agda-spa/Utils.agda
Danila Fedorin f21ebdcf46 Start working on the evaluation operation.
Proving monotonicity is the main hurdle here.

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-03-10 18:13:01 -07:00

71 lines
3.5 KiB
Agda
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

module Utils where
open import Agda.Primitive using () renaming (_⊔_ to _⊔_)
open import Data.Nat using (; suc)
open import Data.List using (List; []; _∷_; _++_) renaming (map to mapˡ)
open import Data.List.Membership.Propositional using (_∈_)
open import Data.List.Relation.Unary.All using (All; []; _∷_; map)
open import Data.List.Relation.Unary.Any using (Any; here; there) -- TODO: re-export these with nicer names from map
open import Function.Definitions using (Injective)
open import Relation.Binary.PropositionalEquality using (_≡_; sym; refl)
open import Relation.Nullary using (¬_)
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
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
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 : ) (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)