## Code Snippet: Lemma Definition ### Overview The image shows a code snippet, likely from a formal verification system like Isabelle or Coq. It defines a lemma named `condRho_of_translate`. The code specifies types, measure spaces, functions, and a simplification rule. ### Components/Axes * **Keywords**: `lemma`, `Type*`, `MeasureSpace`, `Finset`, `fun`, `condRho`, `simp`, `only`, `by` * **Variables**: `Ω`, `S`, `X`, `Y`, `A`, `s`, `ω` * **Operators**: `→`, `+`, `:=` ### Detailed Analysis or ### Content Details The code snippet can be transcribed as follows: ``` lemma condRho_of_translate {Ω S : Type*} [MeasureSpace Ω] (X : Ω → G) (Y : Ω → S) (A : Finset G) (s:G) : condRho (fun ω ↦ X ω + s) Y A = condRho X Y A := by simp only [condRho, rho_of_translate] ``` Breaking down the code: * `lemma condRho_of_translate`: Declares a lemma named `condRho_of_translate`. * `{Ω S : Type*}`: Introduces type variables `Ω` and `S`, both of type `Type*`. * `[MeasureSpace Ω]`: Assumes that `Ω` is a measure space. * `(X : Ω → G)`: Declares a function `X` from `Ω` to `G`. Note: `G` is not explicitly declared, but is likely a group. * `(Y : Ω → S)`: Declares a function `Y` from `Ω` to `S`. * `(A : Finset G)`: Declares `A` as a finite set (Finset) of elements from `G`. * `(s:G)`: Declares `s` as an element of `G`. * `: condRho (fun ω ↦ X ω + s) Y A = condRho X Y A := by`: States the property to be proven. It involves a function `condRho` applied to different arguments. The left side applies `condRho` to a function that maps `ω` to `X ω + s`, along with `Y` and `A`. The right side applies `condRho` to `X`, `Y`, and `A`. The `:= by` indicates that the proof follows. * `simp only [condRho, rho_of_translate]`: Indicates that the proof is done by simplification, using the definitions of `condRho` and `rho_of_translate`. ### Key Observations * The code uses mathematical notation within a programming context. * The lemma relates `condRho` applied to a translated function with `condRho` applied to the original function. * The proof is a simple simplification. ### Interpretation The lemma `condRho_of_translate` likely expresses a property about the invariance of some conditional density or measure (`condRho`) under translation. The translation is represented by adding `s` to the function `X ω`. The simplification step suggests that the definitions of `condRho` and `rho_of_translate` are sufficient to prove this invariance. The code is written in a style common to interactive theorem provers, where the system checks the validity of the proof steps.

\n ## Screenshot: Code Snippet ### Overview The image is a screenshot of a dark-themed code editor displaying a line of code, likely from a formal verification or mathematical software environment. There are three colored dots in the top-left corner. ### Components/Axes There are no axes or traditional chart components. The visible elements are: * **Code:** A single line of code. * **Colored Dots:** Three dots positioned in the top-left corner: Red, Orange, and Green (from left to right). ### Detailed Analysis or Content Details The code snippet reads: ``` lemma condRho_of_translate {Ω S : Type*} [MeasureSpace Ω] (X : Ω → G) (Y : Ω → S) (A : Finset G) (s:G) : condRho (fun ω → X ω + s) Y A = condRho X Y A := by simp only [condRho, rho_of_translate] ``` The code appears to define a lemma named `condRho_of_translate`. It involves types `Type`, `MeasureSpace`, `Finset`, and functions `condRho`, `rho_of_translate`. The code uses symbolic notation common in formal mathematics and computer science. ### Key Observations * The code is syntactically correct, suggesting it's from a well-defined programming or formal language. * The presence of `lemma` indicates this is likely part of a proof or theorem within a formal system. * The use of `simp only` suggests a simplification step in a proof. ### Interpretation The code snippet defines a lemma relating the conditional expectation (`condRho`) of a function after a translation. Specifically, it states that the conditional expectation of `Xω + s` with respect to `A` and `Y` is equal to the conditional expectation of `Xω` with respect to `A` and `Y`. This is a standard result in measure theory and probability. The `simp only` command indicates that the proof of this lemma relies on simplifying the expression using the definitions of `condRho` and `rho_of_translate`. The colored dots in the top-left corner are likely status indicators or visual cues within the code editor. Without further context, their exact meaning is unclear, but they could represent: * Compilation status (Red: Error, Orange: Warning, Green: Success) * Debugging state * Version control status

