{-# OPTIONS --safe #-}
module Cubical.Algebra.RingSolver.RawAlgebra where
open import Cubical.Foundations.Prelude
open import Cubical.Data.Sigma
open import Cubical.Data.Nat using (ℕ)
open import Cubical.Data.Int renaming (_+_ to _+ℤ_ ; _·_ to _·ℤ_ ; -_ to -ℤ_ ; _-_ to _-ℤ_ ; +Assoc to +ℤAssoc ; +Comm to +ℤComm ; -DistL· to -ℤDistL·ℤ)
open import Cubical.Algebra.RingSolver.AlmostRing hiding (⟨_⟩)
open import Cubical.Algebra.RingSolver.RawRing renaming (⟨_⟩ to ⟨_⟩ᵣ)
open import Cubical.Algebra.RingSolver.IntAsRawRing
open import Cubical.Algebra.RingSolver.CommRingAsAlmostRing
open import Cubical.Algebra.CommRing
open import Cubical.Algebra.Ring
private
variable
ℓ ℓ′ : Level
record RawAlgebra (R : RawRing ℓ) (ℓ′ : Level) : Type (ℓ-suc (ℓ-max ℓ ℓ′)) where
constructor rawalgebra
field
Carrier : Type ℓ′
scalar : ⟨ R ⟩ᵣ → Carrier
0r : Carrier
1r : Carrier
_+_ : Carrier → Carrier → Carrier
_·_ : Carrier → Carrier → Carrier
-_ : Carrier → Carrier
infixl 8 _·_
infixl 7 -_
infixl 6 _+_
⟨_⟩ : {R : RawRing ℓ} → RawAlgebra R ℓ′ → Type ℓ′
⟨_⟩ = RawAlgebra.Carrier
module _ (R : CommRing ℓ) where
open CommRingStr (snd R)
open Cubical.Algebra.Ring.RingTheory (CommRing→Ring R)
scalarℕ : ℕ → (fst R)
scalarℕ ℕ.zero = 0r
scalarℕ (ℕ.suc ℕ.zero) = 1r
scalarℕ (ℕ.suc (ℕ.suc n)) = 1r + scalarℕ (ℕ.suc n)
scalar : ℤ → (fst R)
scalar (pos k) = scalarℕ k
scalar (negsuc k) = - scalarℕ (ℕ.suc k)
-DistScalar : (k : ℤ)
→ scalar (-ℤ k) ≡ - (scalar k)
-DistScalar (pos ℕ.zero) = sym 0Selfinverse
-DistScalar (pos (ℕ.suc n)) = refl
-DistScalar (negsuc n) = sym (-Idempotent _)
lemmaSuc : (k : ℤ)
→ scalar (sucℤ k) ≡ 1r + scalar k
lemmaSuc (pos ℕ.zero) = sym (+Rid _)
lemmaSuc (pos (ℕ.suc ℕ.zero)) = refl
lemmaSuc (pos (ℕ.suc (ℕ.suc n))) = refl
lemmaSuc (negsuc ℕ.zero) = sym (+Rinv _)
lemmaSuc (negsuc (ℕ.suc n)) =
scalar (negsuc n) ≡⟨ sym (+Lid (scalar (negsuc n))) ⟩
0r + scalar (negsuc n) ≡[ i ]⟨ +Rinv 1r (~ i) + scalar (negsuc n) ⟩
(1r - 1r) + scalar (negsuc n) ≡⟨ sym (+Assoc _ _ _) ⟩
1r + (- 1r + - scalar (pos (ℕ.suc n))) ≡[ i ]⟨ 1r + -Dist 1r (scalar (pos (ℕ.suc n))) i ⟩
1r + -(1r + scalar (pos (ℕ.suc n))) ≡⟨ refl ⟩
1r + -(scalar (pos (ℕ.suc (ℕ.suc n)))) ≡⟨ refl ⟩
1r + scalar (negsuc (ℕ.suc n)) ∎
lemmaPred : (k : ℤ)
