This commit is contained in:
parent
531b33442d
commit
043b3ebe74
5 changed files with 357 additions and 60 deletions
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@ -8,6 +8,7 @@ import {
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mkdirSync,
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mkdtempSync,
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readFileSync,
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existsSync,
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readdirSync,
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copyFileSync,
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writeFileSync,
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@ -35,6 +36,8 @@ const remarkAgda: RemarkPlugin = ({ base, publicDir }: Options) => {
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const tempDir = mkdtempSync(join(tmpdir(), "agdaRender."));
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const agdaOutDir = join(tempDir, "output");
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const agdaOutFilename = parse(path).base.replace(/\.lagda.md/, ".md");
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const agdaOutFile = join(agdaOutDir, agdaOutFilename);
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mkdirSync(agdaOutDir, { recursive: true });
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const childOutput = spawnSync(
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@ -44,18 +47,32 @@ const remarkAgda: RemarkPlugin = ({ base, publicDir }: Options) => {
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`--html-dir=${agdaOutDir}`,
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"--highlight-occurrences",
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"--html-highlight=code",
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"--allow-unsolved-metas",
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path,
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],
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{},
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);
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// TODO: Handle child output
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console.error("--AGDA OUTPUT--");
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console.error(childOutput);
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console.error(childOutput.stdout?.toString());
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console.error(childOutput.stderr?.toString());
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console.error("--AGDA OUTPUT--");
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if (childOutput.error || !existsSync(agdaOutFile)) {
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throw new Error(
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`Agda error:
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Stdout:
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${childOutput.stdout}
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Stderr:
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${childOutput.stderr}`,
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{
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cause: childOutput.error,
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},
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);
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}
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// // TODO: Handle child output
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// console.error("--AGDA OUTPUT--");
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// console.error(childOutput);
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// console.error(childOutput.stdout?.toString());
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// console.error(childOutput.stderr?.toString());
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// console.error("--AGDA OUTPUT--");
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const referencedFiles = new Set();
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for (const file of readdirSync(agdaOutDir)) {
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@ -87,11 +104,9 @@ const remarkAgda: RemarkPlugin = ({ base, publicDir }: Options) => {
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}
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}
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const filename = parse(path).base.replace(/\.lagda.md/, ".md");
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const htmlname = parse(path).base.replace(/\.lagda.md/, ".html");
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const fullOutputPath = join(agdaOutDir, filename);
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const doc = readFileSync(fullOutputPath);
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const doc = readFileSync(agdaOutFile);
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// This is the post-processed markdown with HTML code blocks replacing the Agda code blocks
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const tree2 = fromMarkdown(doc);
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@ -4,22 +4,24 @@ module Prelude where
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open import Agda.Primitive
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module 𝟘 where
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private
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variable
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l : Level
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data ⊥ : Set where
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rec-⊥ : {A : Set} → ⊥ → A
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rec-⊥ ()
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¬_ : Set → Set
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¬ A = A → ⊥
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open 𝟘 public
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module 𝟙 where
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data ⊤ : Set where
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tt : ⊤
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open 𝟙 public
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module 𝟚 where
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data Bool : Set where
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true : Bool
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false : Bool
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open 𝟚 public
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id : {l : Level} {A : Set l} → A → A
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id x = x
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@ -39,6 +41,19 @@ open Nat public
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infix 4 _≡_
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data _≡_ {l} {A : Set l} : (a b : A) → Set l where
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instance refl : {x : A} → x ≡ x
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{-# BUILTIN EQUALITY _≡_ #-}
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ap : {A B : Set l} → (f : A → B) → {x y : A} → x ≡ y → f x ≡ f y
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ap f refl = refl
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sym : {A : Set l} {x y : A} → x ≡ y → y ≡ x
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sym refl = refl
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trans : {A : Set l} {x y z : A} → x ≡ y → y ≡ z → x ≡ z
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trans refl refl = refl
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infixl 10 _∙_
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_∙_ = trans
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transport : {l₁ l₂ : Level} {A : Set l₁} {x y : A}
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→ (P : A → Set l₂)
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@ -72,3 +87,9 @@ _∘_ : {A B C : Set} (g : B → C) → (f : A → B) → A → C
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_∼_ : {A B : Set} (f g : A → B) → Set
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_∼_ {A} f g = (x : A) → f x ≡ g x
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postulate
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funExt : {l : Level} {A B : Set l}
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→ {f g : A → B}
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→ ((x : A) → f x ≡ g x)
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→ f ≡ g
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@ -1,5 +1,5 @@
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---
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title: "Formally proving true ≢ false in cubical Agda"
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title: "Formally proving true ≢ false in Homotopy Type Theory with Agda"
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slug: "proving-true-from-false"
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date: 2023-04-21
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tags: ["type-theory", "agda"]
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@ -12,12 +12,7 @@ math: true
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These are some imports that are required for code on this page to work properly.