## Code Snippet: Lean 4 Mathematical Lemma ### Overview The image displays a screenshot of a code editor window (dark theme, macOS-style traffic light buttons in the top-left) containing a single mathematical lemma written in the Lean 4 proof assistant language. The lemma concerns the invariance of a conditional rho function (`condRho`) under translation of one of its arguments. ### Components/Axes * **Window Frame**: A dark gray (`#2d2d2d` approximate) rectangular window with rounded corners, centered on a light gray (`#b0b8c4` approximate) background. * **Window Controls**: Three circular buttons in the top-left corner of the window, from left to right: red, yellow, green. * **Code Content**: Monospaced text with syntax highlighting. The primary text color is a light gray/off-white. Keywords (`lemma`, `by`, `simp`) are highlighted in a pink/magenta color. Type names (`Type*`, `MeasureSpace`, `Finset`, `G`, `S`) are in a light blue. The proof tactic `simp` and its arguments are in a purple/violet shade. ### Detailed Analysis / Content Details The code is a single, complete Lean 4 lemma and its proof. The transcription is as follows: ```lean lemma condRho_of_translate {Ω S : Type*} [MeasureSpace Ω] (X : Ω → G) (Y : Ω → S) (A : Finset G) (s:G) : condRho (fun ω ↦ X ω + s) Y A = condRho X Y A := by simp only [condRho, rho_of_translate] ``` **Breakdown of the Statement:** 1. **Lemma Name**: `condRho_of_translate` 2. **Implicit Parameters**: `{Ω S : Type*}` - Declares `Ω` and `S` as arbitrary types. 3. **Type Class Instance**: `[MeasureSpace Ω]` - Assumes `Ω` is equipped with a measure space structure. 4. **Function Parameters**: * `(X : Ω → G)`: A function `X` from `Ω` to an additive group `G`. * `(Y : Ω → S)`: A function `Y` from `Ω` to type `S`. 5. **Additional Parameters**: * `(A : Finset G)`: A finite subset `A` of the group `G`. * `(s:G)`: An element `s` of the group `G`. 6. **Main Claim (Theorem)**: `condRho (fun ω ↦ X ω + s) Y A = condRho X Y A` * This states that the conditional rho function (`condRho`) applied to the *translated* function `(fun ω ↦ X ω + s)` (which adds the constant `s` to each output of `X`) is equal to `condRho` applied to the original function `X`, given the same `Y` and `A`. 7. **Proof**: `:= by simp only [condRho, rho_of_translate]` * The proof is a single tactic call. It uses the simplifier (`simp`) with only the specified lemmas: the definition of `condRho` and a lemma named `rho_of_translate`. This suggests the result follows directly from unfolding definitions and applying a previously proven lemma about rho and translation. ### Key Observations * **Syntactic Precision**: The code uses precise Lean 4 syntax, including the Unicode arrow `→`, the maps-to symbol `↦`, and the colon `:` for type annotations. * **Proof Brevity**: The proof is extremely concise, indicating the result is likely a straightforward consequence of more fundamental definitions and lemmas within the formalized theory. * **Contextual Clues**: The names `condRho`, `rho_of_translate`, `MeasureSpace`, and `Finset` strongly suggest this lemma is part of a formalization of measure theory or probability theory, possibly involving group actions or invariant measures. ### Interpretation This image captures a formal mathematical statement and its machine-checked proof. The lemma asserts a specific **invariance property**: the value of the `condRho` function does not change when its first functional argument (`X`) is uniformly translated by a group element `s`. This is a common type of lemma in areas like ergodic theory or the study of invariant measures, where properties are expected to be unchanged under group actions. The proof strategy (`simp only [condRho, rho_of_translate]`) reveals the logical dependency: the property is not proven from first principles here but is derived by combining the definition of `condRho` with a pre-existing lemma (`rho_of_translate`) that likely establishes a more general or foundational relationship between rho functions and translation. This exemplifies the hierarchical and reusable nature of formalized mathematics. The screenshot itself serves as a record of this verified mathematical fact within a specific digital environment.

