diff --git a/compiler/rustc_data_structures/src/graph/scc/mod.rs b/compiler/rustc_data_structures/src/graph/scc/mod.rs
index 508a084b311f5..7099ca7eb88c2 100644
--- a/compiler/rustc_data_structures/src/graph/scc/mod.rs
+++ b/compiler/rustc_data_structures/src/graph/scc/mod.rs
@@ -9,6 +9,7 @@ use crate::fx::FxHashSet;
 use crate::graph::vec_graph::VecGraph;
 use crate::graph::{DirectedGraph, GraphSuccessors, WithNumEdges, WithNumNodes, WithSuccessors};
 use rustc_index::vec::{Idx, IndexVec};
+use std::cmp::Ord;
 use std::ops::Range;
 
 #[cfg(test)]
@@ -38,7 +39,7 @@ struct SccData<S: Idx> {
     all_successors: Vec<S>,
 }
 
-impl<N: Idx, S: Idx> Sccs<N, S> {
+impl<N: Idx, S: Idx + Ord> Sccs<N, S> {
     pub fn new(graph: &(impl DirectedGraph<Node = N> + WithNumNodes + WithSuccessors)) -> Self {
         SccsConstruction::construct(graph)
     }
@@ -85,7 +86,7 @@ impl<N: Idx, S: Idx> DirectedGraph for Sccs<N, S> {
     type Node = S;
 }
 
-impl<N: Idx, S: Idx> WithNumNodes for Sccs<N, S> {
+impl<N: Idx, S: Idx + Ord> WithNumNodes for Sccs<N, S> {
     fn num_nodes(&self) -> usize {
         self.num_sccs()
     }
@@ -103,7 +104,7 @@ impl<'graph, N: Idx, S: Idx> GraphSuccessors<'graph> for Sccs<N, S> {
     type Iter = std::iter::Cloned<std::slice::Iter<'graph, S>>;
 }
 
-impl<N: Idx, S: Idx> WithSuccessors for Sccs<N, S> {
+impl<N: Idx, S: Idx + Ord> WithSuccessors for Sccs<N, S> {
     fn successors(&self, node: S) -> <Self as GraphSuccessors<'_>>::Iter {
         self.successors(node).iter().cloned()
     }
diff --git a/compiler/rustc_data_structures/src/graph/vec_graph/mod.rs b/compiler/rustc_data_structures/src/graph/vec_graph/mod.rs
index 5d9bc1b2e5168..3d91bcade59a4 100644
--- a/compiler/rustc_data_structures/src/graph/vec_graph/mod.rs
+++ b/compiler/rustc_data_structures/src/graph/vec_graph/mod.rs
@@ -1,3 +1,5 @@
+use std::cmp::Ord;
+
 use crate::graph::{DirectedGraph, GraphSuccessors, WithNumEdges, WithNumNodes, WithSuccessors};
 use rustc_index::vec::{Idx, IndexVec};
 
@@ -17,7 +19,7 @@ pub struct VecGraph<N: Idx> {
     edge_targets: Vec<N>,
 }
 
-impl<N: Idx> VecGraph<N> {
+impl<N: Idx + Ord> VecGraph<N> {
     pub fn new(num_nodes: usize, mut edge_pairs: Vec<(N, N)>) -> Self {
         // Sort the edges by the source -- this is important.
         edge_pairs.sort();
@@ -100,7 +102,7 @@ impl<'graph, N: Idx> GraphSuccessors<'graph> for VecGraph<N> {
     type Iter = std::iter::Cloned<std::slice::Iter<'graph, N>>;
 }
 
-impl<N: Idx> WithSuccessors for VecGraph<N> {
+impl<N: Idx + Ord> WithSuccessors for VecGraph<N> {
     fn successors(&self, node: N) -> <Self as GraphSuccessors<'_>>::Iter {
         self.successors(node).iter().cloned()
     }
diff --git a/compiler/rustc_hir/src/definitions.rs b/compiler/rustc_hir/src/definitions.rs
index ed7afcc07b101..d813c887eee9a 100644
--- a/compiler/rustc_hir/src/definitions.rs
+++ b/compiler/rustc_hir/src/definitions.rs
@@ -101,7 +101,11 @@ impl DefPathTable {
 pub struct Definitions {
     table: DefPathTable,
 