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```agda
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{-# OPTIONS --cubical #-}
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open import Cubical.Foundations.Prelude
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open import Prelude hiding (_≢_; _≡_; refl; transport)
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infix 4 _≢_
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_≢_ : ∀ {A : Set} → A → A → Set
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x ≢ y = ¬ (x ≡ y)
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open import Prelude
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```
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</details>
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@ -53,9 +48,8 @@ left side so it becomes judgmentally equal to the right:
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- suc (suc (suc zero))
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- 3
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However, in cubical Agda, naively using `refl` with the inverse statement
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doesn't work. I've commented it out so the code on this page can continue to
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compile.
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However, in Agda, naively using `refl` with the inverse statement doesn't work.
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I've commented it out so the code on this page can continue to compile.
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```
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-- true≢false = refl
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@ -88,7 +82,7 @@ The strategy here is we define some kind of "type-map". Every time we see
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`false`, we'll map it to empty.
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```
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bool-map : Bool → Type
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bool-map : Bool → Set
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bool-map true = ⊤
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bool-map false = ⊥
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```
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@ -98,26 +92,29 @@ over a path (the path supposedly given to us as the witness that true ≢ false)
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will produce a function from the inhabited type we chose to the empty type!
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```
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true≢false p = transport (λ i → bool-map (p i)) tt
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true≢false p = transport bool-map p tt
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```
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I used `true` here, but I could equally have just used anything else:
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```
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bool-map2 : Bool → Type
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bool-map2 : Bool → Set
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bool-map2 true = 1 ≡ 1
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bool-map2 false = ⊥
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true≢false2 : true ≢ false
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true≢false2 p = transport (λ i → bool-map2 (p i)) refl
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true≢false2 p = transport bool-map2 p refl
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```
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## Note on proving divergence on equivalent values
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Let's make sure this isn't broken by trying to apply this to something that's
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actually true:
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> [!NOTE]
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> Update: some of these have been commented out since regular Agda doesn't support higher inductive types
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```
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Let's make sure this isn't broken by trying to apply this to something that's
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actually true, like this higher inductive type:
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```text
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data NotBool : Type where
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true1 : NotBool
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true2 : NotBool
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@ -128,8 +125,7 @@ In this data type, we have a path over `true1` and `true2` that is a part of the
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definition of the `NotBool` type. Since this is an intrinsic equality, we can't
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map `true1` and `true2` to divergent types. Let's see what happens:
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```
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{-# NON_COVERING #-}
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```text
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notbool-map : NotBool → Type
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notbool-map true1 = ⊤
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notbool-map true2 = ⊥
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@ -12,7 +12,7 @@ First off, a demonstration. Here's the code for function application over a path
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```
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open import Agda.Primitive
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open import Prelude
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open import Prelude hiding (ap)
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ap : {l1 l2 : Level} {A : Set l1} {B : Set l2} {x y : A}
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→ (f : A → B)
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@ -1,5 +1,5 @@
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---
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title: "Boolean equivalences"
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title: "Boolean equivalences in HoTT"
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slug: 2024-06-28-boolean-equivalences
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date: 2024-06-28T21:37:04.299Z
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tags: ["agda", "type-theory", "hott"]
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@ -15,25 +15,41 @@ $$
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(2 \simeq 2) \simeq 2
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$$
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This problem is exercise 2.13 of the [Homotopy Type Theory book][book]. If you
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are planning to attempt this problem, spoilers ahead!
