module Tools.Nat where
open import Agda.Builtin.Nat using (Nat; _+_; _*_; _==_) public
open import Agda.Builtin.Nat using (zero; suc)
import Data.Fin as F
import Data.Fin.Properties as FP
open import Data.Nat.Base
open Data.Nat.Base using (_≤_; _<_; _⊔_; _⊓_; >-nonZero; nonZero) public
open _≤_ public
open import Data.Nat.DivMod
open Data.Nat.DivMod using (_/_; m/n*n≤m) public
open import Data.Nat.Properties
open Data.Nat.Properties
using (_≟_; _<?_; ≤-total;
+-identityʳ; +-assoc; +-comm; +-0-isCommutativeMonoid;
*-identityˡ; *-identityʳ; *-assoc; *-comm; *-zeroʳ;
*-1-isCommutativeMonoid;
m*n≡0⇒m≡0∨n≡0;
⊔-identityʳ; ⊔-assoc; ⊔-comm; ⊔-idem; m≥n⇒m⊔n≡m; m⊔n≡m⇒n≤m;
⊓-assoc; ⊓-comm;
+-distribˡ-⊔; *-distribˡ-+; *-distribˡ-⊔;
⊓-distribʳ-⊔; ⊔-distribʳ-⊓;
⊔-absorbs-⊓; ⊓-absorbs-⊔;
≤-refl; ≤-reflexive; ≤-trans; ≤-antisym; module ≤-Reasoning;
n≮n;
+-mono-≤; m≤m+n; m≤n+m; 0<1+n; n≤1+n;
*-mono-≤; m≤m*n; m≤n*m;
m≤m⊔n; m≤n⊔m;
m<n⊓o⇒m<n; m<n⊓o⇒m<o; ⊓-pres-m<;
m⊓n≤m⊔n)
public
open import Data.Nat.Show using (show) public
open import Relation.Binary using (Tri)
open Tri
open import Tools.Bool
open import Tools.Empty
open import Tools.Function
open import Tools.Level using (Level; lzero)
open import Tools.Product as Σ
open import Tools.PropositionalEquality
open import Tools.Relation
open import Tools.Sum as ⊎ using (_⊎_; inj₁; inj₂)
pattern 1+ n = suc n
pattern 2+ n = 1+ (1+ n)
pattern 3+ n = 1+ (1+ (1+ n))
private variable
a : Level
k m n o : Nat
p : Nat → Bool
Nat-set : Is-set Nat
Nat-set {x = zero} {y = zero} {x = refl} {y = refl} = refl
Nat-set {x = suc m} {y = suc n} {x = p} {y = q} =
$⟨ Nat-set {x = m} {y = n} ⟩
cong pred p ≡ cong pred q →⟨ cong (cong suc) ⟩
cong suc (cong pred p) ≡ cong suc (cong pred q) →⟨ (λ hyp → trans (sym (lemma _)) (trans hyp (lemma _))) ⟩
p ≡ q □
where
lemma : (p : suc m ≡ suc n) → cong suc (cong pred p) ≡ p
lemma refl = refl
⊓≤⇔≤⊎≤ : m ⊓ n ≤ o ⇔ (m ≤ o ⊎ n ≤ o)
⊓≤⇔≤⊎≤ {m = m} {n = n} {o = o} =
(λ m⊓n≤o →
case ≤-total m n of λ where
(inj₁ m≤n) → inj₁ $ begin
m ≤⟨ ⊓-glb ≤-refl m≤n ⟩
m ⊓ n ≤⟨ m⊓n≤o ⟩
o ∎
(inj₂ n≤m) → inj₂ $ begin
n ≤⟨ ⊓-glb n≤m ≤-refl ⟩
m ⊓ n ≤⟨ m⊓n≤o ⟩
o ∎)
, (λ where
(inj₁ m≤o) → begin
m ⊓ n ≤⟨ m⊓n≤m _ _ ⟩
m ≤⟨ m≤o ⟩
o ∎
(inj₂ n≤o) → begin
m ⊓ n ≤⟨ m⊓n≤n _ _ ⟩
n ≤⟨ n≤o ⟩
o ∎)
where
