{-# OPTIONS --cubical-compatible --safe #-}
module Data.List.Relation.Unary.Sorted.TotalOrder.Properties where
open import Data.List.Base
open import Data.List.Relation.Unary.All using (All)
open import Data.List.Relation.Unary.AllPairs using (AllPairs)
open import Data.List.Relation.Unary.Linked as Linked
using (Linked; []; [-]; _∷_; _∷′_; head′; tail)
import Data.List.Relation.Unary.Linked.Properties as Linked
import Data.List.Relation.Binary.Equality.Setoid as Equality
import Data.List.Relation.Binary.Sublist.Setoid as Sublist
import Data.List.Relation.Binary.Sublist.Setoid.Properties as SublistProperties
import Data.List.Relation.Binary.Permutation.Setoid as Permutation
import Data.List.Relation.Binary.Permutation.Setoid.Properties as PermutationProperties
import Data.List.Relation.Binary.Pointwise as Pointwise
open import Data.List.Relation.Unary.Sorted.TotalOrder as Sorted hiding (head)
open import Data.Fin.Base as Fin hiding (_<_; _≤_)
import Data.Fin.Properties as Fin
open import Data.Fin.Permutation
open import Data.Product using (_,_)
open import Data.Maybe.Base using (just; nothing)
open import Data.Maybe.Relation.Binary.Connected using (Connected; just)
open import Data.Nat.Base as ℕ using (ℕ; z≤n; s≤s; zero; suc)
import Data.Nat.Properties as ℕ
open import Function.Base using (_∘_; const)
open import Function.Bundles using (Inverse)
open import Function.Consequences.Propositional using (inverseʳ⇒injective)
open import Level using (Level)
open import Relation.Binary.Core using (_Preserves_⟶_; Rel)
open import Relation.Binary.Bundles using (TotalOrder; DecTotalOrder; Preorder)
import Relation.Binary.Properties.TotalOrder as TotalOrderProperties
import Relation.Binary.Reasoning.PartialOrder as PosetReasoning
open import Relation.Unary using (Pred; Decidable)
open import Relation.Nullary using (contradiction)
open import Relation.Nullary.Decidable using (yes; no)
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
private
variable
a b p ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level
module _ (O : TotalOrder a ℓ₁ ℓ₂) where
open TotalOrder O
AllPairs⇒Sorted : ∀ {xs} → AllPairs _≤_ xs → Sorted O xs
AllPairs⇒Sorted = Linked.AllPairs⇒Linked
Sorted⇒AllPairs : ∀ {xs} → Sorted O xs → AllPairs _≤_ xs
Sorted⇒AllPairs = Linked.Linked⇒AllPairs trans
module _ (O₁ : TotalOrder a ℓ₁ ℓ₂) (O₂ : TotalOrder a ℓ₁ ℓ₂) where
private
module O₁ = TotalOrder O₁
module O₂ = TotalOrder O₂
map⁺ : ∀ {f xs} → f Preserves O₁._≤_ ⟶ O₂._≤_ →
Sorted O₁ xs → Sorted O₂ (map f xs)
map⁺ pres xs↗ = Linked.map⁺ (Linked.map pres xs↗)
map⁻ : ∀ {f xs} → (∀ {x y} → f x O₂.≤ f y → x O₁.≤ y) →
Sorted O₂ (map f xs) → Sorted O₁ xs
map⁻ pres fxs↗ = Linked.map pres (Linked.map⁻ fxs↗)
module _ (O : TotalOrder a ℓ₁ ℓ₂) where
open TotalOrder O
++⁺ : ∀ {xs ys} → Sorted O xs → Connected _≤_ (last xs) (head ys) →
Sorted O ys → Sorted O (xs ++ ys)
++⁺ = Linked.++⁺
module _ (O : TotalOrder a ℓ₁ ℓ₂) where
open TotalOrder O
applyUpTo⁺₁ : ∀ f n → (∀ {i} → suc i ℕ.< n → f i ≤ f (suc i)) →
Sorted O (applyUpTo f n)
