## Logical Rules: Type Systems and Properties ### Overview The image presents a collection of logical rules, likely related to type systems or program semantics. These rules are expressed using mathematical notation and logical symbols, defining relationships between types, expressions, and contexts. The rules are grouped into categories such as "Cast Universal Properties", "Type Universal Properties", and "Error Properties". ### Components/Axes **1. Cast Universal Properties:** - Located at the top of the image. - Subdivided into "Bound" and "Best" categories. - Contains rules involving type casting or subtyping, denoted by symbols like "⊢" and "⊏". **2. Retract Axiom:** - Located below "Cast Universal Properties". - Presents axioms related to retracting or unwrapping types. **3. Type Universal Properties:** - Located in the center of the image. - Organized in a table format with columns for "Type", "β" (beta reduction), and "η" (eta conversion). - Lists rules for various types such as +, 0, ×, 1, U, F, →, &, and ⊤. **4. Error Properties:** - Located at the bottom of the image. - Contains rules related to error handling or type checking in erroneous situations. - Includes labels "ERRBOT" and "STKSTRICT". ### Detailed Analysis or ### Content Details **Cast Universal Properties:** * **Up (Bound):** x: A ⊢ x ⊏ (A' ⇦ A) x: A ⊏ A' * **Up (Best):** x ⊏ x': A ⊏ A' ⊢ (A' ⇦ A) x ⊏ x': A' * **Down (Bound):** •: B' ⊢ (B ⇦ B') • ⊏: B ⊏ B' * **Down (Best):** • ⊏: B ⊏ B' ⊢ (B ⇦ B') •: B **Retract Axiom:** * x: A ⊢ (FA ⇐ F?) (ret (? ⇐ A) x)) ⊏ ret x: FA * x: UB ⊢ (B ⇐) (force (U¿ ⊏ UB) x)) ⊏ force x: B **Type Universal Properties:** | Type | β AND THE LIST GOES ON!