Symbol Reference: let x ↀ t in 𝛂 / [x/t] — Explicit Capture-Avoiding Substitution Operator

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Formula Display:
Proof Trees:

let x ↀ t in 𝛂 / [x/t] \(\mathsf{let}\; x \leftarrow t \;\mathsf{in}\; \alpha \; / \; [x/t]\) — Explicit Capture-Avoiding Substitution Operator

Denotes the capture-avoiding substitution of term t for free occurrences of variable x inside formula or expression 𝛂.

Field Mathematical & Kernel Specification (MKTD1 / MKTD2)
Symbol
let x ↀ t in 𝛂 / [x/t]
\(\mathsf{let}\; x \leftarrow t \;\mathsf{in}\; \alpha \; / \; [x/t]\)
Category Syntactic Hygiene & Substitution
SML AST Constructor Kernel AST Symbol / Sort Constructor in typetheory.ML (Class_p / Truth_p / EXP / UCol)
Constructor Arguments Typed arguments as specified by the Signature / Sort field above
Metatype / Signature Variable β†’ Class β†’ Fm β†’ Fm
Binder Status See Category & Signature above
Defining Kernel Entries (ML Labels) See Related Symbols & Kernel Entries below.
LaTeX Form \mathsf{let}\; x \leftarrow t \;\mathsf{in}\; \alpha \; / \; [x/t]
How to Understand It

In Obelisk (Entries RulForallInstantiation (2.02), RulExistsAbstraction (2.05), AxmTarskiUniverses (3.70), and Section 4 Entries DefMetaConecessitation (4.03)–AxmMetaClassComprehension (4.06)), substitution is represented explicitly in the AST as let x ↀ t in 𝛂 (or 𝛂[x/t]):

  • Rather than silently mutating AST nodes in meta-code, the kernel tracks substitution explicitly and discharges it via the verified metalogical substitution laws (Entries DefMetaConecessitation (4.03)–AxmMetaClassComprehension (4.06)) subject to the FFI hygiene guard.
Canonical Law / Usage (let x ↀ t in (𝛂 β‡’ 𝛃)) ⇔ ((let x ↀ t in 𝛂) β‡’ (let x ↀ t in 𝛃))
Related Symbols & Kernel Entries