Added Agda code for some HITs

Agda-HITs/Imperative/Syntax.agda

@@ -0,0 +1,66 @@
+{-# OPTIONS --without-K --rewriting #-}
+
+open import HoTT
+
+module Syntax where
+
+data Maybe (A : Set) : Set where
+  Just : A -> Maybe A
+  Nothing : Maybe A
+
+eqN : Nat -> Nat -> Bool
+eqN 0 0 = true
+eqN 0 _ = false
+eqN (S _) 0 = false
+eqN (S n) (S m) = eqN n m
+
+-- first coordinate represents the variable x_i, second the value
+State : Set
+State = List (Nat × Nat)
+
+_[_:==_] : State -> Nat -> Nat -> State
+nil [ x :== n ] = (x , n) :: nil
+((y , m) :: s) [ x :== n ] =
+  if eqN x y
+    then (x , n) :: s
+    else ((y , m) :: (s [ x :== n ]))
+
+equals : State -> Nat -> Nat -> Set
+equals nil _ _ = Empty
+equals ((x , n) :: s) y m =
+  if eqN x y
+    then
+      if eqN n m
+        then Unit
+        else Empty
+    else equals s y m
+
+unequals : State -> Nat -> Nat -> Set
+unequals nil _ _ = Unit
+unequals ((x , n) :: s) y m =
+  if eqN x y
+    then
+      if eqN n m
+        then Empty
+        else Unit
+    else unequals s y m
+
+defined : State -> Nat -> Set
+defined nil y = Empty
+defined ((x , n) :: s) y =
+  if eqN x y
+    then Unit
+    else defined s y
+
+undefined : State -> Nat -> Set
+undefined nil y = Unit
+undefined ((x , n) :: s) y =
+  if eqN x y
+    then Empty
+    else undefined s y
+
+data Syntax : Set where
+  skip : Syntax
+  _:=_ : Nat -> Nat -> Syntax
+  conc : Syntax -> Syntax -> Syntax
+  while_==_do_ : Nat -> Nat -> Syntax -> Syntax