Strict Affine Index Maps, Composition and Finite-Prefix Evidence
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261107-r07-subsequence-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=0≥0 and positive integer index_step=1. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(0, 1, k(j)=0+1j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (0, 1, k(j)=0+1j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
2. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=1≥0 and positive integer index_step=2. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(1, 2, k(j)=1+2j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (1, 2, k(j)=1+2j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
3. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=2≥0 and positive integer index_step=3. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(2, 3, k(j)=2+3j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (2, 3, k(j)=2+3j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
4. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=3≥0 and positive integer index_step=4. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(3, 4, k(j)=3+4j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (3, 4, k(j)=3+4j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
5. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=4≥0 and positive integer index_step=5. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(4, 5, k(j)=4+5j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (4, 5, k(j)=4+5j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
6. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=5≥0 and positive integer index_step=1. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(5, 1, k(j)=5+1j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (5, 1, k(j)=5+1j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
7. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=6≥0 and positive integer index_step=2. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(6, 2, k(j)=6+2j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (6, 2, k(j)=6+2j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
8. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=7≥0 and positive integer index_step=3. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(7, 3, k(j)=7+3j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (7, 3, k(j)=7+3j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
9. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=8≥0 and positive integer index_step=4. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(8, 4, k(j)=8+4j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (8, 4, k(j)=8+4j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
10. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=9≥0 and positive integer index_step=5. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(9, 5, k(j)=9+5j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (9, 5, k(j)=9+5j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
11. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=10≥0 and positive integer index_step=1. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(10, 1, k(j)=10+1j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (10, 1, k(j)=10+1j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
12. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=11≥0 and positive integer index_step=2. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(11, 2, k(j)=11+2j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (11, 2, k(j)=11+2j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
13. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=12≥0 and positive integer index_step=3. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(12, 3, k(j)=12+3j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (12, 3, k(j)=12+3j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
14. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=13≥0 and positive integer index_step=4. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(13, 4, k(j)=13+4j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (13, 4, k(j)=13+4j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
15. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=14≥0 and positive integer index_step=5. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(14, 5, k(j)=14+5j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (14, 5, k(j)=14+5j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
16. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=15≥0 and positive integer index_step=1. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(15, 1, k(j)=15+1j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (15, 1, k(j)=15+1j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
17. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=16≥0 and positive integer index_step=2. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(16, 2, k(j)=16+2j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (16, 2, k(j)=16+2j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
18. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=17≥0 and positive integer index_step=3. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(17, 3, k(j)=17+3j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (17, 3, k(j)=17+3j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
19. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=18≥0 and positive integer index_step=4. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(18, 4, k(j)=18+4j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (18, 4, k(j)=18+4j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
20. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=19≥0 and positive integer index_step=5. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(19, 5, k(j)=19+5j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (19, 5, k(j)=19+5j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
21. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=20≥0 and positive integer index_step=1. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(20, 1, k(j)=20+1j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (20, 1, k(j)=20+1j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
22. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=21≥0 and positive integer index_step=2. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(21, 2, k(j)=21+2j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (21, 2, k(j)=21+2j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
23. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=22≥0 and positive integer index_step=3. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(22, 3, k(j)=22+3j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (22, 3, k(j)=22+3j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
24. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=23≥0 and positive integer index_step=4. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(23, 4, k(j)=23+4j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (23, 4, k(j)=23+4j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
25. A Positive Affine Index Step Gives Strict Increase for Every Natural Index
Use k(j)=index_start+index_step*j on natural j starting at0, with index_start=24≥0 and positive integer index_step=5. Give (index_start, index_step, index_formula, strictly_increasing, certificate).
