diff --git a/milli/src/search/new/graph_based_ranking_rule.rs b/milli/src/search/new/graph_based_ranking_rule.rs
index 72c839f2a..bbe4099bc 100644
--- a/milli/src/search/new/graph_based_ranking_rule.rs
+++ b/milli/src/search/new/graph_based_ranking_rule.rs
@@ -45,7 +45,7 @@ use super::interner::MappedInterner;
 use super::logger::SearchLogger;
 use super::query_graph::QueryNode;
 use super::ranking_rule_graph::{
-    DeadEndPathCache, EdgeConditionDocIdsCache, ProximityGraph, RankingRuleGraph,
+    ConditionDocIdsCache, DeadEndPathCache, ProximityGraph, RankingRuleGraph,
     RankingRuleGraphTrait, TypoGraph,
 };
 use super::small_bitmap::SmallBitmap;
@@ -85,12 +85,12 @@ pub struct GraphBasedRankingRuleState<G: RankingRuleGraphTrait> {
     /// The current graph
     graph: RankingRuleGraph<G>,
     /// Cache to retrieve the docids associated with each edge
-    edge_conditions_cache: EdgeConditionDocIdsCache<G>,
+    conditions_cache: ConditionDocIdsCache<G>,
     /// Cache used to optimistically discard paths that resolve to no documents.
     dead_end_path_cache: DeadEndPathCache<G>,
     /// A structure giving the list of possible costs from each node to the end node,
     /// along with a set of unavoidable edges that must be traversed to achieve that distance.
-    all_distances: MappedInterner<Vec<(u16, SmallBitmap<G::EdgeCondition>)>, QueryNode>,
+    all_distances: MappedInterner<Vec<(u16, SmallBitmap<G::Condition>)>, QueryNode>,
     /// An index in the first element of `all_distances`, giving the cost of the next bucket
     cur_distance_idx: usize,
 }
@@ -101,7 +101,7 @@ pub struct GraphBasedRankingRuleState<G: RankingRuleGraphTrait> {
 fn remove_empty_edges<'ctx, G: RankingRuleGraphTrait>(
     ctx: &mut SearchContext<'ctx>,
     graph: &mut RankingRuleGraph<G>,
-    condition_docids_cache: &mut EdgeConditionDocIdsCache<G>,
+    condition_docids_cache: &mut ConditionDocIdsCache<G>,
     universe: &RoaringBitmap,
     dead_end_path_cache: &mut DeadEndPathCache<G>,
 ) -> Result<()> {
@@ -135,7 +135,7 @@ impl<'ctx, G: RankingRuleGraphTrait> RankingRule<'ctx, QueryGraph> for GraphBase
         query_graph: &QueryGraph,
     ) -> Result<()> {
         let mut graph = RankingRuleGraph::build(ctx, query_graph.clone())?;
-        let mut condition_docids_cache = EdgeConditionDocIdsCache::default();
+        let mut condition_docids_cache = ConditionDocIdsCache::default();
         let mut dead_end_path_cache = DeadEndPathCache::new(&graph.conditions_interner);
 
         // First simplify the graph as much as possible, by computing the docids of all the conditions
@@ -153,7 +153,7 @@ impl<'ctx, G: RankingRuleGraphTrait> RankingRule<'ctx, QueryGraph> for GraphBase
 
         let state = GraphBasedRankingRuleState {
             graph,
-            edge_conditions_cache: condition_docids_cache,
+            conditions_cache: condition_docids_cache,
             dead_end_path_cache,
             all_distances,
             cur_distance_idx: 0,
@@ -181,7 +181,7 @@ impl<'ctx, G: RankingRuleGraphTrait> RankingRule<'ctx, QueryGraph> for GraphBase
         remove_empty_edges(
             ctx,
             &mut state.graph,
-            &mut state.edge_conditions_cache,
+            &mut state.conditions_cache,
             universe,
             &mut state.dead_end_path_cache,
         )?;
@@ -204,7 +204,7 @@ impl<'ctx, G: RankingRuleGraphTrait> RankingRule<'ctx, QueryGraph> for GraphBase
 
