283 lines
20 KiB
Agda
283 lines
20 KiB
Agda
module Lattice.Builder where
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open import Lattice
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open import Equivalence
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open import Utils using (Unique; push; empty; Unique-append; Unique-++⁻ˡ; Unique-++⁻ʳ; All¬-¬Any; ¬Any-map; fins; fins-complete)
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open import Data.Nat as Nat using (ℕ)
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open import Data.Fin as Fin using (Fin; suc; zero; _≟_)
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open import Data.Maybe as Maybe using (Maybe; just; nothing; _>>=_; maybe)
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open import Data.Maybe.Properties using (just-injective)
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open import Data.Unit using (⊤; tt)
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open import Data.List.NonEmpty using (List⁺; tail; toList) renaming (_∷_ to _∷⁺_)
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open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_; mapWith∈ to mapWith∈ˡ)
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open import Data.List.Membership.Propositional.Properties using () renaming (∈-++⁺ʳ to ∈ˡ-++⁺ʳ; ∈-++⁺ˡ to ∈ˡ-++⁺ˡ; ∈-cartesianProductWith⁺ to ∈ˡ-cartesianProductWith⁺)
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open import Data.List.Relation.Unary.Any using (Any; here; there)
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open import Data.List.Relation.Unary.All using (All; []; _∷_; map; lookup; zipWith; tabulate)
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open import Data.List.Relation.Unary.All.Properties using () renaming (++⁺ to ++ˡ⁺; ++⁻ʳ to ++ˡ⁻ʳ)
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open import Data.List using (List; _∷_; []; cartesianProduct; cartesianProductWith; foldr) renaming (_++_ to _++ˡ_)
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open import Data.List.Properties using () renaming (++-conicalʳ to ++ˡ-conicalʳ; ++-identityʳ to ++ˡ-identityʳ; ++-assoc to ++ˡ-assoc)
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open import Data.Sum using (_⊎_; inj₁; inj₂)
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open import Data.Product using (Σ; _,_; _×_; proj₁; proj₂; uncurry)
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open import Data.Empty using (⊥; ⊥-elim)
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open import Relation.Nullary using (¬_; Dec; yes; no)
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open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; cong; subst)
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open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
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open import Agda.Primitive using (lsuc; Level) renaming (_⊔_ to _⊔ℓ_)
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record Graph : Set where
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constructor mkGraph
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field
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size : ℕ
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Node : Set
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Node = Fin size
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nodes = fins size
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nodes-complete = fins-complete size
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Edge : Set
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Edge = Node × Node
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field
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edges : List Edge
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data Path : Node → Node → Set where
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done : ∀ {n : Node} → Path n n
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step : ∀ {n₁ n₂ n₃ : Node} → (n₁ , n₂) ∈ˡ edges → Path n₂ n₃ → Path n₁ n₃
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data IsDone : ∀ {n₁ n₂} → Path n₁ n₂ → Set where