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> [!NOTE]
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> This problem is exercise 2.13 of the [Homotopy Type Theory book][book]. If you
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> are planning to attempt this problem, spoilers ahead!
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[book]: https://homotopytypetheory.org/book/
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<small>The following line imports some of the definitions used in this post.</small>
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```
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{-# OPTIONS --allow-unsolved-metas #-}
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open import Prelude
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open Σ
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```
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## The problem explained
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With just the notation, the problem may not be clear.
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$2$ represents the set with only 2 elements in it, which is the booleans. The
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With just that notation, the problem may not be clear.
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$2$ represents the set with only two elements in it, which is the booleans. The
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elements of $2$ are $\textrm{true}$ and $\textrm{false}$.
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In this expression, $\simeq$ denotes [_homotopy equivalence_][1] between sets,
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which you can think of as an isomorphism.
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For some function $f : A \rightarrow B$, we say that $f$ is an equivalence if:
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```
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data 𝟚 : Set where
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true : 𝟚
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false : 𝟚
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```
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In the problem statement, $\simeq$ denotes [_homotopy equivalence_][1] between
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sets, which you can think of as basically a fancy isomorphism. For some
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function $f : A \rightarrow B$, we say that $f$ is an equivalence[^equivalence]
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if:
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[^equivalence]: There are other ways to define equivalences. As we'll show, an important
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property that is missed by this definition is that equivalences should be _mere
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propositions_. The reason why this definition falls short of that requirement is
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shown by Theorem 4.1.3 in the [HoTT book][book].
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[1]: https://en.wikipedia.org/wiki/Homotopy#Homotopy_equivalence
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@ -56,7 +72,7 @@ Then, the expression $2 \simeq 2$ simply expresses an equivalence between the
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boolean set to itself. Here's an example of a function that satisfies this equivalence:
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```
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bool-id : Bool → Bool
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bool-id : 𝟚 → 𝟚
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bool-id b = b
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```
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@ -74,9 +90,9 @@ bool-id-eqv = mkEquiv bool-id id-homotopy id-homotopy
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We can package these together into a single definition that combines the
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function along with proof of its equivalence properties using a dependent
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pair[^1]:
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pair[^dependent-pair]:
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[^1]: A dependent pair (or $\Sigma$-type) is like a regular pair $\langle x, y\rangle$, but where $y$ can depend on $x$.
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[^dependent-pair]: A dependent pair (or $\Sigma$-type) is like a regular pair $\langle x, y\rangle$, but where $y$ can depend on $x$.
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For example, $\langle x , \textrm{isPrime}(x) \rangle$.
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In this case it's useful since we can carry the equivalence information along with the function itself.
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This type is rather core to Martin-Löf Type Theory, you can read more about it [here][dependent-pair].
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@ -93,7 +109,7 @@ A ≃ B = Σ (A → B) (λ f → isEquiv f)
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This gives us an equivalence:
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```
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bool-eqv : Bool ≃ Bool
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bool-eqv : 𝟚 ≃ 𝟚
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bool-eqv = bool-id , bool-id-eqv
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```
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@ -106,7 +122,7 @@ equivalences between the boolean type and itself. Here is another possible
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definition:
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```
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bool-neg : Bool → Bool
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bool-neg : 𝟚 → 𝟚
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bool-neg true = false
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bool-neg false = true
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```
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@ -127,7 +143,7 @@ bool-neg-eqv = mkEquiv bool-neg neg-homotopy neg-homotopy
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neg-homotopy true = refl
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neg-homotopy false = refl
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bool-eqv2 : Bool ≃ Bool
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bool-eqv2 : 𝟚 ≃ 𝟚
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bool-eqv2 = bool-neg , bool-neg-eqv
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```
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@ -147,6 +163,255 @@ That's where the main exercise statement comes in: "show that" $(2 \simeq 2)
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\simeq 2$, or in other words, find an inhabitant of this type.