open ≤-Reasoning
≤⊔⇔≤⊎≤ : m ≤ n ⊔ o ⇔ (m ≤ n ⊎ m ≤ o)
≤⊔⇔≤⊎≤ {m = m} {n = n} {o = o} =
(λ m≤n⊔o →
case ≤-total n o of λ where
(inj₁ n≤o) → inj₂ $ begin
m ≤⟨ m≤n⊔o ⟩
n ⊔ o ≤⟨ ⊔-lub n≤o ≤-refl ⟩
o ∎
(inj₂ o≤n) → inj₁ $ begin
m ≤⟨ m≤n⊔o ⟩
n ⊔ o ≤⟨ ⊔-lub ≤-refl o≤n ⟩
n ∎)
, (λ where
(inj₁ m≤n) → begin
m ≤⟨ m≤n ⟩
n ≤⟨ m≤m⊔n _ _ ⟩
n ⊔ o ∎
(inj₂ m≤o) → begin
m ≤⟨ m≤o ⟩
o ≤⟨ m≤n⊔m _ _ ⟩
n ⊔ o ∎)
where
open ≤-Reasoning
*≡1+→0< : m * n ≡ 1+ o → 0 < m × 0 < n
*≡1+→0< {m = 1+ _} {n = 1+ _} = λ _ → s≤s z≤n , s≤s z≤n
*≡1+→0< {m = 1+ m} {n = 0} {o = o} =
m * 0 ≡ 1+ o →⟨ trans (sym (*-zeroʳ m)) ⟩
0 ≡ 1+ o →⟨ (λ ()) ⟩
0 < 1+ m × 0 < 0 □
private
1+*≤+1+*→1+*≤1+* :
k ≤ m →
1+ m * n ≤ k + 1+ m * o →
1+ m * n ≤ 1+ m * o
1+*≤+1+*→1+*≤1+* {m = m} {n = 0} {o = o} _ _ =
$⟨ _≤_.z≤n ⟩
0 ≤ 1+ m * o ≡⟨ cong (_≤ 1+ m * o) (sym (*-zeroʳ m)) ⟩→
1+ m * 0 ≤ 1+ m * o □
1+*≤+1+*→1+*≤1+* {k = k} {m = m} {n = 1+ n} {o = 0} k≤m =
1+ m * 1+ n ≤ k + m * 0 ≡⟨ cong₂ _≤_ (*-comm (1+ m) (1+ n)) (cong (k +_) (*-zeroʳ m)) ⟩→
1+ n * 1+ m ≤ k + 0 ≡⟨ cong (_ ≤_) (+-identityʳ _) ⟩→
1+ n * 1+ m ≤ k →⟨ idᶠ ⟩
1+ m + n * 1+ m ≤ k →⟨ m+n≤o⇒m≤o _ ⟩
m < k →⟨ flip ≤-trans k≤m ⟩
m < m →⟨ <-irrefl refl ⟩
⊥ →⟨ ⊥-elim ⟩
1+ m * 1+ n ≤ m * 0 □
1+*≤+1+*→1+*≤1+* {k = k} {m = m} {n = 1+ n} {o = 1+ o} k≤m =
1+ m * 1+ n ≤ k + 1+ m * 1+ o ≡⟨ cong₂ _≤_ (*-suc (1+ m) _) (cong (k +_) (*-suc (1+ m) _)) ⟩→
1+ m + 1+ m * n ≤ k + (1+ m + 1+ m * o) ≡⟨ cong (1+ m + 1+ m * n ≤_) $
trans (sym (+-assoc k (1+ m) _)) $
trans (cong (_+ 1+ m * o) (+-comm k _)) $
+-assoc (1+ m) _ _ ⟩→
1+ m + 1+ m * n ≤ 1+ m + (k + 1+ m * o) →⟨ +-cancelˡ-≤ (1+ m) (n + m * n) (k + (o + m * o)) ⟩
1+ m * n ≤ k + 1+ m * o →⟨ 1+*≤+1+*→1+*≤1+* k≤m ⟩
1+ m * n ≤ 1+ m * o →⟨ +-mono-≤ (≤-refl {x = 1+ m}) ⟩
1+ m + 1+ m * n ≤ 1+ m + 1+ m * o ≡⟨ sym $ cong₂ _≤_ (*-suc (1+ m) _) (*-suc (1+ m) _) ⟩→
1+ m * 1+ n ≤ 1+ m * 1+ o □
*≤→≤/ : 1+ m * n ≤ o → n ≤ o / 1+ m
*≤→≤/ {m = m} {n = n} {o = o} = helper (o divMod 1+ m)
where
helper :
(d : DivMod o (1+ m)) →
1+ m * n ≤ o → n ≤ d .DivMod.quotient
helper (record { quotient = q; remainder = r; property = refl }) =
1+ m * n ≤ F.toℕ r + q * 1+ m ≡⟨ cong (λ o → 1+ m * n ≤ F.toℕ r + o) (*-comm q _) ⟩→
1+ m * n ≤ F.toℕ r + 1+ m * q →⟨ 1+*≤+1+*→1+*≤1+* (FP.toℕ≤pred[n] r) ⟩
1+ m * n ≤ 1+ m * q →⟨ *-cancelˡ-≤ (1+ m) ⟩
