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R0013-07451 · q-20261012-advance-20261016-r06-injective-0026
Finite types start at 0: Fin(1)={0,…,0}→Fin(1), complete images=[0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.
G001 — Finite types start at 0: Fin(1)={0,…,0}→Fin(1), complete images=[0]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=0, keeping the codomain. Give (original images, original collisions, retained original input labels, later collisions, later injectivity), without renumbering inputs.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(1)
After
image[0]=0∈Fin(1)
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
Fin(1)→Fin(1);original images=[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.
Rule: finite_injectivity_label_scope · Uses: G001, S001
Before
phase=before_remove;all input labels=[0]
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, S002
Before
phase=after_remove;all input labels=[]
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, S003
Before
removed original input=0;retained labels=[]
After
remaining collisions=[];codomain m=1
Conditions: none
Delete one original input label without renumbering survivors;collision pairs keep original labels and the codomain remains unchanged.
Rule: removed_input_labels_not_renumbered · Uses: G001, S004
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], [], [], [], 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: bbe1c30bf8d6b4f76ef5e7e4afdc565d68e963b83db01a193be6359b4cd1c310candidate record SHA-256: 9a162853a8172e9828aecdabebac2b3d29cd698d5a926d5c25efc9b1f327982asemantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED