Translate informal proof of (node) transitivity into formal one.

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

Language.agda

@@ -6,7 +6,7 @@ open import Data.Integer using (ℤ; +_) renaming (_+_ to _+ᶻ_; _-_ to _-ᶻ_)
 open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
 open import Data.Product using (_×_; Σ; _,_; proj₁; proj₂)
 open import Data.Vec using (Vec; foldr; lookup; _∷_; []; _++_; cast)
-open import Data.Vec.Properties using (++-assoc; ++-identityʳ; lookup-++ˡ)
+open import Data.Vec.Properties using (++-assoc; ++-identityʳ; lookup-++ˡ; lookup-cast₁)
 open import Data.Vec.Relation.Binary.Equality.Cast using (cast-is-id)
 open import Data.List using ([]; _∷_; List) renaming (foldr to foldrˡ; map to mapˡ; _++_ to _++ˡ_)
 open import Data.List.Properties using () renaming (++-assoc to ++ˡ-assoc; map-++ to mapˡ-++ˡ; ++-identityʳ to ++ˡ-identityʳ)
@@ -19,7 +19,7 @@ open import Data.Fin.Properties using (suc-injective)
 open import Relation.Binary.PropositionalEquality as Eq using (subst; cong; _≡_; sym; trans; refl)
 open import Relation.Nullary using (¬_)
 open import Function using (_∘_)
-open Eq.≡-Reasoning using (begin_; step-≡; _∎)
+open Eq.≡-Reasoning
 
 open import Lattice
 open import Utils using (Unique; Unique-map; empty; push; x∈xs⇒fx∈fxs; _⊗_; _,_)
@@ -121,36 +121,39 @@ module Graphs where
                         e ∈ˡ (Graph.edges g₁) →
                         (↑ˡ-Edge e n) ∈ˡ (subst (λ m → List (Fin m × Fin m)) sg₂≡sg₁+n (Graph.edges g₂))
 
+    ⊆-trans : ∀ {g₁ g₂ g₃ : Graph} → g₁ ⊆ g₂ → g₂ ⊆ g₃ → g₁ ⊆ g₃
+    ⊆-trans {MkGraph s₁ ns₁ es₁} {MkGraph s₂ ns₂ es₂} {MkGraph s₃ ns₃ es₃}
+            (Mk-⊆ n₁ p₁@refl g₁[]≡g₂[] e∈g₁⇒e∈g₂) (Mk-⊆ n₂ p₂@refl g₂[]≡g₃[] e∈g₂⇒e∈g₃) = record
+        { n = n₁ +ⁿ n₂
+        ; sg₂≡sg₁+n = +-assoc s₁ n₁ n₂
+        ; g₁[]≡g₂[] = λ idx →
+            begin
+              lookup ns₁ idx
+            ≡⟨ g₁[]≡g₂[] _ ⟩
+              lookup (cast p₁ ns₂) (idx ↑ˡ n₁)
+            ≡⟨ lookup-cast₁ p₁ ns₂ _ ⟩
+              lookup ns₂ (castᶠ (sym p₁) (idx ↑ˡ n₁))
+            ≡⟨ g₂[]≡g₃[] _ ⟩
+              lookup (cast p₂ ns₃) ((castᶠ (sym p₁) (idx ↑ˡ n₁)) ↑ˡ n₂)
+            ≡⟨ lookup-cast₁ p₂ _ _ ⟩
+              lookup ns₃ (castᶠ (sym p₂) (((castᶠ (sym p₁) (idx ↑ˡ n₁)) ↑ˡ n₂)))
+            ≡⟨ cong (lookup ns₃) (flatten-casts (sym p₂) (sym p₁) (sym (+-assoc s₁ n₁ n₂)) idx) ⟩
+              lookup ns₃ (castᶠ (sym (+-assoc s₁ n₁ n₂)) (idx ↑ˡ (n₁ +ⁿ n₂)))
+            ≡⟨ sym (lookup-cast₁ (+-assoc s₁ n₁ n₂) _ _) ⟩
+              lookup (cast (+-assoc s₁ n₁ n₂) ns₃) (idx ↑ˡ (n₁ +ⁿ n₂))
+            ∎
+        ; e∈g₁⇒e∈g₂ = {!!}
+        }
+        where
             flatten-casts : ∀ {s₁ s₂ s₃ n₁ n₂ : ℕ}
-                    (p : s₂ +ⁿ n₂ ≡ s₃) (q : s₁ +ⁿ n₁ ≡ s₂) (r : s₁ +ⁿ (n₁ +ⁿ n₂) ≡ s₃)
+                            (p : s₂ +ⁿ n₂ ≡ s₃) (q : s₁ +ⁿ n₁ ≡ s₂)
+                            (r : s₁ +ⁿ (n₁ +ⁿ n₂) ≡ s₃)
                             (idx : Fin s₁) →
                             castᶠ p ((castᶠ q (idx ↑ˡ n₁)) ↑ˡ n₂) ≡ castᶠ r (idx ↑ˡ (n₁ +ⁿ n₂))
             flatten-casts refl refl r zero = refl
             flatten-casts {(suc s₁)} {s₂} {s₃} {n₁} {n₂} refl refl r (suc idx')
                 rewrite flatten-casts refl refl (sym (+-assoc s₁ n₁ n₂)) idx' = refl
 