n ≤ q □
T-== : T (m == n) ⇔ m ≡ n
T-== = ≡ᵇ⇒≡ _ _ , ≡⇒≡ᵇ _ _
∃< : Nat → (Nat → Bool) → Bool
∃< 0 p = false
∃< (1+ n) p = p n ∨ ∃< n p
∃<⇔ : T (∃< n p) ⇔ (∃ λ m → m < n × T (p m))
∃<⇔ {n = 0} {p = p} =
⊥ ⇔⟨ (⊥-elim , λ { (_ , () , _) }) ⟩
(∃ λ m → m < 0 × T (p m)) □⇔
∃<⇔ {n = 1+ n} {p = p} =
T (p n ∨ ∃< n p) ⇔⟨ T-∨ ⟩
T (p n) ⊎ T (∃< n p) ⇔⟨ id⇔ ⊎-cong-⇔ ∃<⇔ ⟩
T (p n) ⊎ (∃ λ m → m < n × T (p m)) ⇔⟨ (λ where
(inj₁ p-n) → n , ≤-refl , p-n
(inj₂ (m , m<n , p-m)) → m , ≤-trans m<n (n≤1+n _) , p-m)
, (λ (m , m<1+n , p-m) →
case <-cmp m n of λ where
(tri< m<n _ _) → inj₂ (m , m<n , p-m)
(tri≈ _ refl _) → inj₁ p-m
(tri> _ _ m>n) → ⊥-elim (m+n≮n _ _ (≤-trans m<1+n m>n)))
⟩
(∃ λ m → m < 1+ n × T (p m)) □⇔
∃≤ : Nat → (Nat → Bool) → Bool
∃≤ n = ∃< (1+ n)
∃≤⇔ : T (∃≤ n p) ⇔ (∃ λ m → m ≤ n × T (p m))
∃≤⇔ {n = n} {p = p} =
T (∃≤ n p) ⇔⟨ id⇔ ⟩
T (∃< (1+ n) p) ⇔⟨ ∃<⇔ ⟩
(∃ λ m → m < 1+ n × T (p m)) ⇔⟨ (Σ-cong-⇔ λ _ → ((λ { (s≤s m≤n) → m≤n }) , s≤s) ×-cong-⇔ id⇔) ⟩
(∃ λ m → m ≤ n × T (p m)) □⇔
∃-least : (ℕ → Set a) → Set a
∃-least P = ∃ λ n → P n × ∀ m → m < n → ¬ P m
∃⇔∃-least : (∃ λ n → T (p n)) ⇔ ∃-least (λ n → T (p n))
∃⇔∃-least = (uncurry λ _ → ∃→∃-least) , Σ.map idᶠ proj₁
where
∃→∃-least : T (p n) → ∃-least (λ n → T (p n))
∃→∃-least {n = 0} ok =
0 , ok , λ _ ()
∃→∃-least {p = p} {n = 1+ n} ok = lemma _ refl
where
lemma :
∀ b → b ≡ p 0 →
∃ λ n → T (p n) × ∀ m → m < n → ¬ T (p m)
lemma true eq = 0 , subst T eq _ , λ _ ()
lemma false eq =
case ∃→∃-least {p = p ∘→ 1+} ok of λ {
(n , ok , least) →
1+ n
, ok
, (λ where
0 _ →
T (p 0) →⟨ subst T (sym eq) ⟩
⊥ □
(1+ m) 1+m<1+n →
T (p (1+ m)) →⟨ least m (+-cancelˡ-< 1 m n 1+m<1+n) ⟩
⊥ □) }
∃-least-propositional :
∀ {p} {P : ℕ → Set p} →
Function-extensionality lzero p →
Function-extensionality p lzero →
(∀ n → Is-proposition (P n)) →
Is-proposition (∃-least P)
∃-least-propositional
fe₁ fe₂ P-prop
{x = n₁ , p₁ , least₁}
{y = n₂ , p₂ , least₂} =
case n₁≡n₂ of λ {
refl →
cong₂ (λ p least → _ , p , least)
(P-prop _)
(Is-proposition-Π fe₁ λ _ →
Is-proposition-Π fe₁ λ _ →
Is-proposition-Π fe₂ λ _ →
⊥-propositional) }
where
n₁≡n₂ : n₁ ≡ n₂
n₁≡n₂ = case <-cmp n₁ n₂ of λ where
(tri< n₁<n₂ _ _) → ⊥-elim (least₂ _ n₁<n₂ p₁)
(tri≈ _ n₁≡n₂ _) → n₁≡n₂
(tri> _ _ n₁>n₂) → ⊥-elim (least₁ _ n₁>n₂ p₂)