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R0017-09338 · q-20261013-advance-20261019-r06-asymptotic-resource-bound-0063
Use natural n≥0, nonnegative integers a=12, b=25, 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=12, b=25, 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).Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
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.
Rule: full_tail_not_finite_benchmark_scope · Uses: G001
Before
a+b for prescribed constant := sum(12, 25)
After
a+b for prescribed constant = 37
Conditions: none
Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
C-a := sub(37, 12)
After
C-a = 25
Conditions: none
Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
C-b := sub(37, 25)
After
C-b = 12
Conditions: none
Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
a=12;b=25;C=max(1,a+b)=37;N=0
After
C(n+1)-(a*n+b)=(25)*n+(12)>=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.
Rule: linear_all_input_nonnegative_residual · Uses: G001, S004
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.(12, 25, 37, 0, Yes, 12n+25≤37(n+1) for all n≥0)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 8b8ca0ed5580b9a94f12f73736678d42569fa29f00a179132d7f17d301ad00d4candidate record SHA-256: 1ebbe8f81b2bf97c0b6a198f0cd9019f6736d3b14639133a41128c6a5a3c0d72semantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED