From fbaa420ed550e9c3e7cdc09d4a8ec22bfbd782a6 Mon Sep 17 00:00:00 2001
From: JSDurand <mmemmew@gmail.com>
Date: Mon, 27 Feb 2023 12:36:41 +0800
Subject: before a major refactor

I decide to adopt a new approach of recording and updating item
derivation forests.  Since this affects a lot of things, I decide to
commit before the refactor, so that I can create a branch for that
refactor.
---
 chain/src/item/default/mod.rs    | 469 ++++++++++++++++++++++++++++++++++---
 chain/src/item/default/splone.rs | 187 ++++++++++++---
 chain/src/item/forest-format.org |  10 +
 chain/src/item/genins.rs         | 494 +++++++++++++++++++++++++++++----------
 chain/src/item/mod.rs            |  85 +++++--
 5 files changed, 1039 insertions(+), 206 deletions(-)
 create mode 100644 chain/src/item/forest-format.org

(limited to 'chain/src/item')

diff --git a/chain/src/item/default/mod.rs b/chain/src/item/default/mod.rs
index 1a00368..22069d2 100644
--- a/chain/src/item/default/mod.rs
+++ b/chain/src/item/default/mod.rs
@@ -3,11 +3,13 @@
 
 use super::*;
 use crate::atom::default::DefaultAtom;
-use grammar::{GrammarLabel, GrammarLabelType};
+use grammar::{GrammarLabel, GrammarLabelType, TNT};
 use graph::{
     builder::BuilderMut, labelled::binary::PLGBuilderMut, Graph, LabelGraph, PLGraph, RedirectGraph,
 };
 
+use std::collections::HashSet;
+
 use core::fmt::Display;
 
 /// The type of errors for forest operations.
@@ -30,6 +32,8 @@ pub enum Error {
     LabelConversion(ForestLabelError),
     /// Trying to split a packed node.
     SplitPack(usize),
+    /// A cloned node should have exactly one parent.
+    InvalidClone(usize),
 }
 
 impl Display for Error {
@@ -43,6 +47,9 @@ impl Display for Error {
             Error::NodeNoLabel(n) => write!(f, "node {n} has no labels, but it should have one"),
             Error::LabelConversion(le) => write!(f, "fail to convert labels: {le}"),
             Error::SplitPack(n) => write!(f, "cannot split the packed node {n}"),
+            Error::InvalidClone(n) => {
+                write!(f, "the cloned node {n} should have exactly one parent")
+            }
         }
     }
 }
@@ -68,7 +75,7 @@ impl From<ForestLabelError> for Error {
 /// A default implementation of forest.
 #[derive(Debug, Clone)]
 pub struct DefaultForest<T: GraphLabel> {
-    graph: PLGraph<T>,
+    pub(crate) graph: PLGraph<T>,
     root: Option<usize>,
 }
 
@@ -262,15 +269,14 @@ impl<T: GraphLabel> Forest<T> for DefaultForest<ForestLabel<T>> {
         // the consideration).
 
         let fragment = fragment.borrow();
+        let fragment_nodes_len = fragment.nodes_len();
 
-        let mut frag_stack = Vec::with_capacity(fragment.nodes_len());
+        let mut frag_stack = Vec::with_capacity(fragment_nodes_len);
 
-        let mut self_stack = Vec::with_capacity(fragment.nodes_len());
+        let mut self_stack = Vec::with_capacity(fragment_nodes_len);
 
-        use std::collections::HashSet;
-
-        let mut frag_seen: HashSet<usize> = HashSet::with_capacity(fragment.nodes_len());
-        let mut self_seen: HashSet<usize> = HashSet::with_capacity(fragment.nodes_len());
+        let mut frag_seen: HashSet<usize> = HashSet::with_capacity(fragment_nodes_len);
+        let mut self_seen: HashSet<usize> = HashSet::with_capacity(fragment_nodes_len);
 
         let frag_root = if let Some(root) = fragment.root() {
             root
@@ -294,25 +300,25 @@ impl<T: GraphLabel> Forest<T> for DefaultForest<ForestLabel<T>> {
 
             if fragment.vertex_label(frag_top)? != self.vertex_label(self_top)? {
                 // not a prefix
-                dbg!(
-                    frag_top,
-                    self_top,
-                    fragment.vertex_label(frag_top)?,
-                    self.vertex_label(self_top)?
-                );
+                // dbg!(
+                //     frag_top,
+                //     self_top,
+                //     fragment.vertex_label(frag_top)?,
+                //     self.vertex_label(self_top)?
+                // );
                 return Ok(false);
             }
 
             let self_children = self.children_of(self_top)?;
             let frag_children = fragment.children_of(frag_top)?;
 
-            if frag_children.len() != self_children.len() {
-                dbg!(
-                    frag_top,
-                    self_top,
-                    fragment.vertex_label(frag_top)?,
-                    self.vertex_label(self_top)?
-                );
+            if frag_children.len() > self_children.len() {
+                // dbg!(
+                //     frag_top,
+                //     self_top,
+                //     fragment.vertex_label(frag_top)?,
+                //     self.vertex_label(self_top)?
+                // );
                 return Ok(false);
             }
 
@@ -320,7 +326,7 @@ impl<T: GraphLabel> Forest<T> for DefaultForest<ForestLabel<T>> {
                 if frag_seen.contains(&frag_child) && self_seen.contains(&self_child) {
                     continue;
                 } else if frag_seen.contains(&frag_child) || self_seen.contains(&self_child) {
-                    dbg!();
+                    // dbg!();
                     return Ok(false);
                 }
 
@@ -352,7 +358,7 @@ impl<T: GraphLabel> Forest<T> for DefaultForest<ForestLabel<T>> {
         let root = if let Some(root) = fragment.root() {
             root
         } else {
-            // Nothing to do to plant an empty forest.
+            // Nothing to plant.
             return Ok(());
         };
 
@@ -393,9 +399,27 @@ impl<T: GraphLabel> Forest<T> for DefaultForest<ForestLabel<T>> {
             return Ok(());
         }
 
-        // First adjoin those nodes and join the edges later.
+        // First adjoin the relevant nodes and join the edges later.
+
+        let mut used_nodes: Vec<bool> = std::iter::repeat(false).take(nodes_len).collect();
+
+        let mut stack = vec![root];
+
+        while let Some(top) = stack.pop() {
+            if used_nodes.get(top).copied() == Some(true) {
+                continue;
+            }
+
+            *used_nodes
+                .get_mut(top)
+                .ok_or(Error::IndexOutOfBounds(top, nodes_len))? = true;
+
+            stack.extend(fragment.children_of(top)?);
+        }
+
+        let used_nodes = used_nodes;
 
-        for node in 0..nodes_len {
+        for node in (0..nodes_len).filter(|node| used_nodes.get(*node).copied() == Some(true)) {
             let label = fragment
                 .vertex_label(node)?
                 .ok_or(Error::NodeNoLabel(node))?;
@@ -582,6 +606,93 @@ impl<T: GraphLabel> PartialEq for DefaultForest<ForestLabel<T>> {
     }
 }
 