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```
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_ : (Bool ≃ Bool) ≃ Bool
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_ = {! !}
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main-theorem : (𝟚 ≃ 𝟚) ≃ 𝟚
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```
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How do we do this? Well, remember what it means to define an equivalence: first
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define a function, then show that it is an equivalence. Since equivalences are
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symmetrical, it doesn't matter which way we start first. I'll choose to first
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define a function $2 \rightarrow 2 \simeq 2$. This is fairly easy to do by
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case-splitting:
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```
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f : 𝟚 → (𝟚 ≃ 𝟚)
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f true = bool-eqv
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f false = bool-eqv2
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```
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This maps `true` to the equivalence derived from the identity function, and
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`false` to the function derived from the negation function.
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Now we need another function in the other direction. We can't case-split on
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functions, but we can certainly case-split on their output. Specifically, we can
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differentiate `id` from `neg` by their behavior when being called on `true`:
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- $\textrm{id}(\textrm{true}) :\equiv \textrm{true}$
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- $\textrm{neg}(\textrm{true}) :\equiv \textrm{false}$
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```
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g : (𝟚 ≃ 𝟚) → 𝟚
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g eqv = (fst eqv) true
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-- same as this:
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-- let (f , f-is-equiv) = eqv in f true
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```
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Hold on. If you know about the cardinality of functions, you'll observe that in
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the space of functions $f : 2 \rightarrow 2$, there are $2^2 = 4$ equivalence
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classes of functions. So if we mapped 4 functions into 2, how could this be
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considered an equivalence?
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A **key observation** here is that we're not mapping from all possible functions
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$f : 2 \rightarrow 2$. We're mapping functions $f : 2 \simeq 2$ that have
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already been proven to be equivalences. This means we can count on them to be
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structure-preserving, and can rule out cases like the function that maps
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everything to true, since it can't possibly have an inverse.
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We'll come back to this later.
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First, let's show that $g \circ f \sim \textrm{id}$. This one is easy, we can
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just case-split. Each of the cases reduces to something that is definitionally
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equal, so we can use `refl`.
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```
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g∘f : (g ∘ f) ∼ id
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g∘f true = refl
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g∘f false = refl
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```
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Now comes the complicated case: proving $f \circ g \sim \textrm{id}$.
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> [!NOTE]
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> Since Agda's comment syntax is `--`, the horizontal lines in the code below
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> are simply a visual way of separating out our proof premises from our proof
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> goals.
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```
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module f∘g-case where
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goal :
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(eqv : 𝟚 ≃ 𝟚)
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--------------------
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→ f (g eqv) ≡ id eqv
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f∘g : (f ∘ g) ∼ id
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f∘g eqv = goal eqv
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```
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Now our goal is to show that for any equivalence $\textrm{eqv} : 2 \simeq 2$,
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applying $f ∘ g$ to it is the same as not doing anything. We can evaluate the
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$g(\textrm{eqv})$ a little bit to give us a more detailed goal:
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```
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goal2 :
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(eqv : 𝟚 ≃ 𝟚)
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--------------------------
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→ f ((fst eqv) true) ≡ eqv
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-- Solving goal with goal2, leaving us to prove goal2
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goal eqv = goal2 eqv
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```
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The problem is if you think about equivalences as encoding some rich data about
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a function, converting it to a boolean and back is sort of like shoving it into
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a lower-resolution domain and then bringing it back. How can we prove that the
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equivalence is still the same equivalence, and as a result proving that there
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are only 2 possible equivalences?
