# Question 7414: Every Distinct-Input Collision and Injectivity

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0013-07414 · q-20261012-advance-20261016-r06-injective-0014



## Original question and givens

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.


- G001 — 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.





## 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-injective](https://ocme.evemisslab.com/en/mko/mko-injective/)





## Step-by-step derivation

S001 · impliesBefore

source index=0∈Fin(5)

After

image[0]=0∈Fin(2)

Conditions: none

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

Step identifiers and dependenciesRule: injective_complete_original_table_row · Uses: G001





S002 · impliesBefore

source index=1∈Fin(5)

After

image[1]=1∈Fin(2)

Conditions: none

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

Step identifiers and dependenciesRule: injective_complete_original_table_row · Uses: G001, S001





S003 · impliesBefore

source index=2∈Fin(5)

After

image[2]=0∈Fin(2)

Conditions: none

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

Step identifiers and dependenciesRule: injective_complete_original_table_row · Uses: G001, S002





S004 · impliesBefore

source index=3∈Fin(5)

After

image[3]=1∈Fin(2)

Conditions: none

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

Step identifiers and dependenciesRule: injective_complete_original_table_row · Uses: G001, S003





S005 · impliesBefore

source index=4∈Fin(5)

After

image[4]=0∈Fin(2)

Conditions: none

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

Step identifiers and dependenciesRule: injective_complete_original_table_row · Uses: G001, S004





S006 · impliesBefore

Fin(5)→Fin(2);original images=[0, 1, 0, 1, 0]

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 dependenciesRule: finite_injectivity_label_scope · Uses: G001, S005





S007 · impliesBefore

phase=original;inputs=(0,1);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 dependenciesRule: injective_collision_pair_audit · Uses: G001, S006





S008 · impliesBefore

phase=original;inputs=(0,2);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 dependenciesRule: injective_collision_pair_audit · Uses: G001, S007





S009 · impliesBefore

phase=original;inputs=(0,3);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 dependenciesRule: injective_collision_pair_audit · Uses: G001, S008





S010 · impliesBefore

phase=original;inputs=(0,4);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 dependenciesRule: injective_collision_pair_audit · Uses: G001, S009





S011 · impliesBefore

phase=original;inputs=(1,2);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 dependenciesRule: injective_collision_pair_audit · Uses: G001, S010





S012 · impliesBefore

phase=original;inputs=(1,3);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 dependenciesRule: injective_collision_pair_audit · Uses: G001, S011





S013 · impliesBefore

phase=original;inputs=(1,4);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 dependenciesRule: injective_collision_pair_audit · Uses: G001, S012





S014 · impliesBefore

phase=original;inputs=(2,3);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 dependenciesRule: injective_collision_pair_audit · Uses: G001, S013





S015 · impliesBefore

phase=original;inputs=(2,4);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 dependenciesRule: injective_collision_pair_audit · Uses: G001, S014





S016 · impliesBefore

phase=original;inputs=(3,4);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 dependenciesRule: injective_collision_pair_audit · Uses: G001, S015





S017 · impliesBefore

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

After

collisions=[[0, 2], [0, 4], [1, 3], [2, 4]];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 dependenciesRule: complete_pairwise_collision_inventory · Uses: G001, S016









## Reverse checks


- 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

([0, 1, 0, 1, 0], [[0, 2], [0, 4], [1, 3], [2, 4]], 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: 547254cb52da68f3a0dcab9d6fc00a30ef6c1d35de2fc0ce17ac6d174b7cf708
- candidate record SHA-256: cb4ec0eae438623a2ab8dcc650f8023621435fe468288cda4fac31822be54a32
- semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59
- question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261016-r06-injective-0014.json)