Bounded sums of arithmetic functions

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

Created on 2022-03-23.
Last modified on 2023-06-10.

module elementary-number-theory.bounded-sums-arithmetic-functions where
Imports 
open import elementary-number-theory.arithmetic-functions
open import elementary-number-theory.natural-numbers
open import elementary-number-theory.nonzero-natural-numbers

open import foundation.coproduct-types
open import foundation.decidable-propositions
open import foundation.decidable-types
open import foundation.function-types
open import foundation.universe-levels

open import ring-theory.rings

Idea

Given a decidable predicate P on the nonzero natural numbers, and a map f from the nonzero natural numbers in P into a ring R, the bounded sum is a summation of the values of f up to an upper bound b.

Definition

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

  restricted-arithmetic-function-Ring :
    {l' : Level} (P : nonzero-ℕ → Decidable-Prop l') → UU (l ⊔ l')
  restricted-arithmetic-function-Ring P =
    (x : nonzero-ℕ) → type-Decidable-Prop (P x) → type-Ring R

  shift-arithmetic-function-Ring :
    type-arithmetic-functions-Ring R → type-arithmetic-functions-Ring R
  shift-arithmetic-function-Ring f = f ∘ succ-nonzero-ℕ

  shift-restricted-arithmetic-function-Ring :
    {l' : Level} (P : nonzero-ℕ → Decidable-Prop l') →
    restricted-arithmetic-function-Ring P →
    restricted-arithmetic-function-Ring (P ∘ succ-nonzero-ℕ)
  shift-restricted-arithmetic-function-Ring P f = f ∘ succ-nonzero-ℕ

  case-one-bounded-sum-arithmetic-function-Ring :
    {l' : Level} → (P : Decidable-Prop l') →
    is-decidable (type-Decidable-Prop P) →
    (type-Decidable-Prop P → type-Ring R) → type-Ring R
  case-one-bounded-sum-arithmetic-function-Ring P (inl x) f = f x
  case-one-bounded-sum-arithmetic-function-Ring P (inr x) f =
    zero-Ring R

  bounded-sum-arithmetic-function-Ring :
    (b : ℕ) {l' : Level} (P : nonzero-ℕ → Decidable-Prop l')
    (f : restricted-arithmetic-function-Ring P) → type-Ring R
  bounded-sum-arithmetic-function-Ring zero-ℕ P f = zero-Ring R
  bounded-sum-arithmetic-function-Ring (succ-ℕ zero-ℕ) P f =
    case-one-bounded-sum-arithmetic-function-Ring
      ( P one-nonzero-ℕ)
      ( is-decidable-Decidable-Prop (P one-nonzero-ℕ))
      ( f one-nonzero-ℕ)
  bounded-sum-arithmetic-function-Ring (succ-ℕ (succ-ℕ b)) P f =
    add-Ring R
      ( case-one-bounded-sum-arithmetic-function-Ring
        ( P one-nonzero-ℕ)
        ( is-decidable-Decidable-Prop (P one-nonzero-ℕ))
        ( f one-nonzero-ℕ))
      ( bounded-sum-arithmetic-function-Ring
        ( succ-ℕ b)
        ( P ∘ succ-nonzero-ℕ)
        ( f ∘ succ-nonzero-ℕ))

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