         let GraphBasedRankingRuleState {
             graph,
-            edge_conditions_cache: condition_docids_cache,
+            conditions_cache: condition_docids_cache,
             dead_end_path_cache,
             all_distances,
             cur_distance_idx: _,
@@ -326,8 +326,8 @@ impl<'ctx, G: RankingRuleGraphTrait> RankingRule<'ctx, QueryGraph> for GraphBase
             let mut used_phrases = HashSet::new();
             for condition in used_conditions.iter() {
                 let condition = graph.conditions_interner.get(condition);
-                used_words.extend(G::words_used_by_edge_condition(ctx, condition)?);
-                used_phrases.extend(G::phrases_used_by_edge_condition(ctx, condition)?);
+                used_words.extend(G::words_used_by_condition(ctx, condition)?);
+                used_phrases.extend(G::phrases_used_by_condition(ctx, condition)?);
             }
             // 2. Remove the unused words and phrases from all the nodes in the graph
             let mut nodes_to_remove = vec![];
diff --git a/milli/src/search/new/logger/detailed.rs b/milli/src/search/new/logger/detailed.rs
index 96f4092b1..1ed992c56 100644
--- a/milli/src/search/new/logger/detailed.rs
+++ b/milli/src/search/new/logger/detailed.rs
@@ -11,7 +11,7 @@ use crate::search::new::query_graph::QueryNodeData;
 use crate::search::new::query_term::{LocatedQueryTerm, QueryTerm};
 use crate::search::new::ranking_rule_graph::{
     DeadEndPathCache, Edge, ProximityCondition, ProximityGraph, RankingRuleGraph,
-    RankingRuleGraphTrait, TypoEdge, TypoGraph,
+    RankingRuleGraphTrait, TypoCondition, TypoGraph,
 };
 use crate::search::new::small_bitmap::SmallBitmap;
 use crate::search::new::{QueryGraph, QueryNode, SearchContext};
@@ -51,10 +51,10 @@ pub enum SearchEvents {
     },
     TypoState {
         graph: RankingRuleGraph<TypoGraph>,
-        paths: Vec<Vec<Interned<TypoEdge>>>,
+        paths: Vec<Vec<Interned<TypoCondition>>>,
         dead_end_path_cache: DeadEndPathCache<TypoGraph>,
         universe: RoaringBitmap,
-        distances: MappedInterner<Vec<(u16, SmallBitmap<TypoEdge>)>, QueryNode>,
+        distances: MappedInterner<Vec<(u16, SmallBitmap<TypoCondition>)>, QueryNode>,
         cost: u16,
     },
     RankingRuleSkipBucket {
@@ -188,10 +188,10 @@ impl SearchLogger<QueryGraph> for DetailedSearchLogger {
     fn log_typo_state(
         &mut self,
         query_graph: &RankingRuleGraph<TypoGraph>,
-        paths_map: &[Vec<Interned<TypoEdge>>],
+        paths_map: &[Vec<Interned<TypoCondition>>],
         dead_end_path_cache: &DeadEndPathCache<TypoGraph>,
         universe: &RoaringBitmap,
-        distances: &MappedInterner<Vec<(u16, SmallBitmap<TypoEdge>)>, QueryNode>,
+        distances: &MappedInterner<Vec<(u16, SmallBitmap<TypoCondition>)>, QueryNode>,
         cost: u16,
     ) {
         self.events.push(SearchEvents::TypoState {
@@ -430,7 +430,7 @@ results.{random} {{
         ctx: &mut SearchContext,
         node_idx: Interned<QueryNode>,
         node: &QueryNode,
-        distances: &[(u16, SmallBitmap<R::EdgeCondition>)],
+        distances: &[(u16, SmallBitmap<R::Condition>)],
         file: &mut File,
     ) {
         match &node.data {
@@ -527,9 +527,9 @@ shape: class"
     fn ranking_rule_graph_d2_description<R: RankingRuleGraphTrait>(
         ctx: &mut SearchContext,
         graph: &RankingRuleGraph<R>,
-        paths: &[Vec<Interned<R::EdgeCondition>>],
+        paths: &[Vec<Interned<R::Condition>>],
         dead_end_paths_cache: &DeadEndPathCache<R>,
-        distances: MappedInterner<Vec<(u16, SmallBitmap<R::EdgeCondition>)>, QueryNode>,
+        distances: MappedInterner<Vec<(u16, SmallBitmap<R::Condition>)>, QueryNode>,
         file: &mut File,
     ) {
         writeln!(file, "direction: right").unwrap();
@@ -596,7 +596,7 @@ shape: class"
     fn condition_d2_description<R: RankingRuleGraphTrait>(
         ctx: &mut SearchContext,
         graph: &RankingRuleGraph<R>,
-        condition_id: Interned<R::EdgeCondition>,
+        condition_id: Interned<R::Condition>,
         file: &mut File,
     ) {
         let condition = graph.conditions_interner.get(condition_id);
@@ -606,14 +606,14 @@ shape: class"
 shape: class
 {}
 }}",
-            R::label_for_edge_condition(ctx, condition).unwrap()
+            R::label_for_condition(ctx, condition).unwrap()
         )
         .unwrap();
     }
     fn paths_d2_description<R: RankingRuleGraphTrait>(
         ctx: &mut SearchContext,
         graph: &RankingRuleGraph<R>,
-        paths: &[Vec<Interned<R::EdgeCondition>>],
+        paths: &[Vec<Interned<R::Condition>>],
         file: &mut File,
     ) {
         for (path_idx, condition_indexes) in paths.iter().enumerate() {
diff --git a/milli/src/search/new/logger/mod.rs b/milli/src/search/new/logger/mod.rs
index 696f65443..ba58d049f 100644
--- a/milli/src/search/new/logger/mod.rs
+++ b/milli/src/search/new/logger/mod.rs
@@ -6,7 +6,8 @@ use roaring::RoaringBitmap;
 use super::interner::{Interned, MappedInterner};
 use super::query_graph::QueryNode;
 use super::ranking_rule_graph::{
-    DeadEndPathCache, ProximityCondition, ProximityGraph, RankingRuleGraph, TypoEdge, TypoGraph,
+    DeadEndPathCache, ProximityCondition, ProximityGraph, RankingRuleGraph, TypoCondition,
+    TypoGraph,
 };
 use super::small_bitmap::SmallBitmap;
 use super::{RankingRule, RankingRuleQueryTrait};
@@ -76,10 +77,10 @@ pub trait SearchLogger<Q: RankingRuleQueryTrait> {
     fn log_typo_state(
         &mut self,
         query_graph: &RankingRuleGraph<TypoGraph>,
-        paths: &[Vec<Interned<TypoEdge>>],
+        paths: &[Vec<Interned<TypoCondition>>],
         dead_end_path_cache: &DeadEndPathCache<TypoGraph>,
         universe: &RoaringBitmap,
-        distances: &MappedInterner<Vec<(u16, SmallBitmap<TypoEdge>)>, QueryNode>,
+        distances: &MappedInterner<Vec<(u16, SmallBitmap<TypoCondition>)>, QueryNode>,
         cost: u16,
     );
 }
@@ -147,10 +148,10 @@ impl<Q: RankingRuleQueryTrait> SearchLogger<Q> for DefaultSearchLogger {
     fn log_typo_state(
         &mut self,
         _query_graph: &RankingRuleGraph<TypoGraph>,
-        _paths: &[Vec<Interned<TypoEdge>>],
+        _paths: &[Vec<Interned<TypoCondition>>],
         _dead_end_path_cache: &DeadEndPathCache<TypoGraph>,
         _universe: &RoaringBitmap,
-        _distances: &MappedInterner<Vec<(u16, SmallBitmap<TypoEdge>)>, QueryNode>,
+        _distances: &MappedInterner<Vec<(u16, SmallBitmap<TypoCondition>)>, QueryNode>,
         _cost: u16,
     ) {
     }
diff --git a/milli/src/search/new/ranking_rule_graph/cheapest_paths.rs b/milli/src/search/new/ranking_rule_graph/cheapest_paths.rs
index a86f1fcb0..bc0e88326 100644
--- a/milli/src/search/new/ranking_rule_graph/cheapest_paths.rs
+++ b/milli/src/search/new/ranking_rule_graph/cheapest_paths.rs
@@ -22,10 +22,10 @@ impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
         &mut self,
         from: Interned<QueryNode>,
         cost: u16,
-        all_distances: &MappedInterner<Vec<(u16, SmallBitmap<G::EdgeCondition>)>, QueryNode>,
+        all_distances: &MappedInterner<Vec<(u16, SmallBitmap<G::Condition>)>, QueryNode>,
         dead_end_path_cache: &mut DeadEndPathCache<G>,
         mut visit: impl FnMut(
-            &[Interned<G::EdgeCondition>],
+            &[Interned<G::Condition>],
             &mut Self,
             &mut DeadEndPathCache<G>,
         ) -> Result<ControlFlow<()>>,
@@ -46,16 +46,16 @@ impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
         &mut self,
         from: Interned<QueryNode>,
         cost: u16,
-        all_distances: &MappedInterner<Vec<(u16, SmallBitmap<G::EdgeCondition>)>, QueryNode>,
+        all_distances: &MappedInterner<Vec<(u16, SmallBitmap<G::Condition>)>, QueryNode>,
         dead_end_path_cache: &mut DeadEndPathCache<G>,
         visit: &mut impl FnMut(
-            &[Interned<G::EdgeCondition>],
+            &[Interned<G::Condition>],
             &mut Self,
             &mut DeadEndPathCache<G>,
         ) -> Result<ControlFlow<()>>,
-        prev_conditions: &mut Vec<Interned<G::EdgeCondition>>,
-        cur_path: &mut SmallBitmap<G::EdgeCondition>,
-        forbidden_conditions: &mut SmallBitmap<G::EdgeCondition>,
+        prev_conditions: &mut Vec<Interned<G::Condition>>,
+        cur_path: &mut SmallBitmap<G::Condition>,
+        forbidden_conditions: &mut SmallBitmap<G::Condition>,
     ) -> Result<bool> {
         let mut any_valid = false;
 
