The natural numbers base k

Content created by Egbert Rijke, Fredrik Bakke, Jonathan Prieto-Cubides, Julian KG, fernabnor and louismntnu.

Created on 2022-01-26.
Last modified on 2024-04-11.

module elementary-number-theory.finitary-natural-numbers where
Imports 
open import elementary-number-theory.addition-natural-numbers
open import elementary-number-theory.congruence-natural-numbers
open import elementary-number-theory.inequality-natural-numbers
open import elementary-number-theory.multiplication-natural-numbers
open import elementary-number-theory.natural-numbers
open import elementary-number-theory.strict-inequality-natural-numbers

open import foundation.action-on-identifications-functions
open import foundation.coproduct-types
open import foundation.dependent-pair-types
open import foundation.empty-types
open import foundation.function-types
open import foundation.identity-types
open import foundation.injective-maps
open import foundation.sets
open import foundation.universe-levels

open import univalent-combinatorics.standard-finite-types

Definition

The finitary natural numbers

data based-ℕ : ℕ → UU lzero where
  constant-based-ℕ : (k : ℕ) → Fin k → based-ℕ k
  unary-op-based-ℕ : (k : ℕ) → Fin k → based-ℕ k → based-ℕ k

Converting a k-ary natural number to a natural number

constant-ℕ : (k : ℕ) → Fin k → ℕ
constant-ℕ k x = nat-Fin k x

unary-op-ℕ : (k : ℕ) → Fin k → ℕ → ℕ
unary-op-ℕ k x n = (k *ℕ (succ-ℕ n)) +ℕ (nat-Fin k x)

convert-based-ℕ : (k : ℕ) → based-ℕ k → ℕ
convert-based-ℕ k (constant-based-ℕ .k x) =
  constant-ℕ k x
convert-based-ℕ k (unary-op-based-ℕ .k x n) =
  unary-op-ℕ k x (convert-based-ℕ k n)

The type of 0-ary natural numbers is empty

is-empty-based-zero-ℕ : is-empty (based-ℕ zero-ℕ)
is-empty-based-zero-ℕ (constant-based-ℕ .zero-ℕ ())
is-empty-based-zero-ℕ (unary-op-based-ℕ .zero-ℕ () n)

The function convert-based-ℕ is injective

cong-unary-op-ℕ :
  (k : ℕ) (x : Fin k) (n : ℕ) →
  cong-ℕ k (unary-op-ℕ k x n) (nat-Fin k x)
cong-unary-op-ℕ (succ-ℕ k) x n =
  concatenate-cong-eq-ℕ
    ( succ-ℕ k)
    { unary-op-ℕ (succ-ℕ k) x n}
    ( translation-invariant-cong-ℕ'
      ( succ-ℕ k)
      ( (succ-ℕ k) *ℕ (succ-ℕ n))
      ( zero-ℕ)
      ( nat-Fin (succ-ℕ k) x)
      ( pair (succ-ℕ n) (commutative-mul-ℕ (succ-ℕ n) (succ-ℕ k))))
    ( left-unit-law-add-ℕ (nat-Fin (succ-ℕ k) x))

Any natural number of the form constant-ℕ k x is strictly less than any natural number of the form unary-op-ℕ k y m

le-constant-unary-op-ℕ :
  (k : ℕ) (x y : Fin k) (m : ℕ) → le-ℕ (constant-ℕ k x) (unary-op-ℕ k y m)
le-constant-unary-op-ℕ k x y m =
  concatenate-le-leq-ℕ {nat-Fin k x} {k} {unary-op-ℕ k y m}
    ( strict-upper-bound-nat-Fin k x)
    ( transitive-leq-ℕ
      ( k)
      ( k *ℕ (succ-ℕ m))
      ( unary-op-ℕ k y m)
      ( leq-add-ℕ (k *ℕ (succ-ℕ m)) (nat-Fin k y))
      ( leq-mul-ℕ m k))

