300 lines
13 KiB
Agda
300 lines
13 KiB
Agda
module Analysis.Sign where
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open import Data.Integer as Int using (ℤ; +_; -[1+_])
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open import Data.Nat as Nat using (ℕ; suc; zero)
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open import Data.Product using (Σ; proj₁; proj₂; _,_)
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open import Data.Sum using (inj₁; inj₂)
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open import Data.Empty using (⊥; ⊥-elim)
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open import Data.Unit using (⊤; tt)
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open import Data.List.Membership.Propositional as MemProp using () renaming (_∈_ to _∈ˡ_)
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open import Relation.Binary.Definitions using (Decidable)
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open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; subst)
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open import Relation.Nullary using (¬_; yes; no)
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open import Language
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open import Lattice
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open import Equivalence
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open import Showable using (Showable; show)
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open import Utils using (_⇒_; _∧_; _∨_)
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open import Analysis.Utils using (eval-combine₂)
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import Analysis.Forward
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data Sign : Set where
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+ : Sign
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- : Sign
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0ˢ : Sign
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instance
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showable : Showable Sign
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showable = record
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{ show = (λ
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{ + → "+"
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; - → "-"
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; 0ˢ → "0"
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})
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}
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-- g for siGn; s is used for strings and i is not very descriptive.
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_≟ᵍ_ : Decidable (_≡_ {_} {Sign})
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_≟ᵍ_ + + = yes refl
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_≟ᵍ_ + - = no (λ ())
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_≟ᵍ_ + 0ˢ = no (λ ())
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_≟ᵍ_ - + = no (λ ())
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_≟ᵍ_ - - = yes refl
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_≟ᵍ_ - 0ˢ = no (λ ())
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_≟ᵍ_ 0ˢ + = no (λ ())
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_≟ᵍ_ 0ˢ - = no (λ ())
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_≟ᵍ_ 0ˢ 0ˢ = yes refl
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instance
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≡-equiv : IsEquivalence Sign _≡_
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≡-equiv = record
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{ ≈-refl = refl
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; ≈-sym = sym
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; ≈-trans = trans
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}
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≡-Decidable-Sign : IsDecidable {_} {Sign} _≡_
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≡-Decidable-Sign = record { R-dec = _≟ᵍ_ }
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-- embelish 'sign' with a top and bottom element.
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open import Lattice.AboveBelow Sign _ as AB
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using ()
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renaming
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( AboveBelow to SignLattice
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; ⊥ to ⊥ᵍ
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; ⊤ to ⊤ᵍ
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; [_] to [_]ᵍ
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; _≈_ to _≈ᵍ_
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; ≈-⊥-⊥ to ≈ᵍ-⊥ᵍ-⊥ᵍ
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; ≈-⊤-⊤ to ≈ᵍ-⊤ᵍ-⊤ᵍ
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; ≈-lift to ≈ᵍ-lift
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; ≈-refl to ≈ᵍ-refl
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)
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-- 'sign' has no underlying lattice structure, so use the 'plain' above-below lattice.
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open AB.Plain 0ˢ using ()
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renaming
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( isLattice to isLatticeᵍ
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; isFiniteHeightLattice to isFiniteHeightLatticeᵍ
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; _≼_ to _≼ᵍ_
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; _⊔_ to _⊔ᵍ_
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; _⊓_ to _⊓ᵍ_
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; ≼-trans to ≼ᵍ-trans
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)
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plus : SignLattice → SignLattice → SignLattice
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plus ⊥ᵍ _ = ⊥ᵍ
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plus _ ⊥ᵍ = ⊥ᵍ
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plus ⊤ᵍ _ = ⊤ᵍ
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plus _ ⊤ᵍ = ⊤ᵍ
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plus [ + ]ᵍ [ + ]ᵍ = [ + ]ᵍ
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plus [ + ]ᵍ [ - ]ᵍ = ⊤ᵍ
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plus [ + ]ᵍ [ 0ˢ ]ᵍ = [ + ]ᵍ
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plus [ - ]ᵍ [ + ]ᵍ = ⊤ᵍ
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plus [ - ]ᵍ [ - ]ᵍ = [ - ]ᵍ
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plus [ - ]ᵍ [ 0ˢ ]ᵍ = [ - ]ᵍ
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plus [ 0ˢ ]ᵍ [ + ]ᵍ = [ + ]ᵍ
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plus [ 0ˢ ]ᵍ [ - ]ᵍ = [ - ]ᵍ
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plus [ 0ˢ ]ᵍ [ 0ˢ ]ᵍ = [ 0ˢ ]ᵍ
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-- this is incredibly tedious: 125 cases per monotonicity proof, and tactics
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-- are hard. postulate for now.
