Intermediate commit. Switch to *-based definition of <=.

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

Language.agda

@@ -1,18 +1,21 @@
 module Language where
 
 open import Data.Nat using (ℕ; suc; pred; _≤_) renaming (_+_ to _+ⁿ_)
-open import Data.Nat.Properties using (m≤n⇒m≤n+o; ≤-reflexive)
+open import Data.Nat.Properties using (m≤n⇒m≤n+o; ≤-reflexive; +-assoc; +-comm)
 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; _∷_; []; _++_)
+open import Data.Vec using (Vec; foldr; lookup; _∷_; []; _++_; cast)
-open import Data.List using ([]; _∷_; List) renaming (foldr to foldrˡ; map to mapˡ)
+open import Data.Vec.Properties using (++-assoc; ++-identityʳ)
+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)
 open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
 open import Data.List.Relation.Unary.All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any as RelAny using ()
-open import Data.Fin using (Fin; suc; zero; fromℕ; inject₁; inject≤; _↑ʳ_) renaming (_≟_ to _≟ᶠ_)
+open import Data.Fin using (Fin; suc; zero; fromℕ; inject₁; inject≤; _↑ʳ_; _↑ˡ_) renaming (_≟_ to _≟ᶠ_)
 open import Data.Fin.Properties using (suc-injective)
-open import Relation.Binary.PropositionalEquality using (subst; cong; _≡_; refl)
+open import Relation.Binary.PropositionalEquality using (subst; cong; _≡_; sym; refl)
 open import Relation.Nullary using (¬_)
 open import Function using (_∘_)
 
@@ -84,6 +87,7 @@ module Graphs where
     open Semantics
 
     record Graph : Set where
+        constructor MkGraph
         field
             size : ℕ
 
@@ -97,47 +101,47 @@ module Graphs where
             nodes : Vec (List BasicStmt) size
             edges : List Edge
 
-    _[_] : ∀ (g : Graph) → Graph.Index g → List BasicStmt
+    Graph-build-≡ : ∀ (g₁ g₂ : Graph) (p : Graph.size g₁ ≡ Graph.size g₂) →
-    _[_] g idx = lookup (Graph.nodes g) idx
+                      (cast p (Graph.nodes g₁) ≡ Graph.nodes g₂) →
+                      (subst (λ s → List (Fin s × Fin s)) p (Graph.edges g₁) ≡ Graph.edges g₂) →
+                    g₁ ≡ g₂
+    Graph-build-≡ g₁ g₂ refl cns₁≡ns₂ refl
+        rewrite cast-is-id refl (Graph.nodes g₁)
+        rewrite cns₁≡ns₂ = refl
+
+
+    ↑ˡ-Edge : ∀ {n} → Fin n × Fin n → ∀ m → Fin (n +ⁿ m) × Fin (n +ⁿ m)
+    ↑ˡ-Edge (idx₁ , idx₂) m = (idx₁ ↑ˡ m , idx₂ ↑ˡ m)
+
+    ↑ʳ-Edge : ∀ {n} m → Fin n × Fin n → Fin (m +ⁿ n) × Fin (m +ⁿ n)
+    ↑ʳ-Edge m (idx₁ , idx₂) = (m ↑ʳ idx₁ , m ↑ʳ idx₂)
+
+    _∙_ : Graph → Graph → Graph
+    _∙_ (MkGraph s₁ ns₁ es₁) (MkGraph s₂ ns₂ es₂) = MkGraph
+        (s₁ +ⁿ s₂)
+        (ns₁ ++ ns₂)
+        (
+            let
+                edges₁ = mapˡ (λ e → ↑ˡ-Edge e s₂) es₁
+                edges₂ = mapˡ (↑ʳ-Edge s₁) es₂
+            in
+                edges₁ ++ˡ edges₂
+        )
+
+    ∙-assoc : ∀ (g₁ g₂ g₃ : Graph) → g₁ ∙ (g₂ ∙ g₃) ≡ (g₁ ∙ g₂) ∙ g₃
+    ∙-assoc = {!!}
+
+    ∙-zero : ∀ (g : Graph) → g ∙ (MkGraph 0 [] []) ≡ g
+    ∙-zero (MkGraph s ns es) = Graph-build-≡ _ _ (+-comm s 0) (++-identityʳ (+-comm s 0) ns) {!!}
 
     _⊆_ : Graph → Graph → Set
-    _⊆_ g₁ g₂ =
+    _⊆_ g₁ g₂ = Σ Graph (λ g' → g₁ ∙ g' ≡ g₂)
-        Σ (Graph.size g₁ ≤ Graph.size g₂) (λ n₁≤n₂ →
-            ( ∀ (idx : Graph.Index g₁) → g₁ [ idx ] ≡ g₂ [ inject≤ idx n₁≤n₂ ]
-            × ∀ (idx₁ idx₂ : Graph.Index g₁) → (idx₁ , idx₂) ∈ˡ (Graph.edges g₁) →
-                (inject≤ idx₁ n₁≤n₂ , inject≤ idx₂ n₁≤n₂) ∈ˡ (Graph.edges g₂)
-            ))
 
-    -- Note: inject≤ doesn't seem to work as nicely with vector lookups.
+    ⊆-refl : ∀ (g : Graph) → g ⊆ g
-    --       The ↑ˡ and ↑ʳ operators are way nicer. Can we reformulate the
+    ⊆-refl g = (MkGraph 0 [] [] , ∙-zero g)
-    --       ⊆ property to use them?
 
