IntervalAnalysis.cpp

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//===-- IntervalAnalysis.cpp - Interval analysis implementation -*- C++ -*-===//
//
// Part of the LLZK Project, under the Apache License v2.0.
// See LICENSE.txt for license information.
// Copyright 2025 Veridise Inc.
// SPDX-License-Identifier: Apache-2.0
//
//===----------------------------------------------------------------------===//
 
#include "llzk/Analysis/IntervalAnalysis.h"
#include "llzk/Util/APIntHelper.h"
#include "llzk/Util/Debug.h"
#include "llzk/Util/StreamHelper.h"
 
#include <llvm/ADT/TypeSwitch.h>
 
using namespace mlir;
 
namespace llzk {
 
using namespace array;
using namespace boolean;
using namespace component;
using namespace constrain;
using namespace felt;
 
/* Field */
 
Field::Field(std::string_view primeStr) : primeMod(llvm::APSInt(primeStr)) {
  halfPrime = (primeMod + felt(1)) / felt(2);
}
 
const Field &Field::getField(const char *fieldName) {
  static llvm::DenseMap<llvm::StringRef, Field> knownFields;
  static std::once_flag fieldsInit;
  std::call_once(fieldsInit, initKnownFields, knownFields);
 
  if (auto it = knownFields.find(fieldName); it != knownFields.end()) {
    return it->second;
  }
  llvm::report_fatal_error("field \"" + mlir::Twine(fieldName) + "\" is unsupported");
}
 
void Field::initKnownFields(llvm::DenseMap<llvm::StringRef, Field> &knownFields) {
  // bn128/254, default for circom
  knownFields.try_emplace(
      "bn128",
      Field("21888242871839275222246405745257275088696311157297823662689037894645226208583")
  );
  knownFields.try_emplace("bn254", knownFields.at("bn128"));
  // 15 * 2^27 + 1, default for zirgen
  knownFields.try_emplace("babybear", Field("2013265921"));
  // 2^64 - 2^32 + 1, used for plonky2
  knownFields.try_emplace("goldilocks", Field("18446744069414584321"));
  // 2^31 - 1, used for Plonky3
  knownFields.try_emplace("mersenne31", Field("2147483647"));
}
 
llvm::APSInt Field::reduce(llvm::APSInt i) const {
  unsigned maxBits = std::max(i.getBitWidth(), bitWidth());
  llvm::APSInt m = (i.extend(maxBits) % prime().extend(maxBits)).trunc(bitWidth());
  if (m.isNegative()) {
    return prime() + m;
  }
  return m;
}
 
llvm::APSInt Field::reduce(unsigned i) const {
  auto ap = llvm::APSInt(llvm::APInt(bitWidth(), i));
  return reduce(ap);
}
 
/* UnreducedInterval */
 
Interval UnreducedInterval::reduce(const Field &field) const {
  if (safeGt(a, b)) {
    return Interval::Empty(field);
  }
  if (safeGe(width(), field.prime())) {
    return Interval::Entire(field);
  }
  auto lhs = field.reduce(a), rhs = field.reduce(b);
  // lhs and rhs are now guaranteed to have the same bitwidth, so we can use
  // built-in functions.
  if ((rhs - lhs).isZero()) {
    return Interval::Degenerate(field, lhs);
  }
 
  const auto &half = field.half();
  if (lhs.ule(rhs)) {
    if (lhs.ult(half) && rhs.ult(half)) {
      return Interval::TypeA(field, lhs, rhs);
    } else if (lhs.ult(half)) {
      return Interval::TypeC(field, lhs, rhs);
    } else {
      return Interval::TypeB(field, lhs, rhs);
    }
  } else {
    if (lhs.uge(half) && rhs.ult(half)) {
      return Interval::TypeF(field, lhs, rhs);
    } else {
      return Interval::Entire(field);
    }
  }
}
 
UnreducedInterval UnreducedInterval::intersect(const UnreducedInterval &rhs) const {
  auto &lhs = *this;
  return UnreducedInterval(safeMax(lhs.a, rhs.a), safeMin(lhs.b, rhs.b));
}
 
UnreducedInterval UnreducedInterval::doUnion(const UnreducedInterval &rhs) const {
  auto &lhs = *this;
  return UnreducedInterval(safeMin(lhs.a, rhs.a), safeMax(lhs.b, rhs.b));
}
 
UnreducedInterval UnreducedInterval::computeLTPart(const UnreducedInterval &rhs) const {
  if (isEmpty() || rhs.isEmpty()) {
    return *this;
  }
  auto one = llvm::APSInt(llvm::APInt(a.getBitWidth(), 1));
  auto bound = expandingSub(rhs.b, one);
  return UnreducedInterval(a, safeMin(b, bound));
}
 
UnreducedInterval UnreducedInterval::computeLEPart(const UnreducedInterval &rhs) const {
  if (isEmpty() || rhs.isEmpty()) {
    return *this;
  }
  return UnreducedInterval(a, safeMin(b, rhs.b));
}
 
UnreducedInterval UnreducedInterval::computeGTPart(const UnreducedInterval &rhs) const {
  if (isEmpty() || rhs.isEmpty()) {
    return *this;
  }
  auto one = llvm::APSInt(llvm::APInt(a.getBitWidth(), 1));
  auto bound = expandingAdd(rhs.a, one);
  return UnreducedInterval(safeMax(a, bound), b);
}
 
UnreducedInterval UnreducedInterval::computeGEPart(const UnreducedInterval &rhs) const {
  if (isEmpty() || rhs.isEmpty()) {
    return *this;
  }
  return UnreducedInterval(safeMax(a, rhs.a), b);
}
 
UnreducedInterval UnreducedInterval::operator-() const { return UnreducedInterval(-b, -a); }
 
UnreducedInterval operator+(const UnreducedInterval &lhs, const UnreducedInterval &rhs) {
  llvm::APSInt low = expandingAdd(lhs.a, rhs.a), high = expandingAdd(lhs.b, rhs.b);
  return UnreducedInterval(low, high);
}
 
UnreducedInterval operator-(const UnreducedInterval &lhs, const UnreducedInterval &rhs) {
  return lhs + (-rhs);
}
 
