# Question 9343: An Explicit All-Input Linear Resource Witness

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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.

R0017-09343 · q-20261013-advance-20261019-r06-asymptotic-resource-bound-0068



## Original question and givens

Use natural n≥0, nonnegative integers a=17, b=35, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).


- G001 — Use natural n≥0, nonnegative integers a=17, b=35, f(n)=a*n+b and g(n)=n+1 in a fixed cost model. Choose C=max(1,a+b), N=0. Give (a, b, C, N, witness valid?, the exact inequality a*n+b≤C*(n+1) for all n≥0).





## 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-asymptotic-resource-bound](https://ocme.evemisslab.com/en/mko/mko-asymptotic-resource-bound/)





## Step-by-step derivation

S001 · impliesBefore

fixed cost model;explicit nonnegative functions on natural n

After

O,Omega and Theta quantify positive constants over a full tail

Conditions: none

Fix the cost model and function family first.O requires C>0 and N with an upper bound for every n≥N;Omega is the corresponding positive lower bound,and Theta requires both.Finite measurements describe only a finite window.

Step identifiers and dependenciesRule: full_tail_not_finite_benchmark_scope · Uses: G001





S002 · calculationBefore

a+b for prescribed constant := sum(17, 35)

After

a+b for prescribed constant = 52

Conditions: none

Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.

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





S003 · calculationBefore

C-a := sub(52, 17)

After

C-a = 35

Conditions: none

Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.

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





S004 · calculationBefore

C-b := sub(52, 35)

After

C-b = 17

Conditions: none

Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.

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





S005 · impliesBefore

a=17;b=35;C=max(1,a+b)=52;N=0

After

C(n+1)-(a*n+b)=(35)*n+(17)>=0 for every natural n

Conditions: none

C≥a and C≥b make both residual coefficients nonnegative for every natural n.This covers all n rather than selected points;when a=b=0,C=1 remains positive.

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









## Reverse checks


- S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S004 · premise_and_result_audit — Recompute C-b with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute C-a with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute a+b for prescribed constant 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

(17, 35, 52, 0, Yes, 17n+35≤52(n+1) for all n≥0)

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: 5dbcb8f46eb3529def808222a01c06314391417e2db5bd567303165457f8ccdc
- candidate record SHA-256: b6d9387a042e56ad5ecc0e826c104bab95147e4921d0ad78cc0c8ac78f4feb94
- semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8
- question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261019-r06-asymptotic-resource-bound-0068.json)