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diff --git a/tests/lean/Demo/Properties.lean b/tests/lean/Demo/Properties.lean
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+import Base
+import Demo.Demo
+open Primitives
+open Result
+
+namespace demo
+
+#check U32.add_spec
+
+-- @[pspec]
+theorem mul2_add1_spec (x : U32) (h : 2 * ↑x + 1 ≤ U32.max)
+  : ∃ y, mul2_add1 x = ret y ∧
+  ↑y = 2 * ↑x + (1 : Int)
+  := by
+  rw [mul2_add1]
+  progress with U32.add_spec as ⟨ i ⟩
+  progress as ⟨ i' ⟩
+  simp; scalar_tac
+
+theorem use_mul2_add1_spec (x : U32) (y : U32) (h : 2 * ↑x + 1 + ↑y ≤ U32.max) :
+  ∃ z, use_mul2_add1 x y = ret z ∧
+  ↑z = 2 * ↑x + (1 : Int) + ↑y := by
+  rw [use_mul2_add1]
+  progress with mul2_add1_spec as ⟨ i ⟩
+  progress as ⟨ i' ⟩
+  simp; scalar_tac
+
+open CList
+
+@[simp] def CList.to_list {α : Type} (x : CList α) : List α :=
+  match x with
+  | CNil => []
+  | CCons hd tl => hd :: tl.to_list
+
+theorem list_nth_spec {T : Type} [Inhabited T] (l : CList T) (i : U32)
+  (h : ↑i < l.to_list.len) :
+  ∃ x, list_nth T l i = ret x ∧
+  x = l.to_list.index ↑i
+  := by
+  rw [list_nth]
+  match l with
+  | CNil =>
+    simp_all; scalar_tac
+  | CCons hd tl =>
+    simp_all
+    if hi: i = 0#u32 then
+      simp_all
+    else
+      simp_all
+      progress as ⟨ i1 ⟩
+      progress as ⟨ x ⟩
+      simp_all
+
+theorem i32_id_spec (x : I32) (h : 0 ≤ x.val) :
+  ∃ y, i32_id x = ret y ∧ x.val = y.val := by
+  rw [i32_id]
+  if hx : x = 0#i32 then
+    simp_all
+  else
+    simp_all
+    progress as ⟨ x1 ⟩
+    progress as ⟨ x2 ⟩
+    progress
+    simp_all
+termination_by x.val.toNat
+decreasing_by scalar_decr_tac
+
+end demo