Added the ring quotient of the image of an arbitrary function. (#1238)

* Added the ring quotient of the image of an arbitrary function.

* used copattern matching for inducedMapPreservesRing

Author: Freek98

Cubical/Algebra/CommRing/Quotient/ImageQuotient.agda

@@ -0,0 +1,137 @@
+{-# OPTIONS --safe #-}
+
+module Cubical.Algebra.CommRing.Quotient.ImageQuotient where
+
+open import Cubical.Foundations.Prelude
+open import Cubical.Foundations.Structure
+open import Cubical.Foundations.Function
+open import Cubical.Data.Sigma
+
+open import Cubical.Algebra.CommRing
+open import Cubical.Algebra.CommRing.Ideal
+open import Cubical.Algebra.Ring.Properties
+open RingTheory
+
+import Cubical.HITs.SetQuotients as SQ
+import Cubical.Algebra.CommRing.Quotient as CQ
+
+module _ {ℓ : Level} (R : CommRing ℓ) {X : Type ℓ} (f : X → ⟨ R ⟩) where
+  module _ where
+    open CommRingStr ⦃...⦄
+    open CQ
+    instance
+     _ = str R
+
+    data generatedIdeal : ⟨ R ⟩ → Type ℓ where
+      single : ∀ x → generatedIdeal (f x)
+      zero   : generatedIdeal 0r
+      add    : ∀ {x y} → generatedIdeal x → generatedIdeal y →
+                         generatedIdeal (x + y)
+      mul    : ∀ {r x} → generatedIdeal x → generatedIdeal (r · x)
+      squash : ∀ {x}   → isProp (generatedIdeal x)
+
+    genIdeal : IdealsIn R
+    genIdeal = makeIdeal (λ r → generatedIdeal r , squash)
+      add zero λ r → mul
+
+    _/Im_ : CommRing ℓ
+    _/Im_ = R / genIdeal
+
+    quotientImageHom : CommRingHom R _/Im_
+    quotientImageHom = quotientHom R genIdeal
+
+    instance
+     _ = str _/Im_
+
+    zeroOnImage : (x : X) → quotientImageHom $cr (f x) ≡ 0r
+    zeroOnImage x = zeroOnIdeal genIdeal _ (single x)
+
+  module _ {ℓ' : Level} {S : CommRing ℓ'} (g : CommRingHom R S)
+    (gfx=0 : ∀ (x : X) → g $cr (f x) ≡ CommRingStr.0r (snd S)) where
+    open CommIdeal R
+    open IsCommRingHom (snd g)
+    open CommRingStr ⦃...⦄
+    instance
+      _ = snd R
+      _ = snd S
+      _ = snd _/Im_
+
+    extendToIdeal : (r : ⟨ R ⟩) → r ∈ genIdeal → g $cr r ≡ 0r
+    extendToIdeal .(f x) (single x) = gfx=0 x
+    extendToIdeal .(0r) zero = pres0
+    extendToIdeal .(r + s) (add {r} {s} r∈I s∈I) =
+      g $cr (r + s )
+        ≡⟨ pres+ r s ⟩
+      (g $cr r) + (g $cr s)
+        ≡⟨ cong (λ a → a + (g $cr s)) (extendToIdeal r r∈I) ⟩
+      0r + (g $cr s)
+        ≡⟨ cong (λ a → 0r + a) (extendToIdeal s s∈I) ⟩
+      0r + 0r
+        ≡⟨ +IdL 0r ⟩
+      0r ∎
+    extendToIdeal .(r · s) (mul {r} {s} s∈I) =
+      (g $cr (r · s))
+        ≡⟨ pres· r s ⟩
+      (g $cr r) · (g $cr s)
+        ≡⟨ cong (λ a → (g $cr r) · a) (extendToIdeal s s∈I) ⟩
+      (g $cr r) · 0r
+        ≡⟨ 0RightAnnihilates (CommRing→Ring S) (g $cr r) ⟩
+      0r ∎
+    extendToIdeal r (squash r∈I0 r∈I1 i) =
+      is-set (g $cr r) 0r (extendToIdeal r r∈I0) (extendToIdeal r r∈I1) i
+
+    inducedMap : ⟨ _/Im_ ⟩ → ⟨ S ⟩
+    inducedMap = SQ.elim (λ x → is-set) (fst g)
+      λ { a b r → equalByDifference (CommRing→Ring S) _ _
+      (
+      (g $cr a - g $cr b)
+        ≡⟨ cong (λ b → g $cr a + b) (sym (pres- b)) ⟩
+      (g $cr a + g $cr (- b))
+        ≡⟨ sym (pres+ a (- b)) ⟩
+      g $cr (a - b)
+        ≡⟨ extendToIdeal _ r ⟩
+      (0r ∎)
+      )
+      }
+
+    open IsCommRingHom
+
+    inducedMapPreservesRing : IsCommRingHom (str _/Im_) inducedMap (str S)
+    pres0 inducedMapPreservesRing =
+      inducedMap 0r
+      ≡⟨ cong inducedMap (pres0 (snd quotientImageHom)) ⟩
+      inducedMap (quotientImageHom $cr 0r)
+      ≡⟨⟩
+      g $cr 0r
+      ≡⟨ pres0 (snd g) ⟩
+      0r ∎
+    pres1 inducedMapPreservesRing =
+      inducedMap 1r
+        ≡⟨ cong inducedMap (pres1 (snd quotientImageHom)) ⟩
+      inducedMap (quotientImageHom $cr 1r)
+        ≡⟨⟩
+      g $cr 1r
+        ≡⟨ pres1 (snd g) ⟩
+      1r ∎
+    pres+ inducedMapPreservesRing =
+      SQ.elimProp2 (λ x y → is-set _ _ ) (pres+ (snd g))
+    pres· inducedMapPreservesRing =
+      SQ.elimProp2 (λ x y → is-set _ _ ) (pres· (snd g))
+    pres- inducedMapPreservesRing =
+      SQ.elimProp  (λ x   → is-set _ _ ) (pres- (snd g))
+
+    inducedHom : CommRingHom _/Im_ S
+    inducedHom = inducedMap , inducedMapPreservesRing
+
+    inducedMapUnique : (h : ⟨ _/Im_ ⟩ → ⟨ S ⟩) →
+                       fst g ≡ h ∘ (fst quotientImageHom)  →
+                       inducedMap ≡ h
+    inducedMapUnique _ = funExt ∘ SQ.elimProp (λ { x → is-set _ _ }) ∘ funExt⁻
+
+    inducedHomUnique : (h : CommRingHom (_/Im_) S) →
+                       (p : g ≡ (h ∘cr quotientImageHom)) →
+                       inducedHom ≡ h
+    inducedHomUnique h p = Σ≡Prop
+                           (λ { x → isPropIsCommRingHom (str _/Im_) x (str S) })
+                           (inducedMapUnique (fst h) (cong fst p))
+