## Screenshot: Functional Programming Theorem Proof ### Overview The image shows a code snippet in a text editor with syntax highlighting. The code defines a theorem in a functional programming or proof assistant environment (likely SciLean/Haskell). Key elements include theorem declaration, type annotations, and proof steps. ### Components/Axes - **Window Controls**: Top-left corner with red (close), yellow (minimize), and green (maximize) dots. - **Code Structure**: - **Theorem Declaration**: `theorem ext (x y : Cont) : (∀ i, x[i] = y[i]) → x = y := by` - **Proof Steps**: - `intro h` - `apply SciLean.ArrayType.get_injective` - `simp only [h]` - **Syntax Highlighting**: - Keywords (`theorem`, `by`, `intro`, `apply`, `simp`) in red. - Variables (`x`, `y`, `i`, `h`) in green. - Operators (`→`, `:=`, `=`) in cyan. - Type annotations (`Cont`, `ArrayType`) in orange. ### Detailed Analysis 1. **Theorem Statement**: - **Parameters**: `x` and `y` are of type `Cont` (likely a continuation type). - **Hypothesis**: `∀ i, x[i] = y[i]` (element-wise equality for all indices `i`). - **Conclusion**: `x = y` (arrays are equal if all elements are equal). - **Proof Method**: Uses `by` to introduce a proof block. 2. **Proof Steps**: - `intro h`: Introduces the hypothesis `h` (the element-wise equality). - `apply SciLean.ArrayType.get_injective`: Applies a function from the SciLean library to prove injectivity of arrays. - `simp only [h]`: Simplifies the proof using the hypothesis `h`. ### Key Observations - The theorem asserts that array equality is equivalent to element-wise equality. - The proof leverages the `get_injective` function from SciLean, suggesting this is part of a formal verification or dependent type system. - Syntax highlighting distinguishes code elements by color, aiding readability. ### Interpretation This code demonstrates a formal proof in a dependent type system (e.g., Lean or SciLean) that two arrays are equal if and only if their corresponding elements are equal. The use of `get_injective` implies the library provides tools for reasoning about array properties. The proof structure (`intro`, `apply`, `simp`) follows standard tactics in proof assistants, where hypotheses are introduced, library functions are applied, and simplification is used to discharge goals. No numerical data or trends are present, as this is a textual proof snippet. The focus is on logical structure and type-theoretic reasoning rather than empirical data.