## Screenshot: Coq Proof Assistant Code
### Overview
The image shows a code snippet from a proof assistant (likely Coq) defining a formal theorem related to quadratic reciprocity. The code uses syntax highlighting with color-coded keywords, variables, and mathematical notation. The theorem is structured with hypotheses, facts, and a conclusion.
### Components/Axes
- **Window Controls**: Top-left corner has three circular buttons (red, yellow, green) typical of macOS window management.
- **Code Structure**:
- **Theorem Declaration**: `theorem quadratic_reciprocity_2 (p q : ℕ) (hp : p ≠ 2) (hq : q ≠ 2)`
- **Parameters**: `p`, `q` (natural numbers)
- **Hypotheses**: `hp` (p ≠ 2), `hq` (q ≠ 2)
- **Facts**:
- `[Fact (Nat.Prime p)]` and `[Fact (Nat.Prime q)]` (asserting primality of `p` and `q`)
- **Equation**:
- `(legendre_sym p q) * (legendre_sym q p) = -1 ^ ((p-1) / 2 * (q-1) / 2)`
- Uses Legendre symbol notation and exponentiation.
- **Conclusion**:
- `exact book.quadratic_reciprocity.quadratic_reciprocity_1 p q hp hq` (invokes a library lemma).
### Detailed Analysis
- **Syntax Highlighting**:
- Keywords (e.g., `theorem`, `Fact`) in **red**.
- Identifiers (e.g., `quadratic_reciprocity_2`, `legendre_sym`) in **yellow**.
- Mathematical symbols (e.g., `ℕ`, `≠`, `^`) in **white**.
- Operators (e.g., `*`, `/`, `=`) in **cyan**.
- **Mathematical Content**:
- The theorem formalizes the **Quadratic Reciprocity Law**, a foundational result in number theory.
- The equation relates the Legendre symbols `(p|q)` and `(q|p)` via a power of `-1` dependent on `(p-1)/2` and `(q-1)/2`.
### Key Observations
- The code enforces that `p` and `q` are primes **not equal to 2** (hypotheses `hp` and `hq`).
- The use of `exact` suggests the theorem is proven by directly applying a pre-existing lemma from the `book.quadratic_reciprocity` library.
- The Legendre symbol multiplication and exponentiation align with the standard statement of quadratic reciprocity.
### Interpretation
This code represents a formal verification of the Quadratic Reciprocity Law in a proof assistant. The theorem ensures that for odd primes `p` and `q`, the product of their Legendre symbols equals `(-1)` raised to a specific power, reflecting the reciprocity relationship. The use of hypotheses (`hp`, `hq`) and facts (`Nat.Prime`) demonstrates how the proof assistant enforces logical dependencies and mathematical axioms. The reference to `book.quadratic_reciprocity_1` indicates modularity, leveraging existing verified results to construct new proofs. This snippet exemplifies how formal methods can rigorously validate classical mathematical theorems.