+impl<T: GraphLabel> Eq for DefaultForest<ForestLabel<T>> {}
+
+impl<T: GraphLabel> DefaultForest<ForestLabel<T>> {
+    /// A convenient helper function to plant a fragment under a
+    /// node, if it has not been planted yet.
+    ///
+    /// To be precise, this function first checks if the node is
+    /// packed; if so, then it checks every of the cloned children, to
+    /// see if the fragment has already been planted.  If none of the
+    /// clones have the fragment as a prefix, we make a new clone and
+    /// plant the fragment there.
+    ///
+    /// On the other hand, if the node is a plain node, then it checks
+    /// if the fragment is a prefix of the node.  If so, do nothing,
+    /// else clone the node and plant the fragment there, unless the
+    /// node has no children yet, in which case just plant the node
+    /// there.
+    ///
+    /// A special case occurs when the node is the root node, in which
+    /// case we clone it only when its degree is larger than one.
+    ///
+    /// Return either the original node or the cloned node at the end.
+    pub(crate) fn plant_if_needed(
+        &mut self,
+        node: usize,
+        mut fragment: Self,
+    ) -> Result<usize, Error> {
+        if fragment.is_empty() {
+            return Ok(node);
+        }
+
+        let node_label = self.vertex_label(node)?.ok_or(Error::NodeNoLabel(node))?;
+
+        if node_label.is_packed() || node_label.clone_index().is_some() {
+            let mut planted = false;
+
+            let mut node = node;
+
+            if node_label.clone_index().is_some() {
+                let mut parents = self.parents_of(node)?;
+
+                assert_eq!(parents.len(), 1);
+
+                node = parents.next().unwrap().node();
+            }
+
+            for clone in self.children_of(node)? {
+                node = clone;
+
+                if self.is_prefix(clone, &fragment)? {
+                    planted = true;
+
+                    break;
+                }
+            }
+
+            if !planted {
+                let clone = self.clone_node(node, 0, false)?;
+
+                fragment.set_root(1)?;
+
+                self.plant(clone, fragment, false)?;
+
+                return Ok(clone);
+            }
+        } else {
+            let clone_threshold = if self.root == Some(node) { 1 } else { 0 };
+
+            let planted = self.is_prefix(node, &fragment)?;
+
+            fragment.set_root(1)?;
+
+            if !planted && self.degree(node)? > clone_threshold {
+                let clone = self.clone_node(node, 0, false)?;
+
+                self.plant(clone, fragment, false)?;
+
+                return Ok(clone);
+            } else if !planted {
+                self.plant(node, fragment, false)?;
+            }
+        }
+
+        Ok(node)
+    }
+}
+
 pub mod splone;
 
 impl<T: GraphLabel> DefaultForest<T> {
@@ -596,9 +707,18 @@ impl<T: GraphLabel> DefaultForest<T> {
 
         Self { graph, root }
     }
-}
 
-impl<T: GraphLabel> Eq for DefaultForest<ForestLabel<T>> {}
+    /// Set the root to the given node.
+    pub fn set_root(&mut self, root: usize) -> Result<(), Error> {
+        if root >= self.nodes_len() {
+            return Err(Error::IndexOutOfBounds(root, self.nodes_len()));
+        }
+
+        self.root = Some(root);
+
+        Ok(())
+    }
+}
 
 impl DefaultForest<ForestLabel<GrammarLabel>> {
     /// Prints the forest without nodes that do not have ending
@@ -613,6 +733,8 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
                 Ok(label) => {
                     if let Some(label) = label {
                         label.label().end().is_none()
+                            || (matches!(self.degree(*node), Ok(0))
+                                && matches!(self.parents_of(*node).map(|iter| iter.len()), Ok(0)))
                     } else {
                         true
                     }
@@ -674,6 +796,299 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
 
