Minimum on the natural numbers

Content created by Fredrik Bakke, Jonathan Prieto-Cubides, Egbert Rijke, Eléonore Mangel and Amélia Liao.

Created on 2022-03-22.
Last modified on 2023-06-10.

module elementary-number-theory.minimum-natural-numbers where
open import elementary-number-theory.addition-natural-numbers
open import elementary-number-theory.inequality-natural-numbers
open import elementary-number-theory.natural-numbers

open import foundation.action-on-identifications-binary-functions
open import foundation.action-on-identifications-functions
open import foundation.coproduct-types
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.unit-type

open import order-theory.greatest-lower-bounds-posets

open import univalent-combinatorics.standard-finite-types


We define the operation of minimum (greatest lower bound) for the natural numbers.


min-ℕ :   (  )
min-ℕ 0 n = 0
min-ℕ (succ-ℕ m) 0 = 0
min-ℕ (succ-ℕ m) (succ-ℕ n) = succ-ℕ (min-ℕ m n)

ap-min-ℕ : {x x' y y' : }  x  x'  y  y'  min-ℕ x y  min-ℕ x' y'
ap-min-ℕ p q = ap-binary min-ℕ p q

min-Fin-ℕ : (n : )  (Fin (succ-ℕ n)  )  
min-Fin-ℕ zero-ℕ f = f (inr star)
min-Fin-ℕ (succ-ℕ n) f = min-ℕ (f (inr star)) (min-Fin-ℕ n  k  f (inl k)))


The minimum of two natural numbers is a greatest lower bound

leq-min-ℕ :
  (k m n : )  k ≤-ℕ m  k ≤-ℕ n  k ≤-ℕ (min-ℕ m n)
leq-min-ℕ zero-ℕ zero-ℕ zero-ℕ H K = star
leq-min-ℕ zero-ℕ zero-ℕ (succ-ℕ n) H K = star
leq-min-ℕ zero-ℕ (succ-ℕ m) zero-ℕ H K = star
leq-min-ℕ zero-ℕ (succ-ℕ m) (succ-ℕ n) H K = star
leq-min-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H K = leq-min-ℕ k m n H K

leq-left-leq-min-ℕ :
  (k m n : )  k ≤-ℕ (min-ℕ m n)  k ≤-ℕ m
leq-left-leq-min-ℕ zero-ℕ zero-ℕ zero-ℕ H = star
leq-left-leq-min-ℕ zero-ℕ zero-ℕ (succ-ℕ n) H = star
leq-left-leq-min-ℕ zero-ℕ (succ-ℕ m) zero-ℕ H = star
leq-left-leq-min-ℕ zero-ℕ (succ-ℕ m) (succ-ℕ n) H = star
leq-left-leq-min-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H =
  leq-left-leq-min-ℕ k m n H

leq-right-leq-min-ℕ :
  (k m n : )  k ≤-ℕ (min-ℕ m n)  k ≤-ℕ n
leq-right-leq-min-ℕ zero-ℕ zero-ℕ zero-ℕ H = star
leq-right-leq-min-ℕ zero-ℕ zero-ℕ (succ-ℕ n) H = star
leq-right-leq-min-ℕ zero-ℕ (succ-ℕ m) zero-ℕ H = star
leq-right-leq-min-ℕ zero-ℕ (succ-ℕ m) (succ-ℕ n) H = star
leq-right-leq-min-ℕ (succ-ℕ k) (succ-ℕ m) (succ-ℕ n) H =
  leq-right-leq-min-ℕ k m n H

is-greatest-lower-bound-min-ℕ :
  (l m : )  is-greatest-binary-lower-bound-Poset ℕ-Poset l m (min-ℕ l m)
is-greatest-lower-bound-min-ℕ l m =
    ( ℕ-Poset)
    ( leq-left-leq-min-ℕ (min-ℕ l m) l m (refl-leq-ℕ (min-ℕ l m)) ,
      leq-right-leq-min-ℕ (min-ℕ l m) l m (refl-leq-ℕ (min-ℕ l m)))
    ( λ x (H , K)  leq-min-ℕ x l m H K)

Associativity of min-ℕ

associative-min-ℕ :
  (x y z : )  min-ℕ (min-ℕ x y) z  min-ℕ x (min-ℕ y z)
associative-min-ℕ zero-ℕ y z = refl
associative-min-ℕ (succ-ℕ x) zero-ℕ zero-ℕ = refl
associative-min-ℕ (succ-ℕ x) zero-ℕ (succ-ℕ z) = refl
associative-min-ℕ (succ-ℕ x) (succ-ℕ y) zero-ℕ = refl
associative-min-ℕ (succ-ℕ x) (succ-ℕ y) (succ-ℕ z) =
  ap succ-ℕ (associative-min-ℕ x y z)

Zero laws for min-ℕ

left-zero-law-min-ℕ : (x : )  min-ℕ 0 x  0
left-zero-law-min-ℕ x = refl

right-zero-law-min-ℕ : (x : )  min-ℕ x 0  0
right-zero-law-min-ℕ zero-ℕ = refl
right-zero-law-min-ℕ (succ-ℕ x) = refl

Commutativity of min-ℕ

commutative-min-ℕ : (x y : )  min-ℕ x y  min-ℕ y x
commutative-min-ℕ zero-ℕ zero-ℕ = refl
commutative-min-ℕ zero-ℕ (succ-ℕ y) = refl
commutative-min-ℕ (succ-ℕ x) zero-ℕ = refl
commutative-min-ℕ (succ-ℕ x) (succ-ℕ y) = ap succ-ℕ (commutative-min-ℕ x y)

min-ℕ is idempotent

idempotent-min-ℕ : (x : )  min-ℕ x x  x
idempotent-min-ℕ zero-ℕ = refl
idempotent-min-ℕ (succ-ℕ x) = ap succ-ℕ (idempotent-min-ℕ x)

Addition distributes over min-ℕ

left-distributive-add-min-ℕ :
  (x y z : )  x +ℕ (min-ℕ y z)  min-ℕ (x +ℕ y) (x +ℕ z)
left-distributive-add-min-ℕ zero-ℕ y z =
  ( left-unit-law-add-ℕ (min-ℕ y z)) 
  ( ap-min-ℕ (inv (left-unit-law-add-ℕ y)) (inv (left-unit-law-add-ℕ z)))
left-distributive-add-min-ℕ (succ-ℕ x) y z =
  ( left-successor-law-add-ℕ x (min-ℕ y z)) 
  ( ( ap succ-ℕ (left-distributive-add-min-ℕ x y z)) 
    ( ap-min-ℕ
      ( inv (left-successor-law-add-ℕ x y))
      ( inv (left-successor-law-add-ℕ x z))))

right-distributive-add-min-ℕ :
  (x y z : )  (min-ℕ x y) +ℕ z  min-ℕ (x +ℕ z) (y +ℕ z)
right-distributive-add-min-ℕ x y z =
  ( commutative-add-ℕ (min-ℕ x y) z) 
  ( ( left-distributive-add-min-ℕ z x y) 
    ( ap-min-ℕ (commutative-add-ℕ z x) (commutative-add-ℕ z y)))

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