# Question 7427: Append Candidate Guard before New Table and Collisions [Return to catalogue page 38](https://ocme.evemisslab.com/en/questions/page/38/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261016-r06-injective-foundations/#q-20261012-advance-20261016-r06-injective-0052) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0013-07427 · q-20261012-advance-20261016-r06-injective-0052 ## Original question and givens 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. ## Definition and scope 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. - [mko-injective](https://ocme.evemisslab.com/en/mko/mko-injective/) ## Step-by-step derivation S001 · impliesBefore 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. Step identifiers and dependenciesRule: finite_injectivity_label_scope · Uses: G001 S002 · impliesBefore 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. Step identifiers and dependenciesRule: append_output_codomain_guard · Uses: G001, S001 S003 · impliesBefore phase=appended;all input labels=[0] After collisions=[];injective=Yes Conditions: none After all x