Question 7744: Source Deletion with Declared Codomain Kept

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0014-07744 · q-20261012-advance-20261016-r07-surjective-0069

Original question and givens

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 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

G001 — 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 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).

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.

mko-surjective

Step-by-step derivation

S001 · implies

Before

n=5;m=3;images=[0, 2, 1, 0, 1]

After

all original inputs and outputs form a typed total table;labels start at0

Conditions: none

First validate the entire original table as a total function from Fin(n) to Fin(m),including cells not used by a query.Only then evaluate preimages for eligible targets and surjectivity.

Step identifiers and dependencies

Rule: finite_surjection_typed_scope · Uses: G001

S002 · implies

Before

phase=original;target=0;source labels=[0, 1, 2, 3, 4];images on labels=[0, 2, 1, 0, 1]

After

fiber=[0, 3];missing=No

Conditions: none

For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty.

Step identifiers and dependencies

Rule: surjection_complete_target_fiber · Uses: G001, S001

S003 · implies

Before

phase=original;target=1;source labels=[0, 1, 2, 3, 4];images on labels=[0, 2, 1, 0, 1]

After

fiber=[2, 4];missing=No

Conditions: none

For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty.

Step identifiers and dependencies

Rule: surjection_complete_target_fiber · Uses: G001, S002

S004 · implies

Before

phase=original;target=2;source labels=[0, 1, 2, 3, 4];images on labels=[0, 2, 1, 0, 1]

After

fiber=[1];missing=No

Conditions: none

For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty.

Step identifiers and dependencies

Rule: surjection_complete_target_fiber · Uses: G001, S003

S005 · implies

Before

phase=original;targets=[0, 1, 2];fibers=[[0, 3], [2, 4], [1]]

After

missing=[];surjective=Yes

Conditions: none

Surjectivity is relative to the declared codomain:every target needs a preimage.An empty codomain has no targets to test and satisfies the condition;an empty domain with a nonempty codomain misses every target.

Step identifiers and dependencies

Rule: surjection_all_declared_targets · Uses: G001, S004

S006 · implies

Before

deleted original input=0;old labels=[0, 1, 2, 3, 4]

After

kept labels=[1, 2, 3, 4];remaining images=[2, 1, 0, 1];codomain Fin(3)

Conditions: none

Delete the specified original input without changing the codomain or renumbering surviving labels.The remaining images list is ordered by retained labels,not a declaration of relabeled inputs.

Step identifiers and dependencies

Rule: delete_input_keeps_original_labels_and_codomain · Uses: G001, S005

S007 · implies

Before

phase=after deletion;target=0;source labels=[1, 2, 3, 4];images on labels=[2, 1, 0, 1]

After

fiber=[3];missing=No

Conditions: none

For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty.

Step identifiers and dependencies

Rule: surjection_complete_target_fiber · Uses: G001, S006

S008 · implies

Before

phase=after deletion;target=1;source labels=[1, 2, 3, 4];images on labels=[2, 1, 0, 1]

After

fiber=[2, 4];missing=No

Conditions: none

For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty.

Step identifiers and dependencies

Rule: surjection_complete_target_fiber · Uses: G001, S007

S009 · implies

Before

phase=after deletion;target=2;source labels=[1, 2, 3, 4];images on labels=[2, 1, 0, 1]

After

fiber=[1];missing=No

Conditions: none

For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty.

Step identifiers and dependencies

Rule: surjection_complete_target_fiber · Uses: G001, S008

S010 · implies

Before

phase=after deletion;targets=[0, 1, 2];fibers=[[3], [2, 4], [1]]

After

missing=[];surjective=Yes

Conditions: none

Step identifiers and dependencies

Rule: surjection_all_declared_targets · Uses: G001, S009

Reverse checks

S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
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

([0, 2, 1, 0, 1], [], [1, 2, 3, 4], [2, 1, 0, 1], [], Yes)

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: dbc64eac6670f722bf7eb72b5d5cd437e89195dfdb483cfe452d944ca71a2d94
candidate record SHA-256: a654f7fd2978454645a679d3443b6bc043b79631165d4e1eff80901ba61cba82
semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77
question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705
Lean: NOT_RUN · human review: NOT_PERFORMED

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