diff --git a/src/Dedekind/Bool.agda b/src/Dedekind/Bool.agda deleted file mode 100644 index eb288fb..0000000 --- a/src/Dedekind/Bool.agda +++ /dev/null @@ -1,8 +0,0 @@ -module Dedekind.Bool where - --- Booleans -data Bool : Set where - true : Bool - false : Bool - - diff --git a/src/Dedekind/Integer.agda b/src/Dedekind/Integer.agda deleted file mode 100644 index 4c9e181..0000000 --- a/src/Dedekind/Integer.agda +++ /dev/null @@ -1,8 +0,0 @@ -module Dedekind.Integer where - -open import Dedekind.Natural using (ℕ) - --- Integers -data ℤ : Set where - pos : ℕ → ℤ - negsuc : ℕ → ℤ diff --git a/src/Dedekind/Natural.agda b/src/Dedekind/Natural.agda deleted file mode 100644 index 2399d68..0000000 --- a/src/Dedekind/Natural.agda +++ /dev/null @@ -1,15 +0,0 @@ -module Dedekind.Natural where - -open import Dedekind.Set using (⊤; ⊥) - --- Natural -data ℕ : Set where - zero : ℕ - suc : ℕ → ℕ - -NonZero : ℕ → Set -NonZero zero = ⊥ -NonZero (suc n) = ⊤ - -gcd : (x y : ℕ) .{{ _ : NonZero x }} .{{ _ : NonZero y }} → ℕ -gcd x y = ? diff --git a/src/Dedekind/Rational.agda b/src/Dedekind/Rational.agda deleted file mode 100644 index 2153f33..0000000 --- a/src/Dedekind/Rational.agda +++ /dev/null @@ -1,12 +0,0 @@ -module Dedekind.Rational where - -open import Dedekind.Bool using (Bool) -open import Dedekind.Integer using (ℤ) -open import Dedekind.Natural using (ℕ; NonZero) - --- Rational -data ℚ : Set where - rat : ℤ → (d : ℕ) .{{ _ : NonZero d }} → ℚ - -_<_ : ℚ → ℚ → Bool -x < y = ? diff --git a/src/Dedekind/Real.agda b/src/Dedekind/Real.agda deleted file mode 100644 index e442b56..0000000 --- a/src/Dedekind/Real.agda +++ /dev/null @@ -1,20 +0,0 @@ --- Dedekind cuts using Cubical Agda -{-# OPTIONS --cubical #-} - -module Dedekind.Real where - -open import Dedekind.Rational using (ℚ; _<_) - --- Dedekind cuts -record ℝ : Set where - constructor real - field - lower : (ℚ → Bool) - closed : ∀ (x y : ℚ) → x < y - --- Operations on reals -_+_ : ℝ → ℝ → ℝ -a + b = real - (λ q → ℝ.lower a q) - --- Properties of reals diff --git a/src/Dedekind/Set.agda b/src/Dedekind/Set.agda deleted file mode 100644 index 45fa565..0000000 --- a/src/Dedekind/Set.agda +++ /dev/null @@ -1,6 +0,0 @@ -module Dedekind.Set where - -record ⊥ : Set where - -record ⊤ : Set where - instance constructor tt diff --git a/src/HottBook/Chapter3.lagda.md b/src/HottBook/Chapter3.lagda.md index 01af651..5daed49 100644 --- a/src/HottBook/Chapter3.lagda.md +++ b/src/HottBook/Chapter3.lagda.md @@ -14,13 +14,20 @@ isSet : (A : Set) → Set isSet A = (x y : A) → (p q : x ≡ y) → p ≡ q ``` -### Example 3.1.1 +### Example 3.1.2 ``` 𝟙-is-Set : isSet 𝟙 𝟙-is-Set tt tt p q = {! !} ``` +### Example 3.1.3 + +``` +⊥-is-Set : isSet ⊥ +⊥-is-Set () () p q +``` + ## 3.2 Propositions as types? ## 3.4 Classical vs. intuitionistic logic