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[ refactor ] make n≢i : n ≢ toℕ i argument to lower₁ irrelevant #2783

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[ refactor ] make n≢i : n ≢ toℕ i argument to lower₁ irrelevant #2783

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617bd62
refactor: make `n≢i : n ≢ toℕ i` argument to `lower₁` irrelevant
jamesmckinna Jul 25, 2025
1f67bef
tightne imports
jamesmckinna Jul 25, 2025
156e93f
second equivalence proof: pick one
jamesmckinna Jul 25, 2025
c7bb4ca
fix: uncommitted proof
jamesmckinna Jul 25, 2025
31633ff
refactor: use `contradiction-irr` instead, following #2785
jamesmckinna Jul 26, 2025
e7b89f9
fix: `CHANGELOG`
jamesmckinna Jul 27, 2025
8cfb3f5
Merge branch 'master' into Fin-lower-properties
jamesmckinna Jul 31, 2025
2b3aa99
Merge branch 'master' into Fin-lower-properties
jamesmckinna Aug 3, 2025
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8 changes: 4 additions & 4 deletions src/Data/Fin/Base.agda
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Expand Up @@ -19,7 +19,7 @@ open import Level using (0ℓ)
open import Relation.Binary.Core using (Rel)
open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≢_; refl; cong)
open import Relation.Binary.Indexed.Heterogeneous.Core using (IRel)
open import Relation.Nullary.Negation.Core using (¬_; contradiction)
open import Relation.Nullary.Negation.Core using (¬_; contradiction-irr)

private
variable
Expand Down Expand Up @@ -116,8 +116,8 @@ inject≤ {n = suc _} (suc i) m≤n = suc (inject≤ i (ℕ.s≤s⁻¹ m≤n))

-- lower₁ "i" _ = "i".

lower₁ : ∀ (i : Fin (suc n)) → n ≢ toℕ i → Fin n
lower₁ {zero} zero ne = contradiction refl ne
lower₁ : ∀ (i : Fin (suc n)) → .(n ≢ toℕ i) → Fin n
lower₁ {zero} zero ne = contradiction-irr refl ne
lower₁ {suc n} zero _ = zero
lower₁ {suc n} (suc i) ne = suc (lower₁ i (ne ∘ cong suc))

Expand Down Expand Up @@ -253,7 +253,7 @@ opposite {suc n} (suc i) = inject₁ (opposite i)
-- McBride's "First-order unification by structural recursion".

punchOut : ∀ {i j : Fin (suc n)} → i ≢ j → Fin n
punchOut {_} {zero} {zero} i≢j = contradiction refl i≢j
punchOut {_} {zero} {zero} i≢j = contradiction-irr refl i≢j
punchOut {_} {zero} {suc j} _ = j
punchOut {suc _} {suc i} {zero} _ = zero
punchOut {suc _} {suc i} {suc j} i≢j = suc (punchOut (i≢j ∘ cong suc))
Expand Down
46 changes: 29 additions & 17 deletions src/Data/Fin/Properties.agda
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Expand Up @@ -46,8 +46,9 @@ open import Relation.Binary.PropositionalEquality.Properties as ≡
using (module ≡-Reasoning)
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.Reflects using (Reflects; invert)
open import Relation.Nullary.Negation.Core
using (¬_; contradiction; contradiction-irr)
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?)
Expand Down Expand Up @@ -506,30 +507,30 @@ inject!-< {suc n} {suc i} (suc k) = s≤s (inject!-< k)
-- lower₁
------------------------------------------------------------------------

toℕ-lower₁ : ∀ i (p : n ≢ toℕ i) → toℕ (lower₁ i p) ≡ toℕ i
toℕ-lower₁ {ℕ.zero} zero p = contradiction refl p
toℕ-lower₁ {ℕ.suc m} zero p = refl
toℕ-lower₁ {ℕ.suc m} (suc i) p = cong ℕ.suc (toℕ-lower₁ i (p ∘ cong ℕ.suc))
toℕ-lower₁ : ∀ i .(n≢i : n ≢ toℕ i) → toℕ (lower₁ i n≢i) ≡ toℕ i
toℕ-lower₁ {ℕ.zero} zero 0≢0 = contradiction-irr refl 0≢0
toℕ-lower₁ {ℕ.suc m} zero _ = refl
toℕ-lower₁ {ℕ.suc m} (suc i) ne = cong ℕ.suc (toℕ-lower₁ i (ne ∘ cong ℕ.suc))

lower₁-injective : ∀ {n≢i : n ≢ toℕ i} {n≢j : n ≢ toℕ j} →
lower₁-injective : ∀ .{n≢i : n ≢ toℕ i} .{n≢j : n ≢ toℕ j} →
lower₁ i n≢i ≡ lower₁ j n≢j → i ≡ j
lower₁-injective {zero} {zero} {_} {n≢i} {_} _ = contradiction refl n≢i
lower₁-injective {zero} {_} {zero} {_} {n≢j} _ = contradiction refl n≢j
lower₁-injective {suc n} {zero} {zero} {_} {_} refl = refl
lower₁-injective {suc n} {suc i} {suc j} {n≢i} {n≢j} eq =
lower₁-injective {zero} {zero} {_} {0≢0} {_} _ = contradiction-irr refl 0≢0
lower₁-injective {zero} {_} {zero} {_} {0≢0} _ = contradiction-irr refl 0≢0
lower₁-injective {suc n} {zero} {zero} {_} {_} _ = refl
lower₁-injective {suc n} {suc i} {suc j} {_} {_} eq =
cong suc (lower₁-injective (suc-injective eq))