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isDone : ∀ {n : Node} → IsDone (done {n})
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_++_ : ∀ {n₁ n₂ n₃} → Path n₁ n₂ → Path n₂ n₃ → Path n₁ n₃
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done ++ p = p
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(step e p₁) ++ p₂ = step e (p₁ ++ p₂)
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++-done : ∀ {n₁ n₂} (p : Path n₁ n₂) → p ++ done ≡ p
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++-done done = refl
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++-done (step e∈edges p) rewrite ++-done p = refl
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++-assoc : ∀ {n₁ n₂ n₃ n₄} (p₁ : Path n₁ n₂) (p₂ : Path n₂ n₃) (p₃ : Path n₃ n₄) →
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(p₁ ++ p₂) ++ p₃ ≡ p₁ ++ (p₂ ++ p₃)
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++-assoc done p₂ p₃ = refl
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++-assoc (step n₁,n∈edges p₁) p₂ p₃ rewrite ++-assoc p₁ p₂ p₃ = refl
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IsDone-++ˡ : ∀ {n₁ n₂ n₃} (p₁ : Path n₁ n₂) (p₂ : Path n₂ n₃) →
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¬ IsDone p₁ → ¬ IsDone (p₁ ++ p₂)
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IsDone-++ˡ done _ done≢done = ⊥-elim (done≢done isDone)
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interior : ∀ {n₁ n₂} → Path n₁ n₂ → List Node
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interior done = []
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interior (step _ done) = []
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interior (step {n₂ = n₂} _ p) = n₂ ∷ interior p
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interior-extend : ∀ {n₁ n₂ n₃} → (p : Path n₁ n₂) → (n₂,n₃∈edges : (n₂ , n₃) ∈ˡ edges) →
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let p' = (p ++ (step n₂,n₃∈edges done))
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in (interior p' ≡ interior p) ⊎ (interior p' ≡ interior p ++ˡ (n₂ ∷ []))
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interior-extend done _ = inj₁ refl
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interior-extend (step n₁,n₂∈edges done) n₂n₃∈edges = inj₂ refl
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interior-extend {n₂ = n₂} (step {n₂ = n} n₁,n∈edges p@(step _ _)) n₂n₃∈edges
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with p ++ (step n₂n₃∈edges done) | interior-extend p n₂n₃∈edges
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... | done | inj₁ []≡intp rewrite sym []≡intp = inj₁ refl
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... | done | inj₂ []=intp++[n₂] with () ← ++ˡ-conicalʳ (interior p) (n₂ ∷ []) (sym []=intp++[n₂])
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... | step _ p | inj₁ IH rewrite IH = inj₁ refl
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... | step _ p | inj₂ IH rewrite IH = inj₂ refl
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interior-++ : ∀ {n₁ n₂ n₃} → (p₁ : Path n₁ n₂) → (p₂ : Path n₂ n₃) →
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¬ IsDone p₁ → ¬ IsDone p₂ →
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interior (p₁ ++ p₂) ≡ interior p₁ ++ˡ (n₂ ∷ interior p₂)
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interior-++ done _ done≢done _ = ⊥-elim (done≢done isDone)
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interior-++ _ done _ done≢done = ⊥-elim (done≢done isDone)
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interior-++ (step _ done) (step _ _) _ _ = refl
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interior-++ (step n₁,n∈edges p@(step n,n'∈edges p')) p₂ _ p₂≢done
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rewrite interior-++ p p₂ (λ {()}) p₂≢done = refl
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SimpleWalkVia : List Node → Node → Node → Set
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SimpleWalkVia ns n₁ n₂ = Σ (Path n₁ n₂) (λ p → Unique (interior p) × All (_∈ˡ ns) (interior p))
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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
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SimpleWalk-extend (p , (Unique-intp , intp⊆ns)) n₂,n₃∈edges w≢n₂ n₂∈ns
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with p ++ (step n₂,n₃∈edges done) | interior-extend p n₂,n₃∈edges
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... | p' | inj₁ intp'≡intp rewrite sym intp'≡intp = (p' , Unique-intp , intp⊆ns)
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... | p' | inj₂ intp'≡intp++[n₂]
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with intp++[n₂]⊆ns ← ++ˡ⁺ intp⊆ns (n₂∈ns ∷ [])
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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))
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∈ˡ-narrow : ∀ {x y : Node} {ys : List Node} → x ∈ˡ (y ∷ ys) → ¬ y ≡ x → x ∈ˡ ys
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∈ˡ-narrow (here refl) x≢y = ⊥-elim (x≢y refl)
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∈ˡ-narrow (there x∈ys) _ = x∈ys
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SplitSimpleWalkViaHelp : ∀ {n n₁ n₂ ns} (nⁱ : Node)
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(w : SimpleWalkVia (n ∷ ns) n₁ n₂)
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(p₁ : Path n₁ nⁱ) (p₂ : Path nⁱ n₂) →
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¬ IsDone p₁ → ¬ IsDone p₂ →
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All (_∈ˡ ns) (interior p₁) →
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proj₁ w ≡ p₁ ++ p₂ →
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(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)
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SplitSimpleWalkViaHelp nⁱ w done _ done≢done _ _ _ = ⊥-elim (done≢done isDone)
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SplitSimpleWalkViaHelp nⁱ w p₁ done _ done≢done _ _ = ⊥-elim (done≢done isDone)
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SplitSimpleWalkViaHelp {n} {ns = ns} nⁱ w@(p , (Unique-intp , intp⊆ns)) p₁@(step _ _) p₂@(step {n₂ = nⁱ'} nⁱ,nⁱ',∈edges p₂') p₁≢done p₂≢done intp₁⊆ns p≡p₁++p₂
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with intp≡intp₁++[n]++intp₂ ← trans (cong interior p≡p₁++p₂) (interior-++ p₁ p₂ p₁≢done p₂≢done)
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with nⁱ∈n∷ns ∷ intp₂⊆n∷ns ← ++ˡ⁻ʳ (interior p₁) (subst (All (_∈ˡ (n ∷ ns))) intp≡intp₁++[n]++intp₂ intp⊆ns)
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with nⁱ ≟ n
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... | yes refl
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with Unique-intp₁ ← Unique-++⁻ˡ (interior p₁) (subst Unique intp≡intp₁++[n]++intp₂ Unique-intp)
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with (push n≢intp₂ Unique-intp₂) ← Unique-++⁻ʳ (interior p₁) (subst Unique intp≡intp₁++[n]++intp₂ Unique-intp)
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= inj₁ (((p₁ , (Unique-intp₁ , intp₁⊆ns)) , (p₂ , (Unique-intp₂ , zipWith (uncurry ∈ˡ-narrow) (intp₂⊆n∷ns , n≢intp₂)))) , sym p≡p₁++p₂)
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... | no nⁱ≢n
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with p₂'
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... | done
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= let
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-- note: copied with below branch. can't use with <- to
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-- share and re-use because the termination checker loses the thread.
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p₁' = (p₁ ++ (step nⁱ,nⁱ',∈edges done))
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n≢nⁱ n≡nⁱ = nⁱ≢n (sym n≡nⁱ)
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intp₁'=intp₁++[nⁱ] = subst (λ xs → interior p₁' ≡ interior p₁ ++ˡ xs) (++ˡ-identityʳ (nⁱ ∷ [])) (interior-++ p₁ (step nⁱ,nⁱ',∈edges done) p₁≢done (λ {()}))
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intp₁++[nⁱ]⊆ns = ++ˡ⁺ intp₁⊆ns (∈ˡ-narrow nⁱ∈n∷ns n≢nⁱ ∷ [])
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intp₁'⊆ns = subst (All (_∈ˡ ns)) (sym intp₁'=intp₁++[nⁱ]) intp₁++[nⁱ]⊆ns
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-- end shared with below branch.
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Unique-intp₁++[nⁱ] = Unique-++⁻ˡ (interior p₁ ++ˡ (nⁱ ∷ [])) (subst Unique (trans intp≡intp₁++[n]++intp₂ (sym (++ˡ-assoc (interior p₁) (nⁱ ∷ []) []))) Unique-intp)
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in inj₂ ((p₁ ++ (step nⁱ,nⁱ',∈edges done) , (subst Unique (sym intp₁'=intp₁++[nⁱ]) Unique-intp₁++[nⁱ] , intp₁'⊆ns)) , sym p≡p₁++p₂)
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... | p₂'@(step _ _)
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= let p₁' = (p₁ ++ (step nⁱ,nⁱ',∈edges done))
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n≢nⁱ n≡nⁱ = nⁱ≢n (sym n≡nⁱ)
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intp₁'=intp₁++[nⁱ] = subst (λ xs → interior p₁' ≡ interior p₁ ++ˡ xs) (++ˡ-identityʳ (nⁱ ∷ [])) (interior-++ p₁ (step nⁱ,nⁱ',∈edges done) p₁≢done (λ {()}))
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intp₁++[nⁱ]⊆ns = ++ˡ⁺ intp₁⊆ns (∈ˡ-narrow nⁱ∈n∷ns n≢nⁱ ∷ [])
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intp₁'⊆ns = subst (All (_∈ˡ ns)) (sym intp₁'=intp₁++[nⁱ]) intp₁++[nⁱ]⊆ns
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p≡p₁'++p₂' = trans p≡p₁++p₂ (sym (++-assoc p₁ (step nⁱ,nⁱ',∈edges done) p₂'))
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in SplitSimpleWalkViaHelp nⁱ' w p₁' p₂' (IsDone-++ˡ _ _ p₁≢done) (λ {()}) intp₁'⊆ns p≡p₁'++p₂'
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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)
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SplitSimpleWalkVia (done , (_ , _)) = inj₂ ((done , (empty , [])) , refl)
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SplitSimpleWalkVia (step n₁,n₂∈edges done , (_ , _)) = inj₂ ((step n₁,n₂∈edges done , (empty , [])) , refl)
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SplitSimpleWalkVia w@(step {n₂ = nⁱ} n₁,nⁱ∈edges p@(step _ _) , (push nⁱ≢intp Unique-intp , nⁱ∈ns ∷ intp⊆ns)) = SplitSimpleWalkViaHelp nⁱ w (step n₁,nⁱ∈edges done) p (λ {()}) (λ {()}) [] refl
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open import Data.List.Membership.DecSetoid (decSetoid {A = Node} _≟_) using () renaming (_∈?_ to _∈ˡ?_)
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splitFromInterior : ∀ {n₁ n₂ n} (p : Path n₁ n₂) → n ∈ˡ (interior p) →
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Σ (Path n n₂) (λ p' → ¬ IsDone p' × (Σ (List Node) λ ns → interior p ≡ ns ++ˡ interior p'))
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splitFromInterior done ()
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splitFromInterior (step _ done) ()
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splitFromInterior (step {n₂ = n'} n₁,n'∈edges p'@(step _ _)) (here refl) = (p' , ((λ {()}) , (n' ∷ [] , refl)))
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splitFromInterior (step {n₂ = n'} n₁,n'∈edges p'@(step _ _)) (there n∈intp')
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with (p'' , (¬IsDone-p'' , (ns , intp'≡ns++intp''))) ← splitFromInterior p' n∈intp'
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rewrite intp'≡ns++intp'' = (p'' , (¬IsDone-p'' , (n' ∷ ns , refl)))
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findCycleHelp : ∀ {n₁ nⁱ n₂} (p : Path n₁ n₂) (p₁ : Path n₁ nⁱ) (p₂ : Path nⁱ n₂) →
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¬ IsDone p₁ → Unique (interior p₁) →
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p ≡ p₁ ++ p₂ →
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(Σ (SimpleWalkVia (proj₁ nodes) n₁ n₂) λ w → proj₁ w ≡ p) ⊎ (Σ Node (λ n → Σ (SimpleWalkVia (proj₁ nodes) n n) λ w → ¬ IsDone (proj₁ w)))
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findCycleHelp p p₁ done ¬IsDonep₁ Unique-intp₁ p≡p₁++done rewrite ++-done p₁ = inj₁ ((p₁ , (Unique-intp₁ , tabulate (λ {x} _ → nodes-complete x))) , sym p≡p₁++done)
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findCycleHelp {nⁱ = nⁱ} p p₁ (step nⁱ,nⁱ'∈edges p₂') ¬IsDone-p₁ Unique-intp₁ p≡p₁++p₂
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with nⁱ ∈ˡ? interior p₁
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... | no nⁱ∉intp₁ =
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let p₁' = p₁ ++ step nⁱ,nⁱ'∈edges done
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intp₁'≡intp₁++[nⁱ] = subst (λ xs → interior p₁' ≡ interior p₁ ++ˡ xs) (++ˡ-identityʳ (nⁱ ∷ [])) (interior-++ p₁ (step nⁱ,nⁱ'∈edges done) ¬IsDone-p₁ (λ {()}))