is-injective-convert-based-ℕ :
  (k : ℕ) → is-injective (convert-based-ℕ k)
is-injective-convert-based-ℕ
  ( succ-ℕ k)
  { constant-based-ℕ .(succ-ℕ k) x}
  { constant-based-ℕ .(succ-ℕ k) y} p =
  ap (constant-based-ℕ (succ-ℕ k)) (is-injective-nat-Fin (succ-ℕ k) p)
is-injective-convert-based-ℕ
  ( succ-ℕ k)
  { constant-based-ℕ .(succ-ℕ k) x}
  { unary-op-based-ℕ .(succ-ℕ k) y m} p =
  ex-falso
    ( neq-le-ℕ
      ( le-constant-unary-op-ℕ (succ-ℕ k) x y (convert-based-ℕ (succ-ℕ k) m))
      ( p))
is-injective-convert-based-ℕ
  ( succ-ℕ k)
  { unary-op-based-ℕ .(succ-ℕ k) x n}
  { constant-based-ℕ .(succ-ℕ k) y} p =
  ex-falso
    ( neq-le-ℕ
      ( le-constant-unary-op-ℕ (succ-ℕ k) y x (convert-based-ℕ (succ-ℕ k) n))
      ( inv p))
is-injective-convert-based-ℕ
  ( succ-ℕ k)
  { unary-op-based-ℕ .(succ-ℕ k) x n}
  { unary-op-based-ℕ .(succ-ℕ k) y m} p with
  is-injective-nat-Fin (succ-ℕ k) {x} {y}
    ( eq-cong-le-ℕ
      ( succ-ℕ k)
      ( nat-Fin (succ-ℕ k) x)
      ( nat-Fin (succ-ℕ k) y)
      ( strict-upper-bound-nat-Fin (succ-ℕ k) x)
      ( strict-upper-bound-nat-Fin (succ-ℕ k) y)
      ( concatenate-cong-eq-cong-ℕ
        { succ-ℕ k}
        { nat-Fin (succ-ℕ k) x}
        { unary-op-ℕ (succ-ℕ k) x (convert-based-ℕ (succ-ℕ k) n)}
        { unary-op-ℕ (succ-ℕ k) y (convert-based-ℕ (succ-ℕ k) m)}
        { nat-Fin (succ-ℕ k) y}
        ( symmetric-cong-ℕ
          ( succ-ℕ k)
          ( unary-op-ℕ (succ-ℕ k) x (convert-based-ℕ (succ-ℕ k) n))
          ( nat-Fin (succ-ℕ k) x)
          ( cong-unary-op-ℕ (succ-ℕ k) x (convert-based-ℕ (succ-ℕ k) n)))
        ( p)
        ( cong-unary-op-ℕ (succ-ℕ k) y (convert-based-ℕ (succ-ℕ k) m))))
... | refl =
  ap
    ( unary-op-based-ℕ (succ-ℕ k) x)
    ( is-injective-convert-based-ℕ (succ-ℕ k)
      ( is-injective-succ-ℕ
        ( is-injective-left-mul-succ-ℕ k
          ( is-injective-right-add-ℕ (nat-Fin (succ-ℕ k) x) p))))

The zero-element of the k+1-ary natural numbers

zero-based-ℕ : (k : ℕ) → based-ℕ (succ-ℕ k)
zero-based-ℕ k = constant-based-ℕ (succ-ℕ k) (zero-Fin k)

The successor function on the k-ary natural numbers

succ-based-ℕ : (k : ℕ) → based-ℕ k → based-ℕ k
succ-based-ℕ (succ-ℕ k) (constant-based-ℕ .(succ-ℕ k) (inl x)) =
  constant-based-ℕ (succ-ℕ k) (succ-Fin (succ-ℕ k) (inl x))
succ-based-ℕ (succ-ℕ k) (constant-based-ℕ .(succ-ℕ k) (inr _)) =
  unary-op-based-ℕ
    (succ-ℕ k) (zero-Fin k) (constant-based-ℕ (succ-ℕ k) (zero-Fin k))
succ-based-ℕ (succ-ℕ k) (unary-op-based-ℕ .(succ-ℕ k) (inl x) n) =
  unary-op-based-ℕ (succ-ℕ k) (succ-Fin (succ-ℕ k) (inl x)) n
succ-based-ℕ (succ-ℕ k) (unary-op-based-ℕ .(succ-ℕ k) (inr x) n) =
  unary-op-based-ℕ (succ-ℕ k) (zero-Fin k) (succ-based-ℕ (succ-ℕ k) n)