@@ -158,7 +158,7 @@ impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
 
     pub fn initialize_distances_with_necessary_edges(
         &self,
-    ) -> MappedInterner<Vec<(u16, SmallBitmap<G::EdgeCondition>)>, QueryNode> {
+    ) -> MappedInterner<Vec<(u16, SmallBitmap<G::Condition>)>, QueryNode> {
         let mut distances_to_end = self.query_graph.nodes.map(|_| vec![]);
         let mut enqueued = SmallBitmap::new(self.query_graph.nodes.len());
 
@@ -173,7 +173,7 @@ impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
         }
 
         while let Some(cur_node) = node_stack.pop_front() {
-            let mut self_distances = BTreeMap::<u16, SmallBitmap<G::EdgeCondition>>::new();
+            let mut self_distances = BTreeMap::<u16, SmallBitmap<G::Condition>>::new();
 
             let cur_node_edges = &self.edges_of_node.get(cur_node);
             for edge_idx in cur_node_edges.iter() {
diff --git a/milli/src/search/new/ranking_rule_graph/mod.rs b/milli/src/search/new/ranking_rule_graph/mod.rs
index fcb9ade3f..02a68b811 100644
--- a/milli/src/search/new/ranking_rule_graph/mod.rs
+++ b/milli/src/search/new/ranking_rule_graph/mod.rs
@@ -19,11 +19,11 @@ mod typo;
 use std::collections::HashSet;
 use std::hash::Hash;
 
-pub use condition_docids_cache::EdgeConditionDocIdsCache;
+pub use condition_docids_cache::ConditionDocIdsCache;
 pub use dead_end_path_cache::DeadEndPathCache;
 pub use proximity::{ProximityCondition, ProximityGraph};
 use roaring::RoaringBitmap;
-pub use typo::{TypoEdge, TypoGraph};
+pub use typo::{TypoCondition, TypoGraph};
 
 use super::interner::{DedupInterner, FixedSizeInterner, Interned, MappedInterner};
 use super::logger::SearchLogger;
@@ -74,48 +74,48 @@ impl<E> PartialEq for Edge<E> {
 pub trait RankingRuleGraphTrait: Sized {
     /// The condition of an edge connecting two query nodes. The condition
     /// should be sufficient to compute the edge's cost and associated document ids
-    /// in [`resolve_edge_condition`](RankingRuleGraphTrait::resolve_edge_condition).
-    type EdgeCondition: Sized + Clone + PartialEq + Eq + Hash;
+    /// in [`resolve_condition`](RankingRuleGraphTrait::resolve_condition).
+    type Condition: Sized + Clone + PartialEq + Eq + Hash;
 
     /// Return the label of the given edge condition, to be used when visualising
     /// the ranking rule graph.
-    fn label_for_edge_condition<'ctx>(
+    fn label_for_condition<'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        edge: &Self::EdgeCondition,
+        condition: &Self::Condition,
     ) -> Result<String>;
 
-    fn words_used_by_edge_condition<'ctx>(
+    fn words_used_by_condition<'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        edge: &Self::EdgeCondition,
+        condition: &Self::Condition,
     ) -> Result<HashSet<Interned<String>>>;
-    fn phrases_used_by_edge_condition<'ctx>(
+
+    fn phrases_used_by_condition<'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        edge: &Self::EdgeCondition,
+        condition: &Self::Condition,
     ) -> Result<HashSet<Interned<Phrase>>>;
 
     /// Compute the document ids associated with the given edge condition,
     /// restricted to the given universe.
-    fn resolve_edge_condition<'ctx>(
+    fn resolve_condition<'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        edge_condition: &Self::EdgeCondition,
+        condition: &Self::Condition,
         universe: &RoaringBitmap,
     ) -> Result<RoaringBitmap>;
 
-    /// Return the cost and condition of the edges going from the previously visited node
-    /// (with [`build_step_visit_source_node`](RankingRuleGraphTrait::build_step_visit_source_node)) to `dest_node`.
+    /// Return the costs and conditions of the edges going from the source node to the destination node
     fn build_edges<'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        conditions_interner: &mut DedupInterner<Self::EdgeCondition>,
+        conditions_interner: &mut DedupInterner<Self::Condition>,
         source_node: &QueryNode,
         dest_node: &QueryNode,
-    ) -> Result<Vec<(u8, Option<Interned<Self::EdgeCondition>>)>>;
+    ) -> Result<Vec<(u8, Option<Interned<Self::Condition>>)>>;
 
