Test dummy derivative

Author: YingboMa

test/structural_transformation/index_reduction.jl

@@ -131,3 +131,22 @@ let pss_pendulum = partial_state_selection(pendulum)
     # This currently selects `T` rather than `x` at top level. Needs tearing priorities to fix.
     @test_broken length(equations(pss_pendulum)) == 3
 end
+
+sys = structural_simplify(pendulum2)
+@test length(equations(sys)) == 5
+@test length(states(sys)) == 5
+
+u0 = [
+  D(x)    => 0.0,
+  D(y)    => 0.0,
+  x       => sqrt(2)/2,
+  y       => sqrt(2)/2,
+  T       => 0.0
+]
+p = [
+    L => 1.0,
+    g => 9.8
+]
+
+prob_auto = DAEProblem(sys,zeros(length(u0)),u0,(0.0,10.0),p)
+@test_nowarn solve(prob_auto, DFBDF())