65 lines
1.9 KiB
Agda
65 lines
1.9 KiB
Agda
-- ignore
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module 1FundamentalGroup.Quest2Solutions where
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open import 1FundamentalGroup.Preambles.P2
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data _⊔_ (A B : Type) : Type where
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inl : A → A ⊔ B
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inr : B → A ⊔ B
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ℤ≡ℕ⊔ℕ : ℤ ≡ ℕ ⊔ ℕ
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ℤ≡ℕ⊔ℕ = isoToPath (iso fun inv rightInv leftInv) where
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fun : ℤ → ℕ ⊔ ℕ
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fun (pos n) = inl n
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fun (negsuc n) = inr n
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inv : ℕ ⊔ ℕ → ℤ
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inv (inl n) = pos n
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inv (inr n) = negsuc n
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rightInv : section fun inv
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rightInv (inl n) = refl
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rightInv (inr n) = refl
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leftInv : retract fun inv
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leftInv (pos n) = refl
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leftInv (negsuc n) = refl
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∙refl : {A : Type} {x y : A} (p : x ≡ y) → p ∙ refl ≡ p
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∙refl = J (λ y p → p ∙ refl ≡ p) refl∙refl
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refl∙ : {A : Type} {x y : A} (p : x ≡ y) → refl ∙ p ≡ p
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refl∙ = J (λ y p → refl ∙ p ≡ p) refl∙refl
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∙sym : {A : Type} {x y : A} (p : x ≡ y) → p ∙ sym p ≡ refl
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∙sym = J (λ y p → p ∙ sym p ≡ refl)
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(
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refl ∙ sym refl
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≡⟨ cong (λ p → refl ∙ p) symRefl ⟩
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refl ∙ refl
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≡⟨ refl∙refl ⟩
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refl ∎)
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sym∙ : {A : Type} {x y : A} (p : x ≡ y) → (sym p) ∙ p ≡ refl
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sym∙ = J (λ y p → (sym p) ∙ p ≡ refl)
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(
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(sym refl) ∙ refl
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≡⟨ cong (λ p → p ∙ refl) symRefl ⟩
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refl ∙ refl
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≡⟨ refl∙refl ⟩
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refl ∎)
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assoc : {A : Type} {w x : A} (p : w ≡ x) {y z : A} (q : x ≡ y) (r : y ≡ z)
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→ (p ∙ q) ∙ r ≡ p ∙ (q ∙ r)
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assoc {A} = J
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-- casing on p
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(λ x p → {y z : A} (q : x ≡ y) (r : y ≡ z) → (p ∙ q) ∙ r ≡ p ∙ (q ∙ r))
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-- reduce to showing when p = refl
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λ q r → (refl ∙ q) ∙ r
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≡⟨ cong (λ p' → p' ∙ r) (refl∙ q) ⟩
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q ∙ r
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≡⟨ sym (refl∙ (q ∙ r)) ⟩
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refl ∙ q ∙ r ∎
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