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felixwellen merged 5 commits into from
Jul 7, 2025
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Decidable reachability for finite quivers #1198

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Quiver reachability
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58 changes: 58 additions & 0 deletions Cubical/Data/FinData/FinSet.agda
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Original file line number Diff line number Diff line change
@@ -0,0 +1,58 @@
{-# OPTIONS --safe #-}
module Cubical.Data.FinData.FinSet where

open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.Function
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Univalence

open import Cubical.Data.Fin
using ()
renaming (Fin to Finℕ)
open import Cubical.Data.FinData
open import Cubical.Data.Nat as Nat
using (ℕ ; _+_)
open import Cubical.Data.Nat.Order
open import Cubical.Data.Sigma
open import Cubical.Data.SumFin as SumFin
using (fzero ; fsuc)
renaming (Fin to SumFin)
open import Cubical.Data.FinSet

open import Cubical.Relation.Nullary

private
variable
ℓ : Level
m n : ℕ

FinFinℕIso : Iso (Fin n) (Finℕ n)
FinFinℕIso = iso
(λ k → toℕ k , toℕ<n k)
(uncurry (fromℕ' _))
(λ (k , k<n) → Σ≡Prop (λ _ → isProp≤) (toFromId' _ k k<n))
(λ k → fromToId' _ k (toℕ<n k))

Fin≃Finℕ : Fin n ≃ Finℕ n
Fin≃Finℕ = isoToEquiv FinFinℕIso

FinΣ≃ : (n : ℕ) (m : FinVec ℕ n) → Σ (Fin n) (Fin ∘ m) ≃ Fin (foldrFin _+_ 0 m)
FinΣ≃ n m =
Σ-cong-equiv SumFin.FinData≃SumFin (λ fn → SumFin.≡→FinData≃SumFin
(congS m (sym (retIsEq (SumFin.FinData≃SumFin .snd) fn))))
∙ₑ SumFin.SumFinΣ≃ n (m ∘ SumFin.SumFin→FinData)
∙ₑ invEquiv (SumFin.≡→FinData≃SumFin (sum≡ n m))
where
sum≡ : (n : ℕ) (m : FinVec ℕ n) →
foldrFin _+_ 0 m ≡ SumFin.totalSum (m ∘ SumFin.SumFin→FinData)
sum≡ = Nat.elim (λ _ → refl) λ n x m → congS (m zero +_) (x (m ∘ suc))

DecΣ : (n : ℕ) →
(P : FinVec (Type ℓ) n) → (∀ k → Dec (P k)) → Dec (Σ (Fin n) P)
DecΣ n P decP = EquivPresDec
(Σ-cong-equiv-fst (invEquiv SumFin.FinData≃SumFin))
(SumFin.DecΣ n (P ∘ SumFin.SumFin→FinData) (decP ∘ SumFin.SumFin→FinData))

isFinSetFinData : ∀ {n} → isFinSet (Fin n)
isFinSetFinData = subst isFinSet (sym SumFin.FinData≡SumFin) isFinSetFin
45 changes: 45 additions & 0 deletions Cubical/Data/FinSet/Properties.agda
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Expand Up @@ -13,10 +13,13 @@ open import Cubical.Foundations.Equiv renaming (_∙ₑ_ to _⋆_)
open import Cubical.HITs.PropositionalTruncation as Prop

open import Cubical.Data.Nat
import Cubical.Data.Nat.Order.Recursive as Ord
import Cubical.Data.Fin as Fin
open import Cubical.Data.Unit
open import Cubical.Data.Bool
open import Cubical.Data.Empty as Empty
open import Cubical.Data.Sigma
open import Cubical.Data.Sum

open import Cubical.Data.Fin.Base renaming (Fin to Finℕ)
open import Cubical.Data.SumFin
Expand Down Expand Up @@ -94,3 +97,45 @@ transpFamily :
{A : Type ℓ}{B : A → Type ℓ'}
→ ((n , e) : isFinOrd A) → (x : A) → B x ≃ B (invEq e (e .fst x))
transpFamily {B = B} (n , e) x = pathToEquiv (λ i → B (retEq e x (~ i)))

isContr→isFinOrd : ∀ {ℓ} → {A : Type ℓ} →
isContr A → isFinOrd A
isContr→isFinOrd isContrA = 1 , isContr→Equiv isContrA isContrSumFin1

isFinSet⊥ : isFinSet ⊥
isFinSet⊥ = isFinSetFin

isFinSetLift :
{L L' : Level} →
{A : Type L} →
isFinSet A → isFinSet (Lift {L}{L'} A)
fst (isFinSetLift {A = A} isFinSetA) = isFinSetA .fst
snd (isFinSetLift {A = A} isFinSetA) =
Prop.elim
{P = λ _ → ∥ Lift A ≃ Fin (isFinSetA .fst) ∥₁}
(λ [a] → isPropPropTrunc )
(λ A≅Fin → ∣ compEquiv (invEquiv (LiftEquiv {A = A})) A≅Fin ∣₁)
(isFinSetA .snd)

EquivPresIsFinOrd : ∀ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} → A ≃ B → isFinOrd A → isFinOrd B
EquivPresIsFinOrd e (_ , p) = _ , compEquiv (invEquiv e) p

isFinOrdFin : ∀ {n} → isFinOrd (Fin n)
isFinOrdFin {n} = n , (idEquiv (Fin n))

isFinOrd⊥ : isFinOrd ⊥
fst isFinOrd⊥ = 0
snd isFinOrd⊥ = idEquiv ⊥

isFinOrdUnit : isFinOrd Unit
isFinOrdUnit =
EquivPresIsFinOrd
(isContr→Equiv isContrSumFin1 isContrUnit) isFinOrdFin

takeFirstFinOrd : ∀ {ℓ} → (A : Type ℓ) →
(the-ord : isFinOrd A) → 0 Ord.< the-ord .fst → A
takeFirstFinOrd A (suc n , the-eq) x =
the-eq .snd .equiv-proof (Fin→SumFin (Fin.fromℕ≤ 0 n x)) .fst .fst

isFinSet⊤ : isFinSet ⊤
isFinSet⊤ = 1 , ∣ invEquiv ⊎-IdR-⊥-≃ ∣₁
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