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Use the faster quotient and remainder in the whole library #1284

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28 changes: 11 additions & 17 deletions Cubical/Algebra/Group/Instances/IntMod.agda
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Original file line number Diff line number Diff line change
Expand Up @@ -95,13 +95,7 @@ snd Bool≅ℤGroup/2 =
→ Path (Fin (suc n)) (-ₘ (1 mod (suc n) , <→<ᵗ (mod< n 1)))
(n mod (suc n) , <→<ᵗ (mod< n n))
-ₘ1-id zero = refl
-ₘ1-id (suc n) =
cong -ₘ_ (FinPathℕ {suc (suc n)} ((1 mod suc (suc n)) , <→<ᵗ (mod< (suc n) 1)) 1
(modIndBase (suc n) 1 (n , +-comm n 2)) .snd)
∙ Σ≡Prop (λ z → isProp<ᵗ {n = z} {m = suc (suc n)})
((+inductionBase (suc n) _
(λ x _ → ((suc (suc n)) ∸ x) mod (suc (suc n))) λ _ x → x) 1
(n , (+-comm n 2)))
-ₘ1-id (suc n) = refl

suc-ₘ1 : (n y : ℕ)
→ ((suc y mod suc n) , <→<ᵗ (mod< n (suc y))) -ₘ (1 mod (suc n) , <→<ᵗ (mod< n 1))
Expand Down Expand Up @@ -143,9 +137,9 @@ suc-ₘ1 (suc n) y =
sym (GroupTheory.invInv (ℤGroup/ (suc n)) _)
∙ cong -ₘ_
(GroupTheory.invDistr (ℤGroup/ (suc n))
(modInd n 1 , <→<ᵗ (mod< n 1)) (-ₘ (modInd n (suc y) , <→<ᵗ (mod< n (suc y))))
∙ cong (_-ₘ (modInd n 1 , <→<ᵗ (mod< n 1)))
(GroupTheory.invInv (ℤGroup/ (suc n)) (modInd n (suc y) , <→<ᵗ (mod< n (suc y))))
(1 mod suc n , <→<ᵗ (mod< n 1)) (-ₘ (suc y mod suc n , <→<ᵗ (mod< n (suc y))))
∙ cong (_-ₘ (1 mod suc n , <→<ᵗ (mod< n 1)))
(GroupTheory.invInv (ℤGroup/ (suc n)) (suc y mod suc n , <→<ᵗ (mod< n (suc y))))
∙ suc-ₘ1 n y)

