mz_expr/

relation.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.

#![warn(missing_docs)]

use std::cmp::{Ordering, max};
use std::collections::{BTreeMap, BTreeSet};
use std::fmt;
use std::fmt::{Display, Formatter};
use std::hash::{DefaultHasher, Hash, Hasher};
use std::num::NonZeroU64;
use std::time::Instant;

use bytesize::ByteSize;
use differential_dataflow::containers::{Columnation, CopyRegion};
use itertools::Itertools;
use mz_lowertest::MzReflect;
use mz_ore::cast::CastFrom;
use mz_ore::collections::CollectionExt;
use mz_ore::id_gen::IdGen;
use mz_ore::metrics::Histogram;
use mz_ore::num::NonNeg;
use mz_ore::soft_assert_no_log;
use mz_ore::stack::RecursionLimitError;
use mz_ore::str::Indent;
use mz_proto::{IntoRustIfSome, ProtoType, RustType, TryFromProtoError};
use mz_repr::adt::numeric::NumericMaxScale;
use mz_repr::explain::text::text_string_at;
use mz_repr::explain::{
    DummyHumanizer, ExplainConfig, ExprHumanizer, IndexUsageType, PlanRenderingContext,
};
use mz_repr::{
    ColumnName, ColumnType, Datum, Diff, GlobalId, IntoRowIterator, RelationType, Row, RowIterator,
    ScalarType,
};
use proptest::prelude::{Arbitrary, BoxedStrategy, any};
use proptest::strategy::{Strategy, Union};
use proptest_derive::Arbitrary;
use serde::{Deserialize, Serialize};

use crate::Id::Local;
use crate::explain::{HumanizedExpr, HumanizerMode};
use crate::relation::func::{AggregateFunc, LagLeadType, TableFunc};
use crate::row::{RowCollection, SortedRowCollectionIter};
use crate::visit::{Visit, VisitChildren};
use crate::{
    EvalError, FilterCharacteristics, Id, LocalId, MirScalarExpr, UnaryFunc, VariadicFunc,
    func as scalar_func,
};

pub mod canonicalize;
pub mod func;
pub mod join_input_mapper;

include!(concat!(env!("OUT_DIR"), "/mz_expr.relation.rs"));

/// A recursion limit to be used for stack-safe traversals of [`MirRelationExpr`] trees.
///
/// The recursion limit must be large enough to accommodate for the linear representation
/// of some pathological but frequently occurring query fragments.
///
/// For example, in MIR we could have long chains of
/// - (1) `Let` bindings,
/// - (2) `CallBinary` calls with associative functions such as `+`
///
/// Until we fix those, we need to stick with the larger recursion limit.
pub const RECURSION_LIMIT: usize = 2048;

/// A trait for types that describe how to build a collection.
pub trait CollectionPlan {
    /// Collects the set of global identifiers from dataflows referenced in Get.
    fn depends_on_into(&self, out: &mut BTreeSet<GlobalId>);

    /// Returns the set of global identifiers from dataflows referenced in Get.
    ///
    /// See [`CollectionPlan::depends_on_into`] to reuse an existing `BTreeSet`.
    fn depends_on(&self) -> BTreeSet<GlobalId> {
        let mut out = BTreeSet::new();
        self.depends_on_into(&mut out);
        out
    }
}

/// An abstract syntax tree which defines a collection.
///
/// The AST is meant to reflect the capabilities of the `differential_dataflow::Collection` type,
/// written generically enough to avoid run-time compilation work.
///
/// `derived_hash_with_manual_eq` was complaining for the wrong reason: This lint exists because
/// it's bad when `Eq` doesn't agree with `Hash`, which is often quite likely if one of them is
/// implemented manually. However, our manual implementation of `Eq` _will_ agree with the derived
/// one. This is because the reason for the manual implementation is not to change the semantics
/// from the derived one, but to avoid stack overflows.
#[allow(clippy::derived_hash_with_manual_eq)]
#[derive(Clone, Debug, Ord, PartialOrd, Serialize, Deserialize, MzReflect, Hash)]
pub enum MirRelationExpr {
    /// A constant relation containing specified rows.
    ///
    /// The runtime memory footprint of this operator is zero.
    ///
    /// When you would like to pattern match on this, consider using `MirRelationExpr::as_const`
    /// instead, which looks behind `ArrangeBy`s. You might want this matching behavior because
    /// constant folding doesn't remove `ArrangeBy`s.
    Constant {
        /// Rows of the constant collection and their multiplicities.
        rows: Result<Vec<(Row, Diff)>, EvalError>,
        /// Schema of the collection.
        typ: RelationType,
    },
    /// Get an existing dataflow.
    ///
    /// The runtime memory footprint of this operator is zero.
    Get {
        /// The identifier for the collection to load.
        #[mzreflect(ignore)]
        id: Id,
        /// Schema of the collection.
        typ: RelationType,
        /// If this is a global Get, this will indicate whether we are going to read from Persist or
        /// from an index, or from a different object in `objects_to_build`. If it's an index, then
        /// how downstream dataflow operations will use this index is also recorded. This is filled
        /// by `prune_and_annotate_dataflow_index_imports`. Note that this is not used by the
        /// lowering to LIR, but is used only by EXPLAIN.
        #[mzreflect(ignore)]
        access_strategy: AccessStrategy,
    },
    /// Introduce a temporary dataflow.
    ///
    /// The runtime memory footprint of this operator is zero.
    Let {
        /// The identifier to be used in `Get` variants to retrieve `value`.
        #[mzreflect(ignore)]
        id: LocalId,
        /// The collection to be bound to `id`.
        value: Box<MirRelationExpr>,
        /// The result of the `Let`, evaluated with `id` bound to `value`.
        body: Box<MirRelationExpr>,
    },
    /// Introduce mutually recursive bindings.
    ///
    /// Each `LocalId` is immediately bound to an initially empty  collection
    /// with the type of its corresponding `MirRelationExpr`. Repeatedly, each
    /// binding is evaluated using the current contents of each other binding,
    /// and is refreshed to contain the new evaluation. This process continues
    /// through all bindings, and repeats as long as changes continue to occur.
    ///
    /// The resulting value of the expression is `body` evaluated once in the
    /// context of the final iterates.
    ///
    /// A zero-binding instance can be replaced by `body`.
    /// A single-binding instance is equivalent to `MirRelationExpr::Let`.
    ///
    /// The runtime memory footprint of this operator is zero.
    LetRec {
        /// The identifiers to be used in `Get` variants to retrieve each `value`.
        #[mzreflect(ignore)]
        ids: Vec<LocalId>,
        /// The collections to be bound to each `id`.
        values: Vec<MirRelationExpr>,
        /// Maximum number of iterations, after which we should artificially force a fixpoint.
        /// (Whether we error or just stop is configured by `LetRecLimit::return_at_limit`.)
        /// The per-`LetRec` limit that the user specified is initially copied to each binding to
        /// accommodate slicing and merging of `LetRec`s in MIR transforms (e.g., `NormalizeLets`).
        #[mzreflect(ignore)]
        limits: Vec<Option<LetRecLimit>>,
        /// The result of the `Let`, evaluated with `id` bound to `value`.
        body: Box<MirRelationExpr>,
    },
    /// Project out some columns from a dataflow
    ///
    /// The runtime memory footprint of this operator is zero.
    Project {
        /// The source collection.
        input: Box<MirRelationExpr>,
        /// Indices of columns to retain.
        outputs: Vec<usize>,
    },
    /// Append new columns to a dataflow
    ///
    /// The runtime memory footprint of this operator is zero.
    Map {
        /// The source collection.
        input: Box<MirRelationExpr>,
        /// Expressions which determine values to append to each row.
        /// An expression may refer to columns in `input` or
        /// expressions defined earlier in the vector
        scalars: Vec<MirScalarExpr>,
    },
    /// Like Map, but yields zero-or-more output rows per input row
    ///
    /// The runtime memory footprint of this operator is zero.
    FlatMap {
        /// The source collection
        input: Box<MirRelationExpr>,
        /// The table func to apply
        func: TableFunc,
        /// The argument to the table func
        exprs: Vec<MirScalarExpr>,
    },
    /// Keep rows from a dataflow where all the predicates are true
    ///
    /// The runtime memory footprint of this operator is zero.
    Filter {
        /// The source collection.
        input: Box<MirRelationExpr>,
        /// Predicates, each of which must be true.
        predicates: Vec<MirScalarExpr>,
    },
    /// Join several collections, where some columns must be equal.
    ///
    /// For further details consult the documentation for [`MirRelationExpr::join`].
    ///
    /// The runtime memory footprint of this operator can be proportional to
    /// the sizes of all inputs and the size of all joins of prefixes.
    /// This may be reduced due to arrangements available at rendering time.
    Join {
        /// A sequence of input relations.
        inputs: Vec<MirRelationExpr>,
        /// A sequence of equivalence classes of expressions on the cross product of inputs.
        ///
        /// Each equivalence class is a list of scalar expressions, where for each class the
        /// intended interpretation is that all evaluated expressions should be equal.
        ///
        /// Each scalar expression is to be evaluated over the cross-product of all records
        /// from all inputs. In many cases this may just be column selection from specific
        /// inputs, but more general cases exist (e.g. complex functions of multiple columns
        /// from multiple inputs, or just constant literals).
        equivalences: Vec<Vec<MirScalarExpr>>,
        /// Join implementation information.
        #[serde(default)]
        implementation: JoinImplementation,
    },
    /// Group a dataflow by some columns and aggregate over each group
    ///
    /// The runtime memory footprint of this operator is at most proportional to the
    /// number of distinct records in the input and output. The actual requirements
    /// can be less: the number of distinct inputs to each aggregate, summed across
    /// each aggregate, plus the output size. For more details consult the code that
    /// builds the associated dataflow.
    Reduce {
        /// The source collection.
        input: Box<MirRelationExpr>,
        /// Column indices used to form groups.
        group_key: Vec<MirScalarExpr>,
        /// Expressions which determine values to append to each row, after the group keys.
        aggregates: Vec<AggregateExpr>,
        /// True iff the input is known to monotonically increase (only addition of records).
        #[serde(default)]
        monotonic: bool,
        /// User hint: expected number of values per group key. Used to optimize physical rendering.
        #[serde(default)]
        expected_group_size: Option<u64>,
    },
    /// Groups and orders within each group, limiting output.
    ///
    /// The runtime memory footprint of this operator is proportional to its input and output.
    TopK {
        /// The source collection.
        input: Box<MirRelationExpr>,
        /// Column indices used to form groups.
        group_key: Vec<usize>,
        /// Column indices used to order rows within groups.
        order_key: Vec<ColumnOrder>,
        /// Number of records to retain
        #[serde(default)]
        limit: Option<MirScalarExpr>,
        /// Number of records to skip
        #[serde(default)]
        offset: usize,
        /// True iff the input is known to monotonically increase (only addition of records).
        #[serde(default)]
        monotonic: bool,
        /// User-supplied hint: how many rows will have the same group key.
        #[serde(default)]
        expected_group_size: Option<u64>,
    },
    /// Return a dataflow where the row counts are negated
    ///
    /// The runtime memory footprint of this operator is zero.
    Negate {
        /// The source collection.
        input: Box<MirRelationExpr>,
    },
    /// Keep rows from a dataflow where the row counts are positive
    ///
    /// The runtime memory footprint of this operator is proportional to its input and output.
    Threshold {
        /// The source collection.
        input: Box<MirRelationExpr>,
    },
    /// Adds the frequencies of elements in contained sets.
    ///
    /// The runtime memory footprint of this operator is zero.
    Union {
        /// A source collection.
        base: Box<MirRelationExpr>,
        /// Source collections to union.
        inputs: Vec<MirRelationExpr>,
    },
    /// Technically a no-op. Used to render an index. Will be used to optimize queries
    /// on finer grain. Each `keys` item represents a different index that should be
    /// produced from the `keys`.
    ///
    /// The runtime memory footprint of this operator is proportional to its input.
    ArrangeBy {
        /// The source collection
        input: Box<MirRelationExpr>,
        /// Columns to arrange `input` by, in order of decreasing primacy
        keys: Vec<Vec<MirScalarExpr>>,
    },
}

impl PartialEq for MirRelationExpr {
    fn eq(&self, other: &Self) -> bool {
        // Capture the result and test it wrt `Ord` implementation in test environments.
        let result = structured_diff::MreDiff::new(self, other).next().is_none();
        mz_ore::soft_assert_eq_no_log!(result, self.cmp(other) == Ordering::Equal);
        result
    }
}
impl Eq for MirRelationExpr {}

impl Arbitrary for MirRelationExpr {
    type Parameters = ();

    fn arbitrary_with((): Self::Parameters) -> Self::Strategy {
        const VEC_LEN: usize = 2;

        // A strategy for generating the leaf cases.
        let leaf = Union::new([
            (
                any::<Result<Vec<(Row, Diff)>, EvalError>>(),
                any::<RelationType>(),
            )
                .prop_map(|(rows, typ)| MirRelationExpr::Constant { rows, typ })
                .boxed(),
            (any::<Id>(), any::<RelationType>(), any::<AccessStrategy>())
                .prop_map(|(id, typ, access_strategy)| MirRelationExpr::Get {
                    id,
                    typ,
                    access_strategy,
                })
                .boxed(),
        ])
        .boxed();

        leaf.prop_recursive(2, 2, 2, |inner| {
            Union::new([
                // Let
                (any::<LocalId>(), inner.clone(), inner.clone())
                    .prop_map(|(id, value, body)| MirRelationExpr::Let {
                        id,
                        value: Box::new(value),
                        body: Box::new(body),
                    })
                    .boxed(),
                // LetRec
                (
                    any::<Vec<LocalId>>(),
                    proptest::collection::vec(inner.clone(), 1..VEC_LEN),
                    any::<Vec<Option<LetRecLimit>>>(),
                    inner.clone(),
                )
                    .prop_map(|(ids, values, limits, body)| MirRelationExpr::LetRec {
                        ids,
                        values,
                        limits,
                        body: Box::new(body),
                    })
                    .boxed(),
                // Project
                (inner.clone(), any::<Vec<usize>>())
                    .prop_map(|(input, outputs)| MirRelationExpr::Project {
                        input: Box::new(input),
                        outputs,
                    })
                    .boxed(),
                // Map
                (inner.clone(), any::<Vec<MirScalarExpr>>())
                    .prop_map(|(input, scalars)| MirRelationExpr::Map {
                        input: Box::new(input),
                        scalars,
                    })
                    .boxed(),
                // FlatMap
                (
                    inner.clone(),
                    any::<TableFunc>(),
                    any::<Vec<MirScalarExpr>>(),
                )
                    .prop_map(|(input, func, exprs)| MirRelationExpr::FlatMap {
                        input: Box::new(input),
                        func,
                        exprs,
                    })
                    .boxed(),
                // Filter
                (inner.clone(), any::<Vec<MirScalarExpr>>())
                    .prop_map(|(input, predicates)| MirRelationExpr::Filter {
                        input: Box::new(input),
                        predicates,
                    })
                    .boxed(),
                // Join
                (
                    proptest::collection::vec(inner.clone(), 1..VEC_LEN),
                    any::<Vec<Vec<MirScalarExpr>>>(),
                    any::<JoinImplementation>(),
                )
                    .prop_map(
                        |(inputs, equivalences, implementation)| MirRelationExpr::Join {
                            inputs,
                            equivalences,
                            implementation,
                        },
                    )
                    .boxed(),
                // Reduce
                (
                    inner.clone(),
                    any::<Vec<MirScalarExpr>>(),
                    any::<Vec<AggregateExpr>>(),
                    any::<bool>(),
                    any::<Option<u64>>(),
                )
                    .prop_map(
                        |(input, group_key, aggregates, monotonic, expected_group_size)| {
                            MirRelationExpr::Reduce {
                                input: Box::new(input),
                                group_key,
                                aggregates,
                                monotonic,
                                expected_group_size,
                            }
                        },
                    )
                    .boxed(),
                // TopK
                (
                    inner.clone(),
                    any::<Vec<usize>>(),
                    any::<Vec<ColumnOrder>>(),
                    any::<Option<MirScalarExpr>>(),
                    any::<usize>(),
                    any::<bool>(),
                    any::<Option<u64>>(),
                )
                    .prop_map(
                        |(
                            input,
                            group_key,
                            order_key,
                            limit,
                            offset,
                            monotonic,
                            expected_group_size,
                        )| MirRelationExpr::TopK {
                            input: Box::new(input),
                            group_key,
                            order_key,
                            limit,
                            offset,
                            monotonic,
                            expected_group_size,
                        },
                    )
                    .boxed(),
                // Negate
                inner
                    .clone()
                    .prop_map(|input| MirRelationExpr::Negate {
                        input: Box::new(input),
                    })
                    .boxed(),
                // Threshold
                inner
                    .clone()
                    .prop_map(|input| MirRelationExpr::Threshold {
                        input: Box::new(input),
                    })
                    .boxed(),
                // Union
                (
                    inner.clone(),
                    proptest::collection::vec(inner.clone(), 1..VEC_LEN),
                )
                    .prop_map(|(base, inputs)| MirRelationExpr::Union {
                        base: Box::new(base),
                        inputs,
                    })
                    .boxed(),
                // ArrangeBy
                (inner.clone(), any::<Vec<Vec<MirScalarExpr>>>())
                    .prop_map(|(input, keys)| MirRelationExpr::ArrangeBy {
                        input: Box::new(input),
                        keys,
                    })
                    .boxed(),
            ])
        })
        .boxed()
    }

    type Strategy = BoxedStrategy<Self>;
}

impl MirRelationExpr {
    /// Reports the schema of the relation.
    ///
    /// This method determines the type through recursive traversal of the
    /// relation expression, drawing from the types of base collections.
    /// As such, this is not an especially cheap method, and should be used
    /// judiciously.
    ///
    /// The relation type is computed incrementally with a recursive post-order
    /// traversal, that accumulates the input types for the relations yet to be
    /// visited in `type_stack`.
    pub fn typ(&self) -> RelationType {
        let mut type_stack = Vec::new();
        #[allow(deprecated)]
        self.visit_pre_post_nolimit(
            &mut |e: &MirRelationExpr| -> Option<Vec<&MirRelationExpr>> {
                match &e {
                    MirRelationExpr::Let { body, .. } => {
                        // Do not traverse the value sub-graph, since it's not relevant for
                        // determining the relation type of Let operators.
                        Some(vec![&*body])
                    }
                    MirRelationExpr::LetRec { body, .. } => {
                        // Do not traverse the value sub-graph, since it's not relevant for
                        // determining the relation type of Let operators.
                        Some(vec![&*body])
                    }
                    _ => None,
                }
            },
            &mut |e: &MirRelationExpr| {
                match e {
                    MirRelationExpr::Let { .. } => {
                        let body_typ = type_stack.pop().unwrap();
                        // Insert a dummy relation type for the value, since `typ_with_input_types`
                        // won't look at it, but expects the relation type of the body to be second.
                        type_stack.push(RelationType::empty());
                        type_stack.push(body_typ);
                    }
                    MirRelationExpr::LetRec { values, .. } => {
                        let body_typ = type_stack.pop().unwrap();
                        // Insert dummy relation types for the values, since `typ_with_input_types`
                        // won't look at them, but expects the relation type of the body to be last.
                        type_stack
                            .extend(std::iter::repeat(RelationType::empty()).take(values.len()));
                        type_stack.push(body_typ);
                    }
                    _ => {}
                }
                let num_inputs = e.num_inputs();
                let relation_type =
                    e.typ_with_input_types(&type_stack[type_stack.len() - num_inputs..]);
                type_stack.truncate(type_stack.len() - num_inputs);
                type_stack.push(relation_type);
            },
        );
        assert_eq!(type_stack.len(), 1);
        type_stack.pop().unwrap()
    }

    /// Reports the schema of the relation given the schema of the input relations.
    ///
    /// `input_types` is required to contain the schemas for the input relations of
    /// the current relation in the same order as they are visited by `try_visit_children`
    /// method, even though not all may be used for computing the schema of the
    /// current relation. For example, `Let` expects two input types, one for the
    /// value relation and one for the body, in that order, but only the one for the
    /// body is used to determine the type of the `Let` relation.
    ///
    /// It is meant to be used during post-order traversals to compute relation
    /// schemas incrementally.
    pub fn typ_with_input_types(&self, input_types: &[RelationType]) -> RelationType {
        let column_types = self.col_with_input_cols(input_types.iter().map(|i| &i.column_types));
        let unique_keys = self.keys_with_input_keys(
            input_types.iter().map(|i| i.arity()),
            input_types.iter().map(|i| &i.keys),
        );
        RelationType::new(column_types).with_keys(unique_keys)
    }

    /// Reports the column types of the relation given the column types of the
    /// input relations.
    ///
    /// This method delegates to `try_col_with_input_cols`, panicking if an `Err`
    /// variant is returned.
    pub fn col_with_input_cols<'a, I>(&self, input_types: I) -> Vec<ColumnType>
    where
        I: Iterator<Item = &'a Vec<ColumnType>>,
    {
        match self.try_col_with_input_cols(input_types) {
            Ok(col_types) => col_types,
            Err(err) => panic!("{err}"),
        }
    }

    /// Reports the column types of the relation given the column types of the input relations.
    ///
    /// `input_types` is required to contain the column types for the input relations of
    /// the current relation in the same order as they are visited by `try_visit_children`
    /// method, even though not all may be used for computing the schema of the
    /// current relation. For example, `Let` expects two input types, one for the
    /// value relation and one for the body, in that order, but only the one for the
    /// body is used to determine the type of the `Let` relation.
    ///
    /// It is meant to be used during post-order traversals to compute column types
    /// incrementally.
    pub fn try_col_with_input_cols<'a, I>(
        &self,
        mut input_types: I,
    ) -> Result<Vec<ColumnType>, String>
    where
        I: Iterator<Item = &'a Vec<ColumnType>>,
    {
        use MirRelationExpr::*;

        let col_types = match self {
            Constant { rows, typ } => {
                let mut col_types = typ.column_types.clone();
                let mut seen_null = vec![false; typ.arity()];
                if let Ok(rows) = rows {
                    for (row, _diff) in rows {
                        for (datum, i) in row.iter().zip_eq(0..typ.arity()) {
                            if datum.is_null() {
                                seen_null[i] = true;
                            }
                        }
                    }
                }
                for (&seen_null, i) in seen_null.iter().zip_eq(0..typ.arity()) {
                    if !seen_null {
                        col_types[i].nullable = false;
                    } else {
                        assert!(col_types[i].nullable);
                    }
                }
                col_types
            }
            Get { typ, .. } => typ.column_types.clone(),
            Project { outputs, .. } => {
                let input = input_types.next().unwrap();
                outputs.iter().map(|&i| input[i].clone()).collect()
            }
            Map { scalars, .. } => {
                let mut result = input_types.next().unwrap().clone();
                for scalar in scalars.iter() {
                    result.push(scalar.typ(&result))
                }
                result
            }
            FlatMap { func, .. } => {
                let mut result = input_types.next().unwrap().clone();
                result.extend(func.output_type().column_types);
                result
            }
            Filter { predicates, .. } => {
                let mut result = input_types.next().unwrap().clone();

                // Set as nonnull any columns where null values would cause
                // any predicate to evaluate to null.
                for column in non_nullable_columns(predicates) {
                    result[column].nullable = false;
                }
                result
            }
            Join { equivalences, .. } => {
                // Concatenate input column types
                let mut types = input_types.flat_map(|cols| cols.to_owned()).collect_vec();
                // In an equivalence class, if any column is non-null, then make all non-null
                for equivalence in equivalences {
                    let col_inds = equivalence
                        .iter()
                        .filter_map(|expr| match expr {
                            MirScalarExpr::Column(col, _name) => Some(*col),
                            _ => None,
                        })
                        .collect_vec();
                    if col_inds.iter().any(|i| !types.get(*i).unwrap().nullable) {
                        for i in col_inds {
                            types.get_mut(i).unwrap().nullable = false;
                        }
                    }
                }
                types
            }
            Reduce {
                group_key,
                aggregates,
                ..
            } => {
                let input = input_types.next().unwrap();
                group_key
                    .iter()
                    .map(|e| e.typ(input))
                    .chain(aggregates.iter().map(|agg| agg.typ(input)))
                    .collect()
            }
            TopK { .. } | Negate { .. } | Threshold { .. } | ArrangeBy { .. } => {
                input_types.next().unwrap().clone()
            }
            Let { .. } => {
                // skip over the input types for `value`.
                input_types.nth(1).unwrap().clone()
            }
            LetRec { values, .. } => {
                // skip over the input types for `values`.
                input_types.nth(values.len()).unwrap().clone()
            }
            Union { .. } => {
                let mut result = input_types.next().unwrap().clone();
                for input_col_types in input_types {
                    for (base_col, col) in result.iter_mut().zip_eq(input_col_types) {
                        *base_col = base_col
                            .union(col)
                            .map_err(|e| format!("{}\nin plan:\n{}", e, self.pretty()))?;
                    }
                }
                result
            }
        };

        Ok(col_types)
    }

    /// Reports the unique keys of the relation given the arities and the unique
    /// keys of the input relations.
    ///
    /// `input_arities` and `input_keys` are required to contain the
    /// corresponding info for the input relations of
    /// the current relation in the same order as they are visited by `try_visit_children`
    /// method, even though not all may be used for computing the schema of the
    /// current relation. For example, `Let` expects two input types, one for the
    /// value relation and one for the body, in that order, but only the one for the
    /// body is used to determine the type of the `Let` relation.
    ///
    /// It is meant to be used during post-order traversals to compute unique keys
    /// incrementally.
    pub fn keys_with_input_keys<'a, I, J>(
        &self,
        mut input_arities: I,
        mut input_keys: J,
    ) -> Vec<Vec<usize>>
    where
        I: Iterator<Item = usize>,
        J: Iterator<Item = &'a Vec<Vec<usize>>>,
    {
        use MirRelationExpr::*;

        let mut keys = match self {
            Constant {
                rows: Ok(rows),
                typ,
            } => {
                let n_cols = typ.arity();
                // If the `i`th entry is `Some`, then we have not yet observed non-uniqueness in the `i`th column.
                let mut unique_values_per_col = vec![Some(BTreeSet::<Datum>::default()); n_cols];
                for (row, diff) in rows {
                    for (i, datum) in row.iter().enumerate() {
                        if datum != Datum::Dummy {
                            if let Some(unique_vals) = &mut unique_values_per_col[i] {
                                let is_dupe = *diff != Diff::ONE || !unique_vals.insert(datum);
                                if is_dupe {
                                    unique_values_per_col[i] = None;
                                }
                            }
                        }
                    }
                }
                if rows.len() == 0 || (rows.len() == 1 && rows[0].1 == Diff::ONE) {
                    vec![vec![]]
                } else {
                    // XXX - Multi-column keys are not detected.
                    typ.keys
                        .iter()
                        .cloned()
                        .chain(
                            unique_values_per_col
                                .into_iter()
                                .enumerate()
                                .filter(|(_idx, unique_vals)| unique_vals.is_some())
                                .map(|(idx, _)| vec![idx]),
                        )
                        .collect()
                }
            }
            Constant { rows: Err(_), typ } | Get { typ, .. } => typ.keys.clone(),
            Threshold { .. } | ArrangeBy { .. } => input_keys.next().unwrap().clone(),
            Let { .. } => {
                // skip over the unique keys for value
                input_keys.nth(1).unwrap().clone()
            }
            LetRec { values, .. } => {
                // skip over the unique keys for value
                input_keys.nth(values.len()).unwrap().clone()
            }
            Project { outputs, .. } => {
                let input = input_keys.next().unwrap();
                input
                    .iter()
                    .filter_map(|key_set| {
                        if key_set.iter().all(|k| outputs.contains(k)) {
                            Some(
                                key_set
                                    .iter()
                                    .map(|c| outputs.iter().position(|o| o == c).unwrap())
                                    .collect(),
                            )
                        } else {
                            None
                        }
                    })
                    .collect()
            }
            Map { scalars, .. } => {
                let mut remappings = Vec::new();
                let arity = input_arities.next().unwrap();
                for (column, scalar) in scalars.iter().enumerate() {
                    // assess whether the scalar preserves uniqueness,
                    // and could participate in a key!

                    fn uniqueness(expr: &MirScalarExpr) -> Option<usize> {
                        match expr {
                            MirScalarExpr::CallUnary { func, expr } => {
                                if func.preserves_uniqueness() {
                                    uniqueness(expr)
                                } else {
                                    None
                                }
                            }
                            MirScalarExpr::Column(c, _name) => Some(*c),
                            _ => None,
                        }
                    }

                    if let Some(c) = uniqueness(scalar) {
                        remappings.push((c, column + arity));
                    }
                }

                let mut result = input_keys.next().unwrap().clone();
                let mut new_keys = Vec::new();
                // Any column in `remappings` could be replaced in a key
                // by the corresponding c. This could lead to combinatorial
                // explosion using our current representation, so we wont
                // do that. Instead, we'll handle the case of one remapping.
                if remappings.len() == 1 {
                    let (old, new) = remappings.pop().unwrap();
                    for key in &result {
                        if key.contains(&old) {
                            let mut new_key: Vec<usize> =
                                key.iter().cloned().filter(|k| k != &old).collect();
                            new_key.push(new);
                            new_key.sort_unstable();
                            new_keys.push(new_key);
                        }
                    }
                    result.append(&mut new_keys);
                }
                result
            }
            FlatMap { .. } => {
                // FlatMap can add duplicate rows, so input keys are no longer
                // valid
                vec![]
            }
            Negate { .. } => {
                // Although negate may have distinct records for each key,
                // the multiplicity is -1 rather than 1. This breaks many
                // of the optimization uses of "keys".
                vec![]
            }
            Filter { predicates, .. } => {
                // A filter inherits the keys of its input unless the filters
                // have reduced the input to a single row, in which case the
                // keys of the input are `()`.
                let mut input = input_keys.next().unwrap().clone();

                if !input.is_empty() {
                    // Track columns equated to literals, which we can prune.
                    let mut cols_equal_to_literal = BTreeSet::new();

                    // Perform union find on `col1 = col2` to establish
                    // connected components of equated columns. Absent any
                    // equalities, this will be `0 .. #c` (where #c is the
                    // greatest column referenced by a predicate), but each
                    // equality will orient the root of the greater to the root
                    // of the lesser.
                    let mut union_find = Vec::new();

                    for expr in predicates.iter() {
                        if let MirScalarExpr::CallBinary {
                            func: crate::BinaryFunc::Eq,
                            expr1,
                            expr2,
                        } = expr
                        {
                            if let MirScalarExpr::Column(c, _name) = &**expr1 {
                                if expr2.is_literal_ok() {
                                    cols_equal_to_literal.insert(c);
                                }
                            }
                            if let MirScalarExpr::Column(c, _name) = &**expr2 {
                                if expr1.is_literal_ok() {
                                    cols_equal_to_literal.insert(c);
                                }
                            }
                            // Perform union-find to equate columns.
                            if let (Some(c1), Some(c2)) = (expr1.as_column(), expr2.as_column()) {
                                if c1 != c2 {
                                    // Ensure union_find has entries up to
                                    // max(c1, c2) by filling up missing
                                    // positions with identity mappings.
                                    while union_find.len() <= std::cmp::max(c1, c2) {
                                        union_find.push(union_find.len());
                                    }
                                    let mut r1 = c1; // Find the representative column of [c1].
                                    while r1 != union_find[r1] {
                                        assert!(union_find[r1] < r1);
                                        r1 = union_find[r1];
                                    }
                                    let mut r2 = c2; // Find the representative column of [c2].
                                    while r2 != union_find[r2] {
                                        assert!(union_find[r2] < r2);
                                        r2 = union_find[r2];
                                    }
                                    // Union [c1] and [c2] by pointing the
                                    // larger to the smaller representative (we
                                    // update the remaining equivalence class
                                    // members only once after this for-loop).
                                    union_find[std::cmp::max(r1, r2)] = std::cmp::min(r1, r2);
                                }
                            }
                        }
                    }

                    // Complete union-find by pointing each element at its representative column.
                    for i in 0..union_find.len() {
                        // Iteration not required, as each prior already references the right column.
                        union_find[i] = union_find[union_find[i]];
                    }

                    // Remove columns bound to literals, and remap columns equated to earlier columns.
                    // We will re-expand remapped columns in a moment, but this avoids exponential work.
                    for key_set in &mut input {
                        key_set.retain(|k| !cols_equal_to_literal.contains(&k));
                        for col in key_set.iter_mut() {
                            if let Some(equiv) = union_find.get(*col) {
                                *col = *equiv;
                            }
                        }
                        key_set.sort();
                        key_set.dedup();
                    }
                    input.sort();
                    input.dedup();

                    // Expand out each key to each of its equivalent forms.
                    // Each instance of `col` can be replaced by any equivalent column.
                    // This has the potential to result in exponentially sized number of unique keys,
                    // and in the future we should probably maintain unique keys modulo equivalence.

                    // First, compute an inverse map from each representative
                    // column `sub` to all other equivalent columns `col`.
                    let mut subs = Vec::new();
                    for (col, sub) in union_find.iter().enumerate() {
                        if *sub != col {
                            assert!(*sub < col);
                            while subs.len() <= *sub {
                                subs.push(Vec::new());
                            }
                            subs[*sub].push(col);
                        }
                    }
                    // For each column, substitute for it in each occurrence.
                    let mut to_add = Vec::new();
                    for (col, subs) in subs.iter().enumerate() {
                        if !subs.is_empty() {
                            for key_set in input.iter() {
                                if key_set.contains(&col) {
                                    let mut to_extend = key_set.clone();
                                    to_extend.retain(|c| c != &col);
                                    for sub in subs {
                                        to_extend.push(*sub);
                                        to_add.push(to_extend.clone());
                                        to_extend.pop();
                                    }
                                }
                            }
                        }
                        // No deduplication, as we cannot introduce duplicates.
                        input.append(&mut to_add);
                    }
                    for key_set in input.iter_mut() {
                        key_set.sort();
                        key_set.dedup();
                    }
                }
                input
            }
            Join { equivalences, .. } => {
                // It is important the `new_from_input_arities` constructor is
                // used. Otherwise, Materialize may potentially end up in an
                // infinite loop.
                let input_mapper = crate::JoinInputMapper::new_from_input_arities(input_arities);

                input_mapper.global_keys(input_keys, equivalences)
            }
            Reduce { group_key, .. } => {
                // The group key should form a key, but we might already have
                // keys that are subsets of the group key, and should retain
                // those instead, if so.
                let mut result = Vec::new();
                for key_set in input_keys.next().unwrap() {
                    if key_set
                        .iter()
                        .all(|k| group_key.contains(&MirScalarExpr::column(*k)))
                    {
                        result.push(
                            key_set
                                .iter()
                                .map(|i| {
                                    group_key
                                        .iter()
                                        .position(|k| k == &MirScalarExpr::column(*i))
                                        .unwrap()
                                })
                                .collect::<Vec<_>>(),
                        );
                    }
                }
                if result.is_empty() {
                    result.push((0..group_key.len()).collect());
                }
                result
            }
            TopK {
                group_key, limit, ..
            } => {
                // If `limit` is `Some(1)` then the group key will become
                // a unique key, as there will be only one record with that key.
                let mut result = input_keys.next().unwrap().clone();
                if limit.as_ref().and_then(|x| x.as_literal_int64()) == Some(1) {
                    result.push(group_key.clone())
                }
                result
            }
            Union { base, inputs } => {
                // Generally, unions do not have any unique keys, because
                // each input might duplicate some. However, there is at
                // least one idiomatic structure that does preserve keys,
                // which results from SQL aggregations that must populate
                // absent records with default values. In that pattern,
                // the union of one GET with its negation, which has first
                // been subjected to a projection and map, we can remove
                // their influence on the key structure.
                //
                // If there are A, B, each with a unique `key` such that
                // we are looking at
                //
                //     A.proj(set_containing_key) + (B - A.proj(key)).map(stuff)
                //
                // Then we can report `key` as a unique key.
                //
                // TODO: make unique key structure an optimization analysis
                // rather than part of the type information.
                // TODO: perhaps ensure that (above) A.proj(key) is a
                // subset of B, as otherwise there are negative records
                // and who knows what is true (not expected, but again
                // who knows what the query plan might look like).

                let arity = input_arities.next().unwrap();
                let (base_projection, base_with_project_stripped) =
                    if let MirRelationExpr::Project { input, outputs } = &**base {
                        (outputs.clone(), &**input)
                    } else {
                        // A input without a project is equivalent to an input
                        // with the project being all columns in the input in order.
                        ((0..arity).collect::<Vec<_>>(), &**base)
                    };
                let mut result = Vec::new();
                if let MirRelationExpr::Get {
                    id: first_id,
                    typ: _,
                    ..
                } = base_with_project_stripped
                {
                    if inputs.len() == 1 {
                        if let MirRelationExpr::Map { input, .. } = &inputs[0] {
                            if let MirRelationExpr::Union { base, inputs } = &**input {
                                if inputs.len() == 1 {
                                    if let Some((input, outputs)) = base.is_negated_project() {
                                        if let MirRelationExpr::Get {
                                            id: second_id,
                                            typ: _,
                                            ..
                                        } = input
                                        {
                                            if first_id == second_id {
                                                result.extend(
                                                    input_keys
                                                        .next()
                                                        .unwrap()
                                                        .into_iter()
                                                        .filter(|key| {
                                                            key.iter().all(|c| {
                                                                outputs.get(*c) == Some(c)
                                                                    && base_projection.get(*c)
                                                                        == Some(c)
                                                            })
                                                        })
                                                        .cloned(),
                                                );
                                            }
                                        }
                                    }
                                }
                            }
                        }
                    }
                }
                // Important: do not inherit keys of either input, as not unique.
                result
            }
        };
        keys.sort();
        keys.dedup();
        keys
    }

    /// The number of columns in the relation.
    ///
    /// This number is determined from the type, which is determined recursively
    /// at non-trivial cost.
    ///
    /// The arity is computed incrementally with a recursive post-order
    /// traversal, that accumulates the arities for the relations yet to be
    /// visited in `arity_stack`.
    pub fn arity(&self) -> usize {
        let mut arity_stack = Vec::new();
        #[allow(deprecated)]
        self.visit_pre_post_nolimit(
            &mut |e: &MirRelationExpr| -> Option<Vec<&MirRelationExpr>> {
                match &e {
                    MirRelationExpr::Let { body, .. } => {
                        // Do not traverse the value sub-graph, since it's not relevant for
                        // determining the arity of Let operators.
                        Some(vec![&*body])
                    }
                    MirRelationExpr::LetRec { body, .. } => {
                        // Do not traverse the value sub-graph, since it's not relevant for
                        // determining the arity of Let operators.
                        Some(vec![&*body])
                    }
                    MirRelationExpr::Project { .. } | MirRelationExpr::Reduce { .. } => {
                        // No further traversal is required; these operators know their arity.
                        Some(Vec::new())
                    }
                    _ => None,
                }
            },
            &mut |e: &MirRelationExpr| {
                match &e {
                    MirRelationExpr::Let { .. } => {
                        let body_arity = arity_stack.pop().unwrap();
                        arity_stack.push(0);
                        arity_stack.push(body_arity);
                    }
                    MirRelationExpr::LetRec { values, .. } => {
                        let body_arity = arity_stack.pop().unwrap();
                        arity_stack.extend(std::iter::repeat(0).take(values.len()));
                        arity_stack.push(body_arity);
                    }
                    MirRelationExpr::Project { .. } | MirRelationExpr::Reduce { .. } => {
                        arity_stack.push(0);
                    }
                    _ => {}
                }
                let num_inputs = e.num_inputs();
                let input_arities = arity_stack.drain(arity_stack.len() - num_inputs..);
                let arity = e.arity_with_input_arities(input_arities);
                arity_stack.push(arity);
            },
        );
        assert_eq!(arity_stack.len(), 1);
        arity_stack.pop().unwrap()
    }

    /// Reports the arity of the relation given the schema of the input relations.
    ///
    /// `input_arities` is required to contain the arities for the input relations of
    /// the current relation in the same order as they are visited by `try_visit_children`
    /// method, even though not all may be used for computing the schema of the
    /// current relation. For example, `Let` expects two input types, one for the
    /// value relation and one for the body, in that order, but only the one for the
    /// body is used to determine the type of the `Let` relation.
    ///
    /// It is meant to be used during post-order traversals to compute arities
    /// incrementally.
    pub fn arity_with_input_arities<I>(&self, mut input_arities: I) -> usize
    where
        I: Iterator<Item = usize>,
    {
        use MirRelationExpr::*;

        match self {
            Constant { rows: _, typ } => typ.arity(),
            Get { typ, .. } => typ.arity(),
            Let { .. } => {
                input_arities.next();
                input_arities.next().unwrap()
            }
            LetRec { values, .. } => {
                for _ in 0..values.len() {
                    input_arities.next();
                }
                input_arities.next().unwrap()
            }
            Project { outputs, .. } => outputs.len(),
            Map { scalars, .. } => input_arities.next().unwrap() + scalars.len(),
            FlatMap { func, .. } => input_arities.next().unwrap() + func.output_arity(),
            Join { .. } => input_arities.sum(),
            Reduce {
                input: _,
                group_key,
                aggregates,
                ..
            } => group_key.len() + aggregates.len(),
            Filter { .. }
            | TopK { .. }
            | Negate { .. }
            | Threshold { .. }
            | Union { .. }
            | ArrangeBy { .. } => input_arities.next().unwrap(),
        }
    }

    /// The number of child relations this relation has.
    pub fn num_inputs(&self) -> usize {
        let mut count = 0;

        self.visit_children(|_| count += 1);

        count
    }

    /// Constructs a constant collection from specific rows and schema, where
    /// each row will have a multiplicity of one.
    pub fn constant(rows: Vec<Vec<Datum>>, typ: RelationType) -> Self {
        let rows = rows.into_iter().map(|row| (row, Diff::ONE)).collect();
        MirRelationExpr::constant_diff(rows, typ)
    }

    /// Constructs a constant collection from specific rows and schema, where
    /// each row can have an arbitrary multiplicity.
    pub fn constant_diff(rows: Vec<(Vec<Datum>, Diff)>, typ: RelationType) -> Self {
        for (row, _diff) in &rows {
            for (datum, column_typ) in row.iter().zip(typ.column_types.iter()) {
                assert!(
                    datum.is_instance_of(column_typ),
                    "Expected datum of type {:?}, got value {:?}",
                    column_typ,
                    datum
                );
            }
        }
        let rows = Ok(rows
            .into_iter()
            .map(move |(row, diff)| (Row::pack_slice(&row), diff))
            .collect());
        MirRelationExpr::Constant { rows, typ }
    }

    /// If self is a constant, return the value and the type, otherwise `None`.
    /// Looks behind `ArrangeBy`s.
    pub fn as_const(&self) -> Option<(&Result<Vec<(Row, Diff)>, EvalError>, &RelationType)> {
        match self {
            MirRelationExpr::Constant { rows, typ } => Some((rows, typ)),
            MirRelationExpr::ArrangeBy { input, .. } => input.as_const(),
            _ => None,
        }
    }

    /// If self is a constant, mutably return the value and the type, otherwise `None`.
    /// Looks behind `ArrangeBy`s.
    pub fn as_const_mut(
        &mut self,
    ) -> Option<(&mut Result<Vec<(Row, Diff)>, EvalError>, &mut RelationType)> {
        match self {
            MirRelationExpr::Constant { rows, typ } => Some((rows, typ)),
            MirRelationExpr::ArrangeBy { input, .. } => input.as_const_mut(),
            _ => None,
        }
    }

    /// If self is a constant error, return the error, otherwise `None`.
    /// Looks behind `ArrangeBy`s.
    pub fn as_const_err(&self) -> Option<&EvalError> {
        match self {
            MirRelationExpr::Constant { rows: Err(e), .. } => Some(e),
            MirRelationExpr::ArrangeBy { input, .. } => input.as_const_err(),
            _ => None,
        }
    }

    /// Checks if `self` is the single element collection with no columns.
    pub fn is_constant_singleton(&self) -> bool {
        if let Some((Ok(rows), typ)) = self.as_const() {
            rows.len() == 1 && typ.column_types.len() == 0 && rows[0].1 == Diff::ONE
        } else {
            false
        }
    }

    /// Constructs the expression for getting a local collection.
    pub fn local_get(id: LocalId, typ: RelationType) -> Self {
        MirRelationExpr::Get {
            id: Id::Local(id),
            typ,
            access_strategy: AccessStrategy::UnknownOrLocal,
        }
    }

    /// Constructs the expression for getting a global collection
    pub fn global_get(id: GlobalId, typ: RelationType) -> Self {
        MirRelationExpr::Get {
            id: Id::Global(id),
            typ,
            access_strategy: AccessStrategy::UnknownOrLocal,
        }
    }

    /// Retains only the columns specified by `output`.
    pub fn project(mut self, mut outputs: Vec<usize>) -> Self {
        if let MirRelationExpr::Project {
            outputs: columns, ..
        } = &mut self
        {
            // Update `outputs` to reference base columns of `input`.
            for column in outputs.iter_mut() {
                *column = columns[*column];
            }
            *columns = outputs;
            self
        } else {
            MirRelationExpr::Project {
                input: Box::new(self),
                outputs,
            }
        }
    }

    /// Append to each row the results of applying elements of `scalar`.
    pub fn map(mut self, scalars: Vec<MirScalarExpr>) -> Self {
        if let MirRelationExpr::Map { scalars: s, .. } = &mut self {
            s.extend(scalars);
            self
        } else if !scalars.is_empty() {
            MirRelationExpr::Map {
                input: Box::new(self),
                scalars,
            }
        } else {
            self
        }
    }

    /// Append to each row a single `scalar`.
    pub fn map_one(self, scalar: MirScalarExpr) -> Self {
        self.map(vec![scalar])
    }

    /// Like `map`, but yields zero-or-more output rows per input row
    pub fn flat_map(self, func: TableFunc, exprs: Vec<MirScalarExpr>) -> Self {
        MirRelationExpr::FlatMap {
            input: Box::new(self),
            func,
            exprs,
        }
    }

    /// Retain only the rows satisfying each of several predicates.
    pub fn filter<I>(mut self, predicates: I) -> Self
    where
        I: IntoIterator<Item = MirScalarExpr>,
    {
        // Extract existing predicates
        let mut new_predicates = if let MirRelationExpr::Filter { input, predicates } = self {
            self = *input;
            predicates
        } else {
            Vec::new()
        };
        // Normalize collection of predicates.
        new_predicates.extend(predicates);
        new_predicates.retain(|p| !p.is_literal_true());
        new_predicates.sort();
        new_predicates.dedup();
        // Introduce a `Filter` only if we have predicates.
        if !new_predicates.is_empty() {
            self = MirRelationExpr::Filter {
                input: Box::new(self),
                predicates: new_predicates,
            };
        }

        self
    }

    /// Form the Cartesian outer-product of rows in both inputs.
    pub fn product(mut self, right: Self) -> Self {
        if right.is_constant_singleton() {
            self
        } else if self.is_constant_singleton() {
            right
        } else if let MirRelationExpr::Join { inputs, .. } = &mut self {
            inputs.push(right);
            self
        } else {
            MirRelationExpr::join(vec![self, right], vec![])
        }
    }

    /// Performs a relational equijoin among the input collections.
    ///
    /// The sequence `inputs` each describe different input collections, and the sequence `variables` describes
    /// equality constraints that some of their columns must satisfy. Each element in `variable` describes a set
    /// of pairs  `(input_index, column_index)` where every value described by that set must be equal.
    ///
    /// For example, the pair `(input, column)` indexes into `inputs[input][column]`, extracting the `input`th
    /// input collection and for each row examining its `column`th column.
    ///
    /// # Example
    ///
    /// ```rust
    /// use mz_repr::{Datum, ColumnType, RelationType, ScalarType};
    /// use mz_expr::MirRelationExpr;
    ///
    /// // A common schema for each input.
    /// let schema = RelationType::new(vec![
    ///     ScalarType::Int32.nullable(false),
    ///     ScalarType::Int32.nullable(false),
    /// ]);
    ///
    /// // the specific data are not important here.
    /// let data = vec![Datum::Int32(0), Datum::Int32(1)];
    ///
    /// // Three collections that could have been different.
    /// let input0 = MirRelationExpr::constant(vec![data.clone()], schema.clone());
    /// let input1 = MirRelationExpr::constant(vec![data.clone()], schema.clone());
    /// let input2 = MirRelationExpr::constant(vec![data.clone()], schema.clone());
    ///
    /// // Join the three relations looking for triangles, like so.
    /// //
    /// //     Output(A,B,C) := Input0(A,B), Input1(B,C), Input2(A,C)
    /// let joined = MirRelationExpr::join(
    ///     vec![input0, input1, input2],
    ///     vec![
    ///         vec![(0,0), (2,0)], // fields A of inputs 0 and 2.
    ///         vec![(0,1), (1,0)], // fields B of inputs 0 and 1.
    ///         vec![(1,1), (2,1)], // fields C of inputs 1 and 2.
    ///     ],
    /// );
    ///
    /// // Technically the above produces `Output(A,B,B,C,A,C)` because the columns are concatenated.
    /// // A projection resolves this and produces the correct output.
    /// let result = joined.project(vec![0, 1, 3]);
    /// ```
    pub fn join(inputs: Vec<MirRelationExpr>, variables: Vec<Vec<(usize, usize)>>) -> Self {
        let input_mapper = join_input_mapper::JoinInputMapper::new(&inputs);

        let equivalences = variables
            .into_iter()
            .map(|vs| {
                vs.into_iter()
                    .map(|(r, c)| input_mapper.map_expr_to_global(MirScalarExpr::column(c), r))
                    .collect::<Vec<_>>()
            })
            .collect::<Vec<_>>();

        Self::join_scalars(inputs, equivalences)
    }

    /// Constructs a join operator from inputs and required-equal scalar expressions.
    pub fn join_scalars(
        mut inputs: Vec<MirRelationExpr>,
        equivalences: Vec<Vec<MirScalarExpr>>,
    ) -> Self {
        // Remove all constant inputs that are the identity for join.
        // They neither introduce nor modify any column references.
        inputs.retain(|i| !i.is_constant_singleton());
        MirRelationExpr::Join {
            inputs,
            equivalences,
            implementation: JoinImplementation::Unimplemented,
        }
    }

    /// Perform a key-wise reduction / aggregation.
    ///
    /// The `group_key` argument indicates columns in the input collection that should
    /// be grouped, and `aggregates` lists aggregation functions each of which produces
    /// one output column in addition to the keys.
    pub fn reduce(
        self,
        group_key: Vec<usize>,
        aggregates: Vec<AggregateExpr>,
        expected_group_size: Option<u64>,
    ) -> Self {
        MirRelationExpr::Reduce {
            input: Box::new(self),
            group_key: group_key.into_iter().map(MirScalarExpr::column).collect(),
            aggregates,
            monotonic: false,
            expected_group_size,
        }
    }

    /// Perform a key-wise reduction order by and limit.
    ///
    /// The `group_key` argument indicates columns in the input collection that should
    /// be grouped, the `order_key` argument indicates columns that should be further
    /// used to order records within groups, and the `limit` argument constrains the
    /// total number of records that should be produced in each group.
    pub fn top_k(
        self,
        group_key: Vec<usize>,
        order_key: Vec<ColumnOrder>,
        limit: Option<MirScalarExpr>,
        offset: usize,
        expected_group_size: Option<u64>,
    ) -> Self {
        MirRelationExpr::TopK {
            input: Box::new(self),
            group_key,
            order_key,
            limit,
            offset,
            expected_group_size,
            monotonic: false,
        }
    }

    /// Negates the occurrences of each row.
    pub fn negate(self) -> Self {
        if let MirRelationExpr::Negate { input } = self {
            *input
        } else {
            MirRelationExpr::Negate {
                input: Box::new(self),
            }
        }
    }

    /// Removes all but the first occurrence of each row.
    pub fn distinct(self) -> Self {
        let arity = self.arity();
        self.distinct_by((0..arity).collect())
    }

    /// Removes all but the first occurrence of each key. Columns not included
    /// in the `group_key` are discarded.
    pub fn distinct_by(self, group_key: Vec<usize>) -> Self {
        self.reduce(group_key, vec![], None)
    }

    /// Discards rows with a negative frequency.
    pub fn threshold(self) -> Self {
        if let MirRelationExpr::Threshold { .. } = &self {
            self
        } else {
            MirRelationExpr::Threshold {
                input: Box::new(self),
            }
        }
    }

    /// Unions together any number inputs.
    ///
    /// If `inputs` is empty, then an empty relation of type `typ` is
    /// constructed.
    pub fn union_many(mut inputs: Vec<Self>, typ: RelationType) -> Self {
        // Deconstruct `inputs` as `Union`s and reconstitute.
        let mut flat_inputs = Vec::with_capacity(inputs.len());
        for input in inputs {
            if let MirRelationExpr::Union { base, inputs } = input {
                flat_inputs.push(*base);
                flat_inputs.extend(inputs);
            } else {
                flat_inputs.push(input);
            }
        }
        inputs = flat_inputs;
        if inputs.len() == 0 {
            MirRelationExpr::Constant {
                rows: Ok(vec![]),
                typ,
            }
        } else if inputs.len() == 1 {
            inputs.into_element()
        } else {
            MirRelationExpr::Union {
                base: Box::new(inputs.remove(0)),
                inputs,
            }
        }
    }

    /// Produces one collection where each row is present with the sum of its frequencies in each input.
    pub fn union(self, other: Self) -> Self {
        // Deconstruct `self` and `other` as `Union`s and reconstitute.
        let mut flat_inputs = Vec::with_capacity(2);
        if let MirRelationExpr::Union { base, inputs } = self {
            flat_inputs.push(*base);
            flat_inputs.extend(inputs);
        } else {
            flat_inputs.push(self);
        }
        if let MirRelationExpr::Union { base, inputs } = other {
            flat_inputs.push(*base);
            flat_inputs.extend(inputs);
        } else {
            flat_inputs.push(other);
        }

        MirRelationExpr::Union {
            base: Box::new(flat_inputs.remove(0)),
            inputs: flat_inputs,
        }
    }

    /// Arranges the collection by the specified columns
    pub fn arrange_by(self, keys: &[Vec<MirScalarExpr>]) -> Self {
        MirRelationExpr::ArrangeBy {
            input: Box::new(self),
            keys: keys.to_owned(),
        }
    }

    /// Indicates if this is a constant empty collection.
    ///
    /// A false value does not mean the collection is known to be non-empty,
    /// only that we cannot currently determine that it is statically empty.
    pub fn is_empty(&self) -> bool {
        if let Some((Ok(rows), ..)) = self.as_const() {
            rows.is_empty()
        } else {
            false
        }
    }

    /// If the expression is a negated project, return the input and the projection.
    pub fn is_negated_project(&self) -> Option<(&MirRelationExpr, &[usize])> {
        if let MirRelationExpr::Negate { input } = self {
            if let MirRelationExpr::Project { input, outputs } = &**input {
                return Some((&**input, outputs));
            }
        }
        if let MirRelationExpr::Project { input, outputs } = self {
            if let MirRelationExpr::Negate { input } = &**input {
                return Some((&**input, outputs));
            }
        }
        None
    }

    /// Pretty-print this [MirRelationExpr] to a string.
    pub fn pretty(&self) -> String {
        let config = ExplainConfig::default();
        self.explain(&config, None)
    }

    /// Pretty-print this [MirRelationExpr] to a string using a custom
    /// [ExplainConfig] and an optionally provided [ExprHumanizer].
    pub fn explain(&self, config: &ExplainConfig, humanizer: Option<&dyn ExprHumanizer>) -> String {
        text_string_at(self, || PlanRenderingContext {
            indent: Indent::default(),
            humanizer: humanizer.unwrap_or(&DummyHumanizer),
            annotations: BTreeMap::default(),
            config,
        })
    }

    /// Take ownership of `self`, leaving an empty `MirRelationExpr::Constant` with the optionally
    /// given scalar types. The given scalar types should be `base_eq` with the types that `typ()`
    /// would find. Keys and nullability are ignored in the given `RelationType`, and instead we set
    /// the best possible key and nullability, since we are making an empty collection.
    ///
    /// If `typ` is not given, then this calls `.typ()` (which is possibly expensive) to determine
    /// the correct type.
    pub fn take_safely(&mut self, typ: Option<RelationType>) -> MirRelationExpr {
        if let Some(typ) = &typ {
            soft_assert_no_log!(
                self.typ()
                    .column_types
                    .iter()
                    .zip_eq(typ.column_types.iter())
                    .all(|(t1, t2)| t1.scalar_type.base_eq(&t2.scalar_type))
            );
        }
        let mut typ = typ.unwrap_or_else(|| self.typ());
        typ.keys = vec![vec![]];
        for ct in typ.column_types.iter_mut() {
            ct.nullable = false;
        }
        std::mem::replace(
            self,
            MirRelationExpr::Constant {
                rows: Ok(vec![]),
                typ,
            },
        )
    }

    /// Take ownership of `self`, leaving an empty `MirRelationExpr::Constant` with the given scalar
    /// types. Nullability is ignored in the given `ColumnType`s, and instead we set the best
    /// possible nullability, since we are making an empty collection.
    pub fn take_safely_with_col_types(&mut self, typ: Vec<ColumnType>) -> MirRelationExpr {
        self.take_safely(Some(RelationType::new(typ)))
    }

    /// Take ownership of `self`, leaving an empty `MirRelationExpr::Constant` with an **incorrect** type.
    ///
    /// This should only be used if `self` is about to be dropped or otherwise overwritten.
    pub fn take_dangerous(&mut self) -> MirRelationExpr {
        let empty = MirRelationExpr::Constant {
            rows: Ok(vec![]),
            typ: RelationType::new(Vec::new()),
        };
        std::mem::replace(self, empty)
    }

    /// Replaces `self` with some logic applied to `self`.
    pub fn replace_using<F>(&mut self, logic: F)
    where
        F: FnOnce(MirRelationExpr) -> MirRelationExpr,
    {
        let empty = MirRelationExpr::Constant {
            rows: Ok(vec![]),
            typ: RelationType::new(Vec::new()),
        };
        let expr = std::mem::replace(self, empty);
        *self = logic(expr);
    }

    /// Store `self` in a `Let` and pass the corresponding `Get` to `body`.
    pub fn let_in<Body, E>(self, id_gen: &mut IdGen, body: Body) -> Result<MirRelationExpr, E>
    where
        Body: FnOnce(&mut IdGen, MirRelationExpr) -> Result<MirRelationExpr, E>,
    {
        if let MirRelationExpr::Get { .. } = self {
            // already done
            body(id_gen, self)
        } else {
            let id = LocalId::new(id_gen.allocate_id());
            let get = MirRelationExpr::Get {
                id: Id::Local(id),
                typ: self.typ(),
                access_strategy: AccessStrategy::UnknownOrLocal,
            };
            let body = (body)(id_gen, get)?;
            Ok(MirRelationExpr::Let {
                id,
                value: Box::new(self),
                body: Box::new(body),
            })
        }
    }

    /// Return every row in `self` that does not have a matching row in the first columns of `keys_and_values`, using `default` to fill in the remaining columns
    /// (If `default` is a row of nulls, this is the 'outer' part of LEFT OUTER JOIN)
    pub fn anti_lookup<E>(
        self,
        id_gen: &mut IdGen,
        keys_and_values: MirRelationExpr,
        default: Vec<(Datum, ScalarType)>,
    ) -> Result<MirRelationExpr, E> {
        let (data, column_types): (Vec<_>, Vec<_>) = default
            .into_iter()
            .map(|(datum, scalar_type)| (datum, scalar_type.nullable(datum.is_null())))
            .unzip();
        assert_eq!(keys_and_values.arity() - self.arity(), data.len());
        self.let_in(id_gen, |_id_gen, get_keys| {
            let get_keys_arity = get_keys.arity();
            Ok(MirRelationExpr::join(
                vec![
                    // all the missing keys (with count 1)
                    keys_and_values
                        .distinct_by((0..get_keys_arity).collect())
                        .negate()
                        .union(get_keys.clone().distinct()),
                    // join with keys to get the correct counts
                    get_keys.clone(),
                ],
                (0..get_keys_arity).map(|i| vec![(0, i), (1, i)]).collect(),
            )
            // get rid of the extra copies of columns from keys
            .project((0..get_keys_arity).collect())
            // This join is logically equivalent to
            // `.map(<default_expr>)`, but using a join allows for
            // potential predicate pushdown and elision in the
            // optimizer.
            .product(MirRelationExpr::constant(
                vec![data],
                RelationType::new(column_types),
            )))
        })
    }

    /// Return:
    /// * every row in keys_and_values
    /// * every row in `self` that does not have a matching row in the first columns of
    ///   `keys_and_values`, using `default` to fill in the remaining columns
    /// (This is LEFT OUTER JOIN if:
    /// 1) `default` is a row of null
    /// 2) matching rows in `keys_and_values` and `self` have the same multiplicity.)
    pub fn lookup<E>(
        self,
        id_gen: &mut IdGen,
        keys_and_values: MirRelationExpr,
        default: Vec<(Datum<'static>, ScalarType)>,
    ) -> Result<MirRelationExpr, E> {
        keys_and_values.let_in(id_gen, |id_gen, get_keys_and_values| {
            Ok(get_keys_and_values.clone().union(self.anti_lookup(
                id_gen,
                get_keys_and_values,
                default,
            )?))
        })
    }

    /// True iff the expression contains a `NullaryFunc::MzLogicalTimestamp`.
    pub fn contains_temporal(&self) -> bool {
        let mut contains = false;
        self.visit_scalars(&mut |e| contains = contains || e.contains_temporal());
        contains
    }

    /// Fallible visitor for the [`MirScalarExpr`]s directly owned by this relation expression.
    ///
    /// The `f` visitor should not recursively descend into owned [`MirRelationExpr`]s.
    pub fn try_visit_scalars_mut1<F, E>(&mut self, f: &mut F) -> Result<(), E>
    where
        F: FnMut(&mut MirScalarExpr) -> Result<(), E>,
    {
        use MirRelationExpr::*;
        match self {
            Map { scalars, .. } => {
                for s in scalars {
                    f(s)?;
                }
            }
            Filter { predicates, .. } => {
                for p in predicates {
                    f(p)?;
                }
            }
            FlatMap { exprs, .. } => {
                for expr in exprs {
                    f(expr)?;
                }
            }
            Join {
                inputs: _,
                equivalences,
                implementation,
            } => {
                for equivalence in equivalences {
                    for expr in equivalence {
                        f(expr)?;
                    }
                }
                match implementation {
                    JoinImplementation::Differential((_, start_key, _), order) => {
                        if let Some(start_key) = start_key {
                            for k in start_key {
                                f(k)?;
                            }
                        }
                        for (_, lookup_key, _) in order {
                            for k in lookup_key {
                                f(k)?;
                            }
                        }
                    }
                    JoinImplementation::DeltaQuery(paths) => {
                        for path in paths {
                            for (_, lookup_key, _) in path {
                                for k in lookup_key {
                                    f(k)?;
                                }
                            }
                        }
                    }
                    JoinImplementation::IndexedFilter(_coll_id, _idx_id, index_key, _) => {
                        for k in index_key {
                            f(k)?;
                        }
                    }
                    JoinImplementation::Unimplemented => {} // No scalar exprs
                }
            }
            ArrangeBy { keys, .. } => {
                for key in keys {
                    for s in key {
                        f(s)?;
                    }
                }
            }
            Reduce {
                group_key,
                aggregates,
                ..
            } => {
                for s in group_key {
                    f(s)?;
                }
                for agg in aggregates {
                    f(&mut agg.expr)?;
                }
            }
            TopK { limit, .. } => {
                if let Some(s) = limit {
                    f(s)?;
                }
            }
            Constant { .. }
            | Get { .. }
            | Let { .. }
            | LetRec { .. }
            | Project { .. }
            | Negate { .. }
            | Threshold { .. }
            | Union { .. } => (),
        }
        Ok(())
    }

    /// Fallible mutable visitor for the [`MirScalarExpr`]s in the [`MirRelationExpr`] subtree
    /// rooted at `self`.
    ///
    /// Note that this does not recurse into [`MirRelationExpr`] subtrees within [`MirScalarExpr`]
    /// nodes.
    pub fn try_visit_scalars_mut<F, E>(&mut self, f: &mut F) -> Result<(), E>
    where
        F: FnMut(&mut MirScalarExpr) -> Result<(), E>,
        E: From<RecursionLimitError>,
    {
        self.try_visit_mut_post(&mut |expr| expr.try_visit_scalars_mut1(f))
    }

    /// Infallible mutable visitor for the [`MirScalarExpr`]s in the [`MirRelationExpr`] subtree
    /// rooted at `self`.
    ///
    /// Note that this does not recurse into [`MirRelationExpr`] subtrees within [`MirScalarExpr`]
    /// nodes.
    pub fn visit_scalars_mut<F>(&mut self, f: &mut F)
    where
        F: FnMut(&mut MirScalarExpr),
    {
        self.try_visit_scalars_mut(&mut |s| {
            f(s);
            Ok::<_, RecursionLimitError>(())
        })
        .expect("Unexpected error in `visit_scalars_mut` call");
    }

    /// Fallible visitor for the [`MirScalarExpr`]s directly owned by this relation expression.
    ///
    /// The `f` visitor should not recursively descend into owned [`MirRelationExpr`]s.
    pub fn try_visit_scalars_1<F, E>(&self, f: &mut F) -> Result<(), E>
    where
        F: FnMut(&MirScalarExpr) -> Result<(), E>,
    {
        use MirRelationExpr::*;
        match self {
            Map { scalars, .. } => {
                for s in scalars {
                    f(s)?;
                }
            }
            Filter { predicates, .. } => {
                for p in predicates {
                    f(p)?;
                }
            }
            FlatMap { exprs, .. } => {
                for expr in exprs {
                    f(expr)?;
                }
            }
            Join {
                inputs: _,
                equivalences,
                implementation,
            } => {
                for equivalence in equivalences {
                    for expr in equivalence {
                        f(expr)?;
                    }
                }
                match implementation {
                    JoinImplementation::Differential((_, start_key, _), order) => {
                        if let Some(start_key) = start_key {
                            for k in start_key {
                                f(k)?;
                            }
                        }
                        for (_, lookup_key, _) in order {
                            for k in lookup_key {
                                f(k)?;
                            }
                        }
                    }
                    JoinImplementation::DeltaQuery(paths) => {
                        for path in paths {
                            for (_, lookup_key, _) in path {
                                for k in lookup_key {
                                    f(k)?;
                                }
                            }
                        }
                    }
                    JoinImplementation::IndexedFilter(_coll_id, _idx_id, index_key, _) => {
                        for k in index_key {
                            f(k)?;
                        }
                    }
                    JoinImplementation::Unimplemented => {} // No scalar exprs
                }
            }
            ArrangeBy { keys, .. } => {
                for key in keys {
                    for s in key {
                        f(s)?;
                    }
                }
            }
            Reduce {
                group_key,
                aggregates,
                ..
            } => {
                for s in group_key {
                    f(s)?;
                }
                for agg in aggregates {
                    f(&agg.expr)?;
                }
            }
            TopK { limit, .. } => {
                if let Some(s) = limit {
                    f(s)?;
                }
            }
            Constant { .. }
            | Get { .. }
            | Let { .. }
            | LetRec { .. }
            | Project { .. }
            | Negate { .. }
            | Threshold { .. }
            | Union { .. } => (),
        }
        Ok(())
    }

    /// Fallible immutable visitor for the [`MirScalarExpr`]s in the [`MirRelationExpr`] subtree
    /// rooted at `self`.
    ///
    /// Note that this does not recurse into [`MirRelationExpr`] subtrees within [`MirScalarExpr`]
    /// nodes.
    pub fn try_visit_scalars<F, E>(&self, f: &mut F) -> Result<(), E>
    where
        F: FnMut(&MirScalarExpr) -> Result<(), E>,
        E: From<RecursionLimitError>,
    {
        self.try_visit_post(&mut |expr| expr.try_visit_scalars_1(f))
    }

    /// Infallible immutable visitor for the [`MirScalarExpr`]s in the [`MirRelationExpr`] subtree
    /// rooted at `self`.
    ///
    /// Note that this does not recurse into [`MirRelationExpr`] subtrees within [`MirScalarExpr`]
    /// nodes.
    pub fn visit_scalars<F>(&self, f: &mut F)
    where
        F: FnMut(&MirScalarExpr),
    {
        self.try_visit_scalars(&mut |s| {
            f(s);
            Ok::<_, RecursionLimitError>(())
        })
        .expect("Unexpected error in `visit_scalars` call");
    }

    /// Clears the contents of `self` even if it's so deep that simply dropping it would cause a
    /// stack overflow in `drop_in_place`.
    ///
    /// Leaves `self` in an unusable state, so this should only be used if `self` is about to be
    /// dropped or otherwise overwritten.
    pub fn destroy_carefully(&mut self) {
        let mut todo = vec![self.take_dangerous()];
        while let Some(mut expr) = todo.pop() {
            for child in expr.children_mut() {
                todo.push(child.take_dangerous());
            }
        }
    }

    /// Computes the size (total number of nodes) and maximum depth of a MirRelationExpr for
    /// debug printing purposes.
    pub fn debug_size_and_depth(&self) -> (usize, usize) {
        let mut size = 0;
        let mut max_depth = 0;
        let mut todo = vec![(self, 1)];
        while let Some((expr, depth)) = todo.pop() {
            size += 1;
            max_depth = max(max_depth, depth);
            todo.extend(expr.children().map(|c| (c, depth + 1)));
        }
        (size, max_depth)
    }

    /// The MirRelationExpr is considered potentially expensive if and only if
    /// at least one of the following conditions is true:
    ///
    ///  - It contains at least one FlatMap or a Reduce operator.
    ///  - It contains at least one MirScalarExpr with a function call.
    ///
    /// !!!WARNING!!!: this method has an HirRelationExpr counterpart. The two
    /// should be kept in sync w.r.t. HIR ⇒ MIR lowering!
    pub fn could_run_expensive_function(&self) -> bool {
        let mut result = false;
        self.visit_pre(|e: &MirRelationExpr| {
            use MirRelationExpr::*;
            use MirScalarExpr::*;
            if let Err(_) = self.try_visit_scalars::<_, RecursionLimitError>(&mut |scalar| {
                result |= match scalar {
                    Column(_, _) | Literal(_, _) | CallUnmaterializable(_) | If { .. } => false,
                    // Function calls are considered expensive
                    CallUnary { .. } | CallBinary { .. } | CallVariadic { .. } => true,
                };
                Ok(())
            }) {
                // Conservatively set `true` if on RecursionLimitError.
                result = true;
            }
            // FlatMap has a table function; Reduce has an aggregate function.
            // Other constructs use MirScalarExpr to run a function
            result |= matches!(e, FlatMap { .. } | Reduce { .. });
        });
        result
    }

    /// Hash to an u64 using Rust's default Hasher. (Which is a somewhat slower, but better Hasher
    /// than what `Hashable::hashed` would give us.)
    pub fn hash_to_u64(&self) -> u64 {
        let mut h = DefaultHasher::new();
        self.hash(&mut h);
        h.finish()
    }
}

// `LetRec` helpers
impl MirRelationExpr {
    /// True when `expr` contains a `LetRec` AST node.
    pub fn is_recursive(self: &MirRelationExpr) -> bool {
        let mut worklist = vec![self];
        while let Some(expr) = worklist.pop() {
            if let MirRelationExpr::LetRec { .. } = expr {
                return true;
            }
            worklist.extend(expr.children());
        }
        false
    }

    /// Return the number of sub-expressions in the tree (including self).
    pub fn size(&self) -> usize {
        let mut size = 0;
        self.visit_pre(|_| size += 1);
        size
    }

    /// Given the ids and values of a LetRec, it computes the subset of ids that are used across
    /// iterations. These are those ids that have a reference before they are defined, when reading
    /// all the bindings in order.
    ///
    /// For example:
    /// ```SQL
    /// WITH MUTUALLY RECURSIVE
    ///     x(...) AS f(z),
    ///     y(...) AS g(x),
    ///     z(...) AS h(y)
    /// ...;
    /// ```
    /// Here, only `z` is returned, because `x` and `y` are referenced only within the same
    /// iteration.
    ///
    /// Note that if a binding references itself, that is also returned.
    pub fn recursive_ids(ids: &[LocalId], values: &[MirRelationExpr]) -> BTreeSet<LocalId> {
        let mut used_across_iterations = BTreeSet::new();
        let mut defined = BTreeSet::new();
        for (binding_id, value) in itertools::zip_eq(ids.iter(), values.iter()) {
            value.visit_pre(|expr| {
                if let MirRelationExpr::Get {
                    id: Local(get_id), ..
                } = expr
                {
                    // If we haven't seen a definition for it yet, then this will refer
                    // to the previous iteration.
                    // The `ids.contains` part of the condition is needed to exclude
                    // those ids that are not really in this LetRec, but either an inner
                    // or outer one.
                    if !defined.contains(get_id) && ids.contains(get_id) {
                        used_across_iterations.insert(*get_id);
                    }
                }
            });
            defined.insert(*binding_id);
        }
        used_across_iterations
    }

    /// Replaces `LetRec` nodes with a stack of `Let` nodes.
    ///
    /// In each `Let` binding, uses of `Get` in `value` that are not at strictly greater
    /// identifiers are rewritten to be the constant collection.
    /// This makes the computation perform exactly "one" iteration.
    ///
    /// This was used only temporarily while developing `LetRec`.
    pub fn make_nonrecursive(self: &mut MirRelationExpr) {
        let mut deadlist = BTreeSet::new();
        let mut worklist = vec![self];
        while let Some(expr) = worklist.pop() {
            if let MirRelationExpr::LetRec {
                ids,
                values,
                limits: _,
                body,
            } = expr
            {
                let ids_values = values
                    .drain(..)
                    .zip(ids)
                    .map(|(value, id)| (*id, value))
                    .collect::<Vec<_>>();
                *expr = body.take_dangerous();
                for (id, mut value) in ids_values.into_iter().rev() {
                    // Remove references to potentially recursive identifiers.
                    deadlist.insert(id);
                    value.visit_pre_mut(|e| {
                        if let MirRelationExpr::Get {
                            id: crate::Id::Local(id),
                            typ,
                            ..
                        } = e
                        {
                            let typ = typ.clone();
                            if deadlist.contains(id) {
                                e.take_safely(Some(typ));
                            }
                        }
                    });
                    *expr = MirRelationExpr::Let {
                        id,
                        value: Box::new(value),
                        body: Box::new(expr.take_dangerous()),
                    };
                }
                worklist.push(expr);
            } else {
                worklist.extend(expr.children_mut().rev());
            }
        }
    }

    /// For each Id `id'` referenced in `expr`, if it is larger or equal than `id`, then record in
    /// `expire_whens` that when `id'` is redefined, then we should expire the information that
    /// we are holding about `id`. Call `do_expirations` with `expire_whens` at each Id
    /// redefinition.
    ///
    /// IMPORTANT: Relies on the numbering of Ids to be what `renumber_bindings` gives.
    pub fn collect_expirations(
        id: LocalId,
        expr: &MirRelationExpr,
        expire_whens: &mut BTreeMap<LocalId, Vec<LocalId>>,
    ) {
        expr.visit_pre(|e| {
            if let MirRelationExpr::Get {
                id: Id::Local(referenced_id),
                ..
            } = e
            {
                // The following check needs `renumber_bindings` to have run recently
                if referenced_id >= &id {
                    expire_whens
                        .entry(*referenced_id)
                        .or_insert_with(Vec::new)
                        .push(id);
                }
            }
        });
    }

    /// Call this function when `id` is redefined. It modifies `id_infos` by removing information
    /// about such Ids whose information depended on the earlier definition of `id`, according to
    /// `expire_whens`. Also modifies `expire_whens`: it removes the currently processed entry.
    pub fn do_expirations<I>(
        redefined_id: LocalId,
        expire_whens: &mut BTreeMap<LocalId, Vec<LocalId>>,
        id_infos: &mut BTreeMap<LocalId, I>,
    ) -> Vec<(LocalId, I)> {
        let mut expired_infos = Vec::new();
        if let Some(expirations) = expire_whens.remove(&redefined_id) {
            for expired_id in expirations.into_iter() {
                if let Some(offer) = id_infos.remove(&expired_id) {
                    expired_infos.push((expired_id, offer));
                }
            }
        }
        expired_infos
    }
}
/// Augment non-nullability of columns, by observing either
/// 1. Predicates that explicitly test for null values, and
/// 2. Columns that if null would make a predicate be null.
pub fn non_nullable_columns(predicates: &[MirScalarExpr]) -> BTreeSet<usize> {
    let mut nonnull_required_columns = BTreeSet::new();
    for predicate in predicates {
        // Add any columns that being null would force the predicate to be null.
        // Should that happen, the row would be discarded.
        predicate.non_null_requirements(&mut nonnull_required_columns);

        /*
        Test for explicit checks that a column is non-null.

        This analysis is ad hoc, and will miss things:

        materialize=> create table a(x int, y int);
        CREATE TABLE
        materialize=> explain with(types) select x from a where (y=x and y is not null) or x is not null;
        Optimized Plan
        --------------------------------------------------------------------------------------------------------
        Explained Query:                                                                                      +
        Project (#0) // { types: "(integer?)" }                                                             +
        Filter ((#0) IS NOT NULL OR ((#1) IS NOT NULL AND (#0 = #1))) // { types: "(integer?, integer?)" }+
        Get materialize.public.a // { types: "(integer?, integer?)" }                                   +
                                                                                  +
        Source materialize.public.a                                                                           +
        filter=(((#0) IS NOT NULL OR ((#1) IS NOT NULL AND (#0 = #1))))                                     +

        (1 row)
        */

        if let MirScalarExpr::CallUnary {
            func: UnaryFunc::Not(scalar_func::Not),
            expr,
        } = predicate
        {
            if let MirScalarExpr::CallUnary {
                func: UnaryFunc::IsNull(scalar_func::IsNull),
                expr,
            } = &**expr
            {
                if let MirScalarExpr::Column(c, _name) = &**expr {
                    nonnull_required_columns.insert(*c);
                }
            }
        }
    }

    nonnull_required_columns
}

impl CollectionPlan for MirRelationExpr {
    // !!!WARNING!!!: this method has an HirRelationExpr counterpart. The two
    // should be kept in sync w.r.t. HIR ⇒ MIR lowering!
    fn depends_on_into(&self, out: &mut BTreeSet<GlobalId>) {
        if let MirRelationExpr::Get {
            id: Id::Global(id), ..
        } = self
        {
            out.insert(*id);
        }
        self.visit_children(|expr| expr.depends_on_into(out))
    }
}

impl MirRelationExpr {
    /// Iterates through references to child expressions.
    pub fn children(&self) -> impl DoubleEndedIterator<Item = &Self> {
        let mut first = None;
        let mut second = None;
        let mut rest = None;
        let mut last = None;

        use MirRelationExpr::*;
        match self {
            Constant { .. } | Get { .. } => (),
            Let { value, body, .. } => {
                first = Some(&**value);
                second = Some(&**body);
            }
            LetRec { values, body, .. } => {
                rest = Some(values);
                last = Some(&**body);
            }
            Project { input, .. }
            | Map { input, .. }
            | FlatMap { input, .. }
            | Filter { input, .. }
            | Reduce { input, .. }
            | TopK { input, .. }
            | Negate { input }
            | Threshold { input }
            | ArrangeBy { input, .. } => {
                first = Some(&**input);
            }
            Join { inputs, .. } => {
                rest = Some(inputs);
            }
            Union { base, inputs } => {
                first = Some(&**base);
                rest = Some(inputs);
            }
        }

        first
            .into_iter()
            .chain(second)
            .chain(rest.into_iter().flatten())
            .chain(last)
    }

    /// Iterates through mutable references to child expressions.
    pub fn children_mut(&mut self) -> impl DoubleEndedIterator<Item = &mut Self> {
        let mut first = None;
        let mut second = None;
        let mut rest = None;
        let mut last = None;

        use MirRelationExpr::*;
        match self {
            Constant { .. } | Get { .. } => (),
            Let { value, body, .. } => {
                first = Some(&mut **value);
                second = Some(&mut **body);
            }
            LetRec { values, body, .. } => {
                rest = Some(values);
                last = Some(&mut **body);
            }
            Project { input, .. }
            | Map { input, .. }
            | FlatMap { input, .. }
            | Filter { input, .. }
            | Reduce { input, .. }
            | TopK { input, .. }
            | Negate { input }
            | Threshold { input }
            | ArrangeBy { input, .. } => {
                first = Some(&mut **input);
            }
            Join { inputs, .. } => {
                rest = Some(inputs);
            }
            Union { base, inputs } => {
                first = Some(&mut **base);
                rest = Some(inputs);
            }
        }

        first
            .into_iter()
            .chain(second)
            .chain(rest.into_iter().flatten())
            .chain(last)
    }

    /// Iterative pre-order visitor.
    pub fn visit_pre<'a, F: FnMut(&'a Self)>(&'a self, mut f: F) {
        let mut worklist = vec![self];
        while let Some(expr) = worklist.pop() {
            f(expr);
            worklist.extend(expr.children().rev());
        }
    }

    /// Iterative pre-order visitor.
    pub fn visit_pre_mut<F: FnMut(&mut Self)>(&mut self, mut f: F) {
        let mut worklist = vec![self];
        while let Some(expr) = worklist.pop() {
            f(expr);
            worklist.extend(expr.children_mut().rev());
        }
    }

    /// Return a vector of references to the subtrees of this expression
    /// in post-visit order (the last element is `&self`).
    pub fn post_order_vec(&self) -> Vec<&Self> {
        let mut stack = vec![self];
        let mut result = vec![];
        while let Some(expr) = stack.pop() {
            result.push(expr);
            stack.extend(expr.children());
        }
        result.reverse();
        result
    }
}

impl VisitChildren<Self> for MirRelationExpr {
    fn visit_children<F>(&self, mut f: F)
    where
        F: FnMut(&Self),
    {
        for child in self.children() {
            f(child)
        }
    }

    fn visit_mut_children<F>(&mut self, mut f: F)
    where
        F: FnMut(&mut Self),
    {
        for child in self.children_mut() {
            f(child)
        }
    }

    fn try_visit_children<F, E>(&self, mut f: F) -> Result<(), E>
    where
        F: FnMut(&Self) -> Result<(), E>,
        E: From<RecursionLimitError>,
    {
        for child in self.children() {
            f(child)?
        }
        Ok(())
    }

    fn try_visit_mut_children<F, E>(&mut self, mut f: F) -> Result<(), E>
    where
        F: FnMut(&mut Self) -> Result<(), E>,
        E: From<RecursionLimitError>,
    {
        for child in self.children_mut() {
            f(child)?
        }
        Ok(())
    }
}

/// Specification for an ordering by a column.
#[derive(
    Arbitrary,
    Debug,
    Clone,
    Copy,
    Eq,
    PartialEq,
    Ord,
    PartialOrd,
    Serialize,
    Deserialize,
    Hash,
    MzReflect,
)]
pub struct ColumnOrder {
    /// The column index.
    pub column: usize,
    /// Whether to sort in descending order.
    #[serde(default)]
    pub desc: bool,
    /// Whether to sort nulls last.
    #[serde(default)]
    pub nulls_last: bool,
}

impl Columnation for ColumnOrder {
    type InnerRegion = CopyRegion<Self>;
}

impl RustType<ProtoColumnOrder> for ColumnOrder {
    fn into_proto(&self) -> ProtoColumnOrder {
        ProtoColumnOrder {
            column: self.column.into_proto(),
            desc: self.desc,
            nulls_last: self.nulls_last,
        }
    }

    fn from_proto(proto: ProtoColumnOrder) -> Result<Self, TryFromProtoError> {
        Ok(ColumnOrder {
            column: proto.column.into_rust()?,
            desc: proto.desc,
            nulls_last: proto.nulls_last,
        })
    }
}

impl<'a, M> fmt::Display for HumanizedExpr<'a, ColumnOrder, M>
where
    M: HumanizerMode,
{
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        // If you modify this, then please also attend to Display for ColumnOrderWithExpr!
        write!(
            f,
            "{} {} {}",
            self.child(&self.expr.column),
            if self.expr.desc { "desc" } else { "asc" },
            if self.expr.nulls_last {
                "nulls_last"
            } else {
                "nulls_first"
            },
        )
    }
}

/// Describes an aggregation expression.
#[derive(
    Arbitrary, Clone, Debug, Eq, PartialEq, Ord, PartialOrd, Serialize, Deserialize, Hash, MzReflect,
)]
pub struct AggregateExpr {
    /// Names the aggregation function.
    pub func: AggregateFunc,
    /// An expression which extracts from each row the input to `func`.
    pub expr: MirScalarExpr,
    /// Should the aggregation be applied only to distinct results in each group.
    #[serde(default)]
    pub distinct: bool,
}

impl RustType<ProtoAggregateExpr> for AggregateExpr {
    fn into_proto(&self) -> ProtoAggregateExpr {
        ProtoAggregateExpr {
            func: Some(self.func.into_proto()),
            expr: Some(self.expr.into_proto()),
            distinct: self.distinct,
        }
    }

    fn from_proto(proto: ProtoAggregateExpr) -> Result<Self, TryFromProtoError> {
        Ok(Self {
            func: proto.func.into_rust_if_some("ProtoAggregateExpr::func")?,
            expr: proto.expr.into_rust_if_some("ProtoAggregateExpr::expr")?,
            distinct: proto.distinct,
        })
    }
}

impl AggregateExpr {
    /// Computes the type of this `AggregateExpr`.
    pub fn typ(&self, column_types: &[ColumnType]) -> ColumnType {
        self.func.output_type(self.expr.typ(column_types))
    }

    /// Returns whether the expression has a constant result.
    pub fn is_constant(&self) -> bool {
        match self.func {
            AggregateFunc::MaxInt16
            | AggregateFunc::MaxInt32
            | AggregateFunc::MaxInt64
            | AggregateFunc::MaxUInt16
            | AggregateFunc::MaxUInt32
            | AggregateFunc::MaxUInt64
            | AggregateFunc::MaxMzTimestamp
            | AggregateFunc::MaxFloat32
            | AggregateFunc::MaxFloat64
            | AggregateFunc::MaxBool
            | AggregateFunc::MaxString
            | AggregateFunc::MaxDate
            | AggregateFunc::MaxTimestamp
            | AggregateFunc::MaxTimestampTz
            | AggregateFunc::MinInt16
            | AggregateFunc::MinInt32
            | AggregateFunc::MinInt64
            | AggregateFunc::MinUInt16
            | AggregateFunc::MinUInt32
            | AggregateFunc::MinUInt64
            | AggregateFunc::MinMzTimestamp
            | AggregateFunc::MinFloat32
            | AggregateFunc::MinFloat64
            | AggregateFunc::MinBool
            | AggregateFunc::MinString
            | AggregateFunc::MinDate
            | AggregateFunc::MinTimestamp
            | AggregateFunc::MinTimestampTz
            | AggregateFunc::Any
            | AggregateFunc::All
            | AggregateFunc::Dummy => self.expr.is_literal(),
            AggregateFunc::Count => self.expr.is_literal_null(),
            _ => self.expr.is_literal_err(),
        }
    }

    /// Returns an expression that computes `self` on a group that has exactly one row.
    /// Instead of performing a `Reduce` with `self`, one can perform a `Map` with the expression
    /// returned by `on_unique`, which is cheaper. (See `ReduceElision`.)
    pub fn on_unique(&self, input_type: &[ColumnType]) -> MirScalarExpr {
        match &self.func {
            // Count is one if non-null, and zero if null.
            AggregateFunc::Count => self
                .expr
                .clone()
                .call_unary(UnaryFunc::IsNull(crate::func::IsNull))
                .if_then_else(
                    MirScalarExpr::literal_ok(Datum::Int64(0), ScalarType::Int64),
                    MirScalarExpr::literal_ok(Datum::Int64(1), ScalarType::Int64),
                ),

            // SumInt16 takes Int16s as input, but outputs Int64s.
            AggregateFunc::SumInt16 => self
                .expr
                .clone()
                .call_unary(UnaryFunc::CastInt16ToInt64(scalar_func::CastInt16ToInt64)),

            // SumInt32 takes Int32s as input, but outputs Int64s.
            AggregateFunc::SumInt32 => self
                .expr
                .clone()
                .call_unary(UnaryFunc::CastInt32ToInt64(scalar_func::CastInt32ToInt64)),

            // SumInt64 takes Int64s as input, but outputs numerics.
            AggregateFunc::SumInt64 => self.expr.clone().call_unary(UnaryFunc::CastInt64ToNumeric(
                scalar_func::CastInt64ToNumeric(Some(NumericMaxScale::ZERO)),
            )),

            // SumUInt16 takes UInt16s as input, but outputs UInt64s.
            AggregateFunc::SumUInt16 => self.expr.clone().call_unary(
                UnaryFunc::CastUint16ToUint64(scalar_func::CastUint16ToUint64),
            ),

            // SumUInt32 takes UInt32s as input, but outputs UInt64s.
            AggregateFunc::SumUInt32 => self.expr.clone().call_unary(
                UnaryFunc::CastUint32ToUint64(scalar_func::CastUint32ToUint64),
            ),

            // SumUInt64 takes UInt64s as input, but outputs numerics.
            AggregateFunc::SumUInt64 => {
                self.expr.clone().call_unary(UnaryFunc::CastUint64ToNumeric(
                    scalar_func::CastUint64ToNumeric(Some(NumericMaxScale::ZERO)),
                ))
            }

            // JsonbAgg takes _anything_ as input, but must output a Jsonb array.
            AggregateFunc::JsonbAgg { .. } => MirScalarExpr::CallVariadic {
                func: VariadicFunc::JsonbBuildArray,
                exprs: vec![
                    self.expr
                        .clone()
                        .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0))),
                ],
            },

            // JsonbAgg takes _anything_ as input, but must output a Jsonb object.
            AggregateFunc::JsonbObjectAgg { .. } => {
                let record = self
                    .expr
                    .clone()
                    .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));
                MirScalarExpr::CallVariadic {
                    func: VariadicFunc::JsonbBuildObject,
                    exprs: (0..2)
                        .map(|i| {
                            record
                                .clone()
                                .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(i)))
                        })
                        .collect(),
                }
            }

            AggregateFunc::MapAgg { value_type, .. } => {
                let record = self
                    .expr
                    .clone()
                    .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));
                MirScalarExpr::CallVariadic {
                    func: VariadicFunc::MapBuild {
                        value_type: value_type.clone(),
                    },
                    exprs: (0..2)
                        .map(|i| {
                            record
                                .clone()
                                .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(i)))
                        })
                        .collect(),
                }
            }

            // StringAgg takes nested records of strings and outputs a string
            AggregateFunc::StringAgg { .. } => self
                .expr
                .clone()
                .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)))
                .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0))),

            // ListConcat and ArrayConcat take a single level of records and output a list containing exactly 1 element
            AggregateFunc::ListConcat { .. } | AggregateFunc::ArrayConcat { .. } => self
                .expr
                .clone()
                .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0))),

            // RowNumber, Rank, DenseRank take a list of records and output a list containing exactly 1 element
            AggregateFunc::RowNumber { .. } => {
                self.on_unique_ranking_window_funcs(input_type, "?row_number?")
            }
            AggregateFunc::Rank { .. } => self.on_unique_ranking_window_funcs(input_type, "?rank?"),
            AggregateFunc::DenseRank { .. } => {
                self.on_unique_ranking_window_funcs(input_type, "?dense_rank?")
            }

            // The input type for LagLead is ((OriginalRow, (InputValue, Offset, Default)), OrderByExprs...)
            AggregateFunc::LagLead { lag_lead, .. } => {
                let tuple = self
                    .expr
                    .clone()
                    .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));

                // Get the overall return type
                let return_type_with_orig_row = self
                    .typ(input_type)
                    .scalar_type
                    .unwrap_list_element_type()
                    .clone();
                let lag_lead_return_type =
                    return_type_with_orig_row.unwrap_record_element_type()[0].clone();

                // Extract the original row
                let original_row = tuple
                    .clone()
                    .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));

                // Extract the encoded args
                let encoded_args =
                    tuple.call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(1)));

                let (result_expr, column_name) =
                    Self::on_unique_lag_lead(lag_lead, encoded_args, lag_lead_return_type.clone());

