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Cleanup finset and modulo (#92)

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Patrick Stevens
2020-01-01 10:14:55 +00:00
committed by GitHub
parent b6ef9b46f2
commit 019a9d9a07
66 changed files with 1154 additions and 1431 deletions
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15
Sets/Cardinality/Infinite/Definition.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K #-}
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import LogicalFormulae
open import Numbers.Naturals.Naturals
open import Numbers.Naturals.Definition
open import Numbers.Naturals.Order
open import Sets.FinSet.Definition
module Sets.Cardinality.Infinite.Definition where
InfiniteSet : {a : _} (A : Set a) → Set a
InfiniteSet A = (n : ℕ) → (f : FinSet n → A) → (Bijection f) → False

32
Sets/Cardinality/Infinite/Examples.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K #-}
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import LogicalFormulae
open import Numbers.Naturals.Naturals
open import Numbers.Naturals.Definition
open import Numbers.Naturals.Order
open import Sets.FinSet.Definition
open import Sets.FinSet.Lemmas
open import Sets.Cardinality.Infinite.Definition
open import Sets.Cardinality.Finite.Lemmas
module Sets.Cardinality.Infinite.Examples where
ℕIsInfinite : InfiniteSet ℕ
ℕIsInfinite n f bij = pigeonhole (le 0 refl) badInj
where
inv : ℕ → FinSet n
inv = Invertible.inverse (bijectionImpliesInvertible bij)
invInj : Injection inv
invInj = Bijection.inj (invertibleImpliesBijection (inverseIsInvertible (bijectionImpliesInvertible bij)))
bumpUp : FinSet n → FinSet (succ n)
bumpUp = intoSmaller (le 0 refl)
bumpUpInj : Injection bumpUp
bumpUpInj = intoSmallerInj (le 0 refl)
nextInj : Injection (toNat {succ n})
nextInj = finsetInjectIntoℕ {succ n}
bad : FinSet (succ n) → FinSet n
bad a = (inv (toNat a))
badInj : Injection bad
badInj = injComp nextInj invInj

15
Sets/Cardinality/Infinite/Lemmas.agda Normal file
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{-# OPTIONS --safe --warning=error --without-K #-}
open import Functions
open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
open import LogicalFormulae
open import Numbers.Naturals.Naturals
open import Numbers.Naturals.Definition
open import Numbers.Naturals.Order
open import Sets.Cardinality.Infinite.Definition
open import Sets.FinSet.Definition
module Sets.Cardinality.Infinite.Lemmas where
finsetNotInfinite : {n : ℕ} → InfiniteSet (FinSet n) → False
finsetNotInfinite {n} isInfinite = isInfinite n id idIsBijective