Danila Fedorin
8a85c4497c
Other than monotonicity of plus and minus, god damn it. Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
287 lines
13 KiB
Agda
287 lines
13 KiB
Agda
module Analysis.Sign where
|
||
|
||
open import Data.String using (String) renaming (_≟_ to _≟ˢ_)
|
||
open import Data.Nat using (suc)
|
||
open import Data.Product using (_×_; proj₁; _,_)
|
||
open import Data.List using (List; _∷_; []; foldr; cartesianProduct; cartesianProductWith)
|
||
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 (¬_; Dec; yes; no)
|
||
open import Data.Unit using (⊤)
|
||
|
||
open import Language
|
||
open import Lattice
|
||
open import Utils using (Pairwise)
|
||
import Lattice.FiniteValueMap
|
||
|
||
data Sign : Set where
|
||
+ : Sign
|
||
- : Sign
|
||
0ˢ : Sign
|
||
|
||
-- g for siGn; s is used for strings and i is not very descriptive.
|
||
_≟ᵍ_ : IsDecidable (_≡_ {_} {Sign})
|
||
_≟ᵍ_ + + = yes refl
|
||
_≟ᵍ_ + - = no (λ ())
|
||
_≟ᵍ_ + 0ˢ = no (λ ())
|
||
_≟ᵍ_ - + = no (λ ())
|
||
_≟ᵍ_ - - = yes refl
|
||
_≟ᵍ_ - 0ˢ = no (λ ())
|
||
_≟ᵍ_ 0ˢ + = no (λ ())
|
||
_≟ᵍ_ 0ˢ - = no (λ ())
|
||
_≟ᵍ_ 0ˢ 0ˢ = yes refl
|
||
|
||
-- embelish 'sign' with a top and bottom element.
|
||
open import Lattice.AboveBelow Sign _≡_ (record { ≈-refl = refl; ≈-sym = sym; ≈-trans = trans }) _≟ᵍ_ as AB
|
||
using ()
|
||
renaming
|
||
( AboveBelow to SignLattice
|
||
; ≈-dec to ≈ᵍ-dec
|
||
; ⊥ to ⊥ᵍ
|
||
; ⊤ to ⊤ᵍ
|
||
; [_] to [_]ᵍ
|
||
; _≈_ to _≈ᵍ_
|
||
; ≈-⊥-⊥ to ≈ᵍ-⊥ᵍ-⊥ᵍ
|
||
; ≈-⊤-⊤ to ≈ᵍ-⊤ᵍ-⊤ᵍ
|
||
; ≈-lift to ≈ᵍ-lift
|
||
; ≈-refl to ≈ᵍ-refl
|
||
)
|
||
-- 'sign' has no underlying lattice structure, so use the 'plain' above-below lattice.
|
||
open AB.Plain 0ˢ using ()
|
||
renaming
|
||
( finiteHeightLattice to finiteHeightLatticeᵍ
|
||
; isLattice to isLatticeᵍ
|
||
; fixedHeight to fixedHeightᵍ
|
||
; _≼_ to _≼ᵍ_
|
||
; _⊔_ to _⊔ᵍ_
|
||
)
|
||
|
||
open IsLattice isLatticeᵍ using ()
|
||
renaming
|
||
( ≼-trans to ≼ᵍ-trans
|
||
)
|
||
|
||
plus : SignLattice → SignLattice → SignLattice
|
||
plus ⊥ᵍ _ = ⊥ᵍ
|
||
plus _ ⊥ᵍ = ⊥ᵍ
|
||
plus ⊤ᵍ _ = ⊤ᵍ
|
||
plus _ ⊤ᵍ = ⊤ᵍ
|
||
plus [ + ]ᵍ [ + ]ᵍ = [ + ]ᵍ
|
||
plus [ + ]ᵍ [ - ]ᵍ = ⊤ᵍ
|
||
plus [ + ]ᵍ [ 0ˢ ]ᵍ = [ + ]ᵍ
|
||
plus [ - ]ᵍ [ + ]ᵍ = ⊤ᵍ
|
||
plus [ - ]ᵍ [ - ]ᵍ = [ - ]ᵍ
|
||
plus [ - ]ᵍ [ 0ˢ ]ᵍ = [ - ]ᵍ
|
||
plus [ 0ˢ ]ᵍ [ + ]ᵍ = [ + ]ᵍ
|
||
plus [ 0ˢ ]ᵍ [ - ]ᵍ = [ - ]ᵍ
|
||
plus [ 0ˢ ]ᵍ [ 0ˢ ]ᵍ = [ 0ˢ ]ᵍ
|
||
|
||
-- this is incredibly tedious: 125 cases per monotonicity proof, and tactics
|
||
-- are hard. postulate for now.
