Add a new 'properties' module

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

Language.agda

@@ -3,6 +3,7 @@ module Language where
 open import Language.Base public
 open import Language.Semantics public
 open import Language.Graphs public
+open import Language.Properties public
 
 open import Data.Fin using (Fin; suc; zero)
 open import Data.Fin.Properties as FinProp using (suc-injective)

Language/Graphs.agda

@@ -1,7 +1,6 @@
 module Language.Graphs where
 
 open import Language.Base
-open import Language.Semantics
 
 open import Data.Fin as Fin using (Fin; suc; zero; _↑ˡ_; _↑ʳ_)
 open import Data.Fin.Properties as FinProp using (suc-injective)
@@ -57,6 +56,7 @@ record _⊆_ (g₁ g₂ : Graph) : Set where
                     e ListMem.∈ (Graph.edges g₁) →
                     (↑ˡ-Edge e n) ListMem.∈ (subst (λ m → List (Fin m × Fin m)) sg₂≡sg₁+n (Graph.edges g₂))
 
+private
     castᵉ : ∀ {n m : ℕ} .(p : n ≡ m) → (Fin n × Fin n) → (Fin m × Fin m)
     castᵉ p (idx₁ , idx₂) = (Fin.cast p idx₁ , Fin.cast p idx₂)
 
@@ -128,20 +128,6 @@ instance
 
 open Relaxable {{...}}
 
-relax-preserves-[]≡ : ∀ (g₁ g₂ : Graph) (g₁⊆g₂ : g₁ ⊆ g₂) (idx : Graph.Index g₁) →
-                      g₁ [ idx ] ≡ g₂ [ relax g₁⊆g₂ idx ]
-relax-preserves-[]≡ g₁ g₂ (Mk-⊆ n refl newNodes nsg₂≡nsg₁++newNodes _) idx
-    rewrite cast-is-id refl (Graph.nodes g₂)
-    with refl ← nsg₂≡nsg₁++newNodes = sym (lookup-++ˡ (Graph.nodes g₁) _ _)
-
-instance
-    NodeEqualsMonotonic : ∀ {bss : List BasicStmt} →
-                          MonotonicPredicate (λ g n → g [ n ] ≡ bss)
-    NodeEqualsMonotonic = record
-        { relaxPredicate = λ g₁ g₂ idx g₁⊆g₂ g₁[idx]≡bss →
-            trans (sym (relax-preserves-[]≡ g₁ g₂ g₁⊆g₂ idx)) g₁[idx]≡bss
-        }
-
 pushBasicBlock : List BasicStmt → MonotonicGraphFunction Graph.Index
 pushBasicBlock bss g =
     ( record
@@ -160,8 +146,8 @@ pushBasicBlock bss g =
       )
     )
 
-pushBasicBlock-works : ∀ (bss : List BasicStmt) → Always (λ g idx → g [ idx ] ≡ bss) (pushBasicBlock bss)
-pushBasicBlock-works bss = MkAlways (λ g → lookup-++ʳ (Graph.nodes g) (bss ∷ []) zero)
+pushEmptyBlock : MonotonicGraphFunction Graph.Index
+pushEmptyBlock = pushBasicBlock []
 
 addEdges : ∀ (g : Graph) → List (Graph.Edge g) → Σ Graph (λ g' → g ⊆ g')
 addEdges (MkGraph s ns es) es' =
@@ -183,9 +169,6 @@ addEdges (MkGraph s ns es) es' =
         }
     )
 
-pushEmptyBlock : MonotonicGraphFunction Graph.Index
-pushEmptyBlock = pushBasicBlock []
-
 buildCfg : Stmt → MonotonicGraphFunction (Graph.Index ⊗ Graph.Index)
 buildCfg ⟨ bs₁ ⟩ = pushBasicBlock (bs₁ ∷ []) map (λ g idx → (idx , idx))
 buildCfg (s₁ then s₂) =

Language/Properties.agda

@@ -0,0 +1,31 @@
+module Language.Properties where
+
+open import Language.Base
+open import Language.Semantics
+open import Language.Graphs
+
+open import MonotonicState _⊆_ ⊆-trans renaming (MonotonicState to MonotonicGraphFunction)
+open Relaxable {{...}}
+
+open import Data.Fin using (zero)
+open import Data.List using (List)
+open import Data.Vec using (_∷_; [])
+open import Data.Vec.Properties using (cast-is-id; lookup-++ˡ; lookup-++ʳ)
+open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans)
+
+relax-preserves-[]≡ : ∀ (g₁ g₂ : Graph) (g₁⊆g₂ : g₁ ⊆ g₂) (idx : Graph.Index g₁) →
+                      g₁ [ idx ] ≡ g₂ [ relax g₁⊆g₂ idx ]
+relax-preserves-[]≡ g₁ g₂ (Mk-⊆ n refl newNodes nsg₂≡nsg₁++newNodes _) idx
+    rewrite cast-is-id refl (Graph.nodes g₂)
+    with refl ← nsg₂≡nsg₁++newNodes = sym (lookup-++ˡ (Graph.nodes g₁) _ _)
+
+instance
+    NodeEqualsMonotonic : ∀ {bss : List BasicStmt} →
+                          MonotonicPredicate (λ g n → g [ n ] ≡ bss)
+    NodeEqualsMonotonic = record
+        { relaxPredicate = λ g₁ g₂ idx g₁⊆g₂ g₁[idx]≡bss →
+            trans (sym (relax-preserves-[]≡ g₁ g₂ g₁⊆g₂ idx)) g₁[idx]≡bss
+        }
+
+pushBasicBlock-works : ∀ (bss : List BasicStmt) → Always (λ g idx → g [ idx ] ≡ bss) (pushBasicBlock bss)
+pushBasicBlock-works bss = MkAlways (λ g → lookup-++ʳ (Graph.nodes g) (bss ∷ []) zero)