69 lines
2.4 KiB
Agda
69 lines
2.4 KiB
Agda
{-# OPTIONS --cubical #-}
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module jasontime where
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open import Cubical.Foundations.Equiv
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open import Cubical.Foundations.Function
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open import Cubical.Foundations.Isomorphism
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open import Cubical.Foundations.Prelude
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open import Cubical.Foundations.Univalence
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open import Data.Product
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open import Relation.Nullary
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open import Function.LeftInverse hiding (id; _∘_)
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open import Relation.Binary.PropositionalEquality hiding (_≡_)
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-- Cribbed from https://stackoverflow.com/a/13441191
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Surjective : ∀ {a b} {A : Set a} {B : Set b} → (A → B) → Set _
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Surjective {A = A} {B = B} f = ∀ (y : B) → ∃ λ (x : A) → f x ≡ y
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Injective : ∀ {a b} {A : Set a} {B : Set b} → (A → B) → Set _
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Injective {A = A} {B = B} f = ∀ (x y : A) → f x ≡ f y → x ≡ y
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-- contrapositive : {A B : Set} → (A → B) → (¬ B → ¬ A)
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-- contrapositive f g = g ∘ f
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--
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-- Prove: f x ≡ f y → x ≡ y
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-- Contra: x ≢ y → f x ≢ f y
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-- contrapositive-statement : {S T : Set} → (f : S → T) → Surjective f → (x y : S) → x ≡ y → f x ≡ f y
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--
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-- wtf : {A : Set} {x y : A} → ¬ (x ≢ y) → x ≡ y
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-- wtf s = ?
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id : {A : Set} → A → A
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id x = x
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surjective-is-right-inverse : {S T : Set} → (f : S → T) → Surjective f → ∃ (λ g → f ∘ g ≡ id)
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surjective-is-right-inverse f sj .fst t = Σ.fst (sj t)
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surjective-is-right-inverse {S} {T} f sj .snd =
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let g = Σ.fst (surjective-is-right-inverse f sj) in
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?
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-- funExt (λ s i →
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-- let t = f s in
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-- let surjectivity = sj t in
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-- let sj-proof = Σ.snd surjectivity in
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-- let g∘f = g ∘ f in
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-- let Fuck! = transport {A = T} {B = T} ? ? in
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-- ?
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-- )
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-- funExt {A = S} {B = λ x i → ?} (λ x → ?) ? ?
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left-inverse-is-injective : {S T : Set} → (g : T → S) → ∃ (λ f → f ∘ g ≡ id) → Injective g
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jasontime-lemma : {S T : Set} → (f : S → T) → Surjective f → ∃ {A = T → S} (λ g → Injective g)
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jasontime-lemma {S} {T} f sj .fst t =
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let rinv = surjective-is-right-inverse f sj in
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let rinv-fn = Σ.fst rinv in
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rinv-fn t
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jasontime-lemma f sj .snd x y p =
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?
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where
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rinv = surjective-is-right-inverse f sj
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g = Σ.fst rinv
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TheLeftInverse : ∃ (λ f → f ∘ g ≡ id)
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TheLeftInverse .fst = ?
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TheLeftInverse .snd = ?
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Fuck! = left-inverse-is-injective ? TheLeftInverse
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rinv-prf = Σ.snd rinv
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left-inv = Σ.fst TheLeftInverse
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