\n ## Screenshot: Lean 4 Theorem Statement ### Overview The image is a screenshot of a code editor or terminal window displaying a single theorem statement written in the Lean 4 proof assistant language. The window has a dark theme with syntax highlighting. The text is presented in a monospaced font against a dark gray background. The window itself is centered on a light gray background and features three colored window control buttons (red, yellow, green) in its top-left corner. ### Components/Axes * **Window Frame:** A dark gray rectangular window with rounded corners and a subtle drop shadow. * **Window Controls:** Three circular buttons in the top-left corner of the window, from left to right: red, yellow, green. * **Code Content:** The primary content is a three-line block of code with syntax highlighting. The colors appear to be: * Keywords (`theorem`, `by`, `simp`, `only`, `aesop`): Pink/Magenta. * Theorem name (`ContCDiffMapFD_eta`): Light blue/cyan. * Type constructors and certain identifiers (`FD[K,n]`, `DFunLike.ext_iff`): Yellow/Orange. * Variables and other text (`f`, `X`, `Y`, `x`): Light gray/white. * Punctuation and operators (`:`, `→`, `(`, `)`, `[`, `]`, `:=`): Light blue/cyan. ### Detailed Analysis / Content Details The code is transcribed precisely below, preserving line breaks and indentation. **Line 1:** `theorem ContCDiffMapFD_eta (f : X →FD[K,n] Y) : (fun x ↦FD[K,n] f x) = f := by` * **Theorem Declaration:** `theorem ContCDiffMapFD_eta` * **Parameter:** `(f : X →FD[K,n] Y)` - Declares a variable `f` which is a function from type `X` to type `Y`, specifically a `FD[K,n]` map (likely denoting a class of continuously differentiable functions of order `n` with `K` as a coefficient or parameter). * **Statement:** `(fun x ↦FD[K,n] f x) = f` - The theorem asserts that the function which takes an input `x` and applies `f` to it (with the `FD[K,n]` annotation) is equal to the original function `f`. This is an "eta-expansion" property. * **Proof Start:** `:= by` - Indicates the beginning of the proof script. **Line 2:** ` simp only [DFunLike.ext_iff]` * **Proof Step:** An indented line containing the tactic `simp only [DFunLike.ext_iff]`. * **Tactic:** `simp` is a simplification tactic. * **Argument:** `only [DFunLike.ext_iff]` instructs the simplifier to use only the lemma `DFunLike.ext_iff`. This lemma likely states that two functions of a `DFunLike` type (a Lean typeclass for dependent function-like types) are equal if and only if they are equal for all inputs. **Line 3:** `aesop` * **Proof Step:** A final, unindented line containing the tactic `aesop`. * **Tactic:** `aesop` is an automated tactic that attempts to solve the goal using a predefined set of rules and strategies. In this context, after applying the simplification rule, `aesop` likely discharges the remaining, now trivial, goal. ### Key Observations 1. **Syntax Highlighting:** The code uses a consistent color scheme to distinguish language keywords, identifiers, and operators, aiding readability. 2. **Proof Structure:** The proof is concise, consisting of two tactics. The first (`simp`) reduces the goal to a simpler form using a specific extensionality lemma, and the second (`aesop`) automatically solves the simplified goal. 3. **Mathematical Content:** The theorem `ContCDiffMapFD_eta` formalizes a basic property of function equality in the context of a specific function type (`FD[K,n]`). It proves that explicitly wrapping a function call in a lambda abstraction is redundant. ### Interpretation This image captures a snippet of formal mathematics being verified in a proof assistant. The theorem and its proof are not just text but machine-checked logical statements. * **What it demonstrates:** The code proves a fundamental "eta-equivalence" property for a custom function type `FD[K,n]`. In simpler terms, it confirms that the system correctly recognizes that the operation `λx. f(x)` is identical to `f` itself for functions of this type. This is a basic sanity check and a necessary lemma for more complex proofs. * **How elements relate:** The theorem statement defines the logical claim. The proof script (`simp only [DFunLike.ext_iff]` followed by `aesop`) provides the machine-verifiable evidence. The use of `DFunLike.ext_iff` suggests the `FD[K,n]` type is built upon Lean's standard dependent function type infrastructure. * **Notable aspects:** The proof's brevity indicates that this property is considered straightforward within the formal system's library. The reliance on `aesop` for the final step shows the power of automation in modern proof assistants for handling routine logical steps once the correct simplification rule is applied. The screenshot itself is likely from a developer's environment, showing work in progress on a formalization project involving analysis or differential geometry.