UnreducedInterval operator*(const UnreducedInterval &lhs, const UnreducedInterval &rhs) {
  auto v1 = expandingMul(lhs.a, rhs.a);
  auto v2 = expandingMul(lhs.a, rhs.b);
  auto v3 = expandingMul(lhs.b, rhs.a);
  auto v4 = expandingMul(lhs.b, rhs.b);
 
  auto minVal = safeMin({v1, v2, v3, v4});
  auto maxVal = safeMax({v1, v2, v3, v4});
 
  return UnreducedInterval(minVal, maxVal);
}
 
bool UnreducedInterval::overlaps(const UnreducedInterval &rhs) const {
  return isNotEmpty() && rhs.isNotEmpty() && safeGe(b, rhs.a) && safeLe(a, rhs.b);
}
 
std::strong_ordering operator<=>(const UnreducedInterval &lhs, const UnreducedInterval &rhs) {
  if (safeLt(lhs.a, rhs.a) || (safeEq(lhs.a, rhs.a) && safeLt(lhs.b, rhs.b))) {
    return std::strong_ordering::less;
  }
  if (safeGt(lhs.a, rhs.a) || (safeEq(lhs.a, rhs.a) && safeGt(lhs.b, rhs.b))) {
    return std::strong_ordering::greater;
  }
  return std::strong_ordering::equal;
}
 
llvm::APSInt UnreducedInterval::width() const {
  llvm::APSInt w;
  if (safeGt(a, b)) {
    // This would be reduced to an empty Interval, so the width is just zero.
    w = llvm::APSInt::getUnsigned(0);
  } else {
    w = expandingSub(b, a)++;
  }
  ensure(safeGe(w, llvm::APSInt::getUnsigned(0)), "cannot have negative width");
  return w;
}
 
bool UnreducedInterval::isEmpty() const { return safeEq(width(), llvm::APSInt::getUnsigned(0)); }
 
/* Interval */
 
UnreducedInterval Interval::toUnreduced() const {
  if (isEmpty()) {
    return UnreducedInterval(field.get().zero(), field.get().zero());
  }
  if (isEntire()) {
    return UnreducedInterval(field.get().zero(), field.get().maxVal());
  }
  return UnreducedInterval(a, b);
}
 
UnreducedInterval Interval::firstUnreduced() const {
  if (is<Type::TypeF>()) {
    return UnreducedInterval(field.get().prime() - a, b);
  }
  return toUnreduced();
}
 
UnreducedInterval Interval::secondUnreduced() const {
  ensure(is<Type::TypeA, Type::TypeB, Type::TypeC>(), "unsupported range type");
  return UnreducedInterval(a - field.get().prime(), b - field.get().prime());
}
 
Interval Interval::join(const Interval &rhs) const {
  auto &lhs = *this;
  ensure(
      lhs.getField() == rhs.getField(), "interval operations across differing fields is unsupported"
  );
  const Field &f = lhs.getField();
 
  // Trivial cases
  if (lhs.isEntire() || rhs.isEntire()) {
    return Interval::Entire(f);
  }
  if (lhs.isEmpty()) {
    return rhs;
  }
  if (rhs.isEmpty()) {
    return lhs;
  }
  if (lhs.isDegenerate() || rhs.isDegenerate()) {
    return lhs.toUnreduced().doUnion(rhs.toUnreduced()).reduce(f);
  }
 
  // More complex cases
  if (areOneOf<
          {Type::TypeA, Type::TypeA}, {Type::TypeB, Type::TypeB}, {Type::TypeC, Type::TypeC},
          {Type::TypeA, Type::TypeC}, {Type::TypeB, Type::TypeC}>(lhs, rhs)) {
    return Interval(rhs.ty, f, std::min(lhs.a, rhs.a), std::max(lhs.b, rhs.b));
  }
  if (areOneOf<{Type::TypeA, Type::TypeB}>(lhs, rhs)) {
    auto lhsUnred = lhs.firstUnreduced();
    auto opt1 = rhs.firstUnreduced().doUnion(lhsUnred);
    auto opt2 = rhs.secondUnreduced().doUnion(lhsUnred);
    if (opt1.width() <= opt2.width()) {
      return opt1.reduce(f);
    }
    return opt2.reduce(f);
  }
  if (areOneOf<{Type::TypeF, Type::TypeF}, {Type::TypeA, Type::TypeF}>(lhs, rhs)) {
    return lhs.firstUnreduced().doUnion(rhs.firstUnreduced()).reduce(f);
  }
  if (areOneOf<{Type::TypeB, Type::TypeF}>(lhs, rhs)) {
    return lhs.secondUnreduced().doUnion(rhs.firstUnreduced()).reduce(f);
  }
  if (areOneOf<{Type::TypeC, Type::TypeF}>(lhs, rhs)) {
    return Interval::Entire(f);
  }
  if (areOneOf<
          {Type::TypeB, Type::TypeA}, {Type::TypeC, Type::TypeA}, {Type::TypeC, Type::TypeB},
          {Type::TypeF, Type::TypeA}, {Type::TypeF, Type::TypeB}, {Type::TypeF, Type::TypeC}>(
          lhs, rhs
      )) {
    return rhs.join(lhs);
  }
  llvm::report_fatal_error("unhandled join case");
  return Interval::Entire(f);
}
 
Interval Interval::intersect(const Interval &rhs) const {
  auto &lhs = *this;
  ensure(
      lhs.getField() == rhs.getField(), "interval operations across differing fields is unsupported"
  );
 
  // Trivial cases
  if (lhs.isEmpty() || rhs.isEmpty()) {
    return Interval::Empty(field.get());
  }
  if (lhs.isEntire()) {
    return rhs;
  }
  if (rhs.isEntire()) {
    return lhs;
  }
  if (lhs.isDegenerate() || rhs.isDegenerate()) {
    return lhs.toUnreduced().intersect(rhs.toUnreduced()).reduce(field.get());
  }
 