|
||
postulate plus-Monoˡ : ∀ (s₂ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (λ s₁ → plus s₁ s₂)
|
||
postulate plus-Monoʳ : ∀ (s₁ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ_ (plus s₁)
|
||
|
||
minus : SignLattice → SignLattice → SignLattice
|
||
minus ⊥ᵍ _ = ⊥ᵍ
|
||
minus _ ⊥ᵍ = ⊥ᵍ
|
||
minus ⊤ᵍ _ = ⊤ᵍ
|
||
minus _ ⊤ᵍ = ⊤ᵍ
|
||
minus [ + ]ᵍ [ + ]ᵍ = ⊤ᵍ
|
||
minus [ + ]ᵍ [ - ]ᵍ = [ + ]ᵍ
|
||
minus [ + ]ᵍ [ 0ˢ ]ᵍ = [ + ]ᵍ
|
||
minus [ - ]ᵍ [ + ]ᵍ = [ - ]ᵍ
|
||
minus [ - ]ᵍ [ - ]ᵍ = ⊤ᵍ
|
||
minus [ - ]ᵍ [ 0ˢ ]ᵍ = [ - ]ᵍ
|
||
minus [ 0ˢ ]ᵍ [ + ]ᵍ = [ - ]ᵍ
|
||
minus [ 0ˢ ]ᵍ [ - ]ᵍ = [ + ]ᵍ
|
||
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)
|
||
|
||
module _ (prog : Program) where
|
||
open Program prog
|
||
|
||
-- The variable -> sign map is a finite value-map with keys strings. Use a bundle to avoid explicitly specifying operators.
|
||
module VariableSignsFiniteMap = Lattice.FiniteValueMap.WithKeys _≟ˢ_ isLatticeᵍ vars
|
||
open VariableSignsFiniteMap
|
||
using ()
|
||
renaming
|
||
( FiniteMap to VariableSigns
|
||
; isLattice to isLatticeᵛ
|
||
; _≈_ to _≈ᵛ_
|
||
; _⊔_ to _⊔ᵛ_
|
||
; _≼_ to _≼ᵛ_
|
||
; ≈₂-dec⇒≈-dec to ≈ᵍ-dec⇒≈ᵛ-dec
|
||
; _∈_ to _∈ᵛ_
|
||
; _∈k_ to _∈kᵛ_
|
||
; _updating_via_ to _updatingᵛ_via_
|
||
; locate to locateᵛ
|
||
; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵛ≼m₂[k]ᵛ
|
||
)
|
||
open IsLattice isLatticeᵛ
|
||
using ()
|
||
renaming
|
||
( ⊔-Monotonicˡ to ⊔ᵛ-Monotonicˡ
|
||
; ⊔-Monotonicʳ to ⊔ᵛ-Monotonicʳ
|
||
; ⊔-idemp to ⊔ᵛ-idemp
|
||
)
|
||
open Lattice.FiniteValueMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight _≟ˢ_ isLatticeᵍ vars-Unique ≈ᵍ-dec _ fixedHeightᵍ
|
||
using ()
|
||
renaming
|
||
( isFiniteHeightLattice to isFiniteHeightLatticeᵛ
|
||
)
|
||
|
||
≈ᵛ-dec = ≈ᵍ-dec⇒≈ᵛ-dec ≈ᵍ-dec
|
||
joinSemilatticeᵛ = IsFiniteHeightLattice.joinSemilattice isFiniteHeightLatticeᵛ
|
||
fixedHeightᵛ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵛ
|
||
⊥ᵛ = proj₁ (proj₁ (proj₁ fixedHeightᵛ))
|
||
|
||
-- Finally, the map we care about is (state -> (variables -> sign)). Bring that in.
