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[ refactor ] Data.Fin.Properties.decFinSubset #2793

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[ refactor ] Data.Fin.Properties.decFinSubset #2793

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fdc2c02
[ refactoring ] decidable `Data.Fin.Properties`
jamesmckinna Jul 30, 2025
2af5d21
refactor: make `variable`s `P`, `Q` local again
jamesmckinna Aug 2, 2025
ef5b6cf
Merge branch 'agda:master' into decFinSubset
jamesmckinna Oct 1, 2025
ee4d349
refactor: use flipped contradiction
jamesmckinna Oct 1, 2025
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64 changes: 38 additions & 26 deletions src/Data/Fin/Properties.agda
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Original file line number Diff line number Diff line change
Expand Up @@ -29,7 +29,7 @@ open import Data.Product.Properties using (,-injective)
open import Data.Product.Algebra using (×-cong)
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]; [_,_]′)
open import Data.Sum.Properties using ([,]-map; [,]-∘)
open import Function.Base using (_∘_; id; _$_; flip)
open import Function.Base using (_∘_; id; _$_; const; λ-; _$-)
open import Function.Bundles using (Injection; _↣_; _⇔_; _↔_; mk⇔; mk↔ₛ′)
open import Function.Definitions using (Injective; Surjective)
open import Function.Consequences.Propositional using (contraInjective)
Expand All @@ -49,19 +49,21 @@ open import Relation.Binary.PropositionalEquality as ≡
using (≡-≟-identity; ≢-≟-identity)
open import Relation.Nullary.Decidable as Dec
using (Dec; _because_; yes; no; _×-dec_; _⊎-dec_; map′)
open import Relation.Nullary.Negation.Core using (¬_; contradiction)
open import Relation.Nullary.Negation.Core
using (¬_; contradiction; contradiction′)
open import Relation.Nullary.Reflects using (invert)
open import Relation.Unary as U
using (U; Pred; Decidable; _⊆_; Satisfiable; Universal)
open import Relation.Unary.Properties using (U?)

private
variable
a : Level
a p q : Level
A : Set a
m n o : ℕ
i j : Fin n


------------------------------------------------------------------------
-- Fin
------------------------------------------------------------------------
Expand Down Expand Up @@ -995,7 +997,7 @@ pinch-injective {i = suc i} {suc j} {suc k} 1+i≢j 1+i≢k eq =
-- Quantification
------------------------------------------------------------------------

module _ {p} {P : Pred (Fin (suc n)) p} where
module _ {P : Pred (Fin (suc n)) p} where

∀-cons : P zero → Π[ P ∘ suc ] → Π[ P ]
∀-cons z s zero = z
Expand All @@ -1017,33 +1019,19 @@ module _ {p} {P : Pred (Fin (suc n)) p} where
⊎⇔∃ : (P zero ⊎ ∃⟨ P ∘ suc ⟩) ⇔ ∃⟨ P ⟩
⊎⇔∃ = mk⇔ [ ∃-here , ∃-there ] ∃-toSum

decFinSubset : ∀ {p q} {P : Pred (Fin n) p} {Q : Pred (Fin n) q} →
Decidable Q → (∀ {i} → Q i → Dec (P i)) → Dec (Q ⊆ P)
decFinSubset {zero} {_} {_} Q? P? = yes λ {}
decFinSubset {suc n} {P = P} {Q} Q? P?
with Q? zero | ∀-cons {P = λ x → Q x → P x}
... | false because [¬Q0] | cons =
map′ (λ f {x} → cons (⊥-elim ∘ invert [¬Q0]) (λ x → f {x}) x)
(λ f {x} → f {suc x})
(decFinSubset (Q? ∘ suc) P?)
... | true because [Q0] | cons =
map′ (uncurry λ P0 rec {x} → cons (λ _ → P0) (λ x → rec {x}) x)
< _$ invert [Q0] , (λ f {x} → f {suc x}) >
(P? (invert [Q0]) ×-dec decFinSubset (Q? ∘ suc) P?)

any? : ∀ {p} {P : Pred (Fin n) p} → Decidable P → Dec (∃ P)
any? {zero} {P = _} P? = no λ { (() , _) }
any? {suc n} {P = P} P? = Dec.map ⊎⇔∃ (P? zero ⊎-dec any? (P? ∘ suc))

all? : ∀ {p} {P : Pred (Fin n) p} → Decidable P → Dec (∀ f → P f)
all? P? = map′ (λ ∀p f → ∀p tt) (λ ∀p {x} _ → ∀p x)
(decFinSubset U? (λ {f} _ → P? f))
any? : ∀ {P : Pred (Fin n) p} → Decidable P → Dec (∃ P)
any? {zero} P? = no λ{ (() , _) }
any? {suc _} P? = Dec.map ⊎⇔∃ (P? zero ⊎-dec any? (P? ∘ suc))

all? : ∀ {P : Pred (Fin n) p} → Decidable P → Dec (∀ i → P i)
all? {zero} P? = yes λ()
all? {suc _} P? = Dec.map ∀-cons-⇔ (P? zero ×-dec all? (P? ∘ suc))

private
-- A nice computational property of `all?`:
-- The boolean component of the result is exactly the
-- obvious fold of boolean tests (`foldr _∧_ true`).
note : ∀ {p} {P : Pred (Fin 3) p} (P? : Decidable P) →
note : ∀ {P : Pred (Fin 3) p} (P? : Decidable P) →
∃ λ z → Dec.does (all? P?) ≡ z
note P? = Dec.does (P? 0F) ∧ Dec.does (P? 1F) ∧ Dec.does (P? 2F) ∧ true
, refl
Expand All @@ -1066,6 +1054,30 @@ private
¬ (∀ i → P i) → (∃ λ i → ¬ P i)
¬∀⟶∃¬ n P P? ¬P = map id proj₁ (¬∀⟶∃¬-smallest n P P? ¬P)

-- Kleisli lifting of Dec over Unary subset relation

decFinSubset : ∀ {P : Pred (Fin n) p} {Q : Pred (Fin n) q} →
Decidable Q → Q ⊆ Dec ∘ P → Dec (Q ⊆ P)
decFinSubset {zero} {_} {_} Q? P? = yes λ{}
decFinSubset {suc _} {P = P} {Q = Q} Q? P? = dec[Q⊆P]
module DecFinSubset where
Q⊆₀P = Q 0F → P 0F
Q⊆ₛP = Q ∘ suc ⊆ P ∘ suc

cons : Q⊆₀P → Q⊆ₛP → Q ⊆ P
cons q₀⊆p₀ qₛ⊆pₛ = ∀-cons {P = Q U.⇒ P} q₀⊆p₀ (λ- qₛ⊆pₛ) $-

ih : Dec Q⊆ₛP
ih = decFinSubset (Q? ∘ suc) P?

Q⊆P⇒Q⊆ₛP : Q ⊆ P → Q⊆ₛP
Q⊆P⇒Q⊆ₛP q⊆p {x} = q⊆p {suc x}

dec[Q⊆P] : Dec (Q ⊆ P)
dec[Q⊆P] with Q? zero
... | no ¬q₀ = map′ (cons (contradiction′ ¬q₀)) Q⊆P⇒Q⊆ₛP ih
... | yes q₀ = map′ (uncurry (cons ∘ const)) < _$ q₀ , Q⊆P⇒Q⊆ₛP > (P? q₀ ×-dec ih)

------------------------------------------------------------------------
-- Properties of functions to and from Fin
------------------------------------------------------------------------
Expand Down