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R0013-07427 · q-20261012-advance-20261016-r06-injective-0052
Finite types start at 0: Fin(0)=∅→Fin(1), complete images=[]. 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=0. 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(0)=∅→Fin(1), complete images=[]. 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=0. 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
Fin(0)→Fin(1);original images=[]
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
Before
append candidate value=0;old codomain Fin(1)
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, S001
Before
phase=appended;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
original n=0;new input label=0;new table=[0]
After
new domain has 1 labels;codomain m=1
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, S003
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.([], Yes, [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: f580beafa4a27ec3d94310215b559d4167a6bd06a0bffb1e010be0ab25afb410candidate record SHA-256: 88e1859de0968fe7e14de61cd1df7fa21dc82a1cd31b8e39df014b93aca50e3fsemantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED