx := mul(25, -29/25)

After

left sample42 mx = -29

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S257





S259 · calculationBefore

left sample42 value := sum(-29, 12)

After

left sample42 value = -17

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S258





S260 · calculationBefore

left sample42 rectangle := mul(-17, 1/50)

After

left sample42 rectangle = -17/50

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S259





S261 · impliesBefore

[-29/25, -17, -17/50]

After

finite left-sum term index=42

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S260





S262 · calculationBefore

left sample43 offset := mul(43, 1/50)

After

left sample43 offset = 43/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S261





S263 · calculationBefore

left sample43 x := sum(-2, 43/50)

After

left sample43 x = -57/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S262





S264 · calculationBefore

left sample43 mx := mul(25, -57/50)

After

left sample43 mx = -57/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S263





S265 · calculationBefore

left sample43 value := sum(-57/2, 12)

After

left sample43 value = -33/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S264





S266 · calculationBefore

left sample43 rectangle := mul(-33/2, 1/50)

After

left sample43 rectangle = -33/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S265





S267 · impliesBefore

[-57/50, -33/2, -33/100]

After

finite left-sum term index=43

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S266





S268 · calculationBefore

left sample44 offset := mul(44, 1/50)

After

left sample44 offset = 22/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S267





S269 · calculationBefore

left sample44 x := sum(-2, 22/25)

After

left sample44 x = -28/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S268





S270 · calculationBefore

left sample44 mx := mul(25, -28/25)

After

left sample44 mx = -28

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S269





S271 · calculationBefore

left sample44 value := sum(-28, 12)

After

left sample44 value = -16

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S270





S272 · calculationBefore

left sample44 rectangle := mul(-16, 1/50)

After

left sample44 rectangle = -8/25

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S271





S273 · impliesBefore

[-28/25, -16, -8/25]

After

finite left-sum term index=44

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S272





S274 · calculationBefore

left sample45 offset := mul(45, 1/50)

After

left sample45 offset = 9/10

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S273





S275 · calculationBefore

left sample45 x := sum(-2, 9/10)

After

left sample45 x = -11/10

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S274





S276 · calculationBefore

left sample45 mx := mul(25, -11/10)

After

left sample45 mx = -55/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S275





S277 · calculationBefore

left sample45 value := sum(-55/2, 12)

After

left sample45 value = -31/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S276





S278 · calculationBefore

left sample45 rectangle := mul(-31/2, 1/50)

After

left sample45 rectangle = -31/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S277





S279 · impliesBefore

[-11/10, -31/2, -31/100]

After

finite left-sum term index=45

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S278





S280 · calculationBefore

left sample46 offset := mul(46, 1/50)

After

left sample46 offset = 23/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S279





S281 · calculationBefore

left sample46 x := sum(-2, 23/25)

After

left sample46 x = -27/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S280





S282 · calculationBefore

left sample46 mx := mul(25, -27/25)

After

left sample46 mx = -27

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S281





S283 · calculationBefore

left sample46 value := sum(-27, 12)

After

left sample46 value = -15

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S282





S284 · calculationBefore

left sample46 rectangle := mul(-15, 1/50)

After

left sample46 rectangle = -3/10

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S283





S285 · impliesBefore

[-27/25, -15, -3/10]

After

finite left-sum term index=46

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S284





S286 · calculationBefore

left sample47 offset := mul(47, 1/50)

After

left sample47 offset = 47/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S285





S287 · calculationBefore

left sample47 x := sum(-2, 47/50)

After

left sample47 x = -53/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S286





S288 · calculationBefore

left sample47 mx := mul(25, -53/50)

After

left sample47 mx = -53/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S287





S289 · calculationBefore

left sample47 value := sum(-53/2, 12)

After

left sample47 value = -29/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S288





S290 · calculationBefore

left sample47 rectangle := mul(-29/2, 1/50)

After

left sample47 rectangle = -29/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S289





S291 · impliesBefore

[-53/50, -29/2, -29/100]

After

finite left-sum term index=47

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S290





S292 · calculationBefore

left sample48 offset := mul(48, 1/50)

After

left sample48 offset = 24/25

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S291





S293 · calculationBefore

left sample48 x := sum(-2, 24/25)

After

left sample48 x = -26/25

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S292





S294 · calculationBefore

left sample48 mx := mul(25, -26/25)

After

left sample48 mx = -26

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S293





S295 · calculationBefore

left sample48 value := sum(-26, 12)

After

left sample48 value = -14

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S294





S296 · calculationBefore

left sample48 rectangle := mul(-14, 1/50)

After

left sample48 rectangle = -7/25

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S295





S297 · impliesBefore

[-26/25, -14, -7/25]

After

finite left-sum term index=48

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S296





S298 · calculationBefore

left sample49 offset := mul(49, 1/50)

After

left sample49 offset = 49/50

Conditions: none

Left samples use i=0,...,N−1,not the final right endpoint.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S297





S299 · calculationBefore

left sample49 x := sum(-2, 49/50)

After

left sample49 x = -51/50

Conditions: none

Add the indexed offset to the start.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S298





S300 · calculationBefore

left sample49 mx := mul(25, -51/50)

After

left sample49 mx = -51/2

Conditions: none

Evaluate the product term of the affine integrand.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S299





S301 · calculationBefore

left sample49 value := sum(-51/2, 12)

After

left sample49 value = -27/2

Conditions: none

Add c to obtain this sample value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S300





S302 · calculationBefore

left sample49 rectangle := mul(-27/2, 1/50)

After

left sample49 rectangle = -27/100

Conditions: none

Multiply by the step to get the signed rectangle contribution,without absolute values.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S301





S303 · impliesBefore

[-51/50, -27/2, -27/100]

After

finite left-sum term index=49

Conditions: none

List exactly one left term per subinterval;these are finite-sum data,not a claimed limit proof.

Step identifiers and dependenciesRule: left_partition_term · Uses: G001, S302





S304 · calculationBefore

finite left sum := sum(-19/25, -3/4, -37/50, -73/100, -18/25, -71/100, -7/10, -69/100, -17/25, -67/100, -33/50, -13/20, -16/25, -63/100, -31/50, -61/100, -3/5, -59/100, -29/50, -57/100, -14/25, -11/20, -27/50, -53/100, -13/25, -51/100, -1/2, -49/100, -12/25, -47/100, -23/50, -9/20, -11/25, -43/100, -21/50, -41/100, -2/5, -39/100, -19/50, -37/100, -9/25, -7/20, -17/50, -33/100, -8/25, -31/100, -3/10, -29/100, -7/25, -27/100)

After

finite left sum = -103/4

Conditions: none

Add all N contributions.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S303





S305 · calculationBefore

affine integral primitive coefficient1 := div(12, 1)

After

affine integral primitive coefficient1 = 12

Conditions: none

The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S304





S306 · calculationBefore

affine integral primitive coefficient2 := div(25, 2)

After

affine integral primitive coefficient2 = 25/2

Conditions: none

The primitive coefficient for c·x^k is c/(k+1),with positive nonzero k+1.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S305





S307 · impliesBefore

[0, 12, 25/2]

After

H′=the supplied polynomial;integrate from -2 to -1

Conditions: none

Termwise differentiation recovers the original coefficients;the integration constant cancels in the endpoint difference.

Step identifiers and dependenciesRule: polynomial_primitive_identity · Uses: G001, S306





S308 · impliesBefore

affine integral H(b) ascending coefficients=[0, 12, 25/2]

After

evaluate at x=-1

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.

Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S307





S309 · calculationBefore

affine integral H(b) Horner multiply before degree1 := mul(25/2, -1)

After

affine integral H(b) Horner multiply before degree1 = -25/2

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S308





S310 · calculationBefore

affine integral H(b) Horner add degree1 := sum(-25/2, 12)

After

affine integral H(b) Horner add degree1 = -1/2

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S309





S311 · calculationBefore

affine integral H(b) Horner multiply before degree0 := mul(-1/2, -1)

After

affine integral H(b) Horner multiply before degree0 = 1/2

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S310





S312 · calculationBefore

affine integral H(b) Horner add degree0 := sum(1/2, 0)

After

affine integral H(b) Horner add degree0 = 1/2

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S311





S313 · impliesBefore

affine integral H(a) ascending coefficients=[0, 12, 25/2]

After

evaluate at x=-2

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.

Step identifiers and dependenciesRule: polynomial_evaluation_order · Uses: G001, S312





S314 · calculationBefore

affine integral H(a) Horner multiply before degree1 := mul(25/2, -2)

After

affine integral H(a) Horner multiply before degree1 = -25

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S313





S315 · calculationBefore

affine integral H(a) Horner add degree1 := sum(-25, 12)

After

affine integral H(a) Horner add degree1 = -13

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S314





S316 · calculationBefore

affine integral H(a) Horner multiply before degree0 := mul(-13, -2)

After

affine integral H(a) Horner multiply before degree0 = 26

Conditions: none

Multiply by the same evaluation point first.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S315





S317 · calculationBefore

affine integral H(a) Horner add degree0 := sum(26, 0)

After

affine integral H(a) Horner add degree0 = 26

Conditions: none

Then add the full coefficient at this position.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S316





S318 · calculationBefore

affine integral oriented moment := sub(1/2, 26)

After

affine integral oriented moment = -51/2

Conditions: none

Subtract the starting primitive value from the ending value;reversing endpoints naturally reverses the sign.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S317





S319 · calculationBefore

integral minus left sum := sub(-51/2, -103/4)

After

integral minus left sum = 1/4

Conditions: none

The requested direction is I−S_N;a negative slope may give a negative difference,not absolute error.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S318





S320 · impliesBefore

I=-51/2;S_N=-103/4;N=50

After

signed finite-sum difference=1/4

Conditions: none

This is the exact difference for finite N,not evidence of a general convergence rate or integrability of arbitrary functions.

Step identifiers and dependenciesRule: finite_sum_vs_integral · Uses: G001, S319









## Reverse checks


- S320 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S319 · premise_and_result_audit — Recompute integral minus left sum with exact rational arithmetic, not rounded decimals.
- S318 · premise_and_result_audit — Recompute affine integral oriented moment with exact rational arithmetic, not rounded decimals.
- S317 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree0 with exact rational arithmetic, not rounded decimals.
- S316 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
- S315 · premise_and_result_audit — Recompute affine integral H(a) Horner add degree1 with exact rational arithmetic, not rounded decimals.
- S314 · premise_and_result_audit — Recompute affine integral H(a) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
- S313 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S312 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree0 with exact rational arithmetic, not rounded decimals.
- S311 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree0 with exact rational arithmetic, not rounded decimals.
- S310 · premise_and_result_audit — Recompute affine integral H(b) Horner add degree1 with exact rational arithmetic, not rounded decimals.
- S309 · premise_and_result_audit — Recompute affine integral H(b) Horner multiply before degree1 with exact rational arithmetic, not rounded decimals.
- S308 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S307 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S306 · premise_and_result_audit — Recompute affine integral primitive coefficient2 with exact rational arithmetic, not rounded decimals.
- S305 · premise_and_result_audit — Recompute affine integral primitive coefficient1 with exact rational arithmetic, not rounded decimals.
- S304 · premise_and_result_audit — Recompute finite left sum with exact rational arithmetic, not rounded decimals.
- S303 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S302 · premise_and_result_audit — Recompute left sample49 rectangle with exact rational arithmetic, not rounded decimals.
- S301 · premise_and_result_audit — Recompute left sample49 value with exact rational arithmetic, not rounded decimals.
- S300 · premise_and_result_audit — Recompute left sample49 mx with exact rational arithmetic, not rounded decimals.
- S299 · premise_and_result_audit — Recompute left sample49 x with exact rational arithmetic, not rounded decimals.
- S298 · premise_and_result_audit — Recompute left sample49 offset with exact rational arithmetic, not rounded decimals.
- S297 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S296 · premise_and_result_audit — Recompute left sample48 rectangle with exact rational arithmetic, not rounded decimals.
- S295 · premise_and_result_audit — Recompute left sample48 value with exact rational arithmetic, not rounded decimals.
- S294 · premise_and_result_audit — Recompute left sample48 mx with exact rational arithmetic, not rounded decimals.
- S293 · premise_and_result_audit — Recompute left sample48 x with exact rational arithmetic, not rounded decimals.
- S292 · premise_and_result_audit — Recompute left sample48 offset with exact rational arithmetic, not rounded decimals.
- S291 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S290 · premise_and_result_audit — Recompute left sample47 rectangle with exact rational arithmetic, not rounded decimals.
- S289 · premise_and_result_audit — Recompute left sample47 value with exact rational arithmetic, not rounded decimals.
- S288 · premise_and_result_audit — Recompute left sample47 mx with exact rational arithmetic, not rounded decimals.
- S287 · premise_and_result_audit — Recompute left sample47 x with exact rational arithmetic, not rounded decimals.
- S286 · premise_and_result_audit — Recompute left sample47 offset with exact rational arithmetic, not rounded decimals.
- S285 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S284 · premise_and_result_audit — Recompute left sample46 rectangle with exact rational arithmetic, not rounded decimals.
- S283 · premise_and_result_audit — Recompute left sample46 value with exact rational arithmetic, not rounded decimals.
- S282 · premise_and_result_audit — Recompute left sample46 mx with exact rational arithmetic, not rounded decimals.
- S281 · premise_and_result_audit — Recompute left sample46 x with exact rational arithmetic, not rounded decimals.
- S280 · premise_and_result_audit — Recompute left sample46 offset with exact rational arithmetic, not rounded decimals.
- S279 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S278 · premise_and_result_audit — Recompute left sample45 rectangle with exact rational arithmetic, not rounded decimals.
- S277 · premise_and_result_audit — Recompute left sample45 value with exact rational arithmetic, not rounded decimals.
- S276 · premise_and_result_audit — Recompute left sample45 mx with exact rational arithmetic, not rounded decimals.
