text:
\section{Introduction}
\label{sect:intro}

The $\l$-calculus is the model underlying functional programming languages and proof assistants. The gap between the model and its incarnations is huge. In particular, the $\l$-calculus does not come with a fixed reduction strategy, while concrete frameworks need one.
A desirable property is that  the reduction which  is implemented terminates on all terms on which $\beta$ reduction has a reduction sequence to normal form. This is guaranteed by a \emph{normalization theorem}.
 Two classic
 examples are  the \emph{leftmost-outermost} and \emph{head} normalization theorems (theorems 13.2.2 and  11.4.8 in Barendregt
 \cite{Barendregt84}). The former states that if the term has a $\beta$-normal form, leftmost-outermost reduction is guaranteed to find it; the latter has a similar but
 subtler statement, roughly head reduction computes a head normal form, if the term has any.

 Another classic theorem for head reduction states that head reduction\xspace approximates the $\beta$-normal form by computing an essential part of every evaluation sequence. The precise formulation is
 a \emph{factorization theorem}:  a  sequence of $\beta$ steps $t\Rew{\beta}^*s$ can always be re-arranged as a  sequence of head steps ($\Rew{h}$) followed by a sequence of non-head steps ($\Rew{\lnot h}$), that is, $t \Rew{h}^* u \Rew{\lnot h}^* s$.
 Both head and leftmost-outermost reductions play a key role in the theory of the $\l$-calculus as presented in
 Barendregt \cite{Barendregt84} or Krivine \cite{krivine1993lambda}.

Variants of the $\l$-calculus abound and  are
continuously introduced: weak, call-by-value, call-by-need, classical, with pattern matching, sharing, non-determinism, probabilistic choice, quantum features, differentiation, \textit{etc}.
So, normalization and factorization theorems need to be studied in many variations. Concepts and techniques to prove these theorems do exist, but they do not have the essential, intuitive structure of other fundamental properties, such as~confluence.

 \paragraph{This paper.} Here we provide a presentation of factorization and
 normalization revisiting a simple technique
 due to Takahashi \cite{DBLP:journals/iandc/Takahashi95}, making it even \emph{simpler} and \emph{more widely applicable}.
 We separate the abstract reasoning from the concrete details of head reduction,
 and apply the revisited proof method to several case studies.
 The presentation is novel and hopefully accessible to anyone familiar with the $\l$-calculus, without  a background in advanced  notions of rewriting theory.

We provide four case studies,
all following the same method.
Two are revisitations of the classic cases of head and leftmost-outermost (shortened to $\ell o$\xspace) reductions. Two are folklore cases. The first is \emph{weak} ({\em i.e.}\xspace out of $\l$-abstractions) \emph{call-by-value} (shortened to {CbV}\xspace) \emph{reduction}
in its non-deterministic presentation.
The second is \emph{least-level} (shortened to \ll) \emph{reduction}, a reduction\xspace coming from the linear logic literature---sometimes called \emph{by levels}---and which is usually presented
using proof nets (see de Carvalho, Pagani and Tortora de Falco \cite{DBLP:journals/tcs/CarvalhoPF11} or Pagani and Tranquilli \cite{DBLP:journals/mscs/PaganiT17}) or calculi related to proof nets (see Terui \cite{DBLP:journals/aml/Terui07} or Accattoli \cite{DBLP:conf/rta/Accattoli12}), rather than in the ordinary $\l$-calculus.
The $\ell o$\xspace and \ll cases are \emph{full\xspace} reductions for $\beta$,
{\em i.e.}\xspace they have the same normal forms as $\beta$.
The head and weak {CbV}\xspace cases are not full\xspace, as they may not compute $\beta$  normal forms.

\paragraph{Takahashi.}
In \cite{DBLP:journals/iandc/Takahashi95}, Takahashi uses the natural inductive notion of \emph{parallel}\footnotemark
\footnotetext{
	The terminology at work in the literature on $\l$-calculus and the rewriting terminology often clash: the former calls \emph{parallel $\beta$ reduction} what the latter calls \emph{multi-step $\beta$ reduction}---parallel reduction in rewriting is something else.}
$\beta$ reduction (which reduces simultaneously a number of $\beta$-redexes; it is also the key concept in Tait and Martin-L\"of's classic proof of confluence of the $\l$-calculus)
to introduce a simple proof technique
for head factorization, from which head normalization follows.
By iterating head factorization, she also obtains leftmost-outermost normalization, via a simple argument on the structure of~terms due to Mitschke \cite{Mitschke79}.

Her technique has
been employed
for various $\l$-calculi because of its simplicity.
Namely, for the $\l$-calculus with $\eta$ by Ishii \cite{DBLP:journals/tcs/Ishii18}, the call-by-value $\l$-calculus by Ronchi Della Rocca and Paolini \cite{DBLP:journals/iandc/PaoliniR04,RonchiPaolini}, the resource $\l$-calculus by Pagani and Tranquilli \cite{DBLP:conf/aplas/PaganiT09}, pattern calculi by Kesner, Lombardi and  {R{\'i}os}\xspace \cite{DBLP:journals/corr/abs-1102-3734}, the shuffling calculus by Guerrieri, Paolini and Ronchi Della Rocca \cite{GuerrieriPR15,Guerrieri15,DBLP:journals/lmcs/GuerrieriPR17}, and it has
been formalized with proof assistants by McKinna and Pollack \cite{DBLP:journals/jar/McKinnaP99} and Crary \cite{Crary09standard}.

\paragraph{Takahashi revisited.} Despite its simplicity, Takahashi's proof \cite{DBLP:journals/iandc/Takahashi95} of factorization relies on substitutivity properties not satisfied by full\xspace reductions such as $\ell o$\xspace and \ll.
Our first contribution is a proof that is independent
of the substitutivity properties of the factorizing reductions.
It relies on a simpler fact, namely the substitutivity of an \emph{indexed} variant $\partobind n$ of parallel $\beta$ reduction $\parRew{\beta}$.
The definition of $\partobind n$ simply decorates the definition of $\parRew{\beta}$ with a natural number $n$ that intuitively corresponds to the number of redexes reduced in parallel by a $\parRew{\beta}$ step.

We
prove factorization theorems for all our four case studies following this simpler scheme. We also highlight  an interesting point:
factorization for the two full\xspace reductions cannot be
obtained directly following Takahashi's method\footnotemark
\footnotetext{\label{f:direct}It can be obtained indirectly, as a corollary of standardization, proved by Takahashi \cite{DBLP:journals/iandc/Takahashi95} using the concrete structure of terms. Thus the proof is not of an abstract nature.}.

\paragraph{From factorization to essential normalization.} The second main  contribution of our  paper is the isolation of
abstract properties  that together with  factorization imply normalization. First of all we abstract head reduction into a generic reduction $\Rew{\esym}$, called \emph{essential}, and non-head reduction $\Rew{\lnot h}$ into a \emph{non-essential} reduction $\Rew{\isym}$. The first additional property for normalization is \emph{persistence}: steps of the factoring reduction $\Rew{\esym}$ cannot be erased by the factored out $\Rew{\isym}$. The second one is a relaxed form of determinism for $\Rew{\esym}$. We show that in such \emph{essential} rewriting systems $\Rew{\esym}$ has a normalization theorem. The argument is abstract, that is, independent of the specific nature of terms. This is in contrast to how Takahashi \cite{DBLP:journals/iandc/Takahashi95} obtains normalization from factorization: her proof is based on an induction over the structure of terms, and cannot then be disentangled by the concrete nature of the rewriting system under study.

\paragraph{Normalizing reductions for $\beta$.} We  apply both our techniques to our case studies of  full\xspace reduction:  $\ell o$\xspace and \ll, obtaining simple proofs that they are normalizing reductions for $\beta$.
Let us point out that $\ell o$\xspace is also---at present---the only known deterministic reduction to $\beta$ normal form whose number of steps is a reasonable cost model, as shown by
Accattoli and Dal Lago \cite{DBLP:journals/corr/AccattoliL16}. Understanding its normalization is one of the motivations at the inception of this work.

\paragraph{Normalization with respect to different notions of results.}
As a further feature, our approach provides for free normalization theorems for reductions that are not full\xspace for the rewrite system in which they live.
Typical examples are head and weak {CbV}\xspace reductions, which do not compute $\beta$ and {CbV}\xspace normal forms, respectively.
These normalization theorems arise naturally in the theory of the $\l$-calculus. For instance, functional programming languages implement only weak notions of reduction, and head reduction (rather than $\ell o$\xspace) is the key notion for the $\l$-definability of computable functions.

We obtain normalization theorems for head and weak {CbV}\xspace reductions.
Catching normalization for non-full\xspace reductions sets our work apart from the recent studies on normalization by Hirokawa, Middeldorp, and Moser \cite{DBLP:conf/rta/HirokawaMM15} and Van Oostrom and Toyama \cite{DBLP:conf/rta/OostromT16}, discussed below among related works.

\paragraph{Factorization, Normalization, Standardization.} In the literature of the $\l$-calculus, normalization for $\losym$\xspace reduction is often obtained as a corollary of the standardization theorem, which roughly states that every reduction sequence can be re-organized as to reduce redexes according to the left-to-right order (Terese \cite{Terese} following Klop \cite{phdklop} and Barendregt \cite{Barendregt84}, for instance). Standardization is a complex and technical result.
Takahashi \cite{DBLP:journals/iandc/Takahashi95}, using Mitschke's argument \cite{Mitschke79} that iterates
head factorization, obtains a simpler proof technique for $\losym$\xspace normalization---and for standardization as well.
Our work refines that approach, abstracts from it and shows that factorization is a general technique for normalization.

\paragraph{Related work.} Factorization is studied in the abstract
{in \cite{DBLP:conf/ctcs/Mellies97,DBLP:conf/rta/Accattoli12}}.
{Melli{\`e}s}\xspace axiomatic approach \cite{DBLP:conf/ctcs/Mellies97} builds on standardization, and encompasses a wide class of rewriting
systems;
in particular, like us, he can deal with non-full\xspace reductions.
{Accattoli} \cite{DBLP:conf/rta/Accattoli12} relies crucially
on terminating hypotheses, absent instead here.

  Hirokawa, Middeldorp, and Moser \cite{DBLP:conf/rta/HirokawaMM15} and Van Oostrom and Toyama \cite{DBLP:conf/rta/OostromT16} study normalizing strategies  via a clean separation between abstract and term rewriting results.
Our approach to normalization is similar to the one used in  \cite{DBLP:conf/rta/HirokawaMM15} to study $\losym$\xspace evaluation for first-order term rewriting systems.
Our essential systems strictly generalize  their  conditions: uniform termination replaces determinism
(two of the strategies we present here are not deterministic) and---crucially---persistence strictly generalizes the property in their Lemma 7.
Conversely, they focus on hyper-normalization and on extending the method to systems in which left-normality is relaxed. We do not deal with these aspects.
Van Oostrom and Toyama's study \cite{DBLP:conf/rta/OostromT16} of
(hyper-)normalization is
based on an elegant and powerful method based on the random descent property and
an ordered notion of commutative diagrams.
Their method and ours are incomparable: we do not rely on (and do not assume) the random descent property (for its definition and uses see van Oostrom \cite{DBLP:conf/rta/Oostrom07})---even if most strategies naturally have that property---and we do
focus on factorization (which they explicitly
avoid),
since we see it as the crucial tool from  which normalization can be obtained.

As already pointed out, a fundamental difference with respect to both works is that we consider a more general notion of normalization for reductions that are not full\xspace, that is not captured by either of those approaches.

In the literature, normalization is also proved from iterated head factorization (Takahashi \cite{DBLP:journals/iandc/Takahashi95} for $\ell o$\xspace, and Terui \cite{DBLP:journals/aml/Terui07} or Accattoli \cite{DBLP:conf/rta/Accattoli12} for \ll on proof nets-like calculi, or Pagani and Tranquilli \cite{DBLP:journals/mscs/PaganiT17} for \ll on differential proof nets), or as a corollary of standardization (Terese \cite{Terese} following Klop \cite{phdklop} and Barendregt \cite{Barendregt84} for $\ell o$\xspace), or using semantic principles such as intersection types (Krivine \cite{krivine1993lambda} for $\ell o$\xspace and de Carvalho, Pagani and Tortora de Falco \cite{DBLP:journals/tcs/CarvalhoPF11} for \ll on proof nets).
Last, Bonelli \emph{et al.} develop a sophisticated proof of normalization for a $\l$-calculus with powerful pattern matching in \cite{DBLP:journals/tcs/BonelliKLR17}.
Our technique differs from them all.

\section{Factorization and Normalization, Abstractly}
\label{sect:abstract-rewriting}

In this section, we study factorization and normalization abstractly, that is,  independently of the specific structure of the objects to be rewritten.

A \emph{rewriting system} (aka abstract reduction system, see Terese \cite[Ch.~1]{Terese})  $\mathcal S$ is a pair $(S, \to)$
consisting of a  set $S$ and a binary
relation $\to \, \subseteq S\times S$ called \emph{reduction}, whose pairs are written  $t \to s$ and called \emph{$\to$-steps}.
A \emph{$\to$-sequence\xspace} from $t$ is a sequence $t \to s \to \dots$ of $\to$-steps; $t \to^k \tmtwo}%{s$ denotes a sequence of $k$ $\to$-steps from $t$ to
	$s$.
As usual, $\to^*$  (resp. $\to^=$) denotes the  transitive-reflexive (resp. reflexive)  closure of $\to$. Given two reductions $\Rew{1}$ and $\Rew{2}$ we use $\Rew{1}\!\cdot\!\Rew{2}$ for their composition, defined as $t \Rew{1} \!\cdot\! \Rew{2} s$ if $t \tooneu\Rew{2} s$ for some $u$.

In this section we focus on a given sub-reduction $\Rew{\esym}$ of $\to$, called \emph{essential}, for which we study factorization and normalization with respect to $\to$. It comes with a second sub-reduction $\Rew{\isym}$, called \emph{inessential}, such that $\Rew{\esym} \cup \Rew{\isym} \,=\, \to$. Despite the notation, $\Rew{\esym}$ and $\Rew{\isym}$ are not required to be disjoint. In general, we write $(S,\{\Rew{a},\Rew{b}\})$ for the rewriting system
$(S,\to)$ where $\to ~=~\Rew{a} \cup \Rew{b}$.

\subsection{Factorization.}

A rewriting system $(S,\{\Rew{\esym},\Rew{\isym}\})$
satisfies \emph{$\Rew{\esym}$-factorization} (also called \emph{postponement of $\Rew{\isym}$ after $\Rew{\esym}$})
if $t \to^* s$ implies that there exists $u$ such that $t\Rew{\esym}^* u \Rew{\isym}^* s$.
Compactly, we write $\to^* \, \subseteq \, \Rew{\esym}^*\!\cdot\!\Rew{\isym}^*$.
In diagrams, see \reffig{diagrams}.a.

\begin{figure}[t]
\centering
\begin{tabular}{cc @{\hspace{.5cm}} |  cc @{\hspace{.5cm}} |  cc @{\hspace{.5cm}} |  cc}
a)&
	\begin{tikzpicture}[ocenter]
	\node (t) {\small $t$};
	\node (dummy) [right of=t] {};
	\node (s1) [right of=dummy] {\small$s$};
	\node (u) [below of=dummy] {\small$u$};
	\draw[->, labelEndAbove=*] (t) to (s1);
	\draw[->,dashed, labelEndLeft=$\esym$, labelEndRight=*] (t) to (u);
	\draw[->, dashed, labelEndRight=${\lnot\extsym}$, labelEndLeft=*] (u) to  (s1);

	\end{tikzpicture}
&
\ b)
&
	\begin{tikzpicture}[ocenter]
	\node (t) {\small $t$};
	\node (s1) [right of=t] {\small$s_1$};
	\node (s2) [below of=t] {\small$s_2$};
	\node (u) [right of=s2] {\small$u$};
	\draw[->, labelEndBelow=${\lnot\extsym}$] (t) to (s1);
	\draw[->, dashed, labelEndLeft=$\esym$, labelEndRight=*] (t) to (s2);
	\draw[->, dashed, labelEndAbove=*, labelEndBelow=${\lnot\extsym}$] (s2) to  (u);
	\draw[->, labelEndRight=$\esym$] (s1) to  (u);
	\end{tikzpicture}
&
\ c)
&
			\begin{tikzpicture}[ocenter]
			\node (t) {\small $t$};
			\node (s1) [right of=t] {\small$s$};
			\node (u) [below of=s1] {\small$u$};
			\draw[->, double, labelEndBelow=${\lnot\extsym}$] (t) to (s1);
			\draw[->, double, dashed, labelEndAbove] (t) to  (u);
			\draw[->, labelEndRight=$\esym$] (s1) to  (u);
			\end{tikzpicture}
&
\ d)
&
			\begin{tikzpicture}[ocenter]
			\node (t) {\small $t$};
			\node (s2) [below of=t] {\small$s$};
			\node (u) [right of=s2] {\small$u$};
			\draw[->, double] (t) to (u);
			\draw[->, dashed, labelEndLeft=$\esym$, labelEndRight=*] (t) to (s2);
			\draw[->, double, dashed, labelEndBelow=${\lnot\extsym}$] (s2) to  (u);

			\end{tikzpicture}

	\end{tabular}
\caption{Diagrams: a) factorization, b) weak postponement, c) merge, d) split.}
\label{fig:diagrams}
\end{figure}

\paragraph{Proving factorization.}

Factorization is a non-trivial rewriting property, because it is \emph{global}, that is, quantified over all reduction sequences from a term.
To be able to  prove factorization, we would like to reduce it
to \emph{local} properties, {\em i.e.}\xspace properties quantified only over one-step reductions from
{a} term.
At first sight it may seem that a local diagram such as the one in \reffig{diagrams}.b would give factorization by a simple induction. Such a diagram however does not allow to infer factorization without further hypotheses---counterexamples can be found in
Barendregt \cite{Barendregt84}.

The following abstract property is a special case for which
a local condition implies factorization.
It was first observed by Hindley \cite{HindleyPhD}.

\begin{lemma}[Hindley, local postponement]
	\label{l:hindley}
	Let $(S,\{\Rew{\esym},\Rew{\isym}\})$ be a rewriting system. If $\Rew{\isym} \!\cdot\! \Rew{\esym} \,\subseteq\, \Rew{\esym}^* \!\cdot\! \Rew{\isym}^=$ then  $\to^* \,\subseteq\, \Rew{\esym}^* \!\cdot\! \Rew{\isym}^*$.
\end{lemma}

\begin{proof}
	The assumption   $\Rew{\isym} \!\cdot\! \Rew{\esym} \,\subseteq\, \Rew{\esym}^* \!\cdot\! \Rew{\isym}^=$ implies (\#) $\Rew{\isym} \!\cdot\! \Rew{\esym}^* \,\subseteq\, \Rew{\esym}^* \!\cdot\! \Rew{\isym}^=$ (indeed, it is immediate to prove that $\Rew{\isym} \!\cdot\! \Rew{\esym}^k \,\subseteq\, \Rew{\esym}^* \!\cdot\! \Rew{\isym}^=$ by induction on $k$).

		We then prove that  $\to^k \,\subseteq\, \Rew{\esym}^* \!\cdot\! \Rew{\isym}^*$, by induction on  $k$.  The case $k=0$ is trivial. Assume $ \to \!\cdot\! \to^{k-1}$. By {\textit{i.h.}}\xspace, $ \to \!\cdot\! \Rew{\esym}^* \!\cdot\! \Rew{\isym}^*$.
		If the first step is $\Rew{\esym}$, the claim is proved. Otherwise,
		by $(\#)$, from  $(\Rew{\isym} \!\cdot\! \Rew{\esym}^*) \cdot\! \Rew{\isym}^*$ we obtain
		$(\Rew{\esym}^* \!\cdot\! \Rew{\isym}^=) \cdot\! \Rew{\isym}^*$.
\end{proof}

Hindley's local condition
is a strong hypothesis for factorization that in general does not hold in $\l$-calculi---not even in the simple case of head reduction\xspace.
However, the property can be applied in combination with another standard technique: switching to \emph{macro} steps that compress $\Rew{\esym}^* $ or $\Rew{\isym}^*$ into just one step, at the price of some light overhead.
This idea is the essence of both Tait--Martin-L\"of's and Takahashi's techniques,  based on \emph{parallel steps}.
The role of parallel steps in
Takahashi \cite{DBLP:journals/iandc/Takahashi95}
is here captured abstractly  by the notion of  \macrostep\ system.

\begin{definition}[Macro-step  system]
\label{def:macrostep}
A rewriting system $\mathcal S=(S, \{\Rew{\esym}, \Rew{\isym}\})$ is a \emph{\macrostep\ system} if there are two
reductions $\parRew{}$ and $\parRew{\neg\esym}$ (called \emph{\macrostep s} and \emph{inessential \macrostep s}, respectively)  such that
	\begin{itemize}
		\item \emph{Macro}:
		$\Rew{\isym} ~\subseteq~ \parRew{\neg\esym}~ \subseteq ~\Rew{\isym}^*$.

		\item \emph{Merge}: if $t \parRew{\neg\esym} \!\cdot\!  \Rew{\esym} \tmthree}%{u $ then $t \parRew{} \tmthree}%{u$. That is, the diagram in \reffig{diagrams}.c holds.

		\item \emph{Split}:  if $t \parRew{} \tmthree}%{u$ then  $t \Rew{\esym}^* \!\cdot\! \parRew{\neg\esym}  \tmthree}%{u$. That is,  the diagram in \reffig{diagrams}.d holds.
	\end{itemize}
\end{definition}
Note that $\parRew{}$ just plays the role of a ``bridge'' between the hypothesis of the merge condition and the conclusion of the split condition---it shall play a crucial role in the concrete proofs in the next sections.
In this paper, concrete instances of $\parRew{}$ and $\parRew{\neg\esym}$ shall be parallel $\beta$ reduction and some of its variants.

\begin{proposition}[Factorization]
	\label{prop:parallelsplit-gives-fact}
	Every \macrostep\ system $(S, \{\Rew{\esym}, \Rew{\isym}\})$ satisfies  $\Rew{\esym}$-factorization.
\end{proposition}
\begin{proof}
	By  Merge
	and Split,
	$\parRew{\neg\esym} \!\cdot\! \Rew{\esym} \,\subseteq\,  \parRew{} \,\subseteq\, \Rew{\esym}^* \!\cdot\! \parRew{\neg\esym} \,\subseteq\, \Rew{\esym}^* \!\cdot\! \parRew{\neg\esym}^=$.
	By Hindley's lemma (\reflemma{hindley}) applied to
	$\Rew{\esym}$ and $\parRew{\neg\esym}$ (rather than $\Rew{\esym}$ and $\Rew{\isym}$),
	we
	obtain
	$(\Rew{\esym} \cup \parRew{\neg\esym})^* \,\subseteq\, \Rew{\esym}^* \!\cdot\!  \parRew{\neg\esym}^*$.

	Since $\Rew{\isym} \,\subseteq\, \parRew{\neg\esym}$, we have $ (\Rew{\esym} \cup \Rew{\isym})^*  \subseteq\, (\Rew{\esym} \cup \parRew{\neg\esym})^* \subseteq\,  \Rew{\esym}^* \!\cdot\! \parRew{\neg\esym}^* $.
	As $\parRew{\neg\esym} \,\subseteq\, \Rew{\isym}^*$, we have  $\Rew{\esym}^* \!\cdot\! \parRew{\neg\esym}^* \, \subseteq \, \Rew{\esym}^* \!\cdot\! \Rew{\isym}^*$.
	Therefore, $\to^* = (\Rew{\esym} \cup \Rew{\isym})^* \subseteq\, \Rew{\esym}^* \!\cdot\! \Rew{\isym}^*$.
\end{proof}

\subsection{Normalization for full\xspace reductions\xspace}
The interest of  factorization comes from the fact that the essential
reduction $\Rew{\esym}$ on which factorization pivots has some good properties.
Here we pinpoint the abstract  properties which make factorization a privileged method to prove normalization; we collect them into the definition of \emph{\essential system} (\refdef{extsys}).

\paragraph{Normal forms and normalization.} Let us recall what normalization is about.
In general, a term may or may not reduce to a normal form.
And if it does, not all reduction sequences necessarily lead to normal form.
A term is
\emph{weakly} or \emph{strongly normalizing}, depending on if it  may or must reduce to normal form.
If a term $t$ is strongly normalizing, any choice of steps will eventually lead to a normal form.
However, if $t$ is weakly normalizing, how do we compute a normal form? This is the problem tackled by \emph{normalization}:  by repeatedly performing \emph{only specific  steps},  a normal form will be computed, provided that $t $ can reduce to~any.

Recall the statement of the $\losym$\xspace normalization theorem: if $t \Rew{\beta}^* \tmthree}%{u$ with $\tmthree}%{u$ $\beta$-normal, then $t$ $\losym$\xspace-reduces to $\tmthree}%{u$.
Observe  a subtlety:
such a formulation relies on  the determinism of $\losym$\xspace reduction.
We give  a more general formulation of normalizing reduction, valid also for non-deterministic reductions.\\

Formally, given a rewriting system $(S, \to)$, \mbox{a term $t \in S$ is:}
\begin{itemize}
	\item \emph{$\to$-normal} (or in \emph{$\to$-normal form})  if $t \not\to$, {\em i.e.}\xspace there are no $\tmtwo}%{s$ such that $t\to \tmtwo}%{s$;
	\item \emph{weakly $\to$-normalizing} if \emph{there exists} a sequence\xspace  $t \to^*s$ with $s$  $\to$-normal;
	\item \emph{strongly $\to$-normalizing}  if there are no infinite $\to$-sequence\xspace{s} from $t$, or
	equivalently, if \emph{all} maximal  $\to$-sequence\xspace{s} from $t$  are finite.
\end{itemize}
We call \emph{$reduction\xspace$ for} $\to$ any $\Rew{\esym} \,\subseteq\, \to$.
It is \emph{full\xspace}  if $ \Rew{\esym}$ and $\to$ have the same normal forms.\footnotemark
\footnotetext{In  rewriting theory, a \emph{full\xspace reduction\xspace for $\to$} is called a \emph{reduction strategy for} $\to$. We prefer not to use the term strategy because it has different meaning  in the $\l$-calculus, where it is a  \emph{deterministic, not necessarily full\xspace}, reduction for $\to$.}

\begin{definition}[Normalizing reduction\xspace]
	A  full\xspace reduction\xspace $\Rew{\esym}$ for $\to$  is  \emph{normalizing} (for $\to$) if, for every term $t$, $t$ is strongly $\Rew{\esym}$-normalizing whenever it is weakly $\to$-normalizing.
\end{definition}
Note that, since
the normalizing reduction\xspace $\Rew{\esym}$ is full\xspace, if  $t$ is strongly $\Rew{\esym}$-normalizing then \emph{every} maximal $\Rew{\esym}$-sequence\xspace from $t$ ends in a $\to$-normal form.

\begin{definition}[\Essential system]
	\label{def:extsys}
	A rewriting system   $(S,\{\Rew{\esym},\Rew{\isym}\})$ is \emph{\essential} if the following conditions hold:
	\begin{enumerate}
		\item \emph{Persistence:} if $t \Rew{\esym} s$ and $t \Rew{\isym} u$, then $u \Rew{\esym} r$ for some $r$.
		\item  \emph{Uniform termination}: if $t$ is weakly $\Rew{\esym}$-normalizing, then it is  strongly \mbox{$\Rew{\esym}$-normalizing}.
		\item  \emph{Terminal factorization}:  if $t\to^* \tmthree}%{u$ and $\tmthree}%{u$ is $\Rew{\esym}$-normal, then $ t\Rew{\esym}^* \!\cdot\! \Rew{\isym}^*\tmthree}%{u$.
	\end{enumerate}
	It is moreover \emph{full\xspace} if $\Rew{\esym}$ is a full\xspace reduction\xspace for $\to$.
\end{definition}
Comments on the definition:
\begin{itemize}
	\item \emph{Persistence}: it
	means that essential steps are out of reach for inessential steps, that cannot erase them.

	The only way of getting rid of essential steps is by reducing them, and so in that sense they are \emph{essential} to normalization.

	\item \emph{From determinism to uniform termination}: as we already said, in general $\Rew{\esym}$ is not deterministic. For normalization, then,

		 it is not enough that there is a sequence\xspace $t \Rew{\esym}^*\tmthree}%{u$ with $\tmthree}%{u$ $\to$-normal (as in the statement of $\losym$\xspace-normalization). We need  to be sure that there are no
		infinite $\Rew{\esym}$-sequence\xspace{s} from $t$.
		This is exactly what is ensured by the uniform termination property.
		Note that if $\Rew{\esym}$ is deterministic (or has the diamond or random descent properties) then it is  uniformly terminating.

	\item \emph{Terminal factorization}: there are two subtleties. First, we need only a weak form of factorization, namely
	factorization is only required for $\to$-sequence\xspace{s} ending in a $\Rew{\esym}$-normal form\footnote{The difference between factorization and its terminal case is relevant for normalization:
		van Oostrom and Toyama \cite[footnote 8]{DBLP:conf/rta/OostromT16} give an example of normalizing full\xspace reduction\xspace for a rewriting system in  which factorization fails
		but terminal factorization holds.}.
	Second, the reader  may
	expect terminal factorization to be required with respect to $\to$-normal rather than $\Rew{\esym}$-normal forms.

	The two notions coincide if $\Rew{\esym}$ is full\xspace, and for the time being we only discuss full\xspace essential systems.

	We discuss the more general case in \refsect{non-complete}.
\end{itemize}

\begin{example} In the $\l$-calculus with $\beta$ reduction, head reduction $\Rew{h}$  and its associated $\Rew{\lnot h}$ reduction (defined in \refsect{head}) form an \essential system. Similarly, leftmost-outermost $\Rew{\losym}$ reduction and its associated $\Rew{\neg \losym}$ reduction (\refsect{leftmost}) form a full\xspace \essential system. Two more examples are in \refsect{weak-cbv} and \refsect{stratified-cbn}.
\end{example}

\begin{theorem}[Essential full\xspace normalization]
	\label{thm:complete-normalization}
	Let
	$(S,\{\Rew{\esym},\Rew{\isym}\})$ be a full\xspace \essential system. Then	$\Rew{\esym}$ is a  normalizing reduction\xspace for $\to$.
\end{theorem}
\begin{proof}
	Let $t$ be a weakly $\to$-normalizing term, {\em i.e.}\xspace $t \to^* \tmthree}%{u$ for some term $\tmthree}%{u$ in $\to$-normal form (and so in $\Rew{\esym}$-normal form, since $\Rew{\esym} \,\subseteq\, \to$).
	\begin{enumerate}

		\item  \emph{Terminal factorization} implies  $t \Rew{\esym}^*\tmtwo}%{s \Rew{\isym}^* \tmthree}%{u$ for some $s$, since $u$ is $\Rew{\esym}$-normal.
		\item Let us show that $\tmtwo}%{s$ is $\Rew{\esym}$-normal: if not, then $\tmtwo}%{s\toer$ for some $r$, and a straightforward induction on the length of $\tmtwo}%{s \Rew{\isym}^*\tmthree}%{u$ iterating \emph{persistence} gives that $\tmthree}%{u \Rew{\esym} p$ for some $p$, against the hypothesis that $\tmthree}%{u$ is $\Rew{\esym}$-normal. Absurd.

		\item By the previous point, $t$ is weakly $\Rew{\esym}$-normalizing. By \emph{uniform termination}, $t$ is strongly $\Rew{\esym}$-normalizing.
		\qedhere
	\end{enumerate}
\end{proof}

\subsection{A more general notion of normalizing reduction\xspace.}
\label{sect:non-complete}
Essential systems actually encompass also important notions of normalization for
reductions that are \emph{not full\xspace},
such as \emph{head normalization}.
These cases arise naturally in the $\l$-calculus literature, where partial notions of result such as head normal forms or values are of common use.
Normalization for non-full\xspace reductions is instead not so common in the rewriting literature outside the $\l$-calculus. This is why, to guide the reader, we presented first the natural case of full\xspace reductions.

