Prove monotonicity of eval

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

Analysis/Sign.agda

@@ -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