Explore related knowledge and questions →
← Question derivation index · Original question batch
R0013-07403 · q-20261012-advance-20261016-r06-injective-0003
Finite types start at 0: Fin(3)={0,…,2}→Fin(3), complete images=[0, 1, 2]. 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(3)={0,…,2}→Fin(3), complete images=[0, 1, 2]. 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.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.
Before
source index=0∈Fin(3)
After
image[0]=0∈Fin(3)
Conditions: none
First type-check every cell of the complete original table,not only cells queried later.
Rule: injective_complete_original_table_row · Uses: G001
Before
source index=1∈Fin(3)
After
image[1]=1∈Fin(3)
Conditions: none
First type-check every cell of the complete original table,not only cells queried later.
Rule: injective_complete_original_table_row · Uses: G001, S001
Before
source index=2∈Fin(3)
After
image[2]=2∈Fin(3)
Conditions: none
First type-check every cell of the complete original table,not only cells queried later.
Rule: injective_complete_original_table_row · Uses: G001, S002
Before
Fin(3)→Fin(3);original images=[0, 1, 2]
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.
Rule: finite_injectivity_label_scope · Uses: G001, S003
Before
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.
Rule: injective_collision_pair_audit · Uses: G001, S004
Before
phase=original;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.
Rule: injective_collision_pair_audit · Uses: G001, S005
Before
phase=original;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.
Rule: injective_collision_pair_audit · Uses: G001, S006
Before
phase=original;all input labels=[0, 1, 2]
After
collisions=[];injective=Yes
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.
Rule: complete_pairwise_collision_inventory · Uses: G001, S007
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.([0, 1, 2], [], Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 62aa8e4accf8cd3a49382535760621a69eef108e0efa6b47f8ec9873d312234ecandidate record SHA-256: 44eb3d440018557dd51720c53ee9e306acf0f5a1f350af99eb9cd08d9e0701eesemantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED