Danila Fedorin
f21ebdcf46
Proving monotonicity is the main hurdle here. Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
71 lines
3.5 KiB
Agda
71 lines
3.5 KiB
Agda
module Utils where
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open import Agda.Primitive using () renaming (_⊔_ to _⊔ℓ_)
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open import Data.Nat using (ℕ; suc)
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open import Data.List using (List; []; _∷_; _++_) renaming (map to mapˡ)
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open import Data.List.Membership.Propositional using (_∈_)
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open import Data.List.Relation.Unary.All using (All; []; _∷_; map)
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open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
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open import Function.Definitions using (Injective)
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open import Relation.Binary.PropositionalEquality using (_≡_; sym; refl)
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open import Relation.Nullary using (¬_)
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data Unique {c} {C : Set c} : List C → Set c where
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empty : Unique []
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push : ∀ {x : C} {xs : List C}
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→ All (λ x' → ¬ x ≡ x') xs
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→ Unique xs
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→ Unique (x ∷ xs)
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Unique-append : ∀ {c} {C : Set c} {x : C} {xs : List C} →
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¬ x ∈ xs → Unique xs → Unique (xs ++ (x ∷ []))
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Unique-append {c} {C} {x} {[]} _ _ = push [] empty
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Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') =
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push (help x'≢) (Unique-append (λ x∈xs' → x∉xs (there x∈xs')) uxs')
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where
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x'≢x : ¬ x' ≡ x
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x'≢x x'≡x = x∉xs (here (sym x'≡x))
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help : {l : List C} → All (λ x'' → ¬ x' ≡ x'') l → All (λ x'' → ¬ x' ≡ x'') (l ++ (x ∷ []))
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help {[]} _ = x'≢x ∷ []
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help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es
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All-≢-map : ∀ {c d} {C : Set c} {D : Set d} (x : C) {xs : List C} (f : C → D) →
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Injective (_≡_ {_} {C}) (_≡_ {_} {D}) f →
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All (λ x' → ¬ x ≡ x') xs → All (λ y' → ¬ (f x) ≡ y') (mapˡ f xs)
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All-≢-map x f f-Injecitve [] = []
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All-≢-map x {x' ∷ xs'} f f-Injecitve (x≢x' ∷ x≢xs') = (λ fx≡fx' → x≢x' (f-Injecitve fx≡fx')) ∷ All-≢-map x f f-Injecitve x≢xs'
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Unique-map : ∀ {c d} {C : Set c} {D : Set d} {l : List C} (f : C → D) →
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Injective (_≡_ {_} {C}) (_≡_ {_} {D}) f →
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Unique l → Unique (mapˡ f l)
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Unique-map {l = []} _ _ _ = empty
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Unique-map {l = x ∷ xs} f f-Injecitve (push x≢xs uxs) = push (All-≢-map x f f-Injecitve x≢xs) (Unique-map f f-Injecitve uxs)
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All¬-¬Any : ∀ {p c} {C : Set c} {P : C → Set p} {l : List C} → All (λ x → ¬ P x) l → ¬ Any P l
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All¬-¬Any {l = x ∷ xs} (¬Px ∷ _) (here Px) = ¬Px Px
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All¬-¬Any {l = x ∷ xs} (_ ∷ ¬Pxs) (there Pxs) = All¬-¬Any ¬Pxs Pxs
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All-single : ∀ {p c} {C : Set c} {P : C → Set p} {c : C} {l : List C} → All P l → c ∈ l → P c
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All-single {c = c} {l = x ∷ xs} (p ∷ ps) (here refl) = p
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All-single {c = c} {l = x ∷ xs} (p ∷ ps) (there c∈xs) = All-single ps c∈xs
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All-x∈xs : ∀ {a} {A : Set a} (xs : List A) → All (λ x → x ∈ xs) xs
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All-x∈xs [] = []
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All-x∈xs (x ∷ xs') = here refl ∷ map there (All-x∈xs xs')
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x∈xs⇒fx∈fxs : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) {x : A} {xs : List A} →
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x ∈ xs → (f x) ∈ mapˡ f xs
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x∈xs⇒fx∈fxs f (here refl) = here refl
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x∈xs⇒fx∈fxs f (there x∈xs') = there (x∈xs⇒fx∈fxs f x∈xs')
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iterate : ∀ {a} {A : Set a} (n : ℕ) → (f : A → A) → A → A
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iterate 0 _ a = a
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iterate (suc n) f a = f (iterate n f a)
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data Pairwise {a} {b} {c} {A : Set a} {B : Set b} (P : A → B → Set c) : List A → List B → Set (a ⊔ℓ b ⊔ℓ c) where
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[] : Pairwise P [] []
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_∷_ : ∀ {x : A} {y : B} {xs : List A} {ys : List B} →
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P x y → Pairwise P xs ys →
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Pairwise P (x ∷ xs) (y ∷ ys)
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