agda-spa/Lattice/Builder.agda

module Lattice.Builder where

open import Lattice
open import Equivalence
open import Utils using (Unique; push; empty; Unique-append; All¬-¬Any; ¬Any-map; fins; fins-complete)
open import Data.Nat as Nat using (ℕ)
open import Data.Fin as Fin using (Fin; suc; zero; _≟_)
open import Data.Maybe as Maybe using (Maybe; just; nothing; _>>=_; maybe)
open import Data.Maybe.Properties using (just-injective)
open import Data.Unit using (⊤; tt)
open import Data.List.NonEmpty using (List⁺; tail; toList) renaming (_∷_ to _∷⁺_)
open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_; mapWith∈ to mapWith∈ˡ)
open import Data.List.Membership.Propositional.Properties using () renaming (∈-++⁺ʳ to ∈ˡ-++⁺ʳ; ∈-++⁺ˡ to ∈ˡ-++⁺ˡ; ∈-cartesianProductWith⁺ to ∈ˡ-cartesianProductWith⁺)
open import Data.List.Relation.Unary.Any using (Any; here; there)
open import Data.List.Relation.Unary.All using (All; []; _∷_; map; lookup)
open import Data.List.Relation.Unary.All.Properties using () renaming (++⁺ to ++ˡ⁺)
open import Data.List using (List; _∷_; []; cartesianProduct; cartesianProductWith; foldr) renaming (_++_ to _++ˡ_)
open import Data.List.Properties using () renaming (++-conicalʳ to ++ˡ-conicalʳ)
open import Data.Sum using (_⊎_; inj₁; inj₂)
open import Data.Product using (Σ; _,_; _×_; proj₁; proj₂)
open import Data.Empty using (⊥; ⊥-elim)
open import Relation.Nullary using (¬_; Dec; yes; no)
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; cong; subst)
open import Agda.Primitive using (lsuc; Level) renaming (_⊔_ to _⊔ℓ_)

record Graph : Set where
    constructor mkGraph
    field
        size : ℕ

    Node : Set
    Node = Fin size

    nodes = fins size

    nodes-complete = fins-complete size

    Edge : Set
    Edge = Node × Node

    field
        edges : List Edge

    data Path : Node → Node → Set where
        done : ∀ {n : Node} → Path n n
        step : ∀ {n₁ n₂ n₃ : Node} →  (n₁ , n₂) ∈ˡ edges → Path n₂ n₃ → Path n₁ n₃

    _++_ : ∀ {n₁ n₂ n₃} → Path n₁ n₂ → Path n₂ n₃ → Path n₁ n₃
    done ++ p = p
    (step e p₁) ++ p₂ = step e (p₁ ++ p₂)

    ++done : ∀ {n₁ n₂} (p : Path n₁ n₂) → p ++ done ≡ p
    ++done done = refl
    ++done (step e∈edges p) rewrite ++done p = refl

    interior : ∀ {n₁ n₂} → Path n₁ n₂ → List Node
    interior done = []
    interior (step _ done) = []
    interior (step {n₂ = n₂} _ p) = n₂ ∷ interior p

    interior-extend : ∀ {n₁ n₂ n₃} → (p : Path n₁ n₂) →  (n₂,n₃∈edges : (n₂ , n₃) ∈ˡ edges) →
                      let p' = (p ++ (step n₂,n₃∈edges done))
                      in (interior p' ≡ interior p) ⊎ (interior p' ≡ interior p ++ˡ (n₂ ∷ []))
    interior-extend done _ = inj₁ refl
    interior-extend (step n₁,n₂∈edges done) n₂n₃∈edges = inj₂ refl
    interior-extend {n₂ = n₂} (step {n₂ = n} n₁,n∈edges p@(step _ _)) n₂n₃∈edges
        with p ++ (step n₂n₃∈edges done) | interior-extend p n₂n₃∈edges
    ...   | done | inj₁ []≡intp rewrite sym []≡intp = inj₁ refl
    ...   | done | inj₂ []=intp++[n₂] with () ← ++ˡ-conicalʳ (interior p) (n₂ ∷ []) (sym []=intp++[n₂])
    ...   | step _ p | inj₁ IH rewrite IH = inj₁ refl
    ...   | step _ p | inj₂ IH rewrite IH = inj₂ refl

    SimpleWalkVia : List Node → Node → Node → Set
    SimpleWalkVia ns n₁ n₂ = Σ (Path n₁ n₂) (λ p → Unique (interior p) × All (λ n → n ∈ˡ ns) (interior p))