     fn log_state(
         graph: &RankingRuleGraph<Self>,
-        paths: &[Vec<Interned<Self::EdgeCondition>>],
+        paths: &[Vec<Interned<Self::Condition>>],
         dead_end_path_cache: &DeadEndPathCache<Self>,
         universe: &RoaringBitmap,
-        distances: &MappedInterner<Vec<(u16, SmallBitmap<Self::EdgeCondition>)>, QueryNode>,
+        distances: &MappedInterner<Vec<(u16, SmallBitmap<Self::Condition>)>, QueryNode>,
         cost: u16,
         logger: &mut dyn SearchLogger<QueryGraph>,
     );
@@ -127,9 +127,9 @@ pub trait RankingRuleGraphTrait: Sized {
 /// but replacing the edges.
 pub struct RankingRuleGraph<G: RankingRuleGraphTrait> {
     pub query_graph: QueryGraph,
-    pub edges_store: FixedSizeInterner<Option<Edge<G::EdgeCondition>>>,
-    pub edges_of_node: MappedInterner<SmallBitmap<Option<Edge<G::EdgeCondition>>>, QueryNode>,
-    pub conditions_interner: FixedSizeInterner<G::EdgeCondition>,
+    pub edges_store: FixedSizeInterner<Option<Edge<G::Condition>>>,
+    pub edges_of_node: MappedInterner<SmallBitmap<Option<Edge<G::Condition>>>, QueryNode>,
+    pub conditions_interner: FixedSizeInterner<G::Condition>,
 }
 impl<G: RankingRuleGraphTrait> Clone for RankingRuleGraph<G> {
     fn clone(&self) -> Self {
@@ -143,7 +143,7 @@ impl<G: RankingRuleGraphTrait> Clone for RankingRuleGraph<G> {
 }
 impl<G: RankingRuleGraphTrait> RankingRuleGraph<G> {
     /// Remove all edges with the given condition
-    pub fn remove_edges_with_condition(&mut self, condition_to_remove: Interned<G::EdgeCondition>) {
+    pub fn remove_edges_with_condition(&mut self, condition_to_remove: Interned<G::Condition>) {
         for (edge_id, edge_opt) in self.edges_store.iter_mut() {
             let Some(edge) = edge_opt.as_mut() else { continue };
             let Some(condition) = edge.condition else { continue };
diff --git a/milli/src/search/new/ranking_rule_graph/path_set.rs b/milli/src/search/new/ranking_rule_graph/path_set.rs
index 19c7d012e..1a87c9604 100644
--- a/milli/src/search/new/ranking_rule_graph/path_set.rs
+++ b/milli/src/search/new/ranking_rule_graph/path_set.rs
@@ -4,7 +4,7 @@
 
 use crate::search::new::interner::Interned;
 
-/// A set of `Vec<Interned<T>>`.
+/// A set of `Vec<Interned<T>>` implemented as a prefix tree.
 pub struct PathSet<T> {
     nodes: Vec<(Interned<T>, Self)>,
     is_end: bool,
diff --git a/milli/src/search/new/ranking_rule_graph/proximity/compute_docids.rs b/milli/src/search/new/ranking_rule_graph/proximity/compute_docids.rs
index cdf167cb0..bf5278f8d 100644
--- a/milli/src/search/new/ranking_rule_graph/proximity/compute_docids.rs
+++ b/milli/src/search/new/ranking_rule_graph/proximity/compute_docids.rs
@@ -6,7 +6,7 @@ use crate::{CboRoaringBitmapCodec, Result};
 
 pub fn compute_docids<'ctx>(
     ctx: &mut SearchContext<'ctx>,
-    edge: &ProximityCondition,
+    condition: &ProximityCondition,
     universe: &RoaringBitmap,
 ) -> Result<RoaringBitmap> {
     let SearchContext {
@@ -18,7 +18,7 @@ pub fn compute_docids<'ctx>(
         phrase_interner,
         term_interner,
     } = ctx;
-    let pairs = match edge {
+    let pairs = match condition {
         ProximityCondition::Term { term } => {
             return term_docids
                 .get_query_term_docids(
diff --git a/milli/src/search/new/ranking_rule_graph/proximity/mod.rs b/milli/src/search/new/ranking_rule_graph/proximity/mod.rs
index d5655e949..9a6080301 100644
--- a/milli/src/search/new/ranking_rule_graph/proximity/mod.rs
+++ b/milli/src/search/new/ranking_rule_graph/proximity/mod.rs
@@ -45,11 +45,11 @@ pub enum ProximityCondition {
 pub enum ProximityGraph {}
 
 impl RankingRuleGraphTrait for ProximityGraph {
-    type EdgeCondition = ProximityCondition;
+    type Condition = ProximityCondition;
 
