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2022-10-03-notes.lagda.md

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+```
+{#- OPTIONS --cubical -#}
+
+----------
+
+A = ℕ
+
+P : A → Type
+P n = Vec n
+
+_+_ 
+zero + y = y
+suc x + y = suc (x + y)
+
+-- x and y are NOT definitionally equal
+x = 1 + 2
+y = 2 + 1
+
+-- 
+p : x ≡ y
+p = +-comm 1 2
+
+
+Identity {A : Type} {x y : A}
+
+elim-unit :
+  (A : Type) →
+
+  -- unit case
+  (a : Unit → A) →
+
+  Unit → A
+
+
+
+elim-nat : (A : Type) → (z : A) → (s : A → A) → ℕ → A
+
+
+elim-nat A f g 5 = a5
+  - a0 = f
+  - a1 = g a0
+  - a2 = g a1
+  - a3 = g a2
+  - a4 = g a3
+  - a5 = g a4
+
+
+
+elim-identity :
+  -- type that the identity type is over
+  (A : Type) → 
+
+  -- output type
+  (C : (x y : A) → (p : x ≡ y) → Type) →
+
+  -- refl case
+  (c : (z : A) → C z z (refl z)) →
+
+  (a b : A) → (p′ : a ≡ b) →
+  C a b p′
+
+
+
+-- In the case of transports
+D is C in the eliminator
+
+P : A → Type
+
+D : (x y : A) → (p : x ≡ y) → Type
+D x y p = P x → P y
+
+transport : (A : Type) → (P : A → Type) → (x y : A) → (p : x ≡ y) → P x → P y
+transport = elim-identity A (λ a b _ → (P a → P b)) (λ a → id) x y p
+
+
+elim-identity A D d (a a : A) (refl a) = d a
+
+
+transport A P x x (refl x)
+  = elim-identity A (λ a b _ → (P a → P b)) (λ x → id) x x (refl x)
+  = (λ x → id) x
+  = id
+
+---
+
+
+-- Goal:
+p* : P (1 + 2) ≡ P (2 + 1)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+----------
+
+loop ∙ loop ∙ loop : S¹
+3 : ℤ
+
+p : {A : Type} {x y : A} → x ≡ y
+-- Path from P base = ℤ to P base
+
+P : S¹ → Type
+P base = ℤ
+P loop = ?
+```
+
+For this week:
+
+- Try to understand Lemma 2.3.4 in HoTT book