         file.write_all(result.as_bytes())
     }
+
+    /// Remove a node by removing all edges connecting with it.
+    pub fn remove_node(&mut self, node_id: usize) -> Result<(), Error> {
+        let mut builder = PLGBuilderMut::from_graph_mut(&mut self.graph);
+
+        let mut to_remove =
+            Vec::with_capacity(builder.degree(node_id)? + builder.parents_of(node_id)?.len());
+
+        to_remove.extend(
+            builder
+                .children_of(node_id)?
+                .map(|target| (node_id, target)),
+        );
+
+        to_remove.extend(
+            builder
+                .parents_of(node_id)?
+                .map(|parent| (parent.node(), node_id)),
+        );
+
+        for (source, target) in to_remove {
+            builder.remove_edge(source, target, |_| true)?;
+        }
+
+        Ok(())
+    }
+
+    /// Find the last child of the given node whose start and end
+    /// positions contain the given position.  If no such child is
+    /// found, return `Ok(None)`.
+    ///
+    /// The returned tuple is of the form (child, index), where
+    /// `child` is the index of the child node in question, and
+    /// `index` means that the child is the `index`-th child of the
+    /// node.
+    pub(crate) fn position_search(
+        &self,
+        node: usize,
+        pos: usize,
+    ) -> Result<Option<(usize, usize)>, Error> {
+        fn range_contains(label: GrammarLabel, pos: usize) -> bool {
+            let start = label.start();
+
+            if let Some(end) = label.end() {
+                (start..end).contains(&pos)
+            } else {
+                (start..).contains(&pos)
+            }
+        }
+
+        let node_label = self
+            .vertex_label(node)?
+            .ok_or(Error::NodeNoLabel(node))?
+            .label();
+
+        if !range_contains(node_label, pos) {
+            return Ok(None);
+        }
+
+        for (index, child) in self.children_of(node)?.enumerate().rev() {
+            let child_label = self
+                .vertex_label(child)?
+                .ok_or(Error::NodeNoLabel(child))?
+                .label();
+
+            if range_contains(child_label, pos) {
+                return Ok(Some((child, index)));
+            }
+        }
+
+        Ok(None)
+    }
+
+    // /// Check if every child already has an end.
+    // fn every_child_is_completed(&self, node_id: usize, atom: &DefaultAtom) -> Result<bool, Error> {
+    //     let children = self.children_of(node_id)?;
+
+    //     if children.len() == 0 {
+    //         return Ok(true);
+    //     }
+
+    //     let mut pos = self
+    //         .vertex_label(node_id)?
+    //         .ok_or(Error::NodeNoLabel(node_id))?
+    //         .label()
+    //         .start();
+
+    //     let mut last_child_label = None;
+
+    //     for child in children {
+    //         let child_label = self
+    //             .vertex_label(child)?
+    //             .ok_or(Error::NodeNoLabel(child))?
+    //             .label();
+
+    //         last_child_label = Some(child_label);
+
+    //         if child_label.start() != pos || child_label.end().is_none() {
+    //             return Ok(false);
+    //         }
+
+    //         pos = child_label.end().unwrap();
+    //     }
+
+    //     if let Some(label) = last_child_label {
+    //         if let Some(rule) = label.label().rule() {
+    //             if !atom
+    //                 .is_accepting(2 * rule + 1)
+    //                 .map_err(|_| Error::IndexOutOfBounds(2 * rule + 1, atom.nodes_len()))?
+    //             {
+    //                 return Ok(false);
+    //             }
+    //         }
+    //     }
+
+    //     Ok(true)
+    // }
+
+    // /// Complete the forest by trying to set the ending position of
+    // /// every node that does not have an end, by the ending position
+    // /// of its last child.
+    // pub fn complete(&mut self, atom: &DefaultAtom) -> Result<(), Error> {
+    //     let mut stack: Vec<_> = self
+    //         .nodes()
+    //         .filter(|node| {
+    //             let label = self.vertex_label(*node).unwrap().unwrap().label();
+
+    //             label.label().rule().is_some() && label.end().is_some()
+    //         })
+    //         .collect();
+
+    //     let mut second_stack: Vec<usize> = Vec::new();
+
+    //     let mut pending_candidates: Vec<usize> = Vec::new();
+
+    //     let nodes_len = self.nodes_len();
+
+    //     let mut seen_nodes: HashSet<usize> = HashSet::with_capacity(nodes_len);
+
+    //     while !stack.is_empty() {
+    //         while let Some(mut top) = stack.pop() {
+    //             if seen_nodes.contains(&top) {
+    //                 continue;
+    //             }
+
+    //             seen_nodes.insert(top);
+
+    //             let top_label = self.vertex_label(top)?.unwrap();
+
+    //             if top_label.clone_index().is_some() {
+    //                 let mut top_parents = self.parents_of(top)?;
+
+    //                 if top_parents.len() != 1 {
+    //                     return Err(Error::InvalidClone(top));
+    //                 }
+
+    //                 top = top_parents.next().unwrap().node();
+    //             }
+
+    //             let rule_parents = self.parents_of(top)?;
+
+    //             let candidates = {
+    //                 let mut result = Vec::with_capacity(rule_parents.len());
+
+    //                 for parent in rule_parents {
+    //                     // for parent in self.parents_of(parent.node())? {
+    //                     // if self.degree(parent.node())? <= parent.edge() + 1 {
+    //                     result.push(parent);
+    //                     // }
+    //                     // }
+    //                 }
+
+    //                 result
+    //             };
+
+    //             for candidate in candidates {
+    //                 if matches!(self.vertex_label(candidate.node())?, Some(label) if label.label().end().is_none())
+    //                 {
+    //                     if self.every_child_is_completed(candidate.node(), atom)? {
+    //                         let last_child = self
+    //                             .nth_child(candidate.node(), self.degree(candidate.node())? - 1)?;
+    //                         let end = self
+    //                             .vertex_label(last_child)?
+    //                             .ok_or(Error::NodeNoLabel(last_child))?
+    //                             .label()
+    //                             .end();
+
+    //                         let sploned_node = self.splone(
+    //                             candidate.node(),
+    //                             end,
+    //                             self.degree(candidate.node())? - 1,
+    //                             true,
+    //                         )?;
+
+    //                         second_stack.push(sploned_node);
+    //                     } else {
+    //                         pending_candidates.push(candidate.node());
+    //                     }
+    //                 } else {
+    //                     second_stack.push(candidate.node());
+    //                 }
+    //             }
+
+    //             let mut new_pending = Vec::with_capacity(pending_candidates.len());
+
+    //             while let Some(node) = pending_candidates.pop() {
+    //                 if self.every_child_is_completed(node, atom)? {
+    //                     let last_edge = self.degree(node)? - 1;
+    //                     let last_child = self.nth_child(node, last_edge)?;
+    //                     let end = self
+    //                         .vertex_label(last_child)?
+    //                         .ok_or(Error::NodeNoLabel(last_child))?
+    //                         .label()
+    //                         .end();
+
+    //                     let sploned_node = self.splone(node, end, last_edge, true)?;
+
+    //                     second_stack.push(sploned_node);
+    //                 } else {
+    //                     new_pending.push(node);
+    //                 }
+    //             }
+
+    //             std::mem::swap(&mut pending_candidates, &mut new_pending);
+    //         }
+
+    //         std::mem::swap(&mut stack, &mut second_stack);
+    //     }
+
+    //     Ok(())
+    // }
+
+    // /// Unconditionally set the label of the node to be the new label.
+    // ///
+    // /// # Warning
+    // ///
+    // /// Use with caution: it does not do anything special, so it is
+    // /// the responsibility of the caller to ensure this operation is
+    // /// legal.
+    // #[allow(dead_code)]
+    // pub(crate) fn set_label(&mut self, node: usize, label: GrammarLabel) -> Result<(), Error> {
+    //     let status = self
+    //         .vertex_label(node)?
+    //         .ok_or(Error::NodeNoLabel(node))?
+    //         .status;
+
+    //     let label = ForestLabel::new(label, status);
+
+    //     let mut builder = PLGBuilderMut::from_graph_mut(&mut self.graph);
+
+    //     builder.set_label(node, label)?;
+
+    //     Ok(())
+    // }
+
+    /// Set the position of every node.
+    pub(crate) fn set_pos(&mut self, pos: usize) -> Result<(), Error> {
+        let mut builder = PLGBuilderMut::from_graph_mut(&mut self.graph);
+
+        for node in 0..builder.nodes_len() {
+            let label = builder
+                .vertex_label(node)?
+                .ok_or(Error::NodeNoLabel(node))?;
+
+            let mut label_label = label.label();
+
+            label_label.set_start(pos);
+
+            if matches!(label_label.label().tnt(), Some(TNT::Ter(_))) {
+                label_label.set_end(pos + 1);
+            } else if builder.degree(node)? == 1 && label_label.label().rule().is_some() {
+                let child = builder.children_of(node)?.next().unwrap();
+
+                if matches!(
+                    builder
+                        .vertex_label(child)?
+                        .ok_or(Error::NodeNoLabel(child))?
+                        .label()
+                        .label()
+                        .tnt(),
+                    Some(TNT::Ter(_))
+                ) {
+                    label_label.set_end(pos + 1);
+                }
+            }
+
+            let new_label = ForestLabel::new(label_label, label.status);
+
+            builder.set_label(node, new_label)?;
+        }
+
+        Ok(())
+    }
 }
 
 /// Print the labels found in the forest, so that we can easily
diff --git a/chain/src/item/default/splone.rs b/chain/src/item/default/splone.rs
index 851968c..237b92a 100644
--- a/chain/src/item/default/splone.rs
+++ b/chain/src/item/default/splone.rs
@@ -1,4 +1,3 @@
-#![allow(dead_code)]
 //! This module implements a function for "closing" and "splitting" a
 //! node in a forest, and hence the name.
 
@@ -20,6 +19,7 @@ fn get_rule_label(label: GrammarLabel) -> Option<usize> {
 }
 
 /// Existing or non-existing label.
+#[derive(Debug, Copy, Clone)]
 enum Eon {
     Ex(usize),
     Nex(ForestLabel<GrammarLabel>),
@@ -62,6 +62,12 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
     /// [`split_node`][DefaultForest::<ForestLabel<GrammarLabel>>::split_node]
     /// for details.
     ///
+    /// # `completingp`
+    ///
+    /// When we are completing the forest at the end, we do not wish
+    /// to keep the nodes at the end, so we pass a flag to inform the
+    /// function of this intention.
+    ///
     /// # Return
     ///
     /// The function returns the new, splitted or cloned node, or the
@@ -90,6 +96,7 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
         node: usize,
         end: Option<usize>,
         edge_index: usize,
+        completingp: bool,
     ) -> Result<usize, Error> {
         let node_label = self.vertex_label(node)?.ok_or(Error::NodeNoLabel(node))?;
 
@@ -130,7 +137,7 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
             || node_label.clone_index().is_some()
             || new_label.clone_index().is_some()
         {
-            return self.split_node(new_label, node, edge_index);
+            return self.split_node(new_label, node, edge_index, completingp);
         }
 
