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R0013-07477 · q-20261012-advance-20261016-r06-injective-0077
Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[2, 0] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).
G001 — Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Original subset=[2, 0] is declared a distinct set of legal source labels. Give (original images, sorted subset, every within-subset collision, restricted injectivity).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(2)
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]=0∈Fin(2)
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]=1∈Fin(2)
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(2);original images=[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.
Rule: finite_injectivity_label_scope · Uses: G001, S003
Before
phase=restriction;inputs=(0,2);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=restriction;all input labels=[0, 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, S005
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, 0, 1], [0, 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: 34ce937f769d514498dd56764452f4f06cbca557b9c61788a6136fc3165f1783candidate record SHA-256: 60e3e96a5f22b9ba92d658f150b5e03d64ff6a41bddf1e2a25d5e3c49d2d071fsemantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED