The binomial theorem for the natural numbers

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

Created on 2023-02-20.
Last modified on 2023-05-16.

module elementary-number-theory.binomial-theorem-natural-numbers where
open import commutative-algebra.binomial-theorem-commutative-semirings

open import elementary-number-theory.addition-natural-numbers
open import elementary-number-theory.commutative-semiring-of-natural-numbers
open import elementary-number-theory.distance-natural-numbers
open import elementary-number-theory.exponentiation-natural-numbers
open import elementary-number-theory.multiplication-natural-numbers
open import elementary-number-theory.natural-numbers

open import foundation.homotopies
open import foundation.identity-types

open import linear-algebra.vectors

open import univalent-combinatorics.standard-finite-types


The binomial theorem for natural numbers asserts that for any two natural numbers x and y and any natural number n, we have

  (x + y)ⁿ = ∑_{0 ≤ i < n+1} (n choose i) xⁱ yⁿ⁻ⁱ.


Binomial sums

binomial-sum-ℕ : (n : ) (f : functional-vec  (succ-ℕ n))  
binomial-sum-ℕ = binomial-sum-Commutative-Semiring ℕ-Commutative-Semiring


Binomial sums of one and two elements

binomial-sum-one-element-ℕ :
  (f : functional-vec  1)  binomial-sum-ℕ 0 f  head-functional-vec 0 f
binomial-sum-one-element-ℕ =
  binomial-sum-one-element-Commutative-Semiring ℕ-Commutative-Semiring

binomial-sum-two-elements-ℕ :
  (f : functional-vec  2) 
  binomial-sum-ℕ 1 f  (f (zero-Fin 1)) +ℕ (f (one-Fin 1))
binomial-sum-two-elements-ℕ =
  binomial-sum-two-elements-Commutative-Semiring ℕ-Commutative-Semiring

Binomial sums are homotopy invariant

htpy-binomial-sum-ℕ :
  (n : ) {f g : functional-vec  (succ-ℕ n)} 
  (f ~ g)  binomial-sum-ℕ n f  binomial-sum-ℕ n g
htpy-binomial-sum-ℕ =
  htpy-binomial-sum-Commutative-Semiring ℕ-Commutative-Semiring

Multiplication distributes over sums

left-distributive-mul-binomial-sum-ℕ :
  (n : ) (x : ) (f : functional-vec  (succ-ℕ n)) 
  x *ℕ (binomial-sum-ℕ n f)  binomial-sum-ℕ n  i  x *ℕ (f i))
left-distributive-mul-binomial-sum-ℕ =
  left-distributive-mul-binomial-sum-Commutative-Semiring ℕ-Commutative-Semiring

right-distributive-mul-binomial-sum-ℕ :
  (n : ) (f : functional-vec  (succ-ℕ n)) (x : ) 
  (binomial-sum-ℕ n f) *ℕ x  binomial-sum-ℕ n  i  (f i) *ℕ x)
right-distributive-mul-binomial-sum-ℕ =


Binomial theorem for the natural numbers

binomial-theorem-ℕ :
  (n : ) (x y : ) 
  power-ℕ n (x +ℕ y) 
  binomial-sum-ℕ n
    ( λ i 
      ( power-ℕ (nat-Fin (succ-ℕ n) i) x) *ℕ
      ( power-ℕ (dist-ℕ (nat-Fin (succ-ℕ n) i) n) y))
binomial-theorem-ℕ =
  binomial-theorem-Commutative-Semiring ℕ-Commutative-Semiring

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