# Sums of elements in rings

Content created by Fredrik Bakke, Jonathan Prieto-Cubides, Egbert Rijke and Maša Žaucer.

Created on 2023-02-20.

module ring-theory.sums-rings where

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

open import foundation.function-types
open import foundation.homotopies
open import foundation.identity-types
open import foundation.universe-levels

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

open import ring-theory.rings
open import ring-theory.sums-semirings

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


## Idea

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

## Definition

sum-Ring :
{l : Level} (R : Ring l) (n : ℕ) → functional-vec-Ring R n → type-Ring R
sum-Ring R = sum-Semiring (semiring-Ring R)


## Properties

### Sums of one and two elements

module _
{l : Level} (R : Ring l)
where

sum-one-element-Ring :
(f : functional-vec-Ring R 1) → sum-Ring R 1 f ＝ head-functional-vec 0 f
sum-one-element-Ring = sum-one-element-Semiring (semiring-Ring R)

sum-two-elements-Ring :
(f : functional-vec-Ring R 2) →
sum-Ring R 2 f ＝ add-Ring R (f (zero-Fin 1)) (f (one-Fin 1))
sum-two-elements-Ring = sum-two-elements-Semiring (semiring-Ring R)


### Sums are homotopy invariant

module _
{l : Level} (R : Ring l)
where

htpy-sum-Ring :
(n : ℕ) {f g : functional-vec-Ring R n} →
(f ~ g) → sum-Ring R n f ＝ sum-Ring R n g
htpy-sum-Ring = htpy-sum-Semiring (semiring-Ring R)


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

module _
{l : Level} (R : Ring l)
where

cons-sum-Ring :
(n : ℕ) (f : functional-vec-Ring R (succ-ℕ n)) →
{x : type-Ring R} → head-functional-vec n f ＝ x →
sum-Ring R (succ-ℕ n) f ＝
add-Ring R (sum-Ring R n (tail-functional-vec n f)) x
cons-sum-Ring = cons-sum-Semiring (semiring-Ring R)

snoc-sum-Ring :
(n : ℕ) (f : functional-vec-Ring R (succ-ℕ n)) →
{x : type-Ring R} → f (zero-Fin n) ＝ x →
sum-Ring R (succ-ℕ n) f ＝
( x)
( sum-Ring R n (f ∘ inr-Fin n))
snoc-sum-Ring = snoc-sum-Semiring (semiring-Ring R)


### Multiplication distributes over sums

module _
{l : Level} (R : Ring l)
where

left-distributive-mul-sum-Ring :
(n : ℕ) (x : type-Ring R)
(f : functional-vec-Ring R n) →
mul-Ring R x (sum-Ring R n f) ＝
sum-Ring R n (λ i → mul-Ring R x (f i))
left-distributive-mul-sum-Ring =
left-distributive-mul-sum-Semiring (semiring-Ring R)

right-distributive-mul-sum-Ring :
(n : ℕ) (f : functional-vec-Ring R n)
(x : type-Ring R) →
mul-Ring R (sum-Ring R n f) x ＝
sum-Ring R n (λ i → mul-Ring R (f i) x)
right-distributive-mul-sum-Ring =
right-distributive-mul-sum-Semiring (semiring-Ring R)


### Interchange law of sums and addition in a semiring

module _
{l : Level} (R : Ring l)
where

(n : ℕ) (f g : functional-vec-Ring R n) →
( sum-Ring R n f)
( sum-Ring R n g) ＝
sum-Ring R n
( add-functional-vec-Ring R n f g)


### Extending a sum of elements in a semiring

module _
{l : Level} (R : Ring l)
where

extend-sum-Ring :
(n : ℕ) (f : functional-vec-Ring R n) →
sum-Ring R
( succ-ℕ n)
( cons-functional-vec-Ring R n (zero-Ring R) f) ＝
sum-Ring R n f
extend-sum-Ring = extend-sum-Semiring (semiring-Ring R)


### Shifting a sum of elements in a semiring

module _
{l : Level} (R : Ring l)
where

shift-sum-Ring :
(n : ℕ) (f : functional-vec-Ring R n) →
sum-Ring R
( succ-ℕ n)
( snoc-functional-vec-Ring R n f
( zero-Ring R)) ＝
sum-Ring R n f
shift-sum-Ring = shift-sum-Semiring (semiring-Ring R)


### A sum of zeroes is zero

module _
{l : Level} (R : Ring l)
where

sum-zero-Ring :
(n : ℕ) → sum-Ring R n (zero-functional-vec-Ring R n) ＝ zero-Ring R
sum-zero-Ring = sum-zero-Semiring (semiring-Ring R)


### Splitting sums

split-sum-Ring :
{l : Level} (R : Ring l)
(n m : ℕ) (f : functional-vec-Ring R (n +ℕ m)) →
sum-Ring R (n +ℕ m) f ＝