-    n≤n+m : ∀ (n m : ℕ) → n ≤ n +ⁿ m
+    ⊆-trans : ∀ {g₁ g₂ g₃ : Graph} → g₁ ⊆ g₂ → g₂ ⊆ g₃ → g₁ ⊆ g₃
-    n≤n+m n m = m≤n⇒m≤n+o m (≤-reflexive (refl {x = n}))
+    ⊆-trans {g₁} {g₂} {g₃} (g₁₂ , refl) (g₂₃ , refl) = ((g₁₂ ∙ g₂₃) , ∙-assoc g₁ g₁₂ g₂₃)
-
-    lookup-++ˡ : ∀ {a} {A : Set a} {n m : ℕ} (xs : Vec A n) (ys : Vec A m)
-                   (idx : Fin n) → lookup xs idx ≡ lookup (xs ++ ys) (inject≤ idx (n≤n+m n m))
-    lookup-++ˡ = {!!}
-
-    pushBasicBlock : List BasicStmt → (g₁ : Graph) → Σ Graph (λ g₂ → Graph.Index g₂ × g₁ ⊆ g₂)
-    pushBasicBlock bss g₁ =
-        let
-            size' = Graph.size g₁ +ⁿ 1
-            size≤size' = n≤n+m (Graph.size g₁) 1
-            inject-Edge = λ (idx₁ , idx₂) → (inject≤ idx₁ size≤size' , inject≤ idx₂ size≤size')
-        in
-            ( record
-                { size = size'
-                ; nodes = Graph.nodes g₁ ++ (bss ∷ [])
-                ; edges = mapˡ inject-Edge (Graph.edges g₁)
-                }
-            , ( (Graph.size g₁) ↑ʳ zero
-              , ( size≤size'
-                , λ idx → lookup-++ˡ (Graph.nodes g₁) (bss ∷ []) idx
-                , λ idx₁ idx₂ e∈es → x∈xs⇒fx∈fxs inject-Edge e∈es
-                )
-              )
-            )
 
     record Relaxable (T : Graph → Set) : Set where
         field relax : ∀ {g₁ g₂ : Graph} → g₁ ⊆ g₂ → T g₁ → T g₂
@@ -145,14 +149,14 @@ module Graphs where
     instance
         IndexRelaxable : Relaxable Graph.Index
         IndexRelaxable = record
-            { relax = λ g₁⊆g₂ idx → inject≤ idx (proj₁ g₁⊆g₂)
+            { relax = λ { (g' , refl) idx → idx ↑ˡ (Graph.size g') }
             }
 
         EdgeRelaxable : Relaxable Graph.Edge
         EdgeRelaxable = record
-            { relax = λ {g₁} {g₂} g₁⊆g₂ (idx₁ , idx₂) →
+            { relax = λ g₁⊆g₂ (idx₁ , idx₂) →
-                ( Relaxable.relax IndexRelaxable {g₁} {g₂} g₁⊆g₂ idx₁
+                ( Relaxable.relax IndexRelaxable g₁⊆g₂ idx₁
-                , Relaxable.relax IndexRelaxable {g₁} {g₂} g₁⊆g₂ idx₂
+                , Relaxable.relax IndexRelaxable g₁⊆g₂ idx₂
                 )
             }
 
@@ -166,7 +170,36 @@ module Graphs where
                 )
             }
 
-    open Relaxable {{...}} public
+    MonotonicGraphFunction : (Graph → Set) → Set
+    MonotonicGraphFunction T = (g₁ : Graph) → Σ Graph (λ g₂ → T g₂ × g₁ ⊆ g₂)
+
+    infixr 2 _⟨⊗⟩_
+    _⟨⊗⟩_ : ∀ {T₁ T₂ : Graph → Set} {{ T₁Relaxable : Relaxable T₁ }} →
+          MonotonicGraphFunction T₁ → MonotonicGraphFunction T₂ →
+          MonotonicGraphFunction (T₁ ⊗ T₂)
+    _⟨⊗⟩_ {{r}} f₁ f₂ g
+        with (g' , (t₁ , g⊆g')) ← f₁ g
+        with (g'' , (t₂ , g'⊆g'')) ← f₂ g' =
+        (g'' , ((Relaxable.relax r g'⊆g'' t₁ , t₂) , ⊆-trans g⊆g' g'⊆g''))
+
+
+    module Construction where
+        pushBasicBlock : List BasicStmt → MonotonicGraphFunction Graph.Index
+        pushBasicBlock bss g₁ =
+            let
+                g' : Graph
+                g' = record
+                    { size = 1
+                    ; nodes = bss ∷ []
+                    ; edges = []
+                    }
+            in
+                (g₁ ∙ g' , (Graph.size g₁ ↑ʳ zero , (g' , refl)))
+
+        pushEmptyBlock : MonotonicGraphFunction Graph.Index
+        pushEmptyBlock = pushBasicBlock []
+
+    -- open Relaxable {{...}} public
 
 open import Lattice.MapSet _≟ˢ_
     renaming

Utils.agda

@@ -69,5 +69,6 @@ data Pairwise {a} {b} {c} {A : Set a} {B : Set b} (P : A → B → Set c) : List
           P x y → Pairwise P xs ys →
           Pairwise P (x ∷ xs) (y ∷ ys)
 
+infixr 2 _⊗_
 data _⊗_ {a p q} {A : Set a} (P : A → Set p) (Q : A → Set q) : A → Set (a ⊔ℓ p ⊔ℓ q) where
     _,_ : ∀ {val : A} → P val → Q val → (P ⊗ Q) val