-    ⊆-trans : ∀ {g₁ g₂ g₃ : Graph} → g₁ ⊆ g₂ → g₂ ⊆ g₃ → g₁ ⊆ g₃
-    ⊆-trans {MkGraph s₁ ns₁ es₁} {MkGraph s₂ ns₂ es₂} {MkGraph s₃ ns₃ es₃}
-            (Mk-⊆ n₁ p₁@refl g₁[]≡g₂[] e∈g₁⇒e∈g₂) (Mk-⊆ n₂ p₂@refl g₂[]≡g₃[] e∈g₂⇒e∈g₃) = record
-        { n = n₁ +ⁿ n₂
-        ; sg₂≡sg₁+n = +-assoc s₁ n₁ n₂
-        ; g₁[]≡g₂[] = {!!}
-        ; e∈g₁⇒e∈g₂ = {!!}
-        --     lookup ns₁ idx ≡ lookup (cast p₁ ns₂) (idx ↑ˡ n) -- by g₁[]≡g₂[]
-
-        --     lookup (cast p₁ ns₂) (idx ↑ˡ n) ≡ lookup ns₂ (Fin.cast (sym p₁) (idx ↑ˡ n)) -- by lookup-cast₁
-        --                                                            -------- s₁ + n₁ → s₂
-        --                                                                     ---------- Fin (s₁ + n₁)
-        --                                                  ----------------------------- Fin s₂
-
-        --     lookup ns₂ (Fin.cast (sym p₁) (idx ↑ˡ n)) ≡ lookup (cast p₂ ns₃) ((Fin.cast (sym p₁) (idx ↑ˡ n₁) ↑ˡ n₂)) -- by g₂[]≡g₃[]
-
-        --     lookup (cast p₂ ns₃) ((Fin.cast (sym p₁) (idx ↑ˡ n₁) ↑ˡ  n₂)) ≡ lookup ns₃ (Fin.cast (sym p₂) ((Fin.cast (sym p₁) (idx ↑ˡ n₁) ↑ˡ  n₂))) -- by lookup-cast₂
-
-        --     lookup ns₃ (Fin.cast (sym p₂) ((Fin.cast (sym p₁) (idx ↑ˡ n₁) ↑ˡ  n₂))) ≡ lookup ns₃ (Fin.cast (sym (+-assoc s₁ n₁ n₂)) (idx ↑ˡ (n₁ + n₂))) -- by flatten-casts
-        --
-        --     lookup ns₃ (Fin.cast (sym (+-assoc s₁ n₁ n₂)) (idx ↑ˡ (n₁ + n₂))) ≡ lookup (cast (+-assoc s₁ n₁ n₂) ns₃) (idx ↑ˡ (n₁ + n₂)) ∎
-        }
 
     record Relaxable (T : Graph → Set) : Set where
         field relax : ∀ {g₁ g₂ : Graph} → g₁ ⊆ g₂ → T g₁ → T g₂