mz_transform/

normalize_lets.rs

// Copyright Materialize, Inc. and contributors. All rights reserved.
//
// Use of this software is governed by the Business Source License
// included in the LICENSE file.
//
// As of the Change Date specified in that file, in accordance with
// the Business Source License, use of this software will be governed
// by the Apache License, Version 2.0.

//! Normalize the structure of `Let` and `LetRec` operators in expressions.
//!
//! Normalization happens in the context of "scopes", corresponding to
//! 1. the expression's root and 2. each instance of a `LetRec` AST node.
//!
//! Within each scope,
//! 1. Each expression is normalized to have all `Let` nodes at the root
//! of the expression, in order of identifier.
//! 2. Each expression assigns a contiguous block of identifiers.
//!
//! The transform may remove some `Let` and `Get` operators, and does not
//! introduce any new operators.
//!
//! The module also publishes the function `renumber_bindings` which can
//! be used to renumber bindings in an expression starting from a provided
//! `IdGen`, which is used to prepare distinct expressions for inlining.

use mz_expr::{MirRelationExpr, visit::Visit};
use mz_ore::assert_none;
use mz_ore::{id_gen::IdGen, stack::RecursionLimitError};
use mz_repr::optimize::OptimizerFeatures;

use crate::{TransformCtx, catch_unwind_optimize};

pub use renumbering::renumber_bindings;

/// Normalize `Let` and `LetRec` structure.
pub fn normalize_lets(
    expr: &mut MirRelationExpr,
    features: &OptimizerFeatures,
) -> Result<(), crate::TransformError> {
    catch_unwind_optimize(|| NormalizeLets::new(false).action(expr, features))
}

/// Install replace certain `Get` operators with their `Let` value.
#[derive(Debug)]
pub struct NormalizeLets {
    /// If `true`, inline MFPs around a Get.
    ///
    /// We want this value to be true for the NormalizeLets call that comes right
    /// before [crate::join_implementation::JoinImplementation] runs because
    /// - JoinImplementation cannot lift MFPs through a Let.
    /// - JoinImplementation can't extract FilterCharacteristics through a Let.
    ///
    /// Generally, though, we prefer to be more conservative in our inlining in
    /// order to be able to better detect CSEs.
    pub inline_mfp: bool,
}

impl NormalizeLets {
    /// Construct a new [`NormalizeLets`] instance with the given `inline_mfp`.
    pub fn new(inline_mfp: bool) -> NormalizeLets {
        NormalizeLets { inline_mfp }
    }
}

impl crate::Transform for NormalizeLets {
    fn name(&self) -> &'static str {
        "NormalizeLets"
    }

    #[mz_ore::instrument(
        target = "optimizer",
        level = "debug",
        fields(path.segment = "normalize_lets")
    )]
    fn actually_perform_transform(
        &self,
        relation: &mut MirRelationExpr,
        ctx: &mut TransformCtx,
    ) -> Result<(), crate::TransformError> {
        let result = self.action(relation, ctx.features);
        mz_repr::explain::trace_plan(&*relation);
        result
    }
}

impl NormalizeLets {
    /// Normalize `Let` and `LetRec` bindings in `relation`.
    ///
    /// Mechanically, `action` first renumbers all bindings, erroring if any shadowing is encountered.
    /// It then promotes all `Let` and `LetRec` expressions to the roots of their expressions, fusing
    /// `Let` bindings into containing `LetRec` bindings, but leaving stacked `LetRec` bindings unfused to each
    /// other (for reasons of correctness). It then considers potential inlining in each `LetRec` scope.
    /// Lastly, it refreshes the types of each `Get` operator, erroring if any scalar types have changed
    /// but updating nullability and keys.
    ///
    /// We then perform a final renumbering.
    pub fn action(
        &self,
        relation: &mut MirRelationExpr,
        features: &OptimizerFeatures,
    ) -> Result<(), crate::TransformError> {
        // Record whether the relation was initially recursive, to confirm that we do not introduce
        // recursion to a non-recursive expression.
        let was_recursive = relation.is_recursive();

        // Renumber all bindings to ensure that identifier order matches binding order.
        // In particular, as we use `BTreeMap` for binding order, we want to ensure that
        // 1. Bindings within a `LetRec` are assigned increasing identifiers, and
        // 2. Bindings across `LetRec`s are assigned identifiers in "visibility order", corresponding to an
        // in-order traversal.
        // TODO: More can and perhaps should be said about "visibility order" and how let promotion is correct.
        renumbering::renumber_bindings(relation, &mut IdGen::default())?;

        // Promote all `Let` and `LetRec` AST nodes to the roots.
        // After this, all non-`LetRec` nodes contain no further `Let` or `LetRec` nodes,
        // placing all `LetRec` nodes around the root, if not always in a single AST node.
        let_motion::promote_let_rec(relation);
        let_motion::assert_no_lets(relation);
        let_motion::assert_letrec_major(relation);

        // Inlining may violate letrec-major form.
        inlining::inline_lets(relation, self.inline_mfp)?;

        // Return to letrec-major form to refresh types.
        let_motion::promote_let_rec(relation);
        support::refresh_types(relation, features)?;

        // Renumber bindings for good measure.
        // Ideally we could skip when `action` is a no-op, but hard to thread that through at the moment.
        renumbering::renumber_bindings(relation, &mut IdGen::default())?;

        // A final bottom-up traversal to normalize the shape of nested LetRec blocks
        relation.try_visit_mut_post(&mut |relation| -> Result<(), RecursionLimitError> {
            // Move a non-recursive suffix of bindings from the end of the LetRec
            // to the LetRec body.
            // This is unsafe when applied to expressions which contain `ArrangeBy`,
            // as if the extracted suffixes reference arrangements they will not be
            // able to access those arrangements from outside the `LetRec` scope.
            // It happens to work at the moment, so we don't touch it but should fix.
            let bindings = let_motion::harvest_nonrec_suffix(relation)?;
            if let MirRelationExpr::LetRec {
                ids: _,
                values: _,
                limits: _,
                body,
            } = relation
            {
                for (id, value) in bindings.into_iter().rev() {
                    **body = MirRelationExpr::Let {
                        id,
                        value: Box::new(value),
                        body: Box::new(body.take_dangerous()),
                    };
                }
            } else {
                for (id, value) in bindings.into_iter().rev() {
                    *relation = MirRelationExpr::Let {
                        id,
                        value: Box::new(value),
                        body: Box::new(relation.take_dangerous()),
                    };
                }
            }

            // Extract `Let` prefixes from `LetRec`, to reveal their non-recursive nature.
            // This assists with hoisting e.g. arrangements out of `LetRec` blocks, a thing
            // we don't promise to do, but it can be helpful to do. This also exposes more
            // AST nodes to non-`LetRec` analyses, which don't always have parity with `LetRec`.
            let bindings = let_motion::harvest_non_recursive(relation);
            for (id, (value, max_iter)) in bindings.into_iter().rev() {
                assert_none!(max_iter);
                *relation = MirRelationExpr::Let {
                    id,
                    value: Box::new(value),
                    body: Box::new(relation.take_dangerous()),
                };
            }

            Ok(())
        })?;

        if !was_recursive && relation.is_recursive() {
            Err(crate::TransformError::Internal(
                "NormalizeLets introduced LetRec to a LetRec-free expression".to_string(),
            ))?;
        }

        Ok(())
    }
}

// Support methods that are unlikely to be useful to other modules.
mod support {

    use std::collections::BTreeMap;

    use itertools::Itertools;

    use mz_expr::{Id, LetRecLimit, LocalId, MirRelationExpr};
    use mz_repr::optimize::OptimizerFeatures;

    pub(super) fn replace_bindings_from_map(
        map: BTreeMap<LocalId, (MirRelationExpr, Option<LetRecLimit>)>,
        ids: &mut Vec<LocalId>,
        values: &mut Vec<MirRelationExpr>,
        limits: &mut Vec<Option<LetRecLimit>>,
    ) {
        let (new_ids, new_values, new_limits) = map_to_3vecs(map);
        *ids = new_ids;
        *values = new_values;
        *limits = new_limits;
    }

    pub(super) fn map_to_3vecs(
        map: BTreeMap<LocalId, (MirRelationExpr, Option<LetRecLimit>)>,
    ) -> (Vec<LocalId>, Vec<MirRelationExpr>, Vec<Option<LetRecLimit>>) {
        let (new_ids, new_values_and_limits): (Vec<_>, Vec<_>) = map.into_iter().unzip();
        let (new_values, new_limits) = new_values_and_limits.into_iter().unzip();
        (new_ids, new_values, new_limits)
    }

    /// Logic mapped across each use of a `LocalId`.
    pub(super) fn for_local_id<F>(expr: &MirRelationExpr, mut logic: F)
    where
        F: FnMut(LocalId),
    {
        expr.visit_pre(|expr| {
            if let MirRelationExpr::Get {
                id: Id::Local(i), ..
            } = expr
            {
                logic(*i);
            }
        });
    }

    /// Populates `counts` with the number of uses of each local identifier in `expr`.
    pub(super) fn count_local_id_uses(
        expr: &MirRelationExpr,
        counts: &mut std::collections::BTreeMap<LocalId, usize>,
    ) {
        for_local_id(expr, |i| *counts.entry(i).or_insert(0) += 1)
    }

    /// Visit `LetRec` stages and determine and update type information for `Get` nodes.
    ///
    /// This method errors if the scalar type information has changed (number of columns, or types).
    /// It only refreshes the nullability and unique key information. As this information can regress,
    /// we do not error if the type weakens, even though that may be something we want to look into.
    ///
    /// The method relies on the `analysis::{UniqueKeys, SqlRelationType}` analyses to improve its type
    /// information for `LetRec` stages.
    pub(super) fn refresh_types(
        expr: &mut MirRelationExpr,
        features: &OptimizerFeatures,
    ) -> Result<(), crate::TransformError> {
        // Assemble type information once for the whole expression.
        use crate::analysis::{DerivedBuilder, SqlRelationType, UniqueKeys};
        let mut builder = DerivedBuilder::new(features);
        builder.require(SqlRelationType);
        builder.require(UniqueKeys);
        let derived = builder.visit(expr);
        let derived_view = derived.as_view();

        // Collect id -> type mappings.
        let mut types = BTreeMap::new();
        let mut todo = vec![(&*expr, derived_view)];
        while let Some((expr, view)) = todo.pop() {
            let ids = match expr {
                MirRelationExpr::Let { id, .. } => std::slice::from_ref(id),
                MirRelationExpr::LetRec { ids, .. } => ids,
                _ => &[],
            };
            if !ids.is_empty() {
                // The `skip(1)` skips the `body` child, and is followed by binding children.
                for (id, view) in ids.iter().rev().zip_eq(view.children_rev().skip(1)) {
                    let cols = view
                        .value::<SqlRelationType>()
                        .expect("SqlRelationType required")
                        .clone()
                        .expect("Expression not well typed");
                    let keys = view
                        .value::<UniqueKeys>()
                        .expect("UniqueKeys required")
                        .clone();
                    types.insert(*id, mz_repr::SqlRelationType::new(cols).with_keys(keys));
                }
            }
            todo.extend(expr.children().rev().zip_eq(view.children_rev()));
        }

        // Install the new types in each `Get`.
        let mut todo = vec![&mut *expr];
        while let Some(expr) = todo.pop() {
            if let MirRelationExpr::Get {
                id: Id::Local(i),
                typ,
                ..
            } = expr
            {
                if let Some(new_type) = types.get(i) {
                    // Assert that the column length has not changed.
                    if !new_type.column_types.len() == typ.column_types.len() {
                        Err(crate::TransformError::Internal(format!(
                            "column lengths do not match: {:?} v {:?}",
                            new_type.column_types, typ.column_types
                        )))?;
                    }
                    // Assert that the column types have not changed.
                    if !new_type
                        .column_types
                        .iter()
                        .zip_eq(typ.column_types.iter())
                        .all(|(t1, t2)| t1.scalar_type.base_eq(&t2.scalar_type))
                    {
                        Err(crate::TransformError::Internal(format!(
                            "scalar types do not match: {:?} v {:?}",
                            new_type.column_types, typ.column_types
                        )))?;
                    }

                    typ.clone_from(new_type);
                } else {
                    panic!("Type not found for: {:?}", i);
                }
            }
            todo.extend(expr.children_mut());
        }
        Ok(())
    }
}

mod let_motion {

    use std::collections::{BTreeMap, BTreeSet};

    use itertools::Itertools;
    use mz_expr::{LetRecLimit, LocalId, MirRelationExpr};
    use mz_ore::stack::RecursionLimitError;

    use crate::normalize_lets::support::replace_bindings_from_map;

    /// Promotes all `Let` and `LetRec` nodes to the roots of their expressions.
    ///
    /// We cannot (without further reasoning) fuse stacked `LetRec` stages, and instead we just promote
    /// `LetRec` to the roots of their expressions (e.g. as children of another `LetRec` stage).
    pub(crate) fn promote_let_rec(expr: &mut MirRelationExpr) {
        // First, promote all `LetRec` nodes above all other nodes.
        let mut worklist = vec![&mut *expr];
        while let Some(mut expr) = worklist.pop() {
            hoist_bindings(expr);
            while let MirRelationExpr::LetRec { values, body, .. } = expr {
                worklist.extend(values.iter_mut().rev());
                expr = body;
            }
        }

        // Harvest any potential `Let` nodes, via a post-order traversal.
        post_order_harvest_lets(expr);
    }

    /// A stand in for the types of bindings we might encounter.
    ///
    /// As we dissolve various `Let` and `LetRec` expressions, a `Binding` will carry
    /// the relevant information as we hoist it to the root of the expression.
    enum Binding {
        // Binding resulting from a `Let` expression.
        Let(LocalId, MirRelationExpr),
        // Bindings resulting from a `LetRec` expression.
        LetRec(Vec<(LocalId, MirRelationExpr, Option<LetRecLimit>)>),
    }

    /// Hoist all exposed bindings to the root of the expression.
    ///
    /// A binding is "exposed" if the path from the root does not cross a LetRec binding.
    /// After the call, the expression should be a linear sequence of bindings, where each
    /// `Let` binding is of a let-free expression. There may be `LetRec` expressions in the
    /// sequence, and their bindings will have hoisted bindings to their root, but not out
    /// of the binding.
    fn hoist_bindings(expr: &mut MirRelationExpr) {
        // Bindings we have extracted but not fully processed.
        let mut worklist = Vec::new();
        // Bindings we have extracted and then fully processed.
        let mut finished = Vec::new();

        extract_bindings(expr, &mut worklist);
        while let Some(mut bind) = worklist.pop() {
            match &mut bind {
                Binding::Let(_id, value) => {
                    extract_bindings(value, &mut worklist);
                }
                Binding::LetRec(_binds) => {
                    // nothing to do here; we cannot hoist letrec bindings and refine
                    // them in an outer loop.
                }
            }
            finished.push(bind);
        }

        // The worklist is empty and finished should contain only LetRec bindings and Let
        // bindings with let-free expressions bound. We need to re-assemble them now in
        // the correct order. The identifiers are "sequential", so we should be able to
        // sort by them, with some care.

        // We only extract non-empty letrec bindings, so it is safe to peek at the first.
        finished.sort_by_key(|b| match b {
            Binding::Let(id, _) => *id,
            Binding::LetRec(binds) => binds[0].0,
        });

        // To match historical behavior we fuse let bindings into adjacent letrec bindings.
        // We could alternately make each a singleton letrec binding (just, non-recursive).
        // We don't yet have a strong opinion on which is most helpful and least harmful.
        // In the absence of any letrec bindings, we form one to house the let bindings.
        let mut ids = Vec::new();
        let mut values = Vec::new();
        let mut limits = Vec::new();
        let mut compact = Vec::new();
        for bind in finished {
            match bind {
                Binding::Let(id, value) => {
                    ids.push(id);
                    values.push(value);
                    limits.push(None);
                }
                Binding::LetRec(binds) => {
                    for (id, value, limit) in binds {
                        ids.push(id);
                        values.push(value);
                        limits.push(limit);
                    }
                    compact.push((ids, values, limits));
                    ids = Vec::new();
                    values = Vec::new();
                    limits = Vec::new();
                }
            }
        }

        // Remaining bindings can either be fused to the prior letrec, or put in their own.
        if let Some((last_ids, last_vals, last_lims)) = compact.last_mut() {
            last_ids.extend(ids);
            last_vals.extend(values);
            last_lims.extend(limits);
        } else if !ids.is_empty() {
            compact.push((ids, values, limits));
        }

        while let Some((ids, values, limits)) = compact.pop() {
            *expr = MirRelationExpr::LetRec {
                ids,
                values,
                limits,
                body: Box::new(expr.take_dangerous()),
            };
        }
    }

    /// Extracts exposed bindings into `bindings`.
    ///
    /// After this call `expr` will contain no let or letrec bindings, though the bindings
    /// it introduces to `bindings` may themselves contain such bindings (and they should
    /// be further processed if the goal is to maximally extract let bindings).
    fn extract_bindings(expr: &mut MirRelationExpr, bindings: &mut Vec<Binding>) {
        let mut todo = vec![expr];
        while let Some(expr) = todo.pop() {
            match expr {
                MirRelationExpr::Let { id, value, body } => {
                    bindings.push(Binding::Let(*id, value.take_dangerous()));
                    *expr = body.take_dangerous();
                    todo.push(expr);
                }
                MirRelationExpr::LetRec {
                    ids,
                    values,
                    limits,
                    body,
                } => {
                    use itertools::Itertools;
                    let binds: Vec<_> = ids
                        .drain(..)
                        .zip_eq(values.drain(..))
                        .zip_eq(limits.drain(..))
                        .map(|((i, v), l)| (i, v, l))
                        .collect();
                    if !binds.is_empty() {
                        bindings.push(Binding::LetRec(binds));
                    }
                    *expr = body.take_dangerous();
                    todo.push(expr);
                }
                _ => {
                    todo.extend(expr.children_mut());
                }
            }
        }
    }

    /// Performs a post-order traversal of the `LetRec` nodes at the root of an expression.
    ///
    /// The traversal is only of the `LetRec` nodes, for which fear of stack exhaustion is nominal.
    fn post_order_harvest_lets(expr: &mut MirRelationExpr) {
        if let MirRelationExpr::LetRec {
            ids,
            values,
            limits,
            body,
        } = expr
        {
            // Only recursively descend through `LetRec` stages.
            for value in values.iter_mut() {
                post_order_harvest_lets(value);
            }

            let mut bindings = BTreeMap::new();
            for ((id, mut value), max_iter) in ids
                .drain(..)
                .zip_eq(values.drain(..))
                .zip_eq(limits.drain(..))
            {
                bindings.extend(harvest_non_recursive(&mut value));
                bindings.insert(id, (value, max_iter));
            }
            bindings.extend(harvest_non_recursive(body));
            replace_bindings_from_map(bindings, ids, values, limits);
        }
    }

    /// Harvest any safe-to-lift non-recursive bindings from a `LetRec`
    /// expression.
    ///
    /// At the moment, we reason that a binding can be lifted without changing
    /// the output if both:
    /// 1. It references no other non-lifted binding bound in `expr`,
    /// 2. It is referenced by no prior non-lifted binding in `expr`.
    ///
    /// The rationale is that (1) ensures that the binding's value does not
    /// change across iterations, and that (2) ensures that all observations of
    /// the binding are after it assumes its first value, rather than when it
    /// could be empty.
    pub(crate) fn harvest_non_recursive(
        expr: &mut MirRelationExpr,
    ) -> BTreeMap<LocalId, (MirRelationExpr, Option<LetRecLimit>)> {
        if let MirRelationExpr::LetRec {
            ids,
            values,
            limits,
            body,
        } = expr
        {
            // Bindings to lift.
            let mut lifted = BTreeMap::<LocalId, (MirRelationExpr, Option<LetRecLimit>)>::new();
            // Bindings to retain.
            let mut retained = BTreeMap::<LocalId, (MirRelationExpr, Option<LetRecLimit>)>::new();

            // All remaining LocalIds bound by the enclosing LetRec.
            let mut id_set = ids.iter().cloned().collect::<BTreeSet<LocalId>>();
            // All LocalIds referenced up to (including) the current binding.
            let mut cannot = BTreeSet::<LocalId>::new();
            // The reference count of the current bindings.
            let mut refcnt = BTreeMap::<LocalId, usize>::new();

            for ((id, value), max_iter) in ids
                .drain(..)
                .zip_eq(values.drain(..))
                .zip_eq(limits.drain(..))
            {
                refcnt.clear();
                super::support::count_local_id_uses(&value, &mut refcnt);

                // LocalIds that have already been referenced cannot be lifted.
                cannot.extend(refcnt.keys().cloned());

                // - The first conjunct excludes bindings that have already been
                //   referenced.
                // - The second conjunct excludes bindings that reference a
                //   LocalId that either defined later or is a known retained.
                if !cannot.contains(&id) && !refcnt.keys().any(|i| id_set.contains(i)) {
                    lifted.insert(id, (value, None)); // Non-recursive bindings don't need a limit
                    id_set.remove(&id);
                } else {
                    retained.insert(id, (value, max_iter));
                }
            }

            replace_bindings_from_map(retained, ids, values, limits);
            if values.is_empty() {
                *expr = body.take_dangerous();
            }

            lifted
        } else {
            BTreeMap::new()
        }
    }

    /// Harvest any safe-to-lower non-recursive suffix of binding from a
    /// `LetRec` expression.
    pub(crate) fn harvest_nonrec_suffix(
        expr: &mut MirRelationExpr,
    ) -> Result<BTreeMap<LocalId, MirRelationExpr>, RecursionLimitError> {
        if let MirRelationExpr::LetRec {
            ids,
            values,
            limits,
            body,
        } = expr
        {
            // Bindings to lower.
            let mut lowered = BTreeMap::<LocalId, MirRelationExpr>::new();

            let rec_ids = MirRelationExpr::recursive_ids(ids, values);

            while ids.last().map(|id| !rec_ids.contains(id)).unwrap_or(false) {
                let id = ids.pop().expect("non-empty ids");
                let value = values.pop().expect("non-empty values");
                let _limit = limits.pop().expect("non-empty limits");

                lowered.insert(id, value); // Non-recursive bindings don't need a limit
            }

            if values.is_empty() {
                *expr = body.take_dangerous();
            }

            Ok(lowered)
        } else {
            Ok(BTreeMap::new())
        }
    }

    pub(crate) fn assert_no_lets(expr: &MirRelationExpr) {
        expr.visit_pre(|expr| {
            assert!(!matches!(expr, MirRelationExpr::Let { .. }));
        });
    }

    /// Asserts that `expr` in "LetRec-major" form.
    ///
    /// This means `expr` is either `LetRec`-free, or a `LetRec` whose values and body are `LetRec`-major.
    pub(crate) fn assert_letrec_major(expr: &MirRelationExpr) {
        let mut todo = vec![expr];
        while let Some(expr) = todo.pop() {
            match expr {
                MirRelationExpr::LetRec {
                    ids: _,
                    values,
                    limits: _,
                    body,
                } => {
                    todo.extend(values.iter());
                    todo.push(body);
                }
                _ => {
                    expr.visit_pre(|expr| {
                        assert!(!matches!(expr, MirRelationExpr::LetRec { .. }));
                    });
                }
            }
        }
    }
}

mod inlining {

    use std::collections::BTreeMap;

    use itertools::Itertools;
    use mz_expr::{Id, LetRecLimit, LocalId, MirRelationExpr};

    use crate::normalize_lets::support::replace_bindings_from_map;

    pub(super) fn inline_lets(
        expr: &mut MirRelationExpr,
        inline_mfp: bool,
    ) -> Result<(), crate::TransformError> {
        let mut worklist = vec![&mut *expr];
        while let Some(expr) = worklist.pop() {
            inline_lets_core(expr, inline_mfp)?;
            // We descend only into `LetRec` nodes, because `promote_let_rec` ensured that all
            // `LetRec` nodes are clustered near the root. This means that we can get to all the
            // `LetRec` nodes by just descending into `LetRec` nodes, as there can't be any other
            // nodes between them.
            if let MirRelationExpr::LetRec {
                ids: _,
                values,
                limits: _,
                body,
            } = expr
            {
                worklist.extend(values);
                worklist.push(body);
            }
        }
        Ok(())
    }

    /// Considers inlining actions to perform for a sequence of bindings and a
    /// following body.
    ///
    /// A let binding may be inlined only in subsequent bindings or in the body;
    /// other bindings should not "immediately" observe the binding, as that
    /// would be a change to the semantics of `LetRec`. For example, it would
    /// not be correct to replace `C` with `A` in the definition of `B` here:
    /// ```ignore
    /// let A = ...;
    /// let B = A - C;
    /// let C = A;
    /// ```
    /// The explanation is that `B` should always be the difference between the
    /// current and previous `A`, and that the substitution of `C` would instead
    /// make it always zero, changing its definition.
    ///
    /// Here a let binding is proposed for inlining if any of the following is true:
    ///  1. It has a single reference across all bindings and the body.
    ///  2. It is a "sufficiently simple" `Get`, determined in part by the
    ///     `inline_mfp` argument.
    ///
    /// We don't need extra checks for `limits`, because
    ///  - `limits` is only relevant when a binding is directly used through a back edge (because
    ///    that is where the rendering puts the `limits` check);
    ///  - when a binding is directly used through a back edge, it can't be inlined anyway.
    ///  - Also note that if a `LetRec` completely disappears at the end of `inline_lets_core`, then
    ///    there was no recursion in it.
    ///
    /// The case of `Constant` binding is handled here (as opposed to
    /// `FoldConstants`) in a somewhat limited manner (see database-issues#5346). Although a
    /// bit weird, constants should also not be inlined into prior bindings as
    /// this does change the behavior from one where the collection is initially
    /// empty to one where it is always the constant.
    ///
    /// Having inlined bindings, many of them may now be dead (with no
    /// transitive references from `body`). These can now be removed. They may
    /// not be exactly those bindings that were inlineable, as we may not always
    /// be able to apply inlining due to ordering (we cannot inline a binding
    /// into one that is not strictly later).
    pub(super) fn inline_lets_core(
        expr: &mut MirRelationExpr,
        inline_mfp: bool,
    ) -> Result<(), crate::TransformError> {
        if let MirRelationExpr::LetRec {
            ids,
            values,
            limits,
            body,
        } = expr
        {
            // Count the number of uses of each local id across all expressions.
            let mut counts = BTreeMap::new();
            for value in values.iter() {
                super::support::count_local_id_uses(value, &mut counts);
            }
            super::support::count_local_id_uses(body, &mut counts);

            // Each binding can reach one of three positions on its inlineability:
            //  1. The binding is used once and is available to be directly taken.
            //  2. The binding is simple enough that it can just be cloned.
            //  3. The binding is not available for inlining.
            let mut inline_offers = BTreeMap::new();

            // Each binding may require the expiration of prior inlining offers.
            // This occurs when an inlined body references the prior iterate of a binding,
            // and inlining it would change the meaning to be the current iterate.
            // Roughly, all inlining offers expire just after the binding of the least
            // identifier they contain that is greater than the bound identifier itself.
            let mut expire_offers = BTreeMap::new();
            let mut expired_offers = Vec::new();

            // For each binding, inline `Get`s and then determine if *it* should be inlined.
            // It is important that we do the substitution in-order and before reasoning
            // about the inlineability of each binding, to ensure that our conclusion about
            // the inlineability of a binding stays put. Specifically,
            //   1. by going in order no substitution will increase the `Get`-count of an
            //      identifier beyond one, as all in values with strictly greater identifiers.
            //   2. by performing the substitution before reasoning, the structure of the value
            //      as it would be substituted is fixed.
            for ((id, mut expr), max_iter) in ids
                .drain(..)
                .zip_eq(values.drain(..))
                .zip_eq(limits.drain(..))
            {
                // Substitute any appropriate prior let bindings.
                inline_lets_helper(&mut expr, &mut inline_offers)?;

                // Determine the first `id'` at which any inlining offer must expire.
                // An inlining offer expires because it references an `id'` that is not yet bound,
                // indicating a reference to the *prior* iterate of that identifier. Inlining the
                // expression once `id'` becomes bound would advance the reference to be the
                // *current* iterate of the identifier.
                MirRelationExpr::collect_expirations(id, &expr, &mut expire_offers);

                // Gets for `id` only occur in later expressions, so this should still be correct.
                let num_gets = counts.get(&id).map(|x| *x).unwrap_or(0);
                // Counts of zero or one lead to substitution; otherwise certain simple structures
                // are cloned in to `Get` operators, and all others emitted as `Let` bindings.
                if num_gets == 0 {
                } else if num_gets == 1 {
                    inline_offers.insert(id, InlineOffer::Take(Some(expr), max_iter));
                } else {
                    let clone_binding = {
                        let stripped_value = if inline_mfp {
                            mz_expr::MapFilterProject::extract_non_errors_from_expr(&expr).1
                        } else {
                            &expr
                        };
                        match stripped_value {
                            MirRelationExpr::Get { .. } | MirRelationExpr::Constant { .. } => true,
                            _ => false,
                        }
                    };

                    if clone_binding {
                        inline_offers.insert(id, InlineOffer::Clone(expr, max_iter));
                    } else {
                        inline_offers.insert(id, InlineOffer::Unavailable(expr, max_iter));
                    }
                }

                // We must now discard any offers that reference `id`, as it is no longer correct
                // to inline such an offer as it would have access to this iteration's binding of
                // `id` rather than the prior iteration's binding of `id`.
                expired_offers.extend(MirRelationExpr::do_expirations(
                    id,
                    &mut expire_offers,
                    &mut inline_offers,
                ));
            }
            // Complete the inlining in `body`.
            inline_lets_helper(body, &mut inline_offers)?;

            // Re-introduce expired offers for the subsequent logic that expects to see them all.
            for (id, offer) in expired_offers.into_iter() {
                inline_offers.insert(id, offer);
            }

            // We may now be able to discard some of `inline_offer` based on the remaining pattern of `Get` expressions.
            // Starting from `body` and working backwards, we can activate bindings that are still required because we
            // observe `Get` expressions referencing them. Any bindings not so identified can be dropped (including any
            // that may be part of a cycle not reachable from `body`).
            let mut let_bindings = BTreeMap::new();
            let mut todo = Vec::new();
            super::support::for_local_id(body, |id| todo.push(id));
            while let Some(id) = todo.pop() {
                if let Some(offer) = inline_offers.remove(&id) {
                    let (value, max_iter) = match offer {
                        InlineOffer::Take(value, max_iter) => (
                            value.ok_or_else(|| {
                                crate::TransformError::Internal(
                                    "Needed value already taken".to_string(),
                                )
                            })?,
                            max_iter,
                        ),
                        InlineOffer::Clone(value, max_iter) => (value, max_iter),
                        InlineOffer::Unavailable(value, max_iter) => (value, max_iter),
                    };
                    super::support::for_local_id(&value, |id| todo.push(id));
                    let_bindings.insert(id, (value, max_iter));
                }
            }

            // If bindings remain we update the `LetRec`, otherwise we remove it.
            if !let_bindings.is_empty() {
                replace_bindings_from_map(let_bindings, ids, values, limits);
            } else {
                *expr = body.take_dangerous();
            }
        }
        Ok(())
    }

    /// Possible states of let binding inlineability.
    enum InlineOffer {
        /// There is a unique reference to this value and given the option it should take this expression.
        Take(Option<MirRelationExpr>, Option<LetRecLimit>),
        /// Any reference to this value should clone this expression.
        Clone(MirRelationExpr, Option<LetRecLimit>),
        /// Any reference to this value should do no inlining of it.
        Unavailable(MirRelationExpr, Option<LetRecLimit>),
    }

