Mathematical proof

Mathematical statement

Given a proposition P to be proved, a proof must derive P within a specified context from accepted premises, definitions, axioms, or earlier results by valid inference.

Representative formula (exact TeX)

P

Assumptions

• The proposition to be proved, the usable premises, and the inference context are explicitly stated.

Symbol scope

• P — proposition to be proved

Intuition, definition, and misconceptions

A proof does not merely accumulate supporting examples; it explains why the conclusion necessarily follows from the stated conditions. One valid counterexample can refute a universal proposition. Finite positive examples cannot establish a general proposition unless the domain is proved finite, the enumeration complete, and every case verified.

A direct proof moves from premises toward a conclusion. A proof by contradiction adds the negation of the conclusion and derives a contradiction. Any method must still respect the rules available in its logical system.

A formal proof submits the statement and each inference step to a formal system. An informal proof must likewise let readers trace its premises, reasoning, and conclusion.

Common mistakes include treating numerical samples, the appearance of a diagram, an appeal to authority, or unexplained intuition as a general proof.

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

• Logical connectives — A proof must identify the proposition to be proved and its logical structure.

Verification boundary

Finite tests or examples without a proof of a finite domain and complete enumeration support only their stated instances. Even exhaustive checking requires review of the completeness argument and theorem statement. If automated search produces a general proof, its theorem statement, premises, inference rules, and proof object must still be checked.

not_formalized

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

Next obligation

Choose a concrete theorem statement and bind its premises, conclusion, Lean proof term, and reproducible checking 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.