  // More complex cases
  if (areOneOf<
          {Type::TypeA, Type::TypeA}, {Type::TypeB, Type::TypeB}, {Type::TypeC, Type::TypeC},
          {Type::TypeA, Type::TypeC}, {Type::TypeB, Type::TypeC}>(lhs, rhs)) {
    auto maxA = std::max(lhs.a, rhs.a);
    auto minB = std::min(lhs.b, rhs.b);
    if (maxA <= minB) {
      return Interval(lhs.ty, field.get(), maxA, minB);
    } else {
      return Interval::Empty(field.get());
    }
  }
  if (areOneOf<{Type::TypeA, Type::TypeB}>(lhs, rhs)) {
    return Interval::Empty(field.get());
  }
  if (areOneOf<{Type::TypeF, Type::TypeF}, {Type::TypeA, Type::TypeF}>(lhs, rhs)) {
    return lhs.firstUnreduced().intersect(rhs.firstUnreduced()).reduce(field.get());
  }
  if (areOneOf<{Type::TypeB, Type::TypeF}>(lhs, rhs)) {
    return lhs.secondUnreduced().intersect(rhs.firstUnreduced()).reduce(field.get());
  }
  if (areOneOf<{Type::TypeC, Type::TypeF}>(lhs, rhs)) {
    auto rhsUnred = rhs.firstUnreduced();
    auto opt1 = lhs.firstUnreduced().intersect(rhsUnred).reduce(field.get());
    auto opt2 = lhs.secondUnreduced().intersect(rhsUnred).reduce(field.get());
    ensure(!opt1.isEntire() && !opt2.isEntire(), "impossible intersection");
    if (opt1.isEmpty()) {
      return opt2;
    }
    if (opt2.isEmpty()) {
      return opt1;
    }
    return opt1.join(opt2);
  }
  if (areOneOf<
          {Type::TypeB, Type::TypeA}, {Type::TypeC, Type::TypeA}, {Type::TypeC, Type::TypeB},
          {Type::TypeF, Type::TypeA}, {Type::TypeF, Type::TypeB}, {Type::TypeF, Type::TypeC}>(
          lhs, rhs
      )) {
    return rhs.intersect(lhs);
  }
  return Interval::Empty(field.get());
}
 
Interval Interval::difference(const Interval &other) const {
  // intersect checks that we're in the same field
  Interval intersection = intersect(other);
  if (intersection.isEmpty()) {
    // There's nothing to remove, so just return this
    return *this;
  }
 
  const Field &f = field.get();
 
  // Trivial cases with a non-empty intersection
  if (isDegenerate() || other.isEntire()) {
    return Interval::Empty(f);
  }
  if (isEntire()) {
    // Since we don't support punching arbitrary holes in ranges, we only reduce
    // entire ranges if other is [0, b] or [a, prime - 1]
    if (other.a == f.zero()) {
      return UnreducedInterval(other.b + f.one(), f.maxVal()).reduce(f);
    }
    if (other.b == f.maxVal()) {
      return UnreducedInterval(f.zero(), other.a - f.one()).reduce(f);
    }
 
    return *this;
  }
 
  // Non-trivial cases
  // - Internal+internal or external+external cases
  if ((is<Type::TypeA, Type::TypeB, Type::TypeC>() &&
       intersection.is<Type::TypeA, Type::TypeB, Type::TypeC>()) ||
      areOneOf<{Type::TypeF, Type::TypeF}>(*this, intersection)) {
    // The intersection needs to be at the end of the interval, otherwise we would
    // split the interval in two, and we aren't set up to support multiple intervals
    // per value.
    if (a != intersection.a && b != intersection.b) {
      return *this;
    }
    // Otherwise, remove the intersection and reduce
    if (a == intersection.a) {
      return UnreducedInterval(intersection.b + f.one(), b).reduce(f);
    }
    // else b == intersection.b
    return UnreducedInterval(a, intersection.a - f.one()).reduce(f);
  }
  // - Mixed internal/external cases. We flip the comparison
  if (isTypeF()) {
    if (a != intersection.b && b != intersection.a) {
      return *this;
    }
    // Otherwise, remove the intersection and reduce
    if (a == intersection.b) {
      return UnreducedInterval(intersection.a + f.one(), b).reduce(f);
    }
    // else b == intersection.a
    return UnreducedInterval(a, intersection.b - f.one()).reduce(f);
  }
 
  // In cases we don't know how to handle, we over-approximate and return
  // the original interval.
  return *this;
}
 
Interval Interval::operator-() const { return (-firstUnreduced()).reduce(field.get()); }
 
Interval operator+(const Interval &lhs, const Interval &rhs) {
  ensure(lhs.field.get() == rhs.field.get(), "cannot add intervals in different fields");
  return (lhs.firstUnreduced() + rhs.firstUnreduced()).reduce(lhs.field.get());
}
 
Interval operator-(const Interval &lhs, const Interval &rhs) { return lhs + (-rhs); }
 
Interval operator*(const Interval &lhs, const Interval &rhs) {
  ensure(lhs.field.get() == rhs.field.get(), "cannot multiply intervals in different fields");
  const auto &field = lhs.field.get();
  auto zeroInterval = Interval::Degenerate(field, field.zero());
  if (lhs == zeroInterval || rhs == zeroInterval) {
    return zeroInterval;
  }
  if (lhs.isEmpty() || rhs.isEmpty()) {
    return Interval::Empty(field);
  }
  if (lhs.isEntire() || rhs.isEntire()) {
    return Interval::Entire(field);
  }
 
  if (Interval::areOneOf<{Interval::Type::TypeB, Interval::Type::TypeB}>(lhs, rhs)) {
    return (lhs.secondUnreduced() * rhs.secondUnreduced()).reduce(field);
  }
  return (lhs.firstUnreduced() * rhs.firstUnreduced()).reduce(field);
}
 
FailureOr<Interval> operator/(const Interval &lhs, const Interval &rhs) {
  ensure(lhs.getField() == rhs.getField(), "cannot divide intervals in different fields");
  const auto &field = rhs.getField();
  if (rhs.width() > field.one()) {
    return Interval::Entire(field);
  }
  if (rhs.a.isZero()) {
    return failure();
  }
  return success(UnreducedInterval(lhs.a / rhs.a, lhs.b / rhs.a).reduce(field));
}
 
Interval operator%(const Interval &lhs, const Interval &rhs) {
  ensure(
      lhs.getField() == rhs.getField(), "interval operations across differing fields is unsupported"
  );
  const auto &field = rhs.getField();
  return UnreducedInterval(field.zero(), rhs.b).reduce(field);
}
 
void Interval::print(mlir::raw_ostream &os) const {
  os << TypeName(ty);
  if (is<Type::Degenerate>()) {
    os << '(' << a << ')';
  } else if (!is<Type::Entire, Type::Empty>()) {
    os << ":[ " << a << ", " << b << " ]";
  }
}
 
/* ExpressionValue */
 
bool ExpressionValue::operator==(const ExpressionValue &rhs) const {
  if (expr == nullptr && rhs.expr == nullptr) {
    return i == rhs.i;
  }
  if (expr == nullptr || rhs.expr == nullptr) {
    return false;
  }
  return i == rhs.i && *expr == *rhs.expr;
}
 
ExpressionValue
intersection(llvm::SMTSolverRef solver, const ExpressionValue &lhs, const ExpressionValue &rhs) {
  Interval res = lhs.i.intersect(rhs.i);
  auto exprEq = solver->mkEqual(lhs.expr, rhs.expr);
  return ExpressionValue(exprEq, res);
}
 
ExpressionValue
add(llvm::SMTSolverRef solver, const ExpressionValue &lhs, const ExpressionValue &rhs) {
  ExpressionValue res;
  res.i = lhs.i + rhs.i;
  res.expr = solver->mkBVAdd(lhs.expr, rhs.expr);
  return res;
}
 
ExpressionValue
sub(llvm::SMTSolverRef solver, const ExpressionValue &lhs, const ExpressionValue &rhs) {
  ExpressionValue res;
  res.i = lhs.i - rhs.i;
  res.expr = solver->mkBVSub(lhs.expr, rhs.expr);
  return res;
}
 
ExpressionValue
mul(llvm::SMTSolverRef solver, const ExpressionValue &lhs, const ExpressionValue &rhs) {
  ExpressionValue res;
  res.i = lhs.i * rhs.i;
  res.expr = solver->mkBVMul(lhs.expr, rhs.expr);
  return res;
}
 
ExpressionValue
div(llvm::SMTSolverRef solver, DivFeltOp op, const ExpressionValue &lhs,
    const ExpressionValue &rhs) {
  ExpressionValue res;
  auto divRes = lhs.i / rhs.i;
  if (failed(divRes)) {
    op->emitWarning(
        "divisor is not restricted to non-zero values, leading to potential divide-by-zero error."
        " Range of division result will be treated as unbounded."
    );
    res.i = Interval::Entire(lhs.getField());
  } else {
    res.i = *divRes;
  }
  res.expr = solver->mkBVUDiv(lhs.expr, rhs.expr);
  return res;
}
 
ExpressionValue
mod(llvm::SMTSolverRef solver, const ExpressionValue &lhs, const ExpressionValue &rhs) {
  ExpressionValue res;
  res.i = lhs.i % rhs.i;
  res.expr = solver->mkBVURem(lhs.expr, rhs.expr);
  return res;
}
 
ExpressionValue
cmp(llvm::SMTSolverRef solver, CmpOp op, const ExpressionValue &lhs, const ExpressionValue &rhs) {
  ExpressionValue res;
  res.i = Interval::Boolean(lhs.getField());
  switch (op.getPredicate()) {
  case FeltCmpPredicate::EQ:
    res.expr = solver->mkEqual(lhs.expr, rhs.expr);
    res.i = lhs.i.intersect(rhs.i);
    break;
  case FeltCmpPredicate::NE:
    res.expr = solver->mkNot(solver->mkEqual(lhs.expr, rhs.expr));
    break;
  case FeltCmpPredicate::LT:
    res.expr = solver->mkBVUlt(lhs.expr, rhs.expr);
    break;
  case FeltCmpPredicate::LE:
    res.expr = solver->mkBVUle(lhs.expr, rhs.expr);
    break;
  case FeltCmpPredicate::GT:
    res.expr = solver->mkBVUgt(lhs.expr, rhs.expr);
    break;
  case FeltCmpPredicate::GE:
    res.expr = solver->mkBVUge(lhs.expr, rhs.expr);
    break;
  }
  return res;
}
 
ExpressionValue fallbackBinaryOp(
    llvm::SMTSolverRef solver, Operation *op, const ExpressionValue &lhs, const ExpressionValue &rhs
) {
  ExpressionValue res;
  res.i = Interval::Entire(lhs.getField());
  res.expr = TypeSwitch<Operation *, llvm::SMTExprRef>(op)
                 .Case<AndFeltOp>([&](AndFeltOp _) { return solver->mkBVAnd(lhs.expr, rhs.expr); })
                 .Case<OrFeltOp>([&](OrFeltOp _) { return solver->mkBVOr(lhs.expr, rhs.expr); })
                 .Case<XorFeltOp>([&](XorFeltOp _) { return solver->mkBVXor(lhs.expr, rhs.expr); })
                 .Case<ShlFeltOp>([&](ShlFeltOp _) { return solver->mkBVShl(lhs.expr, rhs.expr); })
                 .Case<ShrFeltOp>([&](ShrFeltOp _) {
    return solver->mkBVLshr(lhs.expr, rhs.expr);
  }).Default([&](Operation *unsupported) {
    llvm::report_fatal_error(
        "no fallback provided for " + mlir::Twine(unsupported->getName().getStringRef())
    );
    return nullptr;
  });
 
  return res;
}
 
ExpressionValue neg(llvm::SMTSolverRef solver, const ExpressionValue &val) {
  ExpressionValue res;
  res.i = -val.i;
  res.expr = solver->mkBVNeg(val.expr);
  return res;
}
 
ExpressionValue notOp(llvm::SMTSolverRef solver, const ExpressionValue &val) {
  ExpressionValue res;
  const auto &f = val.getField();
  if (val.i.isDegenerate()) {
    if (val.i == Interval::Degenerate(f, f.zero())) {
      res.i = Interval::Degenerate(f, f.one());
    } else {
      res.i = Interval::Degenerate(f, f.zero());
    }
  }
  res.i = Interval::Boolean(f);
  res.expr = solver->mkBVNot(val.expr);
  return res;
}
 
ExpressionValue
fallbackUnaryOp(llvm::SMTSolverRef solver, Operation *op, const ExpressionValue &val) {
  const Field &field = val.getField();
  ExpressionValue res;
  res.i = Interval::Entire(field);
  res.expr = TypeSwitch<Operation *, llvm::SMTExprRef>(op)
                 .Case<InvFeltOp>([&](InvFeltOp _) {
    // The definition of an inverse X^-1 is Y s.t. XY % prime = 1.
    // To create this expression, we create a new symbol for Y and add the
    // XY % prime = 1 constraint to the solver.
    std::string symName = buildStringViaInsertionOp(*op);
    llvm::SMTExprRef invSym = field.createSymbol(solver, symName.c_str());
    llvm::SMTExprRef one = solver->mkBitvector(field.one(), field.bitWidth());
    llvm::SMTExprRef prime = solver->mkBitvector(field.prime(), field.bitWidth());
    llvm::SMTExprRef mult = solver->mkBVMul(val.getExpr(), invSym);
    llvm::SMTExprRef mod = solver->mkBVURem(mult, prime);
    llvm::SMTExprRef constraint = solver->mkEqual(mod, one);
    solver->addConstraint(constraint);
    return invSym;
  }).Default([&](Operation *unsupported) {
    llvm::report_fatal_error(
        "no fallback provided for " + mlir::Twine(op->getName().getStringRef())
    );
    return nullptr;
  });
 
  return res;
}
 
void ExpressionValue::print(mlir::raw_ostream &os) const {
  if (expr) {
    expr->print(os);
  } else {
    os << "<null expression>";
  }
 
  os << " ( interval: " << i << " )";
}
 
/* IntervalAnalysisLattice */
 
ChangeResult IntervalAnalysisLattice::join(const AbstractDenseLattice &other) {
  const auto *rhs = dynamic_cast<const IntervalAnalysisLattice *>(&other);
  if (!rhs) {
    llvm::report_fatal_error("invalid join lattice type");
  }
  ChangeResult res = ChangeResult::NoChange;
  for (auto &[k, v] : rhs->valMap) {
    auto it = valMap.find(k);
    if (it == valMap.end() || it->second != v) {
      valMap[k] = v;
      res |= ChangeResult::Change;
    }
  }
  for (auto &v : rhs->constraints) {
    if (!constraints.contains(v)) {
      constraints.insert(v);
      res |= ChangeResult::Change;
    }
  }
  for (auto &[e, i] : rhs->intervals) {
    auto it = intervals.find(e);
    if (it == intervals.end() || it->second != i) {
      intervals[e] = i;
      res |= ChangeResult::Change;
    }
  }
  return res;
}
 
void IntervalAnalysisLattice::print(mlir::raw_ostream &os) const {
  os << "IntervalAnalysisLattice { ";
  for (auto &[ref, val] : valMap) {
    os << "\n    (valMap) " << ref << " := " << val;
  }
  for (auto &[expr, interval] : intervals) {
    os << "\n    (intervals) ";
    expr->print(os);
    os << " in " << interval;
  }
  if (!valMap.empty()) {
    os << '\n';
  }
  os << '}';
}
 
FailureOr<IntervalAnalysisLattice::LatticeValue> IntervalAnalysisLattice::getValue(Value v) const {
  auto it = valMap.find(v);
  if (it == valMap.end()) {
    return failure();
  }
  return it->second;
}
 
ChangeResult IntervalAnalysisLattice::setValue(Value v, ExpressionValue e) {
  LatticeValue val(e);
  if (valMap[v] == val) {
    return ChangeResult::NoChange;
  }
  valMap[v] = val;
  intervals[e.getExpr()] = e.getInterval();
  return ChangeResult::Change;
}
 
ChangeResult IntervalAnalysisLattice::addSolverConstraint(ExpressionValue e) {
  if (!constraints.contains(e)) {
    constraints.insert(e);
    return ChangeResult::Change;
  }
  return ChangeResult::NoChange;
}
 
FailureOr<Interval> IntervalAnalysisLattice::findInterval(llvm::SMTExprRef expr) const {
  auto it = intervals.find(expr);
  if (it != intervals.end()) {
    return it->second;
  }
  return failure();
}
 
/* IntervalDataFlowAnalysis */

void IntervalDataFlowAnalysis::visitCallControlFlowTransfer(
    CallOpInterface call, dataflow::CallControlFlowAction action,
    const IntervalAnalysisLattice &before, IntervalAnalysisLattice *after
) {
  if (action == dataflow::CallControlFlowAction::EnterCallee) {
    // We skip updating the incoming lattice for function calls,
    // as values are relative to the containing function/struct, so we don't need to pollute
    // the callee with the callers values.
    setToEntryState(after);
  }
  else if (action == dataflow::CallControlFlowAction::ExitCallee) {
    // Get the argument values of the lattice by getting the state as it would
    // have been for the callsite.
    dataflow::AbstractDenseLattice *beforeCall = nullptr;
    if (auto *prev = call->getPrevNode()) {
      beforeCall = getLattice(prev);
    } else {
      beforeCall = getLattice(call->getBlock());
    }
    ensure(beforeCall, "could not get prior lattice");
 
    // The lattice at the return is the lattice before the call
    propagateIfChanged(after, after->join(*beforeCall));
  }
  else if (action == mlir::dataflow::CallControlFlowAction::ExternalCallee) {
    // For external calls, we propagate what information we already have from
    // before the call to after the call, since the external call won't invalidate
    // any of that information. It also, conservatively, makes no assumptions about
    // external calls and their computation, so CDG edges will not be computed over
    // input arguments to external functions.
    join(after, before);
  }
}
 
void IntervalDataFlowAnalysis::visitOperation(
    Operation *op, const Lattice &before, Lattice *after
) {
  ChangeResult changed = after->join(before);
 
  llvm::SmallVector<LatticeValue> operandVals;
 
  auto constrainRefLattice = dataflowSolver.lookupState<ConstrainRefLattice>(op);
  ensure(constrainRefLattice, "failed to get lattice");
 
  for (auto &operand : op->getOpOperands()) {
    auto val = operand.get();
    // First, lookup the operand value in the before state.
    auto priorState = before.getValue(val);
    if (succeeded(priorState)) {
      operandVals.push_back(*priorState);
      continue;
    }
 
    // Else, look up the stored value by constrain ref.
    // We only care about scalar type values, so we ignore composite types, which
    // are currently limited to non-Signal structs and arrays.
    Type valTy = val.getType();
    if (mlir::isa<ArrayType, StructType>(valTy) && !isSignalType(valTy)) {
      operandVals.push_back(LatticeValue());
      continue;
    }
 
    ConstrainRefLatticeValue refSet = constrainRefLattice->getOrDefault(val);
    ensure(refSet.isScalar(), "should have ruled out array values already");
 
    if (refSet.getScalarValue().empty()) {
      // If we can't compute the reference, then there must be some unsupported
      // op the reference analysis cannot handle. We emit a warning and return
      // early, since there's no meaningful computation we can do for this op.
      std::string warning;
      debug::Appender(warning
      ) << "state of "
        << val << " is empty; defining operation is unsupported by constrain ref analysis";
      op->emitWarning(warning);
      propagateIfChanged(after, changed);
      return;
    } else if (!refSet.isSingleValue()) {
      std::string warning;
      debug::Appender(warning) << "operand " << val << " is not a single value " << refSet
                               << ", overapproximating";
      op->emitWarning(warning);
      // Here, we will override the prior lattice value with a new symbol, representing
      // "any" value, then use that value for the operands.
      ExpressionValue anyVal(field.get(), createFeltSymbol(val));
      changed |= after->setValue(val, anyVal);
      operandVals.emplace_back(anyVal);
    } else {
      auto ref = refSet.getSingleValue();
      ExpressionValue exprVal(field.get(), getOrCreateSymbol(ref));
      changed |= after->setValue(val, exprVal);
      operandVals.emplace_back(exprVal);
    }
  }
 