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postulate plus-Monoˡ : ∀ (s₂ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (λ s₁ → plus s₁ s₂)
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postulate plus-Monoʳ : ∀ (s₁ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (plus s₁)
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plus-Mono₂ : Monotonic₂ _≼ᵍ_ _≼ᵍ_ _≼ᵍ_ plus
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plus-Mono₂ = (plus-Monoˡ , plus-Monoʳ)
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minus : SignLattice → SignLattice → SignLattice
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minus ⊥ᵍ _ = ⊥ᵍ
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minus _ ⊥ᵍ = ⊥ᵍ
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minus ⊤ᵍ _ = ⊤ᵍ
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minus _ ⊤ᵍ = ⊤ᵍ
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minus [ + ]ᵍ [ + ]ᵍ = ⊤ᵍ
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minus [ + ]ᵍ [ - ]ᵍ = [ + ]ᵍ
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minus [ + ]ᵍ [ 0ˢ ]ᵍ = [ + ]ᵍ
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minus [ - ]ᵍ [ + ]ᵍ = [ - ]ᵍ
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minus [ - ]ᵍ [ - ]ᵍ = ⊤ᵍ
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minus [ - ]ᵍ [ 0ˢ ]ᵍ = [ - ]ᵍ
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minus [ 0ˢ ]ᵍ [ + ]ᵍ = [ - ]ᵍ
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minus [ 0ˢ ]ᵍ [ - ]ᵍ = [ + ]ᵍ
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minus [ 0ˢ ]ᵍ [ 0ˢ ]ᵍ = [ 0ˢ ]ᵍ
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postulate minus-Monoˡ : ∀ (s₂ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (λ s₁ → minus s₁ s₂)
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postulate minus-Monoʳ : ∀ (s₁ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (minus s₁)
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minus-Mono₂ : Monotonic₂ _≼ᵍ_ _≼ᵍ_ _≼ᵍ_ minus
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minus-Mono₂ = (minus-Monoˡ , minus-Monoʳ)
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⟦_⟧ᵍ : SignLattice → Value → Set
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⟦_⟧ᵍ ⊥ᵍ _ = ⊥
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⟦_⟧ᵍ ⊤ᵍ _ = ⊤
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⟦_⟧ᵍ [ + ]ᵍ v = Σ ℕ (λ n → v ≡ ↑ᶻ (+_ (suc n)))
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⟦_⟧ᵍ [ 0ˢ ]ᵍ v = v ≡ ↑ᶻ (+_ zero)
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⟦_⟧ᵍ [ - ]ᵍ v = Σ ℕ (λ n → v ≡ ↑ᶻ -[1+ n ])
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⟦⟧ᵍ-respects-≈ᵍ : ∀ {s₁ s₂ : SignLattice} → s₁ ≈ᵍ s₂ → ⟦ s₁ ⟧ᵍ ⇒ ⟦ s₂ ⟧ᵍ
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⟦⟧ᵍ-respects-≈ᵍ ≈ᵍ-⊥ᵍ-⊥ᵍ v bot = bot
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⟦⟧ᵍ-respects-≈ᵍ ≈ᵍ-⊤ᵍ-⊤ᵍ v top = top
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⟦⟧ᵍ-respects-≈ᵍ (≈ᵍ-lift { + } { + } refl) v proof = proof
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⟦⟧ᵍ-respects-≈ᵍ (≈ᵍ-lift { - } { - } refl) v proof = proof
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⟦⟧ᵍ-respects-≈ᵍ (≈ᵍ-lift { 0ˢ } { 0ˢ } refl) v proof = proof
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⟦⟧ᵍ-⊔ᵍ-∨ : ∀ {s₁ s₂ : SignLattice} → (⟦ s₁ ⟧ᵍ ∨ ⟦ s₂ ⟧ᵍ) ⇒ ⟦ s₁ ⊔ᵍ s₂ ⟧ᵍ
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⟦⟧ᵍ-⊔ᵍ-∨ {⊥ᵍ} x (inj₂ px₂) = px₂
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⟦⟧ᵍ-⊔ᵍ-∨ {⊤ᵍ} x _ = tt
