Formal statements in a proof assistant

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• 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.