Skip to content

[ refactor ] Axiom.ExcludedMiddle and Axiom.DoubleNegationElimination #2575

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
jamesmckinna merged 1 commit into from
Feb 7, 2025
Merged

[ refactor ] Axiom.ExcludedMiddle and Axiom.DoubleNegationElimination #2575

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
18 changes: 11 additions & 7 deletions src/Axiom/DoubleNegationElimination.agda
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -10,26 +10,30 @@ module Axiom.DoubleNegationElimination where

open import Axiom.ExcludedMiddle
open import Level
open import Relation.Nullary
open import Relation.Nullary.Negation
open import Relation.Nullary.Decidable
open import Relation.Nullary.Decidable.Core
using (decidable-stable; ¬¬-excluded-middle)
open import Relation.Nullary.Negation.Core using (Stable)

private
variable
ℓ : Level

------------------------------------------------------------------------
-- Definition

-- The classical statement of double negation elimination says that
-- if a property is not not true then it is true.

DoubleNegationElimination : (ℓ : Level) → Set (suc ℓ)
DoubleNegationElimination ℓ = {P : Set ℓ} → ¬ ¬ P → P
DoubleNegationElimination : ∀ ℓ → Set (suc ℓ)
DoubleNegationElimination ℓ = {P : Set ℓ} → Stable P

------------------------------------------------------------------------
-- Properties

-- Double negation elimination is equivalent to excluded middle

em⇒dne : ∀ {ℓ} → ExcludedMiddle ℓ → DoubleNegationElimination ℓ
em⇒dne : ExcludedMiddle ℓ → DoubleNegationElimination ℓ
em⇒dne em = decidable-stable em

dne⇒em : ∀ {ℓ} → DoubleNegationElimination ℓ → ExcludedMiddle ℓ
dne⇒em : DoubleNegationElimination ℓ → ExcludedMiddle ℓ
dne⇒em dne = dne ¬¬-excluded-middle
4 changes: 2 additions & 2 deletions src/Axiom/ExcludedMiddle.agda
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -9,13 +9,13 @@
module Axiom.ExcludedMiddle where

open import Level
open import Relation.Nullary
open import Relation.Nullary.Decidable.Core using (Dec)

------------------------------------------------------------------------
-- Definition

-- The classical statement of excluded middle says that every
-- statement/set is decidable (i.e. it either holds or it doesn't hold).

ExcludedMiddle : (ℓ : Level) → Set (suc ℓ)
ExcludedMiddle : ∀ ℓ → Set (suc ℓ)
ExcludedMiddle ℓ = {P : Set ℓ} → Dec P