isHomℤ→Fin : (n : ℕ) → IsGroupHom (snd ℤGroup) (ℤ→Fin n) (snd (ℤGroup/ (suc n)))
Expand All @@ -161,8 +155,8 @@ isHomℤ→Fin n =
∙∙ ℤ→Fin-presinv n (pos (suc x) +ℤ (pos (suc y)))
∙∙ cong -ₘ_ (pos+case (suc x) (pos (suc y)))
∙∙ GroupTheory.invDistr (ℤGroup/ (suc n))
(modInd n (suc x)
, <→<ᵗ (mod< n (suc x))) (modInd n (suc y) , <→<ᵗ (mod< n (suc y)))
(suc x mod suc n
, <→<ᵗ (mod< n (suc x))) (suc y mod suc n , <→<ᵗ (mod< n (suc y)))
∙∙ +ₘ-comm (ℤ→Fin n (negsuc y)) (ℤ→Fin n (negsuc x))}
where
+1case : (y : ℤ) → ℤ→Fin n (1 +ℤ y) ≡ ℤ→Fin n 1 +ₘ ℤ→Fin n y
Expand All @@ -172,7 +166,7 @@ isHomℤ→Fin n =
∙ Σ≡Prop (λ z → isProp<ᵗ {n = z} {suc n}) (mod+mod≡mod (suc n) 1 (suc y))
+1case (negsuc zero) =
Σ≡Prop (λ z → isProp<ᵗ {n = z} {suc n}) refl
∙ sym (GroupStr.·InvR (snd (ℤGroup/ (suc n))) (modInd n 1 , <→<ᵗ (mod< n 1)))
∙ sym (GroupStr.·InvR (snd (ℤGroup/ (suc n))) (1 mod suc n , <→<ᵗ (mod< n 1)))
+1case (negsuc (suc y)) =
Σ≡Prop (λ z → isProp<ᵗ {n = z} {suc n})
(cong fst (cong (ℤ→Fin n) (+Comm 1 (negsuc (suc y))))
Expand All @@ -191,13 +185,13 @@ isHomℤ→Fin n =
cong (ℤ→Fin n) (cong (_+ℤ y) (+Comm (pos (suc x)) 1)
∙ sym (+Assoc 1 (pos (suc x)) y))
∙∙ +1case (pos (suc x) +ℤ y)
∙∙ (cong ((modInd n 1 , <→<ᵗ (mod< n 1)) +ₘ_) (pos+case (suc x) y)
∙∙ sym (+ₘ-assoc (modInd n 1 , <→<ᵗ (mod< n 1))
(modInd n (suc x) , <→<ᵗ (mod< n (suc x))) (ℤ→Fin n y))
∙∙ (cong ((1 mod suc n , <→<ᵗ (mod< n 1)) +ₘ_) (pos+case (suc x) y)
∙∙ sym (+ₘ-assoc (1 mod suc n , <→<ᵗ (mod< n 1))
(suc x mod suc n , <→<ᵗ (mod< n (suc x))) (ℤ→Fin n y))
∙∙ cong (_+ₘ ℤ→Fin n y) (lem x))
where
lem : (x : ℕ)
→ (modInd n 1 , <→<ᵗ (mod< n 1)) +ₘ (modInd n (suc x) , <→<ᵗ (mod< n (suc x)))
→ (1 mod suc n , <→<ᵗ (mod< n 1)) +ₘ (suc x mod suc n , <→<ᵗ (mod< n (suc x)))
≡ ℤ→Fin n (pos (suc (suc x)))
lem x =
Σ≡Prop (λ z → isProp<ᵗ {n = z} {suc n}) (sym (mod+mod≡mod (suc n) 1 (suc x)))
Expand Down
2 changes: 1 addition & 1 deletion Cubical/Algebra/Group/ZAction.agda
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Expand Up @@ -556,7 +556,7 @@ module _ (f : GroupHom ℤGroup ℤGroup) where
∙∙ cong -_ (cong pos (≡remainder+quotient (suc n) (suc x))))) ∣₁})
BijectionIso.surj (ℤHom→ℤ/im≅ℤ/im1 n p) x =
∣ [ pos (fst x) ]
, (Σ≡Prop (λ _ → isProp<ᵗ) (modIndBase n (fst x) (<ᵗ→< (snd x)))) ∣₁
, (Σ≡Prop (λ _ → isProp<ᵗ) (<→mod≡id (fst x) (suc n) (<ᵗ→< (snd x)))) ∣₁

-- main result
ℤ/imIso : (f : GroupHom ℤGroup ℤGroup)
Expand Down
44 changes: 17 additions & 27 deletions Cubical/Data/Fin/Arithmetic.agda
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Expand Up @@ -21,21 +21,8 @@ snd (_+ₘ_ {n = n} x y) = <→<ᵗ
(mod< n ((fst x) + (fst y)))

-ₘ_ : {n : ℕ} → (x : Fin (suc n)) → Fin (suc n)
fst (-ₘ_ {n = n} x) =
(+induction n _ (λ x _ → ((suc n) ∸ x) mod (suc n)) λ _ x → x) (fst x)
snd (-ₘ_ {n = n} x) = <→<ᵗ (lem (fst x))
where
≡<-trans : {x y z : ℕ} → x < y → x ≡ z → z < y
≡<-trans (k , p) q = k , cong (λ x → k + suc x) (sym q) ∙ p

lem : {n : ℕ} (x : ℕ)
→ (+induction n _ _ _) x < suc n
lem {n = n} =
+induction n _
(λ x p → ≡<-trans (mod< n (suc n ∸ x))
(sym (+inductionBase n _ _ _ x p)))
λ x p → ≡<-trans p
(sym (+inductionStep n _ _ _ x))
fst (-ₘ_ {n = n} x) = (suc n ∸ fst x) mod suc n
snd (-ₘ_ {n = n} x) = <→<ᵗ (mod< n (suc n ∸ fst x))