         // replace the label directly
@@ -151,6 +158,11 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
                 .ok_or(Error::NodeNoLabel(parent))?
                 .label();
 
+            if get_rule_label(parent_label).is_none() {
+                self.print_viz("dbg forest.gv").unwrap();
+                panic!("assumption fails");
+            }
+
             assert!(get_rule_label(parent_label).is_some());
             assert_eq!(builder.degree(parent)?, 1);
 
@@ -207,12 +219,13 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
         new_label: ForestLabel<GrammarLabel>,
         mut node: usize,
         edge_index: usize,
+        completingp: bool,
     ) -> Result<usize, Error> {
         let end = new_label.label().end();
 
         let mut builder = PLGBuilderMut::from_graph_mut(&mut self.graph);
 
-        let mut new_node = builder.add_vertex(new_label);
+        let new_node = builder.add_vertex(new_label);
 
         let new_packed = if new_label.clone_index().is_some() {
             let packed = builder
@@ -261,7 +274,13 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
                     .label();
 
                 assert!(get_rule_label(parent_label).is_some());
-                assert_eq!(self.degree(parent.node())?, 1);
+
+                if self.degree(parent.node())? != 1 {
+                    dbg!(parent);
+                    self.print_viz("dbg forest.gv").unwrap();
+
+                    panic!("assumption fails");
+                }
 
                 if let Some(pos) = end {
                     parent_label.set_end(pos);
@@ -276,11 +295,11 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
                 let new_parent = builder.add_vertex(parent_label);
 
                 if let Some(packed) = new_packed {
-                    new_node = packed;
+                    builder.add_edge(new_parent, packed, new_label)?;
+                } else {
+                    builder.add_edge(new_parent, new_node, new_label)?;
                 }
 
-                builder.add_edge(new_parent, new_node, new_label)?;
-
                 result.extend(
                     self.parents_of(parent.node())?
                         .map(|parent_parent| (parent_parent, new_parent)),
@@ -291,21 +310,48 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
         };
 
         for (parent, new_child) in parents {
-            if self.has_same_children_until(
-                parent.node(),
-                parent.node(),
-                parent.edge(),
-                new_child,
-            )? {
-                continue;
-            }
+            if !completingp {
+                if self.has_same_children_until(
+                    parent.node(),
+                    parent.node(),
+                    parent.edge(),
+                    new_child,
+                )? {
+                    continue;
+                }
 
-            // we don't add a child to parent.edge() here.
-            let cloned = self.clone_node(parent.node(), parent.edge(), false)?;
+                // we don't add a child to parent.edge() here.
+                let cloned = self.clone_node(parent.node(), parent.edge(), false)?;
 
-            let mut builder = PLGBuilderMut::from_graph_mut(&mut self.graph);
+                let mut builder = PLGBuilderMut::from_graph_mut(&mut self.graph);
 
-            builder.add_edge(cloned, new_child, new_label)?;
+                builder.add_edge(cloned, new_child, new_label)?;
+            } else {
+                if self.has_same_children_except(
+                    parent.node(),
+                    parent.node(),
+                    parent.edge(),
+                    new_child,
+                )? {
+                    continue;
+                }
+
+                // we don't add a child to parent.edge() here.
+                let cloned = self.clone_node(parent.node(), parent.edge(), false)?;
+
+                let mut builder = PLGBuilderMut::from_graph_mut(&mut self.graph);
+
+                builder.add_edge(cloned, new_child, new_label)?;
+
+                let children_to_add: Vec<_> = builder
+                    .children_of(parent.node())?
+                    .skip(parent.edge() + 1)
+                    .collect();
+
+                for child in children_to_add {
+                    builder.add_edge(cloned, child, new_label)?;
+                }
+            }
         }
 
         Ok(new_node)
@@ -486,6 +532,85 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
             unreachable!("should not examine children of a packed node")
         }
     }
+
+    /// Detect if a node has a branch that has the same children as
+    /// another node, except for a given index, where it points to another
+    /// given node.
+    ///
+    /// # Clones
+    ///
+    /// If `nodea` is a clone, it checks every clone to see if the
+    /// condition is satisfied for some clone.
+    fn has_same_children_except(
+        &self,
+        nodea: usize,
+        nodeb: usize,
+        edge_index: usize,
+        alternative: usize,
+    ) -> Result<bool, Error> {
+        let labela = self.vertex_label(nodea)?.ok_or(Error::NodeNoLabel(nodea))?;
+        let children_b = self.children_of(nodeb)?;
+
+        if children_b.len() < edge_index + 1 {
+            return Err(Error::IndexOutOfBounds(edge_index, children_b.len() - 1));
+        }
+
+        if labela.is_plain() {
+            let children_a = self.children_of(nodea)?;
+
+            if children_a.len() != children_b.len() {
+                return Ok(false);
+            }
+
+            for (index, (childa, childb)) in
+                children_a.zip(children_b).take(edge_index + 1).enumerate()
+            {
+                if index != edge_index {
+                    if childa != childb {
+                        return Ok(false);
+                    }
+                } else if childa != alternative {
+                    return Ok(false);
+                }
+            }
+
+            Ok(true)
+        } else if labela.clone_index().is_some() {
+            let mut parentsa = self.parents_of(nodea)?;
+
+            assert_eq!(parentsa.len(), 1);
+
+            let parenta = parentsa.next().unwrap().node();
+
+            'branch_loop: for branch in self.children_of(parenta)? {
+                let children_a = self.children_of(branch)?;
+                let children_b = children_b.clone();
+
+                if children_a.len() < children_b.len() {
+                    continue 'branch_loop;
+                }
+
+                for (index, (childa, childb)) in
+                    children_a.zip(children_b).take(edge_index + 1).enumerate()
+                {
+                    if index != edge_index {
+                        if childa != childb {
+                            continue 'branch_loop;
+                        }
+                    } else if childa != alternative {
+                        continue 'branch_loop;
+                    }
+                }
+
+                return Ok(true);
+            }
+
+            Ok(false)
+        } else {
+            // a packed node; this should not happen
+            unreachable!("should not examine children of a packed node")
+        }
+    }
 }
 
 #[cfg(test)]
@@ -688,7 +813,7 @@ mod test_splone {
     fn test() -> Result<(), Box<dyn std::error::Error>> {
         let mut test_forest = create_test_forest()?;
 
-        test_forest.splone(6, Some(3), 1)?;
+        test_forest.splone(6, Some(3), 1, false)?;
 
         let answer = splone_6_3_1()?;
 
@@ -698,7 +823,7 @@ mod test_splone {
             panic!("splone(6, Some(3), 1) produced wrong forest");
         }
 
-        test_forest.splone(6, Some(3), 1)?;
+        test_forest.splone(6, Some(3), 1, false)?;
 
         if test_forest != answer {
             test_forest.print_viz("sploned forest.gv")?;
@@ -706,7 +831,7 @@ mod test_splone {
             panic!("repeated splone(6, Some(3), 1) produced wrong forest");
         }
 
-        test_forest.splone(6, Some(2), 0)?;
+        test_forest.splone(6, Some(2), 0, false)?;
 
         let answer = splone_6_2_0()?;
 
@@ -716,7 +841,7 @@ mod test_splone {
             panic!("splone(6, Some(2), 0) produced wrong forest");
         }
 
-        test_forest.splone(6, Some(2), 0)?;
+        test_forest.splone(6, Some(2), 0, false)?;
 
         if test_forest != answer {
             test_forest.print_viz("sploned forest.gv")?;
@@ -724,7 +849,7 @@ mod test_splone {
             panic!("repeated splone(6, Some(2), 0) produced wrong forest");
         }
 
-        test_forest.splone(6, None, 0)?;
+        test_forest.splone(6, None, 0, false)?;
 
         let answer = splone_6_n_0()?;
 
@@ -734,7 +859,7 @@ mod test_splone {
             panic!("splone(6, None, 0) produced wrong forest");
         }
 
-        test_forest.splone(14, None, 0)?;
+        test_forest.splone(14, None, 0, false)?;
 
         if test_forest != answer {
             test_forest.print_viz("sploned forest.gv")?;
@@ -742,7 +867,7 @@ mod test_splone {
             panic!("repeated splone(6, None, 0) produced wrong forest");
         }
 
-        test_forest.splone(4, Some(3), 0)?;
+        test_forest.splone(4, Some(3), 0, false)?;
 
         let answer = splone_4_3_0()?;
 
@@ -752,7 +877,7 @@ mod test_splone {
             panic!("splone(4, Some(3), 0) produced wrong forest");
         }
 
-        test_forest.splone(4, Some(3), 0)?;
+        test_forest.splone(4, Some(3), 0, false)?;
 
         if test_forest != answer {
             test_forest.print_viz("sploned forest.gv")?;
@@ -760,8 +885,8 @@ mod test_splone {
             panic!("repeated splone(4, Some(3), 0) produced wrong forest");
         }
 
-        test_forest.splone(17, Some(3), 0)?;
-        test_forest.splone(15, Some(3), 0)?;
+        test_forest.splone(17, Some(3), 0, false)?;
+        test_forest.splone(15, Some(3), 0, false)?;
 
         let answer = splone_17_3_0_15_3_0()?;
 
@@ -771,8 +896,8 @@ mod test_splone {
             panic!("splone(17, Some(3), 0) - splone(15, Some(3), 0) produced wrong forest");
         }
 
-        test_forest.splone(17, Some(3), 0)?;
-        test_forest.splone(15, Some(3), 0)?;
+        test_forest.splone(17, Some(3), 0, false)?;
+        test_forest.splone(15, Some(3), 0, false)?;
 
         if test_forest != answer {
             test_forest.print_viz("sploned forest.gv")?;
diff --git a/chain/src/item/forest-format.org b/chain/src/item/forest-format.org
new file mode 100644
index 0000000..eb0f150
--- /dev/null
+++ b/chain/src/item/forest-format.org
@@ -0,0 +1,10 @@
+#+TITLE: Format of forests
+#+DATE: [2023-02-13 Lun 22:05]
+#+AUTHOR: Durand
+
+In this document I try to explain the format of the forests returned
+by the parser generator.
+
+* Basic terms
+
+# TODO: Explain
diff --git a/chain/src/item/genins.rs b/chain/src/item/genins.rs
index 4c51d9a..feb45c6 100644
--- a/chain/src/item/genins.rs
+++ b/chain/src/item/genins.rs
@@ -7,19 +7,24 @@
 //! semirings later on.
 
 use super::*;
-use crate::{atom::DefaultAtom, default::Error, item::default::DefaultForest, Edge};
+use crate::{
+    atom::{Atom, DefaultAtom},
+    default::Error,
+    item::default::DefaultForest,
+    Edge,
+};
 use grammar::{Error as GrammarError, GrammarLabel, GrammarLabelType, TNT};
-use graph::Graph;
+#[allow(unused_imports)]
+use graph::{builder::BuilderMut, labelled::binary::PLGBuilderMut, Graph, RedirectGraph};
 
 use core::borrow::Borrow;
-use std::collections::HashSet as Set;
 
 /// Convert an error telling us that an index is out of bounds.
 ///
 /// # Panics
 ///
 /// The function panics if the error is not of the expected kind.
-fn index_out_of_bounds_conversion(ge: GrammarError) -> Error {
+pub(crate) fn index_out_of_bounds_conversion(ge: GrammarError) -> Error {
     match ge {
         GrammarError::IndexOutOfBounds(index, bound) => Error::IndexOutOfBounds(index, bound),
         _ => Error::Invalid,
@@ -105,11 +110,6 @@ pub fn virtual_generate_fragment(
                     Some(TNT::Ter(ter)) if ter == t)
         {
             for child in child_iter {
-                // #[allow(unused_must_use)]
-                // {
-                //     dbg!(non_start, child, atom.query_expansion(non_start, child));
-                // }
-
                 let line: Vec<GrammarLabelType> = atom
                     .query_expansion(non_start, child)
                     .map_err(index_out_of_bounds_conversion)?
@@ -124,7 +124,15 @@ pub fn virtual_generate_fragment(
                 if result.is_empty() {
                     result = generate_fragment(line, pos)?;
                 } else {
-                    result.plant(0, generate_fragment(line, pos)?, false)?;
+                    let mut new_fragment = generate_fragment(line, pos)?;
+
+                    new_fragment.remove_node(0)?;
+
+                    new_fragment.set_root(1)?;
+
+                    let cloned = result.clone_node(0, 0, false)?;
+
+                    result.plant(cloned, new_fragment, false)?;
                 }
             }
         }
@@ -133,90 +141,313 @@ pub fn virtual_generate_fragment(
     Ok(result)
 }
 
+// TODO: Examine `insert_item` again.
+
 impl DefaultForest<ForestLabel<GrammarLabel>> {
     /// Insert an item derivation forest into a recording forest.
     ///
     /// We need the help of other things just for finding the correct
     /// places to insert these item fragments.
-    pub fn insert_item(
+    pub(crate) fn insert_item(
         &mut self,
         label: Edge,
         fragment: impl Borrow<DefaultForest<ForestLabel<GrammarLabel>>>,
-        atom_child_iter: impl Iterator<Item = usize> + ExactSizeIterator,
+        atom_child_iter: impl Iterator<Item = usize> + ExactSizeIterator + Clone,
         atom: &DefaultAtom,
-    ) -> Result<PaSe, Error> {
-        let fragment = fragment.borrow();
+    ) -> Result<PaVi, Error> {
+        let root = if let Some(root) = self.root() {
+            root
+        } else {
+            panic!("the forest must be non-empty when we insert items");
+        };
+
+        let pavi = label.forest_source();
 
-        let pase = label.forest_source();
+        let true_source = label.true_source();
+
+        let fragment = fragment.borrow();
 
         let fragment_root = if let Some(root) = fragment.root() {
             root
         } else {
-            return Err(Error::EmptyItem);
+            unreachable!("empty item");
         };
 
