# Proof term

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

A proof term t has propositional type P in a fixed formal environment and context, and is checked by the kernel.

mko-lean-proof-term · concept · v0.10.0



## Mathematical statement

Fix a Lean version, imports, environment E and local context Γ. If core term t has propositional type P and the kernel accepts Γ⊢t:P, then t is a proof term for P. A tactic log is a construction trace, not a saved, replayable proof-term receipt.





## Representative formula (exact TeX)

T(t)=P





## Assumptions


- P has already been elaborated as a proposition in the same environment; term, context, imports and toolchain are fixed.
- The source pulse did not run Lean builds. Its question bank uses a synthetic small propositional type system and receipt flags, producing no Lean formal Evidence.





## Symbol scope


- T — type assignment
- t — candidate proof term
- P — propositional type





## Intuition, definition, and misconceptions

Under propositions-as-types, a proof of P is a term of type P. Assumptions, function application, pair construction and projections obey context-dependent type constraints.

A successful tactic with no remaining goals generally produces a candidate proof term. A receipt additionally requires saving that term and checking it with the specified kernel and environment.

Kernel acceptance establishes a formal judgment only. Fidelity to the original Chinese statement and inappropriate axioms in imports are separate semantic and trust obligations.

A tactic transcript, zero goals, successful compilation or model self-report is not the same as a recorded kernel-accepted proof term.





## Adopted derivation guide

[Proof term: reading λh. ⟨snd h, fst h⟩ part by part as a proof term for P∧Q→Q∧P](https://ocme.evemisslab.com/en/derivations/derivation-mko-lean-proof-term-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Formal statements in a proof assistant](https://ocme.evemisslab.com/en/mko/mko-lean-formal-statement/) — First fix P and the identity of its environment.
- [Mathematical proof](https://ocme.evemisslab.com/en/mko/mko-proof/) — A proof term is a proof instance in a formal system.





## Verification boundary

No Lean execution occurred in the source pulse; synthetic type inference is not a Lean kernel receipt and does not become 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 the specified theorem is resumed, pin term, type, environment, imports, axiom inventory and kernel receipt.





## 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-proof-term.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-lean-proof-term/index.md)