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De Morgan's laws for Pred #2832

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De Morgan's laws for Pred #2832

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d24ce8c
Add De Morgan's law for Pred
Taneb Sep 26, 2025
0a974c5
Add more of De Morgan's laws
Taneb Sep 26, 2025
b406933
Don't name unused variables
Taneb Sep 26, 2025
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10 changes: 10 additions & 0 deletions CHANGELOG.md
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Original file line number Diff line number Diff line change
Expand Up @@ -103,3 +103,13 @@ Additions to existing modules
```agda
¬¬-η : A → ¬ ¬ A
```

* In Relation.Unary.Properites
```agda
¬∃⟨P⟩⇒Π[∁P] : ¬ ∃⟨ P ⟩ → Π[ ∁ P ]
¬∃⟨P⟩⇒∀[∁P] : ¬ ∃⟨ P ⟩ → ∀[ ∁ P ]
∃⟨∁P⟩⇒¬Π[P] : ∃⟨ ∁ P ⟩ → ¬ Π[ P ]
∃⟨∁P⟩⇒¬∀[P] : ∃⟨ ∁ P ⟩ → ¬ ∀[ P ]
Π[∁P]⇒¬∃[P] : Π[ ∁ P ] → ¬ ∃⟨ P ⟩
∀[∁P]⇒¬∃[P] : ∀[ ∁ P ] → ¬ ∃⟨ P ⟩
```
23 changes: 22 additions & 1 deletion src/Relation/Unary/Properties.agda
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Expand Up @@ -8,7 +8,7 @@

module Relation.Unary.Properties where

open import Data.Product.Base as Product using (_×_; _,_; swap; proj₁; zip′)
open import Data.Product.Base as Product using (_×_; _,_; -,_; swap; proj₁; zip′)
open import Data.Sum.Base using (inj₁; inj₂)
open import Data.Unit.Base using (tt)
open import Function.Base using (id; _$_; _∘_; _∘₂_)
Expand Down Expand Up @@ -52,6 +52,27 @@ U-Universal = λ _ → _
∁U-Empty : Empty {A = A} (∁ U)
∁U-Empty = λ x x∈∁U → x∈∁U _

------------------------------------------------------------------------
-- De Morgan's laws

¬∃⟨P⟩⇒Π[∁P] : ∀ {P : Pred A ℓ} → ¬ ∃⟨ P ⟩ → Π[ ∁ P ]
¬∃⟨P⟩⇒Π[∁P] ¬sat x Px = ¬sat (x , Px)

¬∃⟨P⟩⇒∀[∁P] : ∀ {P : Pred A ℓ} → ¬ ∃⟨ P ⟩ → ∀[ ∁ P ]
¬∃⟨P⟩⇒∀[∁P] ¬sat Px = ¬sat (-, Px)

∃⟨∁P⟩⇒¬Π[P] : ∀ {P : Pred A ℓ} → ∃⟨ ∁ P ⟩ → ¬ Π[ P ]
∃⟨∁P⟩⇒¬Π[P] (x , ¬Px) ΠP = ¬Px (ΠP x)

∃⟨∁P⟩⇒¬∀[P] : ∀ {P : Pred A ℓ} → ∃⟨ ∁ P ⟩ → ¬ ∀[ P ]
∃⟨∁P⟩⇒¬∀[P] (_ , ¬Px) ∀P = ¬Px ∀P

Π[∁P]⇒¬∃[P] : ∀ {P : Pred A ℓ} → Π[ ∁ P ] → ¬ ∃⟨ P ⟩
Π[∁P]⇒¬∃[P] Π∁P (x , Px) = Π∁P x Px

∀[∁P]⇒¬∃[P] : ∀ {P : Pred A ℓ} → ∀[ ∁ P ] → ¬ ∃⟨ P ⟩
∀[∁P]⇒¬∃[P] ∀∁P (_ , Px) = ∀∁P Px

------------------------------------------------------------------------
-- Subset properties

Expand Down