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{-# OPTIONS --without-K --safe #-}
module FinMerge.Properties where
open import Data.Empty using (⊥-elim)
open import Data.Fin using (Fin; fromℕ<; toℕ; #_; lower₁)
open import Data.Fin.Properties using (¬Fin0)
open import Data.Nat using (ℕ; _+_; _≤_; _<_; z<s; pred; z≤n; s≤s)
open import Data.Nat.Properties using (≤-trans; <⇒≢)
open import Data.Product using (_,_; proj₁; proj₂; Σ-syntax)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong; cong-app; sym; _≢_)
open import Relation.Binary.PropositionalEquality.Properties using (module ≡-Reasoning)
open import Data.Maybe.Base using (Maybe; nothing; just; fromMaybe)
open import Function using (id; _∘_)
open import Util using (_<_<_; _<_≤_; toℕ<; less; equal; greater; compare)
open import FinMerge using (merge; pluck; glue-once; glue-iter)
private
variable
m n : ℕ
not-n : {x : Fin (ℕ.suc n)} → toℕ x < m ≤ n → n ≢ toℕ x
not-n (x<m , m≤n) n≡x = <⇒≢ (≤-trans x<m m≤n) (sym n≡x)
pluck-<
: {x : Fin (ℕ.suc n)}
→ (m≤n : m ≤ n)
→ (x<m : toℕ x < m)
→ pluck m≤n x ≡ just (lower₁ x (not-n (x<m , m≤n)))
pluck-< {_} {_} {Fin.zero} (_≤_.s≤s m≤n) (_≤_.s≤s x<m) = refl
pluck-< {_} {_} {Fin.suc x} (_≤_.s≤s m≤n) (_≤_.s≤s x<m) = ≡
where
open ≡-Reasoning
≡ : pluck (_≤_.s≤s m≤n) (Fin.suc x)
≡ just (lower₁ (Fin.suc x) (not-n (s≤s x<m , s≤s m≤n)))
≡ = begin
pluck (_≤_.s≤s m≤n) (Fin.suc x) ≡⟨⟩
Data.Maybe.Base.map Fin.suc (pluck m≤n x)
≡⟨ cong (Data.Maybe.Base.map Fin.suc) (pluck-< m≤n x<m) ⟩
Data.Maybe.Base.map Fin.suc (just (lower₁ x (not-n (x<m , m≤n)))) ≡⟨⟩
just (Fin.suc (lower₁ x (not-n (x<m , m≤n)))) ≡⟨⟩
just (lower₁ (Fin.suc x) (not-n (s≤s x<m , s≤s m≤n))) ∎
pluck-≡
: {x : Fin (ℕ.suc n)}
→ (m≤n : m ≤ n)
→ (x≡m : toℕ x ≡ m)
→ pluck m≤n x ≡ nothing
pluck-≡ {_} {_} {Fin.zero} z≤n x≡m = refl
pluck-≡ {_} {_} {Fin.suc x} (s≤s m≤n) refl = ≡
where
open ≡-Reasoning
≡ : pluck (s≤s m≤n) (Fin.suc x) ≡ nothing
≡ = begin
pluck (s≤s m≤n) (Fin.suc x) ≡⟨⟩
Data.Maybe.Base.map Fin.suc (pluck m≤n x)
≡⟨ cong (Data.Maybe.Base.map Fin.suc) (pluck-≡ m≤n refl) ⟩
Data.Maybe.Base.map Fin.suc nothing ≡⟨⟩
nothing ∎
i-to-i
: {i j : Fin (ℕ.suc n)} (i<j≤n@(i<j , j≤n) : toℕ i < toℕ j ≤ n)
→ merge i<j≤n i ≡ lower₁ i (not-n i<j≤n)
i-to-i {i = i} (lt , le) = ≡
where
open ≡-Reasoning
≡ : merge (lt , le) i ≡ lower₁ i (not-n (lt , le))
≡ = begin
merge (lt , le) i ≡⟨⟩
fromMaybe (fromℕ< (≤-trans lt le)) (pluck le i)
≡⟨ cong (fromMaybe (fromℕ< (≤-trans lt le))) (pluck-< le lt) ⟩
fromMaybe (fromℕ< (≤-trans lt le)) (just (lower₁ i (not-n (lt , le)))) ≡⟨⟩
lower₁ i (not-n (lt , le)) ∎
lemma₁
: {i j : Fin (ℕ.suc n)} ((lt , le) : toℕ i < toℕ j ≤ n)
→ fromℕ< (≤-trans lt le) ≡ lower₁ i (not-n (lt , le))
