From 166142b14c5f250f4ef4b48bb17f0b880a4985bf Mon Sep 17 00:00:00 2001 From: Jlh18 Date: Mon, 27 Sep 2021 14:01:09 +0100 Subject: [PATCH] added side quest on different defs of empty --- 1FundamentalGroup/Quest0SideQuests/Empty.agda | 24 ++++++++++++++ .../Quest0SideQuests/EmptySolutions.agda | 30 ++++++++++++++++++ .../Quest0SideQuests/Empty.agdai | Bin 0 -> 32990 bytes .../Quest0SideQuests/EmptySolutions.agdai | Bin 0 -> 32963 bytes 4 files changed, 54 insertions(+) create mode 100644 1FundamentalGroup/Quest0SideQuests/Empty.agda create mode 100644 1FundamentalGroup/Quest0SideQuests/EmptySolutions.agda create mode 100644 _build/2.6.2/agda/1FundamentalGroup/Quest0SideQuests/Empty.agdai create mode 100644 _build/2.6.2/agda/1FundamentalGroup/Quest0SideQuests/EmptySolutions.agdai diff --git a/1FundamentalGroup/Quest0SideQuests/Empty.agda b/1FundamentalGroup/Quest0SideQuests/Empty.agda new file mode 100644 index 0000000..0d70498 --- /dev/null +++ b/1FundamentalGroup/Quest0SideQuests/Empty.agda @@ -0,0 +1,24 @@ +module 1FundamentalGroup.Quest0SideQuests.Empty where + +open import Cubical.Foundations.Prelude +open import Cubical.Data.Empty renaming (rec to ⊥Rec) +open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) + +toEmpty : (A : Type) → Type +toEmpty A = {!!} + +-- why Type₁ +pathEmpty : (A : Type) → Type₁ +pathEmpty A = {!!} + +isoEmpty : (A : Type) → Type +isoEmpty A = {!!} + +toEmpty→isoEmpty : (A : Type) → toEmpty A → isoEmpty A +toEmpty→isoEmpty A f = {!!} + +isoEmpty→pathEmpty : (A : Type) → isoEmpty A → pathEmpty A +isoEmpty→pathEmpty A = {!!} + +pathEmpty→toEmpty : (A : Type) → pathEmpty A → toEmpty A +pathEmpty→toEmpty A p = {!!} diff --git a/1FundamentalGroup/Quest0SideQuests/EmptySolutions.agda b/1FundamentalGroup/Quest0SideQuests/EmptySolutions.agda new file mode 100644 index 0000000..0b3df63 --- /dev/null +++ b/1FundamentalGroup/Quest0SideQuests/EmptySolutions.agda @@ -0,0 +1,30 @@ +module 1FundamentalGroup.Quest0SideQuests.EmptySolutions where + +open import Cubical.Foundations.Prelude +open import Cubical.Data.Empty renaming (rec to ⊥Rec) +open import Cubical.Foundations.Isomorphism renaming (Iso to _≅_) + +toEmpty : (A : Type) → Type +toEmpty A = A → ⊥ + +-- why Type₁ +pathEmpty : (A : Type) → Type₁ +pathEmpty A = A ≡ ⊥ + +isoEmpty : (A : Type) → Type +isoEmpty A = A ≅ ⊥ + +toEmpty→isoEmpty : (A : Type) → toEmpty A → isoEmpty A +toEmpty→isoEmpty A f = iso f ⊥Rec rightInv leftInv where + + rightInv : section f ⊥Rec + rightInv () + + leftInv : retract f ⊥Rec + leftInv x = ⊥Rec (f x) + +isoEmpty→pathEmpty : (A : Type) → isoEmpty A → pathEmpty A +isoEmpty→pathEmpty A = isoToPath + +pathEmpty→toEmpty : (A : Type) → pathEmpty A → toEmpty A +pathEmpty→toEmpty A p = J (λ B q → toEmpty B) (λ x → x) (sym p) diff --git a/_build/2.6.2/agda/1FundamentalGroup/Quest0SideQuests/Empty.agdai b/_build/2.6.2/agda/1FundamentalGroup/Quest0SideQuests/Empty.agdai new file mode 100644 index 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