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R0019-10195 · q-20261007-advance-20261009-r01-fundamental-theorem-calculus-0095
On all R, continuous f has ascending coefficients [7, 20, 0, 4], constant term first. Candidate F has ascending rational coefficients [7/1, 7/1, 10/1, 0/1, 2/1]. Does F′=f satisfy the antiderivative premise for the endpoint formula?
G001 — On all R, continuous f has ascending coefficients [7, 20, 0, 4], constant term first. Candidate F has ascending rational coefficients [7/1, 7/1, 10/1, 0/1, 2/1]. Does F′=f satisfy the antiderivative premise for the endpoint formula?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
raw integrand f=[7, 20, 0, 4];ascending powers on all R
After
normalized coefficients=[7/1, 20/1, 0/1, 4/1];declared continuous real polynomial
Conditions: none
The declared integrand is a continuous real polynomial on all R,and endpoints are oriented.Use FTC only under these supplied sufficient hypotheses;finite checks are not a new general formal FTC proof.
Rule: declared_FTC_continuity_hypothesis · Uses: G001
Before
raw candidate F=[[7, 1], [7, 1], [10, 1], [0, 1], [2, 1]]
After
normalized candidate coefficients=[7/1, 7/1, 10/1, 0/1, 2/1]
Conditions: none
Every original denominator must be positive.Normalize to exact rationals while retaining the supplied candidate,not replacing it by a constructed primitive.
Rule: candidate_primitive_original_denominators · Uses: G001, S001
Before
candidate F derivative coefficient0 := mul(1, 7)
After
candidate F derivative coefficient0 = 7
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
candidate F:source power=1;coefficient=7/1
After
derivative power=0;coefficient=7/1
Conditions: none
Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term.
Rule: polynomial_derivative_coefficient_rule · Uses: G001, S003
Before
candidate F derivative coefficient1 := mul(2, 10)
After
candidate F derivative coefficient1 = 20
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
candidate F:source power=2;coefficient=10/1
After
derivative power=1;coefficient=20/1
Conditions: none
Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term.
Rule: polynomial_derivative_coefficient_rule · Uses: G001, S005
Before
candidate F derivative coefficient2 := mul(3, 0)
After
candidate F derivative coefficient2 = 0
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
candidate F:source power=3;coefficient=0/1
After
derivative power=2;coefficient=0/1
Conditions: none
Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term.
Rule: polynomial_derivative_coefficient_rule · Uses: G001, S007
Before
candidate F derivative coefficient3 := mul(4, 2)
After
candidate F derivative coefficient3 = 8
Conditions: none
Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
candidate F:source power=4;coefficient=2/1
After
derivative power=3;coefficient=8/1
Conditions: none
Multiply the coefficient by k and lower its power by1.The constant contributes nothing;retain every higher nonzero term.
Rule: polynomial_derivative_coefficient_rule · Uses: G001, S009
Before
power=0;candidate derivative coefficient=7/1;target f coefficient=7/1
After
coefficient equality=Yes
Conditions: none
Compare F′ and f coefficient by coefficient through the full larger degree;one matching point does not establish the primitive premise.
Rule: FTC_candidate_full_derivative_identity · Uses: G001, S010
Before
power=1;candidate derivative coefficient=20/1;target f coefficient=20/1
After
coefficient equality=Yes
Conditions: none
Compare F′ and f coefficient by coefficient through the full larger degree;one matching point does not establish the primitive premise.
Rule: FTC_candidate_full_derivative_identity · Uses: G001, S011
Before
power=2;candidate derivative coefficient=0/1;target f coefficient=0/1
After
coefficient equality=Yes
Conditions: none
Compare F′ and f coefficient by coefficient through the full larger degree;one matching point does not establish the primitive premise.
Rule: FTC_candidate_full_derivative_identity · Uses: G001, S012
Before
power=3;candidate derivative coefficient=8/1;target f coefficient=4/1
After
coefficient equality=No
Conditions: none
Compare F′ and f coefficient by coefficient through the full larger degree;one matching point does not establish the primitive premise.
Rule: FTC_candidate_full_derivative_identity · Uses: G001, S013
Before
candidate derivative=[7/1, 20/1, 0/1, 8/1];integrand=[7/1, 20/1, 0/1, 4/1]
After
this candidate satisfies F'=f=No
Conditions: none
Failure rejects this F only,not other primitives.The candidate constant is free;coefficient identity is the requested endpoint-formula qualification.
Rule: failed_primitive_candidate_not_all_primitives · Uses: G001, S014
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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.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 candidate F derivative coefficient3 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 candidate F derivative coefficient2 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 candidate F derivative coefficient1 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 candidate F derivative coefficient0 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.No
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: 547dca706d19b2ea344338bab964174e39dd399241f95a1e7b0ad2f398443f63candidate record SHA-256: 424129b33c8866083ac0c7f02a6627c4efee1ff590c8cb6d95bdb4e660fe7a69semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419Lean: NOT_RUN · human review: NOT_PERFORMED