flux_infer/

fixpoint_encoding.rs

//! Encoding of the refinement tree into a fixpoint constraint.

use std::{
    collections::{HashMap, HashSet},
    hash::Hash,
    iter,
    ops::Range,
};

use fixpoint::AdtId;
use flux_common::{
    bug,
    cache::QueryCache,
    dbg,
    index::{IndexGen, IndexVec},
    span_bug, tracked_span_bug,
};
use flux_config::{self as config};
use flux_errors::Errors;
use flux_macros::DebugAsJson;
use flux_middle::{
    FixpointQueryKind,
    def_id::{FluxDefId, MaybeExternId},
    def_id_to_string,
    global_env::GlobalEnv,
    metrics::{self, Metric, TimingKind},
    pretty::{NestedString, PrettyCx, PrettyNested},
    queries::QueryResult,
    query_bug,
    rty::{
        self, ESpan, EarlyReftParam, GenericArgsExt, InternalFuncKind, Lambda, List,
        NameProvenance, PrettyMap, PrettyVar, SpecFuncKind, VariantIdx, fold::TypeFoldable as _,
    },
};
use itertools::Itertools;
use liquid_fixpoint::{
    FixpointResult, FixpointStatus, KVarBind, SmtSolver,
    parser::{FromSexp, ParseError},
    sexp::Parser,
};
use rustc_data_structures::{
    fx::{FxIndexMap, FxIndexSet},
    unord::{UnordMap, UnordSet},
};
use rustc_hir::def_id::{DefId, LocalDefId};
use rustc_index::newtype_index;
use rustc_infer::infer::TyCtxtInferExt as _;
use rustc_middle::ty::TypingMode;
use rustc_span::{DUMMY_SP, Span, Symbol};
use rustc_type_ir::{BoundVar, DebruijnIndex};
use serde::{Deserialize, Deserializer, Serialize};

use crate::{
    fixpoint_encoding::fixpoint::FixpointTypes, fixpoint_qualifiers::FIXPOINT_QUALIFIERS,
    lean_encoding::LeanEncoder, projections::structurally_normalize_expr,
};

pub mod decoding;

pub mod fixpoint {
    use std::fmt;

    use flux_middle::{def_id::FluxDefId, rty::EarlyReftParam};
    use liquid_fixpoint::{FixpointFmt, Identifier};
    use rustc_abi::VariantIdx;
    use rustc_index::newtype_index;
    use rustc_middle::ty::ParamConst;
    use rustc_span::Symbol;

    newtype_index! {
        #[orderable]
        pub struct KVid {}
    }

    impl Identifier for KVid {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            write!(f, "k{}", self.as_u32())
        }
    }

    newtype_index! {
        pub struct LocalVar {}
    }

    newtype_index! {
        pub struct GlobalVar {}
    }

    newtype_index! {
        /// Unique id assigned to each [`rty::AdtSortDef`] that needs to be encoded
        /// into fixpoint
        pub struct AdtId {}
    }

    #[derive(Hash, Copy, Clone, Debug, PartialEq, Eq)]
    pub enum Var {
        Underscore,
        Global(GlobalVar, Option<FluxDefId>),
        Local(LocalVar),
        DataCtor(AdtId, VariantIdx),
        TupleCtor { arity: usize },
        TupleProj { arity: usize, field: u32 },
        DataProj { adt_id: AdtId, /* variant_idx: VariantIdx, */ field: u32 },
        UIFRel(BinRel),
        Param(EarlyReftParam),
        ConstGeneric(ParamConst),
    }

    impl From<GlobalVar> for Var {
        fn from(v: GlobalVar) -> Self {
            Self::Global(v, None)
        }
    }

    impl From<LocalVar> for Var {
        fn from(v: LocalVar) -> Self {
            Self::Local(v)
        }
    }

    impl Identifier for Var {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            match self {
                Var::Global(v, None) => write!(f, "c{}", v.as_u32()),
                Var::Global(v, Some(did)) => write!(f, "f${}${}", did.name(), v.as_u32()),
                Var::Local(v) => write!(f, "a{}", v.as_u32()),
                Var::DataCtor(adt_id, variant_idx) => {
                    write!(f, "mkadt{}${}", adt_id.as_u32(), variant_idx.as_u32())
                }
                Var::TupleCtor { arity } => write!(f, "mktuple{arity}"),
                Var::TupleProj { arity, field } => write!(f, "tuple{arity}${field}"),
                Var::DataProj { adt_id, field } => write!(f, "fld{}${field}", adt_id.as_u32()),
                Var::UIFRel(BinRel::Gt) => write!(f, "gt"),
                Var::UIFRel(BinRel::Ge) => write!(f, "ge"),
                Var::UIFRel(BinRel::Lt) => write!(f, "lt"),
                Var::UIFRel(BinRel::Le) => write!(f, "le"),
                // these are actually not necessary because equality is interpreted for all sorts
                Var::UIFRel(BinRel::Eq) => write!(f, "eq"),
                Var::UIFRel(BinRel::Ne) => write!(f, "ne"),
                Var::Underscore => write!(f, "_$"), // To avoid clashing with `_` used for `app (_ bv_op n)` for parametric SMT ops
                Var::ConstGeneric(param) => {
                    write!(f, "constgen${}${}", param.name, param.index)
                }
                Var::Param(param) => {
                    write!(f, "reftgen${}${}", param.name, param.index)
                }
            }
        }
    }

    #[derive(Clone, Hash, Debug, PartialEq, Eq)]
    pub enum DataSort {
        Tuple(usize),
        Adt(AdtId),
        User(FluxDefId),
    }

    impl Identifier for DataSort {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            match self {
                DataSort::Tuple(arity) => {
                    write!(f, "Tuple{arity}")
                }
                DataSort::Adt(adt_id) => {
                    write!(f, "Adt{}", adt_id.as_u32())
                }
                // There's no way to declare opaque sorts in the fixpoint horn syntax so we encode user
                // declared opaque sorts as integers. Well-formedness should ensure values of these
                // sorts are used "opaquely", i.e., the only values of these sorts are variables.
                DataSort::User(..) => {
                    write!(f, "int")
                }
            }
        }
    }

    #[derive(Hash, Clone, Debug)]
    pub struct SymStr(pub Symbol);

    #[cfg(feature = "rust-fixpoint")]
    impl FixpointFmt for SymStr {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            write!(f, "{}", self.0)
        }
    }

    #[cfg(not(feature = "rust-fixpoint"))]
    impl FixpointFmt for SymStr {
        fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
            write!(f, "\"{}\"", self.0)
        }
    }

    liquid_fixpoint::declare_types! {
        type Sort = DataSort;
        type KVar = KVid;
        type Var = Var;
        type String = SymStr;
        type Tag = super::TagIdx;
    }
    pub use fixpoint_generated::*;
}

/// A type to represent Solutions for KVars
pub type Solution = FxIndexMap<rty::KVid, rty::Binder<rty::Expr>>;
pub type FixpointSolution = (Vec<(fixpoint::Var, fixpoint::Sort)>, fixpoint::Expr);
pub type ClosedSolution = (Vec<(fixpoint::Var, fixpoint::Sort)>, FixpointSolution);

/// A very explicit representation of [`Solution`] for debugging/tracing/serialization ONLY.
#[derive(Serialize, DebugAsJson)]
pub struct SolutionTrace(Vec<KvarSolutionTrace>);

#[derive(Serialize, DebugAsJson)]
pub struct KvarSolutionTrace {
    pub name: String,
    pub args: Vec<String>,
    pub body: NestedString,
}

impl KvarSolutionTrace {
    pub fn new(cx: &PrettyCx, kvid: rty::KVid, bind_expr: &rty::Binder<rty::Expr>) -> Self {
        let mut args = Vec::new();
        let body = cx
            .nested_with_bound_vars("", bind_expr.vars(), Some("".to_string()), |prefix| {
                for arg in prefix.split(',').map(|s| s.trim().to_string()) {
                    args.push(arg);
                }
                bind_expr.skip_binder_ref().fmt_nested(cx)
            })
            .unwrap();

        KvarSolutionTrace { name: format!("{kvid:?}"), args, body }
    }
}

impl SolutionTrace {
    pub fn new<T>(genv: GlobalEnv, ans: &Answer<T>) -> Self {
        let cx = &PrettyCx::default(genv);
        let res = ans
            .solutions()
            .map(|(kvid, bind_expr)| KvarSolutionTrace::new(cx, *kvid, bind_expr))
            .collect();
        SolutionTrace(res)
    }
}

pub struct ParsedResult {
    pub status: FixpointStatus<TagIdx>,
    pub solution: FxIndexMap<fixpoint::KVid, FixpointSolution>,
    pub non_cut_solution: FxIndexMap<fixpoint::KVid, FixpointSolution>,
}

#[derive(Debug, Clone, Default)]
pub struct Answer<Tag> {
    pub errors: Vec<Tag>,
    pub cut_solution: Solution,
    pub non_cut_solution: Solution,
}

impl<Tag> Answer<Tag> {
    pub fn trivial() -> Self {
        Self {
            errors: Vec::new(),
            cut_solution: FxIndexMap::default(),
            non_cut_solution: FxIndexMap::default(),
        }
    }

    pub fn solutions(&self) -> impl Iterator<Item = (&rty::KVid, &rty::Binder<rty::Expr>)> {
        self.cut_solution.iter().chain(self.non_cut_solution.iter())
    }
}

newtype_index! {
    #[debug_format = "TagIdx({})"]
    pub struct TagIdx {}
}

impl Serialize for TagIdx {
    fn serialize<S: serde::Serializer>(&self, serializer: S) -> Result<S::Ok, S::Error> {
        self.as_u32().serialize(serializer)
    }
}

impl<'de> Deserialize<'de> for TagIdx {
    fn deserialize<D: Deserializer<'de>>(deserializer: D) -> Result<Self, D::Error> {
        let idx = usize::deserialize(deserializer)?;
        Ok(TagIdx::from_u32(idx as u32))
    }
}

/// Keep track of all the data sorts that we need to define in fixpoint to encode the constraint.
#[derive(Default)]
pub struct SortEncodingCtxt {
    /// Set of all the tuple arities that need to be defined
    tuples: UnordSet<usize>,
    /// Set of all the [`AdtDefSortDef`](rty::AdtSortDef) that need to be declared as
    /// Fixpoint data-decls
    adt_sorts: FxIndexSet<DefId>,
    /// Set of all opaque types that need to be defined
    user_sorts: FxIndexSet<FluxDefId>,
}

impl SortEncodingCtxt {
    pub fn sort_to_fixpoint(&mut self, sort: &rty::Sort) -> fixpoint::Sort {
        match sort {
            rty::Sort::Int => fixpoint::Sort::Int,
            rty::Sort::Real => fixpoint::Sort::Real,
            rty::Sort::Bool => fixpoint::Sort::Bool,
            rty::Sort::Str => fixpoint::Sort::Str,
            rty::Sort::Char => fixpoint::Sort::Int,
            rty::Sort::BitVec(size) => fixpoint::Sort::BitVec(Box::new(bv_size_to_fixpoint(*size))),
            rty::Sort::App(rty::SortCtor::User(def_id), args) => {
                self.declare_opaque_sort(*def_id);
                let args = args.iter().map(|s| self.sort_to_fixpoint(s)).collect_vec();
                fixpoint::Sort::App(
                    fixpoint::SortCtor::Data(fixpoint::DataSort::User(*def_id)),
                    args,
                )
            }
            // There's no way to declare opaque sorts in the fixpoint horn syntax so we encode type
            // parameter sorts and (unormalizable) type alias sorts as integers. Well-formedness
            // should ensure values of these sorts are used "opaquely", i.e., the only values of
            // these sorts are variables.
            rty::Sort::Param(_)
            | rty::Sort::Alias(rty::AliasKind::Opaque | rty::AliasKind::Projection, ..) => {
                fixpoint::Sort::Int
            }
            rty::Sort::App(rty::SortCtor::Set, args) => {
                let args = args.iter().map(|s| self.sort_to_fixpoint(s)).collect_vec();
                fixpoint::Sort::App(fixpoint::SortCtor::Set, args)
            }
            rty::Sort::App(rty::SortCtor::Map, args) => {
                let args = args.iter().map(|s| self.sort_to_fixpoint(s)).collect_vec();
                fixpoint::Sort::App(fixpoint::SortCtor::Map, args)
            }
            rty::Sort::App(rty::SortCtor::Adt(sort_def), args) => {
                if let Some(variant) = sort_def.opt_struct_variant() {
                    let sorts = variant.field_sorts(args);
                    // do not generate 1-tuples
                    if let [sort] = &sorts[..] {
                        self.sort_to_fixpoint(sort)
                    } else {
                        let adt_id = self.declare_adt(sort_def.did());
                        let ctor = fixpoint::SortCtor::Data(fixpoint::DataSort::Adt(adt_id));
                        let args = args.iter().map(|s| self.sort_to_fixpoint(s)).collect_vec();
                        fixpoint::Sort::App(ctor, args)
                    }
                } else {
                    debug_assert!(args.is_empty());
                    let adt_id = self.declare_adt(sort_def.did());
                    fixpoint::Sort::App(
                        fixpoint::SortCtor::Data(fixpoint::DataSort::Adt(adt_id)),
                        vec![],
                    )
                }
            }
            rty::Sort::Tuple(sorts) => {
                // do not generate 1-tuples
                if let [sort] = &sorts[..] {
                    self.sort_to_fixpoint(sort)
                } else {
                    self.declare_tuple(sorts.len());
                    let ctor = fixpoint::SortCtor::Data(fixpoint::DataSort::Tuple(sorts.len()));
                    let args = sorts.iter().map(|s| self.sort_to_fixpoint(s)).collect();
                    fixpoint::Sort::App(ctor, args)
                }
            }
            rty::Sort::Func(sort) => self.func_sort_to_fixpoint(sort).into_sort(),
            rty::Sort::Var(k) => fixpoint::Sort::Var(k.index()),
            rty::Sort::Err
            | rty::Sort::Infer(_)
            | rty::Sort::Loc
            | rty::Sort::Alias(rty::AliasKind::Free, _) => {
                tracked_span_bug!("unexpected sort `{sort:?}`")
            }
        }
    }

    fn func_sort_to_fixpoint(&mut self, fsort: &rty::PolyFuncSort) -> fixpoint::FunSort {
        let params = fsort.params().len();
        let fsort = fsort.skip_binders();
        let output = self.sort_to_fixpoint(fsort.output());
        fixpoint::FunSort {
            params,
            inputs: fsort
                .inputs()
                .iter()
                .map(|s| self.sort_to_fixpoint(s))
                .collect(),
            output,
        }
    }

    fn declare_tuple(&mut self, arity: usize) {
        self.tuples.insert(arity);
    }

    pub fn declare_opaque_sort(&mut self, def_id: FluxDefId) {
        self.user_sorts.insert(def_id);
    }

    pub fn declare_adt(&mut self, did: DefId) -> AdtId {
        if let Some(idx) = self.adt_sorts.get_index_of(&did) {
            AdtId::from_usize(idx)
        } else {
            let adt_id = AdtId::from_usize(self.adt_sorts.len());
            self.adt_sorts.insert(did);
            adt_id
        }
    }

    pub fn user_sorts_to_fixpoint(&self, genv: GlobalEnv) -> Vec<fixpoint::SortDecl> {
        self.user_sorts
            .iter()
            .map(|sort| {
                let param_count = genv.sort_decl_param_count(sort);
                fixpoint::SortDecl { name: fixpoint::DataSort::User(*sort), vars: param_count }
            })
            .collect()
    }

    fn append_adt_decls(
        &mut self,
        genv: GlobalEnv,
        decls: &mut Vec<fixpoint::DataDecl>,
    ) -> QueryResult {
        // We iterate until we have processed all adt sorts because processing one adt sort may
        // discover new adt sorts to process (e.g., if an adt field has an adt sort).
        let mut idx = 0;
        while let Some(adt_def_id) = self.adt_sorts.get_index(idx) {
            let adt_id = AdtId::from_usize(idx);
            let adt_sort_def = genv.adt_sort_def_of(adt_def_id)?;
            decls.push(fixpoint::DataDecl {
                name: fixpoint::DataSort::Adt(adt_id),
                vars: adt_sort_def.param_count(),
                ctors: adt_sort_def
                    .variants()
                    .iter_enumerated()
                    .map(|(idx, variant)| {
                        let name = fixpoint::Var::DataCtor(adt_id, idx);
                        let fields = variant
                            .field_sorts_instantiate_identity()
                            .iter()
                            .enumerate()
                            .map(|(i, sort)| {
                                fixpoint::DataField {
                                    name: fixpoint::Var::DataProj { adt_id, field: i as u32 },
                                    sort: self.sort_to_fixpoint(sort),
                                }
                            })
                            .collect_vec();
                        fixpoint::DataCtor { name, fields }
                    })
                    .collect(),
            });
            idx += 1;
        }
        Ok(())
    }

    fn append_tuple_decls(tuples: &UnordSet<usize>, decls: &mut Vec<fixpoint::DataDecl>) {
        decls.extend(
            tuples
                .items()
                .into_sorted_stable_ord()
                .into_iter()
                .map(|arity| {
                    fixpoint::DataDecl {
                        name: fixpoint::DataSort::Tuple(*arity),
                        vars: *arity,
                        ctors: vec![fixpoint::DataCtor {
                            name: fixpoint::Var::TupleCtor { arity: *arity },
                            fields: (0..(*arity as u32))
                                .map(|field| {
                                    fixpoint::DataField {
                                        name: fixpoint::Var::TupleProj { arity: *arity, field },
                                        sort: fixpoint::Sort::Var(field as usize),
                                    }
                                })
                                .collect(),
                        }],
                    }
                }),
        );
    }

    pub fn encode_data_decls(&mut self, genv: GlobalEnv) -> QueryResult<Vec<fixpoint::DataDecl>> {
        let mut decls = vec![];
        self.append_adt_decls(genv, &mut decls)?;
        Self::append_tuple_decls(&self.tuples, &mut decls);
        Ok(decls)
    }
}

fn bv_size_to_fixpoint(size: rty::BvSize) -> fixpoint::Sort {
    match size {
        rty::BvSize::Fixed(size) => fixpoint::Sort::BvSize(size),
        rty::BvSize::Param(_var) => {
            // I think we could encode the size as a sort variable, but this would require some care
            // because smtlib doesn't really support parametric sizes. Fixpoint is probably already
            // too liberal about this and it'd be easy to make it crash.
            // fixpoint::Sort::Var(var.index)
            bug!("unexpected parametric bit-vector size")
        }
        rty::BvSize::Infer(_) => bug!("unexpected infer variable for bit-vector size"),
    }
}

pub type FunDeclMap = FxIndexMap<FluxDefId, fixpoint::Var>;
type ConstMap<'tcx> = FxIndexMap<ConstKey<'tcx>, fixpoint::ConstDecl>;

#[derive(Eq, Hash, PartialEq, Clone)]
enum ConstKey<'tcx> {
    RustConst(DefId),
    Alias(FluxDefId, rustc_middle::ty::GenericArgsRef<'tcx>),
    Lambda(Lambda),
    PrimOp(rty::BinOp),
    Cast(rty::Sort, rty::Sort),
}

#[derive(Clone)]
pub enum Backend {
    Fixpoint,
    Lean,
}

pub struct FixpointCtxt<'genv, 'tcx, T: Eq + Hash> {
    comments: Vec<String>,
    genv: GlobalEnv<'genv, 'tcx>,
    kvars: KVarGen,
    scx: SortEncodingCtxt,
    kcx: KVarEncodingCtxt,
    ecx: ExprEncodingCtxt<'genv, 'tcx>,
    tags: IndexVec<TagIdx, T>,
    tags_inv: UnordMap<T, TagIdx>,
}

pub type FixQueryCache = QueryCache<FixpointResult<TagIdx>>;

impl<'genv, 'tcx, Tag> FixpointCtxt<'genv, 'tcx, Tag>
where
    Tag: std::hash::Hash + Eq + Copy,
{
    pub fn new(
        genv: GlobalEnv<'genv, 'tcx>,
        def_id: MaybeExternId,
        kvars: KVarGen,
        backend: Backend,
    ) -> Self {
        Self {
            comments: vec![],
            kvars,
            scx: SortEncodingCtxt::default(),
            genv,
            ecx: ExprEncodingCtxt::new(genv, Some(def_id), backend),
            kcx: Default::default(),
            tags: IndexVec::new(),
            tags_inv: Default::default(),
        }
    }

    pub(crate) fn create_task(
        &mut self,
        def_id: MaybeExternId,
        constraint: fixpoint::Constraint,
        scrape_quals: bool,
        solver: SmtSolver,
    ) -> QueryResult<fixpoint::Task> {
        let kvars = self.kcx.encode_kvars(&self.kvars, &mut self.scx);
        let define_funs = self.ecx.define_funs(def_id, &mut self.scx)?;

        let qualifiers = self
            .ecx
            .qualifiers_for(def_id.local_id(), &mut self.scx)?
            .into_iter()
            .chain(FIXPOINT_QUALIFIERS.iter().cloned())
            .collect();

        // Assuming values should happen after all encoding is done so we are sure we've collected
        // all constants.
        let constraint = self.ecx.assume_const_values(constraint, &mut self.scx)?;

        let constants = self.ecx.const_env.const_map.values().cloned().collect_vec();

        // The rust fixpoint implementation does not yet support polymorphic functions.
        // For now we avoid including these by default so that cases where they are not needed can work.
        // Should be removed when support is added.
        #[cfg(not(feature = "rust-fixpoint"))]
        let constants = constants
            .into_iter()
            .chain(fixpoint::BinRel::INEQUALITIES.into_iter().map(|rel| {
                // ∀a. a -> a -> bool
                let sort = fixpoint::Sort::mk_func(
                    1,
                    [fixpoint::Sort::Var(0), fixpoint::Sort::Var(0)],
                    fixpoint::Sort::Bool,
                );
                fixpoint::ConstDecl { name: fixpoint::Var::UIFRel(rel), sort, comment: None }
            }))
            .collect();

        // We are done encoding expressions. Check if there are any errors.
        self.ecx.errors.to_result()?;

        let task = fixpoint::Task {
            comments: self.comments.clone(),
            constants,
            kvars,
            define_funs,
            constraint,
            qualifiers,
            scrape_quals,
            solver,
            data_decls: self.scx.encode_data_decls(self.genv)?,
        };

        if config::dump_constraint() {
            dbg::dump_item_info(self.genv.tcx(), def_id.resolved_id(), "smt2", &task).unwrap();
        }

        Ok(task)
    }

    pub(crate) fn run_task(
        &mut self,
        cache: &mut FixQueryCache,
        def_id: MaybeExternId,
        kind: FixpointQueryKind,
        task: &fixpoint::Task,
    ) -> QueryResult<ParsedResult> {
        let result = Self::run_task_with_cache(self.genv, task, def_id.resolved_id(), kind, cache);

        if config::dump_checker_trace_info()
            || self.genv.proven_externally(def_id.local_id()).is_some()
        {
            Ok(ParsedResult {
                status: result.status,
                solution: self.parse_kvar_solutions(&result.solution),
                non_cut_solution: self.parse_kvar_solutions(&result.non_cuts_solution),
            })
        } else {
            Ok(ParsedResult {
                status: result.status,
                solution: FxIndexMap::default(),
                non_cut_solution: FxIndexMap::default(),
            })
        }
    }

    pub(crate) fn result_to_answer(&mut self, result: ParsedResult) -> Answer<Tag> {
        let def_span = self.ecx.def_span();
        let errors = match result.status {
            FixpointStatus::Safe(_) => vec![],
            FixpointStatus::Unsafe(_, errors) => {
                metrics::incr_metric(Metric::CsError, errors.len() as u32);
                errors
                    .into_iter()
                    .map(|err| self.tags[err.tag])
                    .unique()
                    .collect_vec()
            }
            FixpointStatus::Crash(err) => span_bug!(def_span, "fixpoint crash: {err:?}"),
        };

        let cut_solution = result
            .solution
            .into_iter()
            .map(|(kvid, sol)| (kvid, self.fixpoint_to_solution(&sol)))
            .collect_vec();

        let non_cut_solution = result
            .non_cut_solution
            .into_iter()
            .map(|(kvid, sol)| (kvid, self.fixpoint_to_solution(&sol)))
            .collect_vec();

        Answer {
            errors,
            cut_solution: self.kcx.group_kvar_solution(cut_solution),
            non_cut_solution: self.kcx.group_kvar_solution(non_cut_solution),
        }
    }

    fn parse_kvar_solutions(
        &mut self,
        kvar_binds: &[KVarBind],
    ) -> FxIndexMap<fixpoint::KVid, FixpointSolution> {
        kvar_binds
            .iter()
            .map(|b| (parse_kvid(&b.kvar), self.parse_kvar_solution(&b.val)))
            .collect()
    }

    fn parse_kvar_solution(&mut self, expr: &str) -> FixpointSolution {
        // 1. convert str -> sexp
        let mut sexp_parser = Parser::new(expr);
        let sexp = match sexp_parser.parse() {
            Ok(sexp) => sexp,
            Err(err) => {
                tracked_span_bug!("cannot parse sexp: {expr:?}: {err:?}");
            }
        };
        let mut fun_decl_map = HashMap::new();
        for (def_id, var) in &self.ecx.const_env.fun_decl_map {
            let fixpoint::Var::Global(idx, _) = var else {
                bug!("non global var encountered for function")
            };
            fun_decl_map.insert(idx.index(), *def_id);
        }
        // 2. convert sexp -> (binds, Expr<fixpoint_encoding::Types>)
        let mut sexp_ctx =
            SexpParseCtxt::new(&mut self.ecx.local_var_env, &fun_decl_map).into_wrapper();
        sexp_ctx.parse_solution(&sexp).unwrap_or_else(|err| {
            tracked_span_bug!("failed to parse solution sexp {sexp:?}: {err:?}");
        })
    }

    pub fn generate_lean_files(
        self,
        def_id: MaybeExternId,
        task: fixpoint::Task,
        kvar_solutions: KVarSolutions,
    ) -> QueryResult {
        // FIXME(nilehmann) opaque sorts should be part of the task.
        let opaque_sorts = self.scx.user_sorts_to_fixpoint(self.genv);
        let sort_deps =
            SortDeps { opaque_sorts, data_decls: task.data_decls, adt_map: self.scx.adt_sorts };

        LeanEncoder::encode(
            self.genv,
            def_id,
            self.ecx.local_var_env.pretty_var_map,
            sort_deps,
            task.define_funs,
            task.kvars,
            task.constraint,
            kvar_solutions,
        )
        .map_err(|err| query_bug!("could not encode constraint: {err:?}"))
    }

    fn run_task_with_cache(
        genv: GlobalEnv,
        task: &fixpoint::Task,
        def_id: DefId,
        kind: FixpointQueryKind,
        cache: &mut FixQueryCache,
    ) -> FixpointResult<TagIdx> {
        let key = kind.task_key(genv.tcx(), def_id);

        let hash = task.hash_with_default();

        if config::is_cache_enabled()
            && let Some(result) = cache.lookup(&key, hash)
        {
            metrics::incr_metric_if(kind.is_body(), Metric::FnCached);
            return result.clone();
        }
        let result = metrics::time_it(TimingKind::FixpointQuery(def_id, kind), || {
            task.run()
                .unwrap_or_else(|err| tracked_span_bug!("failed to run fixpoint: {err}"))
        });

        if config::is_cache_enabled() {
            cache.insert(key, hash, result.clone());
        }

        result
    }

    fn tag_idx(&mut self, tag: Tag) -> TagIdx
    where
        Tag: std::fmt::Debug,
    {
        *self.tags_inv.entry(tag).or_insert_with(|| {
            let idx = self.tags.push(tag);
            self.comments.push(format!("Tag {idx}: {tag:?}"));
            idx
        })
    }

    pub(crate) fn with_early_param(&mut self, param: &EarlyReftParam) {
        self.ecx
            .local_var_env
            .pretty_var_map
            .set(PrettyVar::Param(*param), Some(param.name));
    }

    pub(crate) fn with_name_map<R>(
        &mut self,
        name: rty::Name,
        provenance: NameProvenance,
        f: impl FnOnce(&mut Self, fixpoint::LocalVar) -> R,
    ) -> R {
        let fresh = self.ecx.local_var_env.insert_fvar_map(name, provenance);
        let r = f(self, fresh);
        self.ecx.local_var_env.remove_fvar_map(name);
        r
    }

    pub(crate) fn sort_to_fixpoint(&mut self, sort: &rty::Sort) -> fixpoint::Sort {
        self.scx.sort_to_fixpoint(sort)
    }

    pub(crate) fn var_to_fixpoint(&self, var: &rty::Var) -> fixpoint::Var {
        self.ecx.var_to_fixpoint(var)
    }

    /// Encodes an expression in head position as a [`fixpoint::Constraint`] "peeling out"
    /// implications and foralls.
    ///
    /// [`fixpoint::Constraint`]: liquid_fixpoint::Constraint
    pub(crate) fn head_to_fixpoint(
        &mut self,
        expr: &rty::Expr,
        mk_tag: impl Fn(Option<ESpan>) -> Tag + Copy,
    ) -> QueryResult<fixpoint::Constraint>
    where
        Tag: std::fmt::Debug,
    {
        match expr.kind() {
            rty::ExprKind::BinaryOp(rty::BinOp::And, ..) => {
                // avoid creating nested conjunctions
                let cstrs = expr
                    .flatten_conjs()
                    .into_iter()
                    .map(|e| self.head_to_fixpoint(e, mk_tag))
                    .try_collect()?;
                Ok(fixpoint::Constraint::conj(cstrs))
            }
            rty::ExprKind::BinaryOp(rty::BinOp::Imp, e1, e2) => {
                let (bindings, assumption) = self.assumption_to_fixpoint(e1)?;
                let cstr = self.head_to_fixpoint(e2, mk_tag)?;
                Ok(fixpoint::Constraint::foralls(bindings, mk_implies(assumption, cstr)))
            }
            rty::ExprKind::KVar(kvar) => {
                let mut bindings = vec![];
                let pred = self.kvar_to_fixpoint(kvar, &mut bindings)?;
                Ok(fixpoint::Constraint::foralls(bindings, fixpoint::Constraint::Pred(pred, None)))
            }
            rty::ExprKind::ForAll(pred) => {
                self.ecx
                    .local_var_env
                    .push_layer_with_fresh_names(pred.vars().len());
                let cstr = self.head_to_fixpoint(pred.as_ref().skip_binder(), mk_tag)?;
                let vars = self.ecx.local_var_env.pop_layer();

                let bindings = iter::zip(vars, pred.vars())
                    .map(|(var, kind)| {
                        fixpoint::Bind {
                            name: var.into(),
                            sort: self.scx.sort_to_fixpoint(kind.expect_sort()),
                            pred: fixpoint::Pred::TRUE,
                        }
                    })
                    .collect_vec();

                Ok(fixpoint::Constraint::foralls(bindings, cstr))
            }
            _ => {
                let tag_idx = self.tag_idx(mk_tag(expr.span()));
                let pred = fixpoint::Pred::Expr(self.ecx.expr_to_fixpoint(expr, &mut self.scx)?);
                Ok(fixpoint::Constraint::Pred(pred, Some(tag_idx)))
            }
        }
    }

    /// Encodes an expression in assumptive position as a [`fixpoint::Pred`]. Returns the encoded
    /// predicate and a list of bindings produced by ANF-ing kvars.
    ///
    /// [`fixpoint::Pred`]: liquid_fixpoint::Pred
    pub(crate) fn assumption_to_fixpoint(
        &mut self,
        pred: &rty::Expr,
    ) -> QueryResult<(Vec<fixpoint::Bind>, fixpoint::Pred)> {
        let mut bindings = vec![];
        let mut preds = vec![];
        self.assumption_to_fixpoint_aux(pred, &mut bindings, &mut preds)?;
        Ok((bindings, fixpoint::Pred::and(preds)))
    }

    /// Auxiliary function to merge nested conjunctions in a single predicate
    fn assumption_to_fixpoint_aux(
        &mut self,
        expr: &rty::Expr,
        bindings: &mut Vec<fixpoint::Bind>,
        preds: &mut Vec<fixpoint::Pred>,
    ) -> QueryResult {
        match expr.kind() {
            rty::ExprKind::BinaryOp(rty::BinOp::And, e1, e2) => {
                self.assumption_to_fixpoint_aux(e1, bindings, preds)?;
                self.assumption_to_fixpoint_aux(e2, bindings, preds)?;
            }
            rty::ExprKind::KVar(kvar) => {
                preds.push(self.kvar_to_fixpoint(kvar, bindings)?);
            }
            rty::ExprKind::ForAll(_) => {
                // If a forall appears in assumptive position replace it with true. This is sound
                // because we are weakening the context, i.e., anything true without the assumption
                // should remain true after adding it. Note that this relies on the predicate
                // appearing negatively. This is guaranteed by the surface syntax because foralls
                // can only appear at the top-level in a requires clause.
                preds.push(fixpoint::Pred::TRUE);
            }
            _ => {
                preds.push(fixpoint::Pred::Expr(self.ecx.expr_to_fixpoint(expr, &mut self.scx)?));
            }
        }
        Ok(())
    }

    fn kvar_to_fixpoint(
        &mut self,
        kvar: &rty::KVar,
        bindings: &mut Vec<fixpoint::Bind>,
    ) -> QueryResult<fixpoint::Pred> {
        let decl = self.kvars.get(kvar.kvid);
        let kvids = self.kcx.declare(kvar.kvid, decl, &self.ecx.backend);

        let all_args = self.ecx.exprs_to_fixpoint(&kvar.args, &mut self.scx)?;

        // Fixpoint doesn't support kvars without arguments, which we do generate sometimes. To get
        // around it, we encode `$k()` as ($k 0), or more precisely `(forall ((x int) (= x 0)) ... ($k x)`
        // after ANF-ing.
        if all_args.is_empty() {
            let fresh = self.ecx.local_var_env.fresh_name();
            let var = fixpoint::Var::Local(fresh);
            bindings.push(fixpoint::Bind {
                name: fresh.into(),
                sort: fixpoint::Sort::Int,
                pred: fixpoint::Pred::Expr(fixpoint::Expr::eq(
                    fixpoint::Expr::Var(var),
                    fixpoint::Expr::int(0),
                )),
            });
            return Ok(fixpoint::Pred::KVar(kvids.start, vec![fixpoint::Expr::Var(var)]));
        }

        let kvars = kvids
            .enumerate()
            .map(|(i, kvid)| {
                let args = all_args[i..].to_vec();
                fixpoint::Pred::KVar(kvid, args)
            })
            .collect_vec();

        Ok(fixpoint::Pred::And(kvars))
    }
}

fn const_to_fixpoint(cst: rty::Constant) -> fixpoint::Expr {
    match cst {
        rty::Constant::Int(i) => {
            if i.is_negative() {
                fixpoint::Expr::Neg(Box::new(fixpoint::Constant::Numeral(i.abs()).into()))
            } else {
                fixpoint::Constant::Numeral(i.abs()).into()
            }
        }
        rty::Constant::Real(r) => fixpoint::Constant::Real(r.0).into(),
        rty::Constant::Bool(b) => fixpoint::Constant::Boolean(b).into(),
        rty::Constant::Char(c) => fixpoint::Constant::Numeral(u128::from(c)).into(),
        rty::Constant::Str(s) => fixpoint::Constant::String(fixpoint::SymStr(s)).into(),
        rty::Constant::BitVec(i, size) => fixpoint::Constant::BitVec(i, size).into(),
    }
}

/// During encoding into fixpoint we generate multiple fixpoint kvars per kvar in flux. A
/// [`KVarEncodingCtxt`] is used to keep track of the state needed for this.
///
/// See [`KVarEncoding`]
#[derive(Default)]
struct KVarEncodingCtxt {
    /// A map from a [`rty::KVid`] to the range of [`fixpoint::KVid`]s that will be used to
    /// encode it.
    ranges: FxIndexMap<rty::KVid, Range<fixpoint::KVid>>,
}

impl KVarEncodingCtxt {
    /// Declares that a kvar has to be encoded into fixpoint and assigns a range of
    /// [`fixpoint::KVid`]'s to it.
    fn declare(
        &mut self,
        kvid: rty::KVid,
        decl: &KVarDecl,
        backend: &Backend,
    ) -> Range<fixpoint::KVid> {
        // The start of the next range
        let start = self
            .ranges
            .last()
            .map_or(fixpoint::KVid::from_u32(0), |(_, r)| r.end);

        self.ranges
            .entry(kvid)
            .or_insert_with(|| {
                let single_encoding = matches!(decl.encoding, KVarEncoding::Single)
                    || matches!(backend, Backend::Lean);
                if single_encoding {
                    start..start + 1
                } else {
                    let n = usize::max(decl.self_args, 1);
                    start..start + n
                }
            })
            .clone()
    }

    fn encode_kvars(&self, kvars: &KVarGen, scx: &mut SortEncodingCtxt) -> Vec<fixpoint::KVarDecl> {
        self.ranges
            .iter()
            .flat_map(|(orig, range)| {
                let mut all_sorts = kvars
                    .get(*orig)
                    .sorts
                    .iter()
                    .map(|s| scx.sort_to_fixpoint(s))
                    .collect_vec();

                // See comment in `kvar_to_fixpoint`
                if all_sorts.is_empty() {
                    all_sorts = vec![fixpoint::Sort::Int];
                }

                range.clone().enumerate().map(move |(i, kvid)| {
                    let sorts = all_sorts[i..].to_vec();
                    fixpoint::KVarDecl::new(kvid, sorts, format!("orig: {:?}", orig))
                })
            })
            .collect()
    }

    /// For each [`rty::KVid`] `$k`, this function collects all predicates associated
    /// with the [`fixpoint::KVid`]s that encode `$k` and combines them into a single
    /// predicate by conjoining them.
    ///
    /// A group (i.e., a combined predicate) is included in the result only if *all*
    /// [`fixpoint::KVid`]s in the encoding range of `$k` are present in the input.
    fn group_kvar_solution(
        &self,
        mut items: Vec<(fixpoint::KVid, rty::Binder<rty::Expr>)>,
    ) -> FxIndexMap<rty::KVid, rty::Binder<rty::Expr>> {
        let mut map = FxIndexMap::default();

        items.sort_by_key(|(kvid, _)| *kvid);
        items.reverse();

        for (orig, range) in &self.ranges {
            let mut preds = vec![];
            while let Some((_, t)) = items.pop_if(|(k, _)| range.contains(k)) {
                preds.push(t);
            }
            // We only put it in the map if the entire range is present.
            if preds.len() == range.end.as_usize() - range.start.as_usize() {
                let vars = preds[0].vars().clone();
                let conj = rty::Expr::and_from_iter(
                    preds
                        .into_iter()
                        .enumerate()
                        .map(|(i, e)| e.skip_binder().shift_horizontally(i)),
                );
                map.insert(*orig, rty::Binder::bind_with_vars(conj, vars));
            }
        }
        map
    }
}

/// Environment used to map from [`rty::Var`] to a [`fixpoint::LocalVar`].
struct LocalVarEnv {
    local_var_gen: IndexGen<fixpoint::LocalVar>,
    fvars: UnordMap<rty::Name, fixpoint::LocalVar>,
    /// Layers of late bound variables
    layers: Vec<Vec<fixpoint::LocalVar>>,
    /// While it might seem like the signature should be
    /// [`UnordMap<fixpoint::LocalVar, rty::Var>`], we encode the arguments to
    /// kvars (which can be arbitrary expressions) as local variables; thus we
    /// need to keep the output as an [`rty::Expr`] to reflect this.
    reverse_map: UnordMap<fixpoint::LocalVar, rty::Expr>,
    pretty_var_map: PrettyMap<fixpoint::LocalVar>,
}

impl LocalVarEnv {
    fn new() -> Self {
        Self {
            local_var_gen: IndexGen::new(),
            fvars: Default::default(),
            layers: Vec::new(),
            reverse_map: Default::default(),
            pretty_var_map: PrettyMap::new(),
        }
    }

    // This doesn't require to be mutable because `IndexGen` uses atomics, but we make it mutable
    // to better declare the intent.
    fn fresh_name(&mut self) -> fixpoint::LocalVar {
        self.local_var_gen.fresh()
    }

    fn insert_fvar_map(
        &mut self,
        name: rty::Name,
        provenance: NameProvenance,
    ) -> fixpoint::LocalVar {
        let fresh = self.fresh_name();
        self.fvars.insert(name, fresh);
        self.reverse_map.insert(fresh, rty::Expr::fvar(name));
        self.pretty_var_map
            .set(PrettyVar::Local(fresh), provenance.opt_symbol());
        fresh
    }

    fn remove_fvar_map(&mut self, name: rty::Name) {
        self.fvars.remove(&name);
    }

    /// Push a layer of bound variables assigning a fresh [`fixpoint::LocalVar`] to each one
    fn push_layer_with_fresh_names(&mut self, count: usize) {
        let layer = (0..count).map(|_| self.fresh_name()).collect();
        self.layers.push(layer);
        // FIXME: (ck) what to put in reverse_map here?
    }

    fn push_layer(&mut self, layer: Vec<fixpoint::LocalVar>) {
        self.layers.push(layer);
    }

    fn pop_layer(&mut self) -> Vec<fixpoint::LocalVar> {
        self.layers.pop().unwrap()
    }

    fn get_fvar(&self, name: rty::Name) -> Option<fixpoint::LocalVar> {
        self.fvars.get(&name).copied()
    }

    fn get_late_bvar(&self, debruijn: DebruijnIndex, var: BoundVar) -> Option<fixpoint::LocalVar> {
        let depth = self.layers.len().checked_sub(debruijn.as_usize() + 1)?;
        self.layers[depth].get(var.as_usize()).copied()
    }
}

pub struct KVarGen {
    kvars: IndexVec<rty::KVid, KVarDecl>,
    /// If true, generate dummy [holes] instead of kvars. Used during shape mode to avoid generating
    /// unnecessary kvars.
    ///
    /// [holes]: rty::ExprKind::Hole
    dummy: bool,
}

impl KVarGen {
    pub(crate) fn new(dummy: bool) -> Self {
        Self { kvars: IndexVec::new(), dummy }
    }

    fn get(&self, kvid: rty::KVid) -> &KVarDecl {
        &self.kvars[kvid]
    }

    /// Generate a fresh [kvar] under several layers of [binders]. Each layer may contain any kind
    /// of bound variable, but variables that are not of kind [`BoundVariableKind::Refine`] will
    /// be filtered out.
    ///
    /// The variables bound in the last layer (last element of the `binders` slice) is expected to
    /// have only [`BoundVariableKind::Refine`] and all its elements are used as the [self arguments].
    /// The rest of the binders are appended to the `scope`.
    ///
    /// Note that the returned expression will have escaping variables and it is up to the caller to
    /// put it under an appropriate number of binders.
    ///
    /// Prefer using [`InferCtxt::fresh_kvar`] when possible.
    ///
    /// [binders]: rty::Binder
    /// [kvar]: rty::KVar
    /// [`InferCtxt::fresh_kvar`]: crate::infer::InferCtxt::fresh_kvar
    /// [self arguments]: rty::KVar::self_args
    /// [`BoundVariableKind::Refine`]: rty::BoundVariableKind::Refine
    pub fn fresh(
        &mut self,
        binders: &[rty::BoundVariableKinds],
        scope: impl IntoIterator<Item = (rty::Var, rty::Sort)>,
        encoding: KVarEncoding,
    ) -> rty::Expr {
        if self.dummy {
            return rty::Expr::hole(rty::HoleKind::Pred);
        }

        let args = itertools::chain(
            binders.iter().rev().enumerate().flat_map(|(level, vars)| {
                let debruijn = DebruijnIndex::from_usize(level);
                vars.iter()
                    .cloned()
                    .enumerate()
                    .flat_map(move |(idx, var)| {
                        if let rty::BoundVariableKind::Refine(sort, _, kind) = var {
                            let br = rty::BoundReft { var: BoundVar::from_usize(idx), kind };
                            Some((rty::Var::Bound(debruijn, br), sort))
                        } else {
                            None
                        }
                    })
            }),
            scope,
        );
        let [.., last] = binders else {
            return self.fresh_inner(0, [], encoding);
        };
        let num_self_args = last
            .iter()
            .filter(|var| matches!(var, rty::BoundVariableKind::Refine(..)))
            .count();
        self.fresh_inner(num_self_args, args, encoding)
    }

    fn fresh_inner<A>(&mut self, self_args: usize, args: A, encoding: KVarEncoding) -> rty::Expr
    where
        A: IntoIterator<Item = (rty::Var, rty::Sort)>,
    {
        // asset last one has things
        let mut sorts = vec![];
        let mut exprs = vec![];

        let mut flattened_self_args = 0;
        for (i, (var, sort)) in args.into_iter().enumerate() {
            let is_self_arg = i < self_args;
            let var = var.to_expr();
            sort.walk(|sort, proj| {
                if !matches!(sort, rty::Sort::Loc) {
                    flattened_self_args += is_self_arg as usize;
                    sorts.push(sort.clone());
                    exprs.push(rty::Expr::field_projs(&var, proj));
                }
            });
        }

        let kvid = self
            .kvars
            .push(KVarDecl { self_args: flattened_self_args, sorts, encoding });

        let kvar = rty::KVar::new(kvid, flattened_self_args, exprs);
        rty::Expr::kvar(kvar)
    }
}