  // Now, the way we update is dependent on the type of the operation.
  if (!isConsideredOp(op)) {
    op->emitWarning("unconsidered operation type, analysis may be incomplete");
  }
 
  if (isConstOp(op)) {
    auto constVal = getConst(op);
    auto expr = createConstBitvectorExpr(constVal);
    ExpressionValue latticeVal(field.get(), expr, constVal);
    changed |= after->setValue(op->getResult(0), latticeVal);
  } else if (isArithmeticOp(op)) {
    ensure(operandVals.size() <= 2, "arithmetic op with the wrong number of operands");
    ExpressionValue result;
    if (operandVals.size() == 2) {
      result = performBinaryArithmetic(op, operandVals[0], operandVals[1]);
    } else {
      result = performUnaryArithmetic(op, operandVals[0]);
    }
 
    changed |= after->setValue(op->getResult(0), result);
  } else if (EmitEqualityOp emitEq = mlir::dyn_cast<EmitEqualityOp>(op)) {
    ensure(operandVals.size() == 2, "constraint op with the wrong number of operands");
    Value lhsVal = emitEq.getLhs(), rhsVal = emitEq.getRhs();
    ExpressionValue lhsExpr = operandVals[0].getScalarValue();
    ExpressionValue rhsExpr = operandVals[1].getScalarValue();
 
    ExpressionValue constraint = intersection(smtSolver, lhsExpr, rhsExpr);
    // Update the LHS and RHS to the same value, but restricted intervals
    // based on the constraints
    changed |= applyInterval(emitEq, after, lhsVal, constraint.getInterval());
    changed |= applyInterval(emitEq, after, rhsVal, constraint.getInterval());
    changed |= after->addSolverConstraint(constraint);
  } else if (AssertOp assertOp = mlir::dyn_cast<AssertOp>(op)) {
    ensure(operandVals.size() == 1, "assert op with the wrong number of operands");
    // assert enforces that the operand is true. So we apply an interval of [1, 1]
    // to the operand.
    changed |= applyInterval(
        assertOp, after, assertOp.getCondition(),
        Interval::Degenerate(field.get(), field.get().one())
    );
    // Also add the solver constraint that the expression must be true.
    auto assertExpr = operandVals[0].getScalarValue();
    changed |= after->addSolverConstraint(assertExpr);
  } else if (auto readf = mlir::dyn_cast<FieldReadOp>(op);
             readf && isSignalType(readf.getComponent().getType())) {
    // The reg value read from the signal type is equal to the value of the Signal
    // struct overall.
    changed |= after->setValue(readf.getVal(), operandVals[0].getScalarValue());
  } else if (!isReadOp(op)          /* We do not need to explicitly handle read ops
                      since they are resolved at the operand value step where constrain refs are
                      queries (with the exception of the Signal struct, see above). */
             && !isReturnOp(op)     /* We do not currently handle return ops as the analysis
                 is currently limited to constrain functions, which return no value. */
             && !isDefinitionOp(op) /* The analysis ignores definition ops. */
             &&
             !mlir::isa<CreateStructOp>(op) /* We do not need to analyze the creation of structs. */
  ) {
    op->emitWarning("unhandled operation, analysis may be incomplete");
  }
 
  propagateIfChanged(after, changed);
}
 
llvm::SMTExprRef IntervalDataFlowAnalysis::getOrCreateSymbol(const ConstrainRef &r) {
  auto it = refSymbols.find(r);
  if (it != refSymbols.end()) {
    return it->second;
  }
  auto sym = createFeltSymbol(r);
  refSymbols[r] = sym;
  return sym;
}
 
llvm::SMTExprRef IntervalDataFlowAnalysis::createFeltSymbol(const ConstrainRef &r) const {
  return createFeltSymbol(buildStringViaPrint(r).c_str());
}
 
llvm::SMTExprRef IntervalDataFlowAnalysis::createFeltSymbol(Value v) const {
  return createFeltSymbol(buildStringViaPrint(v).c_str());
}
 
llvm::SMTExprRef IntervalDataFlowAnalysis::createFeltSymbol(const char *name) const {
  return field.get().createSymbol(smtSolver, name);
}
 
llvm::APSInt IntervalDataFlowAnalysis::getConst(Operation *op) const {
  ensure(isConstOp(op), "op is not a const op");
 
  llvm::APInt fieldConst =
      TypeSwitch<Operation *, llvm::APInt>(op)
          .Case<FeltConstantOp>([&](FeltConstantOp feltConst) {
    llvm::APSInt constOpVal(feltConst.getValueAttr().getValue());
    return field.get().reduce(constOpVal);
  })
          .Case<arith::ConstantIndexOp>([&](arith::ConstantIndexOp indexConst) {
    return llvm::APInt(field.get().bitWidth(), indexConst.value());
  })
          .Case<arith::ConstantIntOp>([&](arith::ConstantIntOp intConst) {
    return llvm::APInt(field.get().bitWidth(), intConst.value());
  }).Default([&](Operation *illegalOp) {
    std::string err;
    debug::Appender(err) << "unhandled getConst case: " << *illegalOp;
    llvm::report_fatal_error(Twine(err));
    return llvm::APInt();
  });
  return llvm::APSInt(fieldConst);
}
 