-        let pos = fragment
+        let fragment_root_label = fragment
             .vertex_label(fragment_root)?
-            .ok_or(Error::NodeNoLabel(fragment_root))?
-            .label()
-            .start();
+            .ok_or(Error::NodeNoLabel(fragment_root))?;
+
+        let pos = fragment_root_label.label().start();
+
+        if pavi != true_source {
+            dbg!(label, pos);
+
+            // Completing from true_source up to the pavi
+            let true_source_node = match true_source {
+                PaVi::Parent(node, edge) => {
+                    let nth_child = self.nth_child(node, edge)?;
+                    let nth_child_label = self
+                        .vertex_label(nth_child)?
+                        .ok_or(Error::NodeNoLabel(nth_child))?;
+
+                    let nth_child_degree = self.degree(nth_child)?;
+                    let nth_child_last = if nth_child_degree > 0 {
+                        nth_child_degree - 1
+                    } else {
+                        0
+                    };
+
+                    if matches!(nth_child_label.label().label().tnt(), Some(TNT::Non(_)))
+                        && !nth_child_label.is_packed()
+                    {
+                        let sploned = self.splone(nth_child, Some(pos), nth_child_last, false)?;
+
+                        sploned
+                    } else {
+                        nth_child
+                    }
+                }
+                PaVi::Virtual(nt, t, mut node) => {
+                    let node_label_start = self
+                        .vertex_label(node)?
+                        .ok_or(Error::NodeNoLabel(node))?
+                        .label()
+                        .start();
+
+                    let reduction_fragment = atom.generate_virtual_frags(nt, t, None);
+
+                    // FIXME: We shall "close" this fragment as well.
+
+                    for frag in reduction_fragment.into_iter().flatten() {
+                        let mut frag = frag.clone();
+                        frag.set_pos(node_label_start)?;
+
+                        // NOTE: The function `plant_if_needed`
+                        // assumes that we want to plant the fragment
+                        // as the first child of the node.  This
+                        // assumption holds in this case, but not in
+                        // general.
+
+                        node = self.plant_if_needed(node, frag)?;
+                    }
+
+                    node
+                }
+                PaVi::Empty => {
+                    dbg!();
+                    root
+                }
+            };
+
+            let true_source_pos = self
+                .vertex_label(true_source_node)?
+                .ok_or(Error::NodeNoLabel(true_source_node))?
+                .label()
+                .start();
+
+            let top_node = match pavi {
+                PaVi::Parent(node, edge) => self.nth_child(node, edge)?,
+                PaVi::Virtual(_nt, _t, _node) => {
+                    dbg!(label);
+
+                    self.print_viz("dbg forest.gv").unwrap();
+                    fragment.print_viz("dbg fragment.gv").unwrap();
+
+                    unreachable!("I do not think this is possible")
+                }
+                PaVi::Empty => {
+                    unreachable!("The empty segment should not need to be reduced")
+                }
+            };
+
+            let mut stack = vec![vec![(top_node, 0)]];
+            let mut result_stack = Vec::new();
+
+            while let Some(mut top_stack) = stack.pop() {
+                let (node, _) = top_stack.pop().unwrap();
+
+                let node_label = self.vertex_label(node)?.ok_or(Error::NodeNoLabel(node))?;
+
+                if node_label.is_packed() {
+                    for node in self.children_of(node)? {
+                        let search_child = self.position_search(node, true_source_pos)?;
+
+                        if let Some((child, index)) = search_child {
+                            let mut top_stack = top_stack.clone();
+                            // Fix the previous element
+                            top_stack.push((node, index));
+
+                            if child == true_source_node {
+                                result_stack.push(top_stack);
+                            } else {
+                                top_stack.push((child, 0));
+
+                                stack.push(top_stack);
+                            }
+                        }
+                    }
+                } else {
+                    let search_child = self.position_search(node, true_source_pos)?;
+
+                    if let Some((child, index)) = search_child {
+                        top_stack.push((node, index));
+
+                        if child == true_source_node {
+                            result_stack.push(top_stack);
+                        } else {
+                            top_stack.push((child, 0));
+
+                            stack.push(top_stack);
+                        }
+                    }
+                }
+            }
+
+            // FIXME: We have to change the pavi as well, otherwise we
+            // are going to use the wrong branch for planting later.
+
+            for stack in result_stack {
+                // dbg!(&stack);
+                // self.print_viz("dbg forest.gv").unwrap();
+                // panic!("test");
+
+                let mut first_time = true;
+
+                for (node, index) in stack.into_iter().rev() {
+                    if matches!(
+                        self.vertex_label(node)?
+                            .ok_or(Error::NodeNoLabel(node))?
+                            .label()
+                            .label()
+                            .tnt(),
+                        Some(TNT::Non(_))
+                    ) {
+                        let sploned = self.splone(node, Some(pos), index, false)?;
+
+                        if !first_time {
+                            let last_index = self.degree(sploned)? - 1;
+
+                            let last_child = self.nth_child(sploned, last_index)?;
+
+                            let last_child_label = self
+                                .vertex_label(last_child)?
+                                .ok_or(Error::NodeNoLabel(last_child))?
+                                .label();
+
+                            if last_child_label.end() != Some(pos) {
+                                let closed_label = GrammarLabel::new_closed(
+                                    last_child_label.label(),
+                                    last_child_label.start(),
+                                    pos,
+                                );
+
+                                let closed_label_id = self
+                                    .query_label(closed_label.into())
+                                    .expect("last child closed label not found");
+
+                                let mut builder = PLGBuilderMut::from_graph_mut(&mut self.graph);
 
-        // Ensure the last node in the PaSe is a terminal or a
+                                builder.redirect(sploned, last_index, closed_label_id)?;
+                            }
+                        }
+
+                        first_time = false;
+                    }
+                }
+            }
+        }
+
+        // Ensure the last node in the PaVi is a terminal or a
         // non-terminal node, as an extra safety guard during
         // development.
         #[cfg(debug_assertions)]
         {
-            if let Some(parent) = pase.parent() {
-                assert!(matches!(
-                    self.vertex_label(self.nth_child(parent.node(), parent.edge())?),
-                    Ok(Some(label))
-                        if label.label().label().tnt().is_some()));
-            } else if let Some((_, leaf)) = pase.segment() {
-                assert!(matches!(
-                    self.vertex_label(leaf),
-                    Ok(Some(label))
-                        if label.label().label().tnt().is_some()));
-            } else {
-                unreachable!("a pase is neither a parent nor a segment");
+            match pavi {
+                PaVi::Parent(node, edge) => {
+                    assert!(matches!(
+                        self.vertex_label(self.nth_child(node, edge)?),
+                        Ok(Some(label))
+                            if label.label().label().tnt().is_some()));
+                }
+                PaVi::Virtual(nt, t, node) => {
+                    if !matches!(
+                        self.vertex_label(node),
+                        Ok(Some(label))
+                            if matches!(
+                                label.label().label().tnt(),
+                                Some(TNT::Non(_))))
+                    {
+                        dbg!(node, self.vertex_label(node)?, pavi);
+
+                        self.print_viz("dbg forest.gv").unwrap();
+
+                        panic!("assumption fails");
+                    }
+
+                    if nt >= atom.non_num() {
+                        dbg!();
+                        return Err(Error::IndexOutOfBounds(nt, atom.non_num()));
+                    }
+
+                    if t >= atom.ter_num() {
+                        dbg!();
+                        return Err(Error::IndexOutOfBounds(t, atom.ter_num()));
+                    }
+                }
+                PaVi::Empty => {}
             }
         }
 
-        let mut is_empty_segment = false;
+        let is_empty_segment = pavi.is_empty();
 
         let mut parents: Vec<Parent> = {
             let mut result = Vec::new();
 
-            if let Some(parent) = pase.parent() {
-                result.push(parent);
-            } else {
-                let (root, leaf) = pase.segment().unwrap();
-                let mut seen_nodes = Set::new();
+            match pavi {
+                PaVi::Parent(node, edge) => {
+                    result.push(Parent::new(node, edge));
+                }
+                PaVi::Virtual(nt, t, node) => {
+                    let node_label = self.vertex_label(node)?.ok_or(Error::NodeNoLabel(node))?;
 
-                let mut stack = if root == leaf {
-                    // an empty segment means the root
-                    is_empty_segment = true;
+                    for atom_child in atom_child_iter.clone() {
+                        for rule in atom.trace(nt, t, atom_child).into_iter().flatten() {
+                            let virtual_frag = atom.generate_virtual_frags(nt, t, Some(rule));
 
-                    result.push(Parent::new(root, 0));
-                    vec![]
-                } else {
-                    vec![root]
-                };
+                            if let Some(frag) = virtual_frag {
+                                let mut frag = (*frag.get(0).unwrap()).clone();
 
-                while let Some(top) = stack.pop() {
-                    if seen_nodes.contains(&top) {
-                        continue;
-                    }
+                                frag.set_pos(node_label.label().start())?;
 
-                    seen_nodes.insert(top);
+                                let frag_nodes_len = frag.nodes_len();
 
-                    for (index, child) in self.children_of(top)?.enumerate() {
-                        if child == leaf {
-                            result.push(Parent::new(top, index));
-                        } else {
-                            stack.push(child);
+                                assert!(frag_nodes_len > 1);
+
+                                let last_but_one_label = frag
+                                    .vertex_label(frag_nodes_len - 2)?
+                                    .ok_or(Error::NodeNoLabel(frag_nodes_len - 2))?;
+
+                                // NOTE: The function
+                                // `plant_if_needed` assumes that we
+                                // want to plant the fragment as the
+                                // first child of the node.  This
+                                // assumption holds in this case, but
+                                // not in general.
+
+                                self.plant_if_needed(node, frag)?;
+
+                                let rule_label_pos = self
+                                    .query_label(last_but_one_label)
+                                    .expect("the forest was wrongly planted");
+
+                                result.push(Parent::new(rule_label_pos, 0));
+                            }
                         }
                     }
                 }
+                PaVi::Empty => {
+                    result.push(Parent::new(root, 0));
+                }
             }
 
             result
@@ -251,6 +482,7 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
 
         for atom_child in atom_child_iter {
             // dbg!(label.label(), atom_child);
+
             // Find reduction information.
             let reduction_info = atom
                 .query_reduction(label.label(), atom_child)
@@ -272,7 +504,8 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
                             .label(),
                         GrammarLabelType::TNT(TNT::Non(nt)) if nt == *reduction_nt
                     ) {
-                        let sploned_node = self.splone(node.node(), Some(pos), node.edge())?;
+                        let sploned_node =
+                            self.splone(node.node(), Some(pos), node.edge(), false)?;
 
                         node_label = self
                             .vertex_label(sploned_node)?
@@ -282,16 +515,15 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
                             let mut parent_iter = self.parents_of(sploned_node)?;
 
                             #[cfg(debug_assertions)]
-                            {
-                                assert_eq!(parent_iter.len(), 1);
-
-                                assert!(self
-                                    .vertex_label(sploned_node)?
-                                    .ok_or(Error::NodeNoLabel(sploned_node))?
-                                    .is_packed());
-                            }
+                            assert_eq!(parent_iter.len(), 1);
 
                             node = parent_iter.next().unwrap();
+
+                            #[cfg(debug_assertions)]
+                            assert!(self
+                                .vertex_label(node.node())?
+                                .ok_or(Error::NodeNoLabel(node.node()))?
+                                .is_packed());
                         } else {
                             node = Parent::new(sploned_node, node.edge());
                         }
@@ -337,19 +569,29 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
                     atom_child,
                     parents,
                     reduction_info,
-                    atom.query_reduction(label.label(), atom_child).unwrap()
+                    atom.query_reduction(label.label(), atom_child).unwrap(),
+                    is_empty_segment,
+                    atom.trace(0, 3, atom_child)
+                        .into_iter()
+                        .flatten()
+                        .collect::<Vec<_>>(),
                 );
 
                 self.print_viz("dbg forest.gv").unwrap();
 
                 #[cfg(test)]
-                genins_test::print_labels(atom, self.borrow()).unwrap();
+                crate::item::default::print_labels(atom, self.borrow()).unwrap();
 
                 return Err(Error::CannotPlant);
             }
 
-            for parent in stack.into_iter() {
-                let sploned_node = self.splone(parent.node(), None, parent.edge())?;
+            if pos == 4 && matches!(true_source, PaVi::Virtual(_, _, _)) {
+                dbg!(&stack, reduction_info, true_source, pavi);
+                self.print_viz(&format!("pos4ib.gv")).unwrap();
+            }
+
+            for parent in stack {
+                let sploned_node = self.splone(parent.node(), None, parent.edge(), false)?;
 
                 self.plant(sploned_node, fragment, non_empty)?;
 
@@ -357,24 +599,22 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
             }
         }
 
-        // If the iterator is empty, just plant at the specified
-        // child.
+        // If the iterator is empty, assert the fragment has length
+        // one, and do not plant anything.
         if !non_empty {
-            for parent in parents.into_iter() {
-                let nth_child = self.nth_child(parent.node(), parent.edge())?;
-
-                self.plant(nth_child, fragment, non_empty)?;
-
-                non_empty = true;
-            }
+            assert_eq!(fragment.nodes_len(), 1);
         }
 
         let result = if fragment.nodes_len() == 2 {
-            let root_label = fragment.vertex_label(0)?.unwrap();
-            let leaf_label = fragment.vertex_label(1)?.unwrap();
+            let root_label = fragment_root_label;
+            let leaf_label = fragment
+                .vertex_label(1 - fragment_root)?
+                .ok_or(Error::NodeNoLabel(1 - fragment_root))?;
 
             // it has been planted, so should be safe.
-            let node = self.query_label(root_label).unwrap();
+            let node = self
+                .query_label(root_label)
+                .expect("root label was not found");
 
             let edge = {
                 let mut result = None;
@@ -394,16 +634,52 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
                 }
             };
             // dbg!(node, edge, root_label, leaf_label);
-            PaSe::Parent(node, edge)
+            PaVi::Parent(node, edge)
         } else {
-            let root_label = fragment.vertex_label(0)?.unwrap();
-            let leaf_label = fragment.vertex_label(fragment.nodes_len() - 1)?.unwrap();
-            // dbg!(root_label, leaf_label);
-
-            PaSe::Segment(
-                self.query_label(root_label).unwrap(),
-                self.query_label(leaf_label).unwrap(),
-            )
+            assert_eq!(
+                fragment.nodes_len(),
+                1,
+                "a virtual fragment should consist of a single terminal node."
+            );
+
+            let root_label = fragment_root_label;
+
+            let pavi_parent = pavi.parent().expect(
+                "When we insert a virtual fragment, the forest_source of
+            the label must be a parent.",
+            );
+
+            let nth_child = self.nth_child(pavi_parent.node(), pavi_parent.edge())?;
+
+            let nth_child_label = self
+                .vertex_label(nth_child)?
+                .ok_or(Error::NodeNoLabel(nth_child))?
+                .label()
+                .label();
+
+            let error_str = "When we insert a virtual fragment, the \
+                             forest source of the label must point to \
+                             a non-terminal node";
+
+            let nt = match nth_child_label.tnt().expect(error_str) {
+                TNT::Non(nt) => nt,
+                _ => {
+                    dbg!(nth_child, nth_child_label);
+
+                    panic!("{error_str}");
+                }
+            };
+
+            let t = match root_label.label().label().tnt().unwrap() {
+                TNT::Ter(t) => t,
+                _ => {
+                    dbg!(root_label);
+
+                    panic!("a virtual fragment should consist of a single terminal node")
+                }
+            };
+
+            PaVi::Virtual(nt, t, nth_child)
         };
 