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⟦⟧ᵍ-⊔ᵍ-∨ {[ s₁ ]ᵍ} {[ s₂ ]ᵍ} x px
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with s₁ ≟ᵍ s₂
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... | no _ = tt
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... | yes refl
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with px
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... | inj₁ px₁ = px₁
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... | inj₂ px₂ = px₂
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⟦⟧ᵍ-⊔ᵍ-∨ {[ s₁ ]ᵍ} {⊥ᵍ} x (inj₁ px₁) = px₁
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⟦⟧ᵍ-⊔ᵍ-∨ {[ s₁ ]ᵍ} {⊤ᵍ} x _ = tt
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s₁≢s₂⇒¬s₁∧s₂ : ∀ {s₁ s₂ : Sign} → ¬ s₁ ≡ s₂ → ∀ {v} → ¬ ((⟦ [ s₁ ]ᵍ ⟧ᵍ ∧ ⟦ [ s₂ ]ᵍ ⟧ᵍ) v)
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s₁≢s₂⇒¬s₁∧s₂ { + } { + } +≢+ _ = ⊥-elim (+≢+ refl)
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s₁≢s₂⇒¬s₁∧s₂ { + } { - } _ ((n , refl) , (m , ()))
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s₁≢s₂⇒¬s₁∧s₂ { + } { 0ˢ } _ ((n , refl) , ())
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s₁≢s₂⇒¬s₁∧s₂ { 0ˢ } { + } _ (refl , (m , ()))
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s₁≢s₂⇒¬s₁∧s₂ { 0ˢ } { 0ˢ } +≢+ _ = ⊥-elim (+≢+ refl)
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s₁≢s₂⇒¬s₁∧s₂ { 0ˢ } { - } _ (refl , (m , ()))
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s₁≢s₂⇒¬s₁∧s₂ { - } { + } _ ((n , refl) , (m , ()))
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s₁≢s₂⇒¬s₁∧s₂ { - } { 0ˢ } _ ((n , refl) , ())
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s₁≢s₂⇒¬s₁∧s₂ { - } { - } +≢+ _ = ⊥-elim (+≢+ refl)
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⟦⟧ᵍ-⊓ᵍ-∧ : ∀ {s₁ s₂ : SignLattice} → (⟦ s₁ ⟧ᵍ ∧ ⟦ s₂ ⟧ᵍ) ⇒ ⟦ s₁ ⊓ᵍ s₂ ⟧ᵍ
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⟦⟧ᵍ-⊓ᵍ-∧ {⊥ᵍ} x (bot , _) = bot
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⟦⟧ᵍ-⊓ᵍ-∧ {⊤ᵍ} x (_ , px₂) = px₂
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⟦⟧ᵍ-⊓ᵍ-∧ {[ s₁ ]ᵍ} {[ s₂ ]ᵍ} x (px₁ , px₂)
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with s₁ ≟ᵍ s₂
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... | no s₁≢s₂ = s₁≢s₂⇒¬s₁∧s₂ s₁≢s₂ (px₁ , px₂)
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... | yes refl = px₁
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⟦⟧ᵍ-⊓ᵍ-∧ {[ g₁ ]ᵍ} {⊥ᵍ} x (_ , bot) = bot
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⟦⟧ᵍ-⊓ᵍ-∧ {[ g₁ ]ᵍ} {⊤ᵍ} x (px₁ , _) = px₁
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instance
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latticeInterpretationᵍ : LatticeInterpretation isLatticeᵍ
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latticeInterpretationᵍ = record
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{ ⟦_⟧ = ⟦_⟧ᵍ
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; ⟦⟧-respects-≈ = ⟦⟧ᵍ-respects-≈ᵍ
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; ⟦⟧-⊔-∨ = ⟦⟧ᵍ-⊔ᵍ-∨
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; ⟦⟧-⊓-∧ = ⟦⟧ᵍ-⊓ᵍ-∧
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}
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module WithProg (prog : Program) where
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open Program prog
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open import Analysis.Forward.Lattices SignLattice prog
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open import Analysis.Forward.Evaluation SignLattice prog
