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Question 7455: Input Deletion with Original Collision Labels Retained

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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R0013-07455 · q-20261012-advance-20261016-r06-injective-0030

Original question and givens

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.

  • G001 — 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.

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

source index=0∈Fin(5)

After

image[0]=1∈Fin(3)

Conditions: none

First type-check every cell of the complete original table,not only cells queried later.

Step identifiers and dependencies

Rule: injective_complete_original_table_row · Uses: G001

S002 · implies

Before

source index=1∈Fin(5)

After

image[1]=0∈Fin(3)

Conditions: none

First type-check every cell of the complete original table,not only cells queried later.

Step identifiers and dependencies

Rule: injective_complete_original_table_row · Uses: G001, S001

S003 · implies

Before

source index=2∈Fin(5)

After

image[2]=0∈Fin(3)

Conditions: none

First type-check every cell of the complete original table,not only cells queried later.

Step identifiers and dependencies

Rule: injective_complete_original_table_row · Uses: G001, S002

S004 · implies

Before

source index=3∈Fin(5)

After

image[3]=0∈Fin(3)

Conditions: none

First type-check every cell of the complete original table,not only cells queried later.

Step identifiers and dependencies

Rule: injective_complete_original_table_row · Uses: G001, S003

S005 · implies

Before

source index=4∈Fin(5)

After

image[4]=1∈Fin(3)

Conditions: none

First type-check every cell of the complete original table,not only cells queried later.

Step identifiers and dependencies

Rule: injective_complete_original_table_row · Uses: G001, S004

S006 · implies

Before

Fin(5)→Fin(3);original images=[1, 0, 0, 0, 1]

After

injectivity forbids equal outputs from distinct original input labels

Conditions: none

Injectivity forbids equal outputs for distinct inputs;the empty function is vacuously injective,which is not surjectivity.Type-check the whole original table and retain original labels under restriction or deletion.

Step identifiers and dependencies

Rule: finite_injectivity_label_scope · Uses: G001, S005

S007 · implies

Before

phase=before_remove;inputs=(0,1);outputs=(1,0)

After

collision=No

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S006

S008 · implies

Before

phase=before_remove;inputs=(0,2);outputs=(1,0)

After

collision=No

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S007

S009 · implies

Before

phase=before_remove;inputs=(0,3);outputs=(1,0)

After

collision=No

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S008

S010 · implies

Before

phase=before_remove;inputs=(0,4);outputs=(1,1)

After

collision=Yes

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S009

S011 · implies

Before

phase=before_remove;inputs=(1,2);outputs=(0,0)

After

collision=Yes

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S010

S012 · implies

Before

phase=before_remove;inputs=(1,3);outputs=(0,0)

After

collision=Yes

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S011

S013 · implies

Before

phase=before_remove;inputs=(1,4);outputs=(0,1)

After

collision=No

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S012

S014 · implies

Before

phase=before_remove;inputs=(2,3);outputs=(0,0)

After

collision=Yes

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S013

S015 · implies

Before

phase=before_remove;inputs=(2,4);outputs=(0,1)

After

collision=No

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S014

S016 · implies

Before

phase=before_remove;inputs=(3,4);outputs=(0,1)

After

collision=No

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S015

S017 · implies

Before

phase=before_remove;all input labels=[0, 1, 2, 3, 4]

After

collisions=[[0, 4], [1, 2], [1, 3], [2, 3]];injective=No

Conditions: none

After all x<y pairs are checked,an empty collision list is equivalent to injectivity on this finite restriction.With0 or1 input,there are no distinct-input pairs.

Step identifiers and dependencies

Rule: complete_pairwise_collision_inventory · Uses: G001, S016

S018 · implies

Before

phase=after_remove;inputs=(0,2);outputs=(1,0)

After

collision=No

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S017

S019 · implies

Before

phase=after_remove;inputs=(0,3);outputs=(1,0)

After

collision=No

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S018

S020 · implies

Before

phase=after_remove;inputs=(0,4);outputs=(1,1)

After

collision=Yes

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S019

S021 · implies

Before

phase=after_remove;inputs=(2,3);outputs=(0,0)

After

collision=Yes

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S020

S022 · implies

Before

phase=after_remove;inputs=(2,4);outputs=(0,1)

After

collision=No

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S021

S023 · implies

Before

phase=after_remove;inputs=(3,4);outputs=(0,1)

After

collision=No

Conditions: none

Check each distinct original-label pair once;equal output values do not merge the two inputs.Deduplicating outputs must not erase a collision.

Step identifiers and dependencies

Rule: injective_collision_pair_audit · Uses: G001, S022

S024 · implies

Before

phase=after_remove;all input labels=[0, 2, 3, 4]

After

collisions=[[0, 4], [2, 3]];injective=No

Conditions: none

After all x<y pairs are checked,an empty collision list is equivalent to injectivity on this finite restriction.With0 or1 input,there are no distinct-input pairs.

Step identifiers and dependencies

Rule: complete_pairwise_collision_inventory · Uses: G001, S023

S025 · implies

Before

removed original input=1;retained labels=[0, 2, 3, 4]

After

remaining collisions=[[0, 4], [2, 3]];codomain m=3

Conditions: none

Delete one original input label without renumbering survivors;collision pairs keep original labels and the codomain remains unchanged.

Step identifiers and dependencies

Rule: removed_input_labels_not_renumbered · Uses: G001, S024

Reverse checks

  • S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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

([1, 0, 0, 0, 1], [[0, 4], [1, 2], [1, 3], [2, 3]], [0, 2, 3, 4], [[0, 4], [2, 3]], No)

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: 92fc7a7c0b0c940632cd3a3497647cb35a75f4d2f42c8adb3065ad14df462985
  • candidate record SHA-256: 61634f7df1ae6e466079de1ee5ab3851cd3900b5de00c41a9bee5527f2daf7ba
  • semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59
  • question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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