DanilaFe/agda-spa
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Add formalization of 'traces through graph'

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2024-04-20 21:36:58 -07:00
parent 6e3f06ca5d
commit e218d1b7a3
2 changed files with 33 additions and 4 deletions
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13
Language/Semantics.agda
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@ -5,7 +5,7 @@ open import Language.Base
open import Data.Integer using (ℤ; +_) renaming (_+_ to _+ᶻ_; _-_ to _-ᶻ_) open import Data.Integer using (ℤ; +_) renaming (_+_ to _+ᶻ_; _-_ to _-ᶻ_)
open import Data.Product using (_×_; _,_) open import Data.Product using (_×_; _,_)
open import Data.String using (String) open import Data.String using (String)
open import Data.List using (List; _∷_) open import Data.List as List using (List)
open import Data.Nat using (ℕ) open import Data.Nat using (ℕ)
open import Relation.Nullary using (¬_) open import Relation.Nullary using (¬_)
open import Relation.Binary.PropositionalEquality using (_≡_) open import Relation.Binary.PropositionalEquality using (_≡_)
@ -17,8 +17,8 @@ Env : Set
Env = List (String × Value) Env = List (String × Value)
data _∈_ : (String × Value) → Env → Set where data _∈_ : (String × Value) → Env → Set where
here : ∀ (s : String) (v : Value) (ρ : Env) → (s , v) ∈ ((s , v) ∷ ρ) here : ∀ (s : String) (v : Value) (ρ : Env) → (s , v) ∈ ((s , v) List.∷ ρ)
there : ∀ (s s' : String) (v v' : Value) (ρ : Env) → ¬ (s ≡ s') → (s , v) ∈ ρ → (s , v) ∈ ((s' , v') ∷ ρ) there : ∀ (s s' : String) (v v' : Value) (ρ : Env) → ¬ (s ≡ s') → (s , v) ∈ ρ → (s , v) ∈ ((s' , v') List.∷ ρ)
data _,_⇒ᵉ_ : Env → Expr → Value → Set where data _,_⇒ᵉ_ : Env → Expr → Value → Set where
⇒ᵉ-ℕ : ∀ (ρ : Env) (n : ℕ) → ρ , (# n) ⇒ᵉ (↑ᶻ (+ n)) ⇒ᵉ-ℕ : ∀ (ρ : Env) (n : ℕ) → ρ , (# n) ⇒ᵉ (↑ᶻ (+ n))
@ -33,7 +33,12 @@ data _,_⇒ᵉ_ : Env → Expr → Value → Set where
data _,_⇒ᵇ_ : Env → BasicStmt → Env → Set where data _,_⇒ᵇ_ : Env → BasicStmt → Env → Set where
⇒ᵇ-noop : ∀ (ρ : Env) → ρ , noop ⇒ᵇ ρ ⇒ᵇ-noop : ∀ (ρ : Env) → ρ , noop ⇒ᵇ ρ
⇒ᵇ-← : ∀ (ρ : Env) (x : String) (e : Expr) (v : Value) → ⇒ᵇ-← : ∀ (ρ : Env) (x : String) (e : Expr) (v : Value) →
ρ , e ⇒ᵉ v → ρ , (x ← e) ⇒ᵇ ((x , v) ∷ ρ) ρ , e ⇒ᵉ v → ρ , (x ← e) ⇒ᵇ ((x , v) List.∷ ρ)
data _,_⇒ᵇˢ_ : Env → List BasicStmt → Env → Set where
[] : ∀ {ρ : Env} → ρ , List.[] ⇒ᵇˢ ρ
_∷_ : ∀ {ρ₁ ρ₂ ρ₃ : Env} {bs : BasicStmt} {bss : List BasicStmt} →
ρ₁ , bs ⇒ᵇ ρ₂ → ρ₂ , bss ⇒ᵇˢ ρ₃ → ρ₁ , (bs List.∷ bss) ⇒ᵇˢ ρ₃
data _,_⇒ˢ_ : Env → Stmt → Env → Set where data _,_⇒ˢ_ : Env → Stmt → Env → Set where
⇒ˢ-⟨⟩ : ∀ (ρ₁ ρ₂ : Env) (bs : BasicStmt) → ⇒ˢ-⟨⟩ : ∀ (ρ₁ ρ₂ : Env) (bs : BasicStmt) →

24
Language/Traces.agda Normal file
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@ -0,0 +1,24 @@
module Language.Traces where
open import Language.Base public
open import Language.Semantics public
open import Language.Graphs public
open import Data.Product using (_,_)
open import Data.List.Membership.Propositional as MemProp using ()
module _ {g : Graph} where
open Graph g using (Index; edges)
data Trace : Index → Index → Env → Env → Set where
Trace-single : ∀ {ρ₁ ρ₂ : Env} {idx : Index} →
ρ₁ , (g [ idx ]) ⇒ᵇˢ ρ₂ → Trace idx idx ρ₁ ρ₂
Trace-edge : ∀ {ρ₁ ρ₂ ρ₃ : Env} {idx₁ idx₂ idx₃ : Index} →
ρ₁ , (g [ idx₁ ]) ⇒ᵇˢ ρ₂ → (idx₁ , idx₂) MemProp.∈ edges →
Trace idx₂ idx₃ ρ₂ ρ₃ → Trace idx₁ idx₃ ρ₁ ρ₃
_++⟨_⟩_ : ∀ {idx₁ idx₂ idx₃ idx₄ : Index} {ρ₁ ρ₂ ρ₃ : Env} →
Trace idx₁ idx₂ ρ₁ ρ₂ → (idx₂ , idx₃) MemProp.∈ edges →
Trace idx₃ idx₄ ρ₂ ρ₃ → Trace idx₁ idx₄ ρ₁ ρ₃
_++⟨_⟩_ (Trace-single ρ₁⇒ρ₂) idx₂→idx₃ tr = Trace-edge ρ₁⇒ρ₂ idx₂→idx₃ tr
_++⟨_⟩_ (Trace-edge ρ₁⇒ρ₂ idx₁→idx' tr') idx₂→idx₃ tr = Trace-edge ρ₁⇒ρ₂ idx₁→idx' (tr' ++⟨ idx₂→idx₃ ⟩ tr)