------------------------------------------------------------------------
-- inject₁ and lower₁

inject₁-lower₁ : ∀ (i : Fin (suc n)) (n≢i : n ≢ toℕ i) →
inject₁-lower₁ : ∀ (i : Fin (suc n)) .(n≢i : n ≢ toℕ i) →
inject₁ (lower₁ i n≢i) ≡ i
inject₁-lower₁ {zero} zero 0≢0 = contradiction refl 0≢0
inject₁-lower₁ {zero} zero 0≢0 = contradiction-irr refl 0≢0
inject₁-lower₁ {suc n} zero _ = refl
inject₁-lower₁ {suc n} (suc i) n+1≢i+1 =
cong suc (inject₁-lower₁ i (n+1≢i+1 ∘ cong suc))

lower₁-inject₁′ : ∀ (i : Fin n) (n≢i : n ≢ toℕ (inject₁ i)) →
lower₁-inject₁′ : ∀ (i : Fin n) .(n≢i : n ≢ toℕ (inject₁ i)) →
lower₁ (inject₁ i) n≢i ≡ i
lower₁-inject₁′ zero _ = refl
lower₁-inject₁′ (suc i) n+1≢i+1 =
Expand All @@ -539,15 +540,15 @@ lower₁-inject₁ : ∀ (i : Fin n) →
lower₁ (inject₁ i) (toℕ-inject₁-≢ i) ≡ i
lower₁-inject₁ i = lower₁-inject₁′ i (toℕ-inject₁-≢ i)

lower₁-irrelevant : ∀ (i : Fin (suc n)) (n≢i₁ n≢i₂ : n ≢ toℕ i) →
lower₁-irrelevant : ∀ (i : Fin (suc n)) .(n≢i₁ n≢i₂ : n ≢ toℕ i) →
lower₁ i n≢i₁ ≡ lower₁ i n≢i₂
lower₁-irrelevant {zero} zero 0≢0 _ = contradiction refl 0≢0
lower₁-irrelevant {zero} zero 0≢0 _ = contradiction-irr refl 0≢0
lower₁-irrelevant {suc n} zero _ _ = refl
lower₁-irrelevant {suc n} (suc i) _ _ =
cong suc (lower₁-irrelevant i _ _)

inject₁≡⇒lower₁≡ : ∀ {i : Fin n} {j : Fin (ℕ.suc n)} →
(n≢j : n ≢ toℕ j) → inject₁ i ≡ j → lower₁ j n≢j ≡ i
.(n≢j : n ≢ toℕ j) → inject₁ i ≡ j → lower₁ j n≢j ≡ i
inject₁≡⇒lower₁≡ n≢j i≡j = inject₁-injective (trans (inject₁-lower₁ _ n≢j) (sym i≡j))

------------------------------------------------------------------------
Expand All @@ -561,6 +562,17 @@ lower-injective {n = suc n} zero zero eq = refl
lower-injective {n = suc n} (suc i) (suc j) eq =
cong suc (lower-injective i j (suc-injective eq))

lower₁≗lower : ∀ (i : Fin (suc n)) .(n≢i : n ≢ toℕ i) →
lower₁ i n≢i ≡ lower i (ℕ.≤∧≢⇒< (toℕ≤pred[n]′ i) (n≢i ∘ sym))
lower₁≗lower {n = zero} zero 0≢0 = contradiction-irr refl 0≢0
lower₁≗lower {n = suc _ } zero _ = refl
lower₁≗lower {n = suc _ } (suc i) ne = cong suc (lower₁≗lower i (ne ∘ cong suc))

lower≗lower₁ : ∀ (i : Fin (suc n)) .(i<n : toℕ i ℕ.< n) →
lower i i<n ≡ lower₁ i (ℕ.<⇒≢ i<n ∘ sym)
lower≗lower₁ {n = suc _ } zero _ = refl
lower≗lower₁ {n = suc _ } (suc i) lt = cong suc (lower≗lower₁ i (ℕ.s<s⁻¹ lt))

------------------------------------------------------------------------
-- inject≤
------------------------------------------------------------------------
Expand Down
4 changes: 2 additions & 2 deletions src/Relation/Nullary/Negation/Core.agda
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Expand Up @@ -47,11 +47,11 @@ _¬-⊎_ = [_,_]
------------------------------------------------------------------------
-- Uses of negation

contradiction-irr : .A → ¬ A → Whatever
contradiction-irr : .A → .(¬ A) → Whatever
contradiction-irr a ¬a = ⊥-elim-irr (¬a a)

contradiction : A → ¬ A → Whatever
contradiction a = contradiction-irr a
contradiction a ¬a = contradiction-irr a ¬a

contradiction₂ : A ⊎ B → ¬ A → ¬ B → Whatever
contradiction₂ (inj₁ a) ¬a ¬b = contradiction a ¬a
Expand Down