Extract common parts of forward analyses into Forward.agda

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>

Analysis/Forward.agda

@@ -0,0 +1,187 @@
+open import Language
+open import Lattice
+
+module Analysis.Forward
+    {L : Set} {h}
+    {_≈ˡ_ : L → L → Set} {_⊔ˡ_ : L → L → L} {_⊓ˡ_ : L → L → L}
+    (isFiniteHeightLatticeˡ : IsFiniteHeightLattice L h _≈ˡ_ _⊔ˡ_ _⊓ˡ_)
+    (≈ˡ-dec : IsDecidable _≈ˡ_) 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 Utils using (Pairwise)
+import Lattice.FiniteValueMap
+
+open IsFiniteHeightLattice isFiniteHeightLatticeˡ
+    using ()
+    renaming
+        ( isLattice to isLatticeˡ
+        ; fixedHeight to fixedHeightˡ
+        ; _≼_ to _≼ˡ_
+        )
+
+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 VariableValuesFiniteMap = Lattice.FiniteValueMap.WithKeys _≟ˢ_ isLatticeˡ vars
+    open VariableValuesFiniteMap
+        using ()
+        renaming
+            ( FiniteMap to VariableValues
+            ; isLattice to isLatticeᵛ
+            ; _≈_ to _≈ᵛ_
+            ; _⊔_ to _⊔ᵛ_
+            ; _≼_ to _≼ᵛ_
+            ; ≈₂-dec⇒≈-dec to ≈ˡ-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
+            ; ∈k-dec to ∈k-decᵛ
+            ; all-equal-keys to all-equal-keysᵛ
+            )
+        public
+    open IsLattice isLatticeᵛ
+        using ()
+        renaming
+            ( ⊔-Monotonicˡ to ⊔ᵛ-Monotonicˡ
+            ; ⊔-Monotonicʳ to ⊔ᵛ-Monotonicʳ
+            ; ⊔-idemp to ⊔ᵛ-idemp
+            )
+    open Lattice.FiniteValueMap.IterProdIsomorphism.WithUniqueKeysAndFixedHeight _≟ˢ_ isLatticeˡ vars-Unique ≈ˡ-dec _ fixedHeightˡ
+        using ()
+        renaming
+            ( isFiniteHeightLattice to isFiniteHeightLatticeᵛ
+            )
+
+    ≈ᵛ-dec = ≈ˡ-dec⇒≈ᵛ-dec ≈ˡ-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
+            )
+        public
+    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 → VariableValues
+    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.
+    module WithEvaluator (eval : Expr → VariableValues → L)
+                         (eval-Mono : ∀ (e : Expr) → Monotonic _≼ᵛ_ _≼ˡ_ (eval e)) where
+
+        -- For a particular evaluation function, we need to perform an evaluation
+        -- for an assignment, and update the corresponding key. Use Exercise 4.26's
+        -- generalized update to set the single key's value.
+
+        private module _ (k : String) (e : Expr) where
+            open VariableValuesFiniteMap.GeneralizedUpdate vars isLatticeᵛ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) (λ _ → eval e) (λ _ {vs₁} {vs₂} vs₁≼vs₂ → eval-Mono e {vs₁} {vs₂} vs₁≼vs₂) (k ∷ [])
+                renaming
+                    ( f' to updateVariablesFromExpression
+                    ; f'-Monotonic to updateVariablesFromExpression-Mono
+                    )
+                public
+
+        states-in-Map : ∀ (s : State) (sv : StateVariables) → s ∈kᵐ sv
+        states-in-Map s sv@(m , ksv≡states) rewrite ksv≡states = states-complete s
+
+        -- The per-state update function makes use of the single-key setter,
+        -- updateVariablesFromExpression, for the case where the statement
+        -- is an assignment.
+        --
+        -- This per-state function adjusts the variables in that state,
+        -- also monotonically; we derive the for-each-state update from
+        -- the Exercise 4.26 again.
+
+        updateVariablesForState : State → StateVariables → VariableValues
+        updateVariablesForState s sv
+            with code s
+        ...   | k ← e =
+                let
+                    (vs , s,vs∈sv) = locateᵐ {s} {sv} (states-in-Map s sv)
+                in
+                    updateVariablesFromExpression k e vs
+
+        updateVariablesForState-Monoʳ : ∀ (s : State) → Monotonic _≼ᵐ_ _≼ᵛ_ (updateVariablesForState s)
+        updateVariablesForState-Monoʳ s {sv₁} {sv₂} sv₁≼sv₂
+            with code s
+        ...   | k ← e =
+                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 {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
+                )
+
+        -- Finally, the whole sign analysis consists of getting the 'join'
+        -- of all incoming states, then applying the per-state evaluation
+        -- function. This is just a composition, and is trivially monotonic.
+
+        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)
+
+        -- The fixed point of the 'analyze' function is our final goal.
+        open import Fixedpoint ≈ᵐ-dec isFiniteHeightLatticeᵐ analyze (λ {m₁} {m₂} m₁≼m₂ → analyze-Mono {m₁} {m₂} m₁≼m₂)
+            using ()
+            renaming (aᶠ to result)
+            public

