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Question 15890: A Complete Finite List Gives an At-Most-Countable Certificate

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0030-15890 · q-20261014-advance-20261031-r01-at-most-countable-set-0015

Original question and givens

Complete nonempty duplicate-free finite integer list elements=[15, -16, 68] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

  • G001 — Complete nonempty duplicate-free finite integer list elements=[15, -16, 68] in original list order. Give (elements,injection_pairs,injective?,finite_implies_at_most_countable). Each injection pair is [element,zero_based_position].

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

DECLARED complete finite integer set in original list order=[15, -16, 68];length=3

After

nonempty duplicate-free registered family;original order retained;empty-set countability not denied

Conditions: none

Completeness is the declared set premise,not inferred for an unknown universe from samples.Keep order and reject duplicates rather than sorting or deduplicating.The family's empty-list rejection is an interface limit,not denial of empty-set countability.

Step identifiers and dependencies

Rule: countable_complete_finite_list_premise_not_sampled_subset · Uses: G001

S002 · implies

Before

original element=15;unique original position=0

After

injection pair element_to_position=[15, 0]

Conditions: none

Send each element to its unique zero-based position.Equal positions mean the same list entry,so this is injective into the naturals.The direction is element to position,opposite to enumeration.

Step identifiers and dependencies

Rule: finite_countability_injection_direction · Uses: G001, S001

S003 · implies

Before

original element=-16;unique original position=1

After

injection pair element_to_position=[-16, 1]

Conditions: none

Send each element to its unique zero-based position.Equal positions mean the same list entry,so this is injective into the naturals.The direction is element to position,opposite to enumeration.

Step identifiers and dependencies

Rule: finite_countability_injection_direction · Uses: G001, S002

S004 · implies

Before

original element=68;unique original position=2

After

injection pair element_to_position=[68, 2]

Conditions: none

Send each element to its unique zero-based position.Equal positions mean the same list entry,so this is injective into the naturals.The direction is element to position,opposite to enumeration.

Step identifiers and dependencies

Rule: finite_countability_injection_direction · Uses: G001, S003

Reverse checks

  • S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([15, -16, 68], [[15, 0], [-16, 1], [68, 2]], Yes, finite_implies_at_most_countable)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: d8c7e93c891f16680761414d719ac741085d87df59ae51eed7eb734f72fd5f95
  • candidate record SHA-256: fa348c5ee1eef3bb46988ae8ae26f257e8e1f1c003fe3b3f8aa20a7e625b54ef
  • semantic review receipt SHA-256: 73bc08066ffb1a81eacb2d4e288d03f522aeefd3949fa0d97da41b484ed737bc
  • question-first witness SHA-256: ac120452a208b257b8270f38351a09dcd832cdbc8d9db6b69933f83e43cc9336
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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