Let us first discuss head reduction:  $\Rew{h}$  is deterministic and not full\xspace with respect to $\Rew{\beta}$, as its normal forms may not be $\Rew{\beta}$-normal forms. The well-known property of interest is \emph{head normalization} (Cor.~11.4.8 in Barendregt's book \cite{Barendregt84}):
 \begin{center}
			 If $t \Rew{\beta}^* s$ and $s$ is head normal\footnotemark
			 \footnotetext{``$t$ has a head normal form'' is the usual formulation for ``$t \Rew{\beta}^* s$ for some $s$ that is head normal''. We prefer the latter to avoid the ambiguity of the former about the reduction leading from $t$ to one of its head normal forms ($\Rew{\beta}^*$ or $\Rew{h}^*$?).}
			 then $\Rew{h}$ terminates on $t$.
	\end{center}
\commCF{The above is Barendregt statement. Takahashi statement is:
 \begin{center}
 	 \begin{center}
 			if $t$ has a head normal form, then $t \Rew{h}^* \tmthree}%{u$ for some $\tmthree}%{u$ in head normal form.
 	\end{center}
	if $t$ has a head normal form, iff the $\Rew{h}$-reduction from $t$ terminates.
\end{center}
}
 The statement has two subtleties. First,
 $t$ may $\Rew{\beta}$-reduce to a term  in $\Rew{h}$-normal form in many different ways,
 possibly without using $\Rew{h}$, so that
 the hypotheses may not imply that $\Rew{h}$ terminates.
 Second,
 the conclusion is ``\emph{$\Rew{h}$ terminates on $t$}'' and not $t\Rew{h}^*s$, because in general
 the maximal $\Rew{h}$-sequence from $t$ may end in a term $u\neqs$.
 For instance, let $I=\la y y$: then
	 $I(x(II))\Rew{\beta} I(x I)\Rew{\beta} x I$ is a $\Rew{\beta}$-sequence\xspace to head normal form, and yet
	 the maximal $\Rew{h}$-sequence $I(x(II))\Rew{h} x(II)$ ends in a different term.

Now, let us abstract from head normalization, taking into account that in general the essential reduction\xspace $\Rew{\esym}$---unlike
head reduction\xspace---may not be deterministic, and so we
ask for strong $\Rew{\esym}$-normalization rather than for $\Rew{\esym}$-termination.

\begin{theorem}[Essential normalization]
	\label{thm:par-normalization}
	Let
	$(S,\{\Rew{\esym},\Rew{\isym}\})$ be an  \essential system. If
	$t \to^* u$ and $u$ is $\Rew{\esym}$-normal, then $t$ is strongly $\Rew{\esym}$-normalizing.
\end{theorem}
\begin{proof} Exactly as for \refthm{complete-normalization}, fullness\xspace is not used in that proof.
\end{proof}

In the next section we shall apply \refthm{par-normalization} to head reduction and obtain the head normalization theorem we started with. Another example of a normalization theorem for a non-full\xspace reduction is
in \refsect{weak-cbv}.
Note that the full\xspace variant of the theorem (\refthm{complete-normalization}) is in fact an instance of the general one (\refthm{par-normalization}).

\section{\texorpdfstring{The $\l$-Calculus}{The lambda-Calculus}}
\label{sect:lambda}

This short section recalls basic definitions and properties of the $\l$-calculus and introduces the indexed variant of parallel $\beta$.

The set $\Lambda$ of \emph{terms} of the $\l$-calculus is  given by the following grammar:
\[ \begin{array}{c@{\hspace{.5cm}}rll}
	    	    \textsc{Terms} & t,s,u,r &::=& x \mid  \la\vart \mid ts
    \end{array}\]
We use the usual notions of free and bound variables, $t \isub\vars$ for the meta-level capture-avoiding substitution of $s$ for the free occurrences of $x$ in $t$, and $\sizeptx$ for the number of free occurrences of $x$ in $t$.
The definition of $\beta$ reduction $\Rew{\beta}$ is:
\[
\begin{array}{c@{\ \hspace*{0.35cm}}c@{\ \hspace*{0.35cm}}c@{\ \hspace*{0.35cm}}c}
 \multicolumn{4}{c}{\textsc{$\beta$ reduction} }
 \\[5pt]
	\AxiomC{}

	\UnaryInfC{$(\la\vart) s \Rew{\beta} t \isubx{s}$}
	\DisplayProof
	&
	\AxiomC{$t \Rew{\beta} t'$}

	\UnaryInfC{$t  s \Rew{\beta} t'  s$}
	\DisplayProof
 &

	\AxiomC{$t \Rew{\beta} t'$}

	\UnaryInfC{$\la\vart\ \Rew{\beta} \la\vart' $}
	\DisplayProof
	&
	\AxiomC{$t \Rew{\beta} t'$}

	\UnaryInfC{$s  t \Rew{\beta} s  t'$}
	\DisplayProof
\end{array}\]
Let us recall two basic substitutivity properties of $\beta$ reduction.
\begin{enumerate}
    \item \label{p:tob-subs-one}
    \emph{Left substitutivity of $\Rew{\beta}$}:
    if $t \Rew{\beta} t'$ then $t\isub\vars \Rew{\beta} t'\isub\vars$.
    \item \label{p:tob-subs-two}
    \emph{Right substitutivity of $\Rew{\beta}$}:
    if $s \Rew{\beta} s'$ then $t\isub\vars \Rew{\beta}^* t\isubx{s'}$. It is possible to spell out the number of $\Rew{\beta}$-steps, which is exactly the number of free occurrences of $x$ in $t$, that is, $t\isub\vars \Rew{\beta}^{\sizeptx} t\isubx{s'}$.
\end{enumerate}
\paragraph{Parallel $\beta$ reduction.} Parallel $\beta$-reduction $\parRew{\beta}$ is defined by:
\begin{center}
		\def4pt{1pt}
$\begin{array}{c@{\ \hspace*{0.35cm}}c@{\ \hspace*{0.35cm}}c@{\ \hspace*{0.35cm}}c}
 \multicolumn{4}{c}{\textsc{Parallel $\beta$ reduction}}
 \\[5pt]
\AxiomC{}
\UnaryInfC{$x \parRew{\beta} x$}
\DisplayProof
&
\AxiomC{$t \parRew{\beta} t'$}
\UnaryInfC{$\la \vart \parRew{\beta}  \la x t'$}
\DisplayProof
&
\AxiomC{$t \parRew{\beta} t'$}
\AxiomC{$s \parRew{\beta} s'$}
\BinaryInfC{$t s \parRew{\beta}  t' s'$}
\DisplayProof
&
	\AxiomC{$t \parRew{\beta}  t'$}
	\AxiomC{$s \parRew{\beta} s'$}
\BinaryInfC{$(\lax t)s \parRew{\beta} t'\isub x {s'}$}
\DisplayProof
\end{array}$
	\def4pt{4pt}
\end{center}
Tait--Martin-L\"of's proof of the confluence of $\Rew{\beta}$ relies on the
diamond property of $\parRew{\beta}$\footnotemark,
\footnotetext{Namely, if $s_1 \; \mbox{}_{\beta}{\Leftarrow}\  t \parRew{\beta} s_2$
	then there exists $u$ such that $s_1 \parRew{\beta} u \; \mbox{}_{\beta}{\Leftarrow}\  s_2$.} in turn based on the
following property (see Takahashi~\cite[p.~1]{DBLP:journals/iandc/Takahashi95})
\begin{center}
	 \emph{Substitutivity of $\parRew{\beta}$}:
	if  $t \parRew{\beta} t'$ and  $s \parRew{\beta} s'$ then $t\isub\vars \parRew{\beta} t'\isubx{s'}$.
\end{center}
While the diamond property of $\parRew{\beta}$ does not play a role for factorization, one of the contributions of this work is a new proof technique for factorization relying on the substitutivity property of an indexed refinement of $\parRew{\beta}$.

\paragraph{Indexed parallel $\beta$ reduction.}
The new \emph{indexed} version $\partobind n$ of parallel $\beta$ reduction $\parRew{\beta}$ is equipped
  with  a natural number $n$ which is, roughly, the number of redexes reduced in parallel by a $\parRew{\beta}$; more precisely, $n$ is the length of a particular way of sequentializing the redexes reduced by $\parRew{\beta}$.
  The definition of $\partobind{n}$ is as follows (note that erasing the index one obtains exactly $\parRew{\beta}$, so that $\parRew{\beta} \,=\, \bigcup_{n \in \mathbb{N}} \partobind{n}$):
\begin{center}
		\def4pt{1pt}
$\begin{array}{c@{\ \hspace*{0.35cm}}c@{\ \hspace*{0.35cm}}c@{\ \hspace*{0.35cm}}c}
 \multicolumn{4}{c}{\textsc{Indexed parallel $\beta$ reduction}}
 \\[5pt]
\AxiomC{}
\UnaryInfC{$x \partobind 0 x$}
\DisplayProof
&
\AxiomC{$t \partobind n t'$}
\UnaryInfC{$\la \vart \partobind n  \la x t'$}
\DisplayProof
&
\AxiomC{$t \partobind n t'$}
\AxiomC{$s \partobind m s'$}
\BinaryInfC{$t s \partobindlong {n + m} t' s'$}
\DisplayProof
&
	\AxiomC{$t \partobind n t'$}
	\AxiomC{$s \partobind m s'$}
\BinaryInfC{$(\lax t)s \partobindlong {n + \sizep{t'}x \cdot m +1} t'\isub x {s'}$}
\DisplayProof
\end{array}$
	\def4pt{4pt}
\end{center}

The intuition behind the last clause
is: $(\lax t)s$ reduces to $t'\isub x {s'}$ by
\begin{enumerate}
\item first reducing $(\lax t)s$ to $t\isub x {s}$ ($1$ step);
\item then reducing in $t\isub x {s}$ the $n$ steps corresponding to the sequence\xspace $t \partobind n t'$, obtaining $t'\isub x {s}$;
\item then reducing $s$ to $s'$ for every occurrence of $x$ in $t'$ replaced by $s$, that is, $m$ steps $\sizep{t'}x$ times, obtaining $t'\isub x {s'}$.
\end{enumerate}
Points 2 and 3 hold because of the substitutivity properties of $\beta$ reduction.

It is easily seen that $\partobind 0$ is
the identity relation on terms.
Moreover, $\Rew{\beta} \ =\ \partobind 1$, and $\partobind n \ \subseteq \ \Rew{\beta}^n$, as expected.
The substitutivity of $\partobind n$ is proved by simply indexing the proof of substitutivity of $\parRew{\beta}$.

\begin{lemma}[Substitutivity of $\partobind n$]
\label{l:partobind-subs}
If $t \partobind n t'$ and $s \partobind m s'$, then $t \isub\vars \partobind{k} t' \isubx{s'}$ where $k = n + \sizep{t'}x\cdot m$.
\end{lemma}

\begin{proof}
	By induction on the derivation of $t \partobind n t'$.
	Consider its last rule.
	Cases:
	\begin{itemize}
		\item \emph{Variable}: two sub-cases
		\begin{itemize}
			\item $t = x$: then $t = x \partobind 0 x = t'$ then $t \isub\vars = x\isub\vars = s \partobind m s' = x \isubx{s'} = t' \isubx{s'}$ that satisfies the statement because $n + \sizep{t'}x\cdot m = 0+1\cdot m = m$.
			\item $t = y$: then $t = y \partobind 0 y = t'$ and $t \isub\vars = y\isub\vars = y \partobind 0 y = y \isubx{s'} = t' \isubx{s'}$ that satisfies the statement because $n + \sizep{t'}x\cdot m =
			0 + 0 \cdot m = 0$.
		\end{itemize}

		\item \emph{Abstraction}, {\em i.e.}\xspace $t = \la{y}{u} \partobind{n} \la{y}{u'} = t'$ because $u \partobind{n} u'$;
		we can suppose without loss of generality that $y \neq x$ and $y$ is not free in $s$ (and hence in $s'$), so $\sizep{u'}{x} = \sizep{t'}{x}$ and $t\isub{x}{s} = \la{y}(u\isub{x}{s})$ and $t'\isub{x}{s'} = \la{y}(u'\isub{x}{s'})$.
		By {\textit{i.h.}}\xspace, $u\isub{x}{s} \partobindlong{n +  \sizep{u'}{x} \cdot m} u'\isub{x}{s'}$, thus
		\begin{align*}
		\AxiomC{$u\isub{x}{s} \partobindlong{n +  \sizep{u'}{x} \cdot m} u'\isub{x}{s'}$}

		\UnaryInfC{$t\isub{x}{s} = \la {y}u\isub{x}{s} \partobindlong{n +  \sizep{t'}{x} \cdot m} \la{y}u'\isub{x}{s'} = t'\isub{x}{s'}$}
		\DisplayProof\,.
		\end{align*}

		\item \emph{Application}, {\em i.e.}\xspace
		$t = u r \partobind{n} u' r'= t'$ with $u \partobind {n_u} u'$, $r \partobind {n_r} r'$ and $n = n_u + n_r$.
		By {\textit{i.h.}}\xspace, $u\isub\vars \partobindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{s'}$ and $r\isub\vars \partobindlong {n_r +\sizep{r'}x \cdot m} r'\isubx{s'}$. Then
		\[
			\def4pt{1pt}
			\def\hskip .2in{\hskip .5in}
			\AxiomC{$u\isub\vars \partobindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{s'}$}
		\AxiomC{$r\isub\vars \partobindlong {n_r +\sizep{r'}x \cdot m} r'\isubx{s'}$}

		\BinaryInfC{$t\isub\vars = u\isub\vars r\isub\vars \partobind{k} u'\isubx{s'} r'\isubx{s'} = t'\isubx{s'}$}
		\DisplayProof
						\def\hskip .2in{\hskip .2in}
			\def4pt{4pt}
			\]
	where $k = n_u + \sizep{u'}x \!\cdot m + n_r + \sizep{r'}x \!\cdot m = n + (\sizep{u'}x + \sizep{r'}x)\cdot m = n + \sizep{t'}x \!\cdot m$.

		\item \emph{$\beta$-step}, {\em i.e.}\xspace\
		$t = (\lay u)r \partobind{n} u'\isub y {r'} = t'$
		with $u \partobind {n_u} u'$, $r \partobind {n_r} r'$ and $n = n_u + \sizep{u'}y \cdot n_r +1$.
		We can assume without loss of generality that $y\neq x$ and $y$ is not free in $s$ (and so in $s'$),
		hence $\sizep{t'}{x} = \sizep{u'\isuby {r'}}x = \sizep{u'}{x} + \sizep{u'}{y}\cdot\sizep{r'}{x}$ and $\sizep{u'\isub{x}{s'}}{y} = \sizep{u'}{y}$ and $t\isub\vars = (\la\vartwou\isub\vars) (r\isub\vars)$ and $t'\isubx{s'} = u'\isubx{s'} \isuby{r'\isubx{s'}}$.

		By {\textit{i.h.}}\xspace, $u\isub\vars \partobindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{s'}$ and $r\isub\vars \partobindlong {n_r +\sizep{r'}x \cdot m} r'\isubx{s'}$.
		Then
		\[
					\def\hskip .2in{\hskip .5in}
		\AxiomC{$u\isub\vars \partobindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{s'}$}
		\AxiomC{$r\isub\vars \partobindlong {n_r +\sizep{r'}x \cdot m} r'\isubx{s'}$}

		\BinaryInfC{$t\isub\vars = (\la\vartwou\isub\vars) (r\isub\vars)
		\partobind{k}
		 u'\isubx{s'} \isuby{r'\isubx{s'}}= t'\isubx{s'}$}
		\DisplayProof
					\def\hskip .2in{\hskip .2in}
		\]

		where
		$k = n_u +\sizep{u'}x \cdot m +\sizep{u'}y\cdot(n_r +\sizep{r'}x \cdot m)+1
		=
		n_u + \sizep{u'}y\cdot n_r + 1 + \sizep{u'}x \cdot m + \sizep{u'}y\cdot \sizep{r'}x \cdot m = n + \sizep{t'}x\cdot m.$
		\qedhere
	\end{itemize}
\end{proof}

\section{Head Reduction, Essentially}
\label{sect:head}
We here revisit Takahashi's study \cite{DBLP:journals/iandc/Takahashi95} of head reduction.
We apply the abstract schema for essential reductions developed in \refsect{abstract-rewriting}, which is the same schema used by Takahashi, but we provide a simpler proof technique for one of the required properties (split).
First of all, head reduction $\Rew{h}$
(our essential reduction here)
and its associated inessential reduction $\Rew{\lnot h}$ are defined by:
\[\begin{array}{c}
\begin{array}{c@{\hspace{.5cm}} c@{\hspace{.5cm}} cccc}
 \multicolumn{3}{c}{\textsc{Head reduction}}
 \\[5pt]
	\AxiomC{}

	\UnaryInfC{$(\lax t) s \Rew{h} t\isub\vars$}
	\DisplayProof
	&
	\AxiomC{$t \Rew{h} s$}
	\AxiomC{$t\neq\la\vart'$}

	\BinaryInfC{$t u \Rew{h} s u$}
	\DisplayProof
	&
	\AxiomC{$t \Rew{h} s$}

	\UnaryInfC{$\la\vart \Rew{h} \la\vars$}
	\DisplayProof
\end{array}
\\[18pt]
\begin{array}{c@{\hspace{.5cm}} c@{\hspace{.5cm}} c@{\hspace{.5cm}} c}
\multicolumn{4}{c}{\textsc{$\neg$Head reduction}}
\\[5pt]
\AxiomC{$t \Rew{\beta} t'$}
		\UnaryInfC{$(\lax t) s \Rew{\lnot h} (\lax t') s$}
		\DisplayProof
		&
	\AxiomC{$t \Rew{\beta} t'$}

	\UnaryInfC{$s t  \Rew{\lnot h}  s t'$}
	\DisplayProof
	&
	\AxiomC{$t \Rew{\lnot h} t'$}

	\UnaryInfC{$\la\vart \Rew{\lnot h} \la\vart'$}
	\DisplayProof
	&
		\AxiomC{$t \Rew{\lnot h} t'$}

	\UnaryInfC{$t s \Rew{\lnot h} t' s $}
	\DisplayProof \,.
\end{array}
\end{array}\]
Note that $\Rew{\beta} \, = \, \Rew{h} \cup \Rew{\lnot h}$ but $\Rew{h}$ and $\Rew{\lnot h}$ are not disjoint: $I (II) \Rew{h} II$ and $I(II) \Rew{\lnot h} II$ with $I = \la{z}{z}$.
Indeed,  $I(II)$ contains two distinct redexes, one is $I(II)$ and is fired by $\Rew{h}$, the other one is $II$ and is fired by $\Rew{\lnot h}$;
coincidentally, the two reductions lead to the same term.

 As for Takahashi, a parallel $\neg$head step
$t \parRew{\neg h} s$ is a parallel step $t \parRew{\beta} s$ such that $t \Rew{\lnot h}^* s$. We give explicitly the inference rules for $\parRew{\neg h}$:
\begin{center}
\textsc{Parallel $\neg$head reduction}
\def4pt{1pt}
\def\hskip .2in{\hskip .1in}
\begin{align*}
		\AxiomC{}
		\UnaryInfC{$x \parRew{\neg h} x$}
		\DisplayProof
		\qquad
		\AxiomC{$t \parRew{\beta} t'$}
		\AxiomC{$s \parRew{\beta} s'$}
		\BinaryInfC{$(\lax t) s \parRew{\neg h} (\lax t') s'$}
		\DisplayProof
		\qquad
		\AxiomC{$t \parRew{\neg h} t'$}
		\UnaryInfC{$\la\vart \parRew{\neg h} \la\vart'$}
		\DisplayProof
		\qquad
		\AxiomC{$t \parRew{\neg h} t'$}
		\AxiomC{$s \parRew{\beta} s'$}
		\BinaryInfC{$t s \parRew{\neg h} t' s'$}
		\DisplayProof
	\end{align*}
	\def4pt{4pt}
	\def\hskip .2in{\hskip .2in}
\end{center}
Easy inductions show that $\Rew{\lnot h} \,\subseteq\, \parRew{\neg h} \,\subseteq\, \Rew{\lnot h}^*$.  It is immediate that $\Rew{h}$-normal terms are head normal forms
in the sense of Barendregt \cite[Def. 2.2.11]{Barendregt84}. We do not describe the shape of head normal forms.
Our proofs never use it,
unlike Takahashi's ones. This fact stresses the abstract nature of our proof method.

\paragraph{Head factorization.} We show that $\Rew{h}$ induces a \macrostep\ system, with respect to $\Rew{\lnot h}$, $\parRew{\beta}$, and $\parRew{\neg h}$, to obtain $\Rew{h}$-factorization by \refprop{parallelsplit-gives-fact}.

Therefore, we need to prove merge and split.
Merge is  easily verified by  induction on $t \partonoths$.
The interesting part is the proof of the split property, that in the concrete case of head reduction becomes: if $t \parRew{\beta} s$ then $t \Rew{h}^* \!\cdot\! \parRew{\neg h} s$.
This is obtained as a consequence of the following easy \emph{indexed split} property based on the indexed variant of parallel $\beta$. The original argument of Takahashi \cite{DBLP:journals/iandc/Takahashi95} is more involved, we discuss it after the new proof.

\newcounter{prop:macro-head}
\addtocounter{prop:macro-head}{\value{proposition}}
\begin{proposition}[Head macro-step system]
\label{prop:macro-head}
\hfill
\begin{enumerate}
	\item\label{p:macro-head-merge} \emph{Merge}: if $t \parRew{\neg h} \!\cdot\! \Rew{h} u$ then $t \parRew{\beta} u$.

 	\item\label{p:macro-head-indexed-split} \emph{Indexed split}: if $t \partobind n s$ then $t \partonoths$, or $n>0$ and $t \Rew{h} \!\cdot\! \partobind {n-1} s$.

 	\item\label{p:macro-head-split} \emph{Split}: if $t \parRew{\beta} s$ then $t \Rew{h}^* \!\cdot\! \partonoths$.
\end{enumerate}
That is, $(\Lambda, \{ \Rew{h}, \Rew{\lnot h} \})$ is a \macrostep\ system with respect to $\parRew{\beta}$ and $\parRew{\neg h}$.
\end{proposition}

\begin{proof}
\begin{enumerate}
 	\item Easy induction on $t \partonoths$.
 	Details are in \SLV{\cite{long}.}{the Appendix, p.~\pageref{propappendix:macro-head}. }

\item By induction on $t \partobind n s$.
We freely use the fact that if $t \partobind n s$ then $t \parRew{\beta} s$. Cases:
\begin{itemize}
	\item \emph{Variable}: $t = x \partobind 0 x = s$. Then $t = x \partonothx = s$.

	\item \emph{Abstraction}: $t = \la \vart' \partobind n  \la x s' = s$
	with $t' \partobind n s'$. It follows from the~{\textit{i.h.}}\xspace

	\item \emph{Application}:
 $t = r p \partobind{n} r' p' = s$ with $r \partobind {n_1} r'$, $p \partobind {n_2} p'$ and $n = n_1 + n_2$.
There are only two subcases:
		\begin{itemize}
			\item either $r p \parRew{\neg h} r' p'$, and then the claim holds;
			\item or $r p \not\parRew{\neg h} r' p'$, and then neither $r \parRew{\neg h} r'$ nor $r$ is an abstraction (otherwise $r p \parRew{\neg h} r' p'$).
			By {\textit{i.h.}}\xspace applied to $r \partobind{n_1} r'$,
			$n_1>0$ and there exists $r''$ such that $r \Rew{h} r'' \partobind {n_1-1} r'$.
			Thus, $t = r p \Rew{h} r'' p$ and
			\[\AxiomC{$r'' \partobindlong {n_1-1} r'$}
			\AxiomC{$p \partobind {n_2} p'$}
			\BinaryInfC{$ r'' p \partobindlong {n_1-1 + n_2} r' p' = s$}
			\DisplayProof
			\ .
			\]
		\end{itemize}

	\item \emph{$\beta$-step}:
	$t = (\lax u)r \partobind{n} u'\isub x {r'} = s$ with $u \partobind {n_1} u'$, $r \partobind {n_2} r''$ and $n = n_1 + \sizep{u'}x \cdot n_2 +1 > 0$.
	We have $t = (\lax u)r \Rew{h} u\isub\varr$ and by substitutivity of $\partobind n$ (\reflemma{partobind-subs}) $u\isub\varr \partobindlong{n_1 + \sizep{u'}x \cdot n_2} u' \isubx{r'} = s$.

\end{itemize}
 	\item If $t \parRew{\beta} s$ then $t \partobind n s$ for some $n$. We prove the statement by induction $n$.
 	By \emph{indexed split} (\refpoint{macro-head-indexed-split}), there are only two cases:
	\begin{itemize}
	  \item \emph{$t \partonoths$}. This is an instance of the statement (since $\Rew{h}^*$ is reflexive).

	  \item $n>0$ and there exists $r$ such that $t \Rew{h} r \partobind {n-1} s$.
	  By {\textit{i.h.}}\xspace applied to $r \partobind{n-1} s$, there is $u$ such that $r \Rew{h}^* u \parRew{\neg h} s$, and so $t \Rew{h}^* u \partonoths$.
	  \qedhere
	\end{itemize}
\end{enumerate}
\end{proof}

\begin{theorem}[Head factorization]
\label{thm:head-fact}
 If $t \Rew{\beta}^* \tmthree}%{u$ then $t\Rew{h}^* \!\cdot\! \Rew{\lnot h}^* \tmthree}%{u$.
\end{theorem}

\paragraph{Head normalization.} We
{show} that $(\Lambda, \{ \Rew{h}, \Rew{\lnot h} \})$ is an \essential system (\refdef{extsys});
{thus} the essential normalization theorem (\refthm{par-normalization}) provides normalization.
We already proved \emph{factorization} (\refthm{head-fact}, hence terminal factorization).
We {verify}
persistence and determinism (which implies uniform termination) of $\Rew{h}$.

\newcounter{prop:essential-head}
\addtocounter{prop:essential-head}{\value{proposition}}
\begin{proposition}[Head essential system]
\label{prop:essential-head}
\NoteProof{propappendix:essential-head}
\begin{enumerate}
 \item\label{p:essential-head-persistence} \emph{Persistence}: if $t \Rew{h} s$ and $t \Rew{\lnot h} u$ then $u \Rew{h} r$ for some $r$.
 \item\label{p:essential-head-determinism} \emph{Determinism}: if $t \Rew{h} s_1$ and $t\Rew{h} s_2$ then $s_1 = s_2$.
\end{enumerate}
Then, $(\Lambda, \set{ \Rew{h}, \Rew{\lnot h}})$ is an essential system.
\end{proposition}

\begin{theorem}[Head normalization]
		If $t \Rew{\beta}^* s$ and $s$ is a $\Rew{h}$-normal form, then $\Rew{h}$ terminates on $t$.
\end{theorem}

\subsection{Comparison with Takahashi's proof of the split property.}\label{sect:Takahashi} Our technique differs from Takahashi's in that it is built on simpler properties:
it exploits directly the substitutivity of $\parRew{\beta}$, which is instead not used by Takahashi.
Takahashi's original argument \cite{DBLP:journals/iandc/Takahashi95} for the split property (\emph{if $t \parRew{\beta} s$ then $t\Rew{h}^* \cdot \parRew{\neg h}$}, what she calls the \emph{main lemma}) is by induction on the (concrete) definition of $\parRew{\beta}$ and relies on two substitutivity properties of $\Rew{h}$ and $\parRew{\neg h}$. Looking at them as the reductions $\Rew{\esym}$ and $\Rew{\isym}$ of an essential system, these properties are:
  \begin{itemize}
    \item \emph{Left substitutivity of $\Rew{\esym}$}: if $u \Rew{\esym} q$ then $u \isub\varr \Rew{\esym} q \isub\varr$;
    \item \emph{Left substitutivity of $\parRew{\neg\esym}$}: if $u \parRew{\neg\esym} q$ then $u \isub\varr \parRew{\neg\esym} q \isub\varr$.
  \end{itemize}
From them, left substitutivity of the composed reduction $\Rew{\esym}^* \!\cdot\! \parRew{\neg\esym}$ easily follows. That is, Takahashi's proof of the split property  is by induction on $t \parRew{} s$ using left substitutivity of $\Rew{\esym}^*\cdot\parRew{\neg\esym}$ for the inductive case.

We
exploit the substitutivity of $\partoind n$ instead of left substitutivity of $\Rew{\esym}$ and $\parRew{\neg\esym}$.
It holds for a larger number of essential systems  because $\partoind{n}$ is simply a decoration of $\parRew{}$, which is substitutive \emph{by design}.
There are important systems where Takahashi's hypotheses do not hold.
One such case is $\ell o$\xspace reduction (\refsect{leftmost})---\emph{the} normalizing
	reduction of the $\l$-calculus---we discuss the failure of left substitutivity for $\ell o$\xspace on p.~\pageref{thm:LO-fact-norm};
	another notable case is \ll reduction  (\refsect{stratified-cbn}); both  are full\xspace reductions for~$\beta$.

Let us
point out where the idea behind our approach stems from.
In a sense, Takahashi's proof works by chance: the
split hypothesis is about a \emph{parallel} step $\parRew{\beta}$ but then the key fact used in the proof, left substitutivity of $\Rew{h}^* \!\cdot\! \parRew{\neg h}$, does no longer stay in the borders of the parallel step, since
the prefix $\Rew{h}^*$ is an arbitrary long sequence\xspace that may reduce \emph{created} steps. Our proof scheme instead only focuses on the (expected)
 substitutivity of $\partoind n$, independently of creations.

\section{\texorpdfstring{The Call-by-Value $\l$-Calculus}{The Call-by-Value lambda-Calculus}}
\label{sect:cbv-calculus}

In this short section, we introduce Plotkin's  call-by-value
$\l$-calculus \cite{DBLP:journals/tcs/Plotkin75},
where
 $\beta$ reduction fires only when the argument is a value. In the next section we define \emph{weak} reduction and prove factorization and normalization theorems using the essential technique, exactly as done in the previous section for head reduction.

The set $\Lambda$ of terms is  the same as in \refsect{lambda}. Values,  call-by-value ({CbV}\xspace) $\beta$-reduction $\Rew{\betav}$, and
{CbV}\xspace indexed parallel reduction $\partobvind{n}$ are defined as follows:
 \begin{center}
 $\begin{array}{c@{\hspace{.5cm}}rll}
\textsc{Values} & v &::=& x \mid  \la\vart
\end{array}$
\\[6pt]
$\begin{array}{c@{\ \hspace*{0.35cm}}c@{\ \hspace*{0.35cm}}c@{\ \hspace*{0.35cm}}c}
 \multicolumn{4}{c}{\textsc{$\beta_\val$ reduction}}
 \\[5pt]
	\AxiomC{$v$ value}

 	\UnaryInfC{$(\la\vart) v \Rew{\betav} t \isubx{v}$}
 	\DisplayProof
 	&
 		\AxiomC{$t \Rew{\betav} t'$}

 	\UnaryInfC{$\la\vart\ \Rew{\betav} \la\vart' $}
 	\DisplayProof
 	&
 	\AxiomC{$t \Rew{\betav} t'$}

 \UnaryInfC{$t  s \Rew{\betav} t'  s$}
 \DisplayProof
 	&
 	\AxiomC{$t \Rew{\betav} t'$}

 	\UnaryInfC{$s  t \Rew{\betav} s  t'$}
 	\DisplayProof
\end{array}$
\end{center}
\begin{center}

		\textsc{Indexed parallel $\beta_\val$ reduction}\\[5pt]
	\small
	\def4pt{1pt}
	$\begin{array}{c@{\ \hspace*{0.35cm}}c@{\ \hspace*{0.35cm}}c@{\ \hspace*{0.35cm}}c}
		\AxiomC{}

		\UnaryInfC{$x \partobvind 0 x$}
		\DisplayProof
		&
		\AxiomC{$t \partobvind n t'$}

		\UnaryInfC{$\la \vart \partobvind n  \la x t'$}
		\DisplayProof
        &
		\AxiomC{$t \partobvind n t'$}
		\AxiomC{$s \partobvind m s'$}

		\BinaryInfC{$t s \partobvindlong {n + m} t' s'$}
		\DisplayProof
		&
		\AxiomC{$t \partobvind n t'$}
		\AxiomC{$v \partobvind m v'$}

		\BinaryInfC{$(\lax t)v \partobvindlong {n + \sizep{t'}{x} \cdot m +1} t'\isub x {v'}$}
		\DisplayProof
	\end{array}$
	\def4pt{4pt}
\end{center}
The only difference with the usual parallel $\beta$ (defined in \refsect{lambda}) is the requirement that the argument is a value in the last rule.
As before, the non-indexed parallel reduction $\parRew{\betav}$ is simply obtained by  erasing the index, so that $\parRew{\betav} \,=\, \bigcup_{n \in \mathbb{N}} \partobvind{n}$. Similarly, it is easily seen that $\partobvind{0}$ is the identity relation on terms, $\Rew{\betav} \,=\, \partobvind 1$ and $\partobvind n \,\subseteq\, \Rew{\betav}^n$.
Substitutivity of $\partobvind n$ is proved exactly as for $\partobind n$ (\reflemma{partobind-subs}).