#[derive(Clone)]
struct KVarDecl {
    self_args: usize,
    sorts: Vec<rty::Sort>,
    encoding: KVarEncoding,
}

/// How an [`rty::KVar`] is encoded in the fixpoint constraint
#[derive(Clone, Copy)]
pub enum KVarEncoding {
    /// Generate a single kvar appending the self arguments and the scope, i.e.,
    /// a kvar `$k(a0, ...)[b0, ...]` becomes `$k(a0, ..., b0, ...)` in the fixpoint constraint.
    Single,
    /// Generate a conjunction of kvars, one per argument in [`rty::KVar::args`].
    /// Concretely, a kvar `$k(a0, a1, ..., an)[b0, ...]` becomes
    /// `$k0(a0, a1, ..., an, b0, ...) ∧ $k1(a1, ..., an, b0, ...) ∧ ... ∧ $kn(an, b0, ...)`
    Conj,
}

impl std::fmt::Display for TagIdx {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        write!(f, "{}", self.as_u32())
    }
}

impl std::str::FromStr for TagIdx {
    type Err = std::num::ParseIntError;

    fn from_str(s: &str) -> Result<Self, Self::Err> {
        Ok(Self::from_u32(s.parse()?))
    }
}

#[derive(Default)]
struct ConstEnv<'tcx> {
    const_map: ConstMap<'tcx>,
    const_map_rev: HashMap<fixpoint::GlobalVar, ConstKey<'tcx>>,
    global_var_gen: IndexGen<fixpoint::GlobalVar>,
    fun_decl_map: FunDeclMap,
}

impl<'tcx> ConstEnv<'tcx> {
    fn get_or_insert(
        &mut self,
        key: ConstKey<'tcx>,
        // make_name: impl FnOnce() -> fixpoint::GlobalVar,
        make_const_decl: impl FnOnce(fixpoint::GlobalVar) -> fixpoint::ConstDecl,
    ) -> &fixpoint::ConstDecl {
        self.const_map.entry(key.clone()).or_insert_with(|| {
            let global_name = self.global_var_gen.fresh();
            self.const_map_rev.insert(global_name, key);
            make_const_decl(global_name)
        })
    }
}

pub struct ExprEncodingCtxt<'genv, 'tcx> {
    genv: GlobalEnv<'genv, 'tcx>,
    local_var_env: LocalVarEnv,
    const_env: ConstEnv<'tcx>,
    errors: Errors<'genv>,
    /// Id of the item being checked. This is a [`MaybeExternId`] because we may be encoding
    /// invariants for an extern spec on an enum.
    def_id: Option<MaybeExternId>,
    infcx: rustc_infer::infer::InferCtxt<'tcx>,
    backend: Backend,
}

pub struct KVarSolutions {
    pub cut_solutions: FxIndexMap<fixpoint::KVid, ClosedSolution>,
    pub non_cut_solutions: FxIndexMap<fixpoint::KVid, ClosedSolution>,
}

impl KVarSolutions {
    pub(crate) fn closed_solutions(
        variable_sorts: HashMap<fixpoint::Var, fixpoint::Sort>,
        cut_solutions: FxIndexMap<fixpoint::KVid, FixpointSolution>,
        non_cut_solutions: FxIndexMap<fixpoint::KVid, FixpointSolution>,
    ) -> Self {
        let closed_cut_solutions = cut_solutions
            .into_iter()
            .map(|(k, v)| (k, (vec![], v)))
            .collect();

        let mut closed_non_cut_solutions: FxIndexMap<fixpoint::KVid, ClosedSolution> =
            FxIndexMap::default();
        for (kvid, mut solution) in non_cut_solutions {
            let bound_vars: HashSet<&_> = solution.0.iter().map(|(var, _)| var).collect();
            let vars = solution.1.free_vars();
            let free_vars = vars.iter().filter(|var| {
                !bound_vars.contains(var)
                    && !matches!(
                        var,
                        fixpoint::Var::DataCtor(..)
                            | fixpoint::Var::Global(..)
                            | fixpoint::Var::DataProj { .. }
                    )
            });
            let free_var_subs = free_vars
                .map(|fvar| (*fvar, variable_sorts.get(fvar).unwrap().clone()))
                .collect();
            solution.1.var_sorts_to_int();
            closed_non_cut_solutions.insert(kvid, (free_var_subs, solution));
        }
        KVarSolutions {
            cut_solutions: closed_cut_solutions,
            non_cut_solutions: closed_non_cut_solutions,
        }
    }

    pub(crate) fn is_empty(&self) -> bool {
        self.cut_solutions.is_empty() && self.non_cut_solutions.is_empty()
    }
}

#[derive(Debug)]
pub struct SortDeps {
    pub opaque_sorts: Vec<fixpoint::SortDecl>,
    pub data_decls: Vec<fixpoint::DataDecl>,
    pub adt_map: FxIndexSet<DefId>,
}

impl<'genv, 'tcx> ExprEncodingCtxt<'genv, 'tcx> {
    pub fn new(
        genv: GlobalEnv<'genv, 'tcx>,
        def_id: Option<MaybeExternId>,
        backend: Backend,
    ) -> Self {
        Self {
            genv,
            local_var_env: LocalVarEnv::new(),
            const_env: Default::default(),
            errors: Errors::new(genv.sess()),
            def_id,
            infcx: genv
                .tcx()
                .infer_ctxt()
                .with_next_trait_solver(true)
                .build(TypingMode::non_body_analysis()),
            backend,
        }
    }

    fn def_span(&self) -> Span {
        self.def_id
            .map_or(DUMMY_SP, |def_id| self.genv.tcx().def_span(def_id))
    }

    fn var_to_fixpoint(&self, var: &rty::Var) -> fixpoint::Var {
        match var {
            rty::Var::Free(name) => {
                self.local_var_env
                    .get_fvar(*name)
                    .unwrap_or_else(|| {
                        span_bug!(self.def_span(), "no entry found for name: `{name:?}`")
                    })
                    .into()
            }
            rty::Var::Bound(debruijn, breft) => {
                self.local_var_env
                    .get_late_bvar(*debruijn, breft.var)
                    .unwrap_or_else(|| {
                        span_bug!(self.def_span(), "no entry found for late bound var: `{breft:?}`")
                    })
                    .into()
            }
            rty::Var::ConstGeneric(param) => fixpoint::Var::ConstGeneric(*param),
            rty::Var::EarlyParam(param) => fixpoint::Var::Param(*param),
            rty::Var::EVar(_) => {
                span_bug!(self.def_span(), "unexpected evar: `{var:?}`")
            }
        }
    }

    fn variant_to_fixpoint(
        &self,
        scx: &mut SortEncodingCtxt,
        enum_def_id: &DefId,
        idx: VariantIdx,
    ) -> fixpoint::Var {
        let adt_id = scx.declare_adt(*enum_def_id);
        fixpoint::Var::DataCtor(adt_id, idx)
    }

    fn struct_fields_to_fixpoint(
        &mut self,
        did: &DefId,
        flds: &[rty::Expr],
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<fixpoint::Expr> {
        // do not generate 1-tuples
        if let [fld] = flds {
            self.expr_to_fixpoint(fld, scx)
        } else {
            let adt_id = scx.declare_adt(*did);
            let ctor = fixpoint::Expr::Var(fixpoint::Var::DataCtor(adt_id, VariantIdx::ZERO));
            let args = flds
                .iter()
                .map(|fld| self.expr_to_fixpoint(fld, scx))
                .try_collect()?;
            Ok(fixpoint::Expr::App(Box::new(ctor), None, args, None))
        }
    }

    fn fields_to_fixpoint(
        &mut self,
        flds: &[rty::Expr],
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<fixpoint::Expr> {
        // do not generate 1-tuples
        if let [fld] = flds {
            self.expr_to_fixpoint(fld, scx)
        } else {
            scx.declare_tuple(flds.len());
            let ctor = fixpoint::Expr::Var(fixpoint::Var::TupleCtor { arity: flds.len() });
            let args = flds
                .iter()
                .map(|fld| self.expr_to_fixpoint(fld, scx))
                .try_collect()?;
            Ok(fixpoint::Expr::App(Box::new(ctor), None, args, None))
        }
    }

    fn internal_func_to_fixpoint(
        &mut self,
        internal_func: &InternalFuncKind,
        sort_args: &[rty::SortArg],
        args: &[rty::Expr],
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<fixpoint::Expr> {
        match internal_func {
            InternalFuncKind::Val(op) => {
                let func = fixpoint::Expr::Var(self.define_const_for_prim_op(op, scx));
                let args = self.exprs_to_fixpoint(args, scx)?;
                Ok(fixpoint::Expr::App(Box::new(func), None, args, None))
            }
            InternalFuncKind::Rel(op) => {
                let expr = if let Some(prim_rel) = self.genv.prim_rel_for(op)? {
                    prim_rel.body.replace_bound_refts(args)
                } else {
                    rty::Expr::tt()
                };
                self.expr_to_fixpoint(&expr, scx)
            }
            InternalFuncKind::Cast => {
                let [rty::SortArg::Sort(from), rty::SortArg::Sort(to)] = &sort_args else {
                    span_bug!(self.def_span(), "unexpected cast")
                };
                match from.cast_kind(to) {
                    rty::CastKind::Identity => self.expr_to_fixpoint(&args[0], scx),
                    rty::CastKind::BoolToInt => {
                        Ok(fixpoint::Expr::IfThenElse(Box::new([
                            self.expr_to_fixpoint(&args[0], scx)?,
                            fixpoint::Expr::int(1),
                            fixpoint::Expr::int(0),
                        ])))
                    }
                    rty::CastKind::IntoUnit => self.expr_to_fixpoint(&rty::Expr::unit(), scx),
                    rty::CastKind::Uninterpreted => {
                        let func = fixpoint::Expr::Var(self.define_const_for_cast(from, to, scx));
                        let args = self.exprs_to_fixpoint(args, scx)?;
                        Ok(fixpoint::Expr::App(Box::new(func), None, args, None))
                    }
                }
            }
        }
    }

    fn structurally_normalize_expr(&self, expr: &rty::Expr) -> QueryResult<rty::Expr> {
        if let Some(def_id) = self.def_id {
            structurally_normalize_expr(self.genv, def_id.resolved_id(), &self.infcx, expr)
        } else {
            Ok(expr.clone())
        }
    }

    fn expr_to_fixpoint(
        &mut self,
        expr: &rty::Expr,
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<fixpoint::Expr> {
        let expr = self.structurally_normalize_expr(expr)?;
        let e = match expr.kind() {
            rty::ExprKind::Var(var) => fixpoint::Expr::Var(self.var_to_fixpoint(var)),
            rty::ExprKind::Constant(c) => const_to_fixpoint(*c),
            rty::ExprKind::BinaryOp(op, e1, e2) => self.bin_op_to_fixpoint(op, e1, e2, scx)?,
            rty::ExprKind::UnaryOp(op, e) => self.un_op_to_fixpoint(*op, e, scx)?,
            rty::ExprKind::FieldProj(e, proj) => self.proj_to_fixpoint(e, *proj, scx)?,
            rty::ExprKind::Tuple(flds) => self.fields_to_fixpoint(flds, scx)?,
            rty::ExprKind::Ctor(rty::Ctor::Struct(did), flds) => {
                self.struct_fields_to_fixpoint(did, flds, scx)?
            }
            rty::ExprKind::IsCtor(def_id, variant_idx, e) => {
                let ctor = self.variant_to_fixpoint(scx, def_id, *variant_idx);
                let e = self.expr_to_fixpoint(e, scx)?;
                fixpoint::Expr::IsCtor(ctor, Box::new(e))
            }
            rty::ExprKind::Ctor(rty::Ctor::Enum(did, idx), args) => {
                let ctor = self.variant_to_fixpoint(scx, did, *idx);
                let args = self.exprs_to_fixpoint(args, scx)?;
                fixpoint::Expr::App(Box::new(fixpoint::Expr::Var(ctor)), None, args, None)
            }
            rty::ExprKind::ConstDefId(did) => {
                let var = self.define_const_for_rust_const(*did, scx);
                fixpoint::Expr::Var(var)
            }
            rty::ExprKind::App(func, sort_args, args) => {
                if let rty::ExprKind::InternalFunc(func) = func.kind() {
                    self.internal_func_to_fixpoint(func, sort_args, args, scx)?
                } else {
                    let func = self.expr_to_fixpoint(func, scx)?;
                    let sort_args = self.sort_args_to_fixpoint(sort_args, scx);
                    let args = self.exprs_to_fixpoint(args, scx)?;
                    fixpoint::Expr::App(Box::new(func), Some(sort_args), args, None)
                }
            }
            rty::ExprKind::IfThenElse(p, e1, e2) => {
                fixpoint::Expr::IfThenElse(Box::new([
                    self.expr_to_fixpoint(p, scx)?,
                    self.expr_to_fixpoint(e1, scx)?,
                    self.expr_to_fixpoint(e2, scx)?,
                ]))
            }
            rty::ExprKind::Alias(alias_reft, args) => {
                let sort = self.genv.sort_of_assoc_reft(alias_reft.assoc_id)?;
                let sort = sort.instantiate_identity();
                let func =
                    fixpoint::Expr::Var(self.define_const_for_alias_reft(alias_reft, sort, scx));
                let args = args
                    .iter()
                    .map(|expr| self.expr_to_fixpoint(expr, scx))
                    .try_collect()?;
                fixpoint::Expr::App(Box::new(func), None, args, None)
            }
            rty::ExprKind::Abs(lam) => {
                let var = self.define_const_for_lambda(lam, scx);
                fixpoint::Expr::Var(var)
            }
            rty::ExprKind::Let(init, body) => {
                debug_assert_eq!(body.vars().len(), 1);
                let init = self.expr_to_fixpoint(init, scx)?;

                self.local_var_env.push_layer_with_fresh_names(1);
                let body = self.expr_to_fixpoint(body.skip_binder_ref(), scx)?;
                let vars = self.local_var_env.pop_layer();

                fixpoint::Expr::Let(vars[0].into(), Box::new([init, body]))
            }
            rty::ExprKind::GlobalFunc(SpecFuncKind::Thy(itf)) => fixpoint::Expr::ThyFunc(*itf),
            rty::ExprKind::GlobalFunc(SpecFuncKind::Def(def_id)) => {
                fixpoint::Expr::Var(self.declare_fun(*def_id))
            }
            rty::ExprKind::Exists(expr) => {
                let expr = self.body_to_fixpoint(expr, scx)?;
                fixpoint::Expr::Exists(expr.0, Box::new(expr.1))
            }
            rty::ExprKind::Hole(..)
            | rty::ExprKind::KVar(_)
            | rty::ExprKind::Local(_)
            | rty::ExprKind::PathProj(..)
            | rty::ExprKind::ForAll(_)
            | rty::ExprKind::InternalFunc(_) => {
                span_bug!(self.def_span(), "unexpected expr: `{expr:?}`")
            }
            rty::ExprKind::BoundedQuant(kind, rng, body) => {
                let exprs = (rng.start..rng.end).map(|i| {
                    let arg = rty::Expr::constant(rty::Constant::from(i));
                    body.replace_bound_reft(&arg)
                });
                let expr = match kind {
                    flux_middle::fhir::QuantKind::Forall => rty::Expr::and_from_iter(exprs),
                    flux_middle::fhir::QuantKind::Exists => rty::Expr::or_from_iter(exprs),
                };
                self.expr_to_fixpoint(&expr, scx)?
            }
        };
        Ok(e)
    }

    fn sort_args_to_fixpoint(
        &mut self,
        sort_args: &[rty::SortArg],
        scx: &mut SortEncodingCtxt,
    ) -> Vec<fixpoint::Sort> {
        sort_args
            .iter()
            .map(|s_arg| self.sort_arg_to_fixpoint(s_arg, scx))
            .collect()
    }

    fn sort_arg_to_fixpoint(
        &mut self,
        sort_arg: &rty::SortArg,
        scx: &mut SortEncodingCtxt,
    ) -> fixpoint::Sort {
        match sort_arg {
            rty::SortArg::Sort(sort) => scx.sort_to_fixpoint(sort),
            rty::SortArg::BvSize(sz) => bv_size_to_fixpoint(*sz),
        }
    }

    fn exprs_to_fixpoint<'b>(
        &mut self,
        exprs: impl IntoIterator<Item = &'b rty::Expr>,
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<Vec<fixpoint::Expr>> {
        exprs
            .into_iter()
            .map(|e| self.expr_to_fixpoint(e, scx))
            .try_collect()
    }

    fn proj_to_fixpoint(
        &mut self,
        e: &rty::Expr,
        proj: rty::FieldProj,
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<fixpoint::Expr> {
        let arity = proj.arity(self.genv)?;
        // we encode 1-tuples as the single element inside so no projection necessary here
        if arity == 1 {
            self.expr_to_fixpoint(e, scx)
        } else {
            let proj = match proj {
                rty::FieldProj::Tuple { arity, field } => {
                    scx.declare_tuple(arity);
                    fixpoint::Var::TupleProj { arity, field }
                }
                rty::FieldProj::Adt { def_id, field } => {
                    let adt_id = scx.declare_adt(def_id);
                    fixpoint::Var::DataProj { adt_id, field }
                }
            };
            let proj = fixpoint::Expr::Var(proj);
            Ok(fixpoint::Expr::App(
                Box::new(proj),
                None,
                vec![self.expr_to_fixpoint(e, scx)?],
                None,
            ))
        }
    }

    fn un_op_to_fixpoint(
        &mut self,
        op: rty::UnOp,
        e: &rty::Expr,
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<fixpoint::Expr> {
        match op {
            rty::UnOp::Not => Ok(fixpoint::Expr::Not(Box::new(self.expr_to_fixpoint(e, scx)?))),
            rty::UnOp::Neg => Ok(fixpoint::Expr::Neg(Box::new(self.expr_to_fixpoint(e, scx)?))),
        }
    }

    fn bv_rel_to_fixpoint(&self, rel: &fixpoint::BinRel) -> fixpoint::Expr {
        let itf = match rel {
            fixpoint::BinRel::Gt => fixpoint::ThyFunc::BvUgt,
            fixpoint::BinRel::Ge => fixpoint::ThyFunc::BvUge,
            fixpoint::BinRel::Lt => fixpoint::ThyFunc::BvUlt,
            fixpoint::BinRel::Le => fixpoint::ThyFunc::BvUle,
            _ => span_bug!(self.def_span(), "not a bitvector relation!"),
        };
        fixpoint::Expr::ThyFunc(itf)
    }

    fn set_op_to_fixpoint(&self, op: &rty::BinOp) -> fixpoint::Expr {
        let itf = match op {
            rty::BinOp::Sub(_) => fixpoint::ThyFunc::SetDif,
            rty::BinOp::BitAnd(_) => fixpoint::ThyFunc::SetCap,
            rty::BinOp::BitOr(_) => fixpoint::ThyFunc::SetCup,
            _ => span_bug!(self.def_span(), "not a set operation!"),
        };
        fixpoint::Expr::ThyFunc(itf)
    }

    fn bv_op_to_fixpoint(&self, op: &rty::BinOp) -> fixpoint::Expr {
        let itf = match op {
            rty::BinOp::Add(_) => fixpoint::ThyFunc::BvAdd,
            rty::BinOp::Sub(_) => fixpoint::ThyFunc::BvSub,
            rty::BinOp::Mul(_) => fixpoint::ThyFunc::BvMul,
            rty::BinOp::Div(_) => fixpoint::ThyFunc::BvUdiv,
            rty::BinOp::Mod(_) => fixpoint::ThyFunc::BvUrem,
            rty::BinOp::BitAnd(_) => fixpoint::ThyFunc::BvAnd,
            rty::BinOp::BitOr(_) => fixpoint::ThyFunc::BvOr,
            rty::BinOp::BitXor(_) => fixpoint::ThyFunc::BvXor,
            rty::BinOp::BitShl(_) => fixpoint::ThyFunc::BvShl,
            rty::BinOp::BitShr(_) => fixpoint::ThyFunc::BvLshr,
            _ => span_bug!(self.def_span(), "not a bitvector operation!"),
        };
        fixpoint::Expr::ThyFunc(itf)
    }

    fn bin_op_to_fixpoint(
        &mut self,
        op: &rty::BinOp,
        e1: &rty::Expr,
        e2: &rty::Expr,
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<fixpoint::Expr> {
        let op = match op {
            rty::BinOp::Eq => {
                return Ok(fixpoint::Expr::Atom(
                    fixpoint::BinRel::Eq,
                    Box::new([self.expr_to_fixpoint(e1, scx)?, self.expr_to_fixpoint(e2, scx)?]),
                ));
            }
            rty::BinOp::Ne => {
                return Ok(fixpoint::Expr::Atom(
                    fixpoint::BinRel::Ne,
                    Box::new([self.expr_to_fixpoint(e1, scx)?, self.expr_to_fixpoint(e2, scx)?]),
                ));
            }
            rty::BinOp::Gt(sort) => {
                return self.bin_rel_to_fixpoint(sort, fixpoint::BinRel::Gt, e1, e2, scx);
            }
            rty::BinOp::Ge(sort) => {
                return self.bin_rel_to_fixpoint(sort, fixpoint::BinRel::Ge, e1, e2, scx);
            }
            rty::BinOp::Lt(sort) => {
                return self.bin_rel_to_fixpoint(sort, fixpoint::BinRel::Lt, e1, e2, scx);
            }
            rty::BinOp::Le(sort) => {
                return self.bin_rel_to_fixpoint(sort, fixpoint::BinRel::Le, e1, e2, scx);
            }
            rty::BinOp::And => {
                return Ok(fixpoint::Expr::And(vec![
                    self.expr_to_fixpoint(e1, scx)?,
                    self.expr_to_fixpoint(e2, scx)?,
                ]));
            }
            rty::BinOp::Or => {
                return Ok(fixpoint::Expr::Or(vec![
                    self.expr_to_fixpoint(e1, scx)?,
                    self.expr_to_fixpoint(e2, scx)?,
                ]));
            }
            rty::BinOp::Imp => {
                return Ok(fixpoint::Expr::Imp(Box::new([
                    self.expr_to_fixpoint(e1, scx)?,
                    self.expr_to_fixpoint(e2, scx)?,
                ])));
            }
            rty::BinOp::Iff => {
                return Ok(fixpoint::Expr::Iff(Box::new([
                    self.expr_to_fixpoint(e1, scx)?,
                    self.expr_to_fixpoint(e2, scx)?,
                ])));
            }

            // Bit vector operations
            rty::BinOp::Add(rty::Sort::BitVec(_))
            | rty::BinOp::Sub(rty::Sort::BitVec(_))
            | rty::BinOp::Mul(rty::Sort::BitVec(_))
            | rty::BinOp::Div(rty::Sort::BitVec(_))
            | rty::BinOp::Mod(rty::Sort::BitVec(_))
            | rty::BinOp::BitAnd(rty::Sort::BitVec(_))
            | rty::BinOp::BitOr(rty::Sort::BitVec(_))
            | rty::BinOp::BitXor(rty::Sort::BitVec(_))
            | rty::BinOp::BitShl(rty::Sort::BitVec(_))
            | rty::BinOp::BitShr(rty::Sort::BitVec(_)) => {
                let bv_func = self.bv_op_to_fixpoint(op);
                return Ok(fixpoint::Expr::App(
                    Box::new(bv_func),
                    None,
                    vec![self.expr_to_fixpoint(e1, scx)?, self.expr_to_fixpoint(e2, scx)?],
                    None,
                ));
            }

            // Set operations
            rty::BinOp::Sub(rty::Sort::App(rty::SortCtor::Set, _))
            | rty::BinOp::BitAnd(rty::Sort::App(rty::SortCtor::Set, _))
            | rty::BinOp::BitOr(rty::Sort::App(rty::SortCtor::Set, _)) => {
                let set_func = self.set_op_to_fixpoint(op);
                return Ok(fixpoint::Expr::App(
                    Box::new(set_func),
                    None,
                    vec![self.expr_to_fixpoint(e1, scx)?, self.expr_to_fixpoint(e2, scx)?],
                    None,
                ));
            }

            // Interpreted arithmetic operations
            rty::BinOp::Add(_) => fixpoint::BinOp::Add,
            rty::BinOp::Sub(_) => fixpoint::BinOp::Sub,
            rty::BinOp::Mul(_) => fixpoint::BinOp::Mul,
            rty::BinOp::Div(_) => fixpoint::BinOp::Div,
            rty::BinOp::Mod(_) => fixpoint::BinOp::Mod,

            rty::BinOp::BitAnd(sort)
            | rty::BinOp::BitOr(sort)
            | rty::BinOp::BitXor(sort)
            | rty::BinOp::BitShl(sort)
            | rty::BinOp::BitShr(sort) => {
                bug!("unsupported operation `{op:?}` for sort `{sort:?}`");
            }
        };
        Ok(fixpoint::Expr::BinaryOp(
            op,
            Box::new([self.expr_to_fixpoint(e1, scx)?, self.expr_to_fixpoint(e2, scx)?]),
        ))
    }

    /// A binary relation is encoded as a structurally recursive relation between aggregate sorts.
    /// For "leaf" expressions, we encode them as an interpreted relation if the sort supports it,
    /// otherwise we use an uninterpreted function. For example, consider the following relation
    /// between two tuples of sort `(int, int -> int)`
    /// ```text
    /// (0, λv. v + 1) <= (1, λv. v + 1)
    /// ```
    /// The encoding in fixpoint will be
    ///
    /// ```text
    /// 0 <= 1 && (le (λv. v + 1) (λv. v + 1))
    /// ```
    /// Where `<=` is the (interpreted) less than or equal relation between integers and `le` is
    /// an uninterpreted relation between ([the encoding] of) lambdas.
    ///
    /// [the encoding]: Self::define_const_for_lambda
    fn bin_rel_to_fixpoint(
        &mut self,
        sort: &rty::Sort,
        rel: fixpoint::BinRel,
        e1: &rty::Expr,
        e2: &rty::Expr,
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<fixpoint::Expr> {
        let e = match sort {
            rty::Sort::Int | rty::Sort::Real | rty::Sort::Char => {
                fixpoint::Expr::Atom(
                    rel,
                    Box::new([self.expr_to_fixpoint(e1, scx)?, self.expr_to_fixpoint(e2, scx)?]),
                )
            }
            rty::Sort::BitVec(_) => {
                let e1 = self.expr_to_fixpoint(e1, scx)?;
                let e2 = self.expr_to_fixpoint(e2, scx)?;
                let rel = self.bv_rel_to_fixpoint(&rel);
                fixpoint::Expr::App(Box::new(rel), None, vec![e1, e2], None)
            }
            rty::Sort::Tuple(sorts) => {
                let arity = sorts.len();
                self.apply_bin_rel_rec(sorts, rel, e1, e2, scx, |field| {
                    rty::FieldProj::Tuple { arity, field }
                })?
            }
            rty::Sort::App(rty::SortCtor::Adt(sort_def), args)
                if let Some(variant) = sort_def.opt_struct_variant() =>
            {
                let def_id = sort_def.did();
                let sorts = variant.field_sorts(args);
                self.apply_bin_rel_rec(&sorts, rel, e1, e2, scx, |field| {
                    rty::FieldProj::Adt { def_id, field }
                })?
            }
            _ => {
                let rel = fixpoint::Expr::Var(fixpoint::Var::UIFRel(rel));
                fixpoint::Expr::App(
                    Box::new(rel),
                    None,
                    vec![self.expr_to_fixpoint(e1, scx)?, self.expr_to_fixpoint(e2, scx)?],
                    None,
                )
            }
        };
        Ok(e)
    }

    /// Apply binary relation recursively over aggregate expressions
    fn apply_bin_rel_rec(
        &mut self,
        sorts: &[rty::Sort],
        rel: fixpoint::BinRel,
        e1: &rty::Expr,
        e2: &rty::Expr,
        scx: &mut SortEncodingCtxt,
        mk_proj: impl Fn(u32) -> rty::FieldProj,
    ) -> QueryResult<fixpoint::Expr> {
        Ok(fixpoint::Expr::and(
            sorts
                .iter()
                .enumerate()
                .map(|(idx, s)| {
                    let proj = mk_proj(idx as u32);
                    let e1 = e1.proj_and_reduce(proj);
                    let e2 = e2.proj_and_reduce(proj);
                    self.bin_rel_to_fixpoint(s, rel, &e1, &e2, scx)
                })
                .try_collect()?,
        ))
    }

    /// Declare that the `def_id` of a Flux function (potentially a UIF) needs to be
    /// encoded and assigns a name to it if it hasn't yet been declared. The encoding of the
    /// function body happens in [`Self::define_funs`].
    pub fn declare_fun(&mut self, def_id: FluxDefId) -> fixpoint::Var {
        *self
            .const_env
            .fun_decl_map
            .entry(def_id)
            .or_insert_with(|| {
                let id = self.const_env.global_var_gen.fresh();
                fixpoint::Var::Global(id, Some(def_id))
            })
    }

    /// The logic below is a bit "duplicated" with the `prim_op_sort` in `sortck.rs`;
    /// They are not exactly the same because this is on rty and the other one on fhir.
    /// We should make sure these two remain in sync.
    ///
    /// (NOTE:PrimOpSort) We are somewhat "overloading" the `BinOps`: as we are using them
    /// for (a) interpreted operations on bit vectors AND (b) uninterpreted functions on integers.
    /// So when Binop::BitShr (a) appears in a ExprKind::BinOp, it means bit vectors, but
    /// (b) inside ExprKind::InternalFunc it means int.
    fn prim_op_sort(op: &rty::BinOp, span: Span) -> rty::PolyFuncSort {
        match op {
            rty::BinOp::BitAnd(rty::Sort::Int)
            | rty::BinOp::BitOr(rty::Sort::Int)
            | rty::BinOp::BitXor(rty::Sort::Int)
            | rty::BinOp::BitShl(rty::Sort::Int)
            | rty::BinOp::BitShr(rty::Sort::Int) => {
                let fsort =
                    rty::FuncSort::new(vec![rty::Sort::Int, rty::Sort::Int], rty::Sort::Int);
                rty::PolyFuncSort::new(List::empty(), fsort)
            }
            _ => span_bug!(span, "unexpected prim op: {op:?} in `prim_op_sort`"),
        }
    }

    fn define_const_for_cast(
        &mut self,
        from: &rty::Sort,
        to: &rty::Sort,
        scx: &mut SortEncodingCtxt,
    ) -> fixpoint::Var {
        let key = ConstKey::Cast(from.clone(), to.clone());
        self.const_env
            .get_or_insert(key, |global_name| {
                let fsort = rty::FuncSort::new(vec![from.clone()], to.clone());
                let fsort = rty::PolyFuncSort::new(List::empty(), fsort);
                let sort = scx.func_sort_to_fixpoint(&fsort).into_sort();
                fixpoint::ConstDecl {
                    name: fixpoint::Var::Global(global_name, None),
                    sort,
                    comment: Some(format!("cast uif: ({from:?}) -> {to:?}")),
                }
            })
            .name
    }

    fn define_const_for_prim_op(
        &mut self,
        op: &rty::BinOp,
        scx: &mut SortEncodingCtxt,
    ) -> fixpoint::Var {
        let key = ConstKey::PrimOp(op.clone());
        let span = self.def_span();
        self.const_env
            .get_or_insert(key, |global_name| {
                let sort = scx
                    .func_sort_to_fixpoint(&Self::prim_op_sort(op, span))
                    .into_sort();
                fixpoint::ConstDecl {
                    name: fixpoint::Var::Global(global_name, None),
                    sort,
                    comment: Some(format!("prim op uif: {op:?}")),
                }
            })
            .name
    }

    fn define_const_for_rust_const(
        &mut self,
        def_id: DefId,
        scx: &mut SortEncodingCtxt,
    ) -> fixpoint::Var {
        let key = ConstKey::RustConst(def_id);
        self.const_env
            .get_or_insert(key, |global_name| {
                let sort = self.genv.sort_of_def_id(def_id).unwrap().unwrap();
                fixpoint::ConstDecl {
                    name: fixpoint::Var::Global(global_name, None),
                    sort: scx.sort_to_fixpoint(&sort),
                    comment: Some(format!("rust const: {}", def_id_to_string(def_id))),
                }
            })
            .name
    }

    /// returns the 'constant' UIF for Var used to represent the alias_pred, creating and adding it
    /// to the const_map if necessary
    fn define_const_for_alias_reft(
        &mut self,
        alias_reft: &rty::AliasReft,
        fsort: rty::FuncSort,
        scx: &mut SortEncodingCtxt,
    ) -> fixpoint::Var {
        let tcx = self.genv.tcx();
        let args = alias_reft
            .args
            .to_rustc(tcx)
            .truncate_to(tcx, tcx.generics_of(alias_reft.assoc_id.parent()));
        let key = ConstKey::Alias(alias_reft.assoc_id, args);
        self.const_env
            .get_or_insert(key, |global_name| {
                let comment = Some(format!("alias reft: {alias_reft:?}"));
                let name = fixpoint::Var::Global(global_name, None);
                let fsort = rty::PolyFuncSort::new(List::empty(), fsort);
                let sort = scx.func_sort_to_fixpoint(&fsort).into_sort();
                fixpoint::ConstDecl { name, comment, sort }
            })
            .name
    }