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Note that the function $f$ ultimately still only produces 2 values. That means
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if we want to prove this, we can case-split on $f$'s output. In Agda, this uses
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a syntax known as [with-abstraction]:
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[with-abstraction]: https://agda.readthedocs.io/en/v2.6.4.3-r1/language/with-abstraction.html
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```
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goal3 :
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(eqv : 𝟚 ≃ 𝟚) → (b : 𝟚) → (p : fst eqv true ≡ b)
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------------------------------------------------
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→ f b ≡ eqv
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-- Solving goal2 with goal3, leaving us to prove goal3
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goal2 eqv with b ← (fst eqv) true in p = goal3 eqv b p
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```
|
||||
|
||||
We can now case-split on $b$, which is the output of calling $f$ on the
|
||||
equivalence returned by $g$. This means that for the `true` case, we need to
|
||||
show that $f(b) = \textrm{bool-eqv}$ (which is based on `id`) is equivalent to
|
||||
the equivalence that generated the `true`.
|
||||
|
||||
Let's start with the `id` case; we just need to show that for every equivalence
|
||||
$e$ where running the equivalence function on `true` also returned `true`, $e
|
||||
\equiv f(\textrm{true})$.
|
||||
|
||||
Unfortunately, we don't know if this is true unless our equivalences are _mere
|
||||
propositions_, meaning if two functions are identical, then so are their
|
||||
equivalences.
|
||||
|
||||
$$
|
||||
\textrm{isProp}(P) :\equiv \prod_{x, y: P}(x \equiv y)
|
||||
$$
|
||||
|
||||
<small>Definition 3.3.1 from the [HoTT book][book]</small>
|
||||
|
||||
Applying this to $\textrm{isEquiv}(f)$, we get the property:
|
||||
|
||||
$$
|
||||
\sum_{f : A → B} \left( \prod_{e_1, e_2 : \textrm{isEquiv}(f)} e_1 \equiv e_2 \right)
|
||||
$$
|
||||
|
||||
This proof is shown later in the book, so I will use it here directly without proof[^equiv-isProp]:
|
||||
|
||||
[^equiv-isProp]: I haven't worked that far in the book yet, but this is shown in Theorem 4.2.13 in the [HoTT book][book].
|
||||
I might write a separate post about that when I get there, it seems like an interesting proof as well!
|
||||
|
||||
```
|
||||
postulate
|
||||
equiv-isProp : {A B : Set}
|
||||
→ (e1 e2 : A ≃ B) → (Σ.fst e1 ≡ Σ.fst e2)
|
||||
-----------------------------------------
|
||||
→ e1 ≡ e2
|
||||
```
|
||||
|
||||
Now we're going to need our **key observation** that we made earlier, that
|
||||
equivalences must not map both values to a single one.
|
||||
This way, we can pin the behavior of the function on all inputs by just using
|
||||
its behavior on `true`, since its output on `false` must be _different_.
|
||||
|
||||
We can use a proof that [$\textrm{true} \not\equiv \textrm{false}$][true-not-false] that I've shown previously.
