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Question 7407: Every Distinct-Input Collision and Injectivity

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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R0013-07407 · q-20261012-advance-20261016-r06-injective-0007

Original question and givens

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.

  • G001 — 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.

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(2)

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]=1∈Fin(2)

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(2)

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(2)

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]=0∈Fin(2)

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(2);original images=[1, 1, 0, 0, 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 dependencies

Rule: finite_injectivity_label_scope · Uses: G001, S005

S007 · implies

Before

phase=original;inputs=(0,1);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, S006

S008 · implies

Before

phase=original;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=original;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=original;inputs=(0,4);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 dependencies

Rule: injective_collision_pair_audit · Uses: G001, S009

S011 · implies

Before

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 dependencies

Rule: injective_collision_pair_audit · Uses: G001, S010

S012 · implies

Before

phase=original;inputs=(1,3);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, S011

S013 · implies

Before

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 dependencies

Rule: injective_collision_pair_audit · Uses: G001, S012

S014 · implies

Before

phase=original;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=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 dependencies

Rule: injective_collision_pair_audit · Uses: G001, S014

S016 · implies

Before

phase=original;inputs=(3,4);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 dependencies

Rule: injective_collision_pair_audit · Uses: G001, S015

S017 · implies

Before

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

After

collisions=[[0, 1], [2, 3], [2, 4], [3, 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 dependencies

Rule: 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

([1, 1, 0, 0, 0], [[0, 1], [2, 3], [2, 4], [3, 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: 340534baabc345afd7f7b1a50937907578bb11d3b6c71d37bc1fcdbd10ff2591
  • candidate record SHA-256: 3615e4748021460b82cc02a16a3d1ae4632606b50543e8bb8a4a5526262b96cd
  • semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59
  • question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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