applyUpTo⁺₁ = Linked.applyUpTo⁺₁
applyUpTo⁺₂ : ∀ f n → (∀ i → f i ≤ f (suc i)) →
Sorted O (applyUpTo f n)
applyUpTo⁺₂ = Linked.applyUpTo⁺₂
module _ (O : TotalOrder a ℓ₁ ℓ₂) where
open TotalOrder O
applyDownFrom⁺₁ : ∀ f n → (∀ {i} → suc i ℕ.< n → f (suc i) ≤ f i) →
Sorted O (applyDownFrom f n)
applyDownFrom⁺₁ = Linked.applyDownFrom⁺₁
applyDownFrom⁺₂ : ∀ f n → (∀ i → f (suc i) ≤ f i) →
Sorted O (applyDownFrom f n)
applyDownFrom⁺₂ = Linked.applyDownFrom⁺₂
module _ (DO : DecTotalOrder a ℓ₁ ℓ₂) where
open DecTotalOrder DO using (_≤_; _≤?_) renaming (totalOrder to O)
open TotalOrderProperties O using (≰⇒≥)
private
merge-con : ∀ {v xs ys} →
Connected _≤_ (just v) (head xs) →
Connected _≤_ (just v) (head ys) →
Connected _≤_ (just v) (head (merge _≤?_ xs ys))
merge-con {xs = []} cxs cys = cys
merge-con {xs = x ∷ xs} {[]} cxs cys = cxs
merge-con {xs = x ∷ xs} {y ∷ ys} cxs cys with x ≤? y
... | yes x≤y = cxs
... | no x≰y = cys
merge⁺ : ∀ {xs ys} → Sorted O xs → Sorted O ys → Sorted O (merge _≤?_ xs ys)
merge⁺ {[]} rxs rys = rys
merge⁺ {x ∷ xs} {[]} rxs rys = rxs
merge⁺ {x ∷ xs} {y ∷ ys} rxs rys
with x ≤? y | merge⁺ (Linked.tail rxs) rys
| merge⁺ rxs (Linked.tail rys)
... | yes x≤y | rec | _ = merge-con (head′ rxs) (just x≤y) ∷′ rec
... | no x≰y | _ | rec = merge-con (just (≰⇒≥ x≰y)) (head′ rys) ∷′ rec
S = Preorder.Eq.setoid (DecTotalOrder.preorder DO)
open Sublist S using (_⊆_)
module SP = SublistProperties S
⊆-mergeˡ : ∀ xs ys → xs ⊆ merge _≤?_ xs ys
⊆-mergeˡ = SP.⊆-mergeˡ _≤?_
⊆-mergeʳ : ∀ xs ys → ys ⊆ merge _≤?_ xs ys
⊆-mergeʳ = SP.⊆-mergeʳ _≤?_
module _ (O : TotalOrder a ℓ₁ ℓ₂) {P : Pred _ p} (P? : Decidable P) where
open TotalOrder O
filter⁺ : ∀ {xs} → Sorted O xs → Sorted O (filter P? xs)
filter⁺ = Linked.filter⁺ P? trans
module _ (O : TotalOrder a ℓ₁ ℓ₂) where
open TotalOrder O
lookup-mono-≤ : ∀ {xs} → Sorted O xs →
∀ {i j} → i Fin.≤ j → lookup xs i ≤ lookup xs j
lookup-mono-≤ {x ∷ xs} xs↗ {zero} {zero} z≤n = refl
lookup-mono-≤ {x ∷ xs} xs↗ {zero} {suc j} z≤n = Linked.lookup trans xs↗ (just refl) (suc j)
lookup-mono-≤ {x ∷ xs} xs↗ {suc i} {suc j} (s≤s i≤j) = lookup-mono-≤ (Sorted.tail O {y = x} xs↗) i≤j
module _ (O : TotalOrder a ℓ₁ ℓ₂) where
open TotalOrder O
open Equality Eq.setoid
open Permutation Eq.setoid hiding (refl; trans)
open PermutationProperties Eq.setoid
open PosetReasoning poset
↗↭↗⇒≋ : ∀ {xs ys} → Sorted O xs → Sorted O ys → xs ↭ ys → xs ≋ ys
↗↭↗⇒≋ {xs} {ys} xs↗ ys↗ xs↭ys = Pointwise.lookup⁻
(xs↭ys⇒|xs|≡|ys| xs↭ys)
(λ i≡j → antisym
(↗↭↗⇒≤ (↭-sym xs↭ys) ys↗ xs↗ (≡.sym i≡j))
(↗↭↗⇒≤ xs↭ys xs↗ ys↗ i≡j))
where
↗↭↗⇒≤ : ∀ {xs ys}
(xs↭ys : xs ↭ ys) →
Sorted O xs → Sorted O ys →
∀ {i j} → toℕ i ≡ toℕ j →
lookup ys j ≤ lookup xs i
↗↭↗⇒≤ {xs} {ys} xs↭ys xs↗ ys↗ {i} {j} i≡j
with Fin.injective⇒existsPivot (inverseʳ⇒injective _ (Inverse.inverseʳ (onIndices xs↭ys))) i
... | (k , k≤i , i≤π[k]) = begin
lookup ys j ≤⟨ lookup-mono-≤ O ys↗ (≡.subst (ℕ._≤ _) i≡j i≤π[k]) ⟩
lookup ys (onIndices xs↭ys ⟨$⟩ʳ k) ≈⟨ onIndices-lookup xs↭ys k ⟨
lookup xs k ≤⟨ lookup-mono-≤ O xs↗ k≤i ⟩
lookup xs i ∎