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¬IsDone-p₁' = IsDone-++ˡ p₁ (step nⁱ,nⁱ'∈edges done) ¬IsDone-p₁
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p≡p₁'++p₂' = trans p≡p₁++p₂ (sym (++-assoc p₁ (step nⁱ,nⁱ'∈edges done) p₂'))
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in findCycleHelp p p₁' p₂' ¬IsDone-p₁' (subst Unique (sym intp₁'≡intp₁++[nⁱ]) (Unique-append nⁱ∉intp₁ Unique-intp₁)) p≡p₁'++p₂'
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... | yes nⁱ∈intp₁
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with (pᶜ , (¬IsDone-pᶜ , (ns , intp₁≡ns++intpᶜ))) ← splitFromInterior p₁ nⁱ∈intp₁ =
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let Unique-intp₁' = subst Unique intp₁≡ns++intpᶜ Unique-intp₁
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in inj₂ (nⁱ , ((pᶜ , (Unique-++⁻ʳ ns Unique-intp₁' , tabulate (λ {x} _ → nodes-complete x))) , ¬IsDone-pᶜ))
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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 → ¬ IsDone (proj₁ w)))
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findCycle done = inj₁ ((done , (empty , [])) , refl)
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findCycle (step n₁,n₂∈edges done) = inj₁ ((step n₁,n₂∈edges done , (empty , [])) , refl)
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findCycle p@(step {n₂ = n'} n₁,n'∈edges p'@(step _ _)) = findCycleHelp p (step n₁,n'∈edges done) p' (λ {()}) empty refl
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Adjacency : Set
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Adjacency = ∀ (n₁ n₂ : Node) → List (Path n₁ n₂)
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Adjacency-update : ∀ (n₁ n₂ : Node) → (List (Path n₁ n₂) → List (Path n₁ n₂)) → Adjacency → Adjacency
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Adjacency-update n₁ n₂ f adj n₁' n₂'
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with n₁ ≟ n₁' | n₂ ≟ n₂'
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... | yes refl | yes refl = f (adj n₁ n₂)
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... | _ | _ = adj n₁' n₂'
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Adjecency-append : ∀ {n₁ n₂ : Node} → List (Path n₁ n₂) → Adjacency → Adjacency
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Adjecency-append {n₁} {n₂} ps = Adjacency-update n₁ n₂ (ps ++ˡ_)
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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'
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Adjacency-append-monotonic {adj} {n₁} {n₂} {n₁'} {n₂'} {ps} p∈adj
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with n₁ ≟ n₁' | n₂ ≟ n₂'
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... | yes refl | yes refl = ∈ˡ-++⁺ʳ ps p∈adj
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... | yes refl | no _ = p∈adj
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... | no _ | no _ = p∈adj
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... | no _ | yes _ = p∈adj
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adj⁰ : Adjacency
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adj⁰ n₁ n₂
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with n₁ ≟ n₂
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... | yes refl = done ∷ []
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... | no _ = []
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adj⁰-done : ∀ n → done ∈ˡ adj⁰ n n
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adj⁰-done n
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with n ≟ n
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... | yes refl = here refl
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... | no n≢n = ⊥-elim (n≢n refl)
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seedWithEdges : ∀ (es : List Edge) → (∀ {e} → e ∈ˡ es → e ∈ˡ edges) → Adjacency → Adjacency
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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)))
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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
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seedWithEdges-monotonic {es = []} e∈es⇒e∈edges p∈adj = p∈adj
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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)
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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
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e∈seedWithEdges {es = []} e∈es⇒e∈edges ()
