Move predecessor computation into Graphs

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

Language/Graphs.agda

@@ -7,15 +7,19 @@ open import Data.Fin.Properties as FinProp using (suc-injective)
 open import Data.List as List using (List; []; _∷_)
 open import Data.List.Membership.Propositional as ListMem using ()
 open import Data.List.Membership.Propositional.Properties as ListMemProp using ()
+open import Data.List.Relation.Unary.All using (All; []; _∷_)
+open import Data.List.Relation.Unary.Any as RelAny using ()
 open import Data.Nat as Nat using (ℕ; suc)
 open import Data.Nat.Properties using (+-assoc; +-comm)
-open import Data.Product using (_×_; Σ; _,_)
+open import Data.Product using (_×_; Σ; _,_; proj₁; proj₂)
+open import Data.Product.Properties as ProdProp using ()
 open import Data.Vec using (Vec; []; _∷_; lookup; cast; _++_)
 open import Data.Vec.Properties using (cast-is-id; ++-assoc; lookup-++ˡ; cast-sym; ++-identityʳ; lookup-++ʳ)
 open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym; refl; subst; trans)
+open import Relation.Nullary using (¬_)
 
 open import Lattice
-open import Utils using (x∈xs⇒fx∈fxs; ∈-cartesianProduct)
+open import Utils using (Unique; push; Unique-map; x∈xs⇒fx∈fxs; ∈-cartesianProduct)
 
 record Graph : Set where
     constructor MkGraph
@@ -120,3 +124,40 @@ buildCfg ⟨ bs₁ ⟩ = singleton (bs₁ ∷ [])
 buildCfg (s₁ then s₂) = buildCfg s₁ ↦ buildCfg s₂
 buildCfg (if _ then s₁ else s₂) = singleton [] ↦ (buildCfg s₁ ∙ buildCfg s₂) ↦ singleton []
 buildCfg (while _ repeat s) = loop (buildCfg s)
+
+private
+    z≢sf : ∀ {n : ℕ} (f : Fin n) → ¬ (zero ≡ suc f)
+    z≢sf f ()
+
+    z≢mapsfs : ∀ {n : ℕ} (fs : List (Fin n)) → All (λ sf → ¬ zero ≡ sf) (List.map suc fs)
+    z≢mapsfs [] = []
+    z≢mapsfs (f ∷ fs') = z≢sf f ∷ z≢mapsfs fs'
+
+    finValues : ∀ (n : ℕ) → Σ (List (Fin n)) Unique
+    finValues 0 = ([] , Utils.empty)
+    finValues (suc n') =
+        let
+            (inds' , unids') = finValues n'
+        in
+            ( zero ∷ List.map suc inds'
+            , push (z≢mapsfs inds') (Unique-map suc suc-injective unids')
+            )
+
+    finValues-complete : ∀ (n : ℕ) (f : Fin n) → f ListMem.∈ (proj₁ (finValues n))
+    finValues-complete (suc n') zero = RelAny.here refl
+    finValues-complete (suc n') (suc f') = RelAny.there (x∈xs⇒fx∈fxs suc (finValues-complete n' f'))
+
+module _ (g : Graph) where
+    open import Data.List.Membership.DecPropositional (ProdProp.≡-dec (FinProp._≟_ {Graph.size g}) (FinProp._≟_ {Graph.size g})) using (_∈?_)
+
+    indices : List (Graph.Index g)
+    indices = proj₁ (finValues (Graph.size g))
+
+    indices-complete : ∀ (idx : (Graph.Index g)) → idx ListMem.∈ indices
+    indices-complete = finValues-complete (Graph.size g)
+
+    indices-Unique : Unique indices
+    indices-Unique = proj₂ (finValues (Graph.size g))
+
+    predecessors : (Graph.Index g) → List (Graph.Index g)
+    predecessors idx = List.filter (λ idx' → (idx' , idx) ∈? (Graph.edges g)) indices