Complete Finite-Map Collisions and Original Source Labels
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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-20261012-advance-20261016-r06-injective-foundations · 2026-10-12 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(0)=∅→Fin(0), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([], [], Yes)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], Yes).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
2. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(1)={0,…,0}→Fin(1), complete images=[0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([0], [], Yes)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0], [], Yes).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
3. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(3)={0,…,2}→Fin(3), complete images=[0, 1, 2]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([0, 1, 2], [], Yes)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2], [], Yes).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
4. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 0, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([0, 0, 0, 0, 0], [[0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0], [[0, 1], [0, 2], [0, 3], [0, 4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
5. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 0, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([1, 0, 0, 0, 0], [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 0], [[1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
6. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 1, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([0, 1, 0, 0, 0], [[0, 2], [0, 3], [0, 4], [2, 3], [2, 4], [3, 4]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 0], [[0, 2], [0, 3], [0, 4], [2, 3], [2, 4], [3, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
7. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 1, 0, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([1, 1, 0, 0, 0], [[0, 1], [2, 3], [2, 4], [3, 4]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 0], [[0, 1], [2, 3], [2, 4], [3, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
8. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 0, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([0, 0, 1, 0, 0], [[0, 1], [0, 3], [0, 4], [1, 3], [1, 4], [3, 4]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 0], [[0, 1], [0, 3], [0, 4], [1, 3], [1, 4], [3, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
9. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 0, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([1, 0, 1, 0, 0], [[0, 2], [1, 3], [1, 4], [3, 4]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 0], [[0, 2], [1, 3], [1, 4], [3, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
10. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 1, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([0, 1, 1, 0, 0], [[0, 3], [0, 4], [1, 2], [3, 4]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 0], [[0, 3], [0, 4], [1, 2], [3, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
11. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 1, 1, 0, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([1, 1, 1, 0, 0], [[0, 1], [0, 2], [1, 2], [3, 4]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 0], [[0, 1], [0, 2], [1, 2], [3, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
12. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 0, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([0, 0, 0, 1, 0], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 1, 0], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
13. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 0, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([1, 0, 0, 1, 0], [[0, 3], [1, 2], [1, 4], [2, 4]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 1, 0], [[0, 3], [1, 2], [1, 4], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
14. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 1, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([0, 1, 0, 1, 0], [[0, 2], [0, 4], [1, 3], [2, 4]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 1, 0], [[0, 2], [0, 4], [1, 3], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
15. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 1, 0, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([1, 1, 0, 1, 0], [[0, 1], [0, 3], [1, 3], [2, 4]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 1, 0], [[0, 1], [0, 3], [1, 3], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
16. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 0, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([0, 0, 1, 1, 0], [[0, 1], [0, 4], [1, 4], [2, 3]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 1, 0], [[0, 1], [0, 4], [1, 4], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
17. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 0, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([1, 0, 1, 1, 0], [[0, 2], [0, 3], [1, 4], [2, 3]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 1, 0], [[0, 2], [0, 3], [1, 4], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
18. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 1, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([0, 1, 1, 1, 0], [[0, 4], [1, 2], [1, 3], [2, 3]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 1, 0], [[0, 4], [1, 2], [1, 3], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
19. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 1, 1, 1, 0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([1, 1, 1, 1, 0], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 1, 0], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
20. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([0, 0, 0, 0, 1], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
21. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([1, 0, 0, 0, 1], [[0, 4], [1, 2], [1, 3], [2, 3]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], [[0, 4], [1, 2], [1, 3], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
22. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([0, 1, 0, 0, 1], [[0, 2], [0, 3], [1, 4], [2, 3]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], [[0, 2], [0, 3], [1, 4], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
23. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([1, 1, 0, 0, 1], [[0, 1], [0, 4], [1, 4], [2, 3]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], [[0, 1], [0, 4], [1, 4], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
24. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([0, 0, 1, 0, 1], [[0, 1], [0, 3], [1, 3], [2, 4]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], [[0, 1], [0, 3], [1, 3], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
25. Every Distinct-Input Collision and Injectivity
Finite types start at 0: Fin(5)={0,…,4}→Fin(2), complete images=[1, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all pairs x<x2 with equal outputs, injectivity eligibility), with collision pairs in original-input lexicographic order.
