agda-spa/Analysis/Sign.agda

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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 Function using (_∘_)
open import Language
open import Lattice
open import Utils using (Pairwise)
open import Showable using (Showable; show)
import Lattice.FiniteValueMap
data Sign : Set where
+ : Sign
- : Sign
: Sign
instance
showable : Showable Sign
showable = record
{ show = (λ
{ + "+"
; - "-"
; 0ˢ "0"
})
}
-- 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 WithProg (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
fixedHeightᵐ = IsFiniteHeightLattice.fixedHeight isFiniteHeightLatticeᵐ
-- 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 _ {vs₁} {vs₂} vs₁≼vs₂ eval-Mono e k∈e⇒k∈vars {vs₁} {vs₂} vs₁≼vs₂) (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 (λ x x) (λ a₁≼a₂ a₁≼a₂) updateVariablesForState updateVariablesForState-Monoʳ states
renaming
( f' to updateAll
; f'-Monotonic to updateAll-Mono
)
analyze : StateVariables StateVariables
analyze = updateAll joinAll
analyze-Mono : Monotonic _≼ᵐ_ _≼ᵐ_ analyze
analyze-Mono {sv₁} {sv₂} sv₁≼sv₂ = updateAll-Mono {joinAll sv₁} {joinAll sv₂} (joinAll-Mono {sv₁} {sv₂} sv₁≼sv₂)
open import Fixedpoint ≈ᵐ-dec isFiniteHeightLatticeᵐ analyze (λ {m₁} {m₂} m₁≼m₂ analyze-Mono {m₁} {m₂} m₁≼m₂)
using ()
renaming (aᶠ to signs)
-- Debugging code: print the resulting map.
open import Data.Fin using (suc; zero)
open import Data.Fin.Show using () renaming (show to showFin)
open import Data.Nat.Show using () renaming (show to showNat)
open import Data.String using (_++_)
open import Data.List using () renaming (length to lengthˡ)
output = show signs
-- Debugging code: construct and run a program.
open import Data.Vec using (Vec; _∷_; [])
open import IO
open import Level using (0)
testCode : Vec Stmt _
testCode =
("zero" (# 0))
("pos" ((` "zero") Expr.+ (# 1)))
("neg" ((` "zero") Expr.- (# 1)))
("unknown" ((` "pos") Expr.+ (` "neg")))
[]
testProgram : Program
testProgram = record
{ length = _
; stmts = testCode
}
open WithProg testProgram using (output)
main = run {0} (putStrLn output)