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open import Analysis.Forward.Adapters SignLattice prog
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eval : ∀ (e : Expr) → VariableValues → SignLattice
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eval (e₁ + e₂) vs = plus (eval e₁ vs) (eval e₂ vs)
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eval (e₁ - e₂) vs = minus (eval e₁ vs) (eval e₂ vs)
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eval (` k) vs
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with ∈k-decᵛ k (proj₁ (proj₁ vs))
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... | yes k∈vs = proj₁ (locateᵛ {k} {vs} k∈vs)
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... | no _ = ⊤ᵍ
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eval (# 0) _ = [ 0ˢ ]ᵍ
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eval (# (suc n')) _ = [ + ]ᵍ
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eval-Monoʳ : ∀ (e : Expr) → Monotonic _≼ᵛ_ _≼ᵍ_ (eval e)
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eval-Monoʳ (e₁ + e₂) {vs₁} {vs₂} vs₁≼vs₂ =
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eval-combine₂ (λ {x} {y} {z} → ≼ᵍ-trans {x} {y} {z})
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plus plus-Mono₂ {o₁ = eval e₁ vs₁}
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(eval-Monoʳ e₁ vs₁≼vs₂) (eval-Monoʳ e₂ vs₁≼vs₂)
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eval-Monoʳ (e₁ - e₂) {vs₁} {vs₂} vs₁≼vs₂ =
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eval-combine₂ (λ {x} {y} {z} → ≼ᵍ-trans {x} {y} {z})
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minus minus-Mono₂ {o₁ = eval e₁ vs₁}
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(eval-Monoʳ e₁ vs₁≼vs₂) (eval-Monoʳ e₂ vs₁≼vs₂)
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eval-Monoʳ (` k) {vs₁@((kvs₁ , _) , _)} {vs₂@((kvs₂ , _), _)} vs₁≼vs₂
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with ∈k-decᵛ k kvs₁ | ∈k-decᵛ k kvs₂
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... | yes k∈kvs₁ | yes k∈kvs₂ =
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let
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(v₁ , k,v₁∈vs₁) = locateᵛ {k} {vs₁} k∈kvs₁
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(v₂ , k,v₂∈vs₂) = locateᵛ {k} {vs₂} k∈kvs₂
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in
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m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ vs₁ vs₂ vs₁≼vs₂ k,v₁∈vs₁ k,v₂∈vs₂
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... | yes k∈kvs₁ | no k∉kvs₂ = ⊥-elim (k∉kvs₂ (subst (λ l → k ∈ˡ l) (all-equal-keysᵛ vs₁ vs₂) k∈kvs₁))
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... | no k∉kvs₁ | yes k∈kvs₂ = ⊥-elim (k∉kvs₁ (subst (λ l → k ∈ˡ l) (all-equal-keysᵛ vs₂ vs₁) k∈kvs₂))
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... | no k∉kvs₁ | no k∉kvs₂ = IsLattice.≈-refl isLatticeᵍ
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eval-Monoʳ (# 0) _ = ≈ᵍ-refl
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eval-Monoʳ (# (suc n')) _ = ≈ᵍ-refl
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instance
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SignEval : ExprEvaluator
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SignEval = record { eval = eval; eval-Monoʳ = eval-Monoʳ }
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-- For debugging purposes, print out the result.
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output = show (Analysis.Forward.WithProg.result SignLattice prog)
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-- This should have fewer cases -- the same number as the actual 'plus' above.
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-- But agda only simplifies on first argument, apparently, so we are stuck
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-- listing them all.
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plus-valid : ∀ {g₁ g₂} {z₁ z₂} → ⟦ g₁ ⟧ᵍ (↑ᶻ z₁) → ⟦ g₂ ⟧ᵍ (↑ᶻ z₂) → ⟦ plus g₁ g₂ ⟧ᵍ (↑ᶻ (z₁ Int.+ z₂))
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plus-valid {⊥ᵍ} {_} ⊥ _ = ⊥
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plus-valid {[ + ]ᵍ} {⊥ᵍ} _ ⊥ = ⊥
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plus-valid {[ - ]ᵍ} {⊥ᵍ} _ ⊥ = ⊥
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plus-valid {[ 0ˢ ]ᵍ} {⊥ᵍ} _ ⊥ = ⊥
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plus-valid {⊤ᵍ} {⊥ᵍ} _ ⊥ = ⊥
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plus-valid {⊤ᵍ} {[ + ]ᵍ} _ _ = tt
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plus-valid {⊤ᵍ} {[ - ]ᵍ} _ _ = tt
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plus-valid {⊤ᵍ} {[ 0ˢ ]ᵍ} _ _ = tt
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plus-valid {⊤ᵍ} {⊤ᵍ} _ _ = tt
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plus-valid {[ + ]ᵍ} {[ + ]ᵍ} (n₁ , refl) (n₂ , refl) = (_ , refl)
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plus-valid {[ + ]ᵍ} {[ - ]ᵍ} _ _ = tt
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plus-valid {[ + ]ᵍ} {[ 0ˢ ]ᵍ} (n₁ , refl) refl = (_ , refl)
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plus-valid {[ + ]ᵍ} {⊤ᵍ} _ _ = tt
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plus-valid {[ - ]ᵍ} {[ + ]ᵍ} _ _ = tt
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plus-valid {[ - ]ᵍ} {[ - ]ᵍ} (n₁ , refl) (n₂ , refl) = (_ , refl)
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plus-valid {[ - ]ᵍ} {[ 0ˢ ]ᵍ} (n₁ , refl) refl = (_ , refl)
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plus-valid {[ - ]ᵍ} {⊤ᵍ} _ _ = tt
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plus-valid {[ 0ˢ ]ᵍ} {[ + ]ᵍ} refl (n₂ , refl) = (_ , refl)
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plus-valid {[ 0ˢ ]ᵍ} {[ - ]ᵍ} refl (n₂ , refl) = (_ , refl)
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plus-valid {[ 0ˢ ]ᵍ} {[ 0ˢ ]ᵍ} refl refl = refl
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plus-valid {[ 0ˢ ]ᵍ} {⊤ᵍ} _ _ = tt
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-- Same for this one. It should be easier, but Agda won't simplify.