lemma₁ {ℕ.suc _} {Fin.zero} {Fin.suc _} _ = refl
lemma₁ {ℕ.suc _} {Fin.suc _} {Fin.suc _} (s≤s lt , s≤s le) = cong Fin.suc (lemma₁ (lt , le))
j-to-i
: {i j : Fin (ℕ.suc n)} (i<j≤n@(i<j , j≤n) : toℕ i < toℕ j ≤ n)
→ merge i<j≤n j ≡ lower₁ i (not-n i<j≤n)
j-to-i {i = i} {j = j} (lt , le) = ≡
where
open ≡-Reasoning
≡ : merge (lt , le) j ≡ lower₁ i (not-n (lt , le))
≡ = begin
merge (lt , le) j ≡⟨⟩
fromMaybe (fromℕ< (≤-trans lt le)) (pluck le j)
≡⟨ cong (fromMaybe (fromℕ< (≤-trans lt le))) (pluck-≡ le refl) ⟩
fromMaybe (fromℕ< (≤-trans lt le)) nothing ≡⟨⟩
fromℕ< (≤-trans lt le) ≡⟨ lemma₁ (lt , le) ⟩
lower₁ i (not-n (lt , le)) ∎
merge-i-j
: {i j : Fin (ℕ.suc n)}
→ (i<j≤n : toℕ i < toℕ j ≤ n)
→ merge i<j≤n i ≡ merge i<j≤n j
merge-i-j {_} {i} {j} i<j≤n = ≡
where
open ≡-Reasoning
≡ : merge i<j≤n i ≡ merge i<j≤n j
≡ = begin
merge i<j≤n i ≡⟨ i-to-i i<j≤n ⟩
lower₁ i (not-n i<j≤n) ≡⟨ sym (j-to-i i<j≤n) ⟩
merge i<j≤n j ∎
glue-once-correct
: (f0 g0 : Fin (ℕ.suc n))
→ proj₂ (glue-once f0 g0) f0 ≡ proj₂ (glue-once f0 g0) g0
glue-once-correct {n} f0 g0 with compare f0 g0
... | less (f0<g0 , s≤s g0≤n) = merge-i-j (f0<g0 , g0≤n)
... | equal f0≡g0 = f0≡g0
... | greater (g0<f0 , s≤s f0≤n) = sym (merge-i-j (g0<f0 , f0≤n))
glue-iter-append
: {y : ℕ}
→ (f g : Fin m → Fin y)
→ (h : Fin n → Fin y)
→ Σ[ h′ ∈ (Fin y → Fin (proj₁ (glue-iter f g h))) ] (proj₂ (glue-iter f g h) ≡ h′ ∘ h)
glue-iter-append {ℕ.zero} f g h = id , refl
glue-iter-append {ℕ.suc m} {_} {ℕ.zero} f g h = ⊥-elim (¬Fin0 (f (# 0)))
glue-iter-append {ℕ.suc m} {_} {ℕ.suc y} f g h with
glue-iter-append
(proj₂ (glue-once (f (# 0)) (g (# 0))) ∘ f ∘ Fin.suc)
(proj₂ (glue-once (f (# 0)) (g (# 0))) ∘ g ∘ Fin.suc)
(proj₂ (glue-once (f (# 0)) (g (# 0))) ∘ h)
... | h′ , glue-p∘h≡h′∘p∘h = h′ ∘ proj₂ (glue-once (f (# 0)) (g (# 0))) , glue-p∘h≡h′∘p∘h
lemma₂
: (f g : Fin (ℕ.suc m) → Fin n)
→ let p = proj₂ (glue-iter f g id) in p (f (# 0)) ≡ p (g (# 0))
lemma₂ {_} {ℕ.zero} f g = ⊥-elim (¬Fin0 (f (# 0)))
lemma₂ {_} {ℕ.suc n} f g with
glue-iter-append
(proj₂ (glue-once (f (# 0)) (g (# 0))) ∘ f ∘ Fin.suc)
(proj₂ (glue-once (f (# 0)) (g (# 0))) ∘ g ∘ Fin.suc)
(proj₂ (glue-once (f (# 0)) (g (# 0))))
... | h′ , glue≡h′∘h = ≡
where
open ≡-Reasoning
p = proj₂ (glue-once (f (# 0)) (g (# 0)))
f′ = f ∘ Fin.suc
g′ = g ∘ Fin.suc
≡ : proj₂ (glue-iter (p ∘ f′) (p ∘ g′) p) (f Fin.zero)
≡ proj₂ (glue-iter (p ∘ f′) (p ∘ g′) p) (g Fin.zero)
≡ = begin
proj₂ (glue-iter (p ∘ f′) (p ∘ g′) p) (f Fin.zero)
≡⟨ cong-app glue≡h′∘h (f Fin.zero) ⟩
h′ (p (f Fin.zero)) ≡⟨ cong h′ (glue-once-correct (f (# 0)) (g (# 0))) ⟩
h′ (p (g Fin.zero)) ≡⟨ sym (cong-app glue≡h′∘h (g Fin.zero)) ⟩
proj₂ (glue-iter (p ∘ f′) (p ∘ g′) p) (g Fin.zero) ∎
|