ExpressionValue IntervalDataFlowAnalysis::performBinaryArithmetic(
    Operation *op, const LatticeValue &a, const LatticeValue &b
) {
  ensure(isArithmeticOp(op), "is not arithmetic op");
 
  auto lhs = a.getScalarValue(), rhs = b.getScalarValue();
  ensure(lhs.getExpr(), "cannot perform arithmetic over null lhs smt expr");
  ensure(rhs.getExpr(), "cannot perform arithmetic over null rhs smt expr");
 
  auto res = TypeSwitch<Operation *, ExpressionValue>(op)
                 .Case<AddFeltOp>([&](AddFeltOp _) { return add(smtSolver, lhs, rhs); })
                 .Case<SubFeltOp>([&](SubFeltOp _) { return sub(smtSolver, lhs, rhs); })
                 .Case<MulFeltOp>([&](MulFeltOp _) { return mul(smtSolver, lhs, rhs); })
                 .Case<DivFeltOp>([&](DivFeltOp divOp) { return div(smtSolver, divOp, lhs, rhs); })
                 .Case<ModFeltOp>([&](ModFeltOp _) { return mod(smtSolver, lhs, rhs); })
                 .Case<CmpOp>([&](CmpOp cmpOp) {
    return cmp(smtSolver, cmpOp, lhs, rhs);
  }).Default([&](Operation *unsupported) {
    unsupported->emitWarning(
        "unsupported binary arithmetic operation, defaulting to over-approximated intervals"
    );
    return fallbackBinaryOp(smtSolver, unsupported, lhs, rhs);
  });
 
  ensure(res.getExpr(), "arithmetic produced null smt expr");
  return res;
}
 
ExpressionValue
IntervalDataFlowAnalysis::performUnaryArithmetic(Operation *op, const LatticeValue &a) {
  ensure(isArithmeticOp(op), "is not arithmetic op");
 
  auto val = a.getScalarValue();
  ensure(val.getExpr(), "cannot perform arithmetic over null smt expr");
 
  auto res = TypeSwitch<Operation *, ExpressionValue>(op)
                 .Case<NegFeltOp>([&](NegFeltOp _) { return neg(smtSolver, val); })
                 .Case<NotFeltOp>([&](NotFeltOp _) {
    return notOp(smtSolver, val);
  }).Default([&](Operation *unsupported) {
    unsupported->emitWarning(
        "unsupported unary arithmetic operation, defaulting to over-approximated interval"
    );
    return fallbackUnaryOp(smtSolver, unsupported, val);
  });
 
  ensure(res.getExpr(), "arithmetic produced null smt expr");
  return res;
}
 
ChangeResult IntervalDataFlowAnalysis::applyInterval(
    Operation *originalOp, Lattice *after, Value val, Interval newInterval
) {
  auto latValRes = after->getValue(val);
  if (failed(latValRes)) {
    // visitOperation didn't add val to the lattice, so there's nothing to do
    return ChangeResult::NoChange;
  }
  ExpressionValue newLatticeVal = latValRes->getScalarValue().withInterval(newInterval);
  ChangeResult res = after->setValue(val, newLatticeVal);
  // To allow the dataflow analysis to do its fixed-point iteration, we need to
  // add the new expression to val's lattice as well.
  Lattice *valLattice = nullptr;
  if (auto valOp = val.getDefiningOp()) {
    // Getting the lattice at valOp gives us the "after" lattice, but we want to
    // update the "before" lattice so that the inputs to visitOperation will be
    // changed.
    if (auto prev = valOp->getPrevNode()) {
      valLattice = getOrCreate<Lattice>(prev);
    } else {
      valLattice = getOrCreate<Lattice>(valOp->getBlock());
    }
  } else if (auto blockArg = mlir::dyn_cast<BlockArgument>(val)) {
    valLattice = getOrCreate<Lattice>(blockArg.getOwner());
  } else {
    valLattice = getOrCreate<Lattice>(val);
  }
  ensure(valLattice, "val should have a lattice");
  if (valLattice != after) {
    propagateIfChanged(valLattice, valLattice->setValue(val, newLatticeVal));
  }
 
  // Now we descend into val's operands, if it has any.
  Operation *definingOp = val.getDefiningOp();
  if (!definingOp) {
    return res;
  }
 
  const Field &f = field.get();
 
  // This is a rules-based operation. If we have a rule for a given operation,
  // then we can make some kind of update, otherwise we leave the intervals
  // as is.
  // - First we'll define all the rules so the type switch can be less messy
 
  // cmp.<pred> restricts each side of the comparison if the result is known.
  auto cmpCase = [&](CmpOp cmpOp) {
    // Cmp output range is [0, 1], so in order to do something, we must have newInterval
    // either "true" (1) or "false" (0)
    Interval maxInterval = Interval::Boolean(f);
    ensure(
        newInterval.intersect(maxInterval).isNotEmpty(),
        "new interval for CmpOp outside of allowed boolean range or is empty"
    );
    if (!newInterval.isDegenerate()) {
      // The comparison result is unknown, so we can't update the operand ranges
      return ChangeResult::NoChange;
    }
 
    bool cmpTrue = newInterval.rhs() == f.one();
 
    Value lhs = cmpOp->getOperand(0), rhs = cmpOp->getOperand(1);
    auto lhsLatValRes = after->getValue(lhs), rhsLatValRes = after->getValue(rhs);
    if (failed(lhsLatValRes) || failed(rhsLatValRes)) {
      return ChangeResult::NoChange;
    }
    ExpressionValue lhsExpr = lhsLatValRes->getScalarValue(),
                    rhsExpr = rhsLatValRes->getScalarValue();
 
    Interval newLhsInterval, newRhsInterval;
    const Interval &lhsInterval = lhsExpr.getInterval();
    const Interval &rhsInterval = rhsExpr.getInterval();
 
    FeltCmpPredicate pred = cmpOp.getPredicate();
    // predicate cases
    auto eqCase = [&]() {
      return (pred == FeltCmpPredicate::EQ && cmpTrue) ||
             (pred == FeltCmpPredicate::NE && !cmpTrue);
    };
    auto neCase = [&]() {
      return (pred == FeltCmpPredicate::NE && cmpTrue) ||
             (pred == FeltCmpPredicate::EQ && !cmpTrue);
    };
    auto ltCase = [&]() {
      return (pred == FeltCmpPredicate::LT && cmpTrue) ||
             (pred == FeltCmpPredicate::GE && !cmpTrue);
    };
    auto leCase = [&]() {
      return (pred == FeltCmpPredicate::LE && cmpTrue) ||
             (pred == FeltCmpPredicate::GT && !cmpTrue);
    };
    auto gtCase = [&]() {
      return (pred == FeltCmpPredicate::GT && cmpTrue) ||
             (pred == FeltCmpPredicate::LE && !cmpTrue);
    };
    auto geCase = [&]() {
      return (pred == FeltCmpPredicate::GE && cmpTrue) ||
             (pred == FeltCmpPredicate::LT && !cmpTrue);
    };
 
