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Question 7431: Append Candidate Guard before New Table and Collisions

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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R0013-07431 · q-20261012-advance-20261016-r06-injective-0056

Original question and givens

Finite types start at 0: Fin(8)={0,…,7}→Fin(8), complete images=[0, 1, 2, 3, 4, 5, 6, 7]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=7. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

  • G001 — Finite types start at 0: Fin(8)={0,…,7}→Fin(8), complete images=[0, 1, 2, 3, 4, 5, 6, 7]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. At the new input label n, append candidate value=7. Give (original images, new-value codomain eligibility, new images or outside-codomain, new collisions or not-evaluated, new injectivity or not-evaluated). A valid append can create nine input labels.

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(8)

After

image[0]=0∈Fin(8)

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(8)

After

image[1]=1∈Fin(8)

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(8)

After

image[2]=2∈Fin(8)

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(8)

After

image[3]=3∈Fin(8)

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(8)

After

image[4]=4∈Fin(8)

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

source index=5∈Fin(8)

After

image[5]=5∈Fin(8)

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, S005

S007 · implies

Before

source index=6∈Fin(8)

After

image[6]=6∈Fin(8)

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, S006

S008 · implies

Before

source index=7∈Fin(8)

After

image[7]=7∈Fin(8)

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, S007

S009 · implies

Before

Fin(8)→Fin(8);original images=[0, 1, 2, 3, 4, 5, 6, 7]

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, S008

S010 · implies

Before

append candidate value=7;old codomain Fin(8)

After

qualified=Yes

Conditions: none

Require the new value to lie in the original codomain before constructing a table;an outside value leaves injectivity unevaluated,not False.

Step identifiers and dependencies

Rule: append_output_codomain_guard · Uses: G001, S009

S011 · implies

Before

phase=appended;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 dependencies

Rule: injective_collision_pair_audit · Uses: G001, S010

S012 · implies

Before

phase=appended;inputs=(0,2);outputs=(0,2)

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=appended;inputs=(0,3);outputs=(0,3)

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=appended;inputs=(0,4);outputs=(0,4)

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, S013

S015 · implies

Before

phase=appended;inputs=(0,5);outputs=(0,5)

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=appended;inputs=(0,6);outputs=(0,6)

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=appended;inputs=(0,7);outputs=(0,7)

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, S016

S018 · implies

Before

phase=appended;inputs=(0,8);outputs=(0,7)

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=appended;inputs=(1,2);outputs=(1,2)

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=appended;inputs=(1,3);outputs=(1,3)

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, S019

S021 · implies

Before

phase=appended;inputs=(1,4);outputs=(1,4)

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, S020

S022 · implies

Before

phase=appended;inputs=(1,5);outputs=(1,5)

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=appended;inputs=(1,6);outputs=(1,6)

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=appended;inputs=(1,7);outputs=(1,7)

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, S023

S025 · implies

Before

phase=appended;inputs=(1,8);outputs=(1,7)

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, S024

S026 · implies

Before

phase=appended;inputs=(2,3);outputs=(2,3)

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, S025

S027 · implies

Before

phase=appended;inputs=(2,4);outputs=(2,4)

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, S026

S028 · implies

Before

phase=appended;inputs=(2,5);outputs=(2,5)

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, S027

S029 · implies

Before

phase=appended;inputs=(2,6);outputs=(2,6)

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, S028

S030 · implies

Before

phase=appended;inputs=(2,7);outputs=(2,7)

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, S029

S031 · implies

Before

phase=appended;inputs=(2,8);outputs=(2,7)

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, S030

S032 · implies

Before

phase=appended;inputs=(3,4);outputs=(3,4)

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, S031

S033 · implies

Before

phase=appended;inputs=(3,5);outputs=(3,5)

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, S032

S034 · implies

Before

phase=appended;inputs=(3,6);outputs=(3,6)

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, S033

S035 · implies

Before

phase=appended;inputs=(3,7);outputs=(3,7)

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, S034

S036 · implies

Before

phase=appended;inputs=(3,8);outputs=(3,7)

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, S035

S037 · implies

Before

phase=appended;inputs=(4,5);outputs=(4,5)

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, S036

S038 · implies

Before

phase=appended;inputs=(4,6);outputs=(4,6)

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, S037

S039 · implies

Before

phase=appended;inputs=(4,7);outputs=(4,7)

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, S038

S040 · implies

Before

phase=appended;inputs=(4,8);outputs=(4,7)

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, S039

S041 · implies

Before

phase=appended;inputs=(5,6);outputs=(5,6)

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, S040

S042 · implies

Before

phase=appended;inputs=(5,7);outputs=(5,7)

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, S041

S043 · implies

Before

phase=appended;inputs=(5,8);outputs=(5,7)

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, S042

S044 · implies

Before

phase=appended;inputs=(6,7);outputs=(6,7)

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, S043

S045 · implies

Before

phase=appended;inputs=(6,8);outputs=(6,7)

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, S044

S046 · implies

Before

phase=appended;inputs=(7,8);outputs=(7,7)

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, S045

S047 · implies

Before

phase=appended;all input labels=[0, 1, 2, 3, 4, 5, 6, 7, 8]

After

collisions=[[7, 8]];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, S046

S048 · implies

Before

original n=8;new input label=8;new table=[0, 1, 2, 3, 4, 5, 6, 7, 7]

After

new domain has 9 labels;codomain m=8

Conditions: none

A legal append can extend8 original inputs to9 derived domain labels.Do not reapply the original n≤8 input-slice limit to this derived table,and do not expand the codomain.

Step identifiers and dependencies

Rule: derived_appended_domain_size · Uses: G001, S047

Reverse checks

  • S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S043 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S041 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S033 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S031 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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

([0, 1, 2, 3, 4, 5, 6, 7], Yes, [0, 1, 2, 3, 4, 5, 6, 7, 7], [[7, 8]], 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: 56613c764efc3d619cd7b70d2c907e2ea0eb22b7b37ebe3cdc56cf3007c3591a
  • candidate record SHA-256: 7f4efb6e46b91a7d33eecb06af7a1dabdab9a7e5e10d3c818e5e4254ccfd26f7
  • semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59
  • question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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