-    // FIXME(eddyb) ideally all `LocalDefId`s would be HIR owners.
+    /// Only [`LocalDefId`]s for items and item-like are HIR owners.
+    /// The associated `HirId` has a `local_id` of `0`.
+    /// Generic parameters and closures are also assigned a `LocalDefId` but are not HIR owners.
+    /// Their `HirId`s are defined by their position while lowering the enclosing owner.
+    // FIXME(cjgillot) Some `LocalDefId`s from `use` items are dropped during lowering and lack a `HirId`.
     pub(super) def_id_to_hir_id: IndexVec<LocalDefId, Option<hir::HirId>>,
     /// The reverse mapping of `def_id_to_hir_id`.
     pub(super) hir_id_to_def_id: FxHashMap<hir::HirId, LocalDefId>,
diff --git a/compiler/rustc_hir/src/hir.rs b/compiler/rustc_hir/src/hir.rs
index 64bd32b8ddc79..69572807e7c9d 100644
--- a/compiler/rustc_hir/src/hir.rs
+++ b/compiler/rustc_hir/src/hir.rs
@@ -1203,7 +1203,7 @@ pub enum UnsafeSource {
     UserProvided,
 }
 
-#[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Encodable, Hash, Debug)]
+#[derive(Copy, Clone, PartialEq, Eq, Encodable, Hash, Debug)]
 pub struct BodyId {
     pub hir_id: HirId,
 }
@@ -1980,7 +1980,7 @@ pub struct FnSig<'hir> {
 // The bodies for items are stored "out of line", in a separate
 // hashmap in the `Crate`. Here we just record the hir-id of the item
 // so it can fetched later.
-#[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Encodable, Debug)]
+#[derive(Copy, Clone, PartialEq, Eq, Encodable, Debug)]
 pub struct TraitItemId {
     pub def_id: LocalDefId,
 }
@@ -2043,7 +2043,7 @@ pub enum TraitItemKind<'hir> {
 // The bodies for items are stored "out of line", in a separate
 // hashmap in the `Crate`. Here we just record the hir-id of the item
 // so it can fetched later.
-#[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Encodable, Debug)]
+#[derive(Copy, Clone, PartialEq, Eq, Encodable, Debug)]
 pub struct ImplItemId {
     pub def_id: LocalDefId,
 }
@@ -2644,7 +2644,7 @@ impl<'hir> VariantData<'hir> {
 // The bodies for items are stored "out of line", in a separate
 // hashmap in the `Crate`. Here we just record the hir-id of the item
 // so it can fetched later.
-#[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Encodable, Debug, Hash)]
+#[derive(Copy, Clone, PartialEq, Eq, Encodable, Debug, Hash)]
 pub struct ItemId {
     pub def_id: LocalDefId,
 }
@@ -2883,7 +2883,7 @@ pub enum AssocItemKind {
 // The bodies for items are stored "out of line", in a separate
 // hashmap in the `Crate`. Here we just record the hir-id of the item
 // so it can fetched later.
-#[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Encodable, Debug)]
+#[derive(Copy, Clone, PartialEq, Eq, Encodable, Debug)]
 pub struct ForeignItemId {
     pub def_id: LocalDefId,
 }
diff --git a/compiler/rustc_hir/src/hir_id.rs b/compiler/rustc_hir/src/hir_id.rs
index 39552eb9f3102..1482a96cae316 100644
--- a/compiler/rustc_hir/src/hir_id.rs
+++ b/compiler/rustc_hir/src/hir_id.rs
@@ -11,7 +11,7 @@ use std::fmt;
 /// the `local_id` part of the `HirId` changing, which is a very useful property in
 /// incremental compilation where we have to persist things through changes to
 /// the code base.
-#[derive(Copy, Clone, PartialEq, Eq, Hash, Debug, PartialOrd, Ord)]
+#[derive(Copy, Clone, PartialEq, Eq, Hash, Debug)]
 #[derive(Encodable, Decodable)]
 pub struct HirId {
     pub owner: LocalDefId,
@@ -32,6 +32,10 @@ impl HirId {
     pub fn make_owner(owner: LocalDefId) -> Self {
         Self { owner, local_id: ItemLocalId::from_u32(0) }
     }
+
+    pub fn index(self) -> (usize, usize) {
+        (rustc_index::vec::Idx::index(self.owner), rustc_index::vec::Idx::index(self.local_id))
+    }
 }
 