    /// Substitute `Get{id}` expressions for any proposed expressions.
    ///
    /// The proposed expressions can be proposed either to be taken or cloned.
    fn inline_lets_helper(
        expr: &mut MirRelationExpr,
        inline_offer: &mut BTreeMap<LocalId, InlineOffer>,
    ) -> Result<(), crate::TransformError> {
        let mut worklist = vec![expr];
        while let Some(expr) = worklist.pop() {
            if let MirRelationExpr::Get {
                id: Id::Local(id), ..
            } = expr
            {
                if let Some(offer) = inline_offer.get_mut(id) {
                    // It is important that we *not* continue to iterate
                    // on the contents of `offer`, which has already been
                    // maximally inlined. If we did, we could mis-inline
                    // bindings into bodies that precede them, which would
                    // change the semantics of the expression.
                    match offer {
                        InlineOffer::Take(value, _max_iter) => {
                            *expr = value.take().ok_or_else(|| {
                                crate::TransformError::Internal(format!(
                                    "Value already taken for {:?}",
                                    id
                                ))
                            })?;
                        }
                        InlineOffer::Clone(value, _max_iter) => {
                            *expr = value.clone();
                        }
                        InlineOffer::Unavailable(_, _) => {
                            // Do nothing.
                        }
                    }
                } else {
                    // Presumably a reference to an outer scope.
                }
            } else {
                worklist.extend(expr.children_mut().rev());
            }
        }
        Ok(())
    }
}

mod renumbering {

    use std::collections::BTreeMap;

    use itertools::Itertools;
    use mz_expr::{Id, LocalId, MirRelationExpr};
    use mz_ore::id_gen::IdGen;

    /// Re-assign an identifier to each `Let`.
    ///
    /// Under the assumption that `id_gen` produces identifiers in order, this process
    /// maintains in-orderness of `LetRec` identifiers.
    pub fn renumber_bindings(
        relation: &mut MirRelationExpr,
        id_gen: &mut IdGen,
    ) -> Result<(), crate::TransformError> {
        let mut renaming = BTreeMap::new();
        determine(&*relation, &mut renaming, id_gen)?;
        implement(relation, &renaming)?;
        Ok(())
    }

    /// Performs an in-order traversal of the AST, assigning identifiers as it goes.
    fn determine(
        relation: &MirRelationExpr,
        remap: &mut BTreeMap<LocalId, LocalId>,
        id_gen: &mut IdGen,
    ) -> Result<(), crate::TransformError> {
        // The stack contains pending work as `Result<LocalId, &MirRelationExpr>`, where
        // 1. 'Ok(id)` means the identifier `id` is ready for renumbering,
        // 2. `Err(expr)` means that the expression `expr` needs to be further processed.
        let mut stack: Vec<Result<LocalId, _>> = vec![Err(relation)];
        while let Some(action) = stack.pop() {
            match action {
                Ok(id) => {
                    if remap.contains_key(&id) {
                        Err(crate::TransformError::Internal(format!(
                            "Shadowing of let binding for {:?}",
                            id
                        )))?;
                    } else {
                        remap.insert(id, LocalId::new(id_gen.allocate_id()));
                    }
                }
                Err(expr) => match expr {
                    MirRelationExpr::Let { id, value, body } => {
                        stack.push(Err(body));
                        stack.push(Ok(*id));
                        stack.push(Err(value));
                    }
                    MirRelationExpr::LetRec {
                        ids,
                        values,
                        limits: _,
                        body,
                    } => {
                        stack.push(Err(body));
                        for (id, value) in ids.iter().rev().zip_eq(values.iter().rev()) {
                            stack.push(Ok(*id));
                            stack.push(Err(value));
                        }
                    }
                    _ => {
                        stack.extend(expr.children().rev().map(Err));
                    }
                },
            }
        }
        Ok(())
    }

    fn implement(
        relation: &mut MirRelationExpr,
        remap: &BTreeMap<LocalId, LocalId>,
    ) -> Result<(), crate::TransformError> {
        let mut worklist = vec![relation];
        while let Some(expr) = worklist.pop() {
            match expr {
                MirRelationExpr::Let { id, .. } => {
                    *id = *remap
                        .get(id)
                        .ok_or(crate::TransformError::IdentifierMissing(*id))?;
                }
                MirRelationExpr::LetRec { ids, .. } => {
                    for id in ids.iter_mut() {
                        *id = *remap
                            .get(id)
                            .ok_or(crate::TransformError::IdentifierMissing(*id))?;
                    }
                }
                MirRelationExpr::Get {
                    id: Id::Local(id), ..
                } => {
                    *id = *remap
                        .get(id)
                        .ok_or(crate::TransformError::IdentifierMissing(*id))?;
                }
                _ => {
                    // Remapped identifiers not used in these patterns.
                }
            }
            // The order is not critical, but behave as a stack for clarity.
            worklist.extend(expr.children_mut().rev());
        }
        Ok(())
    }
}