Finite Principal-Filter Images and Direction Guards
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-20261008-advance-20261011-r07-tends-to-relation-foundations · 2026-10-08 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Finite Principal-Filter Image Generator
The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. S=[]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
2. Finite Principal-Filter Image Generator
The complete total map f:Fin(3)→Fin(1) has images=[0, 0, 0] indexed from 0; Fin(3)={0,…,2}. Every table entry must lie in the target type. S=[0, 1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[0]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
3. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[1]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
4. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[0, 1]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
5. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[2]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
6. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[0, 2]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
7. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[1, 2]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
8. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[0, 1, 2]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
9. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[3]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
10. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[0, 3]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
11. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[1, 3]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
12. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[0, 1, 3]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
13. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[2, 3]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2, 3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
14. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[0, 2, 3]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2, 3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
15. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[1, 2, 3]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2, 3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
16. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[0, 1, 2, 3]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2, 3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
17. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[4]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
18. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[0, 4]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
19. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[1, 4]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
20. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[0, 1, 4]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
21. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[2, 4]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
22. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[0, 2, 4]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
23. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[1, 2, 4]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
24. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[0, 1, 2, 4]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
25. Finite Principal-Filter Image Generator
The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[3, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.
View answer and explanation
[3, 4]
Map exactly the elements of S through the legal total map and deduplicate; the empty generator still has an empty image. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [3, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-image · candidate
26. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(3)→Fin(1) has images=[0, 0, 0] indexed from 0; Fin(3)={0,…,2}. Every table entry must lie in the target type. S=[0, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
27. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 0, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[0]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
28. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 0, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[1]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
29. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 0, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[0, 1]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
30. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 1, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[2]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
31. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 1, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[0, 2]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
32. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 1, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[1, 2]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
33. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 1, 0, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[0, 1, 2]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
34. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 0, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[3]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
35. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 0, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[0, 3]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
36. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 0, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[1, 3]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
37. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 0, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[0, 1, 3]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
38. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 1, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[2, 3]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2, 3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
39. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 1, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[0, 2, 3]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2, 3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
40. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 1, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[1, 2, 3]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2, 3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
41. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 1, 1, 0, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[0, 1, 2, 3]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2, 3].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
42. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 0, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[4]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
43. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 0, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[0, 4]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
44. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 0, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[1, 4]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
45. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 0, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[0, 1, 4]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
46. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 1, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[2, 4]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [2, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
47. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[1, 0, 1, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[0, 2, 4]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 2, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
48. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[0, 1, 1, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[1, 2, 4]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [1, 2, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
49. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[1, 1, 1, 0, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[0, 1, 2, 4]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [0, 1, 2, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
50. Finite Principal-Filter Direction Obstructions
The complete total map f:Fin(8)→Fin(2) has images=[0, 0, 0, 1, 1, 0, 0, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3, 4, 5, 6, 7]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
View answer and explanation
[3, 4]
An empty obstruction set means no failure in this finite model; do not reverse the inclusion direction. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [3, 4].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-obstructions · candidate
51. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. S=[]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[], ordered candidates=[-1, 0, 1]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
52. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 4, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
53. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 0, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
54. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[], ordered candidates=[0, 1, 2]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[Yes, Yes, Yes]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, Yes, Yes].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
55. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 4, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
56. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 0, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
57. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 2, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[Yes, Yes, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, Yes, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
58. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 4, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
59. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 0, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
60. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 2, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, Yes, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, Yes, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
61. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 101, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[Yes, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
62. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 0, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
63. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 2, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, Yes, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, Yes, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
64. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 101, 1]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, Yes]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, Yes].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
65. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 4, 1]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[Yes, No, Yes]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, No, Yes].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
66. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 2, 1]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, Yes, Yes]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, Yes, Yes].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
67. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 101, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
68. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 4, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
69. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 0, 102]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[Yes, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
70. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 101, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
71. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 0, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
72. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[1, 2, 5]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[Yes, Yes, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [Yes, Yes, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
73. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[100, 4, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
74. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[3, 0, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, No, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
75. Typed Principal-Filter Obstruction Candidates
The complete total map f:Fin(6)→Fin(2) has images=[0, 1, 1, 0, 0, 1] indexed from 0; Fin(6)={0,…,5}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0], ordered candidates=[0, 2, 0]. For each c, answer whether c∈S and f(c)∉T; do not evaluate f at a candidate outside its type.
View answer and explanation
[No, Yes, No]
Check source typing and S membership before reading f; outside-S candidates and images inside T are not obstructions. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is [No, Yes, No].
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-candidate-mask · candidate
76. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. S=[]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(0)→Fin(0) has after=[]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([], [])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([], []).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
77. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(4)→Fin(3) has images=[0, 0, 1, 2] indexed from 0; Fin(4)={0,…,3}. Every table entry must lie in the target type. S=[0, 1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(3)→Fin(8) has after=[0, 0, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([0], [0])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0], [0]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
78. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([1], [1])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1], [1]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
79. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([0, 1], [0, 1])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1], [0, 1]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
80. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([2], [2])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([2], [2]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
81. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([0, 2], [0, 2])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 2], [0, 2]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
82. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([1, 2], [1, 2])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 2], [1, 2]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
83. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([0, 1, 2], [0, 1, 2])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 2], [0, 1, 2]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
84. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([3], [3])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([3], [3]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
85. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([0, 3], [0, 3])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 3], [0, 3]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
86. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([1, 3], [1, 3])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 3], [1, 3]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
87. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([0, 1, 3], [0, 1, 3])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 3], [0, 1, 3]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
88. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([2, 3], [2, 3])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([2, 3], [2, 3]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
89. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([0, 2, 3], [0, 2, 3])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 2, 3], [0, 2, 3]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
90. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([1, 2, 3], [1, 2, 3])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 2, 3], [1, 2, 3]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
91. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([0, 1, 2, 3], [0, 1, 2, 3])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 2, 3], [0, 1, 2, 3]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
92. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([4], [4])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([4], [4]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
93. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([0, 4], [0, 4])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 4], [0, 4]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
94. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([1, 4], [1, 4])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 4], [1, 4]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
95. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([0, 1, 4], [0, 1, 4])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 4], [0, 1, 4]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
96. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([2, 4], [2, 4])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([2, 4], [2, 4]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
97. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([0, 2, 4], [0, 2, 4])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 2, 4], [0, 2, 4]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
98. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([1, 2, 4], [1, 2, 4])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([1, 2, 4], [1, 2, 4]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
99. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[0, 1, 2, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([0, 1, 2, 4], [0, 1, 2, 4])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([0, 1, 2, 4], [0, 1, 2, 4]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
100. Two Finite Principal-Filter Composition Images
The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[3, 4]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.
View answer and explanation
([3, 4], [3, 4])
Evaluate the composition sourcewise for one route and first deduplicate the intermediate image for the other; all f/g entries must remain total and typed. This is a finite principal-filter model, not a general sequence or point-limit claim. The exact result, retaining the requested order, is ([3, 4], [3, 4]).
Reviewed step-by-step derivation
Tends-to relation · qf-density11-tends-principal-composition-images · candidate
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