The cofibonacci sequence

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-04-08.
Last modified on 2023-09-11.

module elementary-number-theory.cofibonacci where
open import elementary-number-theory.divisibility-natural-numbers
open import elementary-number-theory.equality-natural-numbers
open import elementary-number-theory.fibonacci-sequence
open import elementary-number-theory.inequality-natural-numbers
open import elementary-number-theory.modular-arithmetic-standard-finite-types
open import elementary-number-theory.natural-numbers
open import elementary-number-theory.pisano-periods
open import elementary-number-theory.well-ordering-principle-natural-numbers

open import foundation.action-on-identifications-functions
open import foundation.cartesian-product-types
open import foundation.decidable-types
open import foundation.dependent-pair-types
open import foundation.empty-types
open import foundation.identity-types
open import foundation.transport-along-identifications
open import foundation.universe-levels


The cofibonacci sequence is the unique function G : ℕ → ℕ satisfying the property that

  div-ℕ (G m) n ↔ div-ℕ m (Fibonacci-ℕ n).


The predicate of being a multiple of the m-th cofibonacci number

is-multiple-of-cofibonacci : (m x : )  UU lzero
is-multiple-of-cofibonacci m x =
  is-nonzero-ℕ m  is-nonzero-ℕ x × div-ℕ m (Fibonacci-ℕ x)

is-decidable-is-multiple-of-cofibonacci :
  (m x : )  is-decidable (is-multiple-of-cofibonacci m x)
is-decidable-is-multiple-of-cofibonacci m x =
    ( is-decidable-is-nonzero-ℕ m)
    ( is-decidable-prod
      ( is-decidable-is-nonzero-ℕ x)
      ( is-decidable-div-ℕ m (Fibonacci-ℕ x)))

cofibonacci-multiple : (m : )  Σ  (is-multiple-of-cofibonacci m)
cofibonacci-multiple zero-ℕ = pair zero-ℕ  f  (ex-falso (f refl)))
cofibonacci-multiple (succ-ℕ m) =
    ( pisano-period m)
    ( λ f  pair (is-nonzero-pisano-period m) (div-fibonacci-pisano-period m))

The cofibonacci sequence

least-multiple-of-cofibonacci :
  (m : )  minimal-element-ℕ (is-multiple-of-cofibonacci m)
least-multiple-of-cofibonacci m =
    ( is-multiple-of-cofibonacci m)
    ( is-decidable-is-multiple-of-cofibonacci m)
    ( cofibonacci-multiple m)

cofibonacci :   
cofibonacci m = pr1 (least-multiple-of-cofibonacci m)

is-multiple-of-cofibonacci-cofibonacci :
  (m : )  is-multiple-of-cofibonacci m (cofibonacci m)
is-multiple-of-cofibonacci-cofibonacci m =
  pr1 (pr2 (least-multiple-of-cofibonacci m))

is-lower-bound-cofibonacci :
  (m x : )  is-multiple-of-cofibonacci m x 
  cofibonacci m ≤-ℕ x
is-lower-bound-cofibonacci m =
  pr2 (pr2 (least-multiple-of-cofibonacci m))


The 0-th cofibonacci number is 0

is-zero-cofibonacci-zero-ℕ : is-zero-ℕ (cofibonacci zero-ℕ)
is-zero-cofibonacci-zero-ℕ =
    ( cofibonacci zero-ℕ)
    ( is-lower-bound-cofibonacci zero-ℕ zero-ℕ ( λ f  ex-falso (f refl)))

The cofibonacci sequence is left adjoint to the Fibonacci sequence

forward-is-left-adjoint-cofibonacci :
  (m n : )  div-ℕ (cofibonacci m) n  div-ℕ m (Fibonacci-ℕ n)
forward-is-left-adjoint-cofibonacci zero-ℕ n H =
    ( div-ℕ zero-ℕ)
    ( ap
      ( Fibonacci-ℕ)
      ( inv
        ( is-zero-div-zero-ℕ n
          ( tr  t  div-ℕ t n) is-zero-cofibonacci-zero-ℕ H))))
    ( refl-div-ℕ zero-ℕ)
forward-is-left-adjoint-cofibonacci (succ-ℕ m) zero-ℕ H =
  div-zero-ℕ (succ-ℕ m)
forward-is-left-adjoint-cofibonacci (succ-ℕ m) (succ-ℕ n) H =
    ( succ-ℕ m)
    ( cofibonacci (succ-ℕ m))
    ( succ-ℕ n)
    ( H)
    ( pr2
      ( is-multiple-of-cofibonacci-cofibonacci
        ( succ-ℕ m)
        ( is-nonzero-succ-ℕ m)))

converse-is-left-adjoint-cofibonacci :
  (m n : ℕ) → div-ℕ m (Fibonacci-ℕ n) → div-ℕ (cofibonacci m) n
converse-is-left-adjoint-cofibonacci m n H = {!!}

is-left-adjoint-cofibonacci :
  (m n : ℕ) → div-ℕ (cofibonacci m) n ↔ div-ℕ m (Fibonacci-ℕ n)
is-left-adjoint-cofibonacci m n = {!!}


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