    // new intervals based on case
    if (eqCase()) {
      newLhsInterval = newRhsInterval = lhsInterval.intersect(rhsInterval);
    } else if (neCase()) {
      if (lhsInterval.isDegenerate() && rhsInterval.isDegenerate() && lhsInterval == rhsInterval) {
        // In this case, we know lhs and rhs cannot satisfy this assertion, so they have
        // an empty value range.
        newLhsInterval = newRhsInterval = Interval::Empty(f);
      } else {
        // Leave unchanged
        newLhsInterval = lhsInterval;
        newRhsInterval = rhsInterval;
      }
    } else if (ltCase()) {
      newLhsInterval = lhsInterval.toUnreduced().computeLTPart(rhsInterval.toUnreduced()).reduce(f);
      newRhsInterval = rhsInterval.toUnreduced().computeGEPart(lhsInterval.toUnreduced()).reduce(f);
    } else if (leCase()) {
      newLhsInterval = lhsInterval.toUnreduced().computeLEPart(rhsInterval.toUnreduced()).reduce(f);
      newRhsInterval = rhsInterval.toUnreduced().computeGTPart(lhsInterval.toUnreduced()).reduce(f);
    } else if (gtCase()) {
      newLhsInterval = lhsInterval.toUnreduced().computeGTPart(rhsInterval.toUnreduced()).reduce(f);
      newRhsInterval = rhsInterval.toUnreduced().computeLEPart(lhsInterval.toUnreduced()).reduce(f);
    } else if (geCase()) {
      newLhsInterval = lhsInterval.toUnreduced().computeGEPart(rhsInterval.toUnreduced()).reduce(f);
      newRhsInterval = rhsInterval.toUnreduced().computeLTPart(lhsInterval.toUnreduced()).reduce(f);
    } else {
      cmpOp->emitWarning("unhandled cmp predicate");
      return ChangeResult::NoChange;
    }
 
    // Now we recurse to each operand
    return applyInterval(originalOp, after, lhs, newLhsInterval) |
           applyInterval(originalOp, after, rhs, newRhsInterval);
  };
 
  // If the result of a multiplication is non-zero, then both operands must be
  // non-zero.
  auto mulCase = [&](MulFeltOp mulOp) {
    auto zeroInt = Interval::Degenerate(f, f.zero());
    if (newInterval.intersect(zeroInt).isNotEmpty()) {
      // The multiplication may be zero, so we can't reduce the operands to be non-zero
      return ChangeResult::NoChange;
    }
 
    Value lhs = mulOp->getOperand(0), rhs = mulOp->getOperand(1);
    auto lhsLatValRes = after->getValue(lhs), rhsLatValRes = after->getValue(rhs);
    if (failed(lhsLatValRes) || failed(rhsLatValRes)) {
      return ChangeResult::NoChange;
    }
    ExpressionValue lhsExpr = lhsLatValRes->getScalarValue(),
                    rhsExpr = rhsLatValRes->getScalarValue();
    Interval newLhsInterval = lhsExpr.getInterval().difference(zeroInt);
    Interval newRhsInterval = rhsExpr.getInterval().difference(zeroInt);
    return applyInterval(originalOp, after, lhs, newLhsInterval) |
           applyInterval(originalOp, after, rhs, newRhsInterval);
  };
 
  // We have a special case for the Signal struct: if this value is created
  // from reading a Signal struct's reg field, we also apply the interval to
  // the struct itself.
  auto readfCase = [&](FieldReadOp readfOp) {
    Value comp = readfOp.getComponent();
    if (isSignalType(comp.getType())) {
      return applyInterval(originalOp, after, comp, newInterval);
    }
    return ChangeResult::NoChange;
  };
 
  // - Apply the rules given the op.
  // NOTE: disabling clang-format for this because it makes the last case statement
  // look ugly.
  // clang-format off
  res |= TypeSwitch<Operation *, ChangeResult>(definingOp)
            .Case<CmpOp>([&](CmpOp op) { return cmpCase(op); })
            .Case<MulFeltOp>([&](MulFeltOp op) { return mulCase(op); })
            .Case<FieldReadOp>([&](FieldReadOp op){ return readfCase(op); })
            .Default([&](Operation *_) { return ChangeResult::NoChange; });
  // clang-format on
 
  return res;
}
 
/* StructIntervals */
 
LogicalResult StructIntervals::computeIntervals(
    mlir::DataFlowSolver &solver, mlir::AnalysisManager &am, IntervalAnalysisContext &ctx
) {
  // Get the lattice at the end of the constrain function.
  function::ReturnOp constrainEnd;
  structDef.getConstrainFuncOp().walk([&constrainEnd](function::ReturnOp r) mutable {
    constrainEnd = r;
  });
 
  const IntervalAnalysisLattice *constrainLattice =
      solver.lookupState<IntervalAnalysisLattice>(constrainEnd);
 
  constrainSolverConstraints = constrainLattice->getConstraints();
 
  for (const auto &ref : ConstrainRef::getAllConstrainRefs(structDef)) {
    // We only want to compute intervals for field elements and not composite types,
    // with the exception of the Signal struct.
    if (!ref.isScalar() && !ref.isSignal()) {
      continue;
    }
    // We also don't want to show the interval for a Signal and its internal reg.
    if (auto parentOr = ref.getParentPrefix(); succeeded(parentOr) && parentOr->isSignal()) {
      continue;
    }
    auto symbol = ctx.getSymbol(ref);
    auto constrainInterval = constrainLattice->findInterval(symbol);
    if (succeeded(constrainInterval)) {
      constrainFieldRanges[ref] = *constrainInterval;
    } else {
      constrainFieldRanges[ref] = Interval::Entire(ctx.field);
    }
  }
 
  return success();
}
 
void StructIntervals::print(mlir::raw_ostream &os, bool withConstraints) const {
  os << "StructIntervals { ";
  if (constrainFieldRanges.empty()) {
    os << "}\n";
    return;
  }
 
  for (auto &[ref, interval] : constrainFieldRanges) {
    os << "\n    " << ref << " in " << interval;
  }
 
  if (withConstraints) {
    os << "\n\n    Solver Constraints { ";
    if (constrainSolverConstraints.empty()) {
      os << "}\n";
    } else {
      for (const auto &e : constrainSolverConstraints) {
        os << "\n        ";
        e.getExpr()->print(os);
      }
      os << "\n    }";
    }
  }
 
  os << "\n}\n";
}
 
} // namespace llzk