## Code Snippet: Norm2 Scalar Theorems ### Overview The image presents a code snippet containing two theorem definitions related to the norm2_scalar property, likely within a formal verification or mathematical library environment. The code defines theorems for real scalars and their relationship to the absolute value and square of a variable x. ### Components/Axes The code snippet includes the following components: * **Theorem Definitions:** Two theorems named `norm2_scalar`. * **Type Constraints:** `{R} [RealScalar R] (x : R)` indicating type constraints on the real scalar `R` and variable `x`. * **Equations:** Mathematical equations defining the norm2_scalar property. * **Tactics:** `rw`, `symm`, `simp`, `congr` which are tactics used in interactive theorem proving. * **Library References:** `[SciLean.scalar_norm]` refers to a library function. ### Detailed Analysis or ### Content Details The code snippet contains the following theorems and expressions: 1. **First Theorem:** * `theorem norm2_scalar {R} [RealScalar R] (x : R):` * `||x||₂ [R] = Scalar.abs x := by` * `rw [SciLean.scalar_norm]` This theorem states that the norm2 of `x` with respect to the real scalar `R` is equal to the absolute value of `x`. The `rw` tactic rewrites the goal using the definition from `SciLean.scalar_norm`. 2. **Second Theorem:** * `theorem norm2_scalar {R} [RealScalar R] (x : R):` * `||x||₂² [R] = x^2 := by` * `symm` * `simp [sq]` * `congr` * `simp` This theorem states that the square of the norm2 of `x` with respect to the real scalar `R` is equal to `x` squared. The tactics `symm`, `simp [sq]`, `congr`, and `simp` are used to prove the theorem. ### Key Observations * Both theorems define a relationship between the norm2 of a real scalar and either the absolute value or the square of a variable. * The first theorem uses a rewrite tactic with a library reference, while the second theorem uses a series of simplification and congruence tactics. * The code appears to be written in a language used for formal verification, possibly Lean. ### Interpretation The code snippet demonstrates the definition of two theorems related to the norm2_scalar property. These theorems are likely part of a larger mathematical library or formal verification project. The first theorem establishes a relationship between the norm2 and the absolute value, while the second theorem establishes a relationship between the square of the norm2 and the square of the variable. The tactics used in the proofs suggest an interactive theorem proving environment.