\newcounter{l:tobv-subs}
\addtocounter{l:tobv-subs}{\value{lemma}}
\begin{lemma}[Substitutivity of $\partobvind n$]
\label{l:tobv-subs}
\NoteProof{lappendix:tobv-subs}
\label{p:tobv-subs-ind-par}  If $t \partobvind n t'$ and $v \partobvind m v'$, then $t \isubxv \allowbreak\partobvind{k} t' \isubx{v'}$ where $k = n + \sizep{t'}x\cdot m$.
\end{lemma}

\section{Weak Call-by-Value Reduction, Essentially}
\label{sect:weak-cbv}
The \essential step  we study for the {CbV}\xspace $\l$-calculus is weak {CbV}\xspace reduction $\Rew{\wsym}$, which does not evaluate function bodies (the scope of $\l$-abstractions). Weak  {CbV}\xspace reduction has practical importance, because it is the base of the ML/CAML
family of functional programming languages. We choose it also because it admits the natural and more general \emph{non-deterministic} presentation that follows, even if most of the literature rather presents it in a deterministic way.
\begin{center}
\textsc{Weak {CbV}\xspace reduction} \\[5pt]
$\begin{array}{c@{\ \ \hspace*{0.35cm}}c@{\ \ \hspace*{0.35cm}}c}
 \AxiomC{}
 \UnaryInfC{$(\lax t) v \Rew{\wsym} t\isubxv$}
 \DisplayProof
 &
  \AxiomC{$t \Rew{\wsym} t'$}

 \UnaryInfC{$t  s \Rew{\wsym} t'  s$}
 \DisplayProof
 &
 \AxiomC{$t \Rew{\wsym} t'$}

 \UnaryInfC{$s  t \Rew{\wsym} s  t'$}
 \DisplayProof
\end{array}$
\end{center}
Note that in the case of an
application there is no fixed order in the $\Rew{\wsym}$-reduction of the left and right subterms.
Such a non-determinism is harmless as $\Rew{\wsym}$ satisfies a diamond-like property implying confluence, see
\refpropp{essential-weak}{diamond} below.
It is
well-known that the diamond property implies uniform termination, because it implies
that  all  maximal sequence\xspace{s}  from a term
have the same length. Such a further property is  known as \emph{random descent},
a special form of uniform termination already considered by Newman \cite{Newman} in 1942, see also van~Oostrom~\cite{DBLP:conf/rta/Oostrom07}.

 The inessential reduction $\Rew{\neg \wsym}$ and its parallel version $\parRew{\neg \wsym}$ are defined by:
\begin{center}
	 $\begin{array}{c@{\hspace{.5cm}} c@{\hspace{.5cm}} c@{\hspace{.5cm}} ccc}
 \multicolumn{3}{c}{\textsc{$\neg$Weak reduction}}
 \\[5pt]
 	\AxiomC{$t \Rew{\betav} s$}

 	\UnaryInfC{$\lax t \Rew{\neg \wsym} \lax s $}
 	\DisplayProof
 	&
 		\AxiomC{$t \Rew{\neg \wsym} t'$}

 	\UnaryInfC{$t  s \Rew{\neg \wsym} t'  s$}
 	\DisplayProof
 	&
 	\AxiomC{$t \Rew{\neg \wsym} t'$}

 	\UnaryInfC{$s  t \Rew{\neg \wsym} s  t'$}
 	\DisplayProof
 \end{array}$
\end{center}
\begin{center}
	\textsc{Parallel $\neg$weak reduction}
		\def4pt{1pt}
	\begin{align*}
  \AxiomC{}
    \UnaryInfC{$x \parRew{\neg \wsym} x$}
  \DisplayProof
	\qquad
	\AxiomC{$t \parRew{\betav} t'$}
	\UnaryInfC{$\la\vart \parRew{\neg \wsym} \la\vart'$}
	\DisplayProof
	\qquad
	\AxiomC{$t \parRew{\neg \wsym} t'$}
	\AxiomC{$s \parRew{\neg \wsym} s'$}
	\BinaryInfC{$t s \parRew{\neg \wsym} t' s'$}
	\DisplayProof
\end{align*}
	\def4pt{4pt}
\end{center}
It  is  immediate to check that $\Rew{\betav} \, = \, \Rew{\wsym} \cup \Rew{\neg \wsym}$ and
  $\Rew{\neg \wsym} \,\subseteq\, \parRew{\neg \wsym} \,\subseteq\, \Rew{\neg \wsym}^*$.

\paragraph{Weak {CbV}\xspace factorization.}
We show that  $(\Lambda, \{ \Rew{\wsym}, \Rew{\neg \wsym} \})$ is a \macrostep\ system, with $\parRew{\betav},\parRew{\neg \wsym}$ as \macrostep{s}. Merge and split are proved exactly as in \refsect{head}.

\newcounter{prop:macro-cbv}
\addtocounter{prop:macro-cbv}{\value{proposition}}
\begin{proposition}[Weak {CbV}\xspace macro-step system]
\label{prop:macro-cbv}
\NoteProof{propappendix:macro-cbv}
\begin{enumerate}
 \item\label{p:macro-cbv-merge} \emph{Merge}: if $t \parRew{\neg \wsym}\cdot \Rew{\wsym} u$ then $t \parRew{\betav} u$.
 \item\label{p:macro-cbv-indexed-split} \emph{Indexed split}: if $t \partobvind n s$ then $t \partonotws$, or $n>0$ and $t \Rew{\wsym} \!\cdot\! \partobvind {n-1} s$.
 \item\label{p:macro-cbv-split} \emph{Split}: if $t \parRew{\betav} s$ then $t \Rew{\wsym}^* \!\cdot\! \partonotws$.
\end{enumerate}
That is, $(\Lambda, \set{\Rew{\wsym}, \Rew{\neg \wsym}})$ is a macro-step system with respect to $\parRew{\betav}$ and $\parRew{\neg \wsym}$.
\end{proposition}

\begin{theorem}[Weak {CbV}\xspace factorization]
\label{thm:weak-fact}
 If $t \Rew{\betav}^* s$ then $t\Rew{\wsym}^* \!\cdot\! \Rew{\neg \wsym}^* s$.
\end{theorem}

\paragraph{Plotkin's left reduction.}
 The same argument at work in this section adapts easily to factorization with respect to leftmost weak reduction (used by Plotkin \cite{DBLP:journals/tcs/Plotkin75}), or to rightmost weak reduction, the two natural deterministic variants of $\Rew{\wsym}$.

\paragraph{Weak {CbV}\xspace normalization.}
To obtain a normalization theorem for $\Rew{\wsym}$ via the essential normalization theorem (\refthm{par-normalization}), we need
persistence and uniform termination.
The latter
{follows from} the well-known diamond property of $\Rew{\wsym}$.

\newcounter{essential-weak}
\addtocounter{essential-weak}{\value{proposition}}
\begin{proposition}[Weak {CbV}\xspace essential system]
	\label{prop:essential-weak}
	\NoteProof{propappendix:essential-weak}
	\begin{enumerate}
	 \item\label{p:essential-weak-persistence} \emph{Persistence}: if $t \Rew{\wsym} s$ and $t \Rew{\neg \wsym} u$ then $u \Rew{\wsym} r$ for some $r$.
	 \item\label{p:essential-weak-diamond} \emph{Diamond}: if $s\lRew{\wsym} \!\cdot\! \towu$ with $s \neq u$ then $s \Rew{\wsym} \!\cdot\! \lRew{\wsym} u$.
	\end{enumerate}
Then, $(\Lambda, \set{ \Rew{\wsym}, \Rew{\neg \wsym}})$ is an essential system.
\end{proposition}

\begin{theorem}[Weak {CbV}\xspace normalization]
	\label{thm:cbv-normalization}
		If $t \Rew{\betav}^* s$ and $s$ is a $\Rew{\wsym}$-normal form, then $t$ is strongly $\Rew{\wsym}$-normalizing.
\end{theorem}

{CbV}\xspace is often considered with respect to \emph{closed} terms only.
In such a case the $\Rew{\wsym}$-normal forms are exactly the (closed) values.
Then weak {CbV}\xspace normalization (\refthm{cbv-normalization}) implies the following, analogous to Corollary 1 in Plotkin \cite{DBLP:journals/tcs/Plotkin75} (the  result is there obtained from standardization).

\begin{corollary} Let $t$ be a closed term.
	If $t \Rew{\betav}^* v$ for some value $v$, then every maximal $\Rew{\wsym}$-sequence\xspace from $t$ is finite and ends in a value.
\end{corollary}

\section{Abstract Normalisation}
\label{sect:abstract-rewriting}

\section{Leftmost-Outermost Reduction, Essentially}
\label{sect:leftmost}
Here we apply our technique to leftmost-outermost (shortened to \emph{$\losym$\xspace}) reduction $\Rew{\losym}$, the first example of \emph{full\xspace}
reduction for $\Rew{\beta}$.
The technical development is slightly different from the ones in the previous sections, as factorization relies on persistence.
The same shall happen for the full\xspace \ll reduction of the next section.
It seems to be a feature of full\xspace reductions~for~$\Rew{\beta}$.

\paragraph{$\losym$\xspace and $\neg$$\losym$\xspace reductions.} The definition of $\ell o$\xspace reduction relies on two mutually recursive predicates defining normal and neutral terms (neutral = normal and not an abstraction):
\begin{center}
		\def4pt{2pt}
\textsc{Normal and neutral terms}\\[5pt]
{\small$\begin{array}{c@{\qquad}c@{\qquad}c@{\qquad}c@{\qquad}c}
  \AxiomC{}
  \UnaryInfC{$x$ is neutral}
  \DisplayProof
  &
  \AxiomC{$t$ is neutral}
  \AxiomC{$t$ is normal}
  \BinaryInfC{$ts$ is neutral}
  \DisplayProof
  &
  \AxiomC{$t$ is neutral}
  \UnaryInfC{$t$ is normal}
  \DisplayProof
  &
  \AxiomC{$t$ is normal}
  \UnaryInfC{$\la\vart$ is normal}
  \DisplayProof
\end{array}$ }
	\def4pt{4pt}
\end{center}
Dually, a term is not neutral if it is an abstraction or it is not normal. It is standard that these predicates correctly capture $\beta$ normal forms and neutrality.

The reductions of the $\ell o$\xspace macro-step system are:
 \begin{center}
 $\begin{array}{c@{\hspace{.5cm}} @{\hspace{.5cm}} ccccc}
 \multicolumn{2}{c}{\textsc{$\losym$\xspace reduction}}
 \\[5pt]
	\AxiomC{}

	\UnaryInfC{$(\lax t) s \Rew{\losym} t\isub\vars$}
	\DisplayProof
	&
	\AxiomC{$t \Rew{\losym} s$}
	\AxiomC{$t\neq\la\vart'$}

	\BinaryInfC{$t u \Rew{\losym} s u$}
	\DisplayProof \\\\

	\AxiomC{$t \Rew{\losym} s$}

	\UnaryInfC{$\la\vart \Rew{\losym} \la\vars$}
	\DisplayProof
	&
	  \AxiomC{$u$ is neutral}
	\AxiomC{$t \Rew{\losym} s$}

	\BinaryInfC{$u t \Rew{\losym} u s$}
	\DisplayProof
\end{array}$
\end{center}
\begin{center}
 \begin{tabular}{c}
$\begin{array}{c@{\hspace{.5cm}} @{\hspace{.5cm}} ccccc}
\multicolumn{2}{c}{\textsc{$\neg$$\losym$\xspace reduction}}
\\[5pt]
	\AxiomC{$t \Rew{\beta} t'$}

	\UnaryInfC{$(\lax t) s \Rew{\neg \losym} (\lax t') s$}
	\DisplayProof
	&
	\AxiomC{$t$ is not neutral}
	\AxiomC{$s \Rew{\beta} s'$}

	\BinaryInfC{$t s \Rew{\neg \losym} t s '$}
	\DisplayProof
	\end{array}$
	\\\\
$	\begin{array}{c@{\hspace{.5cm}} @{\hspace{.5cm}} c@{\hspace{.5cm}} @{\hspace{.5cm}} cccc}
	\AxiomC{$t \Rew{\neg \losym} t'$}

	\UnaryInfC{$t s \Rew{\neg \losym} t' s $}
	\DisplayProof
	&
	\AxiomC{$t \Rew{\neg \losym} t'$}

	\UnaryInfC{$s t  \Rew{\neg \losym}  s t'$}
	\DisplayProof
	&
	\AxiomC{$t \Rew{\neg \losym} t'$}

	\UnaryInfC{$\la\vart \Rew{\neg \losym} \la\vart'$}
	\DisplayProof
\end{array}$
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{c}
	\def4pt{1pt}
$\begin{array}{c@{\hspace{.5cm}} @{\hspace{.5cm}} ccccc}
\multicolumn{2}{c}{\textsc{Parallel $\neg$$\losym$\xspace reduction}}
\\[5pt]
  \AxiomC{}
    \UnaryInfC{$x \parRew{\neg \losym} x$}
  \DisplayProof
	&
	\AxiomC{$t$ is not neutral}
	\AxiomC{$t \parRew{\neg \losym} t'$}
	\AxiomC{$s \parRew{\beta} s'$}
	\TrinaryInfC{$t s \parRew{\neg \losym} t' s '$}
	\DisplayProof
\end{array}$
\\\\
\def4pt{1pt}
$\begin{array}{c@{\hspace{.5cm}} @{\hspace{.5cm}} c@{\hspace{.5cm}} @{\hspace{.5cm}} cccc}
	\AxiomC{$t \parRew{\beta} t'$}
	\AxiomC{$s \parRew{\beta} s''$}
	\BinaryInfC{$(\lax t) s \parRew{\neg \losym} (\lax t') s'$}
	\DisplayProof
	&
	\AxiomC{$t \parRew{\neg \losym} t'$}
	\UnaryInfC{$\la\vart \parRew{\neg \losym} \la\vart'$}
	\DisplayProof
	&
	\AxiomC{$t$ neutral}
	\AxiomC{$s \parRew{\neg \losym} s'$}
	\BinaryInfC{$t s \parRew{\neg \losym} t s'$}
	\DisplayProof
\end{array}$
\def4pt{4pt}
\end{tabular}

\end{center}
As
usual, easy inductions show that $\Rew{\beta} \,=\, \Rew{\losym} \cup \Rew{\neg \losym}$ and $\Rew{\neg \losym} \subseteq \parRew{\neg \losym} \subseteq \Rew{\neg \losym}^*$.

Factorization depends on persistence, which is why for $\ell o$\xspace reduction most essential properties are proved before factorization. The proofs are easy inductions.

\newcounter{prop:essential-left}
\addtocounter{prop:essential-left}{\value{proposition}}
\begin{proposition}[$\ell o$\xspace essential properties]
\label{prop:essential-left}
\NoteProof{propappendix:essential-left}
\begin{enumerate}
\item \label{p:essential-left-completeness}
\emph{Fullness\xspace}: if $t \Rew{\beta} s$ then there exists $u$ such that $t \Rew{\losym} u$.
\item\label{p:essential-left-determinism} \emph{Determinism}: if $t \Rew{\losym} s_1$ and $t\Rew{\losym} s_2$ then $s_1 = s_2$.
\item\label{p:essential-left-persistence} \emph{Persistence}: if $t \Rew{\losym} s_1$ and $t \Rew{\neg \losym} s_2$ then $s_2 \Rew{\losym} u$ for some $u$.
\end{enumerate}
\end{proposition}

\newcounter{prop:macro-left}
\addtocounter{prop:macro-left}{\value{proposition}}
\begin{proposition}[$\ell o$\xspace macro-step system]
\label{prop:macro-left}
\NoteProof{propappendix:macro-left}\hfill
\begin{enumerate}
	\item \emph{Merge}: \label{p:macro-left-merge}
if $t \parRew{\neg \losym} \cdot \Rew{\losym} u$ then $t \parRew{\beta} u$.
	\item \label{p:macro-left-indexed-split}
	\emph{Indexed split}: if $t \partobind n s$ then $t \parRew{\neg \losym} s$, or $n>0$ and  $t \Rew{\losym} \!\cdot\! \partobind {n-1} s$.
	\item\label{p:macro-step-split} \emph{Split}: if $t \parRew{\beta} s$ then $t \Rew{\losym}^* \!\cdot\! \parRew{\neg \losym} s$.
	\end{enumerate}
	That is, $(\Lambda, \set{\Rew{\losym}, \Rew{\neg \losym}})$ is a macro-step system with respect to $\parRew{\beta}$ and $\parRew{\neg \losym}$.
\end{proposition}

\begin{proof}
We only show the merge property, and only the case that requires persistence---the rest of the proof is in \SLV{the Appendix of \cite{long}}{the Appendix}.
The proof of the merge property is by induction on $t \partonotlos$. Consider the rule
	\[\AxiomC{$r$ not neutral}
	\AxiomC{$r \parRew{\neg \losym} r'$}
		\AxiomC{$p \parRew{\beta} p'$}

	\TrinaryInfC{$t = r p \parRew{\neg \losym} r' p ' = s$}
	\DisplayProof\ .\]
	Since $r$ is not neutral, it is an abstraction or it is not normal.
	If $r$ is an abstraction this case continues as the easy case of $\parRew{\neg \losym}$ for $\beta$-redexes (see \SLV{the Appendix of \cite{long}}{the Appendix}).
	Otherwise, $r$ is not normal, {\em i.e.}\xspace $r \Rew{\beta} q$. By fullness\xspace $r \Rew{\losym} q'$ for some $q'$, and by persistence (\refpropp{essential-left}{persistence}) $r'\Rew{\losym} r''$ for
	some $r''$. The hypothesis becomes $t = r p \parRew{\neg \losym} r' p ' \Rew{\losym} r'' p' = u$ with $r \parRew{\neg \losym} r'  \Rew{\losym} r''$.
	By {\textit{i.h.}}\xspace, $r \parRew{\beta} r''$. Then,
	\[\AxiomC{$r \parRew{\beta} r''$}
	\AxiomC{$p\parRew{\beta} p'$}

	\BinaryInfC{$t = r p \parRew{\beta} r'' p' = u$}
	\DisplayProof
	\ .
	\qedhere\]
\end{proof}

\paragraph{$\ell o$\xspace split.} As pointed out in \refsect{Takahashi}, Takahashi's proof \cite{DBLP:journals/iandc/Takahashi95} of the split property relies on left substitutivity of head reduction, that is, if $t \Rew{h} s$ then $t \isub\varu \Rew{h} s \isub\varu$ for all terms $u$. Such a property does not hold for $\ell o$\xspace reduction. For instance, $t = x (I y) \Rew{\losym} x y = t'$ but
$t\isubx {\laz z\varthree} = (\laz z\varthree)(Iy) \not\Rew{\losym} (\laz z\varthree) y = t'\isubx {\laz z\varthree}$.
Therefore her proof technique for factorization cannot prove the factorization theorem for $\ell o$\xspace reduction (see also footnote \ref{f:direct}, page \pageref{f:direct}).\medskip

From \refprop{macro-left} it follows the factorization theorem for $\ell o$\xspace reduction, that together with \refprop{essential-left} proves that $(\Lambda, \set{\Rew{\losym}, \Rew{\neg \losym}})$ is an essential system, giving normalization of $\Rew{\losym}$ for $\Rew{\beta}$.
\begin{theorem}
\label{thm:LO-fact-norm}
\hfill
\begin{enumerate}
\item  \emph{$\ell o$\xspace factorization}: if $t \Rew{\beta}^* \tmthree}%{u$ then $t\Rew{\losym}^* \!\cdot\!~ \Rew{\neg \losym}^* \tmthree}%{u$.

\item \emph{$\ell o$\xspace normalization}: $\Rew{\losym}$ is a normalizing reduction for $\Rew{\beta}$.
	\end{enumerate}
\end{theorem}

\section{Least-Level Reduction, Essentially}
\label{sect:stratified-cbn}
\newcommand{\text{ s.t. }}{\text{ s.t. }}
In this section we study another  normalizing full\xspace reduction for $\Rew{\beta}$, namely \emph{least-level} (shortened to \ll) \emph{reduction} $\Rew{x}$, which is non-deterministic.
The intuition is that \ll reduction
 fires a $\beta$-redex of minimal level, where the \emph{level} of a $\beta$-redex is the number of arguments containing it.

The definition of $\Rew{x}$ relies on an indexed variant $\tostratind{k}$ of $\Rew{\beta}$, where $k \in \mathbb{N}$ is the level of the fired $\beta$-redex (do not mix it up with the index of $\partobind n$).
We also define a parallel version $\partostratind{n}$ (with $n \in \mathbb{N} \cup \{\infty\}$) of $\tostratind{k}$, obtained as a decoration of $\parRew{\beta}$, where $n$ is the minimal level of the $\beta$-redexes fired by a $\parRew{\beta}$ step ($\partostratind{\infty}$ does not reduce any $\beta$-redex).
From now on, $\mathbb{N} \cup \{\infty\}$ is considered with its usual order and arithmetic, that is, $\infty + 1 = \infty$.
\begin{center}
	\textsc{ $\beta$ reduction of level $k$}
	\def4pt{1pt}
	\begin{align*}
	\AxiomC{}
	\UnaryInfC{$(\la{x}{t})s \tostratind{0} t \isub{x}{s}$}
	\DisplayProof
	\qquad

	\AxiomC{$t \tostratind{k} t'$}
	\UnaryInfC{$\la\vart \tostratind{k} \la\vart'$}
	\DisplayProof
	\qquad
	\AxiomC{$t \tostratind{k} t'$}
	\UnaryInfC{$t s \tostratind{k} t' s$}
	\DisplayProof
	\qquad
	\AxiomC{$t \tostratind{k} t'$}
	\UnaryInfC{$\tmtwot \tostratind{k\!+\!1} \tmtwot'$}
	\DisplayProof
	\end{align*}
	\def4pt{4pt}
\end{center}
\begin{center}
	\textsc{Parallel $\beta$ reduction of least level $n$}
	\def4pt{1pt}
	\begin{align*}
	\AxiomC{$t \partostratind{k} t'$}
	\AxiomC{$s \partostratind{h} s'$}
	\BinaryInfC{$(\la{x}{t})s \partostratind{0} t' \isub{x}{s'}$}
	\DisplayProof
	\quad

	\AxiomC{$t \partostratind{k} t'$}
	\UnaryInfC{$\la\vart \partostratind{k} \la\vart'$}
	\DisplayProof
	\quad
	\AxiomC{$t \partostratind{k} t'$}
	\AxiomC{$s \partostratind{h} s'$}
	\BinaryInfC{$t s \partostratind{\min\{k,h\!+\!1\}} t' s'$}
	\DisplayProof
	\quad
	\AxiomC{}
	\UnaryInfC{$x \partostratind{\infty} x$}
	\DisplayProof
	\end{align*}
	\def4pt{4pt}
\end{center}
Note  that $t \Rew{\beta} s$ if and only if $t \tostratind{k} s$ for some $k \in \mathbb{N}$.

The \emph{least (reduction) level} $\Deg{t}\in \mathbb{N} \cup \{\infty\}$ of a term $t$ is defined as follows:
\begin{align*}
\Deg{x} &= \infty
&
\Deg{\la{x}{t}} &= \Deg{t}
&
\Deg{ts} &=
\begin{cases}
0 &\textup{if } t = \la{x}{u}
\\
\min \{\Deg{t}, \Deg{s} \!+\! 1 \} &\textup{otherwise.}
\end{cases}
\end{align*}

The definitions of \emph{\ll}, \emph{$\neg$\ll}, and \emph{parallel $\neg$\ll reductions} are:
\begin{center}
\begin{tabular}{r@{\hspace{.5cm}} c@{\hspace{.5cm}} l}
 \textsc{\ll reduction} & $t \Rew{x} s$  & if $t \tostratind{k} s$ with $\degt = k \in \mathbb{N}$;
 \\
 \textsc{$\neg$\ll reduction} & $t \tonotll s$ & if $t \tostratind{k} s$ with $\degt< k \in \mathbb{N}$;
 \\
 \textsc{Parallel $\neg$\ll reduction} &$t \parRew{\lnot\ell\ell} s$ & if $t \partostratind{k} s$ with $k \!=\! \infty$ or $k \!>\! \Deg{t}$.
 \end{tabular}
\end{center}
As
usual, easy inductions show that $\Rew{\beta} = \Rew{\ell\ell} \cup \Rew{\neg \ell\ell}$ and $\Rew{\neg \ell\ell} \subseteq \parRew{\lnot\ell\ell} \subseteq \Rew{\neg \ell\ell}^*$.

\newcounter{prop:deg}
\addtocounter{prop:deg}{\value{proposition}}
\begin{proposition}[Least level properties]
\label{prop:deg}
\NoteProof{propappendix:deg}
Let $t$ be a term.
\begin{enumerate}
	\item\label{p:deg-finite}\emph{Computational meaning of $\ell\ell$}: $\Deg{t} = \inf\{ k \in \mathbb{N} \mid t \tostratind{k} u \text{ for some term } u\}$.
	\item\label{p:deg-invariance-leq} \emph{Monotonicity}: if $t \Rew{\beta} s$ then $\Deg{s} \geq \Deg{t}$.
	\item\label{p:deg-invariance-equal} \emph{Invariance by $\tonotll$}: if $t \tonotll s$ then $\Deg{s} = \Deg{t}$.
\end{enumerate}
\end{proposition}

\refpoint{deg-finite} captures the meaning of the least level, and gives fullness\xspace of $\Rew{\ell\ell}$. In particular, $\Deg{t} = \infty$ if and only if $t$ is $\Rew{\beta}$-normal, since $\inf \, \emptyset = \infty$.
Monotonicity states that $\beta$ steps cannot decrease the least level.
Invariance by $\tonotll$
says that $\tonotll$ cannot change the least level. Essentially, this is persistence.

\newcounter{prop:essential-strat}
\addtocounter{prop:essential-strat}{\value{proposition}}
\begin{proposition}[\ll essential properties]
\label{prop:essential-strat}
\NoteProof{propappendix:essential-strat}\hfill
\begin{enumerate}
\item \label{p:essential-strat-completeness}
\emph{Fullness\xspace}: if $t \Rew{\beta} s$ then $t \Rew{\ell\ell} u$ for some $u$.
\item\label{p:essential-strat-persistence} \emph{Persistence}: if $t \Rew{\ell\ell} s_1$ and $t \Rew{\neg \ell\ell} s_2$ then $s_2 \Rew{\ell\ell} u$ for some $u$.
\item\label{essential-strat-diamond} \emph{Diamond}: if $s\lRew{\ell\ell} \cdot \Rew{\ell\ell}u$ with $s \neq u$ then $s \Rew{\ell\ell} \cdot \lRew{\ell\ell} u$.
\end{enumerate}
\end{proposition}

As for $\ell o$\xspace, merge needs persistence, or, more precisely, invariance by $\tonotll$.

\newcounter{prop:macro-strat}
\addtocounter{prop:macro-strat}{\value{proposition}}
\begin{proposition}[\ll macro-step system]
\hfill\label{prop:macro-strat}
\NoteProof{propappendix:macro-strat}
	\begin{enumerate}

		\item\label{p:macro-strat-merge} \emph{Merge}: if $t \parRew{\lnot\ell\ell} s \Rew{x} u$, then $t \parRew{\beta} u$.
		\item\label{p:macro-strat-indexed-split}\emph{Indexed split}:	if $t \partobind n s$ then $t \parRew{\lnot\ell\ell}s$, or $n>0$ and $t \Rew{x} \cdot \partobindlong {n-1} s$.
		\item\label{p:macro-strat-split} \emph{Split}: if $t \parRew{\beta} s$ then $t \Rew{x}^* \cdot \parRew{\lnot\ell\ell} s$.
	\end{enumerate}
	That is, $(\Lambda, \set{\Rew{\ell\ell}, \Rew{\neg \ell\ell}})$ is a macro-step system with respect to $\parRew{\beta}$ and $\parRew{\lnot\ell\ell}$.
\end{proposition}

\begin{theorem}
\hfill
\begin{enumerate}
\item  \emph{\ll factorization}: if $t \Rew{\beta}^* \tmthree}%{u$ then $t\Rew{\ell\ell}^* \cdot \Rew{\neg \ell\ell}^* \tmthree}%{u$.

\item \emph{\ll normalization}: $\Rew{\ell\ell}$ is a normalizing reduction for $\Rew{\beta}$.
	\end{enumerate}
\end{theorem}

\paragraph{\ll split and $\ell o$\xspace \textit{vs.} \ll.} As for $\ell o$\xspace reduction, left substitutivity does not hold for $\Rew{\ell\ell}$.
Consider $t = x (I y) \Rew{\ell\ell} x y = t'$ where the step has level 1, and $t\isubx {\laz z\varthree} \allowbreak= (\laz z\varthree)(Iy) \not\Rew{\ell\ell} (\laz z\varthree) y = t'\isubx {\laz z\varthree}$ since now there also is a step $(\laz z\varthree)(Iy)\Rew{\ell\ell} (I y)(I y)$ at level 0.

Moreover, \ll and $\ell o$\xspace reductions are incomparable. First, note that
	\emph{$\Rew{\ell\ell} \not\subseteq
	\Rew{\losym}$}: $t = (\la{x}{II})y  \Rew{x} (\la{x}{I})y = s$, because $t \tostratind{0} (\la{x}{I})y$ and $\Deg{t} = 0$, but $t \not\Rew{\losym} s$, indeed $t \Rew{\losym} II$. This fact also shows that $\Rew{\ell\ell}$ is not left–outer in the sense of van Oostrom and Toyama \cite{DBLP:conf/rta/OostromT16}.
	Second, \emph
	$\Rew{\losym} \not\subseteq \Rew{\ell\ell}$}: $t = x (x(II)) (II) \Rew{\losym} x (x I)(II) = s$ but $t \not\Rew{x} s$, indeed $t \tonotll s$ because $t \tostratind{2} s$ and $\Deg{t} = 1$, and $t \Rew{\ell\ell} x (x(II)) I \neq s$.

\section{Preliminary Lemmas}
Notation: $p\to^*q$ indicates a finite sequence from $p$ to $q$.

The lemmas below are elementary properties of $\Rew{\beta}$ (resp. $\Rew{\betav}$)

\begin{lemma}\label{l:basicCbN} If $t\Rew{\beta}\tmtwo}%{s$ by a step which is \emph{not} a $\beta$-redex then  $s$ is an abstraction (resp. an application) \emph{if and only if so} is $t$:
	\begin{enumerate}

		\item  $\tmtwo}%{s=\la x s'$ iff $t=\la x t'$ with  $t'\Rew{\beta} s'$
		\item $\tmtwo}%{s=pq$ iff $t = rs$ with  ($\tmfour}%{r\Rew{\beta} p$ and $\tmtwo}%{s= q$) or ($\tmfour}%{r= p$ and $\tmtwo}%{s\Rew{\beta} q$).
	\end{enumerate}
	So if  $\tmtwo}%{s=pq$ and $t = rs$, then    $p$ is an abstraction iff $r$ is.
\end{lemma}

\begin{lemma}\label{l:basicCbV} If $t\Rew{\betav}\tmtwo}%{s$ by a step which is \emph{not} a $\beta_\val$-redex
	 then  $s$ is an abstraction (resp. an application) \emph{if and only if so} is $t$:
	\begin{enumerate}

		\item $\tmtwo}%{s=\la x s'$ iff $t=\la x t'$, with  $t'\Rew{\betav} s'$
		\item $\tmtwo}%{s=pq$ iff $t = rs$, with  ($\tmfour}%{r\Rew{\betav} p$ and $\tmtwo}%{s= q$) or ($\tmfour}%{r= p$ and $\tmtwo}%{s\Rew{\betav} q$).
	\end{enumerate}
	So  if  $\tmtwo}%{s=pq$ and $t = rs$, then $p$ is an abstraction iff $r$ is; $q$ is a value iff $s$ is.
\end{lemma}

\section{Least-Level Normalization  in CbV}
With the obvious CbV definitions of depth of the $\beta_\val$ step $\tostratind{k}$ and of least-level of a term $\deg t$,
given surface factorization, we obtain normalization.

