agda-spa/Utils.agda

module Utils where

open import Agda.Primitive using () renaming (_⊔_ to _⊔ℓ_)
open import Data.Product as Prod using (Σ; _×_; _,_; proj₁; proj₂)
open import Data.Nat using (ℕ; suc)
open import Data.Fin as Fin using (Fin; suc; zero)
open import Data.Fin.Properties using (suc-injective)
open import Data.List using (List; cartesianProduct; []; _∷_; _++_; foldr; filter) renaming (map to mapˡ)
open import Data.List.Membership.Propositional using (_∈_)
open import Data.List.Membership.Propositional.Properties as ListMemProp using ()
open import Data.List.Relation.Unary.All using (All; []; _∷_; map)
open import Data.List.Relation.Unary.All.Properties using (++⁻ˡ)
open import Data.List.Relation.Unary.Any as Any using (Any; here; there) -- TODO: re-export these with nicer names from map
open import Data.Sum using (_⊎_)
open import Function.Definitions using (Injective)
open import Relation.Binary.PropositionalEquality using (_≡_; sym; refl; cong)
open import Relation.Nullary using (¬_; yes; no)
open import Relation.Unary using (Decidable)

data Unique {c} {C : Set c} : List C → Set c where
    empty : Unique []
    push : ∀ {x : C} {xs : List C}
        → All (λ x' → ¬ x ≡ x') xs
        → Unique xs
        → Unique (x ∷ xs)

Unique-append : ∀ {c} {C : Set c} {x : C} {xs : List C} →
                ¬ x ∈ xs → Unique xs →  Unique (xs ++ (x ∷ []))
Unique-append {c} {C} {x} {[]} _ _ = push [] empty
Unique-append {c} {C} {x} {x' ∷ xs'} x∉xs (push x'≢ uxs') =
    push (help x'≢) (Unique-append (λ x∈xs' → x∉xs (there x∈xs')) uxs')
    where
        x'≢x : ¬ x' ≡ x
        x'≢x x'≡x = x∉xs (here (sym x'≡x))

        help : {l : List C} → All (λ x'' → ¬ x' ≡ x'') l → All (λ x'' → ¬ x' ≡ x'') (l ++ (x ∷ []))
        help {[]} _ = x'≢x ∷ []
        help {e ∷ es} (x'≢e ∷ x'≢es) = x'≢e ∷ help x'≢es

Unique-++⁻ˡ : ∀ {c} {C : Set c} (xs : List C) {ys : List C} → Unique (xs ++ ys) → Unique xs
Unique-++⁻ˡ [] Unique-ys = empty
Unique-++⁻ˡ (x ∷ xs) {ys} (push x≢xs++ys Unique-xs++ys) = push (++⁻ˡ xs {ys = ys} x≢xs++ys) (Unique-++⁻ˡ xs Unique-xs++ys)

Unique-++⁻ʳ : ∀ {c} {C : Set c} (xs : List C) {ys : List C} → Unique (xs ++ ys) → Unique ys
Unique-++⁻ʳ [] Unique-ys = Unique-ys
Unique-++⁻ʳ (x ∷ xs) {ys} (push x≢xs++ys Unique-xs++ys) = Unique-++⁻ʳ xs Unique-xs++ys

All-≢-map : ∀ {c d} {C : Set c} {D : Set d} (x : C) {xs : List C} (f : C → D) →
            Injective (_≡_ {_} {C}) (_≡_ {_} {D}) f →
            All (λ x' → ¬ x ≡ x') xs → All (λ y' → ¬ (f x) ≡ y') (mapˡ f xs)
All-≢-map x f f-Injecitve [] = []
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'

Unique-map : ∀ {c d} {C : Set c} {D : Set d} {l : List C} (f : C → D) →
             Injective (_≡_ {_} {C}) (_≡_ {_} {D}) f →
             Unique l → Unique (mapˡ f l)
Unique-map {l = []} _ _ _ = empty
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)

All¬-¬Any : ∀ {p c} {C : Set c} {P : C → Set p} {l : List C} → All (λ x → ¬ P x) l → ¬ Any P l
All¬-¬Any {l = x ∷ xs} (¬Px ∷ _) (here Px) = ¬Px Px
All¬-¬Any {l = x ∷ xs} (_ ∷ ¬Pxs) (there Pxs) = All¬-¬Any ¬Pxs Pxs

¬Any-map : ∀ {p₁ p₂ c} {C : Set c} {P₁ : C → Set p₁} {P₂ : C → Set p₂} {l : List C} → (∀ {x} → P₁ x → P₂ x) → ¬ Any P₂ l → ¬ Any P₁ l
¬Any-map f ¬Any-P₂ Any-P₁ = ¬Any-P₂ (Any.map f Any-P₁)