View answer and explanation
(24, 5, k(j)=24+5j, Yes, strictly_increasing_for_all_natural_j)
The consecutive index difference is the positive step for every j, supplying a symbolic all-j certificate. Strict increase preserves source order and is stronger than arbitrary injectivity. Retain the source literal index formula. The exact result, retaining the requested order, is (24, 5, k(j)=24+5j, Yes, strictly_increasing_for_all_natural_j).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-index-certificate · candidate
26. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=1+(1)n for n≥0 and index map k(j)=0+1j, with nonnegative start and positive index step. Evaluate j=2. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(2, 2, 3, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (2, 2, 3, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
27. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=2+(2)n for n≥0 and index map k(j)=1+2j, with nonnegative start and positive index step. Evaluate j=3. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(3, 7, 16, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (3, 7, 16, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
28. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=3+(3)n for n≥0 and index map k(j)=2+3j, with nonnegative start and positive index step. Evaluate j=4. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(4, 14, 45, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (4, 14, 45, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
29. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=4+(4)n for n≥0 and index map k(j)=3+4j, with nonnegative start and positive index step. Evaluate j=5. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(5, 23, 96, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (5, 23, 96, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
30. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=5+(5)n for n≥0 and index map k(j)=0+5j, with nonnegative start and positive index step. Evaluate j=6. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(6, 30, 155, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (6, 30, 155, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
31. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=6+(6)n for n≥0 and index map k(j)=1+1j, with nonnegative start and positive index step. Evaluate j=7. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(7, 8, 54, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (7, 8, 54, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
32. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=7+(7)n for n≥0 and index map k(j)=2+2j, with nonnegative start and positive index step. Evaluate j=8. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(8, 18, 133, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (8, 18, 133, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
33. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=8+(1)n for n≥0 and index map k(j)=3+3j, with nonnegative start and positive index step. Evaluate j=9. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(9, 30, 38, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (9, 30, 38, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
34. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=9+(2)n for n≥0 and index map k(j)=0+4j, with nonnegative start and positive index step. Evaluate j=10. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(10, 40, 89, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (10, 40, 89, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
35. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=10+(3)n for n≥0 and index map k(j)=1+5j, with nonnegative start and positive index step. Evaluate j=11. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(11, 56, 178, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (11, 56, 178, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
36. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=11+(4)n for n≥0 and index map k(j)=2+1j, with nonnegative start and positive index step. Evaluate j=12. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(12, 14, 67, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (12, 14, 67, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
37. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=12+(5)n for n≥0 and index map k(j)=3+2j, with nonnegative start and positive index step. Evaluate j=13. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(13, 29, 157, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (13, 29, 157, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
38. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=13+(6)n for n≥0 and index map k(j)=0+3j, with nonnegative start and positive index step. Evaluate j=14. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(14, 42, 265, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (14, 42, 265, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
39. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=14+(7)n for n≥0 and index map k(j)=1+4j, with nonnegative start and positive index step. Evaluate j=15. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(15, 61, 441, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (15, 61, 441, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
40. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=15+(1)n for n≥0 and index map k(j)=2+5j, with nonnegative start and positive index step. Evaluate j=16. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(16, 82, 97, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (16, 82, 97, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
41. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=16+(2)n for n≥0 and index map k(j)=3+1j, with nonnegative start and positive index step. Evaluate j=17. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(17, 20, 56, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (17, 20, 56, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
42. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=17+(3)n for n≥0 and index map k(j)=0+2j, with nonnegative start and positive index step. Evaluate j=18. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(18, 36, 125, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (18, 36, 125, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
43. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=18+(4)n for n≥0 and index map k(j)=1+3j, with nonnegative start and positive index step. Evaluate j=19. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(19, 58, 250, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (19, 58, 250, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
44. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=19+(5)n for n≥0 and index map k(j)=2+4j, with nonnegative start and positive index step. Evaluate j=20. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(20, 82, 429, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (20, 82, 429, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
45. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=20+(6)n for n≥0 and index map k(j)=3+5j, with nonnegative start and positive index step. Evaluate j=21. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(21, 108, 668, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (21, 108, 668, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
46. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=21+(7)n for n≥0 and index map k(j)=0+1j, with nonnegative start and positive index step. Evaluate j=22. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(22, 22, 175, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (22, 22, 175, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
47. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=22+(1)n for n≥0 and index map k(j)=1+2j, with nonnegative start and positive index step. Evaluate j=23. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(23, 47, 69, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (23, 47, 69, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
48. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=23+(2)n for n≥0 and index map k(j)=2+3j, with nonnegative start and positive index step. Evaluate j=24. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(24, 74, 171, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (24, 74, 171, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
49. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=24+(3)n for n≥0 and index map k(j)=3+4j, with nonnegative start and positive index step. Evaluate j=25. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(25, 103, 333, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (25, 103, 333, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
50. A Subsequence Term Uses the Source at Its Increasing Selected Index
Use the entire arithmetic source a_n=25+(4)n for n≥0 and index map k(j)=0+5j, with nonnegative start and positive index step. Evaluate j=26. Give (j, source_index, subsequence_term, certificate).