 \CF{Per fare \textbf{induzione sulle forme normali}, non c'e' bisogno di avere una  speciale grammatica delle forme normali. }\\

We have proved \emph{surface factorization} :
\begin{theorem}[Weak {CbV}\xspace factorization]
	\label{thm:weak-fact0}
	If $t \Rew{\betav}^* s$ then $t\tostratind{0}^* \tmtwo}%{s \tostratind{>0}^* \tmthree}%{u$.
\end{theorem}

\emph{Least-Level Normalization}  follows:
\begin{theorem}[CBV Normalization]
	if $t \Rew{\betav}^* r$ with $r$  $\Rew{\betav}$-normal, then $t\Rew{\ell\ell}^*  \tmfour}%{r$.

\end{theorem}

\begin{proof}	By induction on the term $r$.
	From $t \Rew{\betav}^* r$,
	by  \refthm{weak-fact0}, we have that \begin{center}
		$t\tostratind{0}^* u  \tostratind{>0}^* r$.
	\end{center}

	Cases:

\begin{enumerate}
	\item $r=x$. Then  $u=x$, and trivially  $t\tostratind{0}^* r$.
	\item $r=\l x.r'$. Then $u=\l x. u'$, hence  $u' \Rew{\betav}^*  r'$, where  $ r' $ is $\Rew{\betav}$-normal, and we use the {\textit{i.h.}}\xspace to obtain $u'\Rew{\ell\ell}^* r'$, hence
	$t\tostratind{0}^* u=\l x. u' \Rew{\ell\ell}^* \l x.r'=r $.
	\item $r=r_pr_q$.  The sequence $u  \tostratind{>0}^* r$ never use a $\beta_\val$-rule (since all steps have depth $>0$) therefore
	(by \reflemma{basicCbV}) $u=pq$, with $p\Rew{\betav}^* r_p$ and $q\Rew{\betav}^* r_q$. Since  $r=r_pr_q$ is normal, both $r_p$ and $r_q$ are $\Rew{\betav}$-normal.

	\begin{itemize}

		\item By {\textit{i.h.}}\xspace,
		$p \Rew{\ell\ell}^*  r_p$ and $q \Rew{\ell\ell}^* r_q$.

		\item

		 We  \emph{merge}  the steps in 	$p   \Rew{\ell\ell}^* r_p$ and
		 $q\Rew{\ell\ell}^* r_q$ in increasing order, so to  conclude that $pq\Rew{\ell\ell}^* r_pr_q$, hence 	$t\tostratind{0}^* u= pq\Rew{\ell\ell}^* r_pr_q=r$.

	\end{itemize}

\end{enumerate}
		In all cases, we have proved the claim.
\end{proof}

\newpage

\section{Least-Level Normalization  in CbN}

  \CF{Supponiamo di aver dimostrato surface factorization per CbN...}
\begin{theorem}[Surface CbN\xspace factorization]
	\label{thm:cbn-fact0}
	If $t \Rew{\beta}^* s$ then $t\tobat{0}^* \tmtwo}%{s \tobat{>0}^* \tmthree}%{u$.
\end{theorem}

Least-Level Normalization follows.

\begin{theorem}[CBN Normalization]
	if $t \Rew{\beta}^* r$ with $r$  $\Rew{\beta}$-normal, then $t\Rew{\ell\ell}^*  \tmfour}%{r$.

\end{theorem}

\begin{proof}	By induction on the term $r$.
	From $t \Rew{\beta}^* r$,
	by  \refthm{weak-fact0}, we have that \begin{center}
		$t\tobat{0}^* u  \tobat{>0}^* r$.
	\end{center}

	Cases:

	\begin{enumerate}
		\item $r=x$. Then  $u=x$, and trivially  $t\tobat{0}^* r$.
		\item $r=\l x.r'$. Then $u=\l x. u'$, hence  $u' \Rew{\beta}^*  r'$, where  $ r' $ is $\Rew{\beta}$-normal, and we use the {\textit{i.h.}}\xspace to obtain $u'\Rew{\ell\ell}^* r'$, hence
		$t\tobat{0}^* u=\l x. u' \Rew{\ell\ell}^* \l x.r'=r $.
		\item $r=r_pr_q$.  The sequence $u  \tobat{>0}^* r$ never use a $\beta$-rule, therefore (by \reflemma{basicCbN}), $u=pq$, with $p\Rew{\beta}^* r_p$ and $q\Rew{\beta}^* r_q$.
		Since $r=r_pr_q$ is normal, then  both  $r_p$ and $r_q$ are $\Rew{\beta}$-normal.

		\begin{itemize}

			\item By {\textit{i.h.}}\xspace,
			$p \Rew{\ell\ell}^*  r_p$ and $q \Rew{\ell\ell}^* r_q$.

			\item

			  We   \emph{merge}  the steps of	$p   \Rew{\ell\ell}^* r_p$ and
			$q\Rew{\ell\ell}^* r_q$ in increasing order, so to  conclude that $pq\Rew{\ell\ell}^* r_pr_q$, hence 	$t\tobat{0}^* u= pq\Rew{\ell\ell}^* r_pr_q=r$

		\end{itemize}

	\end{enumerate}
	In all cases, we have proved the claim.
\end{proof}

\newpage

\section{Left-Most Normalization in CbN}

\begin{theorem}[Head  factorization]
	\label{thm:cbn-fact0}
	If $t \Rew{\beta}^* s$ then $t\Rew{h}^* \tmtwo}%{s \Rew{\lnot h}^* \tmthree}%{u$.
\end{theorem}

\begin{theorem}[CBN Normalization]
	if $t \Rew{\beta}^* r$ with $r$  $\Rew{\beta}$-normal, then $t\Rew{\losym}^*  \tmfour}%{r$.

\end{theorem}

\begin{proof}	By induction on the term $r$.
	From $t \Rew{\beta}^* r$,
	by Head Factorization, we have that
	\begin{center}
		$t\Rew{h}^* u  \Rew{\lnot h}^* r$.
	\end{center}

	Cases:

	\begin{enumerate}
		\item $r=x$. Then  $u=x$, and trivially  $t\Rew{h}^* r$.
		\item $r=\l x.r'$. Then $u=\l x. u'$, hence  $u' \Rew{\beta}^*  r'$, where  $ r' $ is $\Rew{\beta}$-normal, and we use the {\textit{i.h.}}\xspace to obtain $u'\Rew{\losym}^* r'$, hence
		$t\Rew{h}^* u=\l x. u' \Rew{\losym}^* \l x.r'=r $.
		\item $r=r_pr_q$.  The sequence $u  \Rew{\lnot h}^* r$ never use a $\beta$-rule, therefore, $u=pq$, with $p\Rew{\beta}^* r_p$ and $q\Rew{\beta}^* r_q$. Observe that (i) $r_p$ and $r_q$ are $\Rew{\beta}$-normal and (ii) $r_p$ is not an abstraction  (because $r_pr_q$ is normal).

		\begin{itemize}

			\item By {\textit{i.h.}}\xspace,
			$p \Rew{\losym}^*  r_p$ and $q \Rew{\losym}^* r_q$.

			\item Because of  (ii) and \reflemma{basicCbN},  no term in the sequence 	$p   \Rew{\losym} p_1 \Rew{\losym}  \dots \Rew{\losym} r_p$ is an abstraction. Hence  $pq \Rew{\losym} p_1q\Rew{\losym} \dots    \Rew{\losym} r_pq$.    Since $r_p$ is normal and not an abstraction, from  $q \Rew{\losym}^* r_q$  we also have $r_pq \Rew{\losym}^* r_pr_q$.
		\end{itemize}

	\end{enumerate}
	In all cases, we have proved the claim.
\end{proof}

\end{document}

\section{Factorization by Level in CbV}
With the obvious CbV definitions of depth of the $\beta_\val$ step $\tostratind{k}$ and of least-level of a term $\deg t$,
given surface factorization, we obtain factorization by level by induction.

\begin{theorem}[Weak {CbV}\xspace factorization]
	\label{thm:weak-fact0}
	If $t \Rew{\betav}^* s$ then $t\tostratind{0}^* \tmtwo}%{s \tostratind{>0}^* \tmthree}%{u$.
\end{theorem}

\begin{theorem}[CBV \emph{\ll factorization}]
 if $t \Rew{\betav}^* r$ then $t\Rew{\esym}^* \cdot \Rew{\isym}^* \tmfour}%{r$.

\end{theorem}

\begin{proof}From $t \Rew{\betav}^* r$,
by  \refthm{weak-fact0}, we have that \begin{center}
	$t\tostratind{0}^* u  \tostratind{>0}^* r$.
\end{center}
We observe that the sequence $u  \tostratind{>0}^* r$ has the  property \#: no step  in the sequence  is a $\beta$-rule.
We   prove by induction on $r$ the following statement, from which the claim follows by taking $k=\deg r$

\begin{center}
	\emph{
	If 	$ u  \Rew{\betav}^* r $ and \# holds, then $u \Rew{\esym}^* \cdot \tostratind{>k}^* r$, for each $k\leq \deg r $, }
\end{center}

	We procede by cases.

\begin{enumerate}
	\item $r=x$. Trivial
	\item $r=\l x.q$ 	Then $u=\l x. p$, hence  $p \Rew{\betav}^*  q$, and we use the {\textit{i.h.}}\xspace.
	\item $r=r_pr_q$.  Because of \#, {\em i.e.}\xspace the sequence never use a $\beta$-rule, therefore, $u=pq$, with $p\Rew{\betav}^* r_p$ and $q\Rew{\betav}^* r_q$.

\begin{itemize}
	\item 	Observe that by def. of $\deg r$, if  $k\leq \deg r $ then $k\leq \deg{ r_p}$ and $k \leq \deg {r_q}$.

	\item 	By  \refthm{weak-fact0}, $p \tostratind{0}^* p' {\tostratind {>0}}^* r_p$ and $q \tostratind{0}^* q' {\tostratind {>0}}^* r_q$

	\item By {\textit{i.h.}}\xspace,
		$p' \Rew{\esym}^* p'' {\tostratind {>k}}^* r_p$ and $q' \Rew{\betav}^* q'' {\tostratind {>k}}^* r_q$.

	\item 	By merging 	$p \tostratind{0}^*p'  \Rew{\esym}^* p''$ and
		$q \tostratind{0}^* q'\Rew{\betav}^* q''$ in increasing order, we conclude that $pq\Rew{\esym}^* p''q''$.

		\item We also have
		 $  p''q'' \tostratind {>k} r_pq'' \tostratind {>k} r_pr_q$.
\end{itemize}
Hence we have proved the statement.

\end{enumerate}

\end{proof}

\SLV{}{
\begin{lemma} 	If $M\parredb N$ then $M\eredb^* \cdot  \iparredb N$

\end{lemma}

\begin{proof}By Lemma \ref{lem:split0}, we have that $M\sred^* U \dparred N$.
	We   prove by induction on $N$ the following, from which the claim follows by taking $k=\deg N$

	\begin{center}
		\emph{	For each $k\leq \deg N$,
			$U\eredb^* \cdot  \parredb \at {>k} N$}
	\end{center}

	We procede by cases.

	\begin{enumerate}
		\item $N=x$. ok
		\item $N=\lam x.S$ Then $U=\lam x. P$, hence  $P\parredb S$, and we use the {\textit{i.h.}}\xspace.
		\item $N=!S$. Then $U= !P$, hence  $P\parredb S$, and we use the {\textit{i.h.}}\xspace.
		\item $N=ST$. Therefore,  $U=PQ$,  $P\parredb S$ and $Q\parredb T$.  By def., if  $k\leq \deg N $ then $k\leq \deg S$ and $k \leq \deg T$.

		By {\textit{i.h.}}\xspace,
		$P \eredb^* P'  \parredb  \at {>k} S$ and $Q \eredb^* Q'  \parredb \at {>k} T$. We conclude that $PQ\eredb^* P'Q'$
		and $  P'Q' \parredb \at {>k} ST$
	\end{enumerate}

\end{proof}
}

\section{Conclusions}
We provide simple proof techniques for factorization and normalization theorems in the $\l$-calculus, simplifying Takahashi's parallel method \cite{DBLP:journals/iandc/Takahashi95},
extending its scope and making it more abstract at the same time.

About the use of parallel reduction,
Takahashi	 claims: ``\textit{once the idea is stated properly, the essential part of the proof is almost over, because the inductive verification of the statement is easy, even mechanical}'' \cite[p.~122]{DBLP:journals/iandc/Takahashi95}.
Our work reinforces this point of view, as our case studies smoothly follow the abstract~schema.

\paragraph{Range of application.}
We apply our method for factorization and
	normalization to two notions of reductions that compute full normal forms:
\begin{itemize}
	\item the classic example of $\ell o$\xspace reduction,

		covered also by the recent techniques by Hirokawa, Middeldorp and Moser \cite{DBLP:conf/rta/HirokawaMM15} and
	van Oostrom and Toyama \cite{DBLP:conf/rta/OostromT16};
	\item
	\ll reduction, which is out of the scope of \cite{DBLP:conf/rta/HirokawaMM15,DBLP:conf/rta/OostromT16} because it is neither deterministic (as required by \cite{DBLP:conf/rta/HirokawaMM15}), nor  left–outer in the sense of \cite{DBLP:conf/rta/OostromT16} (as pointed out here in \refsect{stratified-cbn}).
\end{itemize}

Our approach naturally covers also reductions that do not compute full normal forms, such as head and weak {CbV}\xspace reductions.
	These results are out of reach for
van Oostrom and Toyama's technique \cite{DBLP:conf/rta/OostromT16}, as they clarify in their conclusions.

Because of the minimality of our assumptions, we believe that our method applies to a large variety of other cases and variants of the $\l$-calculus.
\SLV{}{
A key feature of our approach is that it derives normalization from factorization.
However, it is worth noting that factorization is not a necessary condition for normalization.\footnotemark
\footnotetext{For instance, in the weak $\l$-calculus---where weak $\beta$-reduction $\rightarrow_{w\beta}$ does not reduce under abstractions---our technique does not apply because weak head reduction $\Rew{wh}$ ({\em i.e.}\xspace head reduction that does not reduce under abstractions) satisfies a weak head normalization theorem (if $t \rightarrow_{w\beta}^* s$ with $s$ $\Rew{wh}$-normal then $\Rew{wh}$ terminates on $t$) but does not factorize: indeed, given the $\rightarrow_{w\beta}$-sequence $(\la{x}\la{y}x)(II) \rightarrow_{\lnot wh} (\la{x}\la{y}x)I \Rew{wh} \la{y}I$, there is no term $t$ such that $(\la{x}\la{y}x)(II) \Rew{wh}^* t \rightarrow_{\lnot wh}^* \la{y}I$.}
}

\section{Technical appendix: omitted proofs and lemmas}
\label{sect:proofs}

The enumeration of propositions, theorems, lemmas already stated in the body of the article is unchanged.

\input{./proofs/proofs-head}

\input{./proofs/proofs-weak-cbv}

\input{./proofs/proofs-leftmost}

\input{./proofs/proofs-stratified-cbn2}

\section{Proof Appendix}

\begin{lemma}[Substitutivity of $\partobvind n$]
if $t \partobvind n t'$ and $s \partobvind m s'$ where $s$ is a value,  then $t \isub\vars \partobvindlong {n + \sizep{t'}x\cdot m} t' \isubx{s'}$.
\end{lemma}

\begin{proof}
	By induction on the definition of $t \partobvind{n} t'$. Cases:
Cases:
	\begin{enumerate}
	\item \emph{Variable}, {\em i.e.}\xspace $t = x \partobvind{0} x = t'$. Then, $\sizep{t'}x = 1$ and $t\isub{x}{s} = s \partobvind{m} s' = t'\isub{x}{s'}$, where $m = 0 + \sizep{t'}{x} \cdot m$.
	\item \emph{Abstraction}, {\em i.e.}\xspace $t = \la{y}{u} \partobvind{n} \la{y}{u'} = t'$ because $u \partobvind{n} u'$; we can suppose without loss of generality that $y \notin \fv{s} \cup \{x\} \supseteq \fv{s'} \cup \{x\}$, hence $\sizep{u'}{x} = \sizep{t'}{x}$ and $t\isub{x}{s} = \la{y}(u\isub{x}{s})$ and $t'\isub{x}{s'} = \la{y}(u'\isub{x}{s'})$.
	By {\textit{i.h.}}\xspace, $u\isub{x}{s} \partobvindlong{n +  \sizep{u'}{x} \cdot m} u'\isub{x}{s'}$, thus
	\begin{align*}
	\AxiomC{$u\isub{x}{s} \partobvindlong{n +  \sizep{u'}{x} \cdot m} u'\isub{x}{s'}$}

	\UnaryInfC{$t\isub{x}{s} = \la {y}u\isub{x}{s} \partobvindlong{n +  \sizep{t'}{x} \cdot m} \la{y}u'\isub{x}{s'} = t'\isub{x}{s'}$}
	\DisplayProof\,.
	\end{align*}
	\item \emph{Application}, {\em i.e.}\xspace $t = {u}r \partobvind{n_1 + n_2} {u'}r' = t'$ because $u \partobvind{n_1} u'$ and $r \partobvind{n_2} r'$.
	By {\textit{i.h.}}\xspace, $u\isub{x}{s} \partobvindlong{n_1 +  \sizep{u'}{x} \cdot m} u'\isub{x}{s'}$ and $r\isub{x}{s} \partobvindlong{n_2 +  \sizep{r'}{x} \cdot m} r'\isub{x}{s'}$, hence
	\begin{align*}
	\AxiomC{$u\isub{x}{s} \partobvindlong{n_1 +  \sizep{u'}{x} \cdot m} u'\isub{x}{s'}$}
	\AxiomC{$r\isub{x}{s} \partobvindlong{n_2 +  \sizep{r'}{x} \cdot m} r'\isub{x}{s'}$}

	\BinaryInfC{$t\isub{x}{s} = u\isub{x}{s} r\isub{x}{s} \partobvindlong{n +  \sizep{t'}{x} \cdot m} u'\isub{x}{s'} r'\isub{x}{s'} = t'\isub{x}{s'}$}
	\DisplayProof
	\end{align*}
	where $\sizep{t'}{x} = \sizep{u'}{x} + \sizep{r'}{x}$ and $n = n_1 + n_2$.

	\item \emph{$\beta_\val$ step}:
	\[\AxiomC{$u \partobvind {n_1} u'$}
	\AxiomC{$r$ is a value}
	\AxiomC{$r \partobvind {n_2} r'$}
	\TrinaryInfC{$t = (\lay u)r \partobvind{n} u'\isub y {r'} = t'$}
	\DisplayProof\]
	where $n = n_1 + \sizep{u'}{x} \cdot n_2 +1$.
	We can assume without loss of generality that $y \notin \fv{s} \cup \{x\} \supseteq \fv{s'} \cup \{x\}$;
	hence, $\sizep{t'}{x} = \sizep{u'}{x} + \sizep{u'}{y}\sizep{r'}{x} $ and $t\isub{x}{r} = (\la{y}{u}\isub{x}{s})r\isub{x}{s}$ and $t'\isub{x}{r'} = {u'}\isub{y}{r'}\isub{x}{s'} = u'\isub{y}{s'}\isub{x}{r'\isub{y}{s'}}$.
	By {\textit{i.h.}}\xspace, $u\isub{x}{s} \partobvindlong{n_1 + m \cdot \sizep{u'}{x}} u'\isub{x}{s'}$ and $r\isub{x}{s} \partobvindlong{n_2 + m \cdot \sizep{r'}{x}} r'\isub{x}{s'}$.
	Therefore, since $r\isub{x}{s}$ is a value,
	\[\AxiomC{$u\isub{x}{s} \partobvindlong{n_1 + m \cdot \sizep{u'}{x}} u'\isub{x}{s'}$}
	\AxiomC{$r\isub{x}{r} \partobvindlong{n_2 + m \cdot \sizep{r'}{x}} r'\isub{x}{s'}$}
	\BinaryInfC{$t \isub{x}{s} = (\lay u)r \partobvindlong{n_1 + m \cdot \sizep{u'}{x} + (n_2 + m \cdot \sizep{r'}{x})\sizep{u'}{y} + 1} u'\isub y {r'} = t'\isub{x}{s'}$}
	\DisplayProof\]
	where $n_1 + m \cdot \sizep{u'}{x} + (n_2 + m \cdot \sizep{r'}{x})\sizep{u'}{y} + 1 = n + m(\sizep{u'}{x} + \sizep{u'}{y}\cdot \sizep{r'}{x}) = n + m \cdot \sizep{t'}{x}$.
	\qedhere
\end{enumerate}

\end{proof}

\subsection{Omitted proofs  of \refsect{abstract-rewriting} (Factorization and Normalization, Abstractly)}

\begin{theorem}[Essential normalization]
Let $\mathcal S$ be an  \essential system. If $t \to^* s$ and $s$ is a $\Rew{\esym}$-normal form then $t$ is strongly $\Rew{\esym}$-normalizing.
\end{theorem}

\begin{proof}
	Assume $t \to^* \tmthree}%{u$ with $\tmthree}%{u$  $\Rew{\esym}$-normal. Then:
	\begin{enumerate}

		\item  \emph{Terminal factorization} implies  $t \Rew{\esym}^*\tmtwo}%{s \Rew{\isym}^* \tmthree}%{u$.
		\item Let us show that $\tmtwo}%{s$ is $\Rew{\esym}$-normal: if it is not then $\tmtwo}%{s$ a straightforward induction on the length of $\tmtwo}%{s \Rew{\isym}^*\tmthree}%{u$ iterating \emph{persistence} gives that $\tmthree}%{u \Rew{\esym} p$ for some $p$, against the hypothesis that $\tmthree}%{u$ it is $\Rew{\esym}$-normal. Absurd.	We have then proved that that $\tmtwo}%{s$ is $\Rew{\esym}$-normal.

	\item By the previous point $t$ is weakly $\Rew{\esym}$-normalizing. By \emph{uniform termination}, $\Rew{\esym}$ is strongly normalizing. \qedhere
	\end{enumerate}
\end{proof}

\subsection{Omitted proofs  of \refsect{lambda} ($\l$-calculus)}

\subsection{Omitted proofs  of \refsect{head} (head)}

\setcounter{propositionAppendix}{\value{prop:macro-head}}
\begin{propositionAppendix}[Head macro-step system]
	\label{propappendix:macro-head}
	\NoteState{prop:macro-head}
	\hfill
	\begin{enumerate}
		\item\label{pappendix:macro-head-merge} \emph{Merge}: if $t \parRew{\neg h}\cdot \Rew{h} u$ then $t \parRew{\beta} u$.
		\item\label{pappendix:macro-head-indexed-split} \emph{Indexed split}: if $t \partobind n s$ then $t \partonoths$, or $n>0$ and $t \Rew{h} \cdot \partobind {n-1} s$.

		\item\label{pappendix:macro-head-split} \emph{Split}: if $t \parRew{\beta} s$ then $t \Rew{h}^* \cdot \partonoths$.
	\end{enumerate}
	that is, $(\Lambda, \{ \Rew{h}, \Rew{\lnot h} \})$ is a \macrostep\ system with respect to $\parRew{\beta},\parRew{\neg h}$.
\end{propositionAppendix}

\begin{proof}
	\refpoints{macro-head-indexed-split}{macro-head-split} are already proved on p.~\pageref{prop:macro-head}.
	We prove \refpoint{macro-head-merge} by induction on the definition of $t \parRew{\neg h} s$.
	Cases:
	\begin{itemize}
		\item \emph{Variable}: $t = x \parRew{\neg h} x = s$.
		Then, there is no $u$ such that $s \Rew{h} u$.

		\item \emph{Abstraction}: $t = \la \vart' \parRew{\neg h}  \la x s' = s$ because $t' \parRew{\neg h} s'$.
		According to the definition of $s \Rew{h} u$, necessarily $u = \la{x}{u'}$ with $s' \Rew{h} u'$.
		By {\textit{i.h.}}\xspace applied to $t'$, we have $t' \parRew{\beta} u'$ and hence:
		\begin{align*}
			\AxiomC{$t' \parRew{\beta} u'$}
			\UnaryInfC{$t = \la{x}{t'} \parRew{\beta} \la{x}{u'} = u$}
			\DisplayProof
			\,.
		\end{align*}

		\item \emph{Application}:
		\begin{align}
		\label{eq:app}
		\AxiomC{$r \parRew{\neg h} r'$}
		\AxiomC{$p \parRew{\beta} p'$}
		\BinaryInfC{$t = rp \parRew{\neg h} r'p' = s$}
		\DisplayProof
		\end{align}
		Sub-cases:
		\begin{enumerate}
			\item $s = r'p' \Rew{h} r''p' = u$ with $r' \Rew{h} r''$;
			by {\textit{i.h.}}\xspace applied to $r \parRew{\neg h} r' \Rew{h} r''$, we have $r \parRew{\beta} r''$, and so (as $\parRew{\neg h} \,\subseteq\, \parRew{\beta}$)
			\begin{align*}
			\AxiomC{$r \parRew{\beta} r''$}
			\AxiomC{$p \parRew{\beta} p'$}
			\BinaryInfC{$t = rp \parRew{\beta} r''p' = u$}
			\DisplayProof\,;
			\end{align*}

			\item $s = (\la{x}q')p' \Rew{h} q'\isub{x}{p'} = u$, which means that $r' = \la{x}{q'}$ in \eqref{eq:app}.
			According to the definition of $r \parRew{\neg h} r'$, necessarily $r = \la{x}{q}$ with $q \parRew{\neg h} q'$.
			Thus, (as $\parRew{\neg h} \,\subseteq\, \parRew{\beta}$)
			\begin{align*}
			\AxiomC{$\tmsix}%{q\parRew{\beta} \tmsix}%{q'$}
			\AxiomC{$p \parRew{\beta} p'$}
			\BinaryInfC{$t =  (\la {x} q) p \parRew{\beta}  \tmsix}%{q'\isub{x}{p'} = u$}
			\DisplayProof
			\,.
			\end{align*}
		\end{enumerate}
		\item \emph{$\beta$-redex}:
		\begin{align*}
			\AxiomC{$r \parRew{\beta} r'$}
			\AxiomC{$p \parRew{\beta} p'$}
			\BinaryInfC{$t = (\lax r) p \parRew{\neg h} (\lax r') p' = s$}
			\DisplayProof
		\end{align*}
		According to the definition of $s \Rew{h} u$, we have $u = r'\isub{x}{p'}$.
		Hence,
		\[
			\AxiomC{$r \parRew{\beta} r'$}
			\AxiomC{$p \parRew{\beta} p'$}
			\BinaryInfC{$t = (\lax r) p \parRew{\beta} r'\isub{x}{p'} = u$}
			\DisplayProof
			\ . \qedhere
		\]
	\end{itemize}
\end{proof}

\setcounter{propositionAppendix}{\value{prop:essential-head}}
\begin{propositionAppendix}[Head essential system]
	\label{propappendix:essential-head}
\NoteState{prop:essential-head}
\begin{enumerate}
 \item\label{pappendix:essential-head-persistence} \emph{Persistence}: if $t \Rew{h} s$ and $t \Rew{\lnot h} u$ then $u \Rew{h} r$ for some $r$.
 \item\label{pappendix:essential-head-determinism} \emph{Determinism}: if $t \Rew{h} s_1$ and $t\Rew{h} s_2$ then $s_1 = s_2$.
\end{enumerate}
Then, $(\Lambda, \set{ \Rew{\wsym}, \Rew{\neg \wsym}})$ is an essential system.
\end{propositionAppendix}

\begin{proof}\hfill
\begin{enumerate}
 \item \emph{Persistence}: By induction on $t \Rew{h} s$. Cases:
 \begin{itemize}
  \item \emph{Root step}, {\em i.e.}\xspace $t = (\lax p) q \Rew{h} p\isub\varq = s$. Sub-cases:
  \begin{itemize}
    \item \emph{$\Rew{\lnot h}$ in the left sub-term}: $t = (\lax p) q \Rew{\lnot h} (\lax p') q=u$ because $p \Rew{\beta} p'$ or $p \Rew{\lnot h} p'$. In both cases $u = (\lax p') q \Rew{h} p'\isub\varq =:r$.

    \item \emph{$\Rew{\lnot h}$ in the right sub-term}: $t = (\lax p) q \Rew{\lnot h} (\lax p) q'=u$ because $q \Rew{\beta} q'$. Then $u = (\lax p) q' \Rew{h} p\isubx{q'} =:r$.
  \end{itemize}
  \item \emph{Application}: {\em i.e.}\xspace $t = p q \Rew{h} p'q = s$ with $p \Rew{h} p'$ and $p$ not an abstraction. Sub-cases:
  \begin{itemize}
    \item \emph{$\Rew{\lnot h}$ in the left sub-term}: $t = p q \Rew{\lnot h} p'' q=u$ because $p \Rew{\lnot h} p''$. By {\textit{i.h.}}\xspace, $p'' \Rew{h} r'$ for some $r'$. Then $u = p'' q \Rew{h} r'q =:r$.

    \item \emph{$\Rew{\lnot h}$ in the right sub-term}: $t = p q \Rew{\lnot h} p q'=u$ because $q \Rew{\beta} q'$. Then $u = p q' \Rew{h} p'q' =:r$.
  \end{itemize}
  \item \emph{Abstraction}: {\em i.e.}\xspace $t = \lax p \Rew{h} p' = s$. Then $t = \lax p \Rew{\lnot h} \lax''' = u$ for some $p''$ with $p \Rew{\lnot h} p''$. By {\textit{i.h.}}\xspace, $p'' \Rew{h} r'$ for some $r'$. Then $u = \lax''' \Rew{h} \la\varr'$.
 \end{itemize}

	\item \emph{Determinism}: By induction on a derivation with conclusion $t \Rew{h} s_1$.
	Consider its last rule.
	Cases:
	\begin{itemize}
		\item \emph{Abstraction}, {\em i.e.}\xspace $t = \la{x}{t'} \Rew{h} \la{x}{s_1'} = s_1$ because $t' \Rew{h} s_1'$.
		According to the definition of $\Rew{h}$, the only possibility for the last rule of a derivation for $t \Rew{h} s_2$ is
		\begin{align*}
			\AxiomC{$t' \Rew{h} s_2'$}
			\UnaryInfC{$t = \la{x}{t'} \Rew{h} \la{x}{s_2'} = s_2$}
			\DisplayProof
			\ .
		\end{align*}
		By {\textit{i.h.}}\xspace applied to $t'$, we have $s_1' = s_2'$ and hence $s_1 = \la{x}{s_1'} = \la{x}{s_2'} = s_2$.

		\item \emph{Application}, {\em i.e.}\xspace $t = t' t'' \Rew{h} s_1't'' = s_1$ because $t' \Rew{h} s_1'$ and $t' \neq \la{x}{r}$.
		According to the definition of $\Rew{h}$, the the last rule of a derivation for $t \Rew{h} s_2$ can be only
		\begin{align*}
		\AxiomC{$t' \Rew{h} s_2'$}
		\UnaryInfC{$t = t't'' \Rew{h} s_2't'' = s_2$}
		\DisplayProof
		\ .
		\end{align*}
		By {\textit{i.h.}}\xspace applied to $t'$, we have $s_1' = s_2'$ and hence $s_1 = s_1't'' = s_2't'' = s_2$.