- S275 · premise_and_result_audit — Recompute left sample45 x with exact rational arithmetic, not rounded decimals.
- S274 · premise_and_result_audit — Recompute left sample45 offset with exact rational arithmetic, not rounded decimals.
- S273 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S272 · premise_and_result_audit — Recompute left sample44 rectangle with exact rational arithmetic, not rounded decimals.
- S271 · premise_and_result_audit — Recompute left sample44 value with exact rational arithmetic, not rounded decimals.
- S270 · premise_and_result_audit — Recompute left sample44 mx with exact rational arithmetic, not rounded decimals.
- S269 · premise_and_result_audit — Recompute left sample44 x with exact rational arithmetic, not rounded decimals.
- S268 · premise_and_result_audit — Recompute left sample44 offset with exact rational arithmetic, not rounded decimals.
- S267 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S266 · premise_and_result_audit — Recompute left sample43 rectangle with exact rational arithmetic, not rounded decimals.
- S265 · premise_and_result_audit — Recompute left sample43 value with exact rational arithmetic, not rounded decimals.
- S264 · premise_and_result_audit — Recompute left sample43 mx with exact rational arithmetic, not rounded decimals.
- S263 · premise_and_result_audit — Recompute left sample43 x with exact rational arithmetic, not rounded decimals.
- S262 · premise_and_result_audit — Recompute left sample43 offset with exact rational arithmetic, not rounded decimals.
- S261 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S260 · premise_and_result_audit — Recompute left sample42 rectangle with exact rational arithmetic, not rounded decimals.
- S259 · premise_and_result_audit — Recompute left sample42 value with exact rational arithmetic, not rounded decimals.
- S258 · premise_and_result_audit — Recompute left sample42 mx with exact rational arithmetic, not rounded decimals.
- S257 · premise_and_result_audit — Recompute left sample42 x with exact rational arithmetic, not rounded decimals.
- S256 · premise_and_result_audit — Recompute left sample42 offset with exact rational arithmetic, not rounded decimals.
- S255 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S254 · premise_and_result_audit — Recompute left sample41 rectangle with exact rational arithmetic, not rounded decimals.
- S253 · premise_and_result_audit — Recompute left sample41 value with exact rational arithmetic, not rounded decimals.
- S252 · premise_and_result_audit — Recompute left sample41 mx with exact rational arithmetic, not rounded decimals.
- S251 · premise_and_result_audit — Recompute left sample41 x with exact rational arithmetic, not rounded decimals.
- S250 · premise_and_result_audit — Recompute left sample41 offset with exact rational arithmetic, not rounded decimals.
- S249 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S248 · premise_and_result_audit — Recompute left sample40 rectangle with exact rational arithmetic, not rounded decimals.
- S247 · premise_and_result_audit — Recompute left sample40 value with exact rational arithmetic, not rounded decimals.
- S246 · premise_and_result_audit — Recompute left sample40 mx with exact rational arithmetic, not rounded decimals.
- S245 · premise_and_result_audit — Recompute left sample40 x with exact rational arithmetic, not rounded decimals.
- S244 · premise_and_result_audit — Recompute left sample40 offset with exact rational arithmetic, not rounded decimals.
- S243 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S242 · premise_and_result_audit — Recompute left sample39 rectangle with exact rational arithmetic, not rounded decimals.
- S241 · premise_and_result_audit — Recompute left sample39 value with exact rational arithmetic, not rounded decimals.
- S240 · premise_and_result_audit — Recompute left sample39 mx with exact rational arithmetic, not rounded decimals.
- S239 · premise_and_result_audit — Recompute left sample39 x with exact rational arithmetic, not rounded decimals.
- S238 · premise_and_result_audit — Recompute left sample39 offset with exact rational arithmetic, not rounded decimals.
- S237 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S236 · premise_and_result_audit — Recompute left sample38 rectangle with exact rational arithmetic, not rounded decimals.
- S235 · premise_and_result_audit — Recompute left sample38 value with exact rational arithmetic, not rounded decimals.
- S234 · premise_and_result_audit — Recompute left sample38 mx with exact rational arithmetic, not rounded decimals.
- S233 · premise_and_result_audit — Recompute left sample38 x with exact rational arithmetic, not rounded decimals.
- S232 · premise_and_result_audit — Recompute left sample38 offset with exact rational arithmetic, not rounded decimals.
- S231 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S230 · premise_and_result_audit — Recompute left sample37 rectangle with exact rational arithmetic, not rounded decimals.
