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Define interpretation of the sign lattice

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2024-04-30 21:58:41 -07:00
parent be50c76cb1
commit 3859826293
1 changed files with 64 additions and 3 deletions
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67
Analysis/Sign.agda
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@ -1,15 +1,19 @@
module Analysis.Sign where
open import Data.Nat using (suc)
open import Data.Product using (proj₁; _,_)
open import Data.Integer using (ℤ; +_; -[1+_])
open import Data.Nat using (ℕ; suc; zero)
open import Data.Product using (Σ; proj₁; _,_)
open import Data.Sum using (inj₁; inj₂)
open import Data.Empty using (⊥; ⊥-elim)
open import Data.Unit using (⊤; tt)
open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; subst)
open import Relation.Nullary using (yes; no)
open import Relation.Nullary using (¬_; yes; no)
open import Language
open import Lattice
open import Showable using (Showable; show)
open import Utils using (_⇒_; _∧_; _∨_)
import Analysis.Forward
data Sign : Set where
@ -61,6 +65,7 @@ open AB.Plain 0ˢ using ()
; fixedHeight to fixedHeightᵍ
; _≼_ to _≼ᵍ_
; _⊔_ to _⊔ᵍ_
; _⊓_ to _⊓ᵍ_
)
open IsLattice isLatticeᵍ using ()
@ -106,6 +111,62 @@ minus [ 0ˢ ]ᵍ [ 0ˢ ]ᵍ = [ 0ˢ ]ᵍ
postulate minus-Monoˡ : ∀ (s₂ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (λ s₁ → minus s₁ s₂)
postulate minus-Monoʳ : ∀ (s₁ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (minus s₁)
⟦_⟧ᵍ : SignLattice → Value → Set
⟦_⟧ᵍ ⊥ᵍ _ = ⊥
⟦_⟧ᵍ ⊤ᵍ _ = ⊤
⟦_⟧ᵍ [ + ]ᵍ v = Σ ℕ (λ n → v ≡ ↑ᶻ (+_ (suc n)))
⟦_⟧ᵍ [ 0ˢ ]ᵍ v = Σ ℕ (λ n → v ≡ ↑ᶻ (+_ zero))
⟦_⟧ᵍ [ - ]ᵍ v = Σ ℕ (λ n → v ≡ ↑ᶻ -[1+ n ])
⟦⟧ᵍ-respects-≈ᵍ : ∀ {s₁ s₂ : SignLattice} → s₁ ≈ᵍ s₂ → ⟦ s₁ ⟧ᵍ ⇒ ⟦ s₂ ⟧ᵍ
⟦⟧ᵍ-respects-≈ᵍ ≈ᵍ-⊥ᵍ-⊥ᵍ v bot = bot
⟦⟧ᵍ-respects-≈ᵍ ≈ᵍ-⊤ᵍ-⊤ᵍ v top = top
⟦⟧ᵍ-respects-≈ᵍ (≈ᵍ-lift { + } { + } refl) v proof = proof
⟦⟧ᵍ-respects-≈ᵍ (≈ᵍ-lift { - } { - } refl) v proof = proof
⟦⟧ᵍ-respects-≈ᵍ (≈ᵍ-lift { 0ˢ } { 0ˢ } refl) v proof = proof
⟦⟧ᵍ-⊔ᵍ-∨ : ∀ {s₁ s₂ : SignLattice} → (⟦ s₁ ⟧ᵍ ∨ ⟦ s₂ ⟧ᵍ) ⇒ ⟦ s₁ ⊔ᵍ s₂ ⟧ᵍ
⟦⟧ᵍ-⊔ᵍ-∨ {⊥ᵍ} x (inj₂ px₂) = px₂
⟦⟧ᵍ-⊔ᵍ-∨ {⊤ᵍ} x _ = tt
⟦⟧ᵍ-⊔ᵍ-∨ {[ s₁ ]ᵍ} {[ s₂ ]ᵍ} x px
with s₁ ≟ᵍ s₂
... | no _ = tt
... | yes refl
with px
... | inj₁ px₁ = px₁
... | inj₂ px₂ = px₂
⟦⟧ᵍ-⊔ᵍ-∨ {[ s₁ ]ᵍ} {⊥ᵍ} x (inj₁ px₁) = px₁
⟦⟧ᵍ-⊔ᵍ-∨ {[ s₁ ]ᵍ} {⊤ᵍ} x _ = tt
s₁≢s₂⇒¬s₁∧s₂ : ∀ {s₁ s₂ : Sign} → ¬ s₁ ≡ s₂ → ∀ {v} → ¬ ((⟦ [ s₁ ]ᵍ ⟧ᵍ ∧ ⟦ [ s₂ ]ᵍ ⟧ᵍ) v)
s₁≢s₂⇒¬s₁∧s₂ { + } { + } +≢+ _ = ⊥-elim (+≢+ refl)
s₁≢s₂⇒¬s₁∧s₂ { + } { - } _ ((n , refl) , (m , ()))
s₁≢s₂⇒¬s₁∧s₂ { + } { 0ˢ } _ ((n , refl) , (m , ()))
s₁≢s₂⇒¬s₁∧s₂ { 0ˢ } { + } _ ((n , refl) , (m , ()))
s₁≢s₂⇒¬s₁∧s₂ { 0ˢ } { 0ˢ } +≢+ _ = ⊥-elim (+≢+ refl)
s₁≢s₂⇒¬s₁∧s₂ { 0ˢ } { - } _ ((n , refl) , (m , ()))
s₁≢s₂⇒¬s₁∧s₂ { - } { + } _ ((n , refl) , (m , ()))
s₁≢s₂⇒¬s₁∧s₂ { - } { 0ˢ } _ ((n , refl) , (m , ()))
s₁≢s₂⇒¬s₁∧s₂ { - } { - } +≢+ _ = ⊥-elim (+≢+ refl)
⟦⟧ᵍ-⊓ᵍ-∧ : ∀ {s₁ s₂ : SignLattice} → (⟦ s₁ ⟧ᵍ ∧ ⟦ s₂ ⟧ᵍ) ⇒ ⟦ s₁ ⊓ᵍ s₂ ⟧ᵍ
⟦⟧ᵍ-⊓ᵍ-∧ {⊥ᵍ} x (bot , _) = bot
⟦⟧ᵍ-⊓ᵍ-∧ {⊤ᵍ} x (_ , px₂) = px₂
⟦⟧ᵍ-⊓ᵍ-∧ {[ s₁ ]ᵍ} {[ s₂ ]ᵍ} x (px₁ , px₂)
with s₁ ≟ᵍ s₂
... | no s₁≢s₂ = s₁≢s₂⇒¬s₁∧s₂ s₁≢s₂ (px₁ , px₂)
... | yes refl = px₁
⟦⟧ᵍ-⊓ᵍ-∧ {[ g₁ ]ᵍ} {⊥ᵍ} x (_ , bot) = bot
⟦⟧ᵍ-⊓ᵍ-∧ {[ g₁ ]ᵍ} {⊤ᵍ} x (px₁ , _) = px₁
latticeInterpretationᵍ : LatticeInterpretation isLatticeᵍ
latticeInterpretationᵍ = record
{ ⟦_⟧ = ⟦_⟧ᵍ
; ⟦⟧-respects-≈ = ⟦⟧ᵍ-respects-≈ᵍ
; ⟦⟧-⊔-∨ = ⟦⟧ᵍ-⊔ᵍ-∨
; ⟦⟧-⊓-∧ = ⟦⟧ᵍ-⊓ᵍ-∧
}
module WithProg (prog : Program) where
open Program prog