View answer and explanation
(26, 130, 545, a_of_k_j_preserves_original_order)
Compute the selected source index and then a at that index, retaining original sequence order even when the source values decrease. The adapter rejects unsafe Number-range integer products rather than projecting rounded values. The exact result, retaining the requested order, is (26, 130, 545, a_of_k_j_preserves_original_order).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-arithmetic-term · candidate
51. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=0+1i and inner index l(j)=0+1j, with nonnegative starts and positive steps. Evaluate j=1. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(1, 1, 1, 1, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (1, 1, 1, 1, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
52. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=1+2i and inner index l(j)=1+2j, with nonnegative starts and positive steps. Evaluate j=2. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(2, 5, 11, 4, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (2, 5, 11, 4, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
53. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=2+3i and inner index l(j)=2+3j, with nonnegative starts and positive steps. Evaluate j=3. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(3, 11, 35, 9, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (3, 11, 35, 9, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
54. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=0+4i and inner index l(j)=3+4j, with nonnegative starts and positive steps. Evaluate j=4. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(4, 19, 76, 16, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (4, 19, 76, 16, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
55. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=1+5i and inner index l(j)=0+5j, with nonnegative starts and positive steps. Evaluate j=5. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(5, 25, 126, 25, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (5, 25, 126, 25, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
56. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=2+1i and inner index l(j)=1+6j, with nonnegative starts and positive steps. Evaluate j=6. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(6, 37, 39, 6, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (6, 37, 39, 6, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
57. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=0+2i and inner index l(j)=2+1j, with nonnegative starts and positive steps. Evaluate j=7. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(7, 9, 18, 2, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (7, 9, 18, 2, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
58. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=1+3i and inner index l(j)=3+2j, with nonnegative starts and positive steps. Evaluate j=8. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(8, 19, 58, 6, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (8, 19, 58, 6, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
59. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=2+4i and inner index l(j)=0+3j, with nonnegative starts and positive steps. Evaluate j=9. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(9, 27, 110, 12, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (9, 27, 110, 12, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
60. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=0+5i and inner index l(j)=1+4j, with nonnegative starts and positive steps. Evaluate j=10. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(10, 41, 205, 20, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (10, 41, 205, 20, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
61. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=1+1i and inner index l(j)=2+5j, with nonnegative starts and positive steps. Evaluate j=11. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(11, 57, 58, 5, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (11, 57, 58, 5, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
62. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=2+2i and inner index l(j)=3+6j, with nonnegative starts and positive steps. Evaluate j=12. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(12, 75, 152, 12, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (12, 75, 152, 12, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
63. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=0+3i and inner index l(j)=0+1j, with nonnegative starts and positive steps. Evaluate j=13. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(13, 13, 39, 3, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (13, 13, 39, 3, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
64. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=1+4i and inner index l(j)=1+2j, with nonnegative starts and positive steps. Evaluate j=14. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(14, 29, 117, 8, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (14, 29, 117, 8, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
65. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=2+5i and inner index l(j)=2+3j, with nonnegative starts and positive steps. Evaluate j=15. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(15, 47, 237, 15, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (15, 47, 237, 15, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
66. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=0+1i and inner index l(j)=3+4j, with nonnegative starts and positive steps. Evaluate j=16. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(16, 67, 67, 4, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (16, 67, 67, 4, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
67. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=1+2i and inner index l(j)=0+5j, with nonnegative starts and positive steps. Evaluate j=17. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(17, 85, 171, 10, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (17, 85, 171, 10, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
68. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=2+3i and inner index l(j)=1+6j, with nonnegative starts and positive steps. Evaluate j=18. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(18, 109, 329, 18, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (18, 109, 329, 18, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
69. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=0+4i and inner index l(j)=2+1j, with nonnegative starts and positive steps. Evaluate j=19. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(19, 21, 84, 4, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (19, 21, 84, 4, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
70. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=1+5i and inner index l(j)=3+2j, with nonnegative starts and positive steps. Evaluate j=20. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(20, 43, 216, 10, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (20, 43, 216, 10, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
71. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=2+1i and inner index l(j)=0+3j, with nonnegative starts and positive steps. Evaluate j=21. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(21, 63, 65, 3, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (21, 63, 65, 3, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
72. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=0+2i and inner index l(j)=1+4j, with nonnegative starts and positive steps. Evaluate j=22. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(22, 89, 178, 8, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (22, 89, 178, 8, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
73. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=1+3i and inner index l(j)=2+5j, with nonnegative starts and positive steps. Evaluate j=23. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(23, 117, 352, 15, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (23, 117, 352, 15, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
74. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=2+4i and inner index l(j)=3+6j, with nonnegative starts and positive steps. Evaluate j=24. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(24, 147, 590, 24, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (24, 147, 590, 24, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
75. Compose the Inner Index into the Outer Index in That Order
Use outer index k(i)=0+5i and inner index l(j)=0+1j, with nonnegative starts and positive steps. Evaluate j=25. Give (j, inner_index, source_index, composed_step, strictly_increasing, certificate).