_-ₘ_ : {n : ℕ} → (x y : Fin (suc n)) → Fin (suc n)
_-ₘ_ x y = x +ₘ (-ₘ y)
Expand Down Expand Up @@ -63,31 +50,34 @@ snd (_·ₘ_ {n = n} x y) = <→<ᵗ (mod< n (fst x · fst y))
+ₘ-lUnit : {n : ℕ} (x : Fin (suc n)) → 0 +ₘ x ≡ x
+ₘ-lUnit {n = n} (x , p) =
Σ≡Prop (λ z → isProp<ᵗ {n = z} {suc n})
(+inductionBase n _ _ _ x (<ᵗ→< p))
(<→mod≡id x (suc n) (<ᵗ→< p))

+ₘ-rUnit : {n : ℕ} (x : Fin (suc n)) → x +ₘ 0 ≡ x
+ₘ-rUnit x = +ₘ-comm x 0 ∙ (+ₘ-lUnit x)

+ₘ-rCancel : {n : ℕ} (x : Fin (suc n)) → x -ₘ x ≡ 0
+ₘ-rCancel {n = n} x =
Σ≡Prop (λ z → isProp<ᵗ {n = z} {suc n})
(cong (λ z → (fst x + z) mod (suc n))
(+inductionBase n _ _ _ (fst x) (<ᵗ→< (snd x)))
∙∙ sym (mod-rCancel (suc n) (fst x) ((suc n) ∸ (fst x)))
∙∙ cong (_mod (suc n)) (+-comm (fst x) ((suc n) ∸ (fst x)))
∙∙ cong (_mod (suc n))
(≤-∸-+-cancel {m = fst x} {n = suc n}
(<-weaken (<ᵗ→< (snd x))))
∙∙ zero-charac (suc n))
+ₘ-rCancel {n = n} (k , p) =
Σ≡Prop (λ z → isProp<ᵗ {n = z} {suc n}) (
(k + (suc n ∸ k) mod suc n) mod suc n ≡⟨ sym (mod-rCancel (suc n) k _) ⟩
(k + (suc n ∸ k)) mod suc n ≡⟨ cong (_mod suc n) (+-comm k _) ⟩
((suc n ∸ k) + k) mod suc n ≡⟨ cong (_mod suc n) (≤-∸-+-cancel (<-weaken {k} {suc n} (<ᵗ→< p))) ⟩
suc n mod suc n ≡⟨ zero-charac (suc n) ⟩
0 ∎)

+ₘ-lCancel : {n : ℕ} (x : Fin (suc n)) → (-ₘ x) +ₘ x ≡ 0
+ₘ-lCancel {n = n} x = +ₘ-comm (-ₘ x) x ∙ +ₘ-rCancel x

-- -- TODO : Ring laws
-- TODO : Ring laws

private
test₁ : Path (Fin 11) (5 +ₘ 10) 4
test₁ = refl

test₂ : Path (Fin 11) (-ₘ 7 +ₘ 5 +ₘ 10) 8
test₂ = refl

test₃ : Path (Fin 1024) (1022 ·ₘ 1023) 2
test₃ = refl

test₄ : Path (Fin 8192) (-ₘ 32 ·ₘ 64 +ₘ 256) 6400
test₄ = refl
4 changes: 2 additions & 2 deletions Cubical/Data/Nat/GCD.agda
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Original file line number Diff line number Diff line change
Expand Up @@ -19,8 +19,8 @@ open import Cubical.Data.Nat.Properties
open import Cubical.Data.Nat.Order
open import Cubical.Data.Nat.Order.Inductive
open import Cubical.Data.Nat.Mod renaming (
quotient'_/_ to _/_ ; remainder'_/_ to _%_
; ≡remainder'+quotient' to ≡%+·/ ; mod'< to %< )
quotient_/_ to _/_ ; remainder_/_ to _%_
; ≡remainder+quotient to ≡%+·/ ; mod< to %< )
open import Cubical.Data.Nat.Divisibility

open import Cubical.Relation.Nullary
Expand Down
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