113 lines
2.5 KiB
Agda
113 lines
2.5 KiB
Agda
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{-# OPTIONS --without-K #-}
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module Ch6HoTT where
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-- S¹-ind : (P : S¹ → Set) → (b : P base) → (ℓ : transport loop b ≡ b) → (x : S¹) → P x
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-- S¹-ind P b ℓ base = transport ? b
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-- f : A → B
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-- a1 a2 : A
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-- p : a1 ≡ a2
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--
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-- (ap f p) : f a1 ≡ f a2
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-- f : (x : A) → B x
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-- a1 a2 : A
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-- p : a1 ≡ a2
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--
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-- T : B a1 → B a2
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-- T = transport B p
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-- (apd f p) : T (f a1) ≡ f a2
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-- (apd f p) : transport B p (f a1) ≡ f a2
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import Relation.Binary.PropositionalEquality as Eq
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_) renaming (Set to Type) public
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data _≡_ {l : Level} {A : Type l} : A → A → Type l where
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refl : (x : A) → x ≡ x
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-- Path induction
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path-ind : ∀ {u} → {A : Set} →
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(C : (x y : A) → x ≡ y → Set u) →
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(c : (x : A) → C x x (refl x)) →
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{x y : A} → (p : x ≡ y) → C x y p
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path-ind C c {x} (refl x) = c x
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Path : {l : Level} (A : Type l) → A → A → Type l
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Path A x y = x ≡ y
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syntax Path A x y = x ≡ y [ A ]
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id : {ℓ : Level} {A : Set ℓ} → A → A
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id x = x
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-- Transport
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transport : {X : Set} (P : X → Set) {x y : X} → x ≡ y → P x → P y
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transport P (refl x) = id
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ap : {A B : Set} (f : A → B) {x y : A} → x ≡ y → f x ≡ f y
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ap f p = path-ind (λ x y p → f x ≡ f y) (λ x → ?) p
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postulate
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S¹ : Set
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base : S¹
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loop : base ≡ base [ S¹ ]
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-- (apd f p) : transport B p (f a1) ≡ f a2
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-- apd _ loop
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-- f : (x : A) → C x
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-- loop : base ≡ base, base : S¹
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--
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-- apd f loop : transport C loop (f base) ≡ f base
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--
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-- S¹-ind : (C : S¹ → Set)
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-- → (c-base : C base)
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-- → (c-loop : ?)
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-- → (s : S¹) → C s
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--
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-- N-ind : () → (c-zero : C 0) → ... → (n : N) → C n
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--
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-- N-ind 0 : C 0
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--
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-- f : (x : S¹) → C x
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-- f = S¹-ind C _ _
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--
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-- f base : C base
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--
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-- apd (f) loop : (transport C loop (f base)) ≡ f base
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-- (c-loop : (transport C loop (f base)) ≡ f base)
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--
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-- ap f (p₁ ∙ p₂) = (ap f p₁) ∙ (ap f p₂)
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-- (λ _ → A)
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lemma625 : {A : Set} → (a : A) → (p : a ≡ a) → (S¹ → A)
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lemma625 a p s1 = ?
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lemma625prop1 : {A : Set} {a : A} {p : a ≡ a} → (lemma625 a p base ≡ a)
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-- lemma625prop1 = refl
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lemma625prop2 : {A : Set} (a : A) → (p : a ≡ a) → ap (lemma625 a p) loop ≡ p
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-- lemma625prop2 a p = refl
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-- f : S¹ → A
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-- f = lemma625 a p
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--
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-- f-loop : f base ≡ f base
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-- f-loop : ap f loop
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--
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-- f-loop ≡ p
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