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e∈seedWithEdges {es = (n₁' , n₂') ∷ es} e∈es⇒e∈edges (here refl)
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with n₁' ≟ n₁' | n₂' ≟ n₂'
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... | yes refl | yes refl = here refl
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... | no n₁'≢n₁' | _ = ⊥-elim (n₁'≢n₁' refl)
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... | _ | no n₂'≢n₂' = ⊥-elim (n₂'≢n₂' refl)
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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)
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adj¹ : Adjacency
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adj¹ = seedWithEdges edges (λ x → x) adj⁰
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adj¹-adj⁰ : ∀ {n₁ n₂ p} → p ∈ˡ adj⁰ n₁ n₂ → p ∈ˡ adj¹ n₁ n₂
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adj¹-adj⁰ p∈adj⁰ = seedWithEdges-monotonic (λ x → x) p∈adj⁰
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edge∈adj¹ : ∀ {n₁ n₂} (n₁n₂∈edges : (n₁ , n₂) ∈ˡ edges) → (step n₁n₂∈edges done) ∈ˡ adj¹ n₁ n₂
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edge∈adj¹ = e∈seedWithEdges (λ x → x)
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through : Node → Adjacency → Adjacency
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through n adj n₁ n₂ = cartesianProductWith _++_ (adj n₁ n) (adj n n₂) ++ˡ adj n₁ n₂
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through-monotonic : ∀ adj n {n₁ n₂ p} → p ∈ˡ adj n₁ n₂ → p ∈ˡ (through n adj) n₁ n₂
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through-monotonic adj n p∈adjn₁n₂ = ∈ˡ-++⁺ʳ _ p∈adjn₁n₂
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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
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through-++ adj n p₁∈adj p₂∈adj = ∈ˡ-++⁺ˡ (∈ˡ-cartesianProductWith⁺ _++_ p₁∈adj p₂∈adj)
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throughAll : List Node → Adjacency
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throughAll = foldr through adj¹
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throughAll-adj₁ : ∀ {n₁ n₂ p} ns → p ∈ˡ adj¹ n₁ n₂ → p ∈ˡ throughAll ns n₁ n₂
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throughAll-adj₁ [] p∈adj¹ = p∈adj¹
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throughAll-adj₁ (n ∷ ns) p∈adj¹ = through-monotonic (throughAll ns) n (throughAll-adj₁ ns p∈adj¹)
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paths-throughAll : ∀ {n₁ n₂ : Node} (ns : List Node) (w : SimpleWalkVia ns n₁ n₂) → proj₁ w ∈ˡ throughAll ns n₁ n₂
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paths-throughAll {n₁} [] (done , (_ , _)) = adj¹-adj⁰ (adj⁰-done n₁)
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paths-throughAll {n₁} [] (step e∈edges done , (_ , _)) = edge∈adj¹ e∈edges
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paths-throughAll {n₁} [] (step _ (step _ _) , (_ , (() ∷ _)))
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paths-throughAll (n ∷ ns) w
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with SplitSimpleWalkVia w
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... | inj₁ ((w₁ , w₂) , w₁++w₂≡w) rewrite sym w₁++w₂≡w = through-++ (throughAll ns) n (paths-throughAll ns w₁) (paths-throughAll ns w₂)
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... | inj₂ (w' , w'≡w) rewrite sym w'≡w = through-monotonic (throughAll ns) n (paths-throughAll ns w')
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adj : Adjacency
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adj = throughAll (proj₁ nodes)
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NoCycles : Set
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NoCycles = ∀ (n : Node) → All IsDone (adj n n)
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NoCycles⇒adj-complete : NoCycles → ∀ {n₁ n₂} {p : Path n₁ n₂} → p ∈ˡ adj n₁ n₂
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NoCycles⇒adj-complete noCycles {n₁} {n₂} {p}
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with findCycle p
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... | inj₁ (w , w≡p) rewrite sym w≡p = paths-throughAll (proj₁ nodes) w
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... | inj₂ (nᶜ , (wᶜ , wᶜ≢done)) = ⊥-elim (wᶜ≢done (lookup (noCycles nᶜ) (paths-throughAll (proj₁ nodes) wᶜ)))
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