The inverse map of convert-based-ℕ

inv-convert-based-ℕ : (k : ℕ) → ℕ → based-ℕ (succ-ℕ k)
inv-convert-based-ℕ k zero-ℕ =
  zero-based-ℕ k
inv-convert-based-ℕ k (succ-ℕ n) =
  succ-based-ℕ (succ-ℕ k) (inv-convert-based-ℕ k n)

convert-based-succ-based-ℕ :
  (k : ℕ) (x : based-ℕ k) →
  convert-based-ℕ k (succ-based-ℕ k x) ＝ succ-ℕ (convert-based-ℕ k x)
convert-based-succ-based-ℕ (succ-ℕ k) (constant-based-ℕ .(succ-ℕ k) (inl x)) =
  nat-succ-Fin k x
convert-based-succ-based-ℕ
  ( succ-ℕ k) (constant-based-ℕ .(succ-ℕ k) (inr _)) =
  ( ap
    ( λ t → ((succ-ℕ k) *ℕ (succ-ℕ t)) +ℕ t)
    ( is-zero-nat-zero-Fin {k})) ∙
  ( right-unit-law-mul-ℕ (succ-ℕ k))
convert-based-succ-based-ℕ (succ-ℕ k) (unary-op-based-ℕ .(succ-ℕ k) (inl x) n) =
  ap
    ( ((succ-ℕ k) *ℕ (succ-ℕ (convert-based-ℕ (succ-ℕ k) n))) +ℕ_)
    ( nat-succ-Fin k x)
convert-based-succ-based-ℕ
  (succ-ℕ k) (unary-op-based-ℕ .(succ-ℕ k) (inr _) n) =
  ( ap
    ( ( ( succ-ℕ k) *ℕ
        ( succ-ℕ (convert-based-ℕ (succ-ℕ k) (succ-based-ℕ (succ-ℕ k) n))))
          +ℕ_)
    ( is-zero-nat-zero-Fin {k})) ∙
  ( ( ap
      ( ((succ-ℕ k) *ℕ_) ∘ succ-ℕ)
      ( convert-based-succ-based-ℕ (succ-ℕ k) n)) ∙
    ( ( right-successor-law-mul-ℕ
        ( succ-ℕ k)
        ( succ-ℕ (convert-based-ℕ (succ-ℕ k) n))) ∙
      ( commutative-add-ℕ
        ( succ-ℕ k)
        ( (succ-ℕ k) *ℕ (succ-ℕ (convert-based-ℕ (succ-ℕ k) n))))))

is-section-inv-convert-based-ℕ :
  (k n : ℕ) → convert-based-ℕ (succ-ℕ k) (inv-convert-based-ℕ k n) ＝ n
is-section-inv-convert-based-ℕ k zero-ℕ = is-zero-nat-zero-Fin {k}
is-section-inv-convert-based-ℕ k (succ-ℕ n) =
  ( convert-based-succ-based-ℕ (succ-ℕ k) (inv-convert-based-ℕ k n)) ∙
  ( ap succ-ℕ (is-section-inv-convert-based-ℕ k n))

is-retraction-inv-convert-based-ℕ :
  (k : ℕ) (x : based-ℕ (succ-ℕ k)) →
  inv-convert-based-ℕ k (convert-based-ℕ (succ-ℕ k) x) ＝ x
is-retraction-inv-convert-based-ℕ k x =
  is-injective-convert-based-ℕ
    ( succ-ℕ k)
    ( is-section-inv-convert-based-ℕ k (convert-based-ℕ (succ-ℕ k) x))

Recent changes