View answer and explanation
([1, 0, 1, 0, 1], [[0, 2], [0, 4], [1, 3], [2, 4]], No)
Validate the complete table first, then enumerate distinct-input pairs; no collisions means injective, including the empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1], [[0, 2], [0, 4], [1, 3], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-all-collision-pairs · candidate
26. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(1)={0,…,0}→Fin(1), complete images=[0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([0], [], [], [], Yes)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0], [], [], [], Yes).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
27. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([0, 0, 1], [[0, 1]], [1, 2], [], Yes)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1], [[0, 1]], [1, 2], [], Yes).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
28. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(4)={0,…,3}→Fin(2), complete images=[0, 0, 1, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([0, 0, 1, 1], [[0, 1], [2, 3]], [1, 2, 3], [[2, 3]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 1], [[0, 1], [2, 3]], [1, 2, 3], [[2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
29. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([0, 0, 0, 0, 1], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], [1, 2, 3, 4], [[1, 2], [1, 3], [2, 3]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], [1, 2, 3, 4], [[1, 2], [1, 3], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
30. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([1, 0, 0, 0, 1], [[0, 4], [1, 2], [1, 3], [2, 3]], [0, 2, 3, 4], [[0, 4], [2, 3]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], [[0, 4], [1, 2], [1, 3], [2, 3]], [0, 2, 3, 4], [[0, 4], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
31. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([2, 0, 0, 0, 1], [[1, 2], [1, 3], [2, 3]], [0, 1, 3, 4], [[1, 3]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 1], [[1, 2], [1, 3], [2, 3]], [0, 1, 3, 4], [[1, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
32. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([0, 1, 0, 0, 1], [[0, 2], [0, 3], [1, 4], [2, 3]], [0, 1, 2, 4], [[0, 2], [1, 4]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], [[0, 2], [0, 3], [1, 4], [2, 3]], [0, 1, 2, 4], [[0, 2], [1, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
33. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([1, 1, 0, 0, 1], [[0, 1], [0, 4], [1, 4], [2, 3]], [0, 1, 2, 3], [[0, 1], [2, 3]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], [[0, 1], [0, 4], [1, 4], [2, 3]], [0, 1, 2, 3], [[0, 1], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
34. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([2, 1, 0, 0, 1], [[1, 4], [2, 3]], [1, 2, 3, 4], [[1, 4], [2, 3]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 1], [[1, 4], [2, 3]], [1, 2, 3, 4], [[1, 4], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
35. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([0, 2, 0, 0, 1], [[0, 2], [0, 3], [2, 3]], [0, 2, 3, 4], [[0, 2], [0, 3], [2, 3]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 1], [[0, 2], [0, 3], [2, 3]], [0, 2, 3, 4], [[0, 2], [0, 3], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
36. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([1, 2, 0, 0, 1], [[0, 4], [2, 3]], [0, 1, 3, 4], [[0, 4]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 1], [[0, 4], [2, 3]], [0, 1, 3, 4], [[0, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
37. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([2, 2, 0, 0, 1], [[0, 1], [2, 3]], [0, 1, 2, 4], [[0, 1]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 1], [[0, 1], [2, 3]], [0, 1, 2, 4], [[0, 1]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
38. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([0, 0, 1, 0, 1], [[0, 1], [0, 3], [1, 3], [2, 4]], [0, 1, 2, 3], [[0, 1], [0, 3], [1, 3]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], [[0, 1], [0, 3], [1, 3], [2, 4]], [0, 1, 2, 3], [[0, 1], [0, 3], [1, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
39. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([1, 0, 1, 0, 1], [[0, 2], [0, 4], [1, 3], [2, 4]], [1, 2, 3, 4], [[1, 3], [2, 4]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1], [[0, 2], [0, 4], [1, 3], [2, 4]], [1, 2, 3, 4], [[1, 3], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
40. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([2, 0, 1, 0, 1], [[1, 3], [2, 4]], [0, 2, 3, 4], [[2, 4]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 1], [[1, 3], [2, 4]], [0, 2, 3, 4], [[2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
41. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([0, 1, 1, 0, 1], [[0, 3], [1, 2], [1, 4], [2, 4]], [0, 1, 3, 4], [[0, 3], [1, 4]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 1], [[0, 3], [1, 2], [1, 4], [2, 4]], [0, 1, 3, 4], [[0, 3], [1, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
42. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([1, 1, 1, 0, 1], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], [0, 1, 2, 4], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 1], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], [0, 1, 2, 4], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
43. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([2, 1, 1, 0, 1], [[1, 2], [1, 4], [2, 4]], [0, 1, 2, 3], [[1, 2]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 1], [[1, 2], [1, 4], [2, 4]], [0, 1, 2, 3], [[1, 2]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
44. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([0, 2, 1, 0, 1], [[0, 3], [2, 4]], [1, 2, 3, 4], [[2, 4]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 1], [[0, 3], [2, 4]], [1, 2, 3, 4], [[2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
45. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([1, 2, 1, 0, 1], [[0, 2], [0, 4], [2, 4]], [0, 2, 3, 4], [[0, 2], [0, 4], [2, 4]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 1], [[0, 2], [0, 4], [2, 4]], [0, 2, 3, 4], [[0, 2], [0, 4], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
46. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=2, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([2, 2, 1, 0, 1], [[0, 1], [2, 4]], [0, 1, 3, 4], [[0, 1]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 1], [[0, 1], [2, 4]], [0, 1, 3, 4], [[0, 1]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
47. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=3, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([0, 0, 2, 0, 1], [[0, 1], [0, 3], [1, 3]], [0, 1, 2, 4], [[0, 1]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 2, 0, 1], [[0, 1], [0, 3], [1, 3]], [0, 1, 2, 4], [[0, 1]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
48. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=4, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([1, 0, 2, 0, 1], [[0, 4], [1, 3]], [0, 1, 2, 3], [[1, 3]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 0, 1], [[0, 4], [1, 3]], [0, 1, 2, 3], [[1, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
49. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([2, 0, 2, 0, 1], [[0, 2], [1, 3]], [1, 2, 3, 4], [[1, 3]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 2, 0, 1], [[0, 2], [1, 3]], [1, 2, 3, 4], [[1, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
50. Input Deletion with Original Collision Labels Retained
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
View answer and explanation
([0, 1, 2, 0, 1], [[0, 3], [1, 4]], [0, 2, 3, 4], [[0, 3]], No)
Remove only the specified original label and compare equal outputs using all retained original labels. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 0, 1], [[0, 3], [1, 4]], [0, 2, 3, 4], [[0, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-remove-input-collision-change · candidate
51. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(0)=∅→Fin(0), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=0. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([], No, outside-codomain, not-evaluated, not-evaluated)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], No, outside-codomain, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
52. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(0)=∅→Fin(1), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=0. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([], Yes, [0], [], Yes)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], Yes, [0], [], Yes).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
53. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(2)={0,…,1}→Fin(3), complete images=[0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=2. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([0, 1], Yes, [0, 1, 2], [], Yes)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], Yes, [0, 1, 2], [], Yes).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
54. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(2)={0,…,1}→Fin(2), complete images=[0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=0. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([0, 1], Yes, [0, 1, 0], [[0, 2]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], Yes, [0, 1, 0], [[0, 2]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
55. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(2)={0,…,1}→Fin(2), complete images=[0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=-1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([0, 1], No, outside-codomain, not-evaluated, not-evaluated)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1], No, outside-codomain, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
56. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(8)={0,…,7}→Fin(8), complete images=[0, 1, 2, 3, 4, 5, 6, 7]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=7. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([0, 1, 2, 3, 4, 5, 6, 7], Yes, [0, 1, 2, 3, 4, 5, 6, 7, 7], [[7, 8]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], Yes, [0, 1, 2, 3, 4, 5, 6, 7, 7], [[7, 8]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
57. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([0, 0, 0, 0, 1], Yes, [0, 0, 0, 0, 1, 1], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [4, 5]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], Yes, [0, 0, 0, 0, 1, 1], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3], [4, 5]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
58. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=2. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([1, 0, 0, 0, 1], Yes, [1, 0, 0, 0, 1, 2], [[0, 4], [1, 2], [1, 3], [2, 3]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], Yes, [1, 0, 0, 0, 1, 2], [[0, 4], [1, 2], [1, 3], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
59. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=-1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([2, 0, 0, 0, 1], No, outside-codomain, not-evaluated, not-evaluated)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 1], No, outside-codomain, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
60. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=0. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([0, 1, 0, 0, 1], Yes, [0, 1, 0, 0, 1, 0], [[0, 2], [0, 3], [0, 5], [1, 4], [2, 3], [2, 5], [3, 5]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], Yes, [0, 1, 0, 0, 1, 0], [[0, 2], [0, 3], [0, 5], [1, 4], [2, 3], [2, 5], [3, 5]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
61. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([1, 1, 0, 0, 1], Yes, [1, 1, 0, 0, 1, 1], [[0, 1], [0, 4], [0, 5], [1, 4], [1, 5], [2, 3], [4, 5]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], Yes, [1, 1, 0, 0, 1, 1], [[0, 1], [0, 4], [0, 5], [1, 4], [1, 5], [2, 3], [4, 5]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
62. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=2. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([2, 1, 0, 0, 1], Yes, [2, 1, 0, 0, 1, 2], [[0, 5], [1, 4], [2, 3]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 1], Yes, [2, 1, 0, 0, 1, 2], [[0, 5], [1, 4], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
63. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=-1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([0, 2, 0, 0, 1], No, outside-codomain, not-evaluated, not-evaluated)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 1], No, outside-codomain, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
64. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=0. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([1, 2, 0, 0, 1], Yes, [1, 2, 0, 0, 1, 0], [[0, 4], [2, 3], [2, 5], [3, 5]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 1], Yes, [1, 2, 0, 0, 1, 0], [[0, 4], [2, 3], [2, 5], [3, 5]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
65. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([2, 2, 0, 0, 1], Yes, [2, 2, 0, 0, 1, 1], [[0, 1], [2, 3], [4, 5]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 1], Yes, [2, 2, 0, 0, 1, 1], [[0, 1], [2, 3], [4, 5]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
66. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=2. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([0, 0, 1, 0, 1], Yes, [0, 0, 1, 0, 1, 2], [[0, 1], [0, 3], [1, 3], [2, 4]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], Yes, [0, 0, 1, 0, 1, 2], [[0, 1], [0, 3], [1, 3], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
67. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=-1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([1, 0, 1, 0, 1], No, outside-codomain, not-evaluated, not-evaluated)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1], No, outside-codomain, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
68. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=0. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([2, 0, 1, 0, 1], Yes, [2, 0, 1, 0, 1, 0], [[1, 3], [1, 5], [2, 4], [3, 5]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 1], Yes, [2, 0, 1, 0, 1, 0], [[1, 3], [1, 5], [2, 4], [3, 5]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
69. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([0, 1, 1, 0, 1], Yes, [0, 1, 1, 0, 1, 1], [[0, 3], [1, 2], [1, 4], [1, 5], [2, 4], [2, 5], [4, 5]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 1], Yes, [0, 1, 1, 0, 1, 1], [[0, 3], [1, 2], [1, 4], [1, 5], [2, 4], [2, 5], [4, 5]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
70. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=2. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([1, 1, 1, 0, 1], Yes, [1, 1, 1, 0, 1, 2], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 1], Yes, [1, 1, 1, 0, 1, 2], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
71. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=-1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([2, 1, 1, 0, 1], No, outside-codomain, not-evaluated, not-evaluated)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 1], No, outside-codomain, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
72. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=0. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([0, 2, 1, 0, 1], Yes, [0, 2, 1, 0, 1, 0], [[0, 3], [0, 5], [2, 4], [3, 5]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 1], Yes, [0, 2, 1, 0, 1, 0], [[0, 3], [0, 5], [2, 4], [3, 5]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
73. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([1, 2, 1, 0, 1], Yes, [1, 2, 1, 0, 1, 1], [[0, 2], [0, 4], [0, 5], [2, 4], [2, 5], [4, 5]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 1], Yes, [1, 2, 1, 0, 1, 1], [[0, 2], [0, 4], [0, 5], [2, 4], [2, 5], [4, 5]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
74. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=2. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([2, 2, 1, 0, 1], Yes, [2, 2, 1, 0, 1, 2], [[0, 1], [0, 5], [1, 5], [2, 4]], No)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 1], Yes, [2, 2, 1, 0, 1, 2], [[0, 1], [0, 5], [1, 5], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
75. Append Candidate Guard before New Table and Collisions
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=-1. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.