## Screenshot: Code Snippet in Programming Environment ### Overview The image shows a code snippet in a programming environment with syntax highlighting. The code defines a lemma (`condRho_of_translate`) involving measure spaces, functions, and type declarations. The text is color-coded to distinguish keywords, variables, operators, and other elements. ### Components/Axes - **UI Elements**: - Top-left corner: Three circular buttons (red, yellow, green) for window control. - **Code Structure**: - **Lemma Declaration**: `lemma condRho_of_translate {Ω S : Type*} [MeasureSpace Ω] (X : Ω → G) (Y : Ω → S)` - `lemma`: Red (keyword). - `condRho_of_translate`: Yellow (identifier). - `{Ω S : Type*}`: Green (type declaration). - `[MeasureSpace Ω]`: Red (attribute/constraint). - **Function Definitions**: - `(A : Finset G) (s : G) : condRho (fun ω ↦ X ω + s)` - `A : Finset G`: Green (type annotation). - `s : G`: Green (type annotation). - `condRho`: Red (function name). - `fun ω ↦ X ω + s`: Purple (lambda abstraction) and blue (operator `+`). - `Y A := condRho X Y A := by`: - `Y A`: Red (variable/identifier). - `condRho X Y A`: Red (function application). - `by`: Purple (keyword). - **Simplification**: `simp only [condRho, rho_of_translate]` - `simp only`: Red (keyword). - `[condRho, rho_of_translate]`: Green (list of terms). ### Detailed Analysis - **Syntax Highlighting**: - Keywords (`lemma`, `by`, `simp only`): Red. - Type declarations (`Type*`, `Finset G`, `MeasureSpace Ω`): Green. - Function names (`condRho`, `rho_of_translate`): Yellow/Red. - Operators (`→`, `+`, `:=`): Blue/Purple. - Variables/identifiers (`X`, `Y`, `A`, `ω`, `s`): Red/Yellow. - **Code Logic**: - Defines a lemma about a conditional rho function (`condRho`) in a measure space (`Ω`). - `A` is a finite subset of `G`, and `s` is an element of `G`. - `condRho` is a function mapping `ω` to `X ω + s`. - `Y A` is defined as `condRho X Y A` via simplification using `condRho` and `rho_of_translate`. ### Key Observations - The code uses formal type theory syntax (e.g., `Type*`, `Finset`). - The lemma likely relates to measure-preserving transformations or translations between spaces. - The `simp only` clause suggests automated proof simplification using predefined rules. ### Interpretation This code snippet appears to be part of a formal verification system (e.g., Lean, Coq) or a theorem prover. It defines a mathematical lemma about a conditional rho function (`condRho`) in a measure space (`Ω`). The lemma states that under the given type constraints, `Y A` can be simplified to `condRho X Y A` using the `rho_of_translate` rule. The use of `simp only` indicates that the proof relies on specific simplification tactics, common in automated theorem proving. The color-coded syntax highlights the structure and relationships between mathematical objects, functions, and operations. No numerical data, charts, or diagrams are present. The focus is on formal logic and type theory constructs.