Analysis/Sign.agda

@@ -1,21 +1,16 @@
 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.Product using (proj₁; _,_)
 open import Data.Empty using (⊥; ⊥-elim)
-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 Relation.Nullary using (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
+import Analysis.Forward
 
 data Sign : Set where
     + : Sign
@@ -114,101 +109,10 @@ postulate minus-Monoʳ : ∀ (s₁ : SignLattice) → Monotonic _≼ᵍ_ _≼ᵍ
 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 ForwardWithProg = Analysis.Forward.WithProg (record { isLattice = isLatticeᵍ; fixedHeight = fixedHeightᵍ }) ≈ᵍ-dec prog
-    module VariableSignsFiniteMap = Lattice.FiniteValueMap.WithKeys _≟ˢ_ isLatticeᵍ vars
+    open ForwardWithProg
-    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
-            ; all-equal-keys to all-equal-keysᵛ
-            ; ∈k-dec to ∈k-decᵛ
-            )
-    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
+    eval : ∀ (e : Expr) → VariableValues → SignLattice
-    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) → VariableSigns → SignLattice
     eval (e₁ + e₂) vs = plus (eval e₁ vs) (eval e₂ vs)
     eval (e₁ - e₂) vs = minus (eval e₁ vs) (eval e₂ vs)
     eval (` k) vs
@@ -257,50 +161,7 @@ module WithProg (prog : Program) where
     eval-Mono (# 0) _ = ≈ᵍ-refl
     eval-Mono (# (suc n')) _ = ≈ᵍ-refl
 
-    private module _ (k : String) (e : Expr) where
+    open ForwardWithProg.WithEvaluator eval eval-Mono using (result)
-        open VariableSignsFiniteMap.GeneralizedUpdate vars isLatticeᵛ (λ x → x) (λ a₁≼a₂ → a₁≼a₂) (λ _ → eval e) (λ _ {vs₁} {vs₂} vs₁≼vs₂ → eval-Mono e {vs₁} {vs₂} vs₁≼vs₂) (k ∷ [])
-            renaming
-                ( f' to updateVariablesFromExpression
-                ; f'-Monotonic to updateVariablesFromExpression-Mono
-                )
-            public
 
-    updateVariablesForState : State → StateVariables → VariableSigns
+    -- For debugging purposes, print out the result.
-    updateVariablesForState s sv
+    output = show result
-        with code s
-    ...   | k ← e =
-            let
-                (vs , s,vs∈sv) = locateᵐ {s} {sv} (states-in-Map s sv)
-            in
-                updateVariablesFromExpression k e vs
-
-    updateVariablesForState-Monoʳ : ∀ (s : State) → Monotonic _≼ᵐ_ _≼ᵛ_ (updateVariablesForState s)
-    updateVariablesForState-Monoʳ s {sv₁} {sv₂} sv₁≼sv₂
-        with code s
-    ...   | k ← e =
-            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 {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.
-    output = show signs