View answer and explanation
([0, 0, 2, 0, 1], No, outside-codomain, not-evaluated, not-evaluated)
Require the candidate to lie in the original codomain before constructing the extended table or testing its properties; an outside value supplies no new map. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 2, 0, 1], No, outside-codomain, not-evaluated, not-evaluated).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-append-output-qualification · candidate
76. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(0)=∅→Fin(0), complete images=[]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([], [], [], Yes)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([], [], [], Yes).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
77. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[2, 0] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([0, 0, 1], [0, 2], [], Yes)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1], [0, 2], [], Yes).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
78. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[1, 0] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([0, 0, 1], [0, 1], [[0, 1]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1], [0, 1], [[0, 1]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
79. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([0, 0, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 3], [1, 2], [1, 3], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
80. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([1, 0, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 4], [1, 2], [1, 3], [2, 3]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 4], [1, 2], [1, 3], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
81. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([2, 0, 0, 0, 1], [0, 1, 2, 3, 4], [[1, 2], [1, 3], [2, 3]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 0, 0, 1], [0, 1, 2, 3, 4], [[1, 2], [1, 3], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
82. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([0, 1, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [0, 3], [1, 4], [2, 3]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [0, 3], [1, 4], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
83. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([1, 1, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 4], [2, 3]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 4], [1, 4], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
84. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([2, 1, 0, 0, 1], [0, 1, 2, 3, 4], [[1, 4], [2, 3]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 0, 0, 1], [0, 1, 2, 3, 4], [[1, 4], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
85. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([0, 2, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [0, 3], [2, 3]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [0, 3], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
86. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([1, 2, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 4], [2, 3]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 4], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
87. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([2, 2, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [2, 3]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 0, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [2, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
88. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([0, 0, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 3], [1, 3], [2, 4]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 3], [1, 3], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
89. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([1, 0, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 3], [2, 4]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [0, 4], [1, 3], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
90. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([2, 0, 1, 0, 1], [0, 1, 2, 3, 4], [[1, 3], [2, 4]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 1, 0, 1], [0, 1, 2, 3, 4], [[1, 3], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
91. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([0, 1, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 3], [1, 2], [1, 4], [2, 4]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 3], [1, 2], [1, 4], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
92. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([1, 1, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 1, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 2], [0, 4], [1, 2], [1, 4], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
93. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([2, 1, 1, 0, 1], [0, 1, 2, 3, 4], [[1, 2], [1, 4], [2, 4]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 1, 1, 0, 1], [0, 1, 2, 3, 4], [[1, 2], [1, 4], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
94. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([0, 2, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 3], [2, 4]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 3], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
95. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([1, 2, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [0, 4], [2, 4]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 2, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [0, 4], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
96. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([2, 2, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [2, 4]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2, 1, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [2, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
97. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([0, 0, 2, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 3], [1, 3]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0, 2, 0, 1], [0, 1, 2, 3, 4], [[0, 1], [0, 3], [1, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
98. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([1, 0, 2, 0, 1], [0, 1, 2, 3, 4], [[0, 4], [1, 3]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1, 0, 2, 0, 1], [0, 1, 2, 3, 4], [[0, 4], [1, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
99. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([2, 0, 2, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [1, 3]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0, 2, 0, 1], [0, 1, 2, 3, 4], [[0, 2], [1, 3]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
100. Typed Source Restriction and Complete Collision Witnesses
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[0, 1, 2, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[4, 2, 0, 1, 3] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
View answer and explanation
([0, 1, 2, 0, 1], [0, 1, 2, 3, 4], [[0, 3], [1, 4]], No)
Check the entire original table before comparing the restricted original labels; sorting does not merge distinct inputs or invent renumbered ones. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 1, 2, 0, 1], [0, 1, 2, 3, 4], [[0, 3], [1, 4]], No).
Reviewed step-by-step derivation
Injectivity and distinct inputs · qf-density16-inj-restriction-subset-witness · candidate
Canonical question JSONL · English JSONL · Validation report