## Screenshot: Lean Theorem Prover Code Snippet
### Overview
The image shows a code snippet from a Lean theorem prover environment, displaying a formal proof of a mathematical theorem. The code includes type declarations, function definitions, and tactic applications.
### Components/Axes
- **Header**: Three colored dots (red, yellow, green) in the top-left corner, typical of macOS window controls.
- **Code Content**:
- **Theorem Declaration**: `theorem mathd_algebra_270` (pink text).
- **Function Type**: `(f : ℝ → ℝ)` (white text with red `f`).
- **Hypothesis**: `(h₀ : ∀ x, x ≠ -2 → f x = 1 / (x + 2))` (pink `h₀`, white `∀`, red `x`, blue `≠`, yellow `-2`, red `f`, blue `=`, yellow `1`, gray `/`, red `(x + 2)`).
- **Function Application**: `f (f 1) = 3/7` (red `f`, white `(f 1)`, blue `=`, yellow `3/7`).
- **Tactic Application**: `set_option tactic.skipAssignedInstances false in norm_num [h₀]` (pink `set_option`, yellow `tactic.skipAssignedInstances`, red `false`, blue `in`, pink `norm_num`, gray `[h₀]`).
### Detailed Analysis
1. **Theorem Name**: `mathd_algebra_270` (pink text).
2. **Function Definition**: `f` is defined as a function from real numbers to real numbers (`ℝ → ℝ`).
3. **Hypothesis**: `h₀` asserts that for all real numbers `x` not equal to `-2`, `f(x)` equals `1/(x + 2)`.
4. **Function Application**: Applying `f` to `1` yields `3/7`, verified via Lean's normalization tactic.
5. **Tactic**: `set_option tactic.skipAssignedInstances false` disables skipping assigned instances during normalization, ensuring `h₀` is explicitly used.
### Key Observations
- The code uses Lean's syntax for formal proofs, with color-coded syntax highlighting (e.g., pink for keywords, blue for operators).
- The hypothesis `h₀` explicitly excludes `x = -2` to avoid division by zero in `1/(x + 2)`.
- The tactic `norm_num` simplifies numerical expressions, and `[h₀]` ensures the hypothesis is applied.
### Interpretation
This snippet demonstrates a formal proof in Lean where a function `f` is defined to satisfy `f(x) = 1/(x + 2)` for all `x ≠ -2`. The theorem `mathd_algebra_270` likely verifies a property of this function (e.g., `f(1) = 3/7`). The tactic `set_option tactic.skipAssignedInstances false` ensures the hypothesis `h₀` is explicitly referenced during normalization, preventing Lean from ignoring it. This highlights Lean's ability to enforce logical rigor by requiring explicit use of assumptions.
**Note**: No numerical trends or axes are present, as this is a code snippet rather than a chart or diagram. All textual content has been transcribed with color and syntax annotations.
