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R0009-05424 · q-20261008-advance-20261012-r01-mathematical-induction-0049
N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=11, a(k+1)=a(k)+(1), through=8. List a(0) through a(through), inclusive.
G001 — N is indexed from 0; this is an explicitly finite induction protocol, not a universal proof. a(0)=11, a(k+1)=a(k)+(1), through=8. List a(0) through a(through), inclusive.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
natural indices start at0;finite induction protocol
After
index bookkeeping and finite values are not proofs of all P(n)
Conditions: none
Natural indices do not require nonnegative sequence values.Checking a finite range or a successor index does not supply proposition truth or an induction proof over all naturals.
Rule: finite_induction_claim_boundary · Uses: G001
Before
a(0)=11;through=8
After
[11]
Conditions: none
Start with the supplied initial value;through=0 retains only it.
Rule: recurrence_prefix_base · Uses: G001, S001
Before
recurrence next value1 := sum(11, 1)
After
recurrence next value1 = 12
Conditions: none
Add the declared d to the previous value;negative increments or terms are allowed.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
a(0)=11;d=1
After
a(1)=12
Conditions: none
Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.
Rule: finite_recurrence_transition · Uses: G001, S003
Before
recurrence next value2 := sum(12, 1)
After
recurrence next value2 = 13
Conditions: none
Add the declared d to the previous value;negative increments or terms are allowed.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
a(1)=12;d=1
After
a(2)=13
Conditions: none
Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.
Rule: finite_recurrence_transition · Uses: G001, S005
Before
recurrence next value3 := sum(13, 1)
After
recurrence next value3 = 14
Conditions: none
Add the declared d to the previous value;negative increments or terms are allowed.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
a(2)=13;d=1
After
a(3)=14
Conditions: none
Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.
Rule: finite_recurrence_transition · Uses: G001, S007
Before
recurrence next value4 := sum(14, 1)
After
recurrence next value4 = 15
Conditions: none
Add the declared d to the previous value;negative increments or terms are allowed.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
a(3)=14;d=1
After
a(4)=15
Conditions: none
Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.
Rule: finite_recurrence_transition · Uses: G001, S009
Before
recurrence next value5 := sum(15, 1)
After
recurrence next value5 = 16
Conditions: none
Add the declared d to the previous value;negative increments or terms are allowed.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
a(4)=15;d=1
After
a(5)=16
Conditions: none
Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.
Rule: finite_recurrence_transition · Uses: G001, S011
Before
recurrence next value6 := sum(16, 1)
After
recurrence next value6 = 17
Conditions: none
Add the declared d to the previous value;negative increments or terms are allowed.
Rule: exact_rational_arithmetic · Uses: G001, S012
Before
a(5)=16;d=1
After
a(6)=17
Conditions: none
Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.
Rule: finite_recurrence_transition · Uses: G001, S013
Before
recurrence next value7 := sum(17, 1)
After
recurrence next value7 = 18
Conditions: none
Add the declared d to the previous value;negative increments or terms are allowed.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
a(6)=17;d=1
After
a(7)=18
Conditions: none
Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.
Rule: finite_recurrence_transition · Uses: G001, S015
Before
recurrence next value8 := sum(18, 1)
After
recurrence next value8 = 19
Conditions: none
Add the declared d to the previous value;negative increments or terms are allowed.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
a(7)=18;d=1
After
a(8)=19
Conditions: none
Each step uses the latest preceding term,not the initial value again;this constructs the requested finite prefix.
Rule: finite_recurrence_transition · Uses: G001, S017
S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S017 · premise_and_result_audit — Recompute recurrence next value8 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S015 · premise_and_result_audit — Recompute recurrence next value7 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Recompute recurrence next value6 with exact rational arithmetic, not rounded decimals.S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Recompute recurrence next value5 with exact rational arithmetic, not rounded decimals.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Recompute recurrence next value4 with exact rational arithmetic, not rounded decimals.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Recompute recurrence next value3 with exact rational arithmetic, not rounded decimals.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Recompute recurrence next value2 with exact rational arithmetic, not rounded decimals.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute recurrence next value1 with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.[11, 12, 13, 14, 15, 16, 17, 18, 19]
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: 425c3f118b4827387d8db3c56a48618910c383c8d00c1b18686d305f6ad69abbcandidate record SHA-256: eb97587aa7e70b8dcd57f80d40e0ff215dec8a06a9031590f834d9ca43af4c6fsemantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462Lean: NOT_RUN · human review: NOT_PERFORMED