# Formal statements in a proof assistant

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A formal statement is a core term of type Prop obtained by successful elaboration in a fixed version, import environment and local context.

mko-lean-formal-statement · definition · v0.10.0



## Mathematical statement

In a fixed Lean version, with fixed imports, name environment and local context Γ, source syntax s expresses a formal statement if it elaborates successfully to a core term S with Γ⊢S:Prop. A statement and the existence of its checked proof term are separate states.





## Representative formula (exact TeX)

S\in P





## Assumptions


- Lean version, imports, declared names and local context are part of formal-statement identity.
- The source pulse recorded only explicit pipeline receipts while Lean builds were paused by owner instruction; synthetic states are not actual elaboration or kernel checks.





## Symbol scope


- S — elaborated core statement term
- P — represents the proposition sort Prop





## Intuition, definition, and misconceptions

Parsing, elaboration, type checking, construction of a proof term and kernel checking are different stages. A string resembling Lean syntax need not elaborate in the specified environment.

The same syntax can parse or elaborate differently, or fail, when imports, namespaces, notation or local variables differ. A receipt must retain the environment identity.

An elaborated Prop is only a statement to be proved. Claiming its formal proof additionally requires a corresponding proof term and a kernel check.

Successful parsing, successful elaboration, type Prop and a kernel-checked proof are not the same achievement.





## Adopted derivation guide

[Formal statements in a proof assistant: reading a Lean statement layer by layer and separating 'is a statement' from 'is proved'](https://ocme.evemisslab.com/en/derivations/derivation-mko-lean-formal-statement-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Proposition](https://ocme.evemisslab.com/en/mko/mko-proposition/) — The core term representing a formal statement must have propositional type.





## Verification boundary

The source pulse did not execute a Lean build. These exercises check only explicit synthetic pipeline records and state logic; they create and claim no formal Evidence.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

When Lean work for this object is resumed, pin the toolchain/import closure and establish theorem identity, elaboration and kernel receipts for each statement.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/lean-formal-statement.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-lean-formal-statement/index.md)