		\item \emph{$\beta$-rule}, {\em i.e.}\xspace $t = (\la{x}t')t'' \Rew{h} t'\isub{x}{t''} = s_1$.
		According to the definition of $\Rew{h}$, the only possibility for the last rule of a derivation for $t \Rew{h} s_2$ is
		\begin{align*}
			\AxiomC{}
			\UnaryInfC{$t = (\la{x}{t'})t'' \Rew{h} t'\isub{x}{t''} = s_2$}
			\DisplayProof
			&&\text{and hence } s_1 = t'\isub{x}{t''} = s_2.
			\text{\qedhere}
		\end{align*}
	\end{itemize}
\end{enumerate}

\end{proof}

\subsection{Omitted proofs  of \refsect{lambda} ($\l$-calculus)}

\setcounter{lemmaAppendix}{\value{l:partobind-subs}}
\begin{lemmaAppendix}[Substitutivity of $\partobind n$]
	\label{lappendix:partobind-subs}
	\NoteState{l:partobind-subs}
If $t \partobind n t'$ and $s \partobind m s'$ then $t \isub\vars \partobind {n + \sizep{t'}x\cdot m} t' \isubx{s'}$.
\end{lemmaAppendix}

\begin{proof}
By induction on $t \partobind n t'$. Cases:
\begin{itemize}
	\item \emph{Variable}: two sub-cases
	\begin{itemize}
	\item $t = x$: then $t = x \partobind 0 x = t'$ then $t \isub\vars = x\isub\vars = s \partobind m s' = x \isubx{s'} = t' \isubx{s'}$ that satisfies the statement because $n + \sizep{t'}x\cdot m = 0+1\cdot m = m$.
	 \item $t = y$: then $t = y \partobind 0 y = t'$ then $t \isub\vars = y\isub\vars = y \partobind 0 y = y \isubx{s'} = t' \isubx{s'}$ that satisfies the statement because $n + \sizep{t'}x\cdot m = n +0\cdot m=n$.
	 \end{itemize}

	\item \emph{Abstraction}, {\em i.e.}\xspace $t = \la{y}{u} \partobind{n} \la{y}{u'} = t'$ because $u \partobind{n} u'$; we can suppose without loss of generality that $y \neq x$, hence $\sizep{u'}{x} = \sizep{t'}{x}$ and $t\isub{x}{s} = \la{y}(u\isub{x}{s})$ and $t'\isub{x}{s'} = \la{y}(u'\isub{x}{s'})$.
		By {\textit{i.h.}}\xspace, $u\isub{x}{s} \partobindlong{n +  \sizep{u'}{x} \cdot m} u'\isub{x}{s'}$, thus
		\begin{align*}
		\AxiomC{$u\isub{x}{s} \partobindlong{n +  \sizep{u'}{x} \cdot m} u'\isub{x}{s'}$}

		\UnaryInfC{$t\isub{x}{s} = \la {y}u\isub{x}{s} \partobindlong{n +  \sizep{t'}{x} \cdot m} \la{y}u'\isub{x}{s'} = t'\isub{x}{s'}$}
		\DisplayProof\,.
		\end{align*}

	\item \emph{Application}, {\em i.e.}\xspace
	\[\AxiomC{$u \partobind {n_u} u'$}
	\AxiomC{$r \partobind {n_r} r'$}

	\BinaryInfC{$t = u r \partobindlong {n_u + n_r} u' r'= t'$}
	\DisplayProof\]
	with $n = n_u + n_r$. By {\textit{i.h.}}\xspace, $u\isub\vars \partobindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{s'}$ and $r\isub\vars \partobindlong {n_r +\sizep{r'}x \cdot m} r'\isubx{s'}$. Then
	\[\AxiomC{$u\isub\vars \partobindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{s'}$}
	\AxiomC{$r\isub\vars \partobindlong {n_r +\sizep{r'}x \cdot m} r'\isubx{s'}$}

	\BinaryInfC{$t\isub\vars = u\isub\vars r\isub\vars \partobindlong {n_u +\sizep{u'}x \cdot m +n_r +\sizep{r'}x \cdot m} u'\isubx{s'} r'\isubx{s'}= t'\isubx{s'}$}
	\DisplayProof\]
that proves the statement because $$n_u +\sizep{u'}x \cdot m +n_r +\sizep{r'}x \cdot m = n + (\sizep{u'}x + \sizep{r'}x)\cdot m = n + \sizep{t'}x \cdot m$$

\item $\beta$, {\em i.e.}\xspace
	\[\AxiomC{$u \partobind {n_u} u'$}
	\AxiomC{$r \partobind {n_r} r'$}
\BinaryInfC{$t = (\lay u)r \partobindlong {n_u + \sizep{u'}y \cdot n_r +1} u'\isub y {r'} = t'$}
\DisplayProof\]
with $n = n_u + \sizep{u'}y \cdot n_r +1$. We can assume without loss of generality that $y\neq x$,
		hence $\sizep{t'}{x} = \sizep{u'\isuby {r'}}x = \sizep{u'}{x} + \sizep{u'}{y}\cdot\sizep{r'}{x} $ and $t\isub\vars = (\la\vartwou\isub\vars) (r\isub\vars)$ and $t'\isubx{s'} = u'\isubx{s'} \isuby{r'\isubx{s'}}$.

		By {\textit{i.h.}}\xspace, $u\isub\vars \partobindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{s'}$ and $r\isub\vars \partobindlong {n_r +\sizep{r'}x \cdot m} r'\isubx{s'}$. Then
	\[\AxiomC{$u\isub\vars \partobindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{s'}$}
	\AxiomC{$r\isub\vars \partobindlong {n_r +\sizep{r'}x \cdot m} r'\isubx{s'}$}

	\BinaryInfC{$t\isub\vars = (\la\vartwou\isub\vars) (r\isub\vars) \partobindlong {n_u +\sizep{u'}x \cdot m +\sizep{u'}y\cdot(n_r +\sizep{r'}x \cdot m)+1} u'\isubx{s'} \isuby{r'\isubx{s'}}= t'\isubx{s'}$}
\DisplayProof\]

The number on the $\parRew{}$ arrow satisfies
$$n_u +\sizep{u'}x \cdot m +\sizep{u'}y\cdot(n_r +\sizep{r'}x \cdot m)+1
=
\underbrace{n_u + \sizep{u'}y\cdot n_r + 1}_n + \underbrace{\sizep{u'}x \cdot m + \sizep{u'}y\cdot \sizep{r'}x \cdot m}_{\sizep{t'}x\cdot m}$$
\end{itemize}
\end{proof}

\subsection{Omitted proofs  of \refsect{head} (head)}

\setcounter{propositionAppendix}{\value{prop:quant-out-in-split}}
\begin{propositionAppendix}[Head macro-step system]
	\label{propappendix:quant-out-in-split}
	\NoteState{prop:quant-out-in-split}
	\hfill
	\begin{enumerate}
		\item\label{p:quant-out-in-split-merge} \emph{Merge}: if $t \parRew{\neg h}\cdot \Rew{h} u$ then $t \parRew{\beta} u$.
		\item\label{p:quant-out-in-split-indexed} \emph{Indexed split}: if $t \partobind n s$ then $t \partonoths$, or $n>0$ and $t \Rew{h} \cdot \partobind {n-1} s$.

		\item\label{p:quant-out-in-split-split} \emph{Split}: if $t \parRew{\beta} s$ then $t \Rew{h}^* \cdot \partonoths$.
	\end{enumerate}
	that is, $(\Lambda, \{ \Rew{h}, \Rew{\lnot h} \})$ is a \macrostep\ system with respect to $\parRew{\beta},\parRew{\neg h}$.
\end{propositionAppendix}

\begin{proof}
	\refpoints{quant-out-in-split-indexed}{quant-out-in-split-split} are already proved on p.~\pageref{prop:quant-out-in-split}.
	We prove \refpoint{quant-out-in-split-merge} by induction on the definition of $t \parRew{\neg h} s$.
	Cases:
	\begin{itemize}
		\item \emph{Variable}: $t = x \parRew{\neg h} x = s$.
		Then, there is no $u$ such that $s \Rew{h} u$.

		\item \emph{Abstraction}: $t = \la \vart' \parRew{\neg h}  \la x s' = s$ because $t' \parRew{\neg h} s'$.
		According to the definition of $s \Rew{h} u$, necessarily $u = \la{x}{u'}$ with $s' \Rew{h} u'$.
		By {\textit{i.h.}}\xspace applied to $t'$, we have $t' \parRew{\beta} u'$ and hence:
		\begin{align*}
			\AxiomC{$t' \parRew{\beta} u'$}
			\UnaryInfC{$t = \la{x}{t'} \parRew{\beta} \la{x}{u'} = u$}
			\DisplayProof
			\,.
		\end{align*}

		\item \emph{Application}:
		\begin{align}
		\label{eq:app}
		\AxiomC{$r \parRew{\neg h} r'$}
		\AxiomC{$p \parRew{\beta} p'$}
		\BinaryInfC{$t = rp \parRew{\neg h} r'p' = s$}
		\DisplayProof
		\end{align}
		Sub-cases:
		\begin{enumerate}
			\item $s = r'p' \Rew{h} r''p' = u$ with $r' \Rew{h} r''$;
			by {\textit{i.h.}}\xspace applied to $r \parRew{\neg h} r' \Rew{h} r''$, we have $r \parRew{\beta} r''$, and so (as $\parRew{\neg h} \,\subseteq\, \parRew{\beta}$)
			\begin{align*}
			\AxiomC{$r \parRew{\beta} r''$}
			\AxiomC{$p \parRew{\beta} p'$}
			\BinaryInfC{$t = rp \parRew{\beta} r''p' = u$}
			\DisplayProof\,;
			\end{align*}

			\item $s = (\la{x}q')p' \Rew{h} q'\isub{x}{p'} = u$, which means that $r' = \la{x}{q'}$ in \eqref{eq:app}.
			According to the definition of $r \parRew{\neg h} r'$, necessarily $r = \la{x}{q}$ with $q \parRew{\neg h} q'$.
			Thus, (as $\parRew{\neg h} \,\subseteq\, \parRew{\beta}$)
			\begin{align*}
			\AxiomC{$\tmsix}%{q\parRew{\beta} \tmsix}%{q'$}
			\AxiomC{$p \parRew{\beta} p'$}
			\BinaryInfC{$t =  (\la {x} q) p \parRew{\beta}  \tmsix}%{q'\isub{x}{p'} = u$}
			\DisplayProof
			\,.
			\end{align*}
		\end{enumerate}
		\item \emph{$\beta$-redex}:
		\begin{align*}
			\AxiomC{$r \parRew{\beta} r'$}
			\AxiomC{$p \parRew{\beta} p'$}
			\BinaryInfC{$t = (\lax r) p \parRew{\neg h} (\lax r') p' = s$}
			\DisplayProof
		\end{align*}
		According to the definition of $s \Rew{h} u$, we have $u = r'\isub{x}{p'}$.
		Hence,
		\begin{align*}
			\AxiomC{$r \parRew{\beta} r'$}
			\AxiomC{$p \parRew{\beta} p'$}
			\BinaryInfC{$t = (\lax r) p \parRew{\beta} r'\isub{x}{p'} = u$}
			\DisplayProof
		\end{align*}
		\qedhere
	\end{itemize}
\end{proof}

\begin{proposition}[Head essential system]
\begin{enumerate}
 \item \emph{Persistence}: if $t \Rew{h} s$ and $t \Rew{\lnot h} u$ then $u \Rew{h} r$ for some $r$.
 \item \emph{Determinism}: if $t \Rew{h} s_1$ and $t\Rew{h} s_2$ then $s_1 = s_2$.
\end{enumerate}
Then, $(\Lambda, \set{ \Rew{\wsym}, \Rew{\neg \wsym}})$ is an essential system.
\end{proposition}

\begin{proof}
\begin{enumerate}
 \item \emph{Persistence}: by induction on $t \Rew{h} s$. Cases:
 \begin{itemize}
  \item \emph{Root step}, {\em i.e.}\xspace $t = (\lax p) q \Rew{h} p\isub\varq = s$. Sub-cases:
  \begin{itemize}
    \item \emph{$\Rew{\lnot h}$ in the left sub-term}: $t = (\lax p) q \Rew{\lnot h} (\lax p') q=u$ because $p \Rew{\beta} p'$ or $p \Rew{\lnot h} p'$. In both cases $u = (\lax p') q \Rew{h} p'\isub\varq =:r$.

    \item \emph{$\Rew{\lnot h}$ in the right sub-term}: $t = (\lax p) q \Rew{\lnot h} (\lax p) q'=u$ because $q \Rew{\beta} q'$. Then $u = (\lax p) q' \Rew{h} p\isubx{q'} =:r$.
  \end{itemize}
  \item \emph{Application}: {\em i.e.}\xspace $t = p q \Rew{h} p'q = s$ with $p \Rew{h} p'$ and $p$ not an abstraction. Sub-cases:
  \begin{itemize}
    \item \emph{$\Rew{\lnot h}$ in the left sub-term}: $t = p q \Rew{\lnot h} p'' q=u$ because $p \Rew{\lnot h} p''$. By {\textit{i.h.}}\xspace, $p'' \Rew{h} r'$ for some $r'$. Then $u = p'' q \Rew{h} r'q =:r$.

    \item \emph{$\Rew{\lnot h}$ in the right sub-term}: $t = p q \Rew{\lnot h} p q'=u$ because $q \Rew{\beta} q'$. Then $u = p q' \Rew{h} p'q' =:r$.
  \end{itemize}
  \item \emph{Abstraction}: {\em i.e.}\xspace $t = \lax p \Rew{h} p' = s$. Then $t = \lax p \Rew{\lnot h} \lax''' = u$ for some $p''$ with $p \Rew{\lnot h} p''$. By {\textit{i.h.}}\xspace, $p'' \Rew{h} r'$ for some $r'$. Then $u = \lax''' \Rew{h} \la\varr'$.
 \end{itemize}

 \item \emph{Determinism}: \ben{to do}.
\end{enumerate}

\end{proof}

\begin{proof} By induction on $t$.
	We show   that for each possible case  of $t$, persistence holds, and only one reduction applies (\emph{determinism}).
		\begin{itemize}
			\item $t = (\lax p)$. Only the  \emph{root step} applies. Since $t \Rew{\lnot h} u$ we have
			\begin{itemize}
				\item either  $ (\lax p) q \Rew{\lnot h} (\lax p') q=u$.

				\item or $  (\lax p) q \Rew{\lnot h} (\lax p) q'=u$.
			\end{itemize}
		In both cases,  the  \emph{root step}  applies to $u$.

			\item  $t = p q $ with $p$ not an abstraction. Only rule 2. applies, and  $p \Rew{h} p'$.
			Since $t \Rew{\lnot h} u$ we have
			\begin{itemize}
				\item either  $  p q \Rew{\lnot h}  p' q=u$, where $p'$ is not an abstraction

				\item or $  p q \Rew{\lnot h}  p q'=u$, where $p$ is not an abstraction.
			\end{itemize}
					In both cases, rule 2.  applies to $u$.

			\item  $t = \lax p $. Only rule 3. applies, with a premiss $p \Rew{h} p'$. We also have
			  $t = \lax p \Rew{\lnot h} \lax'''$, which gives  $p \Rew{\lnot h} p''$. By {\textit{i.h.}}\xspace, $p'' \Rew{h} r'$ for some $r'$. Then rule 3. applies to  $u = \lax'''$.
		\end{itemize}

\end{proof}

\subsection{Omitted proofs  of \refsect{leftmost} (leftmost-outermost)}

\setcounter{propositionAppendix}{\value{prop:essential-left}}
\begin{propositionAppendix}[$\ell o$\xspace essential properties]
	\label{propappendix:essential-left}
	\NoteState{prop:essential-left}
\hfill
\begin{enumerate}
\item\label{pappendix:essential-left-completeness} \emph{Fullness\xspace}: if $t \Rew{\beta} s$ then there exists $u$ such that $t \Rew{\losym} u$.
\item\label{pappendix:essential-left-determinism} \emph{Determinism}: if $t \Rew{\losym} s_1$ and $t\Rew{\losym} s_2$, then $s_1 = s_2$.
\item\label{pappendix:essential-left-persistence} \emph{Persistence}: if $t \Rew{\losym} s_1$ and $t \Rew{\neg \losym} s_2$, then $s_2 \Rew{\losym} u$ for some $u$.
\end{enumerate}
\end{propositionAppendix}

\begin{proof}\hfill
\begin{enumerate}
	\item \emph{Fullness\xspace}: by induction on $t$. Cases:
  \begin{itemize}
    \item \emph{Variable}, {\em i.e.}\xspace $t = x$: then $t \not\tobs$, and so the statement trivially holds.

    \item \emph{Abstraction}, {\em i.e.}\xspace $t = \la\vart' \Rew{\beta} \la\vars' = s$. It follows by the {\textit{i.h.}}\xspace

    \item \emph{Application}, {\em i.e.}\xspace $t = r p$. Three sub-cases:
    \begin{itemize}
      \item \emph{$r$ is an abstraction}, {\em i.e.}\xspace $r = \la\varq$: then $t = (\la\varq) p \Rew{\losym} q\isub\varq$.

      \item \emph{$r$ is not an abstraction but it is not normal}, {\em i.e.}\xspace $r \Rew{\beta} r'$ for some $r'$: then by {\textit{i.h.}}\xspace $r \Rew{\losym} q$ for some $q$ and so $t = r p \Rew{\losym} \tmsixp$.

      \item \emph{$r$ is neutral}, {\em i.e.}\xspace $t$ is not normal implies $p$ not normal. Then by {\textit{i.h.}}\xspace $p \Rew{\losym} p'$ for some $p'$, and so $t = r p \Rew{\losym} r p'$.
    \end{itemize}
  \end{itemize}

	\item \emph{Determinism}: By induction on a derivation with conclusion $t \Rew{\losym} s_1$.
	Consider its last rule.
	Cases:
	\begin{itemize}
		\item \emph{Abstraction}, {\em i.e.}\xspace $t = \la{x}{t'} \Rew{\losym} \la{x}{s_1'} = s_1$ because $t' \Rew{\losym} s_1'$.
		According to the definition of $\Rew{\losym}$, the last rule of a derivation for $t \Rew{\losym} s_2$ can be only
		\begin{align*}
		\AxiomC{$t' \Rew{\losym} s_2'$}
		\UnaryInfC{$t = \la{x}{t'} \Rew{\losym} \la{x}{s_2'} = s_2$}
		\DisplayProof
		\ .
		\end{align*}
		By {\textit{i.h.}}\xspace applied to $t'$, we have $s_1' = s_2'$ and hence $s_1 = \la{x}{s_1'} = \la{x}{s_2'} = s_2$.

		\item \emph{Application left}, {\em i.e.}\xspace $t = t' t'' \Rew{\losym} s_1't'' = s_1$ because $t' \Rew{\losym} s_1'$ and $t' \neq \la{x}{r}$.
		According to the definition of $\Rew{\losym}$, the last rule of a derivation for $t \Rew{\losym} s_2$ can only be (since $t$ is neither an abstraction nor neutral)
		\begin{align*}
		\AxiomC{$t' \Rew{\losym} s_2'$}
		\UnaryInfC{$t = t't'' \Rew{\losym} s_2't'' = s_2$}
		\DisplayProof
		\ .
		\end{align*}
		By {\textit{i.h.}}\xspace applied to $t'$, we have $s_1' = s_2'$ and hence $s_1 = s_1't'' = s_2't'' = s_2$.

		\item \emph{Application right}, {\em i.e.}\xspace $t = t' t'' \Rew{\losym} t's_1'' = s_1$ because $t'' \Rew{\losym} s_1''$ and $t'$ is neutral.
		According to the definition of $\Rew{\losym}$, the last rule of a derivation for $t \Rew{\losym} s_2$ can only be (since $t$ is normal and not an abstraction)
		\begin{align*}
		\AxiomC{$t'' \Rew{\losym} s_2''$}
		\UnaryInfC{$t = t't'' \Rew{\losym} t's_2'' = s_2$}
		\DisplayProof
		\ .
		\end{align*}
		By {\textit{i.h.}}\xspace applied to $t'$, we have $s_1' = s_2'$ and hence $s_1 = t's_1'' = t's_2'' = s_2$.

		\item \emph{$\beta$-rule}, {\em i.e.}\xspace $t = (\la{x}t')t'' \Rew{\losym} t'\isub{x}{t''} = s_1$.
		According to the definition of $\Rew{\losym}$, the only possibility for the last rule of a derivation for $t \Rew{\losym} s_2$ is
		\begin{align*}
		\AxiomC{}
		\UnaryInfC{$t = (\la{x}{t'})t'' \Rew{\losym} t'\isub{x}{t''} = s_2$}
		\DisplayProof
		&&\text{and hence } s_1 = t'\isub{x}{t''} = s_2.
		\text{\qedhere}
		\end{align*}
	\end{itemize}

	\item \emph{Persistence}: by induction on $t \Rew{\losym} s_1$. Cases:
  \begin{itemize}
    \item \emph{Root}: $t = (\lax r) p \Rew{\losym} r\isubx' = s_1$. Three sub-cases:
    \begin{itemize}
      \item $t = (\lax r) p \Rew{\neg \losym} (\lax r') p  = s_2$ because $r \Rew{\beta} r'$. Then $s_2 = (\lax r') p \Rew{\losym} r'\isubx' =: u$.

      \item $t = (\lax r) p \Rew{\neg \losym} (\lax r') p  = s_2$ because $r \Rew{\neg \losym} r'$. Exactly as the previous one.

      \item $t = (\lax r) p \Rew{\neg \losym} (\lax r) p ' = s_2$ because $p \Rew{\beta} p'$. Then $s_2 = (\lax r) p' \Rew{\losym} r\isubx'' =: u$.
    \end{itemize}

    \item \emph{Abstraction}: $t = \lax r \Rew{\losym} \lax r'= s_1$. Then $t = \lax r \Rew{\neg \losym} \lax r''= s_2$ and the statement follows from the {\textit{i.h.}}\xspace and closure of $\Rew{\losym}$.

    \item \emph{Left of an application}: $t = r p \Rew{\losym} r_1 p = s_1$ with $r \Rew{\losym} r_1$ and $r$ not an abstraction. Two sub-cases:
    \begin{itemize}
      \item $t = r p \Rew{\neg \losym} r_2 p = s_2$ because $r \Rew{\neg \losym} r_2$. Then by {\textit{i.h.}}\xspace there exists $q$ such that $r_2 \Rew{\losym} q$. Note that $\Rew{\neg \losym}$ cannot create a root abstraction (because it never reduces the root redex) so that if $r$ is not an abstraction  then $r_2$ is not an abstraction and $s_2 = r_2 p \Rew{\losym} q p =:u$.

      \item $t = r p \Rew{\neg \losym} r p' = s_2$ by one of the two rules for able to derive it. In both cases $p \Rew{\beta} p'$. Then $s_2 = r p' \Rew{\losym} r_1 p' =:u$ and $s_1 = r_1 p \Rew{\beta} r_1 p' =u$.
    \end{itemize}

    \item \emph{Right of an application}: $t = r p \Rew{\losym} r p_1 = s_1$ with $p \Rew{\losym} p_1$ and $r$ neutral. Then necessarily $t = r p \Rew{\losym} r p_2 = s_2$ with $p \Rew{\neg \losym} p_2$. The statement then follows by the {\textit{i.h.}}\xspace
    \qedhere
  \end{itemize}
\end{enumerate}
\end{proof}

\setcounter{propositionAppendix}{\value{prop:macro-left}}
\begin{propositionAppendix}[$\ell o$\xspace macro-step system]
	\label{propappendix:macro-left}
	\NoteState{prop:macro-left}
\hfill
  \begin{enumerate}
	\item\label{pappendix:macro-left-merge} \emph{Merge}: if $t \parRew{\neg \losym} \cdot \Rew{\losym} u$ then $t \parRew{\beta} u$.
	\item\label{pappendix:macro-left-indexed-split} \emph{Indexed split}: if $t \partobind n s$ then $t \parRew{\neg \losym} s$, or $n>0$ and  $t \Rew{\losym} \cdot \partobind {n-1} s$.
	\item\label{pappendix:macro-left-split} \emph{Split}: if $t \parRew{\beta} s$ then $t \Rew{\losym}^* \cdot \parRew{\neg \losym} s$.
	\end{enumerate}
	That is, $(\Lambda, \set{\Rew{\losym}, \Rew{\neg \losym}})$ is a macro-step system with respect to $\parRew{\beta}$ and $\parRew{\neg \losym}$.
\end{propositionAppendix}

\begin{proof}
\hfill
  \begin{enumerate}
	\item \emph{Merge}: by induction on $t \parRew{\neg \losym} s$. Cases:
\begin{itemize}
	\item Rule
	\[\AxiomC{$r \parRew{\beta} r'$}
	\AxiomC{$p \parRew{\beta} p''$}

	\BinaryInfC{$t = (\lax r) p \parRew{\neg \losym} (\lax r') p' = s$}
	\DisplayProof\]
	Then $(\lax r) p \parRew{\neg \losym} (\lax r') p' \Rew{\losym} r'\isubx{p'} = u$. We simply have:
	\[	\AxiomC{$r \parRew{\beta} r'$}
	\AxiomC{$p \parRew{\beta} p''$}
\BinaryInfC{$t = (\la\varr)p \parRew{\beta} r'\isubx{p'} = u$}
\DisplayProof\]

	\item Rule
	\[\AxiomC{$r$ not neutral}
	\AxiomC{$r \parRew{\neg \losym} r'$}
		\AxiomC{$p \parRew{\beta} p'$}

	\TrinaryInfC{$t = r p \parRew{\neg \losym} r' p ' = s$}
	\DisplayProof\]
	Since $r$ is not neutral, it is an abstraction or it is not normal. If $r$ is an abstraction this case continues goes as the first case. Otherwise, $r$ is not normal, and by persistence (\refpropp{essential-left}{persistence}) $r'$ is not normal.
	Fullness\xspace (\refpropp{essential-left}{completeness}) of $\Rew{\losym}$ gives $r' \Rew{\losym} r''$ for a certain $r''$. The hypothesis becomes $t = r p \parRew{\neg \losym} r' p ' \Rew{\losym} r'' p' = u$.
	By {\textit{i.h.}}\xspace, $r \parRew{\beta} r''$. Then,
	\[\AxiomC{$r \parRew{\beta} r''$}
	\AxiomC{$p\parRew{\beta} p'$}

	\BinaryInfC{$t = r p \parRew{\beta} r'' p' = u$}
	\DisplayProof\]

	\item Rule
	\[\AxiomC{$r \parRew{\neg \losym} r'$}

	\UnaryInfC{$t = \la\varr \parRew{\neg \losym} \la\varr' = s$}
	\DisplayProof \]
	Then $\la\varr \parRew{\neg \losym} \la\varr' \Rew{\losym} \la\varr'' = u$ with $r' \Rew{\losym} r''$. By {\textit{i.h.}}\xspace, $r \parRew{\beta} r''$ and
	\[\AxiomC{$r \parRew{\beta} r'$}
\UnaryInfC{$t = \la \varr \parRew{\beta} \la x r'' = u$}
\DisplayProof\]

	\item Rule
	\[\AxiomC{$r$ neutral }
	\AxiomC{$p \parRew{\neg \losym} p'$}

	\BinaryInfC{$t = r p \parRew{\neg \losym} r p' = s$}
	\DisplayProof  \]
	 Then the hypothesis is $r p \parRew{\neg \losym} r p' \Rew{\losym} r p'' = u$ with $p' \Rew{\losym} p''$. By {\textit{i.h.}}\xspace, $p \parRew{\beta} p''$, and since $\parRew{\beta}$ is reflexive,
		\[\AxiomC{$r \parRew{\beta} r$}
		\AxiomC{$p\parRew{\beta} p''$}

		\BinaryInfC{$t = r p \parRew{\beta} r p'' = u$}
		\DisplayProof\]
\end{itemize}
	\item \emph{Indexed split}: By induction on the definition of $t \partobind n s$. We use freely the fact that if $t \partobind n s$ then $t \parRew{\beta} s$. Cases:
\begin{itemize}
	\item \emph{Variable}: $t = x \partobind 0 x = s$. Then $t = x \parRew{\neg \losym} x = s$.

	\item \emph{Abstraction}: $t = \la \vart' \partobind n  \la x s' = s$ because $t' \partobind n s'$. It follows from the {\textit{i.h.}}\xspace.

	\item \emph{Application}:
	\begin{align*}
	\label{eq:app}
	\AxiomC{$r \partobind {n_1} r'$}
	\AxiomC{$p \partobind {n_2} p'$}
	\BinaryInfC{$t = r p \partobindlong {n_1 + n_2} r' p' = s$}
	\DisplayProof
	\end{align*}
	with $n = n_1 + n_2$. There are only two cases:
	\begin{itemize}
		\item either $r p \parRew{\neg \losym} r' p'$, and then the claim holds;
		\item or $r p \not\parRew{\neg \losym} r' p'$, then the following conditions hold (otherwise $r p \parRew{\neg \losym} r' p'$):
		\begin{enumerate}
			\item $r$ is not an abstraction;
			\item if $r$ is neutral then $p \not\parRew{\neg \losym} p'$;
			\item if $r$ is not neutral then $r \not\parRew{\neg \losym} r'$;
		\end{enumerate}
		So, if $r$ is neutral, then by {\textit{i.h.}}\xspace applied to $p \partobind{n_2} p'$, $n_2 > 0$ and there is $p''$ such that $p \Rew{\losym} p'' \partobind{n_2-1} r'$; thus, $t = r p \Rew{\losym} rp''$ and
		\[
		\AxiomC{$r \partobind {n_1} r'$}
		\AxiomC{$p'' \partobindlong {n_2-1} p'$}
		\BinaryInfC{$r p'' \partobindlong {n_1 + n_2-1} r' p' = s$}
		\DisplayProof
		\ .
		\]

		Otherwise $r$ is not neutral and hence, by {\textit{i.h.}}\xspace applied to $r \partobind{n_1} r'$,
		$n_1>0$ and there exists $r''$ such that $r \Rew{\losym} r'' \partobind {n_1-1} r'$;
		thus, $t = r p \Rew{\losym} r'' p$ and
		\[\AxiomC{$r'' \partobindlong {n_1-1} r'$}
		\AxiomC{$p \partobind {n_2} p'$}
		\BinaryInfC{$r'' p \partobindlong {n_1-1 + n_2} r' p' = s$}
		\DisplayProof
		\ .
		\]
	\end{itemize}

	\item \emph{$\beta$ step}:
		\[\AxiomC{$u \partobind {n_1} u'$}
		\AxiomC{$r \partobind {n_2} r''$}
		\BinaryInfC{$t = (\lax u)r \partobindlong {n_1 + \sizep{s}x \cdot n_2 +1} u'\isub x {r'} = s$}
		\DisplayProof\]
		With $n = n_1 + \sizepsx \cdot n_2 +1 > 0$. We have $t = (\lax u)r \Rew{\losym} u\isub\varr$ and by substitutivity of $\partobind n$ (\reflemma{partobind-subs}) $u\isub\varr \partobindlong{n_1 + \sizepsx \cdot n_2} u' \isubx{r'} = s$.
\end{itemize}

	\item \emph{Split}: exactly as in the head case (\refpropp{macro-head}{split}), using the Indexed Split property for $\ell o$\xspace (\refpoint{macro-left-indexed-split} above).
	\qedhere
	\end{enumerate}
\end{proof}

\subsection{Omitted proofs and lemmas of \refsect{stratified-cbn}}

\setcounter{lemmaAppendix}{\value{l:perpetuity-strat}}
\begin{lemmaAppendix}[Persistence]
\label{lappendix:perpetuity-strat}
\NoteState{l:perpetuity-strat}
	If $t \parRew{\lnot\ell\ell} s$ then $\Deg{s} \leq \Deg{t}$.
\end{lemmaAppendix}

\begin{proof}
	By hypothesis, $t \partostratind{n} s$ for some $n > \Deg{t}$.
	We prove that $\Deg{s} \leq \Deg{t}$ by induction on the definition of $t \partostratind{n} s$.
	Cases:
	\begin{itemize}
		\item \emph{Variable}: $t = x \partostratind{\infty} x = s$.
		Then, $\Deg{t} = \infty = \Deg{s}$.

		\item \emph{Abstraction}: $t = \la \vart' \partostratind{n}  \la x s' = s$ because $t' \partostratind{n} s'$.
		As $n > \Deg{t} = \Deg{t'}$, then $\Deg{s} = \Deg{s'} \leq \Deg{t'} = \Deg{t}$ by {\textit{i.h.}}\xspace

		\item \emph{Application}:
		\begin{align*}
		\AxiomC{$t_0 \partostratind{n_0} s_0$}
		\AxiomC{$t_1 \partostratind{n_1} s_1$}
		\BinaryInfC{$t = t_0 t_1 \partostratind{n} s_0 s_1 = s$}
		\DisplayProof
		\end{align*}
		where $n = \min\{n_0,n_1+1\}$.
		According to the definition of $\Deg{t}$, there are the following sub-cases:
		\begin{enumerate}
			\item $\Deg{t} = \Deg{t_0} \leq \Deg{t_1} + 1$ and $t_0$ is not an abstraction;

			by {\textit{i.h.}}\xspace applied to $t_0$ (since $\Deg{t_0} = \Deg{t} < n \leq n_0$), we have $\Deg{s_0} \leq \Deg{t_0}$, and so (as $\Deg{t_0} \leq \Deg{t_1} + 1$) $\Deg{s} = \min \{\Deg{s_0}, \Deg{s_1} + 1\} \leq \min\{\Deg{t_0}, \Deg{t_1} + 1\} = \Deg{t}$.

			\item $\Deg{t} = \Deg{t_1} + 1 \leq \Deg{t_0}$ and $t_0$ is not an abstraction;

			by {\textit{i.h.}}\xspace applied to $t_1$ (since $\Deg{t_1} + 1 = \Deg{t} < n \leq n_1+1$ and hence $\Deg{t_1} < n_1$), we have $\Deg{s_1} \leq \Deg{t_1}$, and so (as $\Deg{t_1} + 1 \leq \Deg{t_0}$) $\Deg{s} = \min \{\Deg{s_0}, \Deg{s_1} + 1\} \leq \min\{\Deg{t_0}, \Deg{t_1} + 1\} = \Deg{t}$.

			\item $\Deg{t} = 0$ and $t_0$ is an abstraction.
			According to the definition of $t \partostratind{n} s$, since $n > 0$, we have
				\begin{align*}
				\AxiomC{$t_0 \partostratind{n_0} s_0$}
				\UnaryInfC{$\la{x}t_0 \partostratind{n_0} \la{x}s_0$}
				\AxiomC{$t_1 \partostratind{n_1} s_1$}
				\BinaryInfC{$t = (\la{x}t_0)t_1 \partostratind{n} (\la{x}{s_0})s_1 = s$}
				\DisplayProof
				\end{align*}
				where $n = \min\{n_0, n_1+1\}$; therefore $\Deg{s} = 0 = \Deg{t}$.
		\end{enumerate}

		\item \emph{$\beta$-step}:
		\begin{align*}
		\AxiomC{}
		\UnaryInfC{$t = (\la{x}t_0)t_1 \partostratind{0} t_0 \isub{x}{t_1} = s$}
		\DisplayProof
		\end{align*}
		where $0 = n > \Deg{t}$, which is impossible.
		\qedhere
	\end{itemize}
\end{proof}

\begin{propositionAppendix}[Internal-external merge]
	\label{propappendix:i-e-merge-strat}
	\NoteState{prop:i-e-merge-strat}
	\begin{enumerate}
		\item If $t \partostratind{n}s \tostratind{m} u$ with $n > m$, then $t \parRew{\beta} u$.
		\item If $t \parRew{\lnot\ell\ell} s \Rew{x} u$, then $t \parRew{\beta} u$.
	\end{enumerate}
\end{propositionAppendix}

\begin{proof}\hfill
	\begin{enumerate}
	\item By induction on the definition of $t \partostratind{n} s$.
	Cases:
	\begin{itemize}
		\item \emph{Variable}: $t = x \partostratind{\infty} x = s$.
		Then, there is no $u$ such that $s \tostratind{m} u$ for any $m \in \mathbb{N}$.

		\item \emph{Abstraction}: $t = \la \vart' \partostratind{n}  \la x s' = s$ because $t' \partostratind{n} s'$.
		According to the definition of $s \tostratind{m} u$, by necessity $u = \la{x}{u'}$ with $s' \tostratind{m} u'$.
		By {\textit{i.h.}}\xspace, $t' \parRew{\betav} u'$, thus
		\begin{align*}
		\AxiomC{$t' \parRew{\betav} u'$}
		\UnaryInfC{$t = \la{x}{t'} \parRew{\betav} \la{x}{u'} = u$}
		\DisplayProof
		\,.
		\end{align*}

		\item \emph{Application}:
		\begin{align*}
		\AxiomC{$t_0 \partostratind{n_0} s_0$}
		\AxiomC{$t_1 \partostratind{n_1} s_1$}
		\BinaryInfC{$t = t_0 t_1 \partostratind{n} s_0 s_1 = s$}
		\DisplayProof
		\end{align*}
		where $n = \min\{n_0,n_1+1\}$.
		According to the definition of $s \tostratind{m} u$, there are the following sub-cases:
		\begin{enumerate}
			\item $s = s_0s_1 \tostratind{m} u_0s_1 = u$ with $s_0 \tostratind{m} u_0$;
			since $m < n \leq n_0$,
			by {\textit{i.h.}}\xspace applied to $t_0 \partostratind{n_0} s_0 \tostratind{m} u_0$, we have $t_0 \parRew{\betav} u_0$, and so (as $\partostratind{n_1} \,\subseteq\, \parRew{\betav}$)
			\begin{align*}
			\AxiomC{$t_0 \parRew{\betav} u_0$}
			\AxiomC{$t_1 \parRew{\betav} s_1$}
			\BinaryInfC{$t = t_0t_1 \parRew{\betav} u_0s_1 = u$}
			\DisplayProof
			\,;
			\end{align*}
			\item $s = s_0s_1 \tostratind{m} u_0s_1 = u$ with $s_1 \tostratind{m-1} u_1$;
			since $m-1 < n-1 \leq n_1$,
			by {\textit{i.h.}}\xspace applied to $t_1 \partostratind{n_1} s_1 \tostratind{m-1} u_1$, we have $t_1 \parRew{\betav} u_1$, and so (as $t_0 \partostratind{n_0} \,\subseteq\, \parRew{\betav}$)
			\begin{align*}
			\AxiomC{$t_0 \parRew{\betav} s_0$}
			\AxiomC{$t_1 \parRew{\betav} u_1$}
			\BinaryInfC{$t = t_0t_1 \parRew{\betav} s_0u_1 = u$}
			\DisplayProof
			\,;
			\end{align*}

			\item $s = (\la{x}{s_0})s_1 \tostratind{0} s_0\isub{x}{s_1} = u$ with $m = 0$;
			as $n > 0$ then, according to the definition of $t \partostratind{n} s$,
			\begin{align*}
			\AxiomC{$t_0 \partostratind{n_0} s_0$}
			\UnaryInfC{$\la{x}t_0 \partostratind{n_0} \la{x}s_0$}
			\AxiomC{$t_1 \partostratind{n_1} s_1$}
			\BinaryInfC{$t = (\la{x}t_0)t_1 \partostratind{n} (\la{x}{s_0})s_1 = s$}
			\DisplayProof
			\end{align*}
			where $n = \min\{n_0, n_1+1\}$; therefore (as $\partostratind{k} \,\subseteq\, \parRew{\betav}$)
			\begin{align*}
			\AxiomC{$t_0 \parRew{\betav} s_0$}
			\AxiomC{$t_1 \parRew{\betav} s_1$}
			\BinaryInfC{$t = (\la{x}{t_0})t_1 \parRew{\betav} s_0\isub{x}{s_1} = u$}
			\DisplayProof
			\,.
			\end{align*}
		\end{enumerate}
	\end{itemize}

	\item As $t \parRew{\lnot\ell\ell} s \Rew{x} u$, then $t \partostratind{n} s \tostratind{m} u$ for some $n \in \mathbb{N} \cup \{\infty\}$ and $m \in \mathbb{N}$ such that $n > \Deg{t}$ and  $m = \Deg{s}$.
	By persistence (\reflemma{perpetuity-strat}),  $\Deg{t} \geq \Deg{s}$.
	Hence, $n > m$ and we can then merge (by \refpropp{i-e-merge-strat}{ind}) to $t \parRew{\beta} u$.
	\qedhere
	\end{enumerate}
\end{proof}

\begin{lemma}[Decomposition of parallel $\beta$-steps]
\label{l:decomp-strat}
If $t \partobind{n} s$ then $t \partostratind{m} s$ for some $m \in \mathbb{N} \cup \{\infty\}$.
If moreover $n > 0$ then $m \neq \infty$ and $t \tostratind{m} u$ for some term $u$ such that $u \partobindlong{n-1} s$ and $u \partostratind{j} s$ with $j \geq m	$ (possibly $j = \infty$).
\end{lemma}

\begin{proof}
By induction $(n,k) \in \mathbb{N} \times \mathbb{N}$ with the lexicographical order, where $k$ is the length of the derivation of $t \partobind{n} s$.
Cases:
\begin{itemize}
	\item \emph{Variable}: $t = x \partobind{0} x = s$ where $n = 0$.
	Then,
	\begin{align*}
	\AxiomC{}
	\UnaryInfC{$t = x \partostratind{\infty} x = s$}
	\DisplayProof
	\,.
	\end{align*}

	\item \emph{Abstraction}: $t = \la \vart' \partobind{n}  \la x s' = s$ because $t' \partobind{n} s'$.
	By {\textit{i.h.}}\xspace, $t' \partostratind{m} s'$ for some $m \in \mathbb{N} \cup \{\infty\}$.
	Then,
	\begin{align*}
	\AxiomC{$t' \partostratind{m} s'$}
	\UnaryInfC{$t = \la{x}{u'} \partostratind{m} \la{x}{s'} = s$}
	\DisplayProof
	\,.
	\end{align*}
	If moreover $n > 0$, then by {\textit{i.h.}}\xspace, $m \neq \infty$ and $t' \tostratind{m} u'$ for some term $u'$ such that $u' \partobindlong{n-1} s'$ and $u' \partostratind{j} s'$ with $j \geq m$ (possibly $j = \infty$), so
	\begin{align*}
	\AxiomC{$t' \tostratind{m} u'$}
	\UnaryInfC{$t = \la{x}{t'} \tostratind{m} \la{x}{u'} = u$}
	\DisplayProof
	\qquad
	\text{and}
	\qquad
	&
	\AxiomC{$u' \partostratind{j} s'$}
	\UnaryInfC{$u = \la{x}{u'} \partostratind{j} \la{x}{s'} = s$}
	\DisplayProof
	\shortintertext{and}
	\AxiomC{$u' \partobind{n-1} s'$}
	\UnaryInfC{$u = \la{x}{u'} \partobind{n-1} \la{x}{s'} = s$}
	\DisplayProof
	\,.
	\end{align*}

	\item \emph{Application}:
	\begin{align*}
	\AxiomC{$t_0 \partobind{n_0} s_0$}
	\AxiomC{$t_1 \partobind{n_1} s_1$}
	\BinaryInfC{$t = t_0 t_1 \partobind{n} s_0 s_1 = s$}
	\DisplayProof
	\end{align*}
	where $n = n_0 + n_1$.
	By {\textit{i.h.}}\xspace, $t_0 \partostratind{m_0} s_0$ and $t_1 \partostratind{m_1} s_1$ for some $m_0, m_1 \in \mathbb{N} \cup \{\infty\}$.
	Thus,
	\begin{align*}
	\AxiomC{$t_0 \partostratind{m_0} s_0$}
	\AxiomC{$t_1 \partostratind{m_1} s_1$}
	\BinaryInfC{$t = t_0 t_1 \partostratind{m} s_0 s_1 = s$}
	\DisplayProof
	\end{align*}
	where $m = \min\{m_0,m_1+1\}$.
	If moreover $n > 0$, then by {\textit{i.h.}}\xspace, $m_0, m_1 \neq \infty$ and $t_0 \tostratind{m_0} u_0$ and $t_1 \tostratind{n_1} u_1$ for some terms $u_0, u_1$ such that $u_0 \partobindlong{n_0-1} s_0$ and $u_1 \partobindlong{n_1-1} s_1$ and $u_0 \partostratind{j_0} s_0$ and $u_1 \partostratind{j_1} s_1$ with $j_0 \geq m_0$, $j_1 \geq m_1$ (possibly $j_0,j_1 = \infty$).
	There are two possibilities:
	\begin{enumerate}
		\item $m = m_0$; then, $m_0 \leq m_1+1$ and
		\begin{align*}
		\AxiomC{$t_0 \tostratind{m} u_0$}
		\UnaryInfC{$t = t_0t_1 \tostratind{m} u_0t_1 = u$}
		\DisplayProof
		\qquad
		\text{and}
		\qquad
		&
		\AxiomC{$u_0 \partobind{n_0-1} s_0$}
		\AxiomC{$t_1 \partobind{n_1} s_1$}
		\BinaryInfC{$u = u_0t_1 \partobind{n-1} s_0s_1 = s$}
		\DisplayProof
		\shortintertext{and}
		\AxiomC{$u_0 \partostratind{j_0} s_0$}
		\AxiomC{$t_1 \partostratind{m_1} s_1$}
		\BinaryInfC{$u = u_0t_1 \partostratind{j} s_0s_1 = s$}
		\DisplayProof
		\end{align*}
		where $j = \min\{j_0, m_1+1\} \geq m_0 = m$;
		\item $m = m_1+1$; then, $m_1 + 1 \leq m_0$ and
		\begin{align*}
		\AxiomC{$t_1 \tostratind{m-1} u_1$}
		\UnaryInfC{$t = t_0t_1 \tostratind{m} t_0u_1 = u$}
		\DisplayProof
		\qquad
		\text{and}
		\qquad
		&
		\AxiomC{$u_0 \partobind{n_0} s_0$}
		\AxiomC{$t_1 \partobind{n_1-1} s_1$}
		\BinaryInfC{$u = t_0u_1 \partobind{n-1} s_0s_1 = s$}
		\DisplayProof
		\shortintertext{and}
		\AxiomC{$u_0 \partostratind{m_0} s_0$}
		\AxiomC{$t_1 \partostratind{j_1} s_1$}
		\BinaryInfC{$u = t_0u_1 \partostratind{j} s_0s_1 = s$}
		\DisplayProof
		\end{align*}
		where $j = \min\{m_0, j_1+1\} \geq m_1+1 = m$.
	\end{enumerate}

	\item \emph{$\beta$-step}:
	\begin{align*}
	\AxiomC{$t_0 \partobind{n_0} t_0'$}
	\AxiomC{$t_1 \partobind{n_1} t_1'$}
	\BinaryInfC{$t = (\la{x}t_0)t_1 \partobind{n} t_0' \isub{x}{t_1'} = s$}
	\DisplayProof
	\end{align*}
	where $n = n_0 + \sizep{t_0}{x}n_1 + 1 > 0$.
	By {\textit{i.h.}}\xspace, $t_0 \partostratind{m_0} s_0$ and $t_1 \partostratind{m_1} s_1$ for some $m_0, m_1 \in \mathbb{N}$.
	Thus,
	\begin{align*}
	\AxiomC{$t_0 \partostratind{m_0} t_0'$}
	\AxiomC{$t_1 \partostratind{m_1} t_1'$}
	\BinaryInfC{$t = (\la{x}t_0)t_1 \partostratind{0} t_0' \isub{x}{t_1'} = s$}
	\DisplayProof
	\,.
	\end{align*}
	Let $u = t_0\isub{x}{t_1}$.
	By \reflemma{partobind-subs}, $u = t_0\isub{x}{t_1} \partobindlong{n_0 + \sizep{t_0}{x} n_1} t_0'\isub{x}{t_1'} = s$ where $n_0 + \sizep{t_0}{x} n_1 = n - 1$.
	Clearly,
	\begin{align*}
	\AxiomC{}
	\UnaryInfC{$t = (\la{x}t_0)t_1 \tostratind{0} t_0 \isub{x}{t_1} = u$}
	\DisplayProof
	\,.
	\end{align*}
	By {\textit{i.h.}}\xspace (since $n_0 + \sizep{t_0}{x} n_1 = n -1$), $u = t_0\isub{x}{t_1} \partostratind{j} t_0'\isub{x}{t_1'} = s$ for some $j \in \mathbb{N} \cup \{\infty\}$, in particular $j \geq 0$.
	\qedhere
\end{itemize}
\end{proof}

\begin{propositionAppendix}[Stratified indexed split]
	\label{propappendix:quant-out-in-split-strat}
	\NoteState{prop:quant-out-in-split-strat}
	If $t \partobind n s$ then either $t \parRew{\lnot\ell\ell}s$ or $n>0$ and there exists $u$ such that $t \Rew{x} u \partobindlong {n-1} s$.
\end{propositionAppendix}

\begin{proof}
	If $n = 0$, then $t = s$ (because $\partobind{0}$ is the identity relation) and so $t \partostratind{\infty} s$ (because $\partostratind{\infty}$ is the identity relation), hence $t \parRew{\lnot\ell\ell} s$.

	Otherwise $n > 0$ and then, by \reflemma{decomp-strat}, $t \partostratind{k} s$ and $t \tostratind{k} u$ for some $k \in \mathbb{N}$ and some term $u$ such that $u \partobind{n-1} s$.
	There are two cases:
	\begin{enumerate}
		\item either $k > \Deg{t}$ and then $t \parRew{\lnot\ell\ell} s$ (since $t \partostratind{k} s$);
		\item or $k = \Deg{t}$  and so $t \Rew{x} u \partobindlong{n-1} s$.
		\qedhere
	\end{enumerate}
\end{proof}

\begin{corollaryAppendix}[Stratified split]
	\label{coroappendix:e-i-split-strat}
	\NoteState{coro:e-i-split-strat}
	If $t \parRew{\beta} s$ then $t \Rew{x}^* u \parRew{\lnot\ell\ell} s$ for some $u$.
\end{corollaryAppendix}

\begin{proof}
	By hypothesis, $t \partobind{n} s$ for some $n \in \mathbb{N}$.
	We prove the statement by induction on $n \in \mathbb{N}$.

	If $n = 0$, then $t = s$ (because $\partobind{0}$ is the identity relation) and so $t \partostratind{\infty} s$ (because $\partostratind{\infty}$ is the identity relation), hence $t \parRew{\lnot\ell\ell} s$.

	Otherwise $n > 0$ and then, by \reflemma{decomp-strat}, $t \partostratind{k} s$ and $t \tostratind{k} u$ for some $k \in \mathbb{N}$ and some term $u$ such that $u \partobind{n-1} s$.
	There are two cases:
	\begin{enumerate}
		\item either $k > \Deg{t}$ and then $t \parRew{\lnot\ell\ell} s$ (as $t \partostratind{k} s$), so we conclude taking $u = t$;
		\item or $k = \Deg{t}$  and then $t \Rew{x} r \partobindlong{n-1} s$;
		by {\textit{i.h.}}\xspace applied to $r \partobind{n-1} s$, there is a term $u$ such that $r \Rew{x}^* u \parRew{\lnot\ell\ell} s$;
		therefore, $t \Rew{x} r \Rew{x}^* u \parRew{\lnot\ell\ell} s$.
		\qedhere
	\end{enumerate}
\end{proof}

Note that the last proof uses only \reflemma{decomp-strat}, not \refprop{quant-out-in-split-strat}!

\subsection{Omitted proofs and lemmas of \refsect{stratified-cbn}}

\begin{lemma}[Persistence]
\label{l:perpetuity-strat}
	If $t \partostratind{n} s$ and $t \tostratind{m} u$ with $m < n$, then there exists a term $r$ such that $s \tostratind{m} r$.
\end{lemma}

\begin{proof}
	By induction on the definition of $t \partostratind{n} s$.
	Cases:
	\begin{itemize}
		\item \emph{Variable}: $t = x \partostratind{\infty} x = s$.
		This is impossible because there is no $u$ such that $t \tostratind{m} u$ for any $m \in \mathbb{N}$.

		\item \emph{Abstraction}: $t = \la \vart' \partostratind{n}  \la x s' = s$ because $t' \partostratind{n} s'$.
		By definition of $t \tostratind{m} u$, by necessity $u = \la{x}u'$ with $t' \tostratind{m} u'$
		By {\textit{i.h.}}\xspace, $s' \tostratind{m} r'$ for some term $r'$.
		Therefore,
		\begin{align*}
		\AxiomC{$s' \tostratind{m} r'$}
		\UnaryInfC{$s = \la{x}{s'} \tostratind{m} \la{x}{r'} = r$}
		\DisplayProof
		\,.
		\end{align*}

		\item \emph{Application}:
		\begin{align*}
		\AxiomC{$t_0 \partostratind{n_0} s_0$}
		\AxiomC{$t_1 \partostratind{n_1} s_1$}
		\BinaryInfC{$t = t_0 t_1 \partostratind{n} s_0 s_1 = s$}
		\DisplayProof
		\end{align*}
		where $n = \min\{n_0,n_1+1\}$.
		According to the definition of $t \tostratind{m} u$, there are the following sub-cases:
		\begin{enumerate}
			\item $t = t_0t_1 \tostratind{m} u_0t_1 = u$ with $t_0 \tostratind{m} u_0$;
			since $m < n \leq n_0$,
			by {\textit{i.h.}}\xspace applied to $t_0$, we have $s_0 \tostratind{m} r_0$  for some term $r_0$, and so
			\begin{align*}
			\AxiomC{$s_0 \tostratind{m} r_0$}
			\UnaryInfC{$s = s_0s_1 \tostratind{m} r_0s_1 = r$}
			\DisplayProof
			\,;
			\end{align*}
			\item $t = t_0t_1 \tostratind{m} t_0u_1 = u$ with $t_1 \tostratind{m-1} u_1$;
			since $m-1 < n-1 \leq n_1$,
			by {\textit{i.h.}}\xspace applied to $t_1$, we have $s_1 \tostratind{m} r_1$ for some term $r_1$, and so
			\begin{align*}
			\AxiomC{$s_1 \tostratind{m-1} r_1$}
			\UnaryInfC{$s = s_0s_1 \tostratind{m} s_0r_1 = r$}
			\DisplayProof
			\,;
			\end{align*}

			\item $t = (\la{x}{t_0})t_1 \tostratind{0} t_0\isub{x}{t_1} = u$ with $m = 0 < n$.
			According to the definition of $t \partostratind{n} s$,
				\begin{align*}
				\AxiomC{$t_0 \partostratind{n_0} s_0$}
				\UnaryInfC{$\la{x}t_0 \partostratind{n_0} \la{x}s_0$}
				\AxiomC{$t_1 \partostratind{n_1} s_1$}
				\BinaryInfC{$t = (\la{x}t_0)t_1 \partostratind{n} (\la{x}{s_0})s_1 = s$}
				\DisplayProof
				\end{align*}
				where $n = \min\{n_0, n_1+1\}$; therefore
				\begin{align*}
				\AxiomC{}
				\UnaryInfC{$s = (\la{x}s_0)s_1 \tostratind{0} s_0\isub{x}{s_1} = r$}
				\DisplayProof
				\,.
				\end{align*}
		\end{enumerate}

		\item \emph{$\beta$-step}:
		\begin{align*}
		\AxiomC{}
		\UnaryInfC{$t = (\la{x}t_0)t_1 \partostratind{0} t_0 \isub{x}{t_1} = s$}
		\DisplayProof
		\end{align*}
		where $n = 0$. This is impossible because there is no term $u$ such that $t \tostratind{m} u$ with $ m < 0$.
		\qedhere
	\end{itemize}
\end{proof}

\setcounter{propositionAppendix}{\value{prop:i-e-merge-strat}}
\begin{propositionAppendix}[Internal-external merge]
	\label{propappendix:i-e-merge-strat}
	\NoteState{prop:i-e-merge-strat}
	\begin{enumerate}
		\item If $t \partostratind{n}s \tostratind{m} u$ with $n > m$, then $t \parRew{\beta} u$.
		\item If $t \parRew{\lnot\ell\ell} s \Rew{x} u$, then $t \parRew{\beta} u$.
	\end{enumerate}
\end{propositionAppendix}

\begin{proof}\hfill
	\begin{enumerate}
	\item By induction on the definition of $t \partostratind{n} s$.
	Cases:
	\begin{itemize}
		\item \emph{Variable}: $t = x \partostratind{\infty} x = s$.
		Then, there is no $u$ such that $s \tostratind{m} u$ for any $m \in \mathbb{N}$.

		\item \emph{Abstraction}: $t = \la \vart' \partostratind{n}  \la x s' = s$ because $t' \partostratind{n} s'$.
		According to the definition of $s \tostratind{m} u$, by necessity $u = \la{x}{u'}$ with $s' \tostratind{m} u'$.
		By {\textit{i.h.}}\xspace, $t' \parRew{\betav} u'$, thus
		\begin{align*}
		\AxiomC{$t' \parRew{\betav} u'$}
		\UnaryInfC{$t = \la{x}{t'} \parRew{\betav} \la{x}{u'} = u$}
		\DisplayProof
		\,.
		\end{align*}

		\item \emph{Application}:
		\begin{align*}
		\AxiomC{$t_0 \partostratind{n_0} s_0$}
		\AxiomC{$t_1 \partostratind{n_1} s_1$}
		\BinaryInfC{$t = t_0 t_1 \partostratind{n} s_0 s_1 = s$}
		\DisplayProof
		\end{align*}
		where $n = \min\{n_0,n_1+1\}$.
		According to the definition of $s \tostratind{m} u$, there are the following sub-cases:
		\begin{enumerate}
			\item $s = s_0s_1 \tostratind{m} u_0s_1 = u$ with $s_0 \tostratind{m} u_0$;
			since $m < n \leq n_0$,
			by {\textit{i.h.}}\xspace applied to $t_0 \partostratind{n_0} s_0 \tostratind{m} u_0$, we have $t_0 \parRew{\betav} u_0$, and so (as $\partostratind{n_1} \,\subseteq\, \parRew{\betav}$)
			\begin{align*}
			\AxiomC{$t_0 \parRew{\betav} u_0$}
			\AxiomC{$t_1 \parRew{\betav} s_1$}
			\BinaryInfC{$t = t_0t_1 \parRew{\betav} u_0s_1 = u$}
			\DisplayProof
			\,;
			\end{align*}
			\item $s = s_0s_1 \tostratind{m} u_0s_1 = u$ with $s_1 \tostratind{m-1} u_1$;
			since $m-1 < n-1 \leq n_1$,
			by {\textit{i.h.}}\xspace applied to $t_1 \partostratind{n_1} s_1 \tostratind{m-1} u_1$, we have $t_1 \parRew{\betav} u_1$, and so (as $t_0 \partostratind{n_0} \,\subseteq\, \parRew{\betav}$)
			\begin{align*}
			\AxiomC{$t_0 \parRew{\betav} s_0$}
			\AxiomC{$t_1 \parRew{\betav} u_1$}
			\BinaryInfC{$t = t_0t_1 \parRew{\betav} s_0u_1 = u$}
			\DisplayProof
			\,;
			\end{align*}

			\item $s = (\la{x}{s_0})s_1 \tostratind{0} s_0\isub{x}{s_1} = u$ with $m = 0$;
			as $n > 0$ then, according to the definition of $t \partostratind{n} s$,
			\begin{align*}
			\AxiomC{$t_0 \partostratind{n_0} s_0$}
			\UnaryInfC{$\la{x}t_0 \partostratind{n_0} \la{x}s_0$}
			\AxiomC{$t_1 \partostratind{n_1} s_1$}
			\BinaryInfC{$t = (\la{x}t_0)t_1 \partostratind{n} (\la{x}{s_0})s_1 = s$}
			\DisplayProof
			\end{align*}
			where $n = \min\{n_0, n_1+1\}$; therefore (as $\partostratind{k} \,\subseteq\, \parRew{\betav}$)
			\begin{align*}
			\AxiomC{$t_0 \parRew{\betav} s_0$}
			\AxiomC{$t_1 \parRew{\betav} s_1$}
			\BinaryInfC{$t = (\la{x}{t_0})t_1 \parRew{\betav} s_0\isub{x}{s_1} = u$}
			\DisplayProof
			\,.
			\end{align*}
		\end{enumerate}
	\end{itemize}

	\item As $t \parRew{\lnot\ell\ell} s \Rew{x} u$, then $t \partostratind{n} s \tostratind{m} u$ for some $n \in \mathbb{N} \cup \{\infty\}$ and $m \in \mathbb{N}$, where (by definition of $\Rew{x}$) $m$ is minimal for $s$, {\em i.e.}\xspace if $s \tostratind{m'} u'$ then $m'\geq m$, and (by definition of $t \parRew{\lnot\ell\ell} s$), $t \tostratind{k} r$ for term $r$ and some $k < n$.
	By perpetuity (\reflemma{perpetuity-strat}) there exists a term $r'$ such that $s \tostratind{k} r'$, and $k \geq m$ by minimality of $m$.
	Hence, $n > m$ and we can then merge (by \refpropp{i-e-merge-strat}{ind}) to $t \parRew{\beta} u$.
	\qedhere
	\end{enumerate}
\end{proof}

\begin{lemma}[Substitutivity of $\partostratind{}$]
	\label{l:tostrat-subs}
	If $t \partostratind{n} t'$ and $s \partostratind{m} s'$ with $n,m \in \mathbb{N} \cup \{\infty\}$, then $t \isub\vars \partostratind{k} t' \isubx{s'}$ for some $k \in \mathbb{N} \cup \{\infty\}$.
\end{lemma}

\begin{proof}
	By induction on the definition of $t \partostratind{n} s$.
		Cases:
		\begin{itemize}
			\item \emph{Variable}: $t = y \partostratind{\infty} y = t'$.
			There are two sub-cases:
			\begin{enumerate}
				\item either $y = x$ and then $t\isub{x}{s} = s \partostratind{m} s' = t'\isub{x}{s'}$;
				\item or $y \neq x$ and then $t\isub{x}{s} = y \partostratind{\infty} y = t'\isub{x}{s'}$.
			\end{enumerate}

			\item \emph{Abstraction}: $t = \la \vartwou \partostratind{n}  \la y u' = t'$ because $u \partostratind{n} u'$.
			We can suppose without loss of generality that $y \notin \fv{s} \cup \{x\} \supseteq \fv{s'} \cup \{x\}$.
			By {\textit{i.h.}}\xspace, $u \partostratind{j} u'$ for some $j \in \mathbb{N} \cup \{\infty\}$, so
			\begin{align*}
			\AxiomC{$u \partostratind{j} u'$}
			\UnaryInfC{$t\isub{x}{s} = \la{y}{u\isub{x}{s}} \partostratind{j} \la{y}{u'\isub{x}{s'}} = t'\isub{x}{s'}$}
			\DisplayProof\,.
			\end{align*}

			\item \emph{Application}:
			\begin{align*}
			\AxiomC{$t_0 \partostratind{n_0} t'_0$}
			\AxiomC{$t_1 \partostratind{n_1} t'_1$}
			\BinaryInfC{$t = t_0 t_1 \partostratind{n} t'_0 t'_1 = s$}
			\DisplayProof
			\end{align*}
			where $n = \min\{n_0,n_1+1\}$.
			By {\textit{i.h.}}\xspace, $t_0\isub{x}{s} \partostratind{j_0} t_0'\isub{x}{s}$ and $t_1\isub{x}{s} \partostratind{j_1} t_1'\isub{x}{s}$ for some  $j_0, j_1 \in \mathbb{N} \cup \{\infty\}$.
			Therefore,
				\begin{align*}
				\AxiomC{$t_0\isub{x}{s} \partostratind{j_0} t_0'\isub{x}{s}$}
				\AxiomC{$t_1\isub{x}{s} \partostratind{j_1} t_1'\isub{x}{s}$}
				\BinaryInfC{$t\isub{x}{s} = t_0\isub{x}{s}t_1 \isub{x}{s} \partostratind{j} t_0'\isub{x}{s}t_1'\isub{x}{s} = t'$}
				\DisplayProof
				\end{align*}
				where $j = \min\{j_0, j_1+1\}$.

			\item \emph{$\beta$-step}:
			\begin{align*}
				\AxiomC{$t_0 \partostratind{n_0} t_0'$}
				\AxiomC{$t_1 \partostratind{n_1} t_1'$}
				\BinaryInfC{$t = (\la{y}t_0)t_1 \partostratind{0} t_0' \isub{y}{t_1'} = t'$}
				\DisplayProof
			\end{align*}
			We can suppose without loss of generality that $y \notin \fv{s} \cup \{x\} \supseteq \fv{s} \cup \{x\}$, hence $t'\isub{x}{s} = t_0'\isub{y}{t_1'}\isub{x}{s} = t_0'\isub{x}{s}\isub{y}{t_1'\isub{x}{s}}$.
			By {\textit{i.h.}}\xspace, $t_0\isub{x}{s} \partostratind{k_0} t_0'\isub{x}{s}$ and $t_1\isub{x}{s} \partostratind{k_1} t_1'\isub{x}{s}$ for some $k_1, k_2 \in \mathbb{N} \cup \{\infty\}$.
			Therefore,
			\begin{align*}
				\AxiomC{$t_0\isub{x}{s} \partostratind{k_0} t_0'\isub{x}{s}$}
				\AxiomC{$t_1\isub{x}{s} \partostratind{k_1} t_1'\isub{x}{s}$}
				\BinaryInfC{$t\isub{x}{s} = (\la{y}t_0\isub{x}{s})t_1\isub{x}{s} \partostratind{0} t_0'\isub{x}{s} \isub{y}{t_1'\isub{x}{s}} = t'\isub{x}{s}$}
				\DisplayProof\,.
			\end{align*}
			\qedhere
		\end{itemize}
\end{proof}

\begin{lemma}[Decomposition of parallel $\beta$-steps]
\label{l:decomp-strat}
If $t \partobind{n} s$ then $t \partostratind{m} s$ for some $m \in \mathbb{N} \cup \{\infty\}$.
If moreover $n > 0$ then $m \neq \infty$ and $t \tostratind{m} u$ for some term $u$ such that $u \partobindlong{n-1} s$ and $u \partostratind{j} s$ with $j \geq m	$ (possibly $j = \infty$).
\end{lemma}

\begin{proof}
By induction on the definition of $t \partobind{n} s$.
Cases:
\begin{itemize}
	\item \emph{Variable}: $t = x \partobind{0} x = s$.
	Then,
	\begin{align*}
		\AxiomC{}
		\UnaryInfC{$t = x \partostratind{\infty} x = s$}
		\DisplayProof
		\,.
	\end{align*}

	\item \emph{Abstraction}: $t = \la \vart' \partobind{n}  \la x s' = s$ because $t' \partobind{n} s'$.
	By {\textit{i.h.}}\xspace, $t' \partostratind{m} s'$ for some $m \in \mathbb{N} \cup \{\infty\}$.
	Then,
	\begin{align*}
		\AxiomC{$t' \partostratind{m} s'$}
		\UnaryInfC{$t = \la{x}{u'} \partostratind{m} \la{x}{s'} = s$}
		\DisplayProof
		\,.
	\end{align*}
	If moreover $n > 0$, then by {\textit{i.h.}}\xspace, $m \neq \infty$ and $t' \tostratind{m} u'$ for some term $u'$ such that $u' \partobindlong{n-1} s'$ and $u' \partostratind{j} s'$ with $j \geq m$ (possibly $j = \infty$), so
	\begin{align*}
	\AxiomC{$t' \tostratind{m} u'$}
	\UnaryInfC{$t = \la{x}{t'} \tostratind{m} \la{x}{u'} = u$}
	\DisplayProof&
	\qquad
	\text{and}
	\qquad
	\AxiomC{$u' \partostratind{j} s'$}
	\UnaryInfC{$u = \la{x}{u'} \partostratind{j} \la{x}{s'} = s$}
	\DisplayProof
	\\
	\shortintertext{and}
	&\AxiomC{$u' \partobind{n-1} s'$}
	\UnaryInfC{$u = \la{x}{u'} \partobind{n-1} \la{x}{s'} = s$}
	\DisplayProof
	\,.
	\end{align*}

	\item \emph{Application}:
	\begin{align*}
	\AxiomC{$t_0 \partobind{n_0} s_0$}
	\AxiomC{$t_1 \partobind{n_1} s_1$}
	\BinaryInfC{$t = t_0 t_1 \partobind{n} s_0 s_1 = s$}
	\DisplayProof
	\end{align*}
	where $n = n_0 + \cdot n_1$.
	By {\textit{i.h.}}\xspace, $t_0 \partostratind{m_0} s_0$ and $t_1 \partostratind{m_1} s_1$ for some $m_0, m_1 \in \mathbb{N} \cup \{\infty\}$.
	Thus,
	\begin{align*}
	\AxiomC{$t_0 \partostratind{m_0} s_0$}
	\AxiomC{$t_1 \partostratind{m_1} s_1$}
	\BinaryInfC{$t = t_0 t_1 \partostratind{m} s_0 s_1 = s$}
	\DisplayProof
	\end{align*}
	where $m = \min\{m_0,m_1+1\}$.
	If moreover $n > 0$, then by {\textit{i.h.}}\xspace, $m_0, m_1 \neq \infty$ and $t_0 \tostratind{m_0} u_0$ and $t_1 \tostratind{n_1} u_1$ for some terms $u_0, u_1$ such that $u_0 \partobindlong{n_0-1} s_0$ and $u_1 \partobindlong{n_1-1} s_1$ and $u_0 \partostratind{j_0} s_0$ and $u_1 \partostratind{j_1} s_1$ with $j_0 \geq m_0$, $j_1 \geq m_1$ (possibly $j_0,j_1 = \infty$).
	There are two possibilities:
	\begin{enumerate}
		\item $m = m_0$; then, $m_0 \leq m_1+1$ and
		\begin{align*}
		\AxiomC{$t_0 \tostratind{n} u_0$}
		\UnaryInfC{$t = t_0t_1 \tostratind{n} u_0t_1 = u$}
		\DisplayProof
		&
		\qquad
		\text{and}
		\qquad
		\AxiomC{$u_0 \partobind{n_0-1} s_0$}
		\AxiomC{$t_1 \partobind{n_1} s_1$}
		\BinaryInfC{$u = u_0t_1 \partobind{n-1} s_0s_1 = s$}
		\DisplayProof
		\shortintertext{and}
		&
		\AxiomC{$u_0 \partostratind{j_0} s_0$}
		\AxiomC{$t_1 \partostratind{n_1} s_1$}
		\BinaryInfC{$u = u_0t_1 \partostratind{j} s_0s_1 = s$}
		\DisplayProof
		\end{align*}
		where $j = \min\{j_0, m_1+1\} \geq m_0 = m$;
		\item $m = m_1+1$; then, $m_1 + 1 \leq m_0$ and
		\begin{align*}
		\AxiomC{$t_1 \tostratind{m} u_1$}
		\UnaryInfC{$t = t_0t_1 \tostratind{m} t_0u_1 = u$}
		\DisplayProof
		&
		\qquad
		\text{and}
		\qquad
		\AxiomC{$u_0 \partobind{n_0} s_0$}
		\AxiomC{$t_1 \partobind{n_1-1} s_1$}
		\BinaryInfC{$u = t_0u_1 \partobind{n-1} s_0s_1 = s$}
		\DisplayProof
		\\\
		\shortintertext{and}
		&
		\AxiomC{$u_0 \partostratind{m_0} s_0$}
		\AxiomC{$t_1 \partostratind{j_1} s_1$}
		\BinaryInfC{$u = t_0u_1 \partostratind{j} s_0s_1 = s$}
		\DisplayProof
		\end{align*}
		where $j = \min\{m_0, j_1+1\} \geq m_1+1 = m$.
	\end{enumerate}

	\item \emph{$\beta$-step}:
	\begin{align*}
	\AxiomC{$t_0 \partobind{n_0} t_0'$}
	\AxiomC{$t_1 \partobind{n_1} t_1'$}
	\BinaryInfC{$t = (\la{x}t_0)t_1 \partobind{n} t_0' \isub{x}{t_1'} = s$}
	\DisplayProof
	\end{align*}
	where $n = n_0 + \sizep{t_0}{x}n_1 + 1 > 0$.
	By {\textit{i.h.}}\xspace, $t_0 \partostratind{m_0} s_0$ and $t_1 \partostratind{m_1} s_1$ for some $m_0, m_1 \in \mathbb{N}$.
	Thus,
	\begin{align*}
	\AxiomC{$t_0 \partostratind{m_0} t_0'$}
	\AxiomC{$t_1 \partostratind{m_1} t_1'$}
	\BinaryInfC{$t = (\la{x}t_0)t_1 \partostratind{0} t_0' \isub{x}{t_1'} = s$}
	\DisplayProof
	\,.
	\end{align*}
	Let $u = t_0\isub{x}{t_1}$.
	By \reflemma{tob-subs}, $u = t_0\isub{x}{t_1} \partobindlong{n_0 + \sizep{t_0}{x} n_1} t_0'\isub{x}{t_1'} = s$ where $n_0 + \sizep{t_0}{x} n_1 = n - 1$.
	Clearly,
	\begin{align*}
	\AxiomC{}
	\UnaryInfC{$t = (\la{x}t_0)t_1 \tostratind{0} t_0 \isub{x}{t_1} = u$}
	\DisplayProof
	\,.
	\end{align*}
	According to substitutivity (\reflemma{tostrat-subs}), $u = t_0\isub{x}{t_1} \partostratind{j} t_0'\isub{x}{t_1'} = s$ for some $j \in \mathbb{N} \cup \{\infty\}$, in particular $j \geq 0$.
	\qedhere
\end{itemize}
\end{proof}

\setcounter{propositionAppendix}{\value{prop:quant-out-in-split-strat}}
\begin{propositionAppendix}[Indexed external-internal split]
	\label{propappendix:quant-out-in-split-strat}
	\NoteState{prop:quant-out-in-split-strat}
	If $t \partobind n s$ then either $t \parRew{\lnot\ell\ell}s$ or $n>0$ and there exists $u$ such that $t \Rew{x} u \partobindlong {n-1} s$.
\end{propositionAppendix}

\begin{proof}
	If $n = 0$, then $t = s$ (because $\partobind{0}$ is the identity relation) and so $t \partostratind{\infty} s$ (because $\partostratind{\infty}$ is the identity relation), hence $t \parRew{\lnot\ell\ell} s$.

	Otherwise $n > 0$ and then, by \reflemma{decomp-strat}, $t \partostratind{k} s$ and $t \tostratind{k} u$ for some $k \in \mathbb{N}$ and some term $u$ such that $u \partobind{n-1} s$ and $u \partostratind{j} s$ with $j \geq k$.
	There are two cases:
	\begin{enumerate}
		\item either $k$ is not minimal for $t$ ({\em i.e.}\xspace $t \tostratind{k'} u'$ for some $u'$ and some $k' < k$), and then $t \parRew{\lnot\ell\ell} s$ (since $t \partostratind{k} s$);
		\item or $k$ is minimal for $t$ ({\em i.e.}\xspace if $t \tostratind{k'} u'$ for some $u'$ then $k' \geq k$), and so $t \Rew{x} u \partobind{n-1} s$.
		\qedhere
	\end{enumerate}
\end{proof}

\subsection{Omitted proofs and lemmas of \refsect{stratified-cbn} (least-level)}

\setcounter{propositionAppendix}{\value{prop:deg}}
\begin{propositionAppendix}[Least level properties]
	\label{propappendix:deg}
	\NoteState{prop:deg}
\begin{enumerate}
	\item\label{pappendix:deg-finite}\emph{Computational meaning of $\ell\ell$}: $\Deg{t} = \inf\{ k \in \mathbb{N} \mid t \tostratind{k} u \text{ for some term } u\}$.
	\item\label{pappendix:deg-invariance-leq} \emph{Monotonicity}: if $t \Rew{\beta} s$ then $\Deg{s} \geq \Deg{t}$.
	\item\label{pappendix:deg-invariance-equal} \emph{Invariance by $\tonotll$}: if $t \tonotll s$ then $\Deg{s} = \Deg{t}$.
\end{enumerate}
\end{propositionAppendix}

\begin{proof}\hfill
	\begin{enumerate}
		\item
		By induction on $t$.
		For any term $r$, we set $\inf_r = \inf\{ k \in \mathbb{N} \mid r \tostratind{k} u \text{ for some term } u \}$.
		Cases:

		\begin{itemize}
			\item \emph{Variable}, {\em i.e.}\xspace $t$ is a variable. Then, $\Deg{t} = \infty$ and $t$ is $\Rew{\beta}$-normal.

			\item \emph{Abstraction}, {\em i.e.}\xspace $t = \la{x}{s}$.
			Then, $\Deg{t} = \Deg{s}$ and, by {\textit{i.h.}}\xspace, $\Deg{s} = \inf_s$;
			now,
			\begin{align*}
			\AxiomC{$s \tostratind{k} u$}
			\UnaryInfC{$t = \la{x}s \tostratind{k} \la{x}u$}
			\DisplayProof
			\end{align*}
			and there is no other rule for $\tostratind{n}$ whose conclusion is of the form $\la{x}{s} \tostratind{n} p$;
			therefore, $\Deg{t} = \Deg{s} = \inf_s = \inf_t$.

			\item \emph{Application}, {\em i.e.}\xspace $t = t't''$.
			There are two sub-cases:
			\begin{itemize}
				\item $t' = \la{x}s'$, then $\Deg{t} = 0$ and
				\begin{align*}
				\AxiomC{}
				\UnaryInfC{$t = (\la{x}s')t'' \tostratind{0} s'\isub{x}{t''} $}
				\DisplayProof
				\end{align*}
				thus $\inf_t = 0 = \Deg{t}$.
				\item $t'$ is not an abstraction, then $\Deg{t} = \min \{\Deg{t'}, \Deg{t''}+1\}$.
				By {\textit{i.h.}}\xspace, $\Deg{t'} = \inf_{t'}$ and $\Deg{t''} = \inf_{t''}$;
				now,
				\begin{align*}
				\AxiomC{$t' \tostratind{k} s'$}
				\UnaryInfC{$t = t't'' \tostratind{k} s't''$}
				\DisplayProof
				&&\text{and}&&
				\AxiomC{$t'' \tostratind{k} s''$}
				\UnaryInfC{$t = t't'' \tostratind{k+1} t's''$}
				\DisplayProof
				\end{align*}
				and there is no other rule for $\tostratind{n}$ whose conclusion is of the form $t't'' \tostratind{n} p$ (as $t'$ is not an abstraction);
				hence, $\Deg{t} = \min\{\inf_{t'}, \inf_{t''}+1\} = \inf_t$.
			\end{itemize}
		\end{itemize}

		\item \emph{Monotonicity}: by induction on the definition of $t \Rew{\beta} s$.
		Cases:
		\begin{itemize}
			\item \emph{Abstraction}, {\em i.e.}\xspace $t = \la{x}{t'} \Rew{\beta} \la{x}{s'} = s$ because $t' \Rew{\beta} s'$.
			Then, $\Deg{t} = \Deg{t'} \leq \Deg{s'} = \Deg{s}$ by {\textit{i.h.}}\xspace

			\item \emph{Application left}, {\em i.e.}\xspace $t = t' t''  \Rew{\beta} s't'' = s$ because $t' \Rew{\beta} s'$.
			By {\textit{i.h.}}\xspace, $\Deg{t'} \leq \Deg{s'}$.
			Hence, $\Deg{t} = \min \{\Deg{t'}, \Deg{t''} +1\} \leq \min \{ \Deg{s'}, \Deg{t''} + 1\} = \Deg{s}$.

			\item \emph{Application right}, {\em i.e.}\xspace $t = t' t''  \Rew{\beta} t's'' = s$ because $t'' \Rew{\beta} s''$.
			Analogous to the previous case.

			\item \emph{$\beta$-redex}, {\em i.e.}\xspace $t = (\la{x}t')t'' \Rew{\beta} t'\isub{x}{t''} = s$.
			Then, $\Deg{t} = 0 \leq \Deg{s}$.
		\end{itemize}

		\item \emph{Invariance by $\tonotll$}: By hypothesis, $t \tostratind{n} s$ for some $n > \Deg{t}$.
		We proceed by induction on the definition of $t \tostratind{n} s$.
		Cases:
		\begin{itemize}
			\item \emph{Abstraction}: $t = \la \vart' \tostratind{n}  \la x s' = s$ because $t' \tostratind{n} s'$.
			As $n > \Deg{t} = \Deg{t'}$, then $\Deg{s} = \Deg{s'} = \Deg{t'} = \Deg{t}$ by {\textit{i.h.}}\xspace

			\item \emph{Application left}, {\em i.e.}\xspace $t = t' t'' \tostratind{n} s' t'' = s$ because $t' \tostratind{n} s'$.

			According to the definition of $\Deg{t}$, there are the following sub-cases:
			\begin{enumerate}
				\item $\Deg{t} = \Deg{t'} \leq \Deg{t''} + 1$ and $t'$ is not an abstraction;

				by {\textit{i.h.}}\xspace applied to $t'$ (since $\Deg{t'} = \Deg{t} < n$), we have $\Deg{s'} = \Deg{t'}$, and so  $\Deg{s} = \min \{\Deg{s'}, \Deg{t''} + 1\} = \min\{\Deg{t'}, \Deg{t''} + 1\} = \Deg{t}$.

				\item $\Deg{t} = \Deg{t''} + 1 \leq \Deg{t'}$ and $t'$ is not an abstraction;
				by \refpropp{deg}{invariance-leq} (since $\tostratind{n} \subseteq \Rew{\beta}$),
				$\Deg{t'} \leq \Deg{s'}$ and therefore

				$\Deg{s} = \min \{\Deg{s'}, \Deg{t''} + 1\} = \Deg{t''}+1 = \min\{\Deg{t'}, \Deg{t''} + 1\} = \Deg{t}$.

				\item $\Deg{t} = 0$ and $t'= \la{x}u'$.
				According to the definition of $t \tostratind{n} s$, since $n > 0$, we have
				\begin{align*}
				\AxiomC{$u' \tostratind{n} s'$}
				\UnaryInfC{$\la{x}u' \tostratind{n} \la{x}{s'}$}
				\UnaryInfC{$t = (\la{x}u')t'' \tostratind{n} (\la{x}{s'})t'' = s$}
				\DisplayProof
				&&
				\text{or}
				&&
				\AxiomC{$t'' \tostratind{n-1} s''$}
				\UnaryInfC{$t = (\la{x}u')t'' \tostratind{n} (\la{x}{u'})s'' = s$}
				\DisplayProof
				\end{align*}
				therefore $\Deg{s} = 0 = \Deg{t}$.
			\end{enumerate}

			\item \emph{Application right}, {\em i.e.}\xspace $t = t' t'' \tostratind{n} t' s'' = s$ because $t'' \tostratind{n-1} s''$.
			Analogous to the previous case.

			\item \emph{$\beta$-step}, {\em i.e.}\xspace $t = (\la{x}t')t'' \tostratind{0} t' \isub{x}{t''} = s$
			where $0 = n > \Deg{t} = 0$, which is impossible.
			\qedhere
		\end{itemize}
	\end{enumerate}
\end{proof}

\begin{lemma}
	\label{l:not-abs}
	Let $t \Rew{x} s$ with $\Deg{t} > 0$.
	If $t$ is not an abstraction, then $s$ is not an abstraction.
\end{lemma}

\begin{proof}
	By hypothesis, $t = t't''$ and $t'$ is not an abstraction (otherwise $\Deg{t} = 0$).
	Therefore, according to the definition of $\Rew{x}$, there are only two possibilities:
	\begin{enumerate}
		\item either $t = t' t'' \tostratind{k} s't'' = s$ because $t' \tostratind{k} s'$ and $k = \Deg{t}$;
		\item or $t = t't'' \tostratind{k} t' s'' = s$ because $t'' \tostratind{k - 1} s''$ and $k = \Deg{t}$.
	\end{enumerate}
	In both cases, $s$ is not an abstraction.
\end{proof}

\begin{lemma}[Substitutivity by level]
	\label{l:substitutivity-strat}
	If $t \tostratind{k} s$ then $t\isub{x}{u} \tostratind{k} s\isub{x}{u}$.
\end{lemma}

\begin{proof}
		By induction on the definition of $t \tostratind{k} s$.
		Cases:
		\begin{itemize}
			\item \emph{Abstraction}: $t = \la{y}{t'} \tostratind{k} \la{y}{s'} = s$ with $t' \tostratind{k} s'$.
			We can suppose without loss of generality that $y \notin \fv{u} \cup \{x\}$.
			By {\textit{i.h.}}\xspace, $t'\isub{x}{u} \tostratind{k} s'\isub{x}{u}$ and hence $t\isub{x}{u} = \la{y}t'\isub{x}{u} \tostratind{k} \la{y}s'\isub{x}{u} = s\isub{x}{u}$.

			\item \emph{Application left}: $t = t't'' \tostratind{k} s't'' = s$ with $t' \tostratind{k} s'$.
			By {\textit{i.h.}}\xspace, $t'\isub{x}{u} \tostratind{k} s'\isub{x}{u}$ and hence $t\isub{x}{u} = t'\isub{x}{u} t''\isub{x}{u} \tostratind{k} s'\isub{x}{u} t''\isub{x}{u} = s\isub{x}{u}$.

			\item \emph{Application right}: $t = t't'' \tostratind{k} t's'' = s$ with $k > 0$ and $t'' \tostratind{k} s''$.
			Analogous to the previous case.

			\item \emph{$\beta$-redex}: $t = (\la{y}t')t'' \tostratind{0} t'\isub{y}{t''} = s$.
			We can suppose without loss of generality that $y \notin \fv{u} \cup \{x\}$.
			Then, $t\isub{x}{u} = (\la{y}t'\isub{x}{u})t''\isub{x}{u} \tostratind{0} t'\isub{x}{u}\isub{y}{t''\isub{x}{u}} = t'\isub{y}{t''}\isub{x}{u} = s\isub{x}{u}$.
			\qedhere
		\end{itemize}

\end{proof}

\setcounter{propositionAppendix}{\value{prop:essential-strat}}
\begin{propositionAppendix}[\ll essential properties]
	\label{propappendix:essential-strat}
	\NoteState{prop:essential-strat}\hfill
	\begin{enumerate}
		\item\label{pappendix:essential-strat-completeness}
		\emph{Fullness\xspace}: if $t \Rew{\beta} s$ then $t \Rew{\ell\ell} u$ for some $u$.
		\item\label{pappendix:essential-strat-persistence} \emph{Persistence}: if $t \Rew{\ell\ell} s_1$ and $t \Rew{\neg \ell\ell} s_2$ then $s_2 \Rew{\ell\ell} u$ for some $u$.
		\item\label{pappendix:essential-strat-diamond} \emph{Diamond}: if $s\lRew{\ell\ell} \cdot \Rew{\ell\ell}u$ with $s \neq u$ then $s \Rew{\ell\ell} \cdot \lRew{\ell\ell} u$.
	\end{enumerate}
\end{propositionAppendix}

\begin{proof}\hfill
	\begin{enumerate}
		\item \emph{Fullness\xspace}: Since $t$ is not $\Rew{\beta}$-normal, then $\infty \neq \Deg{t} =  \min\{k \in \mathbb{N} \mid t \tostratind{k} r \text{ for some } r\}$ by \refpropp{deg}{finite}.
		Therefore, $t \Rew{x} u$  for some $u$.

		\item \emph{Persistence}: Since $t$ is not $\Rew{\beta}$-normal, $\infty \neq \Deg{t} =  \min\{k \in \mathbb{N} \mid t \tostratind{k} r \text{ for some } r\}$ by \refpropp{deg}{finite}.
		By least level invariance (\refpropp{deg}{invariance-equal}), $\Deg{t} = \Deg{s_2}$ and hence, according to \refpropp{deg}{finite} again, we have $\infty \neq \Deg{s_2} = \min\{k \in \mathbb{N} \mid s_2 \tostratind{k} u \text{ for some } u\}$.
		Therefore, there exists $u$ such that $s_2 \Rew{x} u$.

				\item \emph{Diamond}:
		The idea of the proof is that, when $(\la{x}t)s \Rew{x} t\isub{x}{s}$, the $\beta$-redexes in $s$ can be duplicated in $t\isub{x}{s}$ but they are not at least level and $\Rew{x}$ does not reduce outside the least level.

		Formally, according to the definition of $\Rew{x}$, we have to prove that $s \ltostratind{k} t \tostratind{k} u$ with $k = \Deg{t}$, then $s \tostratind{m} r \ltostratind{n} u$ for some $r$, where $m = \Deg{s}$ and $n = \Deg{u}$.
		To get the right {\textit{i.h.}}\xspace, we prove also that $\Deg{s} = k = \Deg{u}$ (and so $s \tostratind{k} r \ltostratind{k} u$).

		Clearly, $t$ is not a variable, otherwise it would be $\Rew{\beta}$-normal.

		If $t = \la{x}{t'}$, then $s = \la{x}{s'}$ and $u = \la{x}{u'}$ with $s' \ltostratind{k} t' \tostratind{k} u'$ and $s' \neq u'$ and $\Deg{t} = \Deg{t'}$.
		By {\textit{i.h.}}\xspace, $s' \tostratind{m} r \ltostratind{n} u'$ for some term $r$, with $\Deg{s} = \Deg{s'} = k = \Deg{u'} = \Deg{u}$, hence $s = \la{x}s' \tostratind{k} \la{x}{r} \ltostratind{k} \la{x}{u'} = u$.

		Finally, consider $t = t_0t_1$.
		The case where $t_0 = \la{x}{t_2}$ and $s = t_2\isub{x}{t_1} \ltostratind{k} t \tostratind{k} (\la{x}t_2)u_1 = u$ with $t_1 \tostratind{k-1} u_1$ is impossible, because $k = \Deg{t} = 0$.
		The remaining cases for $t = t_0t_1$ are:
		\begin{itemize}
			\item $s = t_0s_1 \ltostratind{k} t = t_0t_1 \tostratind{k} u_0t_1 = u$ with $t_0 \tostratind{k} u_0$  and $t_1 \tostratind{k} s_1$ and $ \Deg{t_0} = \Deg{t} = \Deg{t_1} + 1 = k > 0$.
			Then, $t_0$ is not an abstraction (otherwise $\Deg{t} = 0$) and, by \reflemma{not-abs}, $u_0$ is not an abstraction.
			By \refpropp{deg}{invariance-leq}, $\Deg{s_1} \geq \Deg{t_1}$ and $\Deg{u_0} \geq \Deg{t_0}$.
			Hence, $\Deg{s} = \min\{ \Deg{t_0}, \Deg{s_1}+1 \} = \Deg{t_0} = \Deg{t_1}+1 = \min\{\Deg{u_0}, \Deg{t_1}+1\} = \Deg{u}$ and so $s \tostratind{k} u_0s_1 \ltostratind{k} u$.

			\item $s = s_0 t_1 \ltostratind{k} t = t_0t_1 \tostratind{k} u_0t_1 = u$ with $s_0 \ltostratind{k} t_0 \tostratind{k} u_0$ and $s_0 \neq u_0$ and $k = \Deg{t} = \Deg{t_0} \leq \Deg{t_1}+1$.
			By {\textit{i.h.}}\xspace, there is $r_0$ such that $s_0 \tostratind{k} r_0 \ltostratind{k} u_0$ where $\Deg{s_0} = k = \Deg{u_0}$.
			Thus, $s \tostratind{k} r_0t_1 \ltostratind{k} u$.
			If $s_0$ or $u_0$ is an abstraction, then $\Deg{s} = 0$ or $\Deg{u} = 0$ and hence (by \refpropp{deg}{invariance-leq}) $0 = \Deg{s} \geq \Deg{t}$ or $0 = \Deg{s} \geq \Deg{t}$, so in both cases $k = \Deg{t} = 0 = \Deg{s} = \Deg{u}$.
			Otherwise, $\Deg{s} = \min\{\Deg{s_0}, \Deg{t_1}+1 \} = k = \min\{\Deg{u_0}, \Deg{t_1}+1 \} = \Deg{u}$.

			\item $s = t_0s_1  \ltostratind{k} t = t_0t_1 \tostratind{k} t_0u_0 = u$ with $s_1 \ltostratind{k-1} t_1 \tostratind{k-1} u_1$ and $k = \Deg{t} = \Deg{t_1}+1$.
			Analogous to the previous case.

			\item $s = t_2\isub{x}{t_1} \ltostratind{k} t = (\la{x}t_2)t_1 \tostratind{k} (\la{x}{s_2})t_1  = u$ where $t_0 = \la{x}{t_2}$ and $t_2 \tostratind{k} s_2$ and $k = \Deg{t_2} = \Deg{t} = 0 = \Deg{u}$.
			Thus, $t_2 \tostratind{0} s_2$ and $t \tostratind{0} t_2\isub{x}{t_1}$.
			By substitutivity by level (\reflemma{substitutivity-strat}), $t_2\isub{x}{t_1} \tostratind{0} s_2\isub{x}{t_1}$ and so $\Deg{t_2\isub{x}{t_1}} =0$ by \refpropp{deg}{finite}.

			Therefore $t_2\isub{x}{t_1} \tostratind{0} s_2\isub{x}{t_1} \ltostratind{0} u$.
			\qedhere
		\end{itemize}
	\end{enumerate}
\end{proof}

\begin{lemma}[Merge by level]
	\label{l:merge-by-level}
	If $t \partostratind{n}s \tostratind{m} u$ with $n > m$, then $t \parRew{\beta} u$.
\end{lemma}

\begin{proof}
	By induction on the definition of $t \partostratind{n} s$.
	Cases:
	\begin{itemize}
		\item \emph{Variable}: $t = x \partostratind{\infty} x = s$.
		Then, there is no $u$ such that $s \tostratind{m} u$ for any $m \in \mathbb{N}$.

		\item \emph{Abstraction}: $t = \la \vart' \partostratind{n}  \la x s' = s$ because $t' \partostratind{n} s'$.
		According to the definition of $s \tostratind{m} u$, by necessity $u = \la{x}{u'}$ with $s' \tostratind{m} u'$.
		By {\textit{i.h.}}\xspace, $t' \parRew{} u'$, thus
		\begin{align*}
		\AxiomC{$t' \parRew{} u'$}
		\UnaryInfC{$t = \la{x}{t'} \parRew{} \la{x}{u'} = u$}
		\DisplayProof
		\,.
		\end{align*}

		\item \emph{Application}:
		\begin{align*}
		\AxiomC{$t_0 \partostratind{n_0} s_0$}
		\AxiomC{$t_1 \partostratind{n_1} s_1$}
		\BinaryInfC{$t = t_0 t_1 \partostratind{n} s_0 s_1 = s$}
		\DisplayProof
		\end{align*}
		where $n = \min\{n_0,n_1+1\}$.
		According to the definition of $s \tostratind{m} u$, there are the following sub-cases:
		\begin{enumerate}
			\item $s = s_0s_1 \tostratind{m} u_0s_1 = u$ with $s_0 \tostratind{m} u_0$;
			since $m < n \leq n_0$,
			by {\textit{i.h.}}\xspace applied to $t_0 \partostratind{n_0} s_0 \tostratind{m} u_0$, we have $t_0 \parRew{\beta} u_0$, and so (as $\partostratind{n_1} \,\subseteq\, \parRew{\beta}$)
			\begin{align*}
			\AxiomC{$t_0 \parRew{\beta} u_0$}
			\AxiomC{$t_1 \parRew{\beta} s_1$}
			\BinaryInfC{$t = t_0t_1 \parRew{\beta} u_0s_1 = u$}
			\DisplayProof
			\,;
			\end{align*}
			\item $s = s_0s_1 \tostratind{m} u_0s_1 = u$ with $s_1 \tostratind{m-1} u_1$;
			since $m-1 < n-1 \leq n_1$,
			by {\textit{i.h.}}\xspace applied to $t_1 \partostratind{n_1} s_1 \tostratind{m-1} u_1$, we have $t_1 \parRew{} u_1$, and so (as $ \partostratind{n_0} \,\subseteq\, \parRew{}$)
			\begin{align*}
			\AxiomC{$t_0 \parRew{} s_0$}
			\AxiomC{$t_1 \parRew{} u_1$}
			\BinaryInfC{$t = t_0t_1 \parRew{} s_0u_1 = u$}
			\DisplayProof
			\,;
			\end{align*}

			\item $s = (\la{x}{s_0'})s_1 \tostratind{0} s_0\isub{x}{s_1} = u$ with $s_0 = \la{x}s_0'$ and $m = 0$;
			as $n > 0$ then, according to the definition of $t \partostratind{n} s$,
			\begin{align*}
			\AxiomC{$t_0 \partostratind{n_0} s_0$}
			\UnaryInfC{$\la{x}t_0 \partostratind{n_0} \la{x}s_0$}
			\AxiomC{$t_1 \partostratind{n_1} s_1$}
			\BinaryInfC{$t = (\la{x}t_0)t_1 \partostratind{n} (\la{x}{s_0})s_1 = s$}
			\DisplayProof
			\end{align*}
			where $n = \min\{n_0, n_1+1\}$; therefore (as $\partostratind{k} \,\subseteq\, \parRew{}$)
			\[
			\AxiomC{$t_0 \parRew{} s_0$}
			\AxiomC{$t_1 \parRew{} s_1$}
			\BinaryInfC{$t = (\la{x}{t_0})t_1 \parRew{} s_0\isub{x}{s_1} = u$}
			\DisplayProof
			\,.\qedhere
			\]
		\end{enumerate}
	\end{itemize}
\end{proof}

\setcounter{propositionAppendix}{\value{prop:macro-strat}}
\begin{propositionAppendix}[\ll macro-step system]
	\hfill\label{propappendix:macro-strat}
	\NoteState{prop:macro-strat}
	\begin{enumerate}
		\item\label{pappendix:macro-strat-merge} \emph{Merge}: if $t \parRew{\lnot\ell\ell} s \Rew{x} u$, then $t \parRew{\beta} u$.
		\item\label{pappendix:macro-strat-indexed-split}\emph{Indexed split}:	if $t \partobind n s$ then $t \parRew{\lnot\ell\ell}s$, or $n>0$ and $t \Rew{x} \cdot \partobindlong {n-1} s$.
		\item\label{pappendix:macro-strat-split} \emph{Split}: if $t \parRew{\beta} s$ then $t \Rew{x}^* \cdot \parRew{\lnot\ell\ell}s$.
	\end{enumerate}
	That is, $(\Lambda, \set{\Rew{\ell\ell}, \Rew{\neg \ell\ell}})$ is a macro-step system with respect to $\parRew{\beta}$ and $\parRew{\lnot\ell\ell}$.
\end{propositionAppendix}

\begin{proof}\hfill
	\begin{enumerate}
		\item \emph{Merge}: As $t \parRew{\lnot\ell\ell} s \Rew{x} u$, then $t \partostratind{n} s \tostratind{m} u$ for some $n \in \mathbb{N} \cup \{\infty\}$ and $m \in \mathbb{N}$ such that $n > \Deg{t}$ and  $m = \Deg{s}$.
		Since $\parRew{\lnot\ell\ell} \subseteq \tonotll^*$ and $\tonotll$ cannot change the least level (\refpropp{deg}{invariance-equal}),  $\Deg{t} = \Deg{s}$
		and so $n > m$.
		By Merge by Level (\reflemma{merge-by-level}), $t \parRew{\beta} u$.