|
||
module StateVariablesFiniteMap = Lattice.FiniteValueMap.WithKeys _≟_ isLatticeᵛ states
|
||
open StateVariablesFiniteMap
|
||
using (_[_]; m₁≼m₂⇒m₁[ks]≼m₂[ks])
|
||
renaming
|
||
( FiniteMap to StateVariables
|
||
; isLattice to isLatticeᵐ
|
||
; _∈k_ to _∈kᵐ_
|
||
; locate to locateᵐ
|
||
; _≼_ to _≼ᵐ_
|
||
; ≈₂-dec⇒≈-dec to ≈ᵛ-dec⇒≈ᵐ-dec
|
||
; m₁≼m₂⇒m₁[k]≼m₂[k] to m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ
|
||
)
|
||
open Lattice.FiniteValueMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight _≟_ isLatticeᵛ states-Unique ≈ᵛ-dec _ fixedHeightᵛ
|
||
using ()
|
||
renaming
|
||
( isFiniteHeightLattice to isFiniteHeightLatticeᵐ
|
||
)
|
||
|
||
≈ᵐ-dec = ≈ᵛ-dec⇒≈ᵐ-dec ≈ᵛ-dec
|
||
|
||
-- build up the 'join' function, which follows from Exercise 4.26's
|
||
--
|
||
-- L₁ → (A → L₂)
|
||
--
|
||
-- Construction, with L₁ = (A → L₂), and f = id
|
||
|
||
joinForKey : State → StateVariables → VariableSigns
|
||
joinForKey k states = foldr _⊔ᵛ_ ⊥ᵛ (states [ incoming k ])
|
||
|
||
-- The per-key join is made up of map key accesses (which are monotonic)
|
||
-- and folds using the join operation (also monotonic)
|
||
|
||
joinForKey-Mono : ∀ (k : State) → Monotonic _≼ᵐ_ _≼ᵛ_ (joinForKey k)
|
||
joinForKey-Mono k {fm₁} {fm₂} fm₁≼fm₂ =
|
||
foldr-Mono joinSemilatticeᵛ joinSemilatticeᵛ (fm₁ [ incoming k ]) (fm₂ [ incoming k ]) _⊔ᵛ_ ⊥ᵛ ⊥ᵛ
|
||
(m₁≼m₂⇒m₁[ks]≼m₂[ks] fm₁ fm₂ (incoming k) fm₁≼fm₂)
|
||
(⊔ᵛ-idemp ⊥ᵛ) ⊔ᵛ-Monotonicʳ ⊔ᵛ-Monotonicˡ
|
||
|
||
-- The name f' comes from the formulation of Exercise 4.26.
|
||
|
||
open StateVariablesFiniteMap.GeneralizedUpdate states isLatticeᵐ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) joinForKey joinForKey-Mono states
|
||
renaming
|
||
( f' to joinAll
|
||
; f'-Monotonic to joinAll-Mono
|
||
)
|
||
|
||
-- With 'join' in hand, we need to perform abstract evaluation.
|
||
|
||
vars-in-Map : ∀ (k : String) (vs : VariableSigns) →
|
||
k ∈ˡ vars → k ∈kᵛ vs
|
||
vars-in-Map k vs@(m , kvs≡vars) k∈vars rewrite kvs≡vars = k∈vars
|
||
|
||
states-in-Map : ∀ (s : State) (sv : StateVariables) → s ∈kᵐ sv
|
||
states-in-Map s sv@(m , ksv≡states) rewrite ksv≡states = states-complete s
|
||
|
||
eval : ∀ (e : Expr) → (∀ k → k ∈ᵉ e → k ∈ˡ vars) → VariableSigns → SignLattice
|
||
eval (e₁ + e₂) k∈e⇒k∈vars vs =
|
||
plus (eval e₁ (λ k k∈e₁ → k∈e⇒k∈vars k (in⁺₁ k∈e₁)) vs)
|
||
(eval e₂ (λ k k∈e₂ → k∈e⇒k∈vars k (in⁺₂ k∈e₂)) vs)
|
||
eval (e₁ - e₂) k∈e⇒k∈vars vs =
|
||
minus (eval e₁ (λ k k∈e₁ → k∈e⇒k∈vars k (in⁻₁ k∈e₁)) vs)
|
||
(eval e₂ (λ k k∈e₂ → k∈e⇒k∈vars k (in⁻₂ k∈e₂)) vs)
|
||
eval (` k) k∈e⇒k∈vars vs = proj₁ (locateᵛ {k} {vs} (vars-in-Map k vs (k∈e⇒k∈vars k here)))
|
||
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
|
||
-- TODO: can this be done with less boilerplate?