All-single : ∀ {p c} {C : Set c} {P : C → Set p} {c : C} {l : List C} → All P l → c ∈ l → P c
All-single {c = c} {l = x ∷ xs} (p ∷ ps) (here refl) = p
All-single {c = c} {l = x ∷ xs} (p ∷ ps) (there c∈xs) = All-single ps c∈xs

All-x∈xs : ∀ {a} {A : Set a} (xs : List A) → All (λ x → x ∈ xs) xs
All-x∈xs [] = []
All-x∈xs (x ∷ xs') = here refl ∷ map there (All-x∈xs xs')

x∈xs⇒fx∈fxs : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) {x : A} {xs : List A} →
              x ∈ xs → (f x) ∈ mapˡ f xs
x∈xs⇒fx∈fxs f (here refl) = here refl
x∈xs⇒fx∈fxs f (there x∈xs') = there (x∈xs⇒fx∈fxs f x∈xs')

iterate : ∀ {a} {A : Set a} (n : ℕ) → (f : A → A) → A → A
iterate 0 _ a = a
iterate (suc n) f a = f (iterate n f a)

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
    [] : Pairwise P [] []
    _∷_ : ∀ {x : A} {y : B} {xs : List A} {ys : List B} →
          P x y → Pairwise P xs ys →
          Pairwise P (x ∷ xs) (y ∷ ys)

∈-cartesianProduct : ∀ {a b} {A : Set a} {B : Set b}
                     {x : A} {xs : List A} {y : B} {ys : List B} →
                     x ∈ xs → y ∈ ys → (x Prod., y) ∈ cartesianProduct xs ys
∈-cartesianProduct {x = x} (here refl) y∈ys = ListMemProp.∈-++⁺ˡ (x∈xs⇒fx∈fxs (x Prod.,_) y∈ys)
∈-cartesianProduct {x = x} {xs = x' ∷ _} {ys = ys} (there x∈rest) y∈ys = ListMemProp.∈-++⁺ʳ (mapˡ (x' Prod.,_) ys) (∈-cartesianProduct x∈rest y∈ys)

concat-∈ : ∀ {a} {A : Set a} {x : A} {l : List A} {ls : List (List A)} →
           x ∈ l → l ∈ ls → x ∈ foldr _++_ [] ls
concat-∈ x∈l (here refl) = ListMemProp.∈-++⁺ˡ x∈l
concat-∈ {ls = l' ∷ ls'} x∈l (there l∈ls') = ListMemProp.∈-++⁺ʳ l' (concat-∈ x∈l l∈ls')

filter-++ : ∀ {a p} {A : Set a} (l₁ l₂ : List A) {P : A → Set p} (P? : Decidable P) →
            filter P? (l₁ ++ l₂) ≡ filter P? l₁ ++ filter P? l₂
filter-++ [] l₂ P? = refl
filter-++ (x ∷ xs) l₂ P?
    with P? x
...   | yes _ = cong (x ∷_) (filter-++ xs l₂ P?)
...   | no _ = (filter-++ xs l₂ P?)

_⇒_ : ∀ {a p₁ p₂} {A : Set a} (P : A → Set p₁) (Q : A → Set p₂) →
      Set (a ⊔ℓ p₁ ⊔ℓ p₂)
_⇒_ P Q = ∀ a → P a → Q a

_∨_ : ∀ {a p₁ p₂} {A : Set a} (P : A → Set p₁) (Q : A → Set p₂) →
      A → Set (p₁ ⊔ℓ p₂)
_∨_ P Q a = P a ⊎ Q a

_∧_ : ∀ {a p₁ p₂} {A : Set a} (P : A → Set p₁) (Q : A → Set p₂) →
      A → Set (p₁ ⊔ℓ p₂)
_∧_ P Q a = P a × Q a

it : ∀ {a} {A : Set a} {{_ : A}} → A
it {{x}} = x

z≢sf : ∀ {n : ℕ} (f : Fin n) → ¬ (Fin.zero ≡ Fin.suc f)
z≢sf f ()

z≢mapsfs : ∀ {n : ℕ} (fs : List (Fin n)) → All (λ sf → ¬ zero ≡ sf) (mapˡ suc fs)
z≢mapsfs [] = []
z≢mapsfs (f ∷ fs') = z≢sf f ∷ z≢mapsfs fs'

fins : ∀ (n : ℕ) → Σ (List (Fin n)) Unique
fins 0 = ([] , empty)
fins (suc n') =
    let
        (inds' , unids') = fins n'
    in
        ( zero ∷ mapˡ suc inds'
        , push (z≢mapsfs inds') (Unique-map suc suc-injective unids')
        )

fins-complete : ∀ (n : ℕ) (f : Fin n) → f ∈ (proj₁ (fins n))
fins-complete (suc n') zero = here refl
fins-complete (suc n') (suc f') = there (x∈xs⇒fx∈fxs suc (fins-complete n' f'))