- S229 · premise_and_result_audit — Recompute left sample37 value with exact rational arithmetic, not rounded decimals.
- S228 · premise_and_result_audit — Recompute left sample37 mx with exact rational arithmetic, not rounded decimals.
- S227 · premise_and_result_audit — Recompute left sample37 x with exact rational arithmetic, not rounded decimals.
- S226 · premise_and_result_audit — Recompute left sample37 offset with exact rational arithmetic, not rounded decimals.
- S225 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S224 · premise_and_result_audit — Recompute left sample36 rectangle with exact rational arithmetic, not rounded decimals.
- S223 · premise_and_result_audit — Recompute left sample36 value with exact rational arithmetic, not rounded decimals.
- S222 · premise_and_result_audit — Recompute left sample36 mx with exact rational arithmetic, not rounded decimals.
- S221 · premise_and_result_audit — Recompute left sample36 x with exact rational arithmetic, not rounded decimals.
- S220 · premise_and_result_audit — Recompute left sample36 offset with exact rational arithmetic, not rounded decimals.
- S219 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S218 · premise_and_result_audit — Recompute left sample35 rectangle with exact rational arithmetic, not rounded decimals.
- S217 · premise_and_result_audit — Recompute left sample35 value with exact rational arithmetic, not rounded decimals.
- S216 · premise_and_result_audit — Recompute left sample35 mx with exact rational arithmetic, not rounded decimals.
- S215 · premise_and_result_audit — Recompute left sample35 x with exact rational arithmetic, not rounded decimals.
- S214 · premise_and_result_audit — Recompute left sample35 offset with exact rational arithmetic, not rounded decimals.
- S213 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S212 · premise_and_result_audit — Recompute left sample34 rectangle with exact rational arithmetic, not rounded decimals.
- S211 · premise_and_result_audit — Recompute left sample34 value with exact rational arithmetic, not rounded decimals.
- S210 · premise_and_result_audit — Recompute left sample34 mx with exact rational arithmetic, not rounded decimals.
- S209 · premise_and_result_audit — Recompute left sample34 x with exact rational arithmetic, not rounded decimals.
- S208 · premise_and_result_audit — Recompute left sample34 offset with exact rational arithmetic, not rounded decimals.
- S207 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S206 · premise_and_result_audit — Recompute left sample33 rectangle with exact rational arithmetic, not rounded decimals.
- S205 · premise_and_result_audit — Recompute left sample33 value with exact rational arithmetic, not rounded decimals.
- S204 · premise_and_result_audit — Recompute left sample33 mx with exact rational arithmetic, not rounded decimals.
- S203 · premise_and_result_audit — Recompute left sample33 x with exact rational arithmetic, not rounded decimals.
- S202 · premise_and_result_audit — Recompute left sample33 offset with exact rational arithmetic, not rounded decimals.
- S201 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S200 · premise_and_result_audit — Recompute left sample32 rectangle with exact rational arithmetic, not rounded decimals.
- S199 · premise_and_result_audit — Recompute left sample32 value with exact rational arithmetic, not rounded decimals.
- S198 · premise_and_result_audit — Recompute left sample32 mx with exact rational arithmetic, not rounded decimals.
- S197 · premise_and_result_audit — Recompute left sample32 x with exact rational arithmetic, not rounded decimals.
- S196 · premise_and_result_audit — Recompute left sample32 offset with exact rational arithmetic, not rounded decimals.
- S195 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S194 · premise_and_result_audit — Recompute left sample31 rectangle with exact rational arithmetic, not rounded decimals.
- S193 · premise_and_result_audit — Recompute left sample31 value with exact rational arithmetic, not rounded decimals.
- S192 · premise_and_result_audit — Recompute left sample31 mx with exact rational arithmetic, not rounded decimals.
- S191 · premise_and_result_audit — Recompute left sample31 x with exact rational arithmetic, not rounded decimals.
- S190 · premise_and_result_audit — Recompute left sample31 offset with exact rational arithmetic, not rounded decimals.
- S189 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S188 · premise_and_result_audit — Recompute left sample30 rectangle with exact rational arithmetic, not rounded decimals.
- S187 · premise_and_result_audit — Recompute left sample30 value with exact rational arithmetic, not rounded decimals.
- S186 · premise_and_result_audit — Recompute left sample30 mx with exact rational arithmetic, not rounded decimals.
- S185 · premise_and_result_audit — Recompute left sample30 x with exact rational arithmetic, not rounded decimals.
- S184 · premise_and_result_audit — Recompute left sample30 offset with exact rational arithmetic, not rounded decimals.
