diff options
Diffstat (limited to 'FinMerge/Properties.agda')
| -rw-r--r-- | FinMerge/Properties.agda | 71 |
1 files changed, 25 insertions, 46 deletions
diff --git a/FinMerge/Properties.agda b/FinMerge/Properties.agda index e403d99..79f8686 100644 --- a/FinMerge/Properties.agda +++ b/FinMerge/Properties.agda @@ -1,5 +1,7 @@ +{-# 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) @@ -12,7 +14,7 @@ open import Function using (id; _∘_) open import Util using (_<_<_; _<_≤_; toℕ<; less; equal; greater; compare) -open import FinMerge using (merge; pluck; glue-iter) +open import FinMerge using (merge; pluck; glue-once; glue-iter) private @@ -106,43 +108,41 @@ merge-i-j {_} {i} {j} 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} f g h with compare (f (# 0)) (g (# 0)) -glue-iter-append {ℕ.suc m} f g h | less (f0<g0 , s≤s g0≤n) - with - glue-iter-append - (merge (f0<g0 , g0≤n) ∘ f ∘ Fin.suc) - (merge (f0<g0 , g0≤n) ∘ g ∘ Fin.suc) - (merge (f0<g0 , g0≤n) ∘ h) -... | h′ , glue-p∘h≡h′∘p∘h = h′ ∘ merge (f0<g0 , g0≤n) , glue-p∘h≡h′∘p∘h -glue-iter-append {ℕ.suc m} f g h | equal x = glue-iter-append (f ∘ Fin.suc) (g ∘ Fin.suc) h -glue-iter-append {ℕ.suc m} f g h | greater (g0<f0 , s≤s f0≤n) - with +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 - (merge (g0<f0 , f0≤n) ∘ f ∘ Fin.suc) - (merge (g0<f0 , f0≤n) ∘ g ∘ Fin.suc) - (merge (g0<f0 , f0≤n) ∘ h) -... | h′ , glue-p∘h≡h′∘p∘h = h′ ∘ merge (g0<f0 , f0≤n) , glue-p∘h≡h′∘p∘h + (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 f g with compare (f (# 0)) (g (# 0)) -lemma₂ f g | less (f0<g0 , s≤s g0≤n) - with +lemma₂ {_} {ℕ.zero} f g = ⊥-elim (¬Fin0 (f (# 0))) +lemma₂ {_} {ℕ.suc n} f g with glue-iter-append - (merge (f0<g0 , g0≤n) ∘ f ∘ Fin.suc) - (merge (f0<g0 , g0≤n) ∘ g ∘ Fin.suc) - (merge (f0<g0 , g0≤n)) + (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 = merge (f0<g0 , g0≤n) + 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) @@ -150,27 +150,6 @@ lemma₂ f g | less (f0<g0 , s≤s g0≤n) ≡ = 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′ (merge-i-j (f0<g0 , g0≤n)) ⟩ + 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) ∎ -lemma₂ f g | equal f0≡g0 = cong (proj₂ (glue-iter (f ∘ Fin.suc) (g ∘ Fin.suc) id)) f0≡g0 -lemma₂ f g | greater (g0<f0 , s≤s f0≤n) - with - glue-iter-append - (merge (g0<f0 , f0≤n) ∘ f ∘ Fin.suc) - (merge (g0<f0 , f0≤n) ∘ g ∘ Fin.suc) - (merge (g0<f0 , f0≤n) ∘ id) -... | h′ , glue≡h′∘h = ≡ - where - open ≡-Reasoning - p = merge (g0<f0 , f0≤n) - 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′ (sym (merge-i-j (g0<f0 , f0≤n))) ⟩ - 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) ∎ + proj₂ (glue-iter (p ∘ f′) (p ∘ g′) p) (g Fin.zero) ∎ |