diff --git a/test/Dialect/LLZK_and_Builtin/aux_contract_gaps.llzk b/test/Dialect/LLZK_and_Builtin/aux_contract_gaps.llzk
new file mode 100644
index 0000000000..b2be4756d9
--- /dev/null
+++ b/test/Dialect/LLZK_and_Builtin/aux_contract_gaps.llzk
@@ -0,0 +1,83 @@
+// RUN: llzk-opt -split-input-file --verify-each %s > /dev/null
+
+// The current verifier accepts an auxiliary with no constraining equality.
+module attributes {llzk.lang} {
+  struct.def @UnconstrainedAux {
+    function.def @compute() -> !struct.type<@UnconstrainedAux> {
+      %self = struct.new : !struct.type<@UnconstrainedAux>
+      function.return %self : !struct.type<@UnconstrainedAux>
+    }
+
+    function.def @constrain(%self: !struct.type<@UnconstrainedAux>) {
+      %aux = llzk.aux : !felt.type
+      function.return
+    }
+  }
+}
+
+// -----
+
+// The current verifier does not require a unique constraining equality.
+module attributes {llzk.lang} {
+  struct.def @MultiplyConstrainedAux {
+    function.def @compute(%a: !felt.type, %b: !felt.type) -> !struct.type<@MultiplyConstrainedAux> {
+      %self = struct.new : !struct.type<@MultiplyConstrainedAux>
+      function.return %self : !struct.type<@MultiplyConstrainedAux>
+    }
+
+    function.def @constrain(
+        %self: !struct.type<@MultiplyConstrainedAux>,
+        %a: !felt.type,
+        %b: !felt.type
+    ) {
+      %aux = llzk.aux : !felt.type
+      constrain.eq %aux, %a : !felt.type, !felt.type
+      constrain.eq %aux, %b : !felt.type, !felt.type
+      function.return
+    }
+  }
+}
+
+// -----
+
+// The current verifier accepts an auxiliary constrained on only one branch.
+module attributes {llzk.lang} {
+  struct.def @ConditionallyConstrainedAux {
+    function.def @compute(%a: !felt.type, %cond: i1) -> !struct.type<@ConditionallyConstrainedAux> {
+      %self = struct.new : !struct.type<@ConditionallyConstrainedAux>
+      function.return %self : !struct.type<@ConditionallyConstrainedAux>
+    }
+
+    function.def @constrain(
+        %self: !struct.type<@ConditionallyConstrainedAux>,
+        %a: !felt.type,
+        %cond: i1
+    ) {
+      %aux = llzk.aux : !felt.type
+      scf.if %cond {
+        constrain.eq %aux, %a : !felt.type, !felt.type
+      }
+      function.return
+    }
+  }
+}
+
+// -----
+
+// The current verifier accepts a use before the auxiliary's constraining equality.
+module attributes {llzk.lang} {
+  struct.def @UseBeforeConstraint {
+    function.def @compute(%a: !felt.type) -> !struct.type<@UseBeforeConstraint> {
+      %self = struct.new : !struct.type<@UseBeforeConstraint>
+      function.return %self : !struct.type<@UseBeforeConstraint>
+    }
+
+    function.def @constrain(%self: !struct.type<@UseBeforeConstraint>, %a: !felt.type) {
+      %aux = llzk.aux : !felt.type
+      %square = felt.mul %aux, %aux : !felt.type, !felt.type
+      constrain.eq %square, %a : !felt.type, !felt.type
+      constrain.eq %aux, %a : !felt.type, !felt.type
+      function.return
+    }
+  }
+}
diff --git a/test/Transforms/PolyLowering/poly_lowering_partial_failure_proof.llzk b/test/Transforms/PolyLowering/poly_lowering_partial_failure_proof.llzk
new file mode 100644
index 0000000000..607c7e7ede
--- /dev/null
+++ b/test/Transforms/PolyLowering/poly_lowering_partial_failure_proof.llzk
@@ -0,0 +1,33 @@
+// RUN: not llzk-opt -I %S -llzk-poly-lowering-pass="max-degree=2" -mlir-print-ir-after-failure %s 2>&1 | FileCheck %s
+
+// The pass currently rewrites an earlier equality before discovering the
+// unsupported containment operation later in the same function.
+module attributes {llzk.lang} {
+  struct.def @PartialFailure {
+    function.def @compute(
+        %a: !felt.type,
+        %b: !felt.type,
+        %needle: !felt.type
+    ) -> !struct.type<@PartialFailure> {
+      %self = struct.new : !struct.type<@PartialFailure>
+      function.return %self : !struct.type<@PartialFailure>
+    }
+
+    function.def @constrain(
+        %self: !struct.type<@PartialFailure>,
+        %a: !felt.type,
+        %b: !felt.type,
+        %needle: !felt.type
+    ) {
+      %table = array.new %a, %b : !array.type<2 x !felt.type>
+      %product = felt.mul %a, %b : !felt.type, !felt.type
+      %quartic = felt.mul %product, %product : !felt.type, !felt.type
+      constrain.eq %quartic, %a : !felt.type, !felt.type
+      constrain.in %table, %needle : !array.type<2 x !felt.type>, !felt.type
+      function.return
+    }
+  }
+}
+
+// CHECK: error: EmitContainmentOp is unsupported for now in the lowering pass
+// CHECK: struct.member @__llzk_poly_lowering_pass_aux_member_0 : !felt.type