View answer and explanation
(25, 25, 125, 5, Yes, composition_remains_strictly_increasing)
Compute l(j), then k(l(j)); the composed positive step is outer_step*inner_step. This order is not an arbitrary rearrangement of the source. Unsafe exact-integer results are rejected before English projection. The exact result, retaining the requested order, is (25, 25, 125, 5, Yes, composition_remains_strictly_increasing).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-composed-index · candidate
76. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[0, 1, 2, 3]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([0, 1, 2, 3], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([0, 1, 2, 3], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
77. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[1, 3, 5, 7, 9]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([1, 3, 5, 7, 9], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([1, 3, 5, 7, 9], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
78. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[2, 5, 8, 11, 14, 17]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([2, 5, 8, 11, 14, 17], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([2, 5, 8, 11, 14, 17], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
79. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[3, 7, 11, 15]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([3, 7, 11, 15], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([3, 7, 11, 15], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
80. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[4, 9, 14, 19, 24]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([4, 9, 14, 19, 24], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([4, 9, 14, 19, 24], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
81. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[5, 6, 7, 8, 9, 10]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([5, 6, 7, 8, 9, 10], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([5, 6, 7, 8, 9, 10], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
82. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[6, 8, 10, 12]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([6, 8, 10, 12], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([6, 8, 10, 12], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
83. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[7, 10, 13, 16, 19]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([7, 10, 13, 16, 19], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([7, 10, 13, 16, 19], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
84. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[8, 12, 16, 20, 24, 28]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([8, 12, 16, 20, 24, 28], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([8, 12, 16, 20, 24, 28], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
85. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[9, 14, 19, 24]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([9, 14, 19, 24], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([9, 14, 19, 24], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
86. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[10, 11, 12, 13, 14]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([10, 11, 12, 13, 14], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([10, 11, 12, 13, 14], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
87. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[11, 13, 15, 17, 19, 21]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([11, 13, 15, 17, 19, 21], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([11, 13, 15, 17, 19, 21], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
88. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[12, 15, 18, 21]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([12, 15, 18, 21], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([12, 15, 18, 21], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
89. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[13, 17, 21, 25, 29]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([13, 17, 21, 25, 29], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([13, 17, 21, 25, 29], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
90. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[14, 19, 24, 29, 34, 39]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([14, 19, 24, 29, 34, 39], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([14, 19, 24, 29, 34, 39], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
91. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[15, 16, 17, 18]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([15, 16, 17, 18], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([15, 16, 17, 18], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
92. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[16, 18, 20, 22, 24]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([16, 18, 20, 22, 24], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([16, 18, 20, 22, 24], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
93. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[17, 20, 23, 26, 29, 32]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([17, 20, 23, 26, 29, 32], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([17, 20, 23, 26, 29, 32], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
94. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[18, 22, 26, 30]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([18, 22, 26, 30], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([18, 22, 26, 30], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
95. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[19, 24, 29, 34, 39]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([19, 24, 29, 34, 39], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([19, 24, 29, 34, 39], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
96. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[20, 21, 22, 23, 24, 25]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([20, 21, 22, 23, 24, 25], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([20, 21, 22, 23, 24, 25], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
97. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[21, 23, 25, 27]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([21, 23, 25, 27], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([21, 23, 25, 27], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
98. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[22, 25, 28, 31, 34]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([22, 25, 28, 31, 34], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([22, 25, 28, 31, 34], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
99. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[23, 27, 31, 35, 39, 43]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([23, 27, 31, 35, 39, 43], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([23, 27, 31, 35, 39, 43], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
100. A Finite Increasing List Is Not an Already-Supplied Infinite Certificate
Supplied finite natural-index list=[24, 29, 34, 39]. Give (indices, finite_prefix_increasing, infinite_certificate_supplied, boundary).
No infinite index certificate was supplied; this does not deny that an increasing finite prefix can have an infinite increasing extension.
View answer and explanation
([24, 29, 34, 39], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate)
Check only strict increase of this finite list, then No for an infinite index certificate supplied by that data alone. This does not deny that an increasing finite prefix can be extended; it keeps a finite observation distinct from a stated all-j map. The exact result, retaining the requested order, is ([24, 29, 34, 39], Yes, No, finite_increasing_prefix_not_infinite_subsequence_certificate).
Reviewed step-by-step derivation
Subsequence · qf-mod38-subseq-finite-prefix-boundary · candidate
Canonical question JSONL · English JSONL · Validation report