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      • Agreement of environment sets
      • Closing a code slot under its seven constructors
      • Recognizing well-formed constructor keys
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      • Numerals in a successor-closed ordinal stage
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      • The environment tower
      • The set of all formula codes
      • Describing the closed domain of formula codes
      • Describing the satisfaction table
      • The alphabet of formula codes
      • Reading and validating the satisfaction clauses
      • Pinning recursion on a subcode-closed domain
      • The satisfaction graph formula
      • Uniform satisfaction over all codes
      • A formula for the definable power set
      • A sequence for the constructible hierarchy
      • Coded injections
      • An internal graph of uniform satisfaction
      • Soundness and completeness of the closed code domain
    • Internalizing recursive definitions in L
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      • Graphs of recursive definitions
    • The constructible hierarchy inside L
    • Choice
      • The first stage meeting a set
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      • Canonical names for successor-stage members
      • Well-orders on all stages
      • Formulas for name comparison
      • An internal table of stage orders
      • Adequacy of the stage-order description
      • Adequacy of name comparison
      • The limit-stage order inside L
      • An internal family of earliest-disagreement relations
      • The internal stage-order relation
      • Choice by a transversal
    • WellOrder
      • Strict well-orders and least-element search
    • The constructible universe models ZFC
    • Cardinals and coded injections inside L
    • Turning a definable injection into an internal code
    • Composition and inclusion of coded injections
    • The generalized continuum hypothesis inside L
    • Cantor–Schröder–Bernstein inside L
    • Collapsing a transitive well-founded relation
    • An ordinal L-cardinal above every L-cardinal
    • GCH
      • Assembling GCH from four internal bounds
      • A Δ₀ description of the satisfaction table
      • A Δ₀ description of the definable power set
      • Adequate stages for the GCH argument
      • ω-recursion inside the constructible universe
      • Building and collapsing a Skolem hull
      • A Δ₀ description of the constructible hierarchy
      • Transferring structure through condensation
      • Ordinals below a successor cardinal inject into its base
      • Constructing order types inside L
      • Choosing a cardinal representative for an ordinal
      • The square law for infinite L-cardinals
      • The least-witness map inside a constructible stage
      • Injecting the successor cardinal into the power set
      • Coding finite sequences below an infinite ordinal
      • The counting tools for infinite constructible stages
      • Locating the hull and its collapse inside L
      • Counting a Skolem hull from a counted start
      • Injecting an infinite constructible stage into its index
      • Bounded subsets appear at controlled stages
      • The constructible universe satisfies GCH
module Cubical.Data.Empty.Base where

open import Cubical.Foundations.Prelude

private
  variable
    ℓ ℓ' : Level

data ⊥ : Type₀ where

⊥* : Type ℓ
⊥* = Lift ⊥

rec : {A : Type ℓ} → ⊥ → A
rec ()

rec* : {A : Type ℓ} → ⊥* {ℓ = ℓ'} → A
rec* ()

elim : {A : ⊥ → Type ℓ} → (x : ⊥) → A x
elim ()

elim* : {A : ⊥* {ℓ'} → Type ℓ} → (x : ⊥* {ℓ'}) → A x
elim* ()
Rendered with a generator adapted from the 1lab (AGPL-3.0).
llms.txt
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