|
||
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
|
||
-- TODO: here too -- can this be done with less boilerplate?
|
||
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]ᵛ≼m₂[k]ᵛ vs₁ vs₂ vs₁≼vs₂ k,v₁∈vs₁ k,v₂∈vs₂
|
||
eval-Mono (# 0) _ _ = ≈ᵍ-refl
|
||
eval-Mono (# (suc n')) _ _ = ≈ᵍ-refl
|
||
|
||
private module _ (k : String) (e : Expr) (k∈e⇒k∈vars : ∀ k → k ∈ᵉ e → k ∈ˡ vars) where
|
||
open VariableSignsFiniteMap.GeneralizedUpdate vars isLatticeᵛ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) (λ _ → eval e k∈e⇒k∈vars) (λ _ → eval-Mono e k∈e⇒k∈vars) (k ∷ [])
|
||
renaming
|
||
( f' to updateVariablesFromExpression
|
||
; f'-Monotonic to updateVariablesFromExpression-Mono
|
||
)
|
||
public
|
||
|
||
updateVariablesForState : State → StateVariables → VariableSigns
|
||
updateVariablesForState s sv
|
||
-- More weirdness here. Apparently, capturing the with-equality proof
|
||
-- using 'in p' makes code that reasons about this function (below)
|
||
-- throw ill-typed with-abstraction errors. Instead, make use of the
|
||
-- fact that later with-clauses are generalized over earlier ones to
|
||
-- construct a specialization of vars-complete for (code s).
|
||
with code s | (λ k → vars-complete {k} s)
|
||
... | k ← e | k∈codes⇒k∈vars =
|
||
let
|
||
(vs , s,vs∈sv) = locateᵐ {s} {sv} (states-in-Map s sv)
|
||
in
|
||
updateVariablesFromExpression k e (λ k k∈e → k∈codes⇒k∈vars k (in←₂ k∈e)) vs
|
||
|
||
updateVariablesForState-Monoʳ : ∀ (s : State) → Monotonic _≼ᵐ_ _≼ᵛ_ (updateVariablesForState s)
|
||
updateVariablesForState-Monoʳ s {sv₁} {sv₂} sv₁≼sv₂
|
||
with code s | (λ k → vars-complete {k} s)
|
||
... | k ← e | k∈codes⇒k∈vars =
|
||
let
|
||
(vs₁ , s,vs₁∈sv₁) = locateᵐ {s} {sv₁} (states-in-Map s sv₁)
|
||
(vs₂ , s,vs₂∈sv₂) = locateᵐ {s} {sv₂} (states-in-Map s sv₂)
|
||
vs₁≼vs₂ = m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ sv₁ sv₂ sv₁≼sv₂ s,vs₁∈sv₁ s,vs₂∈sv₂
|
||
in
|
||
updateVariablesFromExpression-Mono k e (λ k k∈e → k∈codes⇒k∈vars k (in←₂ k∈e)) {vs₁} {vs₂} vs₁≼vs₂
|
||
|
||
open StateVariablesFiniteMap.GeneralizedUpdate states isLatticeᵐ joinAll (λ {a₁} {a₂} a₁≼a₂ → joinAll-Mono {a₁} {a₂} a₁≼a₂) updateVariablesForState updateVariablesForState-Monoʳ states
|
||
renaming
|
||
( f' to updateAll
|
||
; f'-Monotonic to updateAll-Mono
|
||
)
|
||
|
||
open import Fixedpoint ≈ᵐ-dec isFiniteHeightLatticeᵐ updateAll (λ {m₁} {m₂} m₁≼m₂ → updateAll-Mono {m₁} {m₂} m₁≼m₂)
|
||
using ()
|
||
renaming (aᶠ to signs)
|