    /// We encode lambdas with uninterpreted constant. Two syntactically equal lambdas will be encoded
    /// with the same constant.
    fn define_const_for_lambda(
        &mut self,
        lam: &rty::Lambda,
        scx: &mut SortEncodingCtxt,
    ) -> fixpoint::Var {
        let key = ConstKey::Lambda(lam.clone());
        self.const_env
            .get_or_insert(key, |global_name| {
                let comment = Some(format!("lambda: {lam:?}"));
                let name = fixpoint::Var::Global(global_name, None);
                let sort = scx
                    .func_sort_to_fixpoint(&lam.fsort().to_poly())
                    .into_sort();
                fixpoint::ConstDecl { name, comment, sort }
            })
            .name
    }

    fn assume_const_values(
        &mut self,
        mut constraint: fixpoint::Constraint,
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<fixpoint::Constraint> {
        // Encoding the value for a constant could in theory define more constants for which
        // we need to assume values, so we iterate until there are no more constants.
        let mut idx = 0;
        while let Some((key, const_)) = self.const_env.const_map.get_index(idx) {
            idx += 1;

            let ConstKey::RustConst(def_id) = key else { continue };
            let info = self.genv.constant_info(def_id)?;
            match info {
                rty::ConstantInfo::Uninterpreted => {}
                rty::ConstantInfo::Interpreted(val, _) => {
                    let e1 = fixpoint::Expr::Var(const_.name);
                    let e2 = self.expr_to_fixpoint(&val, scx)?;
                    let pred = fixpoint::Pred::Expr(e1.eq(e2));
                    constraint = fixpoint::Constraint::ForAll(
                        fixpoint::Bind {
                            name: fixpoint::Var::Underscore,
                            sort: fixpoint::Sort::Int,
                            pred,
                        },
                        Box::new(constraint),
                    );
                }
            }
        }
        Ok(constraint)
    }

    fn qualifiers_for(
        &mut self,
        def_id: LocalDefId,
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<Vec<fixpoint::Qualifier>> {
        self.genv
            .qualifiers_for(def_id)?
            .map(|qual| self.qualifier_to_fixpoint(qual, scx))
            .try_collect()
    }

    fn define_funs(
        &mut self,
        def_id: MaybeExternId,
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<Vec<fixpoint::FunDef>> {
        let reveals: UnordSet<FluxDefId> = self
            .genv
            .reveals_for(def_id.local_id())
            .iter()
            .copied()
            .collect();
        let proven_externally = self.genv.proven_externally(def_id.local_id());
        let mut defs = vec![];

        // Iterate till encoding the body of functions doesn't require any more functions to be encoded.
        let mut idx = 0;
        while let Some((&did, _)) = self.const_env.fun_decl_map.get_index(idx) {
            idx += 1;

            let info = self.genv.normalized_info(did);
            let revealed = reveals.contains(&did);
            let def = if info.uif || (info.hide && !revealed && proven_externally.is_none()) {
                self.fun_decl_to_fixpoint(did, scx)
            } else {
                self.fun_def_to_fixpoint(did, scx)?
            };
            defs.push((info.rank, def));
        }

        // we sort by rank so the definitions go out without any forward dependencies.
        let defs = defs
            .into_iter()
            .sorted_by_key(|(rank, _)| *rank)
            .map(|(_, def)| def)
            .collect();

        Ok(defs)
    }

    fn fun_decl_to_fixpoint(
        &mut self,
        def_id: FluxDefId,
        scx: &mut SortEncodingCtxt,
    ) -> fixpoint::FunDef {
        let name = self.const_env.fun_decl_map[&def_id];
        let sort = scx.func_sort_to_fixpoint(&self.genv.func_sort(def_id));
        fixpoint::FunDef { name, sort, body: None, comment: Some(format!("flux def: {def_id:?}")) }
    }

    pub fn fun_def_to_fixpoint(
        &mut self,
        def_id: FluxDefId,
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<fixpoint::FunDef> {
        let name = *self.const_env.fun_decl_map.get(&def_id).unwrap();
        let body = self.genv.inlined_body(def_id);
        let output = scx.sort_to_fixpoint(self.genv.func_sort(def_id).expect_mono().output());
        let (args, expr) = self.body_to_fixpoint(&body, scx)?;
        let (args, inputs) = args.into_iter().unzip();
        Ok(fixpoint::FunDef {
            name,
            sort: fixpoint::FunSort { params: 0, inputs, output },
            body: Some(fixpoint::FunBody { args, expr }),
            comment: Some(format!("flux def: {def_id:?}")),
        })
    }

    fn body_to_fixpoint(
        &mut self,
        body: &rty::Binder<rty::Expr>,
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<(Vec<(fixpoint::Var, fixpoint::Sort)>, fixpoint::Expr)> {
        self.local_var_env
            .push_layer_with_fresh_names(body.vars().len());

        let expr = self.expr_to_fixpoint(body.as_ref().skip_binder(), scx)?;

        let args: Vec<(fixpoint::Var, fixpoint::Sort)> =
            iter::zip(self.local_var_env.pop_layer(), body.vars())
                .map(|(name, var)| (name.into(), scx.sort_to_fixpoint(var.expect_sort())))
                .collect();

        Ok((args, expr))
    }

    fn qualifier_to_fixpoint(
        &mut self,
        qualifier: &rty::Qualifier,
        scx: &mut SortEncodingCtxt,
    ) -> QueryResult<fixpoint::Qualifier> {
        let (args, body) = self.body_to_fixpoint(&qualifier.body, scx)?;
        let name = qualifier.def_id.name().to_string();
        Ok(fixpoint::Qualifier { name, args, body })
    }
}

fn mk_implies(assumption: fixpoint::Pred, cstr: fixpoint::Constraint) -> fixpoint::Constraint {
    fixpoint::Constraint::ForAll(
        fixpoint::Bind {
            name: fixpoint::Var::Underscore,
            sort: fixpoint::Sort::Int,
            pred: assumption,
        },
        Box::new(cstr),
    )
}

fn parse_kvid(kvid: &str) -> fixpoint::KVid {
    if kvid.starts_with("k")
        && let Some(kvid) = kvid[1..].parse::<u32>().ok()
    {
        fixpoint::KVid::from_u32(kvid)
    } else {
        tracked_span_bug!("unexpected kvar name {kvid}")
    }
}

fn parse_local_var(name: &str) -> Option<fixpoint::Var> {
    if let Some(rest) = name.strip_prefix('a')
        && let Ok(idx) = rest.parse::<u32>()
    {
        return Some(fixpoint::Var::Local(fixpoint::LocalVar::from(idx)));
    }
    None
}

fn parse_global_var(name: &str, fun_decl_map: &HashMap<usize, FluxDefId>) -> Option<fixpoint::Var> {
    if let Some(rest) = name.strip_prefix('c')
        && let Ok(idx) = rest.parse::<u32>()
    {
        return Some(fixpoint::Var::Global(
            fixpoint::GlobalVar::from(idx),
            fun_decl_map.get(&(idx as usize)).copied(),
        ));
    }
    // try parsing as a named global variable
    if let Some(rest) = name.strip_prefix("f$")
        && let parts = rest.split('$').collect::<Vec<_>>()
        && parts.len() == 2
        && let Ok(global_idx) = parts[1].parse::<u32>()
    {
        return Some(fixpoint::Var::Global(
            fixpoint::GlobalVar::from(global_idx),
            fun_decl_map.get(&(global_idx as usize)).copied(),
        ));
    }
    None
}

fn parse_param(name: &str) -> Option<fixpoint::Var> {
    if let Some(rest) = name.strip_prefix("reftgen$")
        && let parts = rest.split('$').collect::<Vec<_>>()
        && parts.len() == 2
        && let Ok(index) = parts[1].parse::<u32>()
    {
        let name = Symbol::intern(parts[0]);
        let param = EarlyReftParam { index, name };
        return Some(fixpoint::Var::Param(param));
    }
    None
}

fn parse_data_proj(name: &str) -> Option<fixpoint::Var> {
    if let Some(rest) = name.strip_prefix("fld")
        && let parts = rest.split('$').collect::<Vec<_>>()
        && parts.len() == 2
        && let Ok(adt_id) = parts[0].parse::<u32>()
        && let Ok(field) = parts[1].parse::<u32>()
    {
        let adt_id = fixpoint::AdtId::from(adt_id);
        return Some(fixpoint::Var::DataProj { adt_id, field });
    }
    None
}

fn parse_data_ctor(name: &str) -> Option<fixpoint::Var> {
    if let Some(rest) = name.strip_prefix("mkadt")
        && let parts = rest.split('$').collect::<Vec<_>>()
        && parts.len() == 2
        && let Ok(adt_id) = parts[0].parse::<u32>()
        && let Ok(variant_idx) = parts[1].parse::<u32>()
    {
        let adt_id = fixpoint::AdtId::from(adt_id);
        let variant_idx = VariantIdx::from(variant_idx);
        return Some(fixpoint::Var::DataCtor(adt_id, variant_idx));
    }
    None
}

struct SexpParseCtxt<'a> {
    local_var_env: &'a mut LocalVarEnv,
    fun_decl_map: &'a HashMap<usize, FluxDefId>,
}

impl<'a> SexpParseCtxt<'a> {
    fn new(
        local_var_env: &'a mut LocalVarEnv,
        fun_decl_map: &'a HashMap<usize, FluxDefId>,
    ) -> Self {
        Self { local_var_env, fun_decl_map }
    }
}

impl FromSexp<FixpointTypes> for SexpParseCtxt<'_> {
    fn fresh_var(&mut self) -> fixpoint::Var {
        fixpoint::Var::Local(self.local_var_env.fresh_name())
    }

    fn kvar(&self, name: &str) -> Result<fixpoint::KVid, ParseError> {
        bug!("TODO: SexpParse: kvar: {name}")
    }

    fn string(&self, s: &str) -> Result<fixpoint::SymStr, ParseError> {
        Ok(fixpoint::SymStr(Symbol::intern(s)))
    }

    fn var(&self, name: &str) -> Result<fixpoint::Var, ParseError> {
        if let Some(var) = parse_local_var(name) {
            return Ok(var);
        }
        if let Some(var) = parse_global_var(name, self.fun_decl_map) {
            return Ok(var);
        }
        if let Some(var) = parse_param(name) {
            return Ok(var);
        }
        if let Some(var) = parse_data_proj(name) {
            return Ok(var);
        }
        if let Some(var) = parse_data_ctor(name) {
            return Ok(var);
        }
        Err(ParseError::err(format!("Unknown variable: {name}")))
    }

    fn sort(&self, name: &str) -> Result<fixpoint::DataSort, ParseError> {
        if let Some(idx) = name.strip_prefix("Adt")
            && let Ok(adt_id) = idx.parse::<u32>()
        {
            return Ok(fixpoint::DataSort::Adt(fixpoint::AdtId::from(adt_id)));
        }
        Err(ParseError::err(format!("Unknown sort: {name}")))
    }
}