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minus-valid : ∀ {g₁ g₂} {z₁ z₂} → ⟦ g₁ ⟧ᵍ (↑ᶻ z₁) → ⟦ g₂ ⟧ᵍ (↑ᶻ z₂) → ⟦ minus g₁ g₂ ⟧ᵍ (↑ᶻ (z₁ Int.- z₂))
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minus-valid {⊥ᵍ} {_} ⊥ _ = ⊥
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minus-valid {[ + ]ᵍ} {⊥ᵍ} _ ⊥ = ⊥
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minus-valid {[ - ]ᵍ} {⊥ᵍ} _ ⊥ = ⊥
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minus-valid {[ 0ˢ ]ᵍ} {⊥ᵍ} _ ⊥ = ⊥
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minus-valid {⊤ᵍ} {⊥ᵍ} _ ⊥ = ⊥
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minus-valid {⊤ᵍ} {[ + ]ᵍ} _ _ = tt
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minus-valid {⊤ᵍ} {[ - ]ᵍ} _ _ = tt
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minus-valid {⊤ᵍ} {[ 0ˢ ]ᵍ} _ _ = tt
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minus-valid {⊤ᵍ} {⊤ᵍ} _ _ = tt
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minus-valid {[ + ]ᵍ} {[ + ]ᵍ} _ _ = tt
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minus-valid {[ + ]ᵍ} {[ - ]ᵍ} (n₁ , refl) (n₂ , refl) = (_ , refl)
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minus-valid {[ + ]ᵍ} {[ 0ˢ ]ᵍ} (n₁ , refl) refl = (_ , refl)
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minus-valid {[ + ]ᵍ} {⊤ᵍ} _ _ = tt
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minus-valid {[ - ]ᵍ} {[ + ]ᵍ} (n₁ , refl) (n₂ , refl) = (_ , refl)
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minus-valid {[ - ]ᵍ} {[ - ]ᵍ} _ _ = tt
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minus-valid {[ - ]ᵍ} {[ 0ˢ ]ᵍ} (n₁ , refl) refl = (_ , refl)
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minus-valid {[ - ]ᵍ} {⊤ᵍ} _ _ = tt
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minus-valid {[ 0ˢ ]ᵍ} {[ + ]ᵍ} refl (n₂ , refl) = (_ , refl)
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minus-valid {[ 0ˢ ]ᵍ} {[ - ]ᵍ} refl (n₂ , refl) = (_ , refl)
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minus-valid {[ 0ˢ ]ᵍ} {[ 0ˢ ]ᵍ} refl refl = refl
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minus-valid {[ 0ˢ ]ᵍ} {⊤ᵍ} _ _ = tt
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eval-valid : IsValidExprEvaluator
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eval-valid (⇒ᵉ-+ ρ e₁ e₂ z₁ z₂ ρ,e₁⇒z₁ ρ,e₂⇒z₂) ⟦vs⟧ρ =
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plus-valid (eval-valid ρ,e₁⇒z₁ ⟦vs⟧ρ) (eval-valid ρ,e₂⇒z₂ ⟦vs⟧ρ)
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eval-valid (⇒ᵉ-- ρ e₁ e₂ z₁ z₂ ρ,e₁⇒z₁ ρ,e₂⇒z₂) ⟦vs⟧ρ =
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minus-valid (eval-valid ρ,e₁⇒z₁ ⟦vs⟧ρ) (eval-valid ρ,e₂⇒z₂ ⟦vs⟧ρ)
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eval-valid {vs} (⇒ᵉ-Var ρ x v x,v∈ρ) ⟦vs⟧ρ
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with ∈k-decᵛ x (proj₁ (proj₁ vs))
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... | yes x∈kvs = ⟦vs⟧ρ (proj₂ (locateᵛ {x} {vs} x∈kvs)) x,v∈ρ
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... | no x∉kvs = tt
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eval-valid (⇒ᵉ-ℕ ρ 0) _ = refl
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eval-valid (⇒ᵉ-ℕ ρ (suc n')) _ = (n' , refl)
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instance
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SignEvalValid : ValidExprEvaluator SignEval latticeInterpretationᵍ
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SignEvalValid = record { valid = eval-valid }
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analyze-correct = Analysis.Forward.WithProg.analyze-correct SignLattice prog tt
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