    SimpleWalk-extend : ∀ {n₁ n₂ n₃ ns} →  (w : SimpleWalkVia ns n₁ n₂) → (n₂ , n₃) ∈ˡ edges →  All (λ nʷ →  ¬ nʷ ≡ n₂) (interior (proj₁ w)) → n₂ ∈ˡ ns → SimpleWalkVia ns n₁ n₃
    SimpleWalk-extend (p , (Unique-intp , intp⊆ns)) n₂,n₃∈edges w≢n₂ n₂∈ns
        with p ++ (step n₂,n₃∈edges done) | interior-extend p n₂,n₃∈edges
    ...   | p' | inj₁ intp'≡intp rewrite sym intp'≡intp = (p' , Unique-intp , intp⊆ns)
    ...   | p' | inj₂ intp'≡intp++[n₂]
            with intp++[n₂]⊆ns ←  ++ˡ⁺ intp⊆ns (n₂∈ns ∷ [])
            rewrite sym intp'≡intp++[n₂] = (p' , (subst Unique (sym intp'≡intp++[n₂]) (Unique-append (¬Any-map sym (All¬-¬Any w≢n₂)) Unique-intp) , intp++[n₂]⊆ns))

    postulate SplitSimpleWalkVia : ∀ {n n₁ n₂ ns} (w : SimpleWalkVia (n ∷ ns) n₁ n₂) →  (Σ (SimpleWalkVia ns n₁ n × SimpleWalkVia ns n n₂) λ (w₁ , w₂) → proj₁ w₁ ++ proj₁ w₂ ≡ proj₁ w) ⊎ (Σ (SimpleWalkVia ns n₁ n₂) λ w' → proj₁ w' ≡ proj₁ w)

    postulate findCycle : ∀ {n₁ n₂} (p : Path n₁ n₂) →  (Σ (SimpleWalkVia (proj₁ nodes) n₁ n₂) λ w → proj₁ w ≡ p) ⊎ (Σ Node (λ n → Σ (SimpleWalkVia (proj₁ nodes) n n) λ w →  ¬ proj₁ w ≡ done))

    Adjacency : Set
    Adjacency = ∀ (n₁ n₂ : Node) → List (Path n₁ n₂)

    Adjacency-update : ∀ (n₁ n₂ : Node) → (List (Path n₁ n₂) → List (Path n₁ n₂)) → Adjacency → Adjacency
    Adjacency-update n₁ n₂ f adj n₁' n₂'
        with n₁ ≟ n₁' | n₂ ≟ n₂'
    ...  | yes refl | yes refl = f (adj n₁ n₂)
    ...  | _        | _        = adj n₁' n₂'

    Adjecency-append : ∀ {n₁ n₂ : Node} → List (Path n₁ n₂) → Adjacency → Adjacency
    Adjecency-append {n₁} {n₂} ps = Adjacency-update n₁ n₂ (ps ++ˡ_)

    Adjacency-append-monotonic : ∀ {adj n₁ n₂ n₁' n₂'} {ps : List (Path n₁ n₂)} {p : Path n₁' n₂'} → p ∈ˡ adj n₁' n₂' → p ∈ˡ Adjecency-append ps adj n₁' n₂'
    Adjacency-append-monotonic {adj} {n₁} {n₂} {n₁'} {n₂'} {ps} p∈adj
        with n₁ ≟ n₁' | n₂ ≟ n₂'
    ...   | yes refl | yes refl = ∈ˡ-++⁺ʳ ps p∈adj
    ...   | yes refl | no _ = p∈adj
    ...   | no _ | no _ = p∈adj
    ...   | no _ | yes _ = p∈adj

    adj⁰ : Adjacency
    adj⁰ n₁ n₂
        with n₁ ≟ n₂
    ... | yes refl = done ∷ []
    ... | no _ = []

    adj⁰-done : ∀ n →  done ∈ˡ adj⁰ n n
    adj⁰-done n
        with n ≟ n
    ... | yes refl = here refl
    ... | no n≢n = ⊥-elim (n≢n refl)

    seedWithEdges : ∀ (es : List Edge) →  (∀ {e} → e ∈ˡ es → e ∈ˡ edges) → Adjacency → Adjacency
    seedWithEdges es e∈es⇒e∈edges adj = foldr (λ ((n₁ , n₂) , n₁n₂∈edges) → Adjacency-update n₁ n₂ ((step n₁n₂∈edges done) ∷_)) adj (mapWith∈ˡ es (λ {e} e∈es →  (e , e∈es⇒e∈edges e∈es)))

    seedWithEdges-monotonic : ∀ {n₁ n₂ es adj} → (e∈es⇒e∈edges : ∀ {e} → e ∈ˡ es → e ∈ˡ edges) → ∀ {p} → p ∈ˡ adj n₁ n₂ → p ∈ˡ seedWithEdges es e∈es⇒e∈edges adj n₁ n₂
    seedWithEdges-monotonic {es = []} e∈es⇒e∈edges p∈adj = p∈adj
    seedWithEdges-monotonic {es = (n₁ , n₂) ∷ es} e∈es⇒e∈edges p∈adj = Adjacency-append-monotonic {ps = step (e∈es⇒e∈edges (here refl)) done ∷ []} (seedWithEdges-monotonic (λ e∈es → e∈es⇒e∈edges (there e∈es)) p∈adj)

