## 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.