diff --git a/xls/passes/basic_simplification_pass.cc b/xls/passes/basic_simplification_pass.cc
index 5e92e5263f..274e5dad74 100644
--- a/xls/passes/basic_simplification_pass.cc
+++ b/xls/passes/basic_simplification_pass.cc
@@ -68,6 +68,66 @@ bool IsBinaryIrreflexiveRelation(Node* node) {
   }
 }
 
+struct EqNeSelectNotOrIncMatch {
+  Node* x;
+  Op cmp;
+};
+
+// Matches either:
+//   (eq|ne)(sel(c, cases=[not(x), add(x, 1)]), 0)
+//   (eq|ne)(0, sel(c, cases=[not(x), add(x, 1)]))
+//
+// Also handles swapped select arms.
+//
+// This is exploiting "not/inc zero-equivalence": the arms `~x` and `x+1` are
+// both equal to zero under the same precondition, namely `x == all_ones`.
+absl::StatusOr<std::optional<EqNeSelectNotOrIncMatch>>
+TryMatchEqOrNeSelectNotOrIncAgainstZero(Node* n) {
+  if (n->op() != Op::kEq && n->op() != Op::kNe) {
+    return std::nullopt;
+  }
+  Node* maybe_sel = nullptr;
+  if (IsLiteralZero(n->operand(0))) {
+    maybe_sel = n->operand(1);
+  } else if (IsLiteralZero(n->operand(1))) {
+    maybe_sel = n->operand(0);
+  } else {
+    return std::nullopt;
+  }
+
+  XLS_ASSIGN_OR_RETURN(std::optional<BinarySelectView> sel,
+                       MatchBinarySelectLike(maybe_sel));
+  if (!sel.has_value()) {
+    return std::nullopt;
+  }
+
+  auto match_add_one = [](Node* node) -> Node* {
+    if (node->op() != Op::kAdd) {
+      return nullptr;
+    }
+    if (IsLiteralUnsignedOne(node->operand(0))) {
+      return node->operand(1);
+    }
+    if (IsLiteralUnsignedOne(node->operand(1))) {
+      return node->operand(0);
+    }
+    return nullptr;
+  };
+
+  // on_false = not(x), on_true = add(x, 1)
+  if (Node* x = match_add_one(sel->on_true);
+      x != nullptr && IsNotOf(sel->on_false, x)) {
+    return EqNeSelectNotOrIncMatch{.x = x, .cmp = n->op()};
+  }
+  // swapped arms
+  if (Node* x = match_add_one(sel->on_false);
+      x != nullptr && IsNotOf(sel->on_true, x)) {
+    return EqNeSelectNotOrIncMatch{.x = x, .cmp = n->op()};
+  }
+
+  return std::nullopt;
+}
+
 // MatchPatterns matches simple tree patterns to find opportunities
 // for simplification.
 //
@@ -94,6 +154,32 @@ absl::StatusOr<bool> MatchPatterns(Node* n) {
     return true;
   }
 
+  // Pattern:
+  //   `(eq|ne)(sel(c, cases=[not(x), add(x, 1)]), 0)` =>
+  //   `(eq|ne)(x, all_ones(width(x)))`
+  //
+  // Why this works (not/inc zero-equivalence):
+  // - `(~x == 0)`  <=>  `(x == all_ones)`
+  // - `(x+1 == 0)` <=>  `(x == all_ones)`   (note: given wraparound arithmetic)
+  // Therefore for `y = sel(c, [~x, x+1])`, `(y != 0)` is exactly `(x !=
+  // all_ones)`, independent of selector `c`.
+  XLS_ASSIGN_OR_RETURN(std::optional<EqNeSelectNotOrIncMatch> not_or_inc_match,
+                       TryMatchEqOrNeSelectNotOrIncAgainstZero(n));
+  if (not_or_inc_match.has_value()) {
+    Node* x = not_or_inc_match->x;
+    XLS_ASSIGN_OR_RETURN(
+        Node * all_ones,
+        n->function_base()->MakeNode<Literal>(
+            n->loc(), Value(Bits::AllOnes(x->BitCountOrDie()))));
+    VLOG(2)
+        << "FOUND: not/inc zero-equivalence: "
+           "(eq|ne)(sel(c, [not(x), add(x, 1)]), 0) => (eq|ne)(x, all_ones)";
+    XLS_RETURN_IF_ERROR(
+        n->ReplaceUsesWithNew<CompareOp>(x, all_ones, not_or_inc_match->cmp)
+            .status());
+    return true;
+  }
+
   // Returns true if all operands of 'node' are the same.
   auto all_operands_same = [&query_engine](Node* node) {
     return std::all_of(node->operands().begin(), node->operands().end(),
diff --git a/xls/passes/basic_simplification_pass_test.cc b/xls/passes/basic_simplification_pass_test.cc
index 39d9130cb7..2c36365ff9 100644
--- a/xls/passes/basic_simplification_pass_test.cc
+++ b/xls/passes/basic_simplification_pass_test.cc
@@ -693,6 +693,32 @@ TEST_F(BasicSimplificationPassTest, SubOfSwappedTwoWaySelectsDoesNotSimplify) {
                      m::Select(m::Param("p"), {m::Param("b"), m::Param("a")})));
 }
 
+TEST_F(BasicSimplificationPassTest, EqNeOfSelectNotOrIncrementAgainstZero) {
+  auto p = CreatePackage();
+  FunctionBuilder fb(TestName(), p.get());
+  BValue c = fb.Param("c", p->GetBitsType(1));
+  BValue x = fb.Param("x", p->GetBitsType(32));
+  BValue nx = fb.Not(x);
+  BValue one = fb.Literal(UBits(1, 32));
+  BValue inc = fb.Add(x, one);
+  BValue s = fb.Select(c, {nx, inc});
+  BValue zero = fb.Literal(UBits(0, 32));
+  BValue eq = fb.Eq(s, zero);
+  BValue ne = fb.Ne(s, zero);
+  BValue r = fb.Tuple({eq, ne});
+  XLS_ASSERT_OK_AND_ASSIGN(Function * f, fb.BuildWithReturnValue(r));
+
+  ScopedVerifyEquivalence sve(f);
+  ASSERT_THAT(Run(p.get()), IsOkAndHolds(true));
+  EXPECT_THAT(
+      f->return_value(),
+      m::Tuple(
+          testing::AnyOf(m::Eq(m::Param("x"), m::Literal(Bits::AllOnes(32))),
+                         m::Eq(m::Literal(Bits::AllOnes(32)), m::Param("x"))),
+          testing::AnyOf(m::Ne(m::Param("x"), m::Literal(Bits::AllOnes(32))),
+                         m::Ne(m::Literal(Bits::AllOnes(32)), m::Param("x")))));
+}
+
 TEST_F(BasicSimplificationPassTest, AddWithZero) {
   auto p = CreatePackage();
   XLS_ASSERT_OK_AND_ASSIGN(Function * f, ParseFunction(R"(