Prove final postulate about cycles in graphs

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>

Lattice/Builder.agda

@@ -12,7 +12,7 @@ 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; zipWith)
+open import Data.List.Relation.Unary.All using (All; []; _∷_; map; lookup; zipWith; tabulate)
 open import Data.List.Relation.Unary.All.Properties using () renaming (++⁺ to ++ˡ⁺; ++⁻ʳ to ++ˡ⁻ʳ)
 open import Data.List using (List; _∷_; []; cartesianProduct; cartesianProductWith; foldr) renaming (_++_ to _++ˡ_)
 open import Data.List.Properties using () renaming (++-conicalʳ to ++ˡ-conicalʳ; ++-identityʳ to ++ˡ-identityʳ; ++-assoc to ++ˡ-assoc)
@@ -21,6 +21,7 @@ open import Data.Product using (Σ; _,_; _×_; proj₁; proj₂; uncurry)
 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 Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 open import Agda.Primitive using (lsuc; Level) renaming (_⊔_ to _⊔ℓ_)
 
 record Graph : Set where
@@ -151,7 +152,40 @@ record Graph : Set where
     SplitSimpleWalkVia (step n₁,n₂∈edges done , (_ , _)) = inj₂ ((step n₁,n₂∈edges done , (empty , [])) , refl)
     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
 
-    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))
+    open import Data.List.Membership.DecSetoid (decSetoid {A = Node} _≟_) using () renaming (_∈?_ to _∈ˡ?_)
+
+    splitFromInterior : ∀ {n₁ n₂ n} (p : Path n₁ n₂) → n ∈ˡ (interior p) →
+                        Σ (Path n n₂) (λ p' → ¬ IsDone p' × (Σ (List Node) λ ns → interior p ≡ ns ++ˡ interior p'))
+    splitFromInterior done ()
+    splitFromInterior (step _ done) ()
+    splitFromInterior (step {n₂ = n'} n₁,n'∈edges p'@(step _ _)) (here refl) = (p' , ((λ {()}) , (n' ∷ [] , refl)))
+    splitFromInterior (step {n₂ = n'} n₁,n'∈edges p'@(step _ _)) (there n∈intp')
+        with (p'' , (¬IsDone-p'' , (ns , intp'≡ns++intp''))) ← splitFromInterior p' n∈intp'
+             rewrite intp'≡ns++intp'' = (p'' , (¬IsDone-p'' , (n' ∷ ns , refl)))
+
+
+    findCycleHelp : ∀ {n₁ nⁱ n₂} (p : Path n₁ n₂) (p₁ : Path n₁ nⁱ) (p₂ : Path nⁱ n₂) →
+                      ¬ IsDone p₁ → Unique (interior p₁) →
+                      p ≡ p₁ ++ p₂ →
+                      (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)))
+    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)
+    findCycleHelp {nⁱ = nⁱ} p p₁ (step nⁱ,nⁱ'∈edges p₂') ¬IsDone-p₁ Unique-intp₁ p≡p₁++p₂
+        with nⁱ ∈ˡ? interior p₁
+    ...   | no nⁱ∉intp₁ =
+            let p₁' = p₁ ++ step nⁱ,nⁱ'∈edges done
+                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 {()}))
+                ¬IsDone-p₁' = IsDone-++ˡ p₁ (step nⁱ,nⁱ'∈edges done) ¬IsDone-p₁
+                p≡p₁'++p₂' = trans p≡p₁++p₂ (sym (++-assoc p₁ (step nⁱ,nⁱ'∈edges done) p₂'))
+            in findCycleHelp p p₁' p₂' ¬IsDone-p₁' (subst Unique (sym intp₁'≡intp₁++[nⁱ]) (Unique-append nⁱ∉intp₁ Unique-intp₁)) p≡p₁'++p₂'
+    ...   | yes nⁱ∈intp₁
+            with (pᶜ , (¬IsDone-pᶜ , (ns , intp₁≡ns++intpᶜ))) ← splitFromInterior p₁ nⁱ∈intp₁ =
+            let Unique-intp₁' = subst Unique intp₁≡ns++intpᶜ Unique-intp₁
+            in inj₂ (nⁱ , ((pᶜ , (Unique-++⁻ʳ ns  Unique-intp₁' , tabulate (λ {x} _ → nodes-complete x))) , ¬IsDone-pᶜ))
+
+    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)))
+    findCycle done = inj₁ ((done , (empty , [])) , refl)
+    findCycle (step n₁,n₂∈edges done) = inj₁ ((step n₁,n₂∈edges done , (empty , [])) , refl)
+    findCycle p@(step {n₂ = n'} n₁,n'∈edges p'@(step _ _)) = findCycleHelp p (step n₁,n'∈edges done) p' (λ {()}) empty refl
 
     Adjacency : Set
     Adjacency = ∀ (n₁ n₂ : Node) → List (Path n₁ n₂)
@@ -239,7 +273,7 @@ record Graph : Set where
     adj = throughAll (proj₁ nodes)
 
     NoCycles : Set
-    NoCycles = ∀ (n : Node) →  All (_≡ done) (adj n n)
+    NoCycles = ∀ (n : Node) →  All IsDone (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}