## Code Snippet: Theorem Definition ### Overview The image shows a code snippet, likely from a formal verification system like Isabelle or Coq. It defines a theorem named `ContCDiffMapFD_eta` and provides a proof using simplification and the `aesop` tactic. ### Components/Axes * **Header:** The top-left of the code block contains three colored circles: red, yellow, and green. These are likely indicators of the code editor's status (e.g., compilation status). * **Code Body:** The main part of the image contains the code itself, formatted with different colors to highlight keywords and variables. ### Detailed Analysis or ### Content Details The code snippet contains the following: * `theorem ContCDiffMapFD_eta (f : X -->FD[K,n] Y) : (fun x -->FD [K,n] f x) = f := by` * This line declares a theorem named `ContCDiffMapFD_eta`. * It states that `f` is a function from `X` to `FD[K,n] Y`. The arrow `-->` likely represents a function type constructor. * The theorem asserts that applying `f` to `x` and then constructing a function from `x` to that result is equivalent to `f` itself. * The `:= by` indicates the start of the proof. * `simp only [DFunLike.ext_iff]` * This line uses the `simp` tactic (likely a simplification tactic) with the `only` modifier. * It specifies that the simplification should only use the `DFunLike.ext_iff` rule, which is likely an extensionality rule for function-like structures. * `aesop` * This line invokes the `aesop` tactic, which is an automated proof search tactic. ### Key Observations * The code uses specific notation (`-->`, `FD[K,n]`) that is likely defined within the context of the formal verification system. * The proof is relatively short, suggesting that the theorem is either straightforward or relies on powerful automated tactics. ### Interpretation The code snippet defines a theorem related to continuous differentiable maps (indicated by `ContCDiffMapFD`). The theorem likely expresses a property of function extensionality in the context of these maps. The proof uses simplification with a specific extensionality rule and then relies on the `aesop` tactic to complete the proof. This suggests that the theorem is provable using standard techniques within the formal verification system.

\n ## Screenshot: Mathematical Theorem Statement ### Overview The image is a screenshot of a dark-themed text editor or terminal window displaying a mathematical theorem statement, likely from a formal verification or proof assistant system. The statement is written in a symbolic notation, and appears to be related to functional differentiation. There are three colored dots in the top-left corner. ### Components/Axes There are no axes or traditional chart components. The visible elements are: * **Colored Dots:** Three dots in the top-left corner, colored red, orange, and green (from left to right). Their purpose is unclear without further context. * **Theorem Statement:** A single line of text containing a theorem name and its formal definition. * **Commands:** Two lines of text below the theorem statement, "simp only [DFunLike.ext_ifff]" and "aesop", which appear to be commands used within the system to simplify or prove the theorem. ### Detailed Analysis or Content Details The theorem statement is: `theorem ContCDiffMapFD_eta (f : X →FD[K,n] Y) : (fun x ↦FD[K,n] f x) = f := by` The commands are: `simp only [DFunLike.ext_ifff]` `aesop` ### Key Observations The theorem name is `ContCDiffMapFD_eta`. The theorem involves a function `f` mapping from `X` to `FD[K,n] Y`. The statement asserts the equality of two expressions involving `f`. The commands suggest a simplification step using `DFunLike.ext_ifff` and a subsequent proof attempt using `aesop`. ### Interpretation The image represents a step in a formal mathematical proof or verification process. The theorem likely deals with continuous differentiation of a function `f` within a specific framework denoted by `FD[K,n]`. The `simp only` command indicates an attempt to simplify the theorem using a specific equality criterion (`DFunLike.ext_ifff`). The `aesop` command suggests the use of an automated theorem prover to attempt to complete the proof. The colored dots in the top-left corner are likely status indicators or visual cues within the environment, but their exact meaning is unknown without additional context. The overall context suggests a highly technical and formal environment for mathematical reasoning.

\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.

## Screenshot: Lean Theorem Definition with Syntax Highlighting ### Overview The image shows a terminal window displaying a Lean theorem definition with syntax highlighting. The code defines a theorem named `ContCDiffMapFD_eta` involving a function `f` and its properties. Key elements include type annotations, function definitions, and a simplification tactic (`simp only`). ### Components/Axes - **Window Controls**: Top-left corner with red, yellow, and green dots (standard macOS window controls). - **Code Structure**: - **Theorem Name**: `theorem ContCDiffMapFD_eta` (red text). - **Parameters**: - `f : X → FD[K,n]` (yellow text). - `Y : (fun x → FD[K,n] x) = f` (purple text). - **Simplification Clause**: `simp only [DFunLike.ext_iff]` (red text). - **Comment**: `aesop` (red text, likely a placeholder or note). - **Syntax Highlighting**: - Keywords (e.g., `theorem`, `simp only`): Red. - Variables/Types (e.g., `X`, `Y`, `FD[K,n]`): Yellow. - Function Definitions (e.g., `fun x → ...`): Purple. - Equality/Assignment (`=`, `:=`): Cyan. - Comments: Green. ### Detailed Analysis 1. **Theorem Definition**: - `theorem ContCDiffMapFD_eta (f : X → FD[K,n]) (Y : (fun x → FD[K,n] x) = f) : ...` - Defines a theorem `ContCDiffMapFD_eta` with two hypotheses: - `f` maps `X` to `FD[K,n]`. - `Y` is a function equal to `f`. - The ellipsis (`...`) indicates the theorem’s body is omitted. 2. **Simplification Tactic**: - `simp only [DFunLike.ext_iff]`: - Applies the `simp` tactic to simplify the proof using only the lemma `DFunLike.ext_iff`. 3. **Comment**: - `aesop`: Likely a placeholder or internal note (e.g., a reference to a person or project). ### Key Observations - The code uses Lean’s type theory syntax, with explicit type annotations and function definitions. - The `simp only` clause restricts simplification to a specific lemma, suggesting a focused proof strategy. - The comment `aesop` is isolated and lacks context, possibly indicating an incomplete or draft state. ### Interpretation This code snippet defines a theorem in Lean, a formal proof assistant, asserting properties of a function `f` under specific conditions. The use of `simp only` indicates the proof relies on a single lemma (`DFunLike.ext_iff`), which may relate to extending functions or properties in a formal system. The comment `aesop` could hint at a collaborative or contextual note, but its meaning is unclear without additional context. The syntax highlighting emphasizes Lean’s structured approach to formalizing mathematical proofs.