Add a new 'properties' module
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -3,6 +3,7 @@ module Language where
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open import Language.Base public
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open import Language.Semantics public
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open import Language.Graphs public
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open import Language.Properties public
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open import Data.Fin using (Fin; suc; zero)
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open import Data.Fin.Properties as FinProp using (suc-injective)
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@ -1,7 +1,6 @@
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module Language.Graphs where
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open import Language.Base
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open import Language.Semantics
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open import Data.Fin as Fin using (Fin; suc; zero; _↑ˡ_; _↑ʳ_)
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open import Data.Fin.Properties as FinProp using (suc-injective)
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@ -57,38 +56,39 @@ record _⊆_ (g₁ g₂ : Graph) : Set where
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e ListMem.∈ (Graph.edges g₁) →
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(↑ˡ-Edge e n) ListMem.∈ (subst (λ m → List (Fin m × Fin m)) sg₂≡sg₁+n (Graph.edges g₂))
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castᵉ : ∀ {n m : ℕ} .(p : n ≡ m) → (Fin n × Fin n) → (Fin m × Fin m)
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castᵉ p (idx₁ , idx₂) = (Fin.cast p idx₁ , Fin.cast p idx₂)
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private
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castᵉ : ∀ {n m : ℕ} .(p : n ≡ m) → (Fin n × Fin n) → (Fin m × Fin m)
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castᵉ p (idx₁ , idx₂) = (Fin.cast p idx₁ , Fin.cast p idx₂)
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↑ˡ-assoc : ∀ {s n₁ n₂} (f : Fin s) (p : s Nat.+ (n₁ Nat.+ n₂) ≡ s Nat.+ n₁ Nat.+ n₂) →
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f ↑ˡ n₁ ↑ˡ n₂ ≡ Fin.cast p (f ↑ˡ (n₁ Nat.+ n₂))
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↑ˡ-assoc zero p = refl
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↑ˡ-assoc {suc s'} {n₁} {n₂} (suc f') p rewrite ↑ˡ-assoc f' (sym (+-assoc s' n₁ n₂)) = refl
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↑ˡ-assoc : ∀ {s n₁ n₂} (f : Fin s) (p : s Nat.+ (n₁ Nat.+ n₂) ≡ s Nat.+ n₁ Nat.+ n₂) →
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f ↑ˡ n₁ ↑ˡ n₂ ≡ Fin.cast p (f ↑ˡ (n₁ Nat.+ n₂))
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↑ˡ-assoc zero p = refl
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↑ˡ-assoc {suc s'} {n₁} {n₂} (suc f') p rewrite ↑ˡ-assoc f' (sym (+-assoc s' n₁ n₂)) = refl
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↑ˡ-Edge-assoc : ∀ {s n₁ n₂} (e : Fin s × Fin s) (p : s Nat.+ (n₁ Nat.+ n₂) ≡ s Nat.+ n₁ Nat.+ n₂) →
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↑ˡ-Edge (↑ˡ-Edge e n₁) n₂ ≡ castᵉ p (↑ˡ-Edge e (n₁ Nat.+ n₂))
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↑ˡ-Edge-assoc (idx₁ , idx₂) p
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rewrite ↑ˡ-assoc idx₁ p
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rewrite ↑ˡ-assoc idx₂ p = refl
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↑ˡ-Edge-assoc : ∀ {s n₁ n₂} (e : Fin s × Fin s) (p : s Nat.+ (n₁ Nat.+ n₂) ≡ s Nat.+ n₁ Nat.+ n₂) →
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↑ˡ-Edge (↑ˡ-Edge e n₁) n₂ ≡ castᵉ p (↑ˡ-Edge e (n₁ Nat.+ n₂))
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↑ˡ-Edge-assoc (idx₁ , idx₂) p
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rewrite ↑ˡ-assoc idx₁ p
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rewrite ↑ˡ-assoc idx₂ p = refl
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↑ˡ-identityʳ : ∀ {s} (f : Fin s) (p : s Nat.+ 0 ≡ s) →
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f ≡ Fin.cast p (f ↑ˡ 0)
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↑ˡ-identityʳ zero p = refl
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↑ˡ-identityʳ {suc s'} (suc f') p rewrite sym (↑ˡ-identityʳ f' (+-comm s' 0)) = refl
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↑ˡ-identityʳ : ∀ {s} (f : Fin s) (p : s Nat.+ 0 ≡ s) →
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f ≡ Fin.cast p (f ↑ˡ 0)
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↑ˡ-identityʳ zero p = refl
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↑ˡ-identityʳ {suc s'} (suc f') p rewrite sym (↑ˡ-identityʳ f' (+-comm s' 0)) = refl
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↑ˡ-Edge-identityʳ : ∀ {s} (e : Fin s × Fin s) (p : s Nat.+ 0 ≡ s) →
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e ≡ castᵉ p (↑ˡ-Edge e 0)
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↑ˡ-Edge-identityʳ (idx₁ , idx₂) p
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rewrite sym (↑ˡ-identityʳ idx₁ p)
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rewrite sym (↑ˡ-identityʳ idx₂ p) = refl
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↑ˡ-Edge-identityʳ : ∀ {s} (e : Fin s × Fin s) (p : s Nat.+ 0 ≡ s) →
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e ≡ castᵉ p (↑ˡ-Edge e 0)
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↑ˡ-Edge-identityʳ (idx₁ , idx₂) p
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rewrite sym (↑ˡ-identityʳ idx₁ p)
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rewrite sym (↑ˡ-identityʳ idx₂ p) = refl
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cast∈⇒∈subst : ∀ {n m : ℕ} (p : n ≡ m) (q : m ≡ n)
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(e : Fin n × Fin n) (es : List (Fin m × Fin m)) →
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castᵉ p e ListMem.∈ es →
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e ListMem.∈ subst (λ m → List (Fin m × Fin m)) q es
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cast∈⇒∈subst refl refl (idx₁ , idx₂) es e∈es
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rewrite FinProp.cast-is-id refl idx₁
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rewrite FinProp.cast-is-id refl idx₂ = e∈es
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cast∈⇒∈subst : ∀ {n m : ℕ} (p : n ≡ m) (q : m ≡ n)
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(e : Fin n × Fin n) (es : List (Fin m × Fin m)) →