→ scalar (predℤ k) ≡ - 1r + scalar k
lemmaPred k = sym(
- 1r + scalar k ≡[ i ]⟨ - 1r + scalar (sucPred k (~ i)) ⟩
- 1r + scalar (sucℤ (predℤ k)) ≡[ i ]⟨ - 1r + lemmaSuc (predℤ k) i ⟩
- 1r + (1r + scalar (predℤ k)) ≡⟨ +Assoc _ _ _ ⟩
(- 1r + 1r) + scalar (predℤ k) ≡[ i ]⟨ +Linv 1r i + scalar (predℤ k) ⟩
0r + scalar (predℤ k) ≡⟨ +Lid _ ⟩
scalar (predℤ k) ∎)
+HomScalar : (k l : ℤ)
→ scalar (k +ℤ l) ≡ (scalar k) + (scalar l)
+HomScalar (pos ℕ.zero) l =
scalar (0 +ℤ l) ≡[ i ]⟨ scalar (sym (pos0+ l) i) ⟩
scalar l ≡⟨ sym (+Lid _) ⟩
0r + scalar l ≡⟨ refl ⟩
scalar 0 + scalar l ∎
+HomScalar (pos (ℕ.suc ℕ.zero)) l =
scalar (1 +ℤ l) ≡[ i ]⟨ scalar (+ℤComm 1 l i) ⟩
scalar (l +ℤ 1) ≡⟨ refl ⟩
scalar (sucℤ l) ≡⟨ lemmaSuc l ⟩
1r + scalar l ≡⟨ refl ⟩
scalar (pos (ℕ.suc ℕ.zero)) + scalar l ∎
+HomScalar (pos (ℕ.suc (ℕ.suc n))) l =
scalar (pos (ℕ.suc (ℕ.suc n)) +ℤ l) ≡⟨ refl ⟩
scalar ((pos (ℕ.suc n) +ℤ 1) +ℤ l) ≡[ i ]⟨ scalar ((+ℤComm (pos (ℕ.suc n)) 1 i) +ℤ l) ⟩
scalar ((1 +ℤ (pos (ℕ.suc n))) +ℤ l) ≡[ i ]⟨ scalar (+ℤAssoc 1 (pos (ℕ.suc n)) l (~ i)) ⟩
scalar (1 +ℤ (pos (ℕ.suc n) +ℤ l)) ≡⟨ +HomScalar (pos (ℕ.suc ℕ.zero)) (pos (ℕ.suc n) +ℤ l) ⟩
scalar 1 + scalar (pos (ℕ.suc n) +ℤ l) ≡⟨ refl ⟩
1r + (scalar (pos (ℕ.suc n) +ℤ l)) ≡[ i ]⟨ 1r + +HomScalar (pos (ℕ.suc n)) l i ⟩
1r + (scalar (pos (ℕ.suc n)) + scalar l) ≡⟨ +Assoc _ _ _ ⟩
(1r + scalar (pos (ℕ.suc n))) + scalar l ≡⟨ refl ⟩
scalar (pos (ℕ.suc (ℕ.suc n))) + scalar l ∎
+HomScalar (negsuc ℕ.zero) l =
scalar (-1 +ℤ l) ≡[ i ]⟨ scalar (+ℤComm -1 l i) ⟩
scalar (l +ℤ -1) ≡⟨ refl ⟩
scalar (predℤ l) ≡⟨ lemmaPred l ⟩
- 1r + scalar l ≡⟨ refl ⟩
scalar -1 + scalar l ∎
+HomScalar (negsuc (ℕ.suc n)) l =
scalar (negsuc (ℕ.suc n) +ℤ l) ≡⟨ refl ⟩
scalar ((negsuc n +ℤ -1) +ℤ l) ≡[ i ]⟨ scalar (+ℤComm (negsuc n) -1 i +ℤ l) ⟩
scalar ((-1 +ℤ negsuc n) +ℤ l) ≡[ i ]⟨ scalar (+ℤAssoc -1 (negsuc n) l (~ i)) ⟩
scalar (-1 +ℤ (negsuc n +ℤ l)) ≡⟨ +HomScalar -1 (negsuc n +ℤ l) ⟩
- 1r + scalar (negsuc n +ℤ l) ≡[ i ]⟨ - 1r + +HomScalar (negsuc n) l i ⟩
- 1r + (scalar (negsuc n) + scalar l) ≡⟨ +Assoc (- 1r) _ _ ⟩
(- 1r + (scalar (negsuc n))) + scalar l ≡⟨ refl ⟩
(- 1r + - scalar (pos (ℕ.suc n))) + scalar l ≡[ i ]⟨ -Dist 1r (scalar (pos (ℕ.suc n))) i + scalar l ⟩
