Prove that evaluation is monotonic and complete sign analysis

Other than monotonicity of plus and minus, god damn it.

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

Analysis/Sign.agda

@@ -103,8 +103,9 @@ 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.
-    open Lattice.FiniteValueMap.WithKeys _≟ˢ_ isLatticeᵍ vars
+    module VariableSignsFiniteMap = Lattice.FiniteValueMap.WithKeys _≟ˢ_ isLatticeᵍ vars
-        using (m₁≼m₂⇒m₁[k]≼m₂[k])
+    open VariableSignsFiniteMap
+        using ()
         renaming
             ( FiniteMap to VariableSigns
             ; isLattice to isLatticeᵛ
@@ -116,6 +117,7 @@ module _ (prog : Program) where
             ; _∈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 ()
@@ -145,13 +147,17 @@ module _ (prog : Program) where
             ; _∈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 isFiniteHeightLatticeᵛ
+            ( 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)
@@ -201,6 +207,7 @@ module _ (prog : Program) where
     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
@@ -214,6 +221,7 @@ module _ (prog : Program) where
                 (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
@@ -230,24 +238,49 @@ module _ (prog : Program) where
             (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
+            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
 
-    updateForState : State → StateVariables → VariableSigns
+    updateVariablesForState : State → StateVariables → VariableSigns
-    updateForState s sv
+    updateVariablesForState s sv
-        with code s in p
+        -- More weirdness here. Apparently, capturing the with-equality proof
-    ...   | k ← e =
+        -- 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)
-                k∈e⇒k∈codes = λ k k∈e → subst (λ stmt → k ∈ᵗ stmt) (sym p) (in←₂ k∈e)
-                k∈e⇒k∈vars = λ k k∈e → vars-complete s (k∈e⇒k∈codes k k∈e)
             in
-                vs updatingᵛ (k ∷ []) via (λ _ → eval e k∈e⇒k∈vars vs)
+                updateVariablesFromExpression k e (λ k k∈e → k∈codes⇒k∈vars k (in←₂ k∈e)) vs
 
-    open StateVariablesFiniteMap.GeneralizedUpdate states isLatticeᵐ joinAll (λ {a₁} {a₂} a₁≼a₂ → joinAll-Mono {a₁} {a₂} a₁≼a₂) updateForState {!!} states
+    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)