 impl fmt::Display for HirId {
@@ -40,6 +44,18 @@ impl fmt::Display for HirId {
     }
 }
 
+impl Ord for HirId {
+    fn cmp(&self, other: &Self) -> std::cmp::Ordering {
+        (self.index()).cmp(&(other.index()))
+    }
+}
+
+impl PartialOrd for HirId {
+    fn partial_cmp(&self, other: &Self) -> Option<std::cmp::Ordering> {
+        Some(self.cmp(&other))
+    }
+}
+
 rustc_data_structures::define_id_collections!(HirIdMap, HirIdSet, HirId);
 rustc_data_structures::define_id_collections!(ItemLocalMap, ItemLocalSet, ItemLocalId);
 
diff --git a/compiler/rustc_index/src/bit_set.rs b/compiler/rustc_index/src/bit_set.rs
index 241cbe234874d..5aa213cb70134 100644
--- a/compiler/rustc_index/src/bit_set.rs
+++ b/compiler/rustc_index/src/bit_set.rs
@@ -675,7 +675,7 @@ impl<T: Idx> SparseBitSet<T> {
 
     fn insert(&mut self, elem: T) -> bool {
         assert!(elem.index() < self.domain_size);
-        let changed = if let Some(i) = self.elems.iter().position(|&e| e >= elem) {
+        let changed = if let Some(i) = self.elems.iter().position(|&e| e.index() >= elem.index()) {
             if self.elems[i] == elem {
                 // `elem` is already in the set.
                 false
@@ -715,6 +715,10 @@ impl<T: Idx> SparseBitSet<T> {
         self.elems.iter()
     }
 
+    bit_relations_inherent_impls! {}
+}
+
+impl<T: Idx + Ord> SparseBitSet<T> {
     fn last_set_in(&self, range: impl RangeBounds<T>) -> Option<T> {
         let mut last_leq = None;
         for e in self.iter() {
@@ -724,8 +728,6 @@ impl<T: Idx> SparseBitSet<T> {
         }
         last_leq
     }
-
-    bit_relations_inherent_impls! {}
 }
 
 /// A fixed-size bitset type with a hybrid representation: sparse when there
@@ -802,7 +804,10 @@ impl<T: Idx> HybridBitSet<T> {
     /// Returns the previous element present in the bitset from `elem`,
     /// inclusively of elem. That is, will return `Some(elem)` if elem is in the
     /// bitset.
-    pub fn last_set_in(&self, range: impl RangeBounds<T>) -> Option<T> {
+    pub fn last_set_in(&self, range: impl RangeBounds<T>) -> Option<T>
+    where
+        T: Ord,
+    {
         match self {
             HybridBitSet::Sparse(sparse) => sparse.last_set_in(range),
             HybridBitSet::Dense(dense) => dense.last_set_in(range),
diff --git a/compiler/rustc_index/src/vec.rs b/compiler/rustc_index/src/vec.rs
index 55ccfd0ad2380..e3c6528b21885 100644
--- a/compiler/rustc_index/src/vec.rs
+++ b/compiler/rustc_index/src/vec.rs
@@ -12,7 +12,7 @@ use std::vec;
 /// Represents some newtyped `usize` wrapper.
 ///
 /// Purpose: avoid mixing indexes for different bitvector domains.
-pub trait Idx: Copy + 'static + Ord + Debug + Hash {
+pub trait Idx: Copy + 'static + Eq + PartialEq + Debug + Hash {
     fn new(idx: usize) -> Self;
 
     fn index(self) -> usize;
diff --git a/compiler/rustc_metadata/src/rmeta/encoder.rs b/compiler/rustc_metadata/src/rmeta/encoder.rs
index a6828e103dc5f..eeb907d01148b 100644
--- a/compiler/rustc_metadata/src/rmeta/encoder.rs
+++ b/compiler/rustc_metadata/src/rmeta/encoder.rs
@@ -1306,7 +1306,7 @@ impl<'a, 'tcx> EncodeContext<'a, 'tcx> {
             })
             .collect::<Vec<_>>();
         // Sort everything to ensure a stable order for diagnotics.
-        keys_and_jobs.sort_by_key(|&(def_id, _, _)| def_id);
+        keys_and_jobs.sort_by_key(|&(def_id, _, _)| def_id.index());
         for (def_id, encode_const, encode_opt) in keys_and_jobs.into_iter() {
             debug_assert!(encode_const || encode_opt);
 
diff --git a/compiler/rustc_middle/src/mir/mono.rs b/compiler/rustc_middle/src/mir/mono.rs
index fd8606e6929e2..1422537cd5060 100644
--- a/compiler/rustc_middle/src/mir/mono.rs
+++ b/compiler/rustc_middle/src/mir/mono.rs
@@ -6,7 +6,7 @@ use rustc_data_structures::fingerprint::Fingerprint;
 use rustc_data_structures::fx::FxHashMap;
 use rustc_data_structures::stable_hasher::{HashStable, StableHasher};
 use rustc_hir::def_id::{CrateNum, DefId, LOCAL_CRATE};
-use rustc_hir::{HirId, ItemId};
+use rustc_hir::ItemId;
 use rustc_query_system::ich::{NodeIdHashingMode, StableHashingContext};
 use rustc_session::config::OptLevel;
 use rustc_span::source_map::Span;
@@ -355,7 +355,7 @@ impl<'tcx> CodegenUnit<'tcx> {
         // The codegen tests rely on items being process in the same order as
         // they appear in the file, so for local items, we sort by node_id first
         #[derive(PartialEq, Eq, PartialOrd, Ord)]
-        pub struct ItemSortKey<'tcx>(Option<HirId>, SymbolName<'tcx>);
+        pub struct ItemSortKey<'tcx>(Option<usize>, SymbolName<'tcx>);
 
         fn item_sort_key<'tcx>(tcx: TyCtxt<'tcx>, item: MonoItem<'tcx>) -> ItemSortKey<'tcx> {
             ItemSortKey(
@@ -366,10 +366,7 @@ impl<'tcx> CodegenUnit<'tcx> {
                             // instances into account. The others don't matter for
                             // the codegen tests and can even make item order
                             // unstable.
-                            InstanceDef::Item(def) => def
-                                .did
-                                .as_local()
-                                .map(|def_id| tcx.hir().local_def_id_to_hir_id(def_id)),
+                            InstanceDef::Item(def) => Some(def.did.index.as_usize()),
                             InstanceDef::VtableShim(..)
                             | InstanceDef::ReifyShim(..)
                             | InstanceDef::Intrinsic(..)
@@ -380,10 +377,10 @@ impl<'tcx> CodegenUnit<'tcx> {
                             | InstanceDef::CloneShim(..) => None,
                         }
                     }
-                    MonoItem::Static(def_id) => {
-                        def_id.as_local().map(|def_id| tcx.hir().local_def_id_to_hir_id(def_id))
+                    MonoItem::Static(def_id) => Some(def_id.index.as_usize()),
+                    MonoItem::GlobalAsm(item_id) => {
+                        Some(item_id.def_id.to_def_id().index.as_usize())
                     }
-                    MonoItem::GlobalAsm(item_id) => Some(item_id.hir_id()),
                 },
                 item.symbol_name(tcx),
             )
diff --git a/compiler/rustc_middle/src/ty/fast_reject.rs b/compiler/rustc_middle/src/ty/fast_reject.rs
index c4043d9698cc1..04011552e31ed 100644
--- a/compiler/rustc_middle/src/ty/fast_reject.rs
+++ b/compiler/rustc_middle/src/ty/fast_reject.rs
@@ -20,7 +20,7 @@ pub type SimplifiedType = SimplifiedTypeGen<DefId>;
 #[derive(Clone, Copy, Debug, PartialEq, Eq, Hash, PartialOrd, Ord, TyEncodable, TyDecodable)]
 pub enum SimplifiedTypeGen<D>
 where
-    D: Copy + Debug + Ord + Eq,
+    D: Copy + Debug + Eq,
 {
     BoolSimplifiedType,
     CharSimplifiedType,
diff --git a/compiler/rustc_mir_transform/src/inline.rs b/compiler/rustc_mir_transform/src/inline.rs
index 8be95b2d95a20..e1f30fef44f99 100644
--- a/compiler/rustc_mir_transform/src/inline.rs
+++ b/compiler/rustc_mir_transform/src/inline.rs
@@ -212,7 +212,8 @@ impl<'tcx> Inliner<'tcx> {
             // a lower `HirId` than the callee. This ensures that the callee will
             // not inline us. This trick only works without incremental compilation.
             // So don't do it if that is enabled.
-            if !self.tcx.dep_graph.is_fully_enabled() && self.hir_id < callee_hir_id {
+            if !self.tcx.dep_graph.is_fully_enabled() && self.hir_id.index() < callee_hir_id.index()
+            {
                 return Ok(());
             }
 