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castᵉ p e ListMem.∈ es →
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e ListMem.∈ subst (λ m → List (Fin m × Fin m)) q es
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cast∈⇒∈subst refl refl (idx₁ , idx₂) es e∈es
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rewrite FinProp.cast-is-id refl idx₁
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rewrite FinProp.cast-is-id refl idx₂ = e∈es
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⊆-trans : ∀ {g₁ g₂ g₃ : Graph} → g₁ ⊆ g₂ → g₂ ⊆ g₃ → g₁ ⊆ g₃
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⊆-trans {MkGraph s₁ ns₁ es₁} {MkGraph s₂ ns₂ es₂} {MkGraph s₃ ns₃ es₃}
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@ -128,20 +128,6 @@ instance
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open Relaxable {{...}}
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relax-preserves-[]≡ : ∀ (g₁ g₂ : Graph) (g₁⊆g₂ : g₁ ⊆ g₂) (idx : Graph.Index g₁) →
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g₁ [ idx ] ≡ g₂ [ relax g₁⊆g₂ idx ]
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relax-preserves-[]≡ g₁ g₂ (Mk-⊆ n refl newNodes nsg₂≡nsg₁++newNodes _) idx
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rewrite cast-is-id refl (Graph.nodes g₂)
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with refl ← nsg₂≡nsg₁++newNodes = sym (lookup-++ˡ (Graph.nodes g₁) _ _)
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instance
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NodeEqualsMonotonic : ∀ {bss : List BasicStmt} →
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MonotonicPredicate (λ g n → g [ n ] ≡ bss)
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NodeEqualsMonotonic = record
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{ relaxPredicate = λ g₁ g₂ idx g₁⊆g₂ g₁[idx]≡bss →
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trans (sym (relax-preserves-[]≡ g₁ g₂ g₁⊆g₂ idx)) g₁[idx]≡bss
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}
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pushBasicBlock : List BasicStmt → MonotonicGraphFunction Graph.Index
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pushBasicBlock bss g =
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( record
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@ -160,8 +146,8 @@ pushBasicBlock bss g =
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)
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)
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pushBasicBlock-works : ∀ (bss : List BasicStmt) → Always (λ g idx → g [ idx ] ≡ bss) (pushBasicBlock bss)
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pushBasicBlock-works bss = MkAlways (λ g → lookup-++ʳ (Graph.nodes g) (bss ∷ []) zero)
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pushEmptyBlock : MonotonicGraphFunction Graph.Index
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pushEmptyBlock = pushBasicBlock []
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addEdges : ∀ (g : Graph) → List (Graph.Edge g) → Σ Graph (λ g' → g ⊆ g')
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addEdges (MkGraph s ns es) es' =
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@ -183,9 +169,6 @@ addEdges (MkGraph s ns es) es' =
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}
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)
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pushEmptyBlock : MonotonicGraphFunction Graph.Index
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pushEmptyBlock = pushBasicBlock []
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buildCfg : Stmt → MonotonicGraphFunction (Graph.Index ⊗ Graph.Index)
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buildCfg ⟨ bs₁ ⟩ = pushBasicBlock (bs₁ ∷ []) map (λ g idx → (idx , idx))
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buildCfg (s₁ then s₂) =
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@ -0,0 +1,31 @@
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module Language.Properties where
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open import Language.Base
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open import Language.Semantics
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open import Language.Graphs
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open import MonotonicState _⊆_ ⊆-trans renaming (MonotonicState to MonotonicGraphFunction)
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open Relaxable {{...}}
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open import Data.Fin using (zero)
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open import Data.List using (List)
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open import Data.Vec using (_∷_; [])
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open import Data.Vec.Properties using (cast-is-id; lookup-++ˡ; lookup-++ʳ)
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open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans)
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relax-preserves-[]≡ : ∀ (g₁ g₂ : Graph) (g₁⊆g₂ : g₁ ⊆ g₂) (idx : Graph.Index g₁) →
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g₁ [ idx ] ≡ g₂ [ relax g₁⊆g₂ idx ]
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relax-preserves-[]≡ g₁ g₂ (Mk-⊆ n refl newNodes nsg₂≡nsg₁++newNodes _) idx
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rewrite cast-is-id refl (Graph.nodes g₂)
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with refl ← nsg₂≡nsg₁++newNodes = sym (lookup-++ˡ (Graph.nodes g₁) _ _)
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instance
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NodeEqualsMonotonic : ∀ {bss : List BasicStmt} →
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MonotonicPredicate (λ g n → g [ n ] ≡ bss)
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NodeEqualsMonotonic = record
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{ relaxPredicate = λ g₁ g₂ idx g₁⊆g₂ g₁[idx]≡bss →
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trans (sym (relax-preserves-[]≡ g₁ g₂ g₁⊆g₂ idx)) g₁[idx]≡bss
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}
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pushBasicBlock-works : ∀ (bss : List BasicStmt) → Always (λ g idx → g [ idx ] ≡ bss) (pushBasicBlock bss)
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pushBasicBlock-works bss = MkAlways (λ g → lookup-++ʳ (Graph.nodes g) (bss ∷ []) zero)
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