(- (1r + scalar (pos (ℕ.suc n)))) + scalar l ≡⟨ refl ⟩
scalar (negsuc (ℕ.suc n)) + scalar l ∎
lemma1 : (n : ℕ)
→ 1r + scalar (pos n) ≡ scalar (pos (ℕ.suc n))
lemma1 ℕ.zero = +Rid _
lemma1 (ℕ.suc k) = refl
lemma-1 : (n : ℕ)
→ - 1r + scalar (negsuc n) ≡ scalar (negsuc (ℕ.suc n))
lemma-1 ℕ.zero = -Dist _ _
lemma-1 (ℕ.suc k) =
- 1r + scalar (negsuc (ℕ.suc k)) ≡⟨ refl ⟩
- 1r + - scalar (pos (ℕ.suc (ℕ.suc k))) ≡⟨ -Dist _ _ ⟩
- (1r + scalar (pos (ℕ.suc (ℕ.suc k)))) ≡⟨ refl ⟩
scalar (negsuc (ℕ.suc (ℕ.suc k))) ∎
·HomScalar : (k l : ℤ)
→ scalar (k ·ℤ l) ≡ scalar k · scalar l
·HomScalar (pos ℕ.zero) l = 0r ≡⟨ sym (0LeftAnnihilates (scalar l)) ⟩ 0r · scalar l ∎
·HomScalar (pos (ℕ.suc n)) l =
scalar (l +ℤ (pos n ·ℤ l)) ≡⟨ +HomScalar l (pos n ·ℤ l) ⟩
scalar l + scalar (pos n ·ℤ l) ≡[ i ]⟨ scalar l + ·HomScalar (pos n) l i ⟩
scalar l + (scalar (pos n) · scalar l) ≡[ i ]⟨ ·Lid (scalar l) (~ i) + (scalar (pos n) · scalar l) ⟩
1r · scalar l + (scalar (pos n) · scalar l) ≡⟨ sym (·Ldist+ 1r _ _) ⟩
(1r + scalar (pos n)) · scalar l ≡[ i ]⟨ lemma1 n i · scalar l ⟩
scalar (pos (ℕ.suc n)) · scalar l ∎
·HomScalar (negsuc ℕ.zero) l =
scalar (-ℤ l) ≡⟨ -DistScalar l ⟩
- scalar l ≡[ i ]⟨ - (·Lid (scalar l) (~ i)) ⟩
- (1r · scalar l) ≡⟨ sym (-DistL· _ _) ⟩
- 1r · scalar l ≡⟨ refl ⟩
scalar (negsuc ℕ.zero) · scalar l ∎
·HomScalar (negsuc (ℕ.suc n)) l =
scalar ((-ℤ l) +ℤ (negsuc n ·ℤ l)) ≡⟨ +HomScalar (-ℤ l) (negsuc n ·ℤ l) ⟩
scalar (-ℤ l) + scalar (negsuc n ·ℤ l) ≡[ i ]⟨ -DistScalar l i + scalar (negsuc n ·ℤ l) ⟩
- scalar l + scalar (negsuc n ·ℤ l) ≡[ i ]⟨ - scalar l + ·HomScalar (negsuc n) l i ⟩
- scalar l + scalar (negsuc n) · scalar l ≡[ i ]⟨ (- ·Lid (scalar l) (~ i))
+ scalar (negsuc n) · scalar l ⟩
- (1r · scalar l) + scalar (negsuc n) · scalar l ≡[ i ]⟨ -DistL· 1r (scalar l) (~ i)
+ scalar (negsuc n) · scalar l ⟩
- 1r · scalar l + scalar (negsuc n) · scalar l ≡⟨ sym (·Ldist+ _ _ _) ⟩
(- 1r + scalar (negsuc n)) · scalar l ≡[ i ]⟨ lemma-1 n i · scalar l ⟩
scalar (negsuc (ℕ.suc n)) · scalar l ∎
CommRing→RawℤAlgebra : CommRing ℓ → RawAlgebra ℤAsRawRing ℓ
CommRing→RawℤAlgebra (R , commringstr 0r 1r _+_ _·_ -_ isCommRing) =
rawalgebra R (scalar ((R , commringstr 0r 1r _+_ _·_ -_ isCommRing))) 0r 1r _+_ _·_ -_