Use instances to simplify printing code

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

Analysis/Sign.agda

@@ -13,6 +13,7 @@ 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
@@ -20,6 +21,16 @@ data Sign : Set where
     - : Sign
     0ˢ : 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
@@ -301,20 +312,7 @@ module WithProg (prog : Program) where
     open import Data.String using (_++_)
     open import Data.List using () renaming (length to lengthˡ)
 
-    showAboveBelow : AB.AboveBelow → String
-    showAboveBelow AB.⊤ = "⊤"
-    showAboveBelow AB.⊥ = "⊥"
-    showAboveBelow (AB.[_] +) = "+"
-    showAboveBelow (AB.[_] -) = "-"
-    showAboveBelow (AB.[_] 0ˢ) = "0"
-
-    showVarSigns : VariableSigns → String
-    showVarSigns ((kvs , _) , _) = "{" ++ foldr (λ (x , y) rest → x ++ " ↦ " ++ showAboveBelow y ++ ", " ++ rest) "" kvs ++ "}"
-
-    showStateVars : StateVariables → String
-    showStateVars ((kvs , _) , _) = "{" ++ foldr (λ (x , y) rest → (showFin x) ++ " ↦ " ++ showVarSigns y ++ ", " ++ rest) "" kvs ++ "}"
-
-    output = showStateVars signs
+    output = show signs
 
 
 -- Debugging code: construct and run a program.