|
||||
|
||||
[true-not-false]: https://mzhang.io/posts/proving-true-from-false/
|
||||
|
||||
```
|
||||
true≢false : true ≢ false
|
||||
-- read: true ≡ false → ⊥
|
||||
true≢false p = bottom-generator p
|
||||
where
|
||||
map : 𝟚 → Set
|
||||
map true = ⊤
|
||||
map false = ⊥
|
||||
|
||||
bottom-generator : true ≡ false → ⊥
|
||||
bottom-generator p = transport map p tt
|
||||
```
|
||||
|
||||
Let's transform this into information about $f$'s output on different inputs:
|
||||
|
||||
```
|
||||
observation : (f : 𝟚 → 𝟚) → isEquiv f → f true ≢ f false
|
||||
-- read: f true ≡ f false → ⊥
|
||||
observation f (mkEquiv g f∘g g∘f) p =
|
||||
let
|
||||
-- Given p : f true ≡ f false
|
||||
-- Proof strategy is to call g on it to get (g ∘ f) true ≡ (g ∘ f) false
|
||||
-- Then, use our equivalence properties to reduce it to true ≡ false
|
||||
-- Then, apply the lemma true≢false we just proved
|
||||
step1 = ap g p
|
||||
step2 = sym (g∘f true) ∙ step1 ∙ (g∘f false)
|
||||
step3 = true≢false step2
|
||||
in step3
|
||||
```
|
||||
|
||||
For convenience, let's rewrite this so that it takes in an arbitrary $b$ and
|
||||
returns the version of the inequality we want:
|
||||
|
||||
```
|
||||
observation' : (f : 𝟚 → 𝟚) → isEquiv f → (b : 𝟚) → f b ≢ f (bool-neg b)
|
||||
-- ^^^^^^^^^^^^^^^^^^^^
|
||||
observation' f eqv true p = observation f eqv p
|
||||
observation' f eqv false p = observation f eqv (sym p)
|
||||
```
|
||||
|
||||
Then, solving the `true` case is simply a matter of using function
|
||||
extensionality (functions are equal if they are point-wise equal) to show that
|
||||
just the function part of the equivalences are identical, and then using this
|
||||
property to prove that the equivalences must be identical as well.
|
||||
|
||||
```
|
||||
goal3 eqv true p = equiv-isProp bool-eqv eqv functions-equal
|
||||
where
|
||||
e = fst eqv
|
||||
|
||||
pointwise-equal : (x : 𝟚) → e x ≡ x
|
||||
pointwise-equal true = p
|
||||
pointwise-equal false with ef ← e false in eq = helper ef eq
|
||||
where
|
||||
helper : (ef : 𝟚) → e false ≡ ef → ef ≡ false
|
||||
helper true eq = rec-⊥ (observation' e (snd eqv) false (eq ∙ sym p))
|
||||
helper false eq = refl
|
||||
|
||||
-- NOTE: fst bool-eqv = id, definitionally
|
||||
functions-equal : id ≡ e
|
||||
functions-equal = sym (funExt pointwise-equal)
|
||||
```
|
||||
|
||||
The `false` case is proved similarly.
|
||||
|
||||
```
|
||||
goal3 eqv false p = equiv-isProp bool-eqv2 eqv functions-equal
|
||||
where
|
||||
e = fst eqv
|
||||
|
||||
pointwise-equal : (x : 𝟚) → e x ≡ bool-neg x
|
||||
pointwise-equal true = p
|
||||
pointwise-equal false with ef ← e false in eq = helper ef eq
|
||||
where
|
||||
helper : (ef : 𝟚) → e false ≡ ef → ef ≡ true
|
||||
helper true eq = refl
|
||||
helper false eq = rec-⊥ (observation' e (snd eqv) true (p ∙ sym eq))
|
||||
|
||||
functions-equal : bool-neg ≡ e
|
||||
functions-equal = sym (funExt pointwise-equal)
|
||||
```
|
||||
|
||||
Putting this all together, we get the property we want to prove:
|
||||
|
||||
```
|
||||
open f∘g-case
|
||||
|
||||
-- main-theorem : (𝟚 ≃ 𝟚) ≃ 𝟚
|
||||
main-theorem = g , mkEquiv f g∘f f∘g
|
||||
```
|
||||
|
||||
Now that Agda's all happy, our work here is done!
|
||||
|
||||
Going through all this taught me a lot about how the basics of equivalences and
|
||||
how to express a lot of different ideas into the type system. Thanks for
|
||||
reading!
|
||||
|
|
|
|||
Loading…
Reference in a new issue