Sums in commutative rings

module commutative-algebra.sums-commutative-rings where

open import commutative-algebra.commutative-rings

open import elementary-number-theory.addition-natural-numbers
open import elementary-number-theory.natural-numbers

open import foundation.coproduct-types
open import foundation.functions
open import foundation.homotopies
open import foundation.identity-types
open import foundation.unit-type
open import foundation.universe-levels

open import linear-algebra.vectors
open import linear-algebra.vectors-on-commutative-rings

open import ring-theory.sums-rings

open import univalent-combinatorics.coproduct-types
open import univalent-combinatorics.standard-finite-types

Idea

The sum operation extends the binary addition operation on a commutative ring A to any family of elements of A indexed by a standard finite type.

Definition

sum-Commutative-Ring :
  {l : Level} (A : Commutative-Ring l) (n : ℕ) →
  (functional-vec-Commutative-Ring A n) → type-Commutative-Ring A
sum-Commutative-Ring A = sum-Ring (ring-Commutative-Ring A)

Properties

Sums of one and two elements

module _
  {l : Level} (A : Commutative-Ring l)
  where

  sum-one-element-Commutative-Ring :
    (f : functional-vec-Commutative-Ring A ) →
    sum-Commutative-Ring A  f ＝ head-functional-vec  f
  sum-one-element-Commutative-Ring =
    sum-one-element-Ring (ring-Commutative-Ring A)

  sum-two-elements-Commutative-Ring :
    (f : functional-vec-Commutative-Ring A ) →
    sum-Commutative-Ring A  f ＝
    add-Commutative-Ring A (f (zero-Fin )) (f (one-Fin ))
  sum-two-elements-Commutative-Ring =
    sum-two-elements-Ring (ring-Commutative-Ring A)

Sums are homotopy invariant

module _
  {l : Level} (A : Commutative-Ring l)
  where

  htpy-sum-Commutative-Ring :
    (n : ℕ) {f g : functional-vec-Commutative-Ring A n} →
    (f ~ g) → sum-Commutative-Ring A n f ＝ sum-Commutative-Ring A n g
  htpy-sum-Commutative-Ring = htpy-sum-Ring (ring-Commutative-Ring A)

Sums are equal to the zero-th term plus the rest

module _
  {l : Level} (A : Commutative-Ring l)
  where

  cons-sum-Commutative-Ring :
    (n : ℕ) (f : functional-vec-Commutative-Ring A (succ-ℕ n)) →
    {x : type-Commutative-Ring A} → head-functional-vec n f ＝ x →
    sum-Commutative-Ring A (succ-ℕ n) f ＝
    add-Commutative-Ring A
      ( sum-Commutative-Ring A n (tail-functional-vec n f)) x
  cons-sum-Commutative-Ring = cons-sum-Ring (ring-Commutative-Ring A)

  snoc-sum-Commutative-Ring :
    (n : ℕ) (f : functional-vec-Commutative-Ring A (succ-ℕ n)) →
    {x : type-Commutative-Ring A} → f (zero-Fin n) ＝ x →
    sum-Commutative-Ring A (succ-ℕ n) f ＝
    add-Commutative-Ring A
      ( x)
      ( sum-Commutative-Ring A n (f ∘ inr-Fin n))
  snoc-sum-Commutative-Ring = snoc-sum-Ring (ring-Commutative-Ring A)

Multiplication distributes over sums

module _
  {l : Level} (A : Commutative-Ring l)
  where

  left-distributive-mul-sum-Commutative-Ring :
    (n : ℕ) (x : type-Commutative-Ring A)
    (f : functional-vec-Commutative-Ring A n) →
    mul-Commutative-Ring A x (sum-Commutative-Ring A n f) ＝
    sum-Commutative-Ring A n (λ i → mul-Commutative-Ring A x (f i))
  left-distributive-mul-sum-Commutative-Ring =
    left-distributive-mul-sum-Ring (ring-Commutative-Ring A)

  right-distributive-mul-sum-Commutative-Ring :
    (n : ℕ) (f : functional-vec-Commutative-Ring A n)
    (x : type-Commutative-Ring A) →
    mul-Commutative-Ring A (sum-Commutative-Ring A n f) x ＝
    sum-Commutative-Ring A n (λ i → mul-Commutative-Ring A (f i) x)
  right-distributive-mul-sum-Commutative-Ring =
    right-distributive-mul-sum-Ring (ring-Commutative-Ring A)

Interchange law of sums and addition in a commutative ring

module _
  {l : Level} (A : Commutative-Ring l)
  where

  interchange-add-sum-Commutative-Ring :
    (n : ℕ) (f g : functional-vec-Commutative-Ring A n) →
    add-Commutative-Ring A
      ( sum-Commutative-Ring A n f)
      ( sum-Commutative-Ring A n g) ＝
    sum-Commutative-Ring A n
      ( add-functional-vec-Commutative-Ring A n f g)
  interchange-add-sum-Commutative-Ring =
    interchange-add-sum-Ring (ring-Commutative-Ring A)

Extending a sum of elements in a commutative ring

module _
  {l : Level} (A : Commutative-Ring l)
  where

  extend-sum-Commutative-Ring :
    (n : ℕ) (f : functional-vec-Commutative-Ring A n) →
    sum-Commutative-Ring A
      ( succ-ℕ n)
      ( cons-functional-vec-Commutative-Ring A n (zero-Commutative-Ring A) f) ＝
    sum-Commutative-Ring A n f
  extend-sum-Commutative-Ring = extend-sum-Ring (ring-Commutative-Ring A)

Shifting a sum of elements in a commutative ring

module _
  {l : Level} (A : Commutative-Ring l)
  where

  shift-sum-Commutative-Ring :
    (n : ℕ) (f : functional-vec-Commutative-Ring A n) →
    sum-Commutative-Ring A
      ( succ-ℕ n)
      ( snoc-functional-vec-Commutative-Ring A n f
        ( zero-Commutative-Ring A)) ＝
    sum-Commutative-Ring A n f
  shift-sum-Commutative-Ring = shift-sum-Ring (ring-Commutative-Ring A)

Splitting sums

split-sum-Commutative-Ring :
  {l : Level} (A : Commutative-Ring l)
  (n m : ℕ) (f : functional-vec-Commutative-Ring A (n +ℕ m)) →
  sum-Commutative-Ring A (n +ℕ m) f ＝
  add-Commutative-Ring A
    ( sum-Commutative-Ring A n (f ∘ inl-coprod-Fin n m))
    ( sum-Commutative-Ring A m (f ∘ inr-coprod-Fin n m))
split-sum-Commutative-Ring A n zero-ℕ f =
  inv (right-unit-law-add-Commutative-Ring A (sum-Commutative-Ring A n f))
split-sum-Commutative-Ring A n (succ-ℕ m) f =
  ( ap
    ( add-Commutative-Ring' A (f(inr star)))
    ( split-sum-Commutative-Ring A n m (f ∘ inl))) ∙
  ( associative-add-Commutative-Ring A _ _ _)

A sum of zeroes is zero

module _
  {l : Level} (A : Commutative-Ring l)
  where

  sum-zero-Commutative-Ring :
    (n : ℕ) →
    sum-Commutative-Ring A n
      ( zero-functional-vec-Commutative-Ring A n) ＝
    zero-Commutative-Ring A
  sum-zero-Commutative-Ring = sum-zero-Ring (ring-Commutative-Ring A)