    e∈seedWithEdges : ∀ {n₁ n₂ es adj} → (e∈es⇒e∈edges : ∀ {e} → e ∈ˡ es → e ∈ˡ edges) → ∀ (n₁n₂∈es : (n₁ , n₂) ∈ˡ es) →  (step (e∈es⇒e∈edges n₁n₂∈es) done) ∈ˡ seedWithEdges es e∈es⇒e∈edges adj n₁ n₂
    e∈seedWithEdges {es = []} e∈es⇒e∈edges ()
    e∈seedWithEdges {es = (n₁' , n₂') ∷ es} e∈es⇒e∈edges (here refl)
        with n₁' ≟ n₁' | n₂' ≟ n₂'
    ...   | yes refl | yes refl = here refl
    ...   | no n₁'≢n₁' | _ = ⊥-elim (n₁'≢n₁' refl)
    ...   | _ | no n₂'≢n₂' = ⊥-elim (n₂'≢n₂' refl)
    e∈seedWithEdges {n₁} {n₂} {es = (n₁' , n₂') ∷ es} {adj} e∈es⇒e∈edges (there n₁n₂∈es) = Adjacency-append-monotonic {ps = step (e∈es⇒e∈edges (here refl)) done ∷ []} (e∈seedWithEdges (λ e∈es → e∈es⇒e∈edges (there e∈es)) n₁n₂∈es)

    adj¹ : Adjacency
    adj¹ = seedWithEdges edges (λ x → x) adj⁰

    adj¹-adj⁰ : ∀ {n₁ n₂ p} → p ∈ˡ adj⁰ n₁ n₂ → p ∈ˡ adj¹ n₁ n₂
    adj¹-adj⁰ p∈adj⁰ = seedWithEdges-monotonic (λ x → x) p∈adj⁰

    edge∈adj¹ : ∀ {n₁ n₂} (n₁n₂∈edges : (n₁ , n₂) ∈ˡ edges) →  (step n₁n₂∈edges done) ∈ˡ adj¹ n₁ n₂
    edge∈adj¹ = e∈seedWithEdges (λ x → x)

    through : Node → Adjacency → Adjacency
    through n adj n₁ n₂ = cartesianProductWith _++_ (adj n₁ n) (adj n n₂) ++ˡ adj n₁ n₂

    through-monotonic : ∀ adj n {n₁ n₂ p} → p ∈ˡ adj n₁ n₂ → p ∈ˡ (through n adj) n₁ n₂
    through-monotonic adj n p∈adjn₁n₂ = ∈ˡ-++⁺ʳ _ p∈adjn₁n₂

    through-++ : ∀ adj n {n₁ n₂} {p₁ : Path n₁ n} {p₂ : Path n n₂} → p₁ ∈ˡ adj n₁ n → p₂ ∈ˡ adj n n₂ → (p₁ ++ p₂) ∈ˡ through n adj n₁ n₂
    through-++ adj n p₁∈adj p₂∈adj = ∈ˡ-++⁺ˡ (∈ˡ-cartesianProductWith⁺ _++_ p₁∈adj p₂∈adj)

    throughAll : List Node → Adjacency
    throughAll = foldr through adj¹

    throughAll-adj₁ : ∀ {n₁ n₂ p} ns → p ∈ˡ adj¹ n₁ n₂ → p ∈ˡ throughAll ns n₁ n₂
    throughAll-adj₁ [] p∈adj¹ = p∈adj¹
    throughAll-adj₁ (n ∷ ns) p∈adj¹ = through-monotonic (throughAll ns) n (throughAll-adj₁ ns p∈adj¹)

    paths-throughAll : ∀ {n₁ n₂ : Node} (ns : List Node) (w : SimpleWalkVia ns n₁ n₂) →  proj₁ w ∈ˡ throughAll ns n₁ n₂
    paths-throughAll {n₁} [] (done , (_ , _)) = adj¹-adj⁰ (adj⁰-done n₁)
    paths-throughAll {n₁} [] (step e∈edges done , (_ , _)) = edge∈adj¹ e∈edges
    paths-throughAll {n₁} [] (step _ (step _ _) , (_ , (() ∷ _)))
    paths-throughAll (n ∷ ns) w
        with SplitSimpleWalkVia w
    ...  | inj₁ ((w₁ , w₂) , w₁++w₂≡w) rewrite sym w₁++w₂≡w = through-++ (throughAll ns) n (paths-throughAll ns w₁) (paths-throughAll ns w₂)
    ...  | inj₂ (w' , w'≡w) rewrite sym w'≡w = through-monotonic (throughAll ns) n (paths-throughAll ns w')

    adj : Adjacency
    adj = throughAll (proj₁ nodes)

    NoCycles : Set
    NoCycles = ∀ (n : Node) →  All (_≡ done) (adj n n)

    NoCycles⇒adj-complete : NoCycles → ∀ {n₁ n₂} {p : Path n₁ n₂} →  p ∈ˡ adj n₁ n₂
    NoCycles⇒adj-complete noCycles {n₁} {n₂} {p}
        with findCycle p
    ...   | inj₁ (w , w≡p) rewrite sym w≡p = paths-throughAll (proj₁ nodes) w
    ...   | inj₂ (nᶜ , (wᶜ , wᶜ≢done)) = ⊥-elim (wᶜ≢done (lookup (noCycles nᶜ) (paths-throughAll (proj₁ nodes) wᶜ)))