\n ## Code Snippet: Theorem Definition ### Overview The image presents a snippet of code, likely from a formal verification or proof assistant system (possibly Lean). It defines a theorem named `schur_convex_xlogx` concerning Schur convexity and the logarithmic function. The code is formatted with line numbers and syntax highlighting. ### Components/Axes The image consists of lines of code. Key elements include: * **Line Numbers:** Numbers 530-536 on the left side. * **Theorem Declaration:** `theorem schur_convex_xlogx {n : ℕ} :` * **Schur Convexity Expression:** `SchurConvex (fun x : Fin n -> ℝ => Σ i, x i * Real.Log (x i)) := by` * **Commands:** `unfold SchurConvex`, `intro x y hx nonnegative hy nonnegative h`, `obtain (h_majorization, h_sum_eq) := h` ### Detailed Analysis or Content Details Here's a line-by-line transcription: * **530:** (Empty line) * **531:** `theorem schur_convex_xlogx {n : ℕ} :` - Defines a theorem named `schur_convex_xlogx`. The `{n : ℕ}` indicates a type parameter `n` which is a natural number. * **532:** `SchurConvex (fun x : Fin n -> ℝ => Σ i, x i * Real.Log (x i)) := by` - States that the function `fun x : Fin n -> ℝ => Σ i, x i * Real.Log (x i)` is Schur convex. `Fin n` likely represents a finite set of size `n`. `ℝ` represents the real numbers. `Σ i, x i * Real.Log (x i)` is a summation over `i` of `x i` multiplied by the natural logarithm of `x i`. The `:= by` indicates the start of the proof. * **533:** `unfold SchurConvex` - Expands the definition of `SchurConvex`. * **534:** `intro x y hx nonnegative hy nonnegative h` - Introduces variables `x` and `y` and hypotheses `hx`, `hy`, and `h`. `nonnegative` likely refers to a condition that the variables are non-negative. * **535:** `obtain (h_majorization, h_sum_eq) := h` - Obtains two hypotheses, `h_majorization` and `h_sum_eq`, from the hypothesis `h`. ### Key Observations The code defines a theorem related to Schur convexity, a concept in mathematical analysis. The theorem involves a function that calculates a weighted sum of logarithms. The proof appears to be in progress, with unfolding of definitions and introduction of variables. ### Interpretation This code snippet is part of a formal proof of a mathematical theorem. The theorem states that a specific function involving logarithms is Schur convex. Schur convexity is a property related to majorization and has applications in optimization and economics. The code uses a formal proof assistant to rigorously verify the theorem. The `obtain` command suggests that the proof is proceeding by deriving new hypotheses from existing ones. The use of type annotations (e.g., `{n : ℕ}`) indicates a strong emphasis on type safety and correctness.