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Add proofs about monotonicity of foldl

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Danila Fedorin 2024-05-08 21:48:41 -07:00
parent ad4592d4d2
commit dd8cdcd10c
1 changed files with 17 additions and 2 deletions
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Lattice.agda
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@ -137,7 +137,7 @@ module _ {a b} {A : Set a} {B : Set b}
const-Mono : ∀ (x : B) → Monotonic _≼₁_ _≼₂_ (λ _ → x) const-Mono : ∀ (x : B) → Monotonic _≼₁_ _≼₂_ (λ _ → x)
const-Mono x _ = ⊔₂-idemp x const-Mono x _ = ⊔₂-idemp x
open import Data.List as List using (List; foldr; _∷_) open import Data.List as List using (List; foldr; foldl; _∷_)
open import Utils using (Pairwise; _∷_) open import Utils using (Pairwise; _∷_)
foldr-Mono : ∀ (l₁ l₂ : List A) (f : A → B → B) (b₁ b₂ : B) → foldr-Mono : ∀ (l₁ l₂ : List A) (f : A → B → B) (b₁ b₂ : B) →
@ -150,13 +150,22 @@ module _ {a b} {A : Set a} {B : Set b}
≼₂-trans (f-Mono₁ (foldr f b₁ xs) x≼y) ≼₂-trans (f-Mono₁ (foldr f b₁ xs) x≼y)
(f-Mono₂ y (foldr-Mono xs ys f b₁ b₂ xs≼ys b₁≼b₂ f-Mono₁ f-Mono₂)) (f-Mono₂ y (foldr-Mono xs ys f b₁ b₂ xs≼ys b₁≼b₂ f-Mono₁ f-Mono₂))
foldl-Mono : ∀ (l₁ l₂ : List A) (f : B → A → B) (b₁ b₂ : B) →
Pairwise _≼₁_ l₁ l₂ → b₁ ≼₂ b₂ →
(∀ a → Monotonic _≼₂_ _≼₂_ (λ b → f b a)) →
(∀ b → Monotonic _≼₁_ _≼₂_ (f b)) →
foldl f b₁ l₁ ≼₂ foldl f b₂ l₂
foldl-Mono List.[] List.[] f b₁ b₂ _ b₁≼b₂ _ _ = b₁≼b₂
foldl-Mono (x ∷ xs) (y ∷ ys) f b₁ b₂ (x≼y ∷ xs≼ys) b₁≼b₂ f-Mono₁ f-Mono₂ =
foldl-Mono xs ys f (f b₁ x) (f b₂ y) xs≼ys (≼₂-trans (f-Mono₁ x b₁≼b₂) (f-Mono₂ b₂ x≼y)) f-Mono₁ f-Mono₂
module _ {a b} {A : Set a} {B : Set b} module _ {a b} {A : Set a} {B : Set b}
{_≈₂_ : B → B → Set b} {_⊔₂_ : B → B → B} {_≈₂_ : B → B → Set b} {_⊔₂_ : B → B → B}
(lB : IsSemilattice B _≈₂_ _⊔₂_) where (lB : IsSemilattice B _≈₂_ _⊔₂_) where
open IsSemilattice lB using () renaming (_≼_ to _≼₂_; ⊔-idemp to ⊔₂-idemp; ≼-trans to ≼₂-trans) open IsSemilattice lB using () renaming (_≼_ to _≼₂_; ⊔-idemp to ⊔₂-idemp; ≼-trans to ≼₂-trans)
open import Data.List as List using (List; foldr; _∷_) open import Data.List as List using (List; foldr; foldl; _∷_)
open import Utils using (Pairwise; _∷_) open import Utils using (Pairwise; _∷_)
foldr-Mono' : ∀ (l : List A) (f : A → B → B) → foldr-Mono' : ∀ (l : List A) (f : A → B → B) →
@ -165,6 +174,12 @@ module _ {a b} {A : Set a} {B : Set b}
foldr-Mono' List.[] f _ b₁≼b₂ = b₁≼b₂ foldr-Mono' List.[] f _ b₁≼b₂ = b₁≼b₂
foldr-Mono' (x ∷ xs) f f-Mono₂ b₁≼b₂ = f-Mono₂ x (foldr-Mono' xs f f-Mono₂ b₁≼b₂) foldr-Mono' (x ∷ xs) f f-Mono₂ b₁≼b₂ = f-Mono₂ x (foldr-Mono' xs f f-Mono₂ b₁≼b₂)
foldl-Mono' : ∀ (l : List A) (f : B → A → B) →
(∀ b → Monotonic _≼₂_ _≼₂_ (λ a → f a b)) →
Monotonic _≼₂_ _≼₂_ (λ b → foldl f b l)
foldl-Mono' List.[] f _ b₁≼b₂ = b₁≼b₂
foldl-Mono' (x ∷ xs) f f-Mono₁ b₁≼b₂ = foldl-Mono' xs f f-Mono₁ (f-Mono₁ x b₁≼b₂)
record IsLattice {a} (A : Set a) record IsLattice {a} (A : Set a)
(_≈_ : A → A → Set a) (_≈_ : A → A → Set a)
(_⊔_ : A → A → A) (_⊔_ : A → A → A)