- S183 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S182 · premise_and_result_audit — Recompute left sample29 rectangle with exact rational arithmetic, not rounded decimals.
- S181 · premise_and_result_audit — Recompute left sample29 value with exact rational arithmetic, not rounded decimals.
- S180 · premise_and_result_audit — Recompute left sample29 mx with exact rational arithmetic, not rounded decimals.
- S179 · premise_and_result_audit — Recompute left sample29 x with exact rational arithmetic, not rounded decimals.
- S178 · premise_and_result_audit — Recompute left sample29 offset with exact rational arithmetic, not rounded decimals.
- S177 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S176 · premise_and_result_audit — Recompute left sample28 rectangle with exact rational arithmetic, not rounded decimals.
- S175 · premise_and_result_audit — Recompute left sample28 value with exact rational arithmetic, not rounded decimals.
- S174 · premise_and_result_audit — Recompute left sample28 mx with exact rational arithmetic, not rounded decimals.
- S173 · premise_and_result_audit — Recompute left sample28 x with exact rational arithmetic, not rounded decimals.
- S172 · premise_and_result_audit — Recompute left sample28 offset with exact rational arithmetic, not rounded decimals.
- S171 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S170 · premise_and_result_audit — Recompute left sample27 rectangle with exact rational arithmetic, not rounded decimals.
- S169 · premise_and_result_audit — Recompute left sample27 value with exact rational arithmetic, not rounded decimals.
- S168 · premise_and_result_audit — Recompute left sample27 mx with exact rational arithmetic, not rounded decimals.
- S167 · premise_and_result_audit — Recompute left sample27 x with exact rational arithmetic, not rounded decimals.
- S166 · premise_and_result_audit — Recompute left sample27 offset with exact rational arithmetic, not rounded decimals.
- S165 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S164 · premise_and_result_audit — Recompute left sample26 rectangle with exact rational arithmetic, not rounded decimals.
- S163 · premise_and_result_audit — Recompute left sample26 value with exact rational arithmetic, not rounded decimals.
- S162 · premise_and_result_audit — Recompute left sample26 mx with exact rational arithmetic, not rounded decimals.
- S161 · premise_and_result_audit — Recompute left sample26 x with exact rational arithmetic, not rounded decimals.
- S160 · premise_and_result_audit — Recompute left sample26 offset with exact rational arithmetic, not rounded decimals.
- S159 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S158 · premise_and_result_audit — Recompute left sample25 rectangle with exact rational arithmetic, not rounded decimals.
- S157 · premise_and_result_audit — Recompute left sample25 value with exact rational arithmetic, not rounded decimals.
- S156 · premise_and_result_audit — Recompute left sample25 mx with exact rational arithmetic, not rounded decimals.
- S155 · premise_and_result_audit — Recompute left sample25 x with exact rational arithmetic, not rounded decimals.
- S154 · premise_and_result_audit — Recompute left sample25 offset with exact rational arithmetic, not rounded decimals.
- S153 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S152 · premise_and_result_audit — Recompute left sample24 rectangle with exact rational arithmetic, not rounded decimals.
- S151 · premise_and_result_audit — Recompute left sample24 value with exact rational arithmetic, not rounded decimals.
- S150 · premise_and_result_audit — Recompute left sample24 mx with exact rational arithmetic, not rounded decimals.
- S149 · premise_and_result_audit — Recompute left sample24 x with exact rational arithmetic, not rounded decimals.
- S148 · premise_and_result_audit — Recompute left sample24 offset with exact rational arithmetic, not rounded decimals.
- S147 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S146 · premise_and_result_audit — Recompute left sample23 rectangle with exact rational arithmetic, not rounded decimals.
- S145 · premise_and_result_audit — Recompute left sample23 value with exact rational arithmetic, not rounded decimals.
- S144 · premise_and_result_audit — Recompute left sample23 mx with exact rational arithmetic, not rounded decimals.
- S143 · premise_and_result_audit — Recompute left sample23 x with exact rational arithmetic, not rounded decimals.
- S142 · premise_and_result_audit — Recompute left sample23 offset 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.





## Result

1/4

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: f9d5dd056a03791cfe650b3b96d8afcde27f070e51be8c93db8ef551efbc068e
- candidate record SHA-256: bb39d070e83df78cd4dead1790a22f00ecb90bfc93f34a0b3cc7bd68dc00922e
- semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29d
- question-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261007-advance-20261009-r06-definite-integral-0075.json)