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R0013-07428 · q-20261012-advance-20261016-r06-injective-0053
Finite types start at 0: Fin(2)={0,…,1}→Fin(3), complete images=[0, 1]. 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=2. 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(2)={0,…,1}→Fin(3), complete images=[0, 1]. 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=2. 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.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(2)
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(2)
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
Fin(2)→Fin(3);original images=[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, S002
Before
append candidate value=2;old codomain Fin(3)
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.
Rule: append_output_codomain_guard · Uses: G001, S003
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.
Rule: injective_collision_pair_audit · Uses: G001, S004
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.
Rule: injective_collision_pair_audit · Uses: G001, S005
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.
Rule: injective_collision_pair_audit · Uses: G001, S006
Before
phase=appended;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
Before
original n=2;new input label=2;new table=[0, 1, 2]
After
new domain has 3 labels;codomain m=3
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.
Rule: derived_appended_domain_size · Uses: G001, S008
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.([0, 1], Yes, [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: bf550fa535cba4b63b6d904b3feab920c033515c9a404435eb9f356748b8695ccandidate record SHA-256: cea8625e09f5d4239c46a004dae480896ef4bc0b4fcecc536bf6880a093f50c4semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED