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[ refactor ] Data.Fin.Properties following #2746 #2782

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[ refactor ] Data.Fin.Properties following #2746 #2782

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8f714db
refactor: following #2746
jamesmckinna Jul 25, 2025
f9e3203
refactor: deploy the `flip contradiction` workaround
jamesmckinna Jul 26, 2025
e858014
refactor: prefer `<⇒notInjective` over `injective⇒≤`
jamesmckinna Jul 26, 2025
cffce87
fix: whitespace
jamesmckinna Jul 26, 2025
92477cf
fix: DRY with `private` specialisation
jamesmckinna Aug 2, 2025
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32 changes: 17 additions & 15 deletions src/Data/Fin/Properties.agda
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Original file line number Diff line number Diff line change
Expand Up @@ -1035,24 +1035,26 @@ pigeonhole : m ℕ.< n → (f : Fin n → Fin m) → ∃₂ λ i j → i < j ×
pigeonhole z<s f = contradiction (f zero) λ()
pigeonhole (s<s m<n@(s≤s _)) f with any? (λ k → f zero ≟ f (suc k))
... | yes (j , f₀≡fⱼ) = zero , suc j , z<s , f₀≡fⱼ
... | no f₀≢fₖ
with i , j , i<j , fᵢ≡fⱼ ← pigeonhole m<n (λ j → punchOut (f₀≢fₖ ∘ (j ,_ )))
= suc i , suc j , s<s i<j , punchOut-injective (f₀≢fₖ ∘ (i ,_)) _ fᵢ≡fⱼ

injective⇒≤ : ∀ {f : Fin m → Fin n} → Injective _≡_ _≡_ f → m ℕ.≤ n
injective⇒≤ {zero} {_} {f} _ = z≤n
injective⇒≤ {suc _} {zero} {f} _ = contradiction (f zero) ¬Fin0
injective⇒≤ {suc _} {suc _} {f} inj = s≤s (injective⇒≤ (λ eq →
suc-injective (inj (punchOut-injective
(contraInjective inj 0≢1+n)
(contraInjective inj 0≢1+n) eq))))
... | no f₀≢fₖ =
let i , j , i<j , fᵢ≡fⱼ = pigeonhole m<n (λ j → punchOut (f₀≢fₖ ∘ (j ,_ )))
in suc i , suc j , s<s i<j , punchOut-injective (f₀≢fₖ ∘ (i ,_)) _ fᵢ≡fⱼ

<⇒notInjective : ∀ {f : Fin m → Fin n} → n ℕ.< m → ¬ (Injective _≡_ _≡_ f)
<⇒notInjective n<m inj = ℕ.≤⇒≯ (injective⇒≤ inj) n<m
<⇒notInjective {f = f} n<m inj =
let i , j , i<j , fᵢ≡fⱼ = pigeonhole n<m f in <-irrefl (inj fᵢ≡fⱼ) i<j

-- specialised to m = suc n

private
notInjective-Fin[1+n]→Fin[n] : ∀ {f : Fin (suc n) → Fin n} → ¬ (Injective _≡_ _≡_ f)
notInjective-Fin[1+n]→Fin[n] {n = n} = <⇒notInjective (ℕ.n<1+n n)

injective⇒≤ : ∀ {f : Fin m → Fin n} → Injective _≡_ _≡_ f → m ℕ.≤ n
injective⇒≤ = ℕ.≮⇒≥ ∘ flip <⇒notInjective

ℕ→Fin-notInjective : ∀ (f : ℕ → Fin n) → ¬ (Injective _≡_ _≡_ f)
ℕ→Fin-notInjective f inj = ℕ.<-irrefl refl
(injective⇒≤ (Comp.injective _≡_ _≡_ _≡_ toℕ-injective inj))
ℕ→Fin-notInjective f =
notInjective-Fin[1+n]→Fin[n] ∘ Comp.injective _≡_ _≡_ _≡_ toℕ-injective

-- Cantor-Schröder-Bernstein for finite sets

Expand All @@ -1067,7 +1069,7 @@ injective⇒existsPivot : ∀ {f : Fin n → Fin m} → Injective _≡_ _≡_ f
injective⇒existsPivot {f = f} f-injective i
with any? (λ j → j ≤? i ×-dec i ≤? f j)
... | yes result = result
... | no ¬result = contradiction (injective⇒≤ f∘inject!-injective) ℕ.1+n≰n
... | no ¬result = flip contradiction notInjective-Fin[1+n]→Fin[n] f∘inject!-injective
where
fj<i : (j : Fin′ (suc i)) → f (inject! j) < i
fj<i j with f (inject! j) <? i
Expand Down