diff --git a/python/dex/interop/jax/apply.py b/python/dex/interop/jax/apply.py
index a698b1969..149c99240 100644
--- a/python/dex/interop/jax/apply.py
+++ b/python/dex/interop/jax/apply.py
@@ -202,7 +202,7 @@ def dex_call_batched(batched_args, batched_dims, func_atom):
   batched_dims_it = iter(batched_dims)
   for binder in native.argument_signature:
     if binder.implicit:
-      batched_args.append("{" + binder.name + "}")
+      batched_args.append("(given (" + binder.name + "))")
     else:
       ty = binder.type.dex_annotation()
       if next(batched_dims_it) is not batching.not_mapped:
@@ -304,7 +304,7 @@ def dex_call_jvp(arg_values, arg_tangents, func_atom):
   # ```
   # \ (given (n1)) (given (n2)) (given (n3)) (p1:ty1) (p2:ty2) (p3:ty3) (t1:ty1) (t2:ty2) (t3:ty3).
   #   linearized = linearize(func_uncurried, (p1, p2, p3))
-  #   snd(linearized((t1, t2, t3)))
+  #   snd(linearized)((t1, t2, t3))
   # ```
   evaluate_linearized = module.eval(
       f"\\ {juxt_string(implicit_args)} {juxt_arg_string(primals, name_to_ty)} {juxt_arg_string(tangents, name_to_ty)}." +
@@ -366,8 +366,8 @@ def dex_call_evaluate_linearized_transpose(cotangents, *args, func_atom):
   # Concretely, if `f` has three primal arguments, `func_atom` should look like:
   # ```
   # \ (given (n0)) (given (n1)) (given (n2)) x0 x1 x2 t0 t1 t2.
-  #   intermediate_linearized = linearize f (x0, x1, x2)
-  #   snd intermediate_linearized (t0, t1, t2)
+  #   intermediate_linearized = linearize(f, (x0, x1, x2))
+  #   snd(intermediate_linearized)((t0, t1, t2))
   # ```
   # In particular, its explicit arguments are assumed to be
   # `num_primals` primal inputs, followed by `num_primals` tangent
@@ -472,12 +472,14 @@ def dex_call_evaluate_linearized_transpose(cotangents, *args, func_atom):
   linearized_inputs = juxt_string(primal_params + tangent_params)
 
   # \ (given (n0)) (given (n1)) (given (n2)) x0 x1 x2 t1 ct.
-  #   transpose_linear (\(t0, t2). linearized x0 x1 x2 t0 t1 t2) ct
+  #   transpose_linear(\<fresh name>.
+  #     (t0, t2) = <fresh name>
+  #     linearized x0 x1 x2 t0 t1 t2)(ct)
   transposed = module.eval(
-      f"\\ {transposed_atom_params}. transpose_linear" +
-      f"(\\ {linear_lambda_param}." +
-      f"\n   {tuple_unpack_string(tangent_name, tangent_inputs)}" +
-      f"\n   {linearized_name} {linearized_inputs})(ct)\n"
+      f"\\ {transposed_atom_params}."
+      f"\n  transpose_linear(\\ {linear_lambda_param}." +
+      f"\n    {tuple_unpack_string(tangent_name, tangent_inputs)}" +
+      f"\n    {linearized_name} {linearized_inputs})(ct)\n"
   )
 
   # Tuple of cotangents relating to linear tangent inputs. In the given
diff --git a/python/tests/jax_test.py b/python/tests/jax_test.py
index c838dd32a..7081505ac 100644
--- a/python/tests/jax_test.py
+++ b/python/tests/jax_test.py
@@ -226,7 +226,7 @@ def sqr(x:(Fin n => Float)) -> Fin n => Float given (n:Nat) =
 
   def test_dex_not_knowing_shape_vmap(self):
     m = dex.Module(dedent("""
-    def sqr(x:(Fin n => Float)) : Fin n => Float given (n:Nat) =
+    def sqr(x:(Fin n => Float)) -> Fin n => Float given (n:Nat) =
       for i. x[i] * x[i]
     """))
     dex_sqr = primitive(m.sqr)
@@ -237,7 +237,7 @@ def sqr(x:(Fin n => Float)) : Fin n => Float given (n:Nat) =
 
   def test_dex_not_knowing_shape_jvp(self):
     m = dex.Module(dedent("""
-    def sqr(x:(Fin n => Float)) : Fin n => Float given (n:Nat) =
+    def sqr(x:(Fin n => Float)) -> Fin n => Float given (n:Nat) =
       for i. x[i] * x[i]
     """))
     dex_sqr = primitive(m.sqr)
@@ -338,7 +338,7 @@ def f_jax(x, y):
 
   def test_interleave_implicit_args_vjp(self):
     f_dex = primitive(dex.eval(
-        r'\(given (n:Nat)) x:((Fin n) => Float) {m} y:((Fin n) => (Fin m) => Float). '
+        r'\(given (n:Nat)) x:((Fin n) => Float) (given (m)) y:((Fin n) => (Fin m) => Float). '
         'for i. x[i] * x[i] + 2.0 * sum(y[i])'))
 
     def f_with_dex(x, y):
diff --git a/src/lib/Inference.hs b/src/lib/Inference.hs
index dc0680a36..e6f8e6774 100644
--- a/src/lib/Inference.hs
+++ b/src/lib/Inference.hs
@@ -476,7 +476,7 @@ instance InfBuilder (InfererM i) where
                 Abs infFrag' $ Abs b' result
             HoistFailure vs -> do
               throw EscapedNameErr $ (pprint vs)
-                ++ "\nFailed to exchage binders in buildAbsInf"
+                ++ "\nFailed to exchange binders in buildAbsInf"
                 ++ "\n" ++ pprint infFrag
     Abs b e <- return ab
     ty' <- zonk ty
diff --git a/tests/uexpr-tests.dx b/tests/uexpr-tests.dx
index 52a2f4f1e..c32378219 100644
--- a/tests/uexpr-tests.dx
+++ b/tests/uexpr-tests.dx
@@ -200,7 +200,7 @@ def passthrough(f:(a)->{|eff} b, x:a) -> {|eff} b
   f : (aa:Type, aa) -> aa = \bb x. myId x
   f Int 1
 > Leaked local variables:[bb]
-> Failed to exchage binders in buildAbsInf
+> Failed to exchange binders in buildAbsInf
 > Pending emissions: 
 > Defaults:  
 > Solver substitution: [(_.3, bb)]