                MirScalarExpr::CallVariadic {
                    func: VariadicFunc::ListCreate {
                        elem_type: return_type_with_orig_row,
                    },
                    exprs: vec![MirScalarExpr::CallVariadic {
                        func: VariadicFunc::RecordCreate {
                            field_names: vec![column_name, ColumnName::from("?record?")],
                        },
                        exprs: vec![result_expr, original_row],
                    }],
                }
            }

            // The input type for FirstValue is ((OriginalRow, InputValue), OrderByExprs...)
            AggregateFunc::FirstValue { window_frame, .. } => {
                let tuple = self
                    .expr
                    .clone()
                    .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));

                // Get the overall return type
                let return_type_with_orig_row = self
                    .typ(input_type)
                    .scalar_type
                    .unwrap_list_element_type()
                    .clone();
                let first_value_return_type =
                    return_type_with_orig_row.unwrap_record_element_type()[0].clone();

                // Extract the original row
                let original_row = tuple
                    .clone()
                    .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));

                // Extract the input value
                let arg = tuple.call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(1)));

                let (result_expr, column_name) = Self::on_unique_first_value_last_value(
                    window_frame,
                    arg,
                    first_value_return_type,
                );

                MirScalarExpr::CallVariadic {
                    func: VariadicFunc::ListCreate {
                        elem_type: return_type_with_orig_row,
                    },
                    exprs: vec![MirScalarExpr::CallVariadic {
                        func: VariadicFunc::RecordCreate {
                            field_names: vec![column_name, ColumnName::from("?record?")],
                        },
                        exprs: vec![result_expr, original_row],
                    }],
                }
            }

            // The input type for LastValue is ((OriginalRow, InputValue), OrderByExprs...)
            AggregateFunc::LastValue { window_frame, .. } => {
                let tuple = self
                    .expr
                    .clone()
                    .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));

                // Get the overall return type
                let return_type_with_orig_row = self
                    .typ(input_type)
                    .scalar_type
                    .unwrap_list_element_type()
                    .clone();
                let last_value_return_type =
                    return_type_with_orig_row.unwrap_record_element_type()[0].clone();

                // Extract the original row
                let original_row = tuple
                    .clone()
                    .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));

                // Extract the input value
                let arg = tuple.call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(1)));

                let (result_expr, column_name) = Self::on_unique_first_value_last_value(
                    window_frame,
                    arg,
                    last_value_return_type,
                );

                MirScalarExpr::CallVariadic {
                    func: VariadicFunc::ListCreate {
                        elem_type: return_type_with_orig_row,
                    },
                    exprs: vec![MirScalarExpr::CallVariadic {
                        func: VariadicFunc::RecordCreate {
                            field_names: vec![column_name, ColumnName::from("?record?")],
                        },
                        exprs: vec![result_expr, original_row],
                    }],
                }
            }

            // The input type for window aggs is ((OriginalRow, InputValue), OrderByExprs...)
            // See an example MIR in `window_func_applied_to`.
            AggregateFunc::WindowAggregate {
                wrapped_aggregate,
                window_frame,
                order_by: _,
            } => {
                // TODO: deduplicate code between the various window function cases.

                let tuple = self
                    .expr
                    .clone()
                    .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));

                // Get the overall return type
                let return_type = self
                    .typ(input_type)
                    .scalar_type
                    .unwrap_list_element_type()
                    .clone();
                let window_agg_return_type = return_type.unwrap_record_element_type()[0].clone();

                // Extract the original row
                let original_row = tuple
                    .clone()
                    .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));

                // Extract the input value
                let arg_expr = tuple.call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(1)));

                let (result, column_name) = Self::on_unique_window_agg(
                    window_frame,
                    arg_expr,
                    input_type,
                    window_agg_return_type,
                    wrapped_aggregate,
                );

                MirScalarExpr::CallVariadic {
                    func: VariadicFunc::ListCreate {
                        elem_type: return_type,
                    },
                    exprs: vec![MirScalarExpr::CallVariadic {
                        func: VariadicFunc::RecordCreate {
                            field_names: vec![column_name, ColumnName::from("?record?")],
                        },
                        exprs: vec![result, original_row],
                    }],
                }
            }

            // The input type is ((OriginalRow, (Arg1, Arg2, ...)), OrderByExprs...)
            AggregateFunc::FusedWindowAggregate {
                wrapped_aggregates,
                order_by: _,
                window_frame,
            } => {
                // Throw away OrderByExprs
                let tuple = self
                    .expr
                    .clone()
                    .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));

                // Extract the original row
                let original_row = tuple
                    .clone()
                    .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));

                // Extract the args of the fused call
                let all_args = tuple.call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(1)));

                let return_type_with_orig_row = self
                    .typ(input_type)
                    .scalar_type
                    .unwrap_list_element_type()
                    .clone();

                let all_func_return_types =
                    return_type_with_orig_row.unwrap_record_element_type()[0].clone();
                let mut func_result_exprs = Vec::new();
                let mut col_names = Vec::new();
                for (idx, wrapped_aggr) in wrapped_aggregates.iter().enumerate() {
                    let arg = all_args
                        .clone()
                        .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(idx)));
                    let return_type =
                        all_func_return_types.unwrap_record_element_type()[idx].clone();
                    let (result, column_name) = Self::on_unique_window_agg(
                        window_frame,
                        arg,
                        input_type,
                        return_type,
                        wrapped_aggr,
                    );
                    func_result_exprs.push(result);
                    col_names.push(column_name);
                }

                MirScalarExpr::CallVariadic {
                    func: VariadicFunc::ListCreate {
                        elem_type: return_type_with_orig_row,
                    },
                    exprs: vec![MirScalarExpr::CallVariadic {
                        func: VariadicFunc::RecordCreate {
                            field_names: vec![
                                ColumnName::from("?fused_window_aggr?"),
                                ColumnName::from("?record?"),
                            ],
                        },
                        exprs: vec![
                            MirScalarExpr::CallVariadic {
                                func: VariadicFunc::RecordCreate {
                                    field_names: col_names,
                                },
                                exprs: func_result_exprs,
                            },
                            original_row,
                        ],
                    }],
                }
            }

            // The input type is ((OriginalRow, (Args1, Args2, ...)), OrderByExprs...)
            AggregateFunc::FusedValueWindowFunc {
                funcs,
                order_by: outer_order_by,
            } => {
                // Throw away OrderByExprs
                let tuple = self
                    .expr
                    .clone()
                    .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));

                // Extract the original row
                let original_row = tuple
                    .clone()
                    .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));

                // Extract the encoded args of the fused call
                let all_encoded_args =
                    tuple.call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(1)));

                let return_type_with_orig_row = self
                    .typ(input_type)
                    .scalar_type
                    .unwrap_list_element_type()
                    .clone();

                let all_func_return_types =
                    return_type_with_orig_row.unwrap_record_element_type()[0].clone();
                let mut func_result_exprs = Vec::new();
                let mut col_names = Vec::new();
                for (idx, func) in funcs.iter().enumerate() {
                    let args_for_func = all_encoded_args
                        .clone()
                        .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(idx)));
                    let return_type_for_func =
                        all_func_return_types.unwrap_record_element_type()[idx].clone();
                    let (result, column_name) = match func {
                        AggregateFunc::LagLead {
                            lag_lead,
                            order_by,
                            ignore_nulls: _,
                        } => {
                            assert_eq!(order_by, outer_order_by);
                            Self::on_unique_lag_lead(lag_lead, args_for_func, return_type_for_func)
                        }
                        AggregateFunc::FirstValue {
                            window_frame,
                            order_by,
                        } => {
                            assert_eq!(order_by, outer_order_by);
                            Self::on_unique_first_value_last_value(
                                window_frame,
                                args_for_func,
                                return_type_for_func,
                            )
                        }
                        AggregateFunc::LastValue {
                            window_frame,
                            order_by,
                        } => {
                            assert_eq!(order_by, outer_order_by);
                            Self::on_unique_first_value_last_value(
                                window_frame,
                                args_for_func,
                                return_type_for_func,
                            )
                        }
                        _ => panic!("unknown function in FusedValueWindowFunc"),
                    };
                    func_result_exprs.push(result);
                    col_names.push(column_name);
                }

                MirScalarExpr::CallVariadic {
                    func: VariadicFunc::ListCreate {
                        elem_type: return_type_with_orig_row,
                    },
                    exprs: vec![MirScalarExpr::CallVariadic {
                        func: VariadicFunc::RecordCreate {
                            field_names: vec![
                                ColumnName::from("?fused_value_window_func?"),
                                ColumnName::from("?record?"),
                            ],
                        },
                        exprs: vec![
                            MirScalarExpr::CallVariadic {
                                func: VariadicFunc::RecordCreate {
                                    field_names: col_names,
                                },
                                exprs: func_result_exprs,
                            },
                            original_row,
                        ],
                    }],
                }
            }

            // All other variants should return the argument to the aggregation.
            AggregateFunc::MaxNumeric
            | AggregateFunc::MaxInt16
            | AggregateFunc::MaxInt32
            | AggregateFunc::MaxInt64
            | AggregateFunc::MaxUInt16
            | AggregateFunc::MaxUInt32
            | AggregateFunc::MaxUInt64
            | AggregateFunc::MaxMzTimestamp
            | AggregateFunc::MaxFloat32
            | AggregateFunc::MaxFloat64
            | AggregateFunc::MaxBool
            | AggregateFunc::MaxString
            | AggregateFunc::MaxDate
            | AggregateFunc::MaxTimestamp
            | AggregateFunc::MaxTimestampTz
            | AggregateFunc::MaxInterval
            | AggregateFunc::MaxTime
            | AggregateFunc::MinNumeric
            | AggregateFunc::MinInt16
            | AggregateFunc::MinInt32
            | AggregateFunc::MinInt64
            | AggregateFunc::MinUInt16
            | AggregateFunc::MinUInt32
            | AggregateFunc::MinUInt64
            | AggregateFunc::MinMzTimestamp
            | AggregateFunc::MinFloat32
            | AggregateFunc::MinFloat64
            | AggregateFunc::MinBool
            | AggregateFunc::MinString
            | AggregateFunc::MinDate
            | AggregateFunc::MinTimestamp
            | AggregateFunc::MinTimestampTz
            | AggregateFunc::MinInterval
            | AggregateFunc::MinTime
            | AggregateFunc::SumFloat32
            | AggregateFunc::SumFloat64
            | AggregateFunc::SumNumeric
            | AggregateFunc::Any
            | AggregateFunc::All
            | AggregateFunc::Dummy => self.expr.clone(),
        }
    }

    /// `on_unique` for ROW_NUMBER, RANK, DENSE_RANK
    fn on_unique_ranking_window_funcs(
        &self,
        input_type: &[ColumnType],
        col_name: &str,
    ) -> MirScalarExpr {
        let list = self
            .expr
            .clone()
            // extract the list within the record
            .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));

        // extract the expression within the list
        let record = MirScalarExpr::CallVariadic {
            func: VariadicFunc::ListIndex,
            exprs: vec![
                list,
                MirScalarExpr::literal_ok(Datum::Int64(1), ScalarType::Int64),
            ],
        };

        MirScalarExpr::CallVariadic {
            func: VariadicFunc::ListCreate {
                elem_type: self
                    .typ(input_type)
                    .scalar_type
                    .unwrap_list_element_type()
                    .clone(),
            },
            exprs: vec![MirScalarExpr::CallVariadic {
                func: VariadicFunc::RecordCreate {
                    field_names: vec![ColumnName::from(col_name), ColumnName::from("?record?")],
                },
                exprs: vec![
                    MirScalarExpr::literal_ok(Datum::Int64(1), ScalarType::Int64),
                    record,
                ],
            }],
        }
    }

    /// `on_unique` for `lag` and `lead`
    fn on_unique_lag_lead(
        lag_lead: &LagLeadType,
        encoded_args: MirScalarExpr,
        return_type: ScalarType,
    ) -> (MirScalarExpr, ColumnName) {
        let expr = encoded_args
            .clone()
            .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(0)));
        let offset = encoded_args
            .clone()
            .call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(1)));
        let default_value =
            encoded_args.call_unary(UnaryFunc::RecordGet(scalar_func::RecordGet(2)));

        // In this case, the window always has only one element, so if the offset is not null and
        // not zero, the default value should be returned instead.
        let value = offset
            .clone()
            .call_binary(
                MirScalarExpr::literal_ok(Datum::Int32(0), ScalarType::Int32),
                crate::BinaryFunc::Eq,
            )
            .if_then_else(expr, default_value);
        let result_expr = offset
            .call_unary(UnaryFunc::IsNull(crate::func::IsNull))
            .if_then_else(MirScalarExpr::literal_null(return_type), value);

        let column_name = ColumnName::from(match lag_lead {
            LagLeadType::Lag => "?lag?",
            LagLeadType::Lead => "?lead?",
        });

        (result_expr, column_name)
    }

    /// `on_unique` for `first_value` and `last_value`
    fn on_unique_first_value_last_value(
        window_frame: &WindowFrame,
        arg: MirScalarExpr,
        return_type: ScalarType,
    ) -> (MirScalarExpr, ColumnName) {
        // If the window frame includes the current (single) row, return its value, null otherwise
        let result_expr = if window_frame.includes_current_row() {
            arg
        } else {
            MirScalarExpr::literal_null(return_type)
        };
        (result_expr, ColumnName::from("?first_value?"))
    }

    /// `on_unique` for window aggregations
    fn on_unique_window_agg(
        window_frame: &WindowFrame,
        arg_expr: MirScalarExpr,
        input_type: &[ColumnType],
        return_type: ScalarType,
        wrapped_aggr: &AggregateFunc,
    ) -> (MirScalarExpr, ColumnName) {
        // If the window frame includes the current (single) row, evaluate the wrapped aggregate on
        // that row. Otherwise, return the default value for the aggregate.
        let result_expr = if window_frame.includes_current_row() {
            AggregateExpr {
                func: wrapped_aggr.clone(),
                expr: arg_expr,
                distinct: false, // We have just one input element; DISTINCT doesn't matter.
            }
            .on_unique(input_type)
        } else {
            MirScalarExpr::literal_ok(wrapped_aggr.default(), return_type)
        };
        (result_expr, ColumnName::from("?window_agg?"))
    }

    /// Returns whether the expression is COUNT(*) or not.  Note that
    /// when we define the count builtin in sql::func, we convert
    /// COUNT(*) to COUNT(true), making it indistinguishable from
    /// literal COUNT(true), but we prefer to consider this as the
    /// former.
    ///
    /// (HIR has the same `is_count_asterisk`.)
    pub fn is_count_asterisk(&self) -> bool {
        self.func == AggregateFunc::Count && self.expr.is_literal_true() && !self.distinct
    }
}

/// Describe a join implementation in dataflow.
#[derive(
    Clone, Debug, Eq, PartialEq, Ord, PartialOrd, Serialize, Deserialize, Hash, MzReflect, Arbitrary,
)]
pub enum JoinImplementation {
    /// Perform a sequence of binary differential dataflow joins.
    ///
    /// The first argument indicates
    /// 1) the index of the starting collection,
    /// 2) if it should be arranged, the keys to arrange it by, and
    /// 3) the characteristics of the starting collection (for EXPLAINing).
    /// The sequence that follows lists other relation indexes, and the key for
    /// the arrangement we should use when joining it in.
    /// The JoinInputCharacteristics are for EXPLAINing the characteristics that
    /// were used for join ordering.
    ///
    /// Each collection index should occur exactly once, either as the starting collection
    /// or somewhere in the list.
    Differential(
        (
            usize,
            Option<Vec<MirScalarExpr>>,
            Option<JoinInputCharacteristics>,
        ),
        Vec<(usize, Vec<MirScalarExpr>, Option<JoinInputCharacteristics>)>,
    ),
    /// Perform independent delta query dataflows for each input.
    ///
    /// The argument is a sequence of plans, for the input collections in order.
    /// Each plan starts from the corresponding index, and then in sequence joins
    /// against collections identified by index and with the specified arrangement key.
    /// The JoinInputCharacteristics are for EXPLAINing the characteristics that were
    /// used for join ordering.
    DeltaQuery(Vec<Vec<(usize, Vec<MirScalarExpr>, Option<JoinInputCharacteristics>)>>),
    /// Join a user-created index with a constant collection to speed up the evaluation of a
    /// predicate such as `(f1 = 3 AND f2 = 5) OR (f1 = 7 AND f2 = 9)`.
    /// This gets translated to a Differential join during MIR -> LIR lowering, but we still want
    /// to represent it in MIR, because the fast path detection wants to match on this.
    ///
    /// Consists of (`<coll_id>`, `<index_id>`, `<index_key>`, `<constants>`)
    IndexedFilter(
        GlobalId,
        GlobalId,
        Vec<MirScalarExpr>,
        #[mzreflect(ignore)] Vec<Row>,
    ),
    /// No implementation yet selected.
    Unimplemented,
}

impl Default for JoinImplementation {
    fn default() -> Self {
        JoinImplementation::Unimplemented
    }
}

impl JoinImplementation {
    /// Returns `true` iff the value is not [`JoinImplementation::Unimplemented`].
    pub fn is_implemented(&self) -> bool {
        match self {
            Self::Unimplemented => false,
            _ => true,
        }
    }

    /// Returns an optional implementation name if the value is not [`JoinImplementation::Unimplemented`].
    pub fn name(&self) -> Option<&'static str> {
        match self {
            Self::Differential(..) => Some("differential"),
            Self::DeltaQuery(..) => Some("delta"),
            Self::IndexedFilter(..) => Some("indexed_filter"),
            Self::Unimplemented => None,
        }
    }
}

/// Characteristics of a join order candidate collection.
///
/// A candidate is described by a collection and a key, and may have various liabilities.
/// Primarily, the candidate may risk substantial inflation of records, which is something
/// that concerns us greatly. Additionally, the candidate may be unarranged, and we would
/// prefer candidates that do not require additional memory. Finally, we prefer lower id
/// collections in the interest of consistent tie-breaking. For more characteristics, see
/// comments on individual fields.
///
/// This has more than one version. `new` instantiates the appropriate version based on a
/// feature flag.
#[derive(
    Eq, PartialEq, Ord, PartialOrd, Debug, Clone, Serialize, Deserialize, Hash, MzReflect, Arbitrary,
)]
pub enum JoinInputCharacteristics {
    /// Old version, with `enable_join_prioritize_arranged` turned off.
    V1(JoinInputCharacteristicsV1),
    /// Newer version, with `enable_join_prioritize_arranged` turned on.
    V2(JoinInputCharacteristicsV2),
}

impl JoinInputCharacteristics {
    /// Creates a new instance with the given characteristics.
    pub fn new(
        unique_key: bool,
        key_length: usize,
        arranged: bool,
        cardinality: Option<usize>,
        filters: FilterCharacteristics,
        input: usize,
        enable_join_prioritize_arranged: bool,
    ) -> Self {
        if enable_join_prioritize_arranged {
            Self::V2(JoinInputCharacteristicsV2::new(
                unique_key,
                key_length,
                arranged,
                cardinality,
                filters,
                input,
            ))
        } else {
            Self::V1(JoinInputCharacteristicsV1::new(
                unique_key,
                key_length,
                arranged,
                cardinality,
                filters,
                input,
            ))
        }
    }