-    fn resolve_edge_condition<'ctx>(
+    fn resolve_condition<'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        condition: &Self::EdgeCondition,
+        condition: &Self::Condition,
         universe: &RoaringBitmap,
     ) -> Result<roaring::RoaringBitmap> {
         compute_docids::compute_docids(ctx, condition, universe)
@@ -57,10 +57,10 @@ impl RankingRuleGraphTrait for ProximityGraph {
 
     fn build_edges<'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        conditions_interner: &mut DedupInterner<Self::EdgeCondition>,
+        conditions_interner: &mut DedupInterner<Self::Condition>,
         source_node: &QueryNode,
         dest_node: &QueryNode,
-    ) -> Result<Vec<(u8, Option<Interned<Self::EdgeCondition>>)>> {
+    ) -> Result<Vec<(u8, Option<Interned<Self::Condition>>)>> {
         build::build_edges(ctx, conditions_interner, source_node, dest_node)
     }
 
@@ -76,11 +76,11 @@ impl RankingRuleGraphTrait for ProximityGraph {
         logger.log_proximity_state(graph, paths, dead_end_path_cache, universe, distances, cost);
     }
 
-    fn label_for_edge_condition<'ctx>(
+    fn label_for_condition<'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        edge: &Self::EdgeCondition,
+        condition: &Self::Condition,
     ) -> Result<String> {
-        match edge {
+        match condition {
             ProximityCondition::Term { term } => {
                 let term = ctx.term_interner.get(*term);
                 Ok(format!("{} : exists", ctx.word_interner.get(term.original)))
@@ -117,11 +117,11 @@ impl RankingRuleGraphTrait for ProximityGraph {
         }
     }
 
-    fn words_used_by_edge_condition<'ctx>(
+    fn words_used_by_condition<'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        edge: &Self::EdgeCondition,
+        condition: &Self::Condition,
     ) -> Result<HashSet<Interned<String>>> {
-        match edge {
+        match condition {
             ProximityCondition::Term { term } => {
                 let term = ctx.term_interner.get(*term);
                 Ok(HashSet::from_iter(term.all_single_words_except_prefix_db()))
@@ -153,11 +153,11 @@ impl RankingRuleGraphTrait for ProximityGraph {
         }
     }
 
-    fn phrases_used_by_edge_condition<'ctx>(
+    fn phrases_used_by_condition<'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        edge: &Self::EdgeCondition,
+        condition: &Self::Condition,
     ) -> Result<HashSet<Interned<Phrase>>> {
-        match edge {
+        match condition {
             ProximityCondition::Term { term } => {
                 let term = ctx.term_interner.get(*term);
                 Ok(HashSet::from_iter(term.all_phrases()))
diff --git a/milli/src/search/new/ranking_rule_graph/typo/mod.rs b/milli/src/search/new/ranking_rule_graph/typo/mod.rs
index b41787ac2..0b81ec0ec 100644
--- a/milli/src/search/new/ranking_rule_graph/typo/mod.rs
+++ b/milli/src/search/new/ranking_rule_graph/typo/mod.rs
@@ -14,19 +14,18 @@ use std::fmt::Write;
 use std::iter::FromIterator;
 
 #[derive(Clone, PartialEq, Eq, Hash)]
-pub struct TypoEdge {
+pub struct TypoCondition {
     term: Interned<QueryTerm>,
-    nbr_typos: u8,
 }
 
 pub enum TypoGraph {}
 
 impl RankingRuleGraphTrait for TypoGraph {
-    type EdgeCondition = TypoEdge;
+    type Condition = TypoCondition;
 
