exercises
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.vscode/settings.json
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.vscode/settings.json
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@ -3,5 +3,5 @@
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"gitdoc.enabled": false,
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"gitdoc.enabled": false,
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"gitdoc.autoCommitDelay": 300000,
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"gitdoc.autoCommitDelay": 300000,
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"gitdoc.commitMessageFormat": "'auto gitdoc commit'",
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"gitdoc.commitMessageFormat": "'auto gitdoc commit'",
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"agdaMode.connection.commandLineOptions": "--no-load-primitives"
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"agdaMode.connection.commandLineOptions": ""
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}
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}
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@ -42,11 +42,21 @@ private
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State and prove funExt for dependent functions `f g : (x : A) → B x`
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State and prove funExt for dependent functions `f g : (x : A) → B x`
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```
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funExtDep : {f g : (x : A) → B x}
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→ ((x : A) → f x ≡ g x)
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→ f ≡ g
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funExtDep p i x = (p x) i
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```
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### Exercise 2 (★)
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### Exercise 2 (★)
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Generalize the type of ap to dependent function `f : (x : A) → B x`
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Generalize the type of ap to dependent function `f : (x : A) → B x`
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(hint: the result should be a `PathP`)
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(hint: the result should be a `PathP`)
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```
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-- apDep :
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```
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## Part II: Some facts about (homotopy) propositions and sets
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## Part II: Some facts about (homotopy) propositions and sets
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@ -65,7 +75,11 @@ Prove
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```agda
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```agda
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isPropΠ : (h : (x : A) → isProp (B x)) → isProp ((x : A) → B x)
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isPropΠ : (h : (x : A) → isProp (B x)) → isProp ((x : A) → B x)
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isPropΠ = {!!}
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isPropΠ h f g i x =
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let
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a = f x
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b = g x
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in h x a b i
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```
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```
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### Exercise 5 (★)
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### Exercise 5 (★)
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```agda
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```agda
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funExt⁻ : {f g : (x : A) → B x} → f ≡ g → ((x : A) → f x ≡ g x)
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funExt⁻ : {f g : (x : A) → B x} → f ≡ g → ((x : A) → f x ≡ g x)
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funExt⁻ = {!!}
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funExt⁻ p x i = p i x
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```
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```
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### Exercise 6 (★★)
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### Exercise 6 (★★)
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@ -83,7 +97,14 @@ Use funExt⁻ to prove isSetΠ:
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```agda
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```agda
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isSetΠ : (h : (x : A) → isSet (B x)) → isSet ((x : A) → B x)
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isSetΠ : (h : (x : A) → isSet (B x)) → isSet ((x : A) → B x)
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isSetΠ = {!!}
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isSetΠ h =
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λ p q → -- p, q : (x : A) → B x
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λ r s → -- r, s : p ≡ q
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λ i j → -- j : p ≡ q, i : r j ≡ s j
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λ x →
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let
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test = funExt⁻ r x
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in {! !}
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```
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```
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### Exercise 7 (★★★): alternative contractibility of singletons
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### Exercise 7 (★★★): alternative contractibility of singletons
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@ -165,6 +186,8 @@ suspUnitChar = {!!}
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Define suspension using the Pushout HIT and prove that it's equal to Susp.
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Define suspension using the Pushout HIT and prove that it's equal to Susp.
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```
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```
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### Exercise 12 (🌶)
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### Exercise 12 (🌶)
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