    /// Turns the instance into a String to be printed in EXPLAIN.
    pub fn explain(&self) -> String {
        match self {
            Self::V1(jic) => jic.explain(),
            Self::V2(jic) => jic.explain(),
        }
    }

    /// Whether the join input described by `self` is arranged.
    pub fn arranged(&self) -> bool {
        match self {
            Self::V1(jic) => jic.arranged,
            Self::V2(jic) => jic.arranged,
        }
    }

    /// Returns the `FilterCharacteristics` for the join input described by `self`.
    pub fn filters(&mut self) -> &mut FilterCharacteristics {
        match self {
            Self::V1(jic) => &mut jic.filters,
            Self::V2(jic) => &mut jic.filters,
        }
    }
}

/// Newer version of `JoinInputCharacteristics`, with `enable_join_prioritize_arranged` turned on.
#[derive(
    Eq, PartialEq, Ord, PartialOrd, Debug, Clone, Serialize, Deserialize, Hash, MzReflect, Arbitrary,
)]
pub struct JoinInputCharacteristicsV2 {
    /// An excellent indication that record count will not increase.
    pub unique_key: bool,
    /// Cross joins are bad.
    /// (`key_length > 0` also implies that it is not a cross join. However, we need to note cross
    /// joins in a separate field, because not being a cross join is more important than `arranged`,
    /// but otherwise `key_length` is less important than `arranged`.)
    pub not_cross: bool,
    /// Indicates that there will be no additional in-memory footprint.
    pub arranged: bool,
    /// A weaker signal that record count will not increase.
    pub key_length: usize,
    /// Estimated cardinality (lower is better)
    pub cardinality: Option<std::cmp::Reverse<usize>>,
    /// Characteristics of the filter that is applied at this input.
    pub filters: FilterCharacteristics,
    /// We want to prefer input earlier in the input list, for stability of ordering.
    pub input: std::cmp::Reverse<usize>,
}

impl JoinInputCharacteristicsV2 {
    /// Creates a new instance with the given characteristics.
    pub fn new(
        unique_key: bool,
        key_length: usize,
        arranged: bool,
        cardinality: Option<usize>,
        filters: FilterCharacteristics,
        input: usize,
    ) -> Self {
        Self {
            unique_key,
            not_cross: key_length > 0,
            arranged,
            key_length,
            cardinality: cardinality.map(std::cmp::Reverse),
            filters,
            input: std::cmp::Reverse(input),
        }
    }

    /// Turns the instance into a String to be printed in EXPLAIN.
    pub fn explain(&self) -> String {
        let mut e = "".to_owned();
        if self.unique_key {
            e.push_str("U");
        }
        // Don't need to print `not_cross`, because that is visible in the printed key.
        // if !self.not_cross {
        //     e.push_str("C");
        // }
        for _ in 0..self.key_length {
            e.push_str("K");
        }
        if self.arranged {
            e.push_str("A");
        }
        if let Some(std::cmp::Reverse(cardinality)) = self.cardinality {
            e.push_str(&format!("|{cardinality}|"));
        }
        e.push_str(&self.filters.explain());
        e
    }
}

/// Old version of `JoinInputCharacteristics`, with `enable_join_prioritize_arranged` turned off.
#[derive(
    Eq, PartialEq, Ord, PartialOrd, Debug, Clone, Serialize, Deserialize, Hash, MzReflect, Arbitrary,
)]
pub struct JoinInputCharacteristicsV1 {
    /// An excellent indication that record count will not increase.
    pub unique_key: bool,
    /// A weaker signal that record count will not increase.
    pub key_length: usize,
    /// Indicates that there will be no additional in-memory footprint.
    pub arranged: bool,
    /// Estimated cardinality (lower is better)
    pub cardinality: Option<std::cmp::Reverse<usize>>,
    /// Characteristics of the filter that is applied at this input.
    pub filters: FilterCharacteristics,
    /// We want to prefer input earlier in the input list, for stability of ordering.
    pub input: std::cmp::Reverse<usize>,
}

impl JoinInputCharacteristicsV1 {
    /// Creates a new instance with the given characteristics.
    pub fn new(
        unique_key: bool,
        key_length: usize,
        arranged: bool,
        cardinality: Option<usize>,
        filters: FilterCharacteristics,
        input: usize,
    ) -> Self {
        Self {
            unique_key,
            key_length,
            arranged,
            cardinality: cardinality.map(std::cmp::Reverse),
            filters,
            input: std::cmp::Reverse(input),
        }
    }

    /// Turns the instance into a String to be printed in EXPLAIN.
    pub fn explain(&self) -> String {
        let mut e = "".to_owned();
        if self.unique_key {
            e.push_str("U");
        }
        for _ in 0..self.key_length {
            e.push_str("K");
        }
        if self.arranged {
            e.push_str("A");
        }
        if let Some(std::cmp::Reverse(cardinality)) = self.cardinality {
            e.push_str(&format!("|{cardinality}|"));
        }
        e.push_str(&self.filters.explain());
        e
    }
}

/// Instructions for finishing the result of a query.
///
/// The primary reason for the existence of this structure and attendant code
/// is that SQL's ORDER BY requires sorting rows (as already implied by the
/// keywords), whereas much of the rest of SQL is defined in terms of unordered
/// multisets. But as it turns out, the same idea can be used to optimize
/// trivial peeks.
///
/// The generic parameters are for accommodating prepared statement parameters in
/// `limit` and `offset`: the planner can hold these fields as HirScalarExpr long enough to call
/// `bind_parameters` on them.
#[derive(Arbitrary, Debug, Clone, Serialize, Deserialize, PartialEq, Eq)]
pub struct RowSetFinishing<L = NonNeg<i64>, O = usize> {
    /// Order rows by the given columns.
    pub order_by: Vec<ColumnOrder>,
    /// Include only as many rows (after offset).
    pub limit: Option<L>,
    /// Omit as many rows.
    pub offset: O,
    /// Include only given columns.
    pub project: Vec<usize>,
}

impl RustType<ProtoRowSetFinishing> for RowSetFinishing {
    fn into_proto(&self) -> ProtoRowSetFinishing {
        ProtoRowSetFinishing {
            order_by: self.order_by.into_proto(),
            limit: self.limit.into_proto(),
            offset: self.offset.into_proto(),
            project: self.project.into_proto(),
        }
    }

    fn from_proto(x: ProtoRowSetFinishing) -> Result<Self, TryFromProtoError> {
        Ok(RowSetFinishing {
            order_by: x.order_by.into_rust()?,
            limit: x.limit.into_rust()?,
            offset: x.offset.into_rust()?,
            project: x.project.into_rust()?,
        })
    }
}

impl<L> RowSetFinishing<L> {
    /// Returns a trivial finishing, i.e., that does nothing to the result set.
    pub fn trivial(arity: usize) -> RowSetFinishing<L> {
        RowSetFinishing {
            order_by: Vec::new(),
            limit: None,
            offset: 0,
            project: (0..arity).collect(),
        }
    }
    /// True if the finishing does nothing to any result set.
    pub fn is_trivial(&self, arity: usize) -> bool {
        self.limit.is_none()
            && self.order_by.is_empty()
            && self.offset == 0
            && self.project.iter().copied().eq(0..arity)
    }
    /// True if the finishing does not require an ORDER BY.
    ///
    /// LIMIT and OFFSET without an ORDER BY _are_ streamable: without an
    /// explicit ordering we will skip an arbitrary bag of elements and return
    /// the first arbitrary elements in the remaining bag. The result semantics
    /// are still correct but maybe surprising for some users.
    pub fn is_streamable(&self, arity: usize) -> bool {
        self.order_by.is_empty() && self.project.iter().copied().eq(0..arity)
    }
}

impl RowSetFinishing<NonNeg<i64>, usize> {
    /// The number of rows needed from before the finishing to evaluate the finishing:
    /// offset + limit.
    ///
    /// If it returns None, then we need all the rows.
    pub fn num_rows_needed(&self) -> Option<usize> {
        self.limit
            .as_ref()
            .map(|l| usize::cast_from(u64::from(l.clone())) + self.offset)
    }
}

impl RowSetFinishing {
    /// Applies finishing actions to a [`RowCollection`], and reports the total
    /// time it took to run.
    ///
    /// Returns a [`SortedRowCollectionIter`] that contains all of the response data, as
    /// well as the size of the response in bytes.
    pub fn finish(
        &self,
        rows: RowCollection,
        max_result_size: u64,
        max_returned_query_size: Option<u64>,
        duration_histogram: &Histogram,
    ) -> Result<(SortedRowCollectionIter, usize), String> {
        let now = Instant::now();
        let result = self.finish_inner(rows, max_result_size, max_returned_query_size);
        let duration = now.elapsed();
        duration_histogram.observe(duration.as_secs_f64());

        result
    }

    /// Implementation for [`RowSetFinishing::finish`].
    fn finish_inner(
        &self,
        rows: RowCollection,
        max_result_size: u64,
        max_returned_query_size: Option<u64>,
    ) -> Result<(SortedRowCollectionIter, usize), String> {
        // How much additional memory is required to make a sorted view.
        let sorted_view_mem = rows.entries().saturating_mul(std::mem::size_of::<usize>());
        let required_memory = rows.byte_len().saturating_add(sorted_view_mem);

        // Bail if creating the sorted view would require us to use too much memory.
        if required_memory > usize::cast_from(max_result_size) {
            let max_bytes = ByteSize::b(max_result_size);
            return Err(format!("result exceeds max size of {max_bytes}",));
        }

        let sorted_view = rows.sorted_view(&self.order_by);
        let mut iter = sorted_view
            .into_row_iter()
            .apply_offset(self.offset)
            .with_projection(self.project.clone());

        if let Some(limit) = self.limit {
            let limit = u64::from(limit);
            let limit = usize::cast_from(limit);
            iter = iter.with_limit(limit);
        };

        // TODO(parkmycar): Re-think how we can calculate the total response size without
        // having to iterate through the entire collection of Rows, while still
        // respecting the LIMIT, OFFSET, and projections.
        //
        // Note: It feels a bit bad always calculating the response size, but we almost
        // always need it to either check the `max_returned_query_size`, or for reporting
        // in the query history.
        let response_size: usize = iter.clone().map(|row| row.data().len()).sum();

        // Bail if we would end up returning more data to the client than they can support.
        if let Some(max) = max_returned_query_size {
            if response_size > usize::cast_from(max) {
                let max_bytes = ByteSize::b(max);
                return Err(format!("result exceeds max size of {max_bytes}"));
            }
        }

        Ok((iter, response_size))
    }
}

/// A [RowSetFinishing] that can be repeatedly applied to batches of updates (in
/// a [RowCollection]) and keeps track of the remaining limit, offset, and cap
/// on query result size.
#[derive(Debug)]
pub struct RowSetFinishingIncremental {
    /// Include only as many rows (after offset).
    pub remaining_limit: Option<usize>,
    /// Omit as many rows.
    pub remaining_offset: usize,
    /// The maximum allowed result size, as requested by the client.
    pub max_returned_query_size: Option<u64>,
    /// Tracks our remaining allowed budget for result size.
    pub remaining_max_returned_query_size: Option<u64>,
    /// Include only given columns.
    pub project: Vec<usize>,
}

impl RowSetFinishingIncremental {
    /// Turns the given [RowSetFinishing] into a [RowSetFinishingIncremental].
    /// Can only be used when [is_streamable](RowSetFinishing::is_streamable) is
    /// `true`.
    ///
    /// # Panics
    ///
    /// Panics if the result is not streamable, that is it has an ORDER BY.
    pub fn new(
        offset: usize,
        limit: Option<NonNeg<i64>>,
        project: Vec<usize>,
        max_returned_query_size: Option<u64>,
    ) -> Self {
        let limit = limit.map(|l| {
            let l = u64::from(l);
            let l = usize::cast_from(l);
            l
        });

        RowSetFinishingIncremental {
            remaining_limit: limit,
            remaining_offset: offset,
            max_returned_query_size,
            remaining_max_returned_query_size: max_returned_query_size,
            project,
        }
    }

    /// Applies finishing actions to the given [`RowCollection`], and reports
    /// the total time it took to run.
    ///
    /// Returns a [`SortedRowCollectionIter`] that contains all of the response
    /// data.
    pub fn finish_incremental(
        &mut self,
        rows: RowCollection,
        max_result_size: u64,
        duration_histogram: &Histogram,
    ) -> Result<SortedRowCollectionIter, String> {
        let now = Instant::now();
        let result = self.finish_incremental_inner(rows, max_result_size);
        let duration = now.elapsed();
        duration_histogram.observe(duration.as_secs_f64());

        result
    }

    fn finish_incremental_inner(
        &mut self,
        rows: RowCollection,
        max_result_size: u64,
    ) -> Result<SortedRowCollectionIter, String> {
        // How much additional memory is required to make a sorted view.
        let sorted_view_mem = rows.entries().saturating_mul(std::mem::size_of::<usize>());
        let required_memory = rows.byte_len().saturating_add(sorted_view_mem);

        // Bail if creating the sorted view would require us to use too much memory.
        if required_memory > usize::cast_from(max_result_size) {
            let max_bytes = ByteSize::b(max_result_size);
            return Err(format!("total result exceeds max size of {max_bytes}",));
        }

        let batch_num_rows = rows.count(0, None);

        let sorted_view = rows.sorted_view(&[]);
        let mut iter = sorted_view
            .into_row_iter()
            .apply_offset(self.remaining_offset)
            .with_projection(self.project.clone());

        if let Some(limit) = self.remaining_limit {
            iter = iter.with_limit(limit);
        };

        self.remaining_offset = self.remaining_offset.saturating_sub(batch_num_rows);
        if let Some(remaining_limit) = self.remaining_limit.as_mut() {
            *remaining_limit -= iter.count();
        }

        // TODO(parkmycar): Re-think how we can calculate the total response size without
        // having to iterate through the entire collection of Rows, while still
        // respecting the LIMIT, OFFSET, and projections.
        //
        // Note: It feels a bit bad always calculating the response size, but we almost
        // always need it to either check the `max_returned_query_size`, or for reporting
        // in the query history.
        let response_size: usize = iter.clone().map(|row| row.data().len()).sum();

        // Bail if we would end up returning more data to the client than they can support.
        if let Some(max) = self.remaining_max_returned_query_size {
            if response_size > usize::cast_from(max) {
                let max_bytes = ByteSize::b(self.max_returned_query_size.expect("known to exist"));
                return Err(format!("total result exceeds max size of {max_bytes}"));
            }
        }

        Ok(iter)
    }
}

/// Compare `left` and `right` using `order`. If that doesn't produce a strict
/// ordering, call `tiebreaker`.
pub fn compare_columns<F>(
    order: &[ColumnOrder],
    left: &[Datum],
    right: &[Datum],
    tiebreaker: F,
) -> Ordering
where
    F: Fn() -> Ordering,
{
    for order in order {
        let cmp = match (&left[order.column], &right[order.column]) {
            (Datum::Null, Datum::Null) => Ordering::Equal,
            (Datum::Null, _) => {
                if order.nulls_last {
                    Ordering::Greater
                } else {
                    Ordering::Less
                }
            }
            (_, Datum::Null) => {
                if order.nulls_last {
                    Ordering::Less
                } else {
                    Ordering::Greater
                }
            }
            (lval, rval) => {
                if order.desc {
                    rval.cmp(lval)
                } else {
                    lval.cmp(rval)
                }
            }
        };
        if cmp != Ordering::Equal {
            return cmp;
        }
    }
    tiebreaker()
}

/// Describe a window frame, e.g. `RANGE UNBOUNDED PRECEDING` or
/// `ROWS BETWEEN 5 PRECEDING AND CURRENT ROW`.
///
/// Window frames define a subset of the partition , and only a subset of
/// window functions make use of the window frame.
#[derive(
    Arbitrary, Debug, Clone, Eq, PartialEq, Ord, PartialOrd, Serialize, Deserialize, Hash, MzReflect,
)]
pub struct WindowFrame {
    /// ROWS, RANGE or GROUPS
    pub units: WindowFrameUnits,
    /// Where the frame starts
    pub start_bound: WindowFrameBound,
    /// Where the frame ends
    pub end_bound: WindowFrameBound,
}

impl Display for WindowFrame {
    fn fmt(&self, f: &mut Formatter<'_>) -> fmt::Result {
        write!(
            f,
            "{} between {} and {}",
            self.units, self.start_bound, self.end_bound
        )
    }
}

impl WindowFrame {
    /// Return the default window frame used when one is not explicitly defined
    pub fn default() -> Self {
        WindowFrame {
            units: WindowFrameUnits::Range,
            start_bound: WindowFrameBound::UnboundedPreceding,
            end_bound: WindowFrameBound::CurrentRow,
        }
    }

    fn includes_current_row(&self) -> bool {
        use WindowFrameBound::*;
        match self.start_bound {
            UnboundedPreceding => match self.end_bound {
                UnboundedPreceding => false,
                OffsetPreceding(0) => true,
                OffsetPreceding(_) => false,
                CurrentRow => true,
                OffsetFollowing(_) => true,
                UnboundedFollowing => true,
            },
            OffsetPreceding(0) => match self.end_bound {
                UnboundedPreceding => unreachable!(),
                OffsetPreceding(0) => true,
                // Any nonzero offsets here will create an empty window
                OffsetPreceding(_) => false,
                CurrentRow => true,
                OffsetFollowing(_) => true,
                UnboundedFollowing => true,
            },
            OffsetPreceding(_) => match self.end_bound {
                UnboundedPreceding => unreachable!(),
                // Window ends at the current row
                OffsetPreceding(0) => true,
                OffsetPreceding(_) => false,
                CurrentRow => true,
                OffsetFollowing(_) => true,
                UnboundedFollowing => true,
            },
            CurrentRow => true,
            OffsetFollowing(0) => match self.end_bound {
                UnboundedPreceding => unreachable!(),
                OffsetPreceding(_) => unreachable!(),
                CurrentRow => unreachable!(),
                OffsetFollowing(_) => true,
                UnboundedFollowing => true,
            },
            OffsetFollowing(_) => match self.end_bound {
                UnboundedPreceding => unreachable!(),
                OffsetPreceding(_) => unreachable!(),
                CurrentRow => unreachable!(),
                OffsetFollowing(_) => false,
                UnboundedFollowing => false,
            },
            UnboundedFollowing => false,
        }
    }
}

impl RustType<ProtoWindowFrame> for WindowFrame {
    fn into_proto(&self) -> ProtoWindowFrame {
        ProtoWindowFrame {
            units: Some(self.units.into_proto()),
            start_bound: Some(self.start_bound.into_proto()),
            end_bound: Some(self.end_bound.into_proto()),
        }
    }

    fn from_proto(proto: ProtoWindowFrame) -> Result<Self, TryFromProtoError> {
        Ok(WindowFrame {
            units: proto.units.into_rust_if_some("ProtoWindowFrame::units")?,
            start_bound: proto
                .start_bound
                .into_rust_if_some("ProtoWindowFrame::start_bound")?,
            end_bound: proto
                .end_bound
                .into_rust_if_some("ProtoWindowFrame::end_bound")?,
        })
    }
}

