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Prove monotonicity of eval

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2024-03-10 20:29:05 -07:00
parent 96f3ceaeb2
commit 8964ba59a1
2 changed files with 50 additions and 3 deletions
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44
Analysis/Sign.agda
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@ -56,6 +56,11 @@ open AB.Plain 0ˢ using ()
; _⊔_ to _⊔ᵍ_
)
open IsLattice isLatticeᵍ using ()
renaming
( ≼-trans to ≼ᵍ-trans
)
plus : SignLattice → SignLattice → SignLattice
plus ⊥ᵍ _ = ⊥ᵍ
plus _ ⊥ᵍ = ⊥ᵍ
@ -99,7 +104,7 @@ module _ (prog : Program) where
-- The variable -> sign map is a finite value-map with keys strings. Use a bundle to avoid explicitly specifying operators.
open Lattice.FiniteValueMap.WithKeys _≟ˢ_ isLatticeᵍ vars
using ()
using (m₁≼m₂⇒m₁[k]≼m₂[k])
renaming
( FiniteMap to VariableSigns
; isLattice to isLatticeᵛ
@ -193,6 +198,43 @@ module _ (prog : Program) where
eval (# 0) _ _ = [ 0ˢ ]ᵍ
eval (# (suc n')) _ _ = [ + ]ᵍ
eval-Mono : ∀ (e : Expr) (k∈e⇒k∈vars : ∀ k → k ∈ᵉ e → k ∈ˡ vars) → Monotonic _≼ᵛ_ _≼ᵍ_ (eval e k∈e⇒k∈vars)
eval-Mono (e₁ + e₂) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
let
k∈e₁⇒k∈vars = λ k k∈e₁ → k∈e⇒k∈vars k (in⁺₁ k∈e₁)
k∈e₂⇒k∈vars = λ k k∈e₂ → k∈e⇒k∈vars k (in⁺₂ k∈e₂)
g₁vs₁ = eval e₁ k∈e₁⇒k∈vars vs₁
g₂vs₁ = eval e₂ k∈e₂⇒k∈vars vs₁
g₁vs₂ = eval e₁ k∈e₁⇒k∈vars vs₂
g₂vs₂ = eval e₂ k∈e₂⇒k∈vars vs₂
in
≼ᵍ-trans
{plus g₁vs₁ g₂vs₁} {plus g₁vs₂ g₂vs₁} {plus g₁vs₂ g₂vs₂}
(plus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ k∈e₁⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
(plus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ k∈e₂⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
eval-Mono (e₁ - e₂) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
let
k∈e₁⇒k∈vars = λ k k∈e₁ → k∈e⇒k∈vars k (in⁻₁ k∈e₁)
k∈e₂⇒k∈vars = λ k k∈e₂ → k∈e⇒k∈vars k (in⁻₂ k∈e₂)
g₁vs₁ = eval e₁ k∈e₁⇒k∈vars vs₁
g₂vs₁ = eval e₂ k∈e₂⇒k∈vars vs₁
g₁vs₂ = eval e₁ k∈e₁⇒k∈vars vs₂
g₂vs₂ = eval e₂ k∈e₂⇒k∈vars vs₂
in
≼ᵍ-trans
{minus g₁vs₁ g₂vs₁} {minus g₁vs₂ g₂vs₁} {minus g₁vs₂ g₂vs₂}
(minus-Monoˡ g₂vs₁ {g₁vs₁} {g₁vs₂} (eval-Mono e₁ k∈e₁⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
(minus-Monoʳ g₁vs₂ {g₂vs₁} {g₂vs₂} (eval-Mono e₂ k∈e₂⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂))
eval-Mono (` k) k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂ =
let
(v₁ , k,v₁∈vs₁) = locateᵛ {k} {vs₁} (vars-in-Map k vs₁ (k∈e⇒k∈vars k here))
(v₂ , k,v₂∈vs₂) = locateᵛ {k} {vs₂} (vars-in-Map k vs₂ (k∈e⇒k∈vars k here))
in
m₁≼m₂⇒m₁[k]≼m₂[k] vs₁ vs₂ vs₁≼vs₂ k,v₁∈vs₁ k,v₂∈vs₂
eval-Mono (# 0) _ _ = ≈ᵍ-refl
eval-Mono (# (suc n')) _ _ = ≈ᵍ-refl
updateForState : State → StateVariables → VariableSigns
updateForState s sv
with code s in p

9
Lattice/FiniteMap.agda
View File

@ -12,7 +12,7 @@ module Lattice.FiniteMap {a b : Level} {A : Set a} {B : Set b}
open IsLattice lB using () renaming (_≼_ to _≼₂_)
open import Lattice.Map ≡-dec-A lB as Map
using (Map; ⊔-equal-keys; ⊓-equal-keys; ∈k-dec; m₁≼m₂⇒m₁[k]≼m₂[k])
using (Map; ⊔-equal-keys; ⊓-equal-keys; ∈k-dec)
renaming
( _≈_ to _≈ᵐ_
; _⊔_ to _⊔ᵐ_
@ -30,6 +30,7 @@ open import Lattice.Map ≡-dec-A lB as Map
; absorb-⊓-⊔ to absorb-⊓ᵐ-⊔ᵐ
; ≈-dec to ≈ᵐ-dec
; _[_] to _[_]ᵐ
; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ
; locate to locateᵐ
; keys to keysᵐ
; _updating_via_ to _updatingᵐ_via_
@ -138,6 +139,10 @@ module WithKeys (ks : List A) where
; isLattice = isLattice
}
m₁≼m₂⇒m₁[k]≼m₂[k] : ∀ (fm₁ fm₂ : FiniteMap) {k : A} {v₁ v₂ : B} →
fm₁ ≼ fm₂ → (k , v₁) ∈ fm₁ → (k , v₂) ∈ fm₂ → v₁ ≼₂ v₂
m₁≼m₂⇒m₁[k]≼m₂[k] (m₁ , _) (m₂ , _) m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂ = m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ m₁ m₂ m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂
module GeneralizedUpdate
{l} {L : Set l}
{_≈ˡ_ : L → L → Set l} {_⊔ˡ_ : L → L → L} {_⊓ˡ_ : L → L → L}
@ -177,7 +182,7 @@ module WithKeys (ks : List A) where
(v₁ , k,v₁∈m₁) = locateᵐ {m = m₁} k∈km₁
(v₂ , k,v₂∈m₂) = locateᵐ {m = m₂} k∈km₂
in
(m₁≼m₂⇒m₁[k]≼m₂[k] m₁ m₂ m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂) ∷ m₁≼m₂⇒m₁[ks]≼m₂[ks] fm₁ fm₂ ks'' m₁≼m₂
(m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ m₁ m₂ m₁≼m₂ k,v₁∈m₁ k,v₂∈m₂) ∷ m₁≼m₂⇒m₁[ks]≼m₂[ks] fm₁ fm₂ ks'' m₁≼m₂
... | no k∉km₁ | no k∉km₂ = m₁≼m₂⇒m₁[ks]≼m₂[ks] fm₁ fm₂ ks'' m₁≼m₂
... | yes k∈km₁ | no k∉km₂ = ⊥-elim (∈k-exclusive fm₁ fm₂ (k∈km₁ , k∉km₂))
... | no k∉km₁ | yes k∈km₂ = ⊥-elim (∈k-exclusive fm₂ fm₁ (k∈km₂ , k∉km₁))