         Ok(result)
@@ -413,40 +689,10 @@ impl DefaultForest<ForestLabel<GrammarLabel>> {
 #[cfg(test)]
 mod genins_test {
     use super::*;
-    #[allow(unused_imports)]
-    use crate::{default::DefaultChain, item::default::leaf, Chain};
+    use crate::item::default::leaf;
 
     use grammar::test_grammar_helper::*;
 
-    pub fn print_labels(
-        atom: impl Borrow<DefaultAtom>,
-        forest: impl Borrow<DefaultForest<ForestLabel<GrammarLabel>>>,
-    ) -> Result<(), Box<dyn std::error::Error>> {
-        let forest = forest.borrow();
-        let atom = atom.borrow();
-
-        for node in forest.nodes() {
-            let label = forest.vertex_label(node)?;
-
-            if let Some(label) = label {
-                let label = label.label.label();
-
-                match label {
-                    GrammarLabelType::TNT(tnt) => {
-                        println!("{tnt} = {}", atom.name_of_tnt(tnt)?);
-                    }
-                    GrammarLabelType::Rule(pos) => {
-                        println!("pos {pos} = {}", atom.rule_pos_string(pos)?);
-                    }
-                }
-            } else {
-                return Err(Error::NodeNoLabel(node).into());
-            }
-        }
-
-        Ok(())
-    }
-
     #[test]
     fn test_generate_fragment() -> Result<(), Box<dyn std::error::Error>> {
         let grammar = new_notes_grammar()?;
@@ -497,7 +743,7 @@ mod genins_test {
             0,
         )?;
 
-        print_labels(&atom, &virtual_fragment)?;
+        crate::item::default::print_labels(&atom, &virtual_fragment)?;
 
         assert_eq!(virtual_fragment, test_fragment);
 
@@ -511,7 +757,7 @@ mod genins_test {
         let test_fragment =
             generate_fragment([TNT::Non(3).into(), 38.into(), TNT::Ter(2).into()], 1)?;
 
-        print_labels(&atom, &virtual_fragment)?;
+        crate::item::default::print_labels(&atom, &virtual_fragment)?;
 
         assert_eq!(virtual_fragment, test_fragment);
 
diff --git a/chain/src/item/mod.rs b/chain/src/item/mod.rs
index 284d640..39d04c7 100644
--- a/chain/src/item/mod.rs
+++ b/chain/src/item/mod.rs
@@ -9,43 +9,64 @@ use graph::{error::Error as GError, GraphLabel, LabelGraph, Parent, ParentsGraph
 
 use core::borrow::Borrow;
 
-/// A parent or a segment.
+/// A parent or a virtual segment.
+///
+/// # Parent
 ///
 /// A parent is a node with an edge index, which represents a certain
 /// edge.
 ///
-/// A segment represents every edge from the root node to the single
-/// terminating node.
-#[derive(Debug, Copy, Clone, Eq, PartialEq, Ord, PartialOrd, Hash)]
-pub enum PaSe {
+/// # Virtual Segment
+///
+/// A virtual segment represents an expansion from a non-terminal by a
+/// terminal.  We do not directly add this segment when we encounter
+/// this expansion at the start because this expansion might contain
+/// multiple derivations some of which will not be used.
+///
+/// If we add the expansion immediately when we encounter it, we have
+/// to later discern and delete those unwanted derivations.  This is
+/// asking for trouble, in my experiences.
+#[derive(Debug, Copy, Clone, Eq, PartialEq, Ord, PartialOrd, Hash, Default)]
+pub enum PaVi {
     /// An edge from a node, as the n-th child.
     Parent(usize, usize),
-    /// A segment from a root to the single terminating node.
-    Segment(usize, usize),
+    /// A virtual segment from a non-terminal by a terminal, rooted at
+    /// a node.
+    ///
+    /// # Tuple elements
+    ///
+    /// The contained tuple is of the form (nt, t, node), which means
+    /// a virtually added node at the `node` representing the
+    /// expansion from the non-terminal `nt` by the terminal `t`.
+    Virtual(usize, usize, usize),
+    /// This is an empty segment that represents the root node.  This
+    /// is a special case for the unit state of the chain-rule
+    /// machine.
+    #[default]
+    Empty,
 }
 
-impl From<Parent> for PaSe {
+impl From<Parent> for PaVi {
     fn from(value: Parent) -> Self {
         Self::Parent(value.node(), value.edge())
     }
 }
 
-impl Default for PaSe {
-    fn default() -> Self {
-        Self::Segment(0, 0)
-    }
-}
-
-impl core::fmt::Display for PaSe {
+impl core::fmt::Display for PaVi {
     fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
         match self {
             Self::Parent(node, edge) => write!(f, "the {edge}-th edge from {node}"),
-            Self::Segment(root, leaf) => write!(f, "a segment from {root} to {leaf}"),
+            Self::Virtual(nt, t, node) => write!(
+                f,
+                "a virtual node for non-terminal {nt} and terminal {t} at node {node}"
+            ),
+            Self::Empty => write!(f, "empty segment at root"),
         }
     }
 }
 
-impl PaSe {
+impl PaVi {
+    /// Get the Parent variant.
     fn parent(self) -> Option<Parent> {
         if let Self::Parent(node, edge) = self {
             Some(Parent::new(node, edge))
@@ -54,12 +75,29 @@ impl PaSe {
         }
     }
 
-    fn segment(self) -> Option<(usize, usize)> {
-        if let Self::Segment(root, leaf) = self {
-            Some((root, leaf))
-        } else {
-            None
-        }
+    // /// Get the "Virtual" variant.
+    // ///
+    // /// # Name
+    // ///
+    // /// We cannot use the name "virtual" since it is a reserved
+    // /// keyword in Rust, so we use its French name.
+    // ///
+    // /// # Tuple elements
+    // ///
+    // /// The returned tuple is of the form (nt, t, node), which means a
+    // /// virtually added node at the `node` representing the expansion
+    // /// from the non-terminal `nt` by the terminal `t`.
+    // fn virtuel(self) -> Option<(usize, usize, usize)> {
+    //     if let Self::Virtual(nt, t, node) = self {
+    //         Some((nt, t, node))
+    //     } else {
+    //         None
+    //     }
+    // }
+
+    /// Is this an empty segment?
+    fn is_empty(self) -> bool {
+        self == Self::Empty
     }
 }
 
@@ -101,7 +139,6 @@ impl<T: GraphLabel> core::fmt::Display for ForestLabel<T> {
 }
 
 /// The type of erros for converting forest labels.
-#[non_exhaustive]
 #[derive(Debug, Copy, Clone, Ord, PartialOrd, Eq, PartialEq)]
 pub enum ForestLabelError {
     /// Try to pack a cloned node.
-- 
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