-    fn resolve_edge_condition<'db_cache, 'ctx>(
+    fn resolve_condition<'db_cache, 'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        edge: &Self::EdgeCondition,
+        condition: &Self::Condition,
         universe: &RoaringBitmap,
     ) -> Result<RoaringBitmap> {
         let SearchContext {
@@ -47,7 +46,7 @@ impl RankingRuleGraphTrait for TypoGraph {
                 word_interner,
                 term_interner,
                 phrase_interner,
-                edge.term,
+                condition.term,
             )?;
 
         Ok(docids)
@@ -55,10 +54,10 @@ impl RankingRuleGraphTrait for TypoGraph {
 
     fn build_edges<'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        conditions_interner: &mut DedupInterner<Self::EdgeCondition>,
+        conditions_interner: &mut DedupInterner<Self::Condition>,
         _from_node: &QueryNode,
         to_node: &QueryNode,
-    ) -> Result<Vec<(u8, Option<Interned<Self::EdgeCondition>>)>> {
+    ) -> Result<Vec<(u8, Option<Interned<Self::Condition>>)>> {
         let SearchContext { term_interner, .. } = ctx;
         match &to_node.data {
             QueryNodeData::Term(LocatedQueryTerm { value, positions }) => {
@@ -121,10 +120,10 @@ impl RankingRuleGraphTrait for TypoGraph {
                     if !new_term.is_empty() {
                         edges.push((
                             nbr_typos as u8 + base_cost,
-                            Some(conditions_interner.insert(TypoEdge {
-                                term: term_interner.insert(new_term),
-                                nbr_typos: nbr_typos as u8,
-                            })),
+                            Some(
+                                conditions_interner
+                                    .insert(TypoCondition { term: term_interner.insert(new_term) }),
+                            ),
                         ))
                     }
                 }
@@ -137,21 +136,21 @@ impl RankingRuleGraphTrait for TypoGraph {
 
     fn log_state(
         graph: &RankingRuleGraph<Self>,
-        paths: &[Vec<Interned<TypoEdge>>],
+        paths: &[Vec<Interned<TypoCondition>>],
         dead_end_path_cache: &DeadEndPathCache<Self>,
         universe: &RoaringBitmap,
-        distances: &MappedInterner<Vec<(u16, SmallBitmap<TypoEdge>)>, QueryNode>,
+        distances: &MappedInterner<Vec<(u16, SmallBitmap<TypoCondition>)>, QueryNode>,
         cost: u16,
         logger: &mut dyn SearchLogger<QueryGraph>,
     ) {
         logger.log_typo_state(graph, paths, dead_end_path_cache, universe, distances, cost);
     }
 
-    fn label_for_edge_condition<'ctx>(
+    fn label_for_condition<'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        edge: &Self::EdgeCondition,
+        condition: &Self::Condition,
     ) -> Result<String> {
-        let TypoEdge { term, nbr_typos: _ } = edge;
+        let TypoCondition { term } = condition;
         let term = ctx.term_interner.get(*term);
         let QueryTerm {
             original: _,
@@ -203,20 +202,20 @@ impl RankingRuleGraphTrait for TypoGraph {
         Ok(s)
     }
 
-    fn words_used_by_edge_condition<'ctx>(
+    fn words_used_by_condition<'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        edge: &Self::EdgeCondition,
+        condition: &Self::Condition,
     ) -> Result<HashSet<Interned<String>>> {
-        let TypoEdge { term, .. } = edge;
+        let TypoCondition { term, .. } = condition;
         let term = ctx.term_interner.get(*term);
         Ok(HashSet::from_iter(term.all_single_words_except_prefix_db()))
     }
 
-    fn phrases_used_by_edge_condition<'ctx>(
+    fn phrases_used_by_condition<'ctx>(
         ctx: &mut SearchContext<'ctx>,
-        edge: &Self::EdgeCondition,
+        condition: &Self::Condition,
     ) -> Result<HashSet<Interned<Phrase>>> {
-        let TypoEdge { term, .. } = edge;
+        let TypoCondition { term, .. } = condition;
         let term = ctx.term_interner.get(*term);
         Ok(HashSet::from_iter(term.all_phrases()))
     }