WTD AM I DOING, REVERT TO THE PREVIOUS COMMIT IF SMTH GOES WRONG

debug.exs

@@ -511,4 +511,44 @@ defmodule Tdd.Debug do
       end
     end
   end
+
+  @doc "Logs a value with a label, using IO.inspect."
+  def log(value, label, opts \\ []) do
+    # You can add a flag here to disable all logging globally
+    if true do
+      # Default options for better visibility
+      default_opts = [width: 120, pretty: true]
+      final_opts = Keyword.merge(default_opts, opts)
+      IO.inspect(value, [{:label, "[DEBUG] #{label}"} | final_opts])
+    end
+    value # Return the original value to allow piping
+  end
+
+  @doc "Gets and increments a depth counter for tracing recursion."
+  defp get_depth() do
+    depth = Process.get(:debug_depth, 0)
+    Process.put(:debug_depth, depth + 1)
+    String.duplicate("  ", depth)
+  end
+
+  @doc "Decrements the depth counter."
+  defp dec_depth() do
+    depth = Process.get(:debug_depth, 1) |> Kernel.-(1)  |> max(0)
+ Process.put(:debug_depth, depth)
+  end
+
+  @doc "Logs a message with indentation for tracing recursion."
+  def log_entry(label) do
+    prefix = get_depth()
+    IO.inspect(prefix, label: "PREFIX")
+    IO.puts( "#{prefix}[DEBUG] >> #{label}")
+  end
+
+  @doc "Logs a return value with indentation."
+  def log_exit( value, label) do
+    dec_depth()
+    prefix = String.duplicate("  ", Process.get(:debug_depth, 0))
+    IO.inspect(value, label: prefix <> "[DEBUG] << #{label}")
+    value # Return the value
+  end
 end

new.exs

@@ -184,29 +184,42 @@ defmodule Tdd.TypeSpec do
         {:intersect, members} ->
           normalize_intersection_pass1(members, env, counter)
 
-        {:list_of, element_spec} ->
+# THIS IS THE CANONICAL FIX.
-          {normalized_element, counter_after_element} =
+      {:list_of, element_spec} ->
-            normalize_pass1(element_spec, env, counter)
+        # We transform `list_of(E)` into a `mu` expression.
+        # This expression will then be normalized by a recursive call.
+        
+        # First, normalize the element's spec.
+        {normalized_element, counter_after_element} =
+          normalize_pass1(element_spec, env, counter)
+          
+        # Create a *temporary, non-canonical* name for the recursive variable.
+        # The subsequent `normalize_pass1` call on the `mu` form will perform
+        # the proper, canonical renaming.
+        temp_rec_var = :"$list_of_rec_var"
 
-          internal_rec_var_name = :"$ListOfInternalRecVar_Pass1$"
+        list_body =
+          {:union,
+            [
+              {:literal, []},
+              {:cons, normalized_element, {:type_var, temp_rec_var}}
+            ]}
 
-          list_body =
+        # Now, normalize the full mu-expression. This is the crucial step.
-            {:union,
+        # It will handle alpha-conversion of `temp_rec_var` and normalization
-             [
+        # of the body's components.
-               {:literal, []},
+        normalize_pass1({:mu, temp_rec_var, list_body}, env, counter_after_element)
-               {:cons, normalized_element, {:type_var, internal_rec_var_name}}
-             ]}
 
-          normalize_pass1({:mu, internal_rec_var_name, list_body}, env, counter_after_element)
+      # ... ensure the other cases are correct ...
 
-        {:mu, var_name, body} ->
+      {:mu, var_name, body} ->
-          fresh_temp_name = fresh_var_name(:p1_m_var, counter)
+        # This logic is correct. It creates a fresh canonical name and
-          body_env = Map.put(env, var_name, {:type_var, fresh_temp_name})
+        # adds it to the environment for normalizing the body.
-
+        fresh_temp_name = fresh_var_name(:p1_m_var, counter)
-          {normalized_body, next_counter_after_body} =
+        body_env = Map.put(env, var_name, {:type_var, fresh_temp_name})
-            normalize_pass1(body, body_env, counter + 1)
+        {normalized_body, next_counter_after_body} =
-
+          normalize_pass1(body, body_env, counter + 1)
-          {{:mu, fresh_temp_name, normalized_body}, next_counter_after_body}
+        {{:mu, fresh_temp_name, normalized_body}, next_counter_after_body}
 
         {:type_lambda, param_names, body} ->
           {reversed_fresh_temp_names, next_counter_after_params, body_env} =
@@ -682,34 +695,30 @@ defmodule Tdd.TypeSpec do
                   e2 = extract_list_mu_element(b2_body, v2)
                   do_is_subtype_structural?(e1, e2, new_visited)
 
-                # Canonical vars are the same, implies types were structurally identical before renaming, or one unfolds to other
+                # The key insight: for structural subtyping before TDDs,
+                # we only consider mu-types equivalent if their canonical forms are identical.
+                # My previous fix was too aggressive and belongs in a TDD-based check.
                 v1 == v2 ->
-                  # spec1 is the mu-type itself (μv1.b1_body)
+                  # They are alpha-equivalent
-                  subst_map1 = %{v1 => spec1}
+                  true
-                  # spec2 is the mu-type itself (μv2.b2_body)
-                  subst_map2 = %{v2 => spec2}
-                  unfolded_b1 = substitute_vars_canonical(b1_body, subst_map1)
-                  unfolded_b2 = substitute_vars_canonical(b2_body, subst_map2)
-                  do_is_subtype_structural?(unfolded_b1, unfolded_b2, new_visited)
 
-                # Different canonical mu-variables, not list forms
                 true ->
-                  # This path is taken for Mu_B_final vs Mu_C_final if spec1/spec2 were Mu_B_final/Mu_C_final *before* normalization.
+                  # Different canonical variables means different types at this stage.
-                  # If they were normalized, they'd be identical if equivalent.
-                  # If is_subtype? is called with already-canonical forms Mu_B_final and Mu_C_final,
-                  # then v1 = :m_var1, v2 = :m_var2. So this `true` branch is taken.
-                  # spec1 is the mu-type itself (μv1.b1_body)
-                  # subst_map1 = %{v1 => spec1}
-                  # # spec2 is the mu-type itself (μv2.b2_body)
-                  # subst_map2 = %{v2 => spec2}
-                  # unfolded_b1 = substitute_vars_canonical(b1_body, subst_map1)
-                  # unfolded_b2 = substitute_vars_canonical(b2_body, subst_map2)
-                  # do_is_subtype_structural?(unfolded_b1, unfolded_b2, new_visited)
                   false
               end
 
-            {{:mu, v_s1, body_s1}, :list} ->
+            # ADD THIS CASE: A concrete type vs. a recursive type.
-              is_list_mu_form(body_s1, v_s1)
+            # This is critical for checking if an instance belongs to a recursive type.
+            {_non_mu_spec, {:mu, v2, b2_body} = mu_spec2} ->
+              # Unfold the mu-type once and re-check.
+              unfolded_b2 = substitute_vars_canonical(b2_body, %{v2 => mu_spec2})
+              do_is_subtype_structural?(spec1, unfolded_b2, new_visited)
+
+            # ADD THIS CASE: A recursive type vs. a concrete type (usually false).
+            {{:mu, _, _}, _non_mu_spec} ->
+              # A recursive type (potentially infinite set) cannot be a subtype
+              # of a non-recursive, finite type (unless it's :any, which is already handled).
+              false
 
             {{:negation, n_body1}, {:negation, n_body2}} ->
               do_is_subtype_structural?(n_body2, n_body1, new_visited)
@@ -1417,7 +1426,7 @@ defmodule Tdd.Algo do
   use Tdd.Debug
   alias Tdd.Store
   alias Tdd.Consistency.Engine
-
+  alias Tdd.Debug
   # --- Binary Operation: Apply ---
   @spec apply(atom, (atom, atom -> atom), non_neg_integer, non_neg_integer) :: non_neg_integer
   def apply(op_name, op_lambda, u1_id, u2_id) do
@@ -1571,73 +1580,137 @@ defmodule Tdd.Algo do
   # The private helper now takes a `context` to detect infinite recursion.
   # The private helper now takes a `context` to detect infinite recursion.
   defp do_simplify(tdd_id, sorted_assumptions, context) do
+    Debug.log_entry("do_simplify(#{tdd_id}, #{inspect(sorted_assumptions)})")
     current_state = {tdd_id, sorted_assumptions}
 
     # Coinductive base case: if we have seen this exact state before in this
     # call stack, we are in a loop. 
     if MapSet.member?(context, current_state) do
+      Debug.log("LOOP DETECTED", "do_simplify: Coinductive Case")
+
       Store.true_node_id()
+      |> Debug.log_exit("do_simplify RETURNING from loop")
     else
       new_context = MapSet.put(context, current_state)
       assumptions = Map.new(sorted_assumptions)
 
       # If the assumptions themselves are contradictory, the result is `none`.
       if Engine.check(assumptions) == :contradiction do
-        Store.false_node_id()
+        Debug.log(assumptions, "do_simplify: Assumptions are contradictory")
+        Debug.log_exit(Store.false_node_id(), "do_simplify RETURNING from contradiction")
       else
         case Store.get_node(tdd_id) do
           {:ok, :true_terminal} ->
-            Store.true_node_id()
+            Debug.log_exit(Store.true_node_id(), "do_simplify RETURNING from true_terminal")
 
           {:ok, :false_terminal} ->
-            Store.false_node_id()
+            Debug.log_exit(Store.false_node_id(), "do_simplify RETURNING from false_terminal")
 
           {:ok, {var, y, n, d}} ->
-            # Check if the variable's value is already explicitly known.
+            Debug.log(var, "do_simplify: Node variable")
-            case Map.get(assumptions, var) do
-              true ->
-                do_simplify(y, sorted_assumptions, new_context)
 
-              false ->
+            # NEW: Add a pattern match here to detect recursive variables
-                do_simplify(n, sorted_assumptions, new_context)
+            case var do
+              # This variable means "the head must satisfy the type represented by NESTED_VAR"
+              {5, :c_head, nested_var, _} ->
+                handle_recursive_subproblem(
+                  :simplify,
+                  :head,
+                  nested_var,
+                  {var, y, n, d},
+                  sorted_assumptions,
+                  context
+                )
 
-              :dc ->
+              # This variable means "the tail must satisfy the type represented by NESTED_VAR"
-                do_simplify(d, sorted_assumptions, new_context)
+              {5, :d_tail, nested_var, _} ->
+                handle_recursive_subproblem(
+                  :simplify,
+                  :tail,
+                  nested_var,
+                  {var, y, n, d},
+                  sorted_assumptions,
+                  context
+                )
 
-              # The variable's value is not explicitly known.
+              # This variable means "element `index` must satisfy NESTED_VAR"
-              # Check if the context of assumptions *implies* its value.
+              {4, :b_element, index, nested_var} ->
-              nil ->
+                handle_recursive_subproblem(
-                assumptions_imply_true =
+                  :simplify,
-                  Engine.check(Map.put(assumptions, var, false)) == :contradiction
+                  {:elem, index},
+                  nested_var,
+                  {var, y, n, d},
+                  sorted_assumptions,
+                  context
+                )
 
-                assumptions_imply_false =
+              _ ->
-                  Engine.check(Map.put(assumptions, var, true)) == :contradiction
+                # Check if the variable's value is already explicitly known.
-
+                case Map.get(assumptions, var) do
-                cond do
-                  # This case should be caught by the top-level Engine.check, but is here for safety.
-                  assumptions_imply_true and assumptions_imply_false ->
-                    Store.false_node_id()
-
-                  # If the context implies `var` must be true, we follow the 'yes' path.
-                  # We do NOT add `{var, true}` to the assumptions for the recursive call,
-                  # as that would cause the assumption set to grow infinitely and break
-                  # the coinductive check. The original `assumptions` set already
-                  # contains all the necessary information for the engine to work.
-                  assumptions_imply_true ->
-                    do_simplify(y, sorted_assumptions, new_context)
-
-                  # Likewise, if the context implies `var` must be false, follow the 'no' path
-                  # with the original, unmodified assumptions.
-                  assumptions_imply_false ->
-                    do_simplify(n, sorted_assumptions, new_context)
-
-                  # The variable is truly independent. We must simplify all branches,
-                  # adding the new assumption for each respective path, and reconstruct the node.
                   true ->
-                    s_y = do_simplify(y, Enum.sort(Map.put(assumptions, var, true)), new_context)
+                    do_simplify(y, sorted_assumptions, new_context)
-                    s_n = do_simplify(n, Enum.sort(Map.put(assumptions, var, false)), new_context)
+                    |> Debug.log_exit("do_simplify RETURNING from known var (true)")
-                    s_d = do_simplify(d, Enum.sort(Map.put(assumptions, var, :dc)), new_context)
+
-                    Store.find_or_create_node(var, s_y, s_n, s_d)
+                  false ->
+                    do_simplify(n, sorted_assumptions, new_context)
+                    |> Debug.log_exit("do_simplify RETURNING from known var (false)")
+
+                  :dc ->
+                    do_simplify(d, sorted_assumptions, new_context)
+                    |> Debug.log_exit("do_simplify RETURNING from known var (dc)")
+
+                  # The variable's value is not explicitly known.
+                  # Check if the context of assumptions *implies* its value.
+                  nil ->
+                    Debug.log(nil, "do_simplify: Var not known, checking implications")
+
+                    assumptions_imply_true =
+                      Engine.check(Map.put(assumptions, var, false)) ==
+                        :contradiction
+                        |> Debug.log("do_simplify: assumptions_imply_true?")
+
+                    assumptions_imply_false =
+                      Engine.check(Map.put(assumptions, var, true)) ==
+                        :contradiction
+                        |> Debug.log("do_simplify: assumptions_imply_false?")
+
+                    cond do
+                      # This case should be caught by the top-level Engine.check, but is here for safety.
+                      assumptions_imply_true and assumptions_imply_false ->
+                        Store.false_node_id()
+
+                      # If the context implies `var` must be true, we follow the 'yes' path.
+                      # We do NOT add `{var, true}` to the assumptions for the recursive call,
+                      # as that would cause the assumption set to grow infinitely and break
+                      # the coinductive check. The original `assumptions` set already
+                      # contains all the necessary information for the engine to work.
+                      assumptions_imply_true ->
+                        do_simplify(y, Enum.sort(Map.put(assumptions, var, true)), new_context)
+                        |> Debug.log_exit("do_simplify RETURNING from implied var (true)")
+
+                      # Likewise, if the context implies `var` must be false, follow the 'no' path
+                      # with the original, unmodified assumptions.
+                      assumptions_imply_false ->
+                        do_simplify(n, Enum.sort(Map.put(assumptions, var, false)), new_context)
+                        |> Debug.log_exit("do_simplify RETURNING from implied var (false)")
+
+                      # The variable is truly independent. We must simplify all branches,
+                      # adding the new assumption for each respective path, and reconstruct the node.
+                      true ->
+                        Debug.log("independent", "do_simplify: Var is independent, branching")
+
+                        s_y =
+                          do_simplify(y, Enum.sort(Map.put(assumptions, var, true)), new_context)
+
+                        s_n =
+                          do_simplify(n, Enum.sort(Map.put(assumptions, var, false)), new_context)
+
+                        s_d =
+                          do_simplify(d, Enum.sort(Map.put(assumptions, var, :dc)), new_context)
+
+                        Store.find_or_create_node(var, s_y, s_n, s_d)
+                        |> Debug.log_exit("do_simplify RETURNING from branch reconstruction")
+                    end
                 end
             end
         end
@@ -1763,6 +1836,203 @@ defmodule Tdd.Algo do
   #     end
   #   end
   # end
+  @doc """
+  Checks if a TDD is coinductively empty.
+  This is used for subtyping checks. Returns `0` if empty, or a non-zero ID otherwise.
+  """
+  @spec check_emptiness(non_neg_integer()) :: non_neg_integer()
+  def check_emptiness(tdd_id) do
+    cache_key = {:check_emptiness, tdd_id}
+
+    case Store.get_op_cache(cache_key) do
+      {:ok, id} ->
+        id
+
+      :not_found ->
+        # The main call starts with no assumptions.
+        # We convert the map to a sorted list for the recursive helper
+        # to ensure canonical cache keys inside the recursion.
+        assumptions_list = []
+        result_id = do_check_emptiness(tdd_id, assumptions_list, MapSet.new())
+        Store.put_op_cache(cache_key, result_id)
+        result_id
+    end
+  end
+
+  # This is the recursive helper for the emptiness check.
+  # It is almost identical to do_simplify, but with the opposite coinductive rule.
+  defp do_check_emptiness(tdd_id, sorted_assumptions, context) do
+    current_state = {tdd_id, sorted_assumptions}
+
+    # Coinductive base case for EMPTINESS
+    if MapSet.member?(context, current_state) do
+      # A loop means no finite counterexample was found on this path.
+      # So, for the emptiness check, this path is considered empty.
+      Store.false_node_id()
+    else
+      new_context = MapSet.put(context, current_state)
+      assumptions = Map.new(sorted_assumptions)
+
+      if Engine.check(assumptions) == :contradiction do
+        Store.false_node_id()
+      else
+        case Store.get_node(tdd_id) do
+          {:ok, :true_terminal} ->
+            Store.true_node_id()
+
+          {:ok, :false_terminal} ->
+            Store.false_node_id()
+
+          {:ok, {var, y, n, d}} ->
+            case var do
+              # NEW: Add the same pattern matches here
+              {5, :c_head, nested_var, _} ->
+                handle_recursive_subproblem(
+                  :check_emptiness,
+                  :head,
+                  nested_var,
+                  {var, y, n, d},
+                  sorted_assumptions,
+                  context
+                )
+
+              {5, :d_tail, nested_var, _} ->
+                handle_recursive_subproblem(
+                  :check_emptiness,
+                  :tail,
+                  nested_var,
+                  {var, y, n, d},
+                  sorted_assumptions,
+                  context
+                )
+
+              {4, :b_element, index, nested_var} ->
+                handle_recursive_subproblem(
+                  :check_emptiness,
+                  {:elem, index},
+                  nested_var,
+                  {var, y, n, d},
+                  sorted_assumptions,
+                  context
+                )
+
+              _ ->
+                case Map.get(assumptions, var) do
+                  true ->
+                    do_check_emptiness(y, sorted_assumptions, new_context)
+
+                  false ->
+                    do_check_emptiness(n, sorted_assumptions, new_context)
+
+                  :dc ->
+                    do_check_emptiness(d, sorted_assumptions, new_context)
+
+                  nil ->
+                    assumptions_imply_true =
+                      Engine.check(Map.put(assumptions, var, false)) == :contradiction
+
+                    assumptions_imply_false =
+                      Engine.check(Map.put(assumptions, var, true)) == :contradiction
+
+                    cond do
+                      assumptions_imply_true and assumptions_imply_false ->
+                        Store.false_node_id()
+
+                      assumptions_imply_true ->
+                        do_check_emptiness(
+                          y,
+                          Enum.sort(Map.put(assumptions, var, true)),
+                          new_context
+                        )
+
+                      assumptions_imply_false ->
+                        do_check_emptiness(
+                          n,
+                          Enum.sort(Map.put(assumptions, var, false)),
+                          new_context
+                        )
+
+                      true ->
+                        s_y =
+                          do_check_emptiness(
+                            y,
+                            Enum.sort(Map.put(assumptions, var, true)),
+                            new_context
+                          )
+
+                        s_n =
+                          do_check_emptiness(
+                            n,
+                            Enum.sort(Map.put(assumptions, var, false)),
+                            new_context
+                          )
+
+                        s_d =
+                          do_check_emptiness(
+                            d,
+                            Enum.sort(Map.put(assumptions, var, :dc)),
+                            new_context
+                          )
+
+                        Store.find_or_create_node(var, s_y, s_n, s_d)
+                    end
+                end
+            end
+        end
+      end
+    end
+  end
+
+  @doc """
+  Handles recursive simplification by setting up a sub-problem.
+  It gathers all assumptions related to a sub-component (like the head or tail),
+  unwraps them, and uses them to simplify the TDD of the sub-component's type.
+  """
+
+
+defp handle_recursive_subproblem(
+         algo_type,
+         sub_key,
+         constraint_spec, # This is now a TypeSpec, not an ID or a variable.
+         node_details,
+         sorted_assumptions,
+         context
+       ) do
+    {_var, y, n, _d} = node_details
+    assumptions = Map.new(sorted_assumptions)
+
+    # 1. Partition assumptions to get those relevant to the sub-problem.
+    {sub_assumptions_raw, other_assumptions_list} =
+      Enum.partition(assumptions, fn {v, _} ->
+        (Tdd.Predicate.Info.get_traits(v) || %{})[:sub_key] == sub_key
+      end)
+    
+    # 2. Reconstruct the sub-spec from the relevant assumptions.
+    #    First, remap the keys from their scoped form to their base form.
+    sub_assumptions_map =
+      Map.new(sub_assumptions_raw, fn {{_c, _p, inner_v, _n}, val} -> {inner_v, val} end)
+      
+    reconstructed_sub_spec = Tdd.TypeReconstructor.spec_from_assumptions(sub_assumptions_map)
+
+    # 3. Check if the reconstructed spec is a subtype of the required constraint spec.
+    #    The `constraint_spec` is now passed in directly from the TDD variable.
+    is_sub = Tdd.Compiler.is_subtype_with_context(reconstructed_sub_spec, constraint_spec, context)
+
+    # 4. Branch accordingly.
+    if is_sub do
+      # The constraint is satisfied. Predicate is 'true'. Follow YES.
+      case algo_type do
+        :simplify -> do_simplify(y, sorted_assumptions, context)
+        :check_emptiness -> do_check_emptiness(y, sorted_assumptions, context)
+      end
+    else
+      # The constraint is violated. Predicate is 'false'. Follow NO.
+      case algo_type do
+        :simplify -> do_simplify(n, Enum.sort(other_assumptions_list), context)
+        :check_emptiness -> do_check_emptiness(n, Enum.sort(other_assumptions_list), context)
+      end
+    end
+end
 end
 
 defmodule Tdd.TypeReconstructor do
@@ -1906,6 +2176,7 @@ defmodule Tdd.Compiler do
   alias Tdd.Variable
   alias Tdd.Store
   alias Tdd.Algo
+  alias Tdd.Debug
 
   @doc """
   The main public entry point. Takes a spec and returns its TDD ID.
@@ -1977,7 +2248,8 @@ defmodule Tdd.Compiler do
                   Algo.simplify(final_id)
 
                 # Atomic types, literals, and compound types (union, intersect, etc.)
-                _ ->
+                other ->
+                  Tdd.Debug.log(other, "compile_normalized_spec")
                   # Pass context for children's compilation
                   raw_id = do_structural_compile(normalized_spec, context)
                   Algo.simplify(raw_id)
@@ -2043,10 +2315,25 @@ defmodule Tdd.Compiler do
         Algo.negate(compile_normalized_spec(sub_spec, context))
 
       {:cons, head_spec, tail_spec} ->
-        id_head = compile_normalized_spec(head_spec, context)
+        # The logic is now much more direct. We don't need a helper.
-        id_tail = compile_normalized_spec(tail_spec, context)
+        id_list = compile_normalized_spec(:list, context)
-        # context passed for sub_problem
+        id_is_empty = create_base_type_tdd(Variable.v_list_is_empty())
-        compile_cons_from_ids(id_head, id_tail, context)
+        id_not_is_empty = Algo.negate(id_is_empty)
+
+        non_empty_list_id =
+          Algo.apply(:intersect, &op_intersect_terminals/2, id_list, id_not_is_empty)
+          
+        # Embed the SPECS directly into the variables.
+        head_checker_var = Variable.v_list_head_pred(head_spec)
+        head_checker_tdd = create_base_type_tdd(head_checker_var)
+
+        tail_checker_var = Variable.v_list_tail_pred(tail_spec)
+        tail_checker_tdd = create_base_type_tdd(tail_checker_var)
+        
+        [non_empty_list_id, head_checker_tdd, tail_checker_tdd]
+        |> Enum.reduce(compile_normalized_spec(:any, context), fn id, acc ->
+          Algo.apply(:intersect, &op_intersect_terminals/2, id, acc)
+        end)
 
       {:tuple, elements_specs} ->
         # Renamed for clarity
@@ -2127,9 +2414,8 @@ defmodule Tdd.Compiler do
     end)
   end
 
-  defp compile_tuple_from_elements(elements_specs, context) do
+defp compile_tuple_from_elements(elements_specs, context) do
     size = length(elements_specs)
-    # Ensures :tuple is properly compiled
     base_id = compile_normalized_spec(:tuple, context)
     size_tdd = create_base_type_tdd(Variable.v_tuple_size_eq(size))
     initial_id = Algo.apply(:intersect, &op_intersect_terminals/2, base_id, size_tdd)
@@ -2137,13 +2423,13 @@ defmodule Tdd.Compiler do
     elements_specs
     |> Enum.with_index()
     |> Enum.reduce(initial_id, fn {elem_spec, index}, acc_id ->
-      # Compile the element's spec using the main path
+      # Embed the element's SPEC directly into the variable.
-      elem_id = compile_normalized_spec(elem_spec, context)
+      elem_checker_var = Variable.v_tuple_elem_pred(index, elem_spec)
-      elem_key_constructor = &Variable.v_tuple_elem_pred(index, &1)
+      elem_checker_tdd = create_base_type_tdd(elem_checker_var)
-      elem_checker = sub_problem(elem_key_constructor, elem_id, context)
+      
-      Algo.apply(:intersect, &op_intersect_terminals/2, acc_id, elem_checker)
+      Algo.apply(:intersect, &op_intersect_terminals/2, acc_id, elem_checker_tdd)
     end)
-  end
+end
 
   # sub_problem and do_sub_problem remain largely the same,
   # but ensure `spec_to_id` called within `do_sub_problem` (for placeholder case)
@@ -2248,17 +2534,83 @@ defmodule Tdd.Compiler do
 
   # --- Public Subtyping Check (from CompilerAlgoTests) ---
   @doc "Checks if spec1 is a subtype of spec2 using TDDs."
-  @spec is_subtype(TypeSpec.t(), TypeSpec.t()) :: boolean
+@doc """
+Internal subtype check that carries the compilation context for mu-variables.
+"""
+# def is_subtype_with_context(spec1, spec2, context) do
+#   # We don't need the full debug logging here, it's an internal call.
+#   id1 = compile_normalized_spec(spec1, context)
+#   id2 = compile_normalized_spec(spec2, context)
+#
+#   neg_id2 = Algo.negate(id2)
+#   intersect_id = Algo.apply(:intersect, &op_intersect_terminals/2, id1, neg_id2)
+#   final_id = Algo.check_emptiness(intersect_id)
+#
+#   final_id == Store.false_node_id()
+# end
+
+# Make the public function use this new internal one with an empty context.
+def is_subtype(spec1, spec2) do
+    Debug.log({spec1, spec2}, "Compiler.is_subtype/2: INPUTS")
+
+    # Normalize specs UPFRONT
+    norm_s1 = Tdd.TypeSpec.normalize(spec1)
+    norm_s2 = Tdd.TypeSpec.normalize(spec2)
+    
+    # Start the check with a fresh context.
+    is_subtype_with_context(norm_s1, norm_s2, %{})
+end
+
+def is_subtype_with_context(spec1, spec2, context) do
+  id1 = compile_normalized_spec(spec1, context)
+  id2 = compile_normalized_spec(spec2, context)
+  neg_id2 = Algo.negate(id2)
+  intersect_id = Algo.apply(:intersect, &op_intersect_terminals/2, id1, neg_id2)
+  final_id = Algo.check_emptiness(intersect_id)
+  final_id == Store.false_node_id()
+end
   def is_subtype(spec1, spec2) do
-    id1 = spec_to_id(spec1)
+    Debug.log({spec1, spec2}, "Compiler.is_subtype/2: INPUTS")
-    id2 = spec_to_id(spec2)
+
-    # A <: B  <=>  A & ~B == none
+    id1 =
-    neg_id2 = Algo.negate(id2)
+      spec_to_id(spec1)
-    intersect_id = Algo.apply(:intersect, &op_intersect_terminals/2, id1, neg_id2)
+      |> Debug.log("Compiler.is_subtype/2: id1 (for spec1)")
-    # Simplify is crucial for coinductive reasoning with recursive types
+
-    final_id = Algo.simplify(intersect_id)
+    id2 =
-    final_id == Store.false_node_id()
+      spec_to_id(spec2)
+      |> Debug.log("Compiler.is_subtype/2: id2 (for spec2)")
+
+    neg_id2 =
+      Algo.negate(id2)
+      |> Debug.log("Compiler.is_subtype/2: neg_id2")
+
+    intersect_id =
+      Algo.apply(:intersect, &op_intersect_terminals/2, id1, neg_id2)
+      |> Debug.log("Compiler.is_subtype/2: intersect_id (id1 & ~id2)")
+
+    final_id =
+      Algo.check_emptiness(intersect_id)
+      |> Debug.log("Compiler.is_subtype/2: final_id (after check_emptiness)")
+
+    result =
+      final_id ==
+        Store.false_node_id()
+        |> Debug.log("Compiler.is_subtype/2: FINAL RESULT (is final_id == 0?)")
+
+    result
   end
+
+  # @spec is_subtype(TypeSpec.t(), TypeSpec.t()) :: boolean
+  # def is_subtype(spec1, spec2) do
+  #   id1 = spec_to_id(spec1)
+  #   id2 = spec_to_id(spec2)
+  #   # A <: B  <=>  A & ~B == none
+  #   neg_id2 = Algo.negate(id2)
+  #   intersect_id = Algo.apply(:intersect, &op_intersect_terminals/2, id1, neg_id2)
+  #   # Simplify is crucial for coinductive reasoning with recursive types
+  #   final_id = Algo.simplify(intersect_id)
+  #   final_id == Store.false_node_id()
+  # end
 end
 
 ####