/// Describe how frame bounds are interpreted
#[derive(
    Arbitrary, Debug, Clone, Eq, PartialEq, Ord, PartialOrd, Serialize, Deserialize, Hash, MzReflect,
)]
pub enum WindowFrameUnits {
    /// Each row is treated as the unit of work for bounds
    Rows,
    /// Each peer group is treated as the unit of work for bounds,
    /// and offset-based bounds use the value of the ORDER BY expression
    Range,
    /// Each peer group is treated as the unit of work for bounds.
    /// Groups is currently not supported, and it is rejected during planning.
    Groups,
}

impl Display for WindowFrameUnits {
    fn fmt(&self, f: &mut Formatter<'_>) -> fmt::Result {
        match self {
            WindowFrameUnits::Rows => write!(f, "rows"),
            WindowFrameUnits::Range => write!(f, "range"),
            WindowFrameUnits::Groups => write!(f, "groups"),
        }
    }
}

impl RustType<proto_window_frame::ProtoWindowFrameUnits> for WindowFrameUnits {
    fn into_proto(&self) -> proto_window_frame::ProtoWindowFrameUnits {
        use proto_window_frame::proto_window_frame_units::Kind::*;
        proto_window_frame::ProtoWindowFrameUnits {
            kind: Some(match self {
                WindowFrameUnits::Rows => Rows(()),
                WindowFrameUnits::Range => Range(()),
                WindowFrameUnits::Groups => Groups(()),
            }),
        }
    }

    fn from_proto(
        proto: proto_window_frame::ProtoWindowFrameUnits,
    ) -> Result<Self, TryFromProtoError> {
        use proto_window_frame::proto_window_frame_units::Kind::*;
        Ok(match proto.kind {
            Some(Rows(())) => WindowFrameUnits::Rows,
            Some(Range(())) => WindowFrameUnits::Range,
            Some(Groups(())) => WindowFrameUnits::Groups,
            None => {
                return Err(TryFromProtoError::missing_field(
                    "ProtoWindowFrameUnits::kind",
                ));
            }
        })
    }
}

/// Specifies [WindowFrame]'s `start_bound` and `end_bound`
///
/// The order between frame bounds is significant, as Postgres enforces
/// some restrictions there.
#[derive(
    Arbitrary, Debug, Clone, Serialize, Deserialize, PartialEq, Eq, Hash, MzReflect, PartialOrd, Ord,
)]
pub enum WindowFrameBound {
    /// `UNBOUNDED PRECEDING`
    UnboundedPreceding,
    /// `<N> PRECEDING`
    OffsetPreceding(u64),
    /// `CURRENT ROW`
    CurrentRow,
    /// `<N> FOLLOWING`
    OffsetFollowing(u64),
    /// `UNBOUNDED FOLLOWING`.
    UnboundedFollowing,
}

impl Display for WindowFrameBound {
    fn fmt(&self, f: &mut Formatter<'_>) -> fmt::Result {
        match self {
            WindowFrameBound::UnboundedPreceding => write!(f, "unbounded preceding"),
            WindowFrameBound::OffsetPreceding(offset) => write!(f, "{} preceding", offset),
            WindowFrameBound::CurrentRow => write!(f, "current row"),
            WindowFrameBound::OffsetFollowing(offset) => write!(f, "{} following", offset),
            WindowFrameBound::UnboundedFollowing => write!(f, "unbounded following"),
        }
    }
}

impl RustType<proto_window_frame::ProtoWindowFrameBound> for WindowFrameBound {
    fn into_proto(&self) -> proto_window_frame::ProtoWindowFrameBound {
        use proto_window_frame::proto_window_frame_bound::Kind::*;
        proto_window_frame::ProtoWindowFrameBound {
            kind: Some(match self {
                WindowFrameBound::UnboundedPreceding => UnboundedPreceding(()),
                WindowFrameBound::OffsetPreceding(offset) => OffsetPreceding(*offset),
                WindowFrameBound::CurrentRow => CurrentRow(()),
                WindowFrameBound::OffsetFollowing(offset) => OffsetFollowing(*offset),
                WindowFrameBound::UnboundedFollowing => UnboundedFollowing(()),
            }),
        }
    }

    fn from_proto(x: proto_window_frame::ProtoWindowFrameBound) -> Result<Self, TryFromProtoError> {
        use proto_window_frame::proto_window_frame_bound::Kind::*;
        Ok(match x.kind {
            Some(UnboundedPreceding(())) => WindowFrameBound::UnboundedPreceding,
            Some(OffsetPreceding(offset)) => WindowFrameBound::OffsetPreceding(offset),
            Some(CurrentRow(())) => WindowFrameBound::CurrentRow,
            Some(OffsetFollowing(offset)) => WindowFrameBound::OffsetFollowing(offset),
            Some(UnboundedFollowing(())) => WindowFrameBound::UnboundedFollowing,
            None => {
                return Err(TryFromProtoError::missing_field(
                    "ProtoWindowFrameBound::kind",
                ));
            }
        })
    }
}

/// Maximum iterations for a LetRec.
#[derive(
    Debug, Clone, Copy, PartialEq, Eq, PartialOrd, Ord, Hash, Serialize, Deserialize, Arbitrary,
)]
pub struct LetRecLimit {
    /// Maximum number of iterations to evaluate.
    pub max_iters: NonZeroU64,
    /// Whether to throw an error when reaching the above limit.
    /// If true, we simply use the current contents of each Id as the final result.
    pub return_at_limit: bool,
}

impl LetRecLimit {
    /// Compute the smallest limit from a Vec of `LetRecLimit`s.
    pub fn min_max_iter(limits: &Vec<Option<LetRecLimit>>) -> Option<u64> {
        limits
            .iter()
            .filter_map(|l| l.as_ref().map(|l| l.max_iters.get()))
            .min()
    }

    /// The default value of `LetRecLimit::return_at_limit` when using the RECURSION LIMIT option of
    /// WMR without ERROR AT or RETURN AT.
    pub const RETURN_AT_LIMIT_DEFAULT: bool = false;
}

impl Display for LetRecLimit {
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
        write!(f, "[recursion_limit={}", self.max_iters)?;
        if self.return_at_limit != LetRecLimit::RETURN_AT_LIMIT_DEFAULT {
            write!(f, ", return_at_limit")?;
        }
        write!(f, "]")
    }
}

/// For a global Get, this indicates whether we are going to read from Persist or from an index.
/// (See comment in MirRelationExpr::Get.)
#[derive(Clone, Debug, Eq, PartialEq, Ord, PartialOrd, Serialize, Deserialize, Hash, Arbitrary)]
pub enum AccessStrategy {
    /// It's either a local Get (a CTE), or unknown at the time.
    /// `prune_and_annotate_dataflow_index_imports` decides it for global Gets, and thus switches to
    /// one of the other variants.
    UnknownOrLocal,
    /// The Get will read from Persist.
    Persist,
    /// The Get will read from an index or indexes: (index id, how the index will be used).
    Index(Vec<(GlobalId, IndexUsageType)>),
    /// The Get will read a collection that is computed by the same dataflow, but in a different
    /// `BuildDesc` in `objects_to_build`.
    SameDataflow,
}

#[cfg(test)]
mod tests {
    use mz_ore::assert_ok;
    use mz_proto::protobuf_roundtrip;
    use mz_repr::explain::text::text_string_at;
    use proptest::prelude::*;

    use crate::explain::HumanizedExplain;

    use super::*;

    proptest! {
        #[mz_ore::test]
        fn column_order_protobuf_roundtrip(expect in any::<ColumnOrder>()) {
            let actual = protobuf_roundtrip::<_, ProtoColumnOrder>(&expect);
            assert_ok!(actual);
            assert_eq!(actual.unwrap(), expect);
        }
    }

    proptest! {
        #[mz_ore::test]
        #[cfg_attr(miri, ignore)] // error: unsupported operation: can't call foreign function `decContextDefault` on OS `linux`
        fn aggregate_expr_protobuf_roundtrip(expect in any::<AggregateExpr>()) {
            let actual = protobuf_roundtrip::<_, ProtoAggregateExpr>(&expect);
            assert_ok!(actual);
            assert_eq!(actual.unwrap(), expect);
        }
    }

    proptest! {
        #[mz_ore::test]
        fn window_frame_units_protobuf_roundtrip(expect in any::<WindowFrameUnits>()) {
            let actual = protobuf_roundtrip::<_, proto_window_frame::ProtoWindowFrameUnits>(&expect);
            assert_ok!(actual);
            assert_eq!(actual.unwrap(), expect);
        }
    }

    proptest! {
        #[mz_ore::test]
        fn window_frame_bound_protobuf_roundtrip(expect in any::<WindowFrameBound>()) {
            let actual = protobuf_roundtrip::<_, proto_window_frame::ProtoWindowFrameBound>(&expect);
            assert_ok!(actual);
            assert_eq!(actual.unwrap(), expect);
        }
    }

    proptest! {
        #[mz_ore::test]
        fn window_frame_protobuf_roundtrip(expect in any::<WindowFrame>()) {
            let actual = protobuf_roundtrip::<_, ProtoWindowFrame>(&expect);
            assert_ok!(actual);
            assert_eq!(actual.unwrap(), expect);
        }
    }

    #[mz_ore::test]
    fn test_row_set_finishing_as_text() {
        let finishing = RowSetFinishing {
            order_by: vec![ColumnOrder {
                column: 4,
                desc: true,
                nulls_last: true,
            }],
            limit: Some(NonNeg::try_from(7).unwrap()),
            offset: Default::default(),
            project: vec![1, 3, 4, 5],
        };

        let mode = HumanizedExplain::new(false);
        let expr = mode.expr(&finishing, None);

        let act = text_string_at(&expr, mz_ore::str::Indent::default);

        let exp = {
            use mz_ore::fmt::FormatBuffer;
            let mut s = String::new();
            write!(&mut s, "Finish");
            write!(&mut s, " order_by=[#4 desc nulls_last]");
            write!(&mut s, " limit=7");
            write!(&mut s, " output=[#1, #3..=#5]");
            writeln!(&mut s, "");
            s
        };

        assert_eq!(act, exp);
    }
}

/// An iterator over AST structures, which calls out nodes in difference.
///
/// The iterators visit two ASTs in tandem, continuing as long as the AST node data matches,
/// and yielding an output pair as soon as the AST nodes do not match. Their intent is to call
/// attention to the moments in the ASTs where they differ, and incidentally a stack-free way
/// to compare two ASTs.
mod structured_diff {

    use super::MirRelationExpr;

    ///  An iterator over structured differences between two `MirRelationExpr` instances.
    pub struct MreDiff<'a> {
        /// Pairs of expressions that must still be compared.
        todo: Vec<(&'a MirRelationExpr, &'a MirRelationExpr)>,
    }

    impl<'a> MreDiff<'a> {
        /// Create a new `MirRelationExpr` structured difference.
        pub fn new(expr1: &'a MirRelationExpr, expr2: &'a MirRelationExpr) -> Self {
            MreDiff {
                todo: vec![(expr1, expr2)],
            }
        }
    }

    impl<'a> Iterator for MreDiff<'a> {
        // Pairs of expressions that do not match.
        type Item = (&'a MirRelationExpr, &'a MirRelationExpr);

        fn next(&mut self) -> Option<Self::Item> {
            while let Some((expr1, expr2)) = self.todo.pop() {
                match (expr1, expr2) {
                    (
                        MirRelationExpr::Constant {
                            rows: rows1,
                            typ: typ1,
                        },
                        MirRelationExpr::Constant {
                            rows: rows2,
                            typ: typ2,
                        },
                    ) => {
                        if rows1 != rows2 || typ1 != typ2 {
                            return Some((expr1, expr2));
                        }
                    }
                    (
                        MirRelationExpr::Get {
                            id: id1,
                            typ: typ1,
                            access_strategy: as1,
                        },
                        MirRelationExpr::Get {
                            id: id2,
                            typ: typ2,
                            access_strategy: as2,
                        },
                    ) => {
                        if id1 != id2 || typ1 != typ2 || as1 != as2 {
                            return Some((expr1, expr2));
                        }
                    }
                    (
                        MirRelationExpr::Let {
                            id: id1,
                            body: body1,
                            value: value1,
                        },
                        MirRelationExpr::Let {
                            id: id2,
                            body: body2,
                            value: value2,
                        },
                    ) => {
                        if id1 != id2 {
                            return Some((expr1, expr2));
                        } else {
                            self.todo.push((body1, body2));
                            self.todo.push((value1, value2));
                        }
                    }
                    (
                        MirRelationExpr::LetRec {
                            ids: ids1,
                            body: body1,
                            values: values1,
                            limits: limits1,
                        },
                        MirRelationExpr::LetRec {
                            ids: ids2,
                            body: body2,
                            values: values2,
                            limits: limits2,
                        },
                    ) => {
                        if ids1 != ids2 || values1.len() != values2.len() || limits1 != limits2 {
                            return Some((expr1, expr2));
                        } else {
                            self.todo.push((body1, body2));
                            self.todo.extend(values1.iter().zip(values2.iter()));
                        }
                    }
                    (
                        MirRelationExpr::Project {
                            outputs: outputs1,
                            input: input1,
                        },
                        MirRelationExpr::Project {
                            outputs: outputs2,
                            input: input2,
                        },
                    ) => {
                        if outputs1 != outputs2 {
                            return Some((expr1, expr2));
                        } else {
                            self.todo.push((input1, input2));
                        }
                    }
                    (
                        MirRelationExpr::Map {
                            scalars: scalars1,
                            input: input1,
                        },
                        MirRelationExpr::Map {
                            scalars: scalars2,
                            input: input2,
                        },
                    ) => {
                        if scalars1 != scalars2 {
                            return Some((expr1, expr2));
                        } else {
                            self.todo.push((input1, input2));
                        }
                    }
                    (
                        MirRelationExpr::Filter {
                            predicates: predicates1,
                            input: input1,
                        },
                        MirRelationExpr::Filter {
                            predicates: predicates2,
                            input: input2,
                        },
                    ) => {
                        if predicates1 != predicates2 {
                            return Some((expr1, expr2));
                        } else {
                            self.todo.push((input1, input2));
                        }
                    }
                    (
                        MirRelationExpr::FlatMap {
                            input: input1,
                            func: func1,
                            exprs: exprs1,
                        },
                        MirRelationExpr::FlatMap {
                            input: input2,
                            func: func2,
                            exprs: exprs2,
                        },
                    ) => {
                        if func1 != func2 || exprs1 != exprs2 {
                            return Some((expr1, expr2));
                        } else {
                            self.todo.push((input1, input2));
                        }
                    }
                    (
                        MirRelationExpr::Join {
                            inputs: inputs1,
                            equivalences: eq1,
                            implementation: impl1,
                        },
                        MirRelationExpr::Join {
                            inputs: inputs2,
                            equivalences: eq2,
                            implementation: impl2,
                        },
                    ) => {
                        if inputs1.len() != inputs2.len() || eq1 != eq2 || impl1 != impl2 {
                            return Some((expr1, expr2));
                        } else {
                            self.todo.extend(inputs1.iter().zip(inputs2.iter()));
                        }
                    }
                    (
                        MirRelationExpr::Reduce {
                            aggregates: aggregates1,
                            input: inputs1,
                            group_key: gk1,
                            monotonic: m1,
                            expected_group_size: egs1,
                        },
                        MirRelationExpr::Reduce {
                            aggregates: aggregates2,
                            input: inputs2,
                            group_key: gk2,
                            monotonic: m2,
                            expected_group_size: egs2,
                        },
                    ) => {
                        if aggregates1 != aggregates2 || gk1 != gk2 || m1 != m2 || egs1 != egs2 {
                            return Some((expr1, expr2));
                        } else {
                            self.todo.push((inputs1, inputs2));
                        }
                    }
                    (
                        MirRelationExpr::TopK {
                            group_key: gk1,
                            order_key: order1,
                            input: input1,
                            limit: l1,
                            offset: o1,
                            monotonic: m1,
                            expected_group_size: egs1,
                        },
                        MirRelationExpr::TopK {
                            group_key: gk2,
                            order_key: order2,
                            input: input2,
                            limit: l2,
                            offset: o2,
                            monotonic: m2,
                            expected_group_size: egs2,
                        },
                    ) => {
                        if order1 != order2
                            || gk1 != gk2
                            || l1 != l2
                            || o1 != o2
                            || m1 != m2
                            || egs1 != egs2
                        {
                            return Some((expr1, expr2));
                        } else {
                            self.todo.push((input1, input2));
                        }
                    }
                    (
                        MirRelationExpr::Negate { input: input1 },
                        MirRelationExpr::Negate { input: input2 },
                    ) => {
                        self.todo.push((input1, input2));
                    }
                    (
                        MirRelationExpr::Threshold { input: input1 },
                        MirRelationExpr::Threshold { input: input2 },
                    ) => {
                        self.todo.push((input1, input2));
                    }
                    (
                        MirRelationExpr::Union {
                            base: base1,
                            inputs: inputs1,
                        },
                        MirRelationExpr::Union {
                            base: base2,
                            inputs: inputs2,
                        },
                    ) => {
                        if inputs1.len() != inputs2.len() {
                            return Some((expr1, expr2));
                        } else {
                            self.todo.push((base1, base2));
                            self.todo.extend(inputs1.iter().zip(inputs2.iter()));
                        }
                    }
                    (
                        MirRelationExpr::ArrangeBy {
                            keys: keys1,
                            input: input1,
                        },
                        MirRelationExpr::ArrangeBy {
                            keys: keys2,
                            input: input2,
                        },
                    ) => {
                        if keys1 != keys2 {
                            return Some((expr1, expr2));
                        } else {
                            self.todo.push((input1, input2));
                        }
                    }
                    _ => {
                        return Some((expr1, expr2));
                    }
                }
            }
            None
        }
    }
}