diff --git a/compiler/rustc_span/src/def_id.rs b/compiler/rustc_span/src/def_id.rs
index 64c2ef30b4d76..d15befbf28730 100644
--- a/compiler/rustc_span/src/def_id.rs
+++ b/compiler/rustc_span/src/def_id.rs
@@ -322,7 +322,7 @@ rustc_data_structures::define_id_collections!(DefIdMap, DefIdSet, DefId);
 /// few cases where we know that only DefIds from the local crate are expected
 /// and a DefId from a different crate would signify a bug somewhere. This
 /// is when LocalDefId comes in handy.
-#[derive(Clone, Copy, PartialEq, Eq, PartialOrd, Ord, Hash)]
+#[derive(Clone, Copy, PartialEq, Eq, Hash)]
 pub struct LocalDefId {
     pub local_def_index: DefIndex,
 }
diff --git a/compiler/rustc_span/src/lib.rs b/compiler/rustc_span/src/lib.rs
index 2934368dfeb89..3bbf2a0e45666 100644
--- a/compiler/rustc_span/src/lib.rs
+++ b/compiler/rustc_span/src/lib.rs
@@ -424,7 +424,7 @@ impl FileName {
 /// `SpanData` is public because `Span` uses a thread-local interner and can't be
 /// sent to other threads, but some pieces of performance infra run in a separate thread.
 /// Using `Span` is generally preferred.
-#[derive(Clone, Copy, Hash, PartialEq, Eq, Ord, PartialOrd)]
+#[derive(Clone, Copy, Hash, PartialEq, Eq)]
 pub struct SpanData {
     pub lo: BytePos,
     pub hi: BytePos,
@@ -434,6 +434,36 @@ pub struct SpanData {
     pub parent: Option<LocalDefId>,
 }
 
+// Order spans by position in the file.
+impl Ord for SpanData {
+    fn cmp(&self, other: &Self) -> Ordering {
+        let SpanData {
+            lo: s_lo,
+            hi: s_hi,
+            ctxt: s_ctxt,
+            // `LocalDefId` does not implement `Ord`.
+            // The other fields are enough to determine in-file order.
+            parent: _,
+        } = self;
+        let SpanData {
+            lo: o_lo,
+            hi: o_hi,
+            ctxt: o_ctxt,
+            // `LocalDefId` does not implement `Ord`.
+            // The other fields are enough to determine in-file order.
+            parent: _,
+        } = other;
+
+        (s_lo, s_hi, s_ctxt).cmp(&(o_lo, o_hi, o_ctxt))
+    }
+}
+
+impl PartialOrd for SpanData {
+    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
+        Some(self.cmp(other))
+    }
+}
+
 impl SpanData {
     #[inline]
     pub fn span(&self) -> Span {