		\item \emph{Indexed split}: By induction on the definition of $t \partobind n s$.
		We freely use the fact that if $t \partobind n s$ then $t \parRew{\beta} s$. Cases:
		\begin{itemize}
			\item \emph{Variable}: $t = x \partobind 0 x = s$.
			Then, $t = x \partonotstratx = s$  since $x \partostratind{\infty} x$.

			\item \emph{Abstraction}: $t = \la \vart' \partobind n  \la x s' = s$ because $t' \partobind n s'$. It follows by the {\textit{i.h.}}\xspace.

			\item \emph{Application}:
			\begin{align}
			\label{eq:app-parallel}
			\AxiomC{$\tmfour}%{r \partobind {n_1} \tmfour}%{r'$}
			\AxiomC{$p \partobind {n_2} p'$}
			\BinaryInfC{$t = r p \partobindlong {n_1 + n_2} r' p' = s$}
			\DisplayProof
			\end{align}
			with $n = n_1 + n_2$.
			There are only two cases:
			\begin{itemize}
				\item either $r p \partonotstrat r' p'$, and then the claim holds;
				\item or $rp \not\parRew{\lnot\ell\ell} r'p'$ and hence any derivation with conclusion $\tmfour}%{r p \partostratind{d} \tmfour}%{r' p'$ is such that $d =\deg{\tmfour}%{r p} \in \mathbb{N}$.
				Let us rewrite the derivation \eqref{eq:app-parallel} replacing $\partobind{n}$ with $\partostratind{k}$: we have\footnote{This is possible because the inference rules for $\partobind{n}$ and $\partostratind{k}$ are the same except for the way they manage their own indexes $n$ and $k$.}
				\[\AxiomC{$\tmfour}%{r \partostratind {d_{\tmfour}%{r}} \tmfour}%{r'$}
				\AxiomC{$p \partostratind{d_{p}} p'$}
				\BinaryInfC{$t = \tmfour}%{r p \partostratind d \tmfour}%{r' p' = s$}
				\DisplayProof\]
				where $d = \min\{d_r, d_p +1 \}$.
				Thus, there are two sub-cases:
				\begin{enumerate}
					\item $d = d_r \leq d_p+1$ and then $d=\deg  {r p}\leq \deg r \leq d_{r}=d $ (the first inequality holds by definition of $\Deg{rp}$), hence $\deg r = d_{r}$;
					we apply the {\textit{i.h.}}\xspace\ to $r \partobind {n_1} r'$ and we have that  $r \parRew{\lnot\ell\ell} r'$, or $n_1>0$ and $r \Rew{x} u_1 \partobind {n_1-1} \tmfour}%{r'$;
					but $r \parRew{\lnot\ell\ell} r'$ is impossible because otherwise $r p \partonotstrat r' p'$ (as $d_r \leq d_p+1$of s);
					therefore, $n_1>0$ and $r \Rew{x} u_1 \partobind {n_1-1} \tmfour}%{r'$, and so $n>0 $ and  $t  = rp  \Rew{x} u_1p  \partobindlong {n_1-1+n_2} r'p' = s$.
					\item $d=d_{p}+1 \leq d_{r}$ and then $d=\deg  {r p}\leq \deg p +1 \leq	  d_{p}+1 =d $, hence $\deg p =
					d_{p}$; we conclude analogously to thee previous sub-case.
				\end{enumerate}
			\end{itemize}

			\item \emph{$\beta$ step}:
			\[\AxiomC{$u \partobind {n_1} u'$}
			\AxiomC{$r \partobind {n_2} r'$}
			\BinaryInfC{$t = (\lax u)r \partobindlong {n_1 + \sizep{u'}x \cdot n_2 +1} u'\isub x {r'} = s$}
			\DisplayProof\]
			With $n = n_1 + \sizep{u'}x \cdot n_2 +1 > 0$. We have $t = (\lax u)r \Rew{x} u\isub\varr$ and by substitutivity of $\partobind n$ (\reflemma{partobind-subs}) $u\isub\varr \partobindlong{n_1 + \sizep{u'}x \cdot n_2} u' \isubx{r'} = s$.
		\end{itemize}

		\item\emph{Split}: Exactly as in the head case (\refpropp{macro-head}{split}), using the Indexed Split property for \ll (\refpoint{macro-strat-indexed-split} above).
		\qedhere
	\end{enumerate}
\end{proof}

\newcommand{\tmfive}{p}

\subsection{Omitted proofs and lemmas of \refsect{stratified-cbn}}
\newcommand{\partonotstrat}{\parRew{\lnot\ell\ell}}

 We show that $\parRew{\lnot\ell\ell}$ does not increase the degree of a term. Hence in particular the same holds for $\tonotstrat$.

\newcounter{l:perpetuity-strat}
\addtocounter{l:perpetuity-strat}{\value{lemma}}
\begin{lemma}[Persistence]
	\label{l:perpetuity-strat}
	\NoteProof{lappendix:perpetuity-strat}
	\begin{enumerate}
		\item If $t \parRew{} s$ then $\Deg{s} \geq \Deg{t}$.
		\item 	If $t \parRew{\lnot\ell\ell} s$ then $\Deg{s} = \Deg{t}$.
	\end{enumerate}
\end{lemma}

\begin{proof}
\textbf{1.} is easily proved by a straightfoward induction on $\parRew{}$.

\textbf{2}. is proved by induction on $t$. 	By hypothesis, $t \partostratind{n} s$ for some $n > \Deg{t}$.
We prove that $\Deg{s} = \Deg{t}$.
Cases:
\begin{itemize}
	\item $t = x \parRew{} x = s$.
	Then, $\Deg{t}  = \Deg{s}$.

	\item $t = \la \vart' \partostratind{n}  \la x s' = s$ with  $t' \partostratind{n} s'$.
	Since $n > \Deg{t} = \Deg{t'}$, by {\textit{i.h.}}\xspace we have  $\Deg{s} = \Deg{s'} = \Deg{t'} = \Deg{t}$.

	\item $t$ is an \emph{application}. We consider two case, according to the definition of $\Deg{t}$

\begin{itemize}

	\item 	Assume $t=(\la{x}t_0)t_1$. We have  $\Deg{t}=0$ and since $t \partostratind{n} s$ with $n > 0$,
	the only possible derivation is the following.

	\begin{align*}
	\AxiomC{$t_0 \parRew{} s_0$}
	\UnaryInfC{$\la{x}t_0 \parRew{} \la{x}s_0$}
	\AxiomC{$t_1 \parRew{} s_1$}
	\BinaryInfC{$t = (\la{x}t_0)t_1 \parRew{} (\la{x}{s_0})s_1 = s$}
	\DisplayProof
	\end{align*}
	Therefore,  $\Deg{s} = 0 = \Deg{t}$.

\item  Assume $t=t_0t_1$, where $t_0$ is not an abstraction. We have
\begin{align*}
\AxiomC{$t_0 \partostratind{n_0} s_0$}
\AxiomC{$t_1 \partostratind{n_1} s_1$}
\BinaryInfC{$t = t_0 t_1 \partostratind{n} s_0 s_1 = s$}
\DisplayProof
\end{align*}
where $n = \min\{n_0,n_1+1\}$ and $ \deg{t} =\min \{\deg {t_0}, \deg {t_1} +1)\}$.
\begin{enumerate}
	\item  If $\deg {t_0}$ is  minimal, then
	$n_0\geq n > \deg {t_0} $. Therefore $t_0 \parRew{\lnot\ell\ell} s_0$ and by the {\textit{i.h.}}\xspace, $\deg{t_0}=\deg{\tmtwo}%{s_0}$.
	We also know (by point 1.) that $\deg {\tmtwo}%{s_1}\geq \deg {t_1}$, hence $ \deg{\tmtwo}%{s} =\min \{\deg {\tmtwo}%{s_0}, \deg {\tmtwo}%{s_1} +1)
	= \deg {\tmtwo}%{s_0}= \deg {t_0} = \deg{t}$.

	\item If $\deg {t_1} +1$ is minimal, then $n_1+1 \geq n> \deg {t_1+1}$, hence $n_1 > \deg{t_1}$.  Therefore $t_1 \parRew{\lnot\ell\ell} s_1$ and by the {\textit{i.h.}}\xspace, $\deg{t_1}=\deg{\tmtwo}%{s_1}.$, and we conclude similarly to the previous case.
\end{enumerate}

\end{itemize}
\end{itemize}

\end{proof}

\setcounter{lemmaAppendix}{\value{l:perpetuity-strat}}
\begin{lemmaAppendix}[Persistence]
\label{lappendix:perpetuity-strat}
\NoteState{l:perpetuity-strat}
	If $t \parRew{\lnot\ell\ell} s$ then $\Deg{s} \leq \Deg{t}$.
\end{lemmaAppendix}

\begin{proof}
	By hypothesis, $t \partostratind{n} s$ for some $n > \Deg{t}$.
	We prove that $\Deg{s} \leq \Deg{t}$ by induction on the definition of $t \partostratind{n} s$.
	Cases:
	\begin{itemize}
		\item \emph{Variable}: $t = x \partostratind{\infty} x = s$.
		Then, $\Deg{t} = \infty = \Deg{s}$.

		\item \emph{Abstraction}: $t = \la \vart' \partostratind{n}  \la x s' = s$ because $t' \partostratind{n} s'$.
		As $n > \Deg{t} = \Deg{t'}$, then $\Deg{s} = \Deg{s'} \leq \Deg{t'} = \Deg{t}$ by {\textit{i.h.}}\xspace

		\item \emph{Application}:
		\begin{align*}
		\AxiomC{$t_0 \partostratind{n_0} s_0$}
		\AxiomC{$t_1 \partostratind{n_1} s_1$}
		\BinaryInfC{$t = t_0 t_1 \partostratind{n} s_0 s_1 = s$}
		\DisplayProof
		\end{align*}
		where $n = \min\{n_0,n_1+1\}$.
		According to the definition of $\Deg{t}$, there are the following sub-cases:
		\begin{enumerate}
			\item $\Deg{t} = \Deg{t_0} \leq \Deg{t_1} + 1$ and $t_0$ is not an abstraction;

			by {\textit{i.h.}}\xspace applied to $t_0$ (since $\Deg{t_0} = \Deg{t} < n \leq n_0$), we have $\Deg{s_0} \leq \Deg{t_0}$, and so (as $\Deg{t_0} \leq \Deg{t_1} + 1$) $\Deg{s} = \min \{\Deg{s_0}, \Deg{s_1} + 1\} \leq \min\{\Deg{t_0}, \Deg{t_1} + 1\} = \Deg{t}$.

			\item $\Deg{t} = \Deg{t_1} + 1 \leq \Deg{t_0}$ and $t_0$ is not an abstraction;

			by {\textit{i.h.}}\xspace applied to $t_1$ (since $\Deg{t_1} + 1 = \Deg{t} < n \leq n_1+1$ and hence $\Deg{t_1} < n_1$), we have $\Deg{s_1} \leq \Deg{t_1}$, and so (as $\Deg{t_1} + 1 \leq \Deg{t_0}$) $\Deg{s} = \min \{\Deg{s_0}, \Deg{s_1} + 1\} \leq \min\{\Deg{t_0}, \Deg{t_1} + 1\} = \Deg{t}$.

			\item $\Deg{t} = 0$ and $t_0$ is an abstraction.
			According to the definition of $t \partostratind{n} s$, since $n > 0$, we have
				\begin{align*}
				\AxiomC{$t_0 \partostratind{n_0} s_0$}
				\UnaryInfC{$\la{x}t_0 \partostratind{n_0} \la{x}s_0$}
				\AxiomC{$t_1 \partostratind{n_1} s_1$}
				\BinaryInfC{$t = (\la{x}t_0)t_1 \partostratind{n} (\la{x}{s_0})s_1 = s$}
				\DisplayProof
				\end{align*}
				where $n = \min\{n_0, n_1+1\}$; therefore $\Deg{s} = 0 = \Deg{t}$.
		\end{enumerate}

		\item \emph{$\beta$-step}:
		\begin{align*}
		\AxiomC{}
		\UnaryInfC{$t = (\la{x}t_0)t_1 \partostratind{0} t_0 \isub{x}{t_1} = s$}
		\DisplayProof
		\end{align*}
		where $0 = n > \Deg{t}$, which is impossible.
		\qedhere
	\end{itemize}
\end{proof}

\begin{propositionAppendix}[Merge]
	\label{propappendix:i-e-merge-strat}
	\NoteState{prop:i-e-merge-strat}
	\ If $t \parRew{\lnot\ell\ell} s \Rew{x} u$, then $t \parRew{\beta} u$.
\end{propositionAppendix}

\begin{proof}\hfill
	We first prove a stronger hypothesis, from which the claim  follows.
	\begin{enumerate}

	\item We prove the following property
	\begin{center}
	(\#) 	If $t \partostratind{n}s \tostratind{m} u$ with $n > m$, then $t \parRew{\beta} u$.
	\end{center}
	by induction on the definition of $t \partostratind{n} s$.
	Cases:
	\begin{itemize}
		\item \emph{Variable}: $t = x \partostratind{\infty} x = s$.
		Then, there is no $u$ such that $s \tostratind{m} u$ for any $m \in \mathbb{N}$.

		\item \emph{Abstraction}: $t = \la \vart' \partostratind{n}  \la x s' = s$ because $t' \partostratind{n} s'$.
		According to the definition of $s \tostratind{m} u$, by necessity $u = \la{x}{u'}$ with $s' \tostratind{m} u'$.
		By {\textit{i.h.}}\xspace, $t' \parRew{\betav} u'$, thus
		\begin{align*}
		\AxiomC{$t' \parRew{\betav} u'$}
		\UnaryInfC{$t = \la{x}{t'} \parRew{\betav} \la{x}{u'} = u$}
		\DisplayProof
		\,.
		\end{align*}

		\item \emph{Application}:
		\begin{align*}
		\AxiomC{$t_0 \partostratind{n_0} s_0$}
		\AxiomC{$t_1 \partostratind{n_1} s_1$}
		\BinaryInfC{$t = t_0 t_1 \partostratind{n} s_0 s_1 = s$}
		\DisplayProof
		\end{align*}
		where $n = \min\{n_0,n_1+1\}$.
		According to the definition of $s \tostratind{m} u$, there are the following sub-cases:
		\begin{enumerate}
			\item $s = s_0s_1 \tostratind{m} u_0s_1 = u$ with $s_0 \tostratind{m} u_0$;
			since $m < n \leq n_0$,
			by {\textit{i.h.}}\xspace applied to $t_0 \partostratind{n_0} s_0 \tostratind{m} u_0$, we have $t_0 \parRew{\betav} u_0$, and so (as $\partostratind{n_1} \,\subseteq\, \parRew{\betav}$)
			\begin{align*}
			\AxiomC{$t_0 \parRew{\betav} u_0$}
			\AxiomC{$t_1 \parRew{\betav} s_1$}
			\BinaryInfC{$t = t_0t_1 \parRew{\betav} u_0s_1 = u$}
			\DisplayProof
			\,;
			\end{align*}
			\item $s = s_0s_1 \tostratind{m} u_0s_1 = u$ with $s_1 \tostratind{m-1} u_1$;
			since $m-1 < n-1 \leq n_1$,
			by {\textit{i.h.}}\xspace applied to $t_1 \partostratind{n_1} s_1 \tostratind{m-1} u_1$, we have $t_1 \parRew{\betav} u_1$, and so (as $t_0 \partostratind{n_0} \,\subseteq\, \parRew{\betav}$)
			\begin{align*}
			\AxiomC{$t_0 \parRew{\betav} s_0$}
			\AxiomC{$t_1 \parRew{\betav} u_1$}
			\BinaryInfC{$t = t_0t_1 \parRew{\betav} s_0u_1 = u$}
			\DisplayProof
			\,;
			\end{align*}

			\item $s = (\la{x}{s_0})s_1 \tostratind{0} s_0\isub{x}{s_1} = u$ with $m = 0$;
			as $n > 0$ then, according to the definition of $t \partostratind{n} s$,
			\begin{align*}
			\AxiomC{$t_0 \partostratind{n_0} s_0$}
			\UnaryInfC{$\la{x}t_0 \partostratind{n_0} \la{x}s_0$}
			\AxiomC{$t_1 \partostratind{n_1} s_1$}
			\BinaryInfC{$t = (\la{x}t_0)t_1 \partostratind{n} (\la{x}{s_0})s_1 = s$}
			\DisplayProof
			\end{align*}
			where $n = \min\{n_0, n_1+1\}$; therefore (as $\partostratind{k} \,\subseteq\, \parRew{\betav}$)
			\begin{align*}
			\AxiomC{$t_0 \parRew{\betav} s_0$}
			\AxiomC{$t_1 \parRew{\betav} s_1$}
			\BinaryInfC{$t = (\la{x}{t_0})t_1 \parRew{\betav} s_0\isub{x}{s_1} = u$}
			\DisplayProof
			\,.
			\end{align*}
		\end{enumerate}
	\end{itemize}

	\item As $t \parRew{\lnot\ell\ell} s \Rew{x} u$, then $t \partostratind{n} s \tostratind{m} u$ for some $n \in \mathbb{N} \cup \{\infty\}$ and $m \in \mathbb{N}$ such that $n > \Deg{t}$ and  $m = \Deg{s}$.
	By (\reflemma{perpetuity-strat},  $\Deg{t} = \Deg{s}$.
	Hence, $n > m$ and we can then merge (by \refpropp{i-e-merge-strat}{ind}) to $t \parRew{\beta} u$.
	\qedhere
	\end{enumerate}
\end{proof}

\begin{lemma}[Decomposition of parallel $\beta$-steps]
\label{l:decomp-strat}
If $t \partobind{n} s$ then $t \partostratind{m} s$ for some $m \in \mathbb{N} \cup \{\infty\}$.
If moreover $n > 0$ then $m \neq \infty$ and $t \tostratind{m} u$ for some term $u$ such that $u \partobindlong{n-1} s$ and $u \partostratind{j} s$ with $j \geq m	$ (possibly $j = \infty$).
\end{lemma}

\begin{proof}
By induction $(n,k) \in \mathbb{N} \times \mathbb{N}$ with the lexicographical order, where $k$ is the length of the derivation of $t \partobind{n} s$.
Cases:
\begin{itemize}
	\item \emph{Variable}: $t = x \partobind{0} x = s$ where $n = 0$.
	Then,
	\begin{align*}
	\AxiomC{}
	\UnaryInfC{$t = x \partostratind{\infty} x = s$}
	\DisplayProof
	\,.
	\end{align*}

	\item \emph{Abstraction}: $t = \la \vart' \partobind{n}  \la x s' = s$ because $t' \partobind{n} s'$.
	By {\textit{i.h.}}\xspace, $t' \partostratind{m} s'$ for some $m \in \mathbb{N} \cup \{\infty\}$.
	Then,
	\begin{align*}
	\AxiomC{$t' \partostratind{m} s'$}
	\UnaryInfC{$t = \la{x}{u'} \partostratind{m} \la{x}{s'} = s$}
	\DisplayProof
	\,.
	\end{align*}
	If moreover $n > 0$, then by {\textit{i.h.}}\xspace, $m \neq \infty$ and $t' \tostratind{m} u'$ for some term $u'$ such that $u' \partobindlong{n-1} s'$ and $u' \partostratind{j} s'$ with $j \geq m$ (possibly $j = \infty$), so
	\begin{align*}
	\AxiomC{$t' \tostratind{m} u'$}
	\UnaryInfC{$t = \la{x}{t'} \tostratind{m} \la{x}{u'} = u$}
	\DisplayProof
	\qquad
	\text{and}
	\qquad
	&
	\AxiomC{$u' \partostratind{j} s'$}
	\UnaryInfC{$u = \la{x}{u'} \partostratind{j} \la{x}{s'} = s$}
	\DisplayProof
	\shortintertext{and}
	\AxiomC{$u' \partobind{n-1} s'$}
	\UnaryInfC{$u = \la{x}{u'} \partobind{n-1} \la{x}{s'} = s$}
	\DisplayProof
	\,.
	\end{align*}

	\item \emph{Application}:
	\begin{align*}
	\AxiomC{$t_0 \partobind{n_0} s_0$}
	\AxiomC{$t_1 \partobind{n_1} s_1$}
	\BinaryInfC{$t = t_0 t_1 \partobind{n} s_0 s_1 = s$}
	\DisplayProof
	\end{align*}
	where $n = n_0 + n_1$.
	By {\textit{i.h.}}\xspace, $t_0 \partostratind{m_0} s_0$ and $t_1 \partostratind{m_1} s_1$ for some $m_0, m_1 \in \mathbb{N} \cup \{\infty\}$.
	Thus,
	\begin{align*}
	\AxiomC{$t_0 \partostratind{m_0} s_0$}
	\AxiomC{$t_1 \partostratind{m_1} s_1$}
	\BinaryInfC{$t = t_0 t_1 \partostratind{m} s_0 s_1 = s$}
	\DisplayProof
	\end{align*}
	where $m = \min\{m_0,m_1+1\}$.
	If moreover $n > 0$, then by {\textit{i.h.}}\xspace, $m_0, m_1 \neq \infty$ and $t_0 \tostratind{m_0} u_0$ and $t_1 \tostratind{n_1} u_1$ for some terms $u_0, u_1$ such that $u_0 \partobindlong{n_0-1} s_0$ and $u_1 \partobindlong{n_1-1} s_1$ and $u_0 \partostratind{j_0} s_0$ and $u_1 \partostratind{j_1} s_1$ with $j_0 \geq m_0$, $j_1 \geq m_1$ (possibly $j_0,j_1 = \infty$).
	There are two possibilities:
	\begin{enumerate}
		\item $m = m_0$; then, $m_0 \leq m_1+1$ and
		\begin{align*}
		\AxiomC{$t_0 \tostratind{m} u_0$}
		\UnaryInfC{$t = t_0t_1 \tostratind{m} u_0t_1 = u$}
		\DisplayProof
		\qquad
		\text{and}
		\qquad
		&
		\AxiomC{$u_0 \partobind{n_0-1} s_0$}
		\AxiomC{$t_1 \partobind{n_1} s_1$}
		\BinaryInfC{$u = u_0t_1 \partobind{n-1} s_0s_1 = s$}
		\DisplayProof
		\shortintertext{and}
		\AxiomC{$u_0 \partostratind{j_0} s_0$}
		\AxiomC{$t_1 \partostratind{m_1} s_1$}
		\BinaryInfC{$u = u_0t_1 \partostratind{j} s_0s_1 = s$}
		\DisplayProof
		\end{align*}
		where $j = \min\{j_0, m_1+1\} \geq m_0 = m$;
		\item $m = m_1+1$; then, $m_1 + 1 \leq m_0$ and
		\begin{align*}
		\AxiomC{$t_1 \tostratind{m-1} u_1$}
		\UnaryInfC{$t = t_0t_1 \tostratind{m} t_0u_1 = u$}
		\DisplayProof
		\qquad
		\text{and}
		\qquad
		&
		\AxiomC{$u_0 \partobind{n_0} s_0$}
		\AxiomC{$t_1 \partobind{n_1-1} s_1$}
		\BinaryInfC{$u = t_0u_1 \partobind{n-1} s_0s_1 = s$}
		\DisplayProof
		\shortintertext{and}
		\AxiomC{$u_0 \partostratind{m_0} s_0$}
		\AxiomC{$t_1 \partostratind{j_1} s_1$}
		\BinaryInfC{$u = t_0u_1 \partostratind{j} s_0s_1 = s$}
		\DisplayProof
		\end{align*}
		where $j = \min\{m_0, j_1+1\} \geq m_1+1 = m$.
	\end{enumerate}

	\item \emph{$\beta$-step}:
	\begin{align*}
	\AxiomC{$t_0 \partobind{n_0} t_0'$}
	\AxiomC{$t_1 \partobind{n_1} t_1'$}
	\BinaryInfC{$t = (\la{x}t_0)t_1 \partobind{n} t_0' \isub{x}{t_1'} = s$}
	\DisplayProof
	\end{align*}
	where $n = n_0 + \sizep{t_0}{x}n_1 + 1 > 0$.
	By {\textit{i.h.}}\xspace, $t_0 \partostratind{m_0} s_0$ and $t_1 \partostratind{m_1} s_1$ for some $m_0, m_1 \in \mathbb{N}$.
	Thus,
	\begin{align*}
	\AxiomC{$t_0 \partostratind{m_0} t_0'$}
	\AxiomC{$t_1 \partostratind{m_1} t_1'$}
	\BinaryInfC{$t = (\la{x}t_0)t_1 \partostratind{0} t_0' \isub{x}{t_1'} = s$}
	\DisplayProof
	\,.
	\end{align*}
	Let $u = t_0\isub{x}{t_1}$.
	By \reflemma{partobind-subs}, $u = t_0\isub{x}{t_1} \partobindlong{n_0 + \sizep{t_0}{x} n_1} t_0'\isub{x}{t_1'} = s$ where $n_0 + \sizep{t_0}{x} n_1 = n - 1$.
	Clearly,
	\begin{align*}
	\AxiomC{}
	\UnaryInfC{$t = (\la{x}t_0)t_1 \tostratind{0} t_0 \isub{x}{t_1} = u$}
	\DisplayProof
	\,.
	\end{align*}
	By {\textit{i.h.}}\xspace (since $n_0 + \sizep{t_0}{x} n_1 = n -1$), $u = t_0\isub{x}{t_1} \partostratind{j} t_0'\isub{x}{t_1'} = s$ for some $j \in \mathbb{N} \cup \{\infty\}$, in particular $j \geq 0$.
	\qedhere
\end{itemize}
\end{proof}

\begin{propositionAppendix}[Stratified indexed split]
	\label{propappendix:quant-out-in-split-strat}
	\NoteState{prop:quant-out-in-split-strat}
	If $t \partobind n s$ then either $t \parRew{\lnot\ell\ell}s$ or $n>0$ and there exists $u$ such that $t \Rew{x} u \partobindlong {n-1} s$.
\end{propositionAppendix}

\begin{proof}
	If $n = 0$, then $t = s$ (because $\partobind{0}$ is the identity relation) and so $t \partostratind{\infty} s$ (because $\partostratind{\infty}$ is the identity relation), hence $t \parRew{\lnot\ell\ell} s$.

	Otherwise $n > 0$ and then, by \reflemma{decomp-strat}, $t \partostratind{k} s$ and $t \tostratind{k} u$ for some $k \in \mathbb{N}$ and some term $u$ such that $u \partobind{n-1} s$.
	There are two cases:
	\begin{enumerate}
		\item either $k > \Deg{t}$ and then $t \parRew{\lnot\ell\ell} s$ (since $t \partostratind{k} s$);
		\item or $k = \Deg{t}$  and so $t \Rew{x} u \partobindlong{n-1} s$.
		\qedhere
	\end{enumerate}
\end{proof}

\begin{corollaryAppendix}[Stratified split]
	\label{coroappendix:e-i-split-strat}
	\NoteState{coro:e-i-split-strat}
	If $t \parRew{\beta} s$ then $t \Rew{x}^* u \parRew{\lnot\ell\ell} s$ for some $u$.
\end{corollaryAppendix}

\begin{proof}
	By hypothesis, $t \partobind{n} s$ for some $n \in \mathbb{N}$.
	We prove the statement by induction on $n \in \mathbb{N}$.

	If $n = 0$, then $t = s$ (because $\partobind{0}$ is the identity relation) and so $t \partostratind{\infty} s$ (because $\partostratind{\infty}$ is the identity relation), hence $t \parRew{\lnot\ell\ell} s$.

	Otherwise $n > 0$ and then, by \reflemma{decomp-strat}, $t \partostratind{k} s$ and $t \tostratind{k} u$ for some $k \in \mathbb{N}$ and some term $u$ such that $u \partobind{n-1} s$.
	There are two cases:
	\begin{enumerate}
		\item either $k > \Deg{t}$ and then $t \parRew{\lnot\ell\ell} s$ (as $t \partostratind{k} s$), so we conclude taking $u = t$;
		\item or $k = \Deg{t}$  and then $t \Rew{x} r \partobindlong{n-1} s$;
		by {\textit{i.h.}}\xspace applied to $r \partobind{n-1} s$, there is a term $u$ such that $r \Rew{x}^* u \parRew{\lnot\ell\ell} s$;
		therefore, $t \Rew{x} r \Rew{x}^* u \parRew{\lnot\ell\ell} s$.
		\qedhere
	\end{enumerate}
\end{proof}

Note that the last proof uses only \reflemma{decomp-strat}, not \refprop{quant-out-in-split-strat}!

=========================

\subsection{STRATIFIED: Proposta di prova per Indexed Split e Split}
\newcommand{\tmfive}{p}
\CF{Scriverei la prova di Indexed Split esattamente come quella per Head. La prova sotto e' cut-and-paste dalla prova di Head;
	ho cambiato solo il caso applicazione.
	Non mi sembra di vedere difficolta' particolari; in un certo senso e' piu' facile, perche' non occorre distinguere se in  $rp$ il termine $r$
	 e' astrazione. \textbf{Ho segnalato in blu il punto chiave}.
	Cosa ne pensate?
 }

\begin{propositionAppendix}[Indexed split]
	\label{propappendix:quant-out-in-split-strat}
	\NoteState{prop:quant-out-in-split-strat}
	If $t \partobind n s$ then either $t\partonotstrat \tmtwo}%{s$ or $n>0$ and there exists $u$ such that $t \Rew{x} \tmthree}%{u \partobindlong {n-1} \tmtwo}%{s$.
\end{propositionAppendix}

\proof
By induction on $t \partobind n s$. We freely use the fact that if $t \partobind n s$ then $t \parRew{\beta} s$. Cases:
\begin{itemize}
	\item \emph{Variable}: $t = x \partobind 0 x = s$. Then $t = x \partonotstratx = s$.

	\item \emph{Abstraction}: $t = \la \vart' \partobind n  \la x s' = s$ because $t' \partobind n s'$. It follows by the {\textit{i.h.}}\xspace.

	\item \emph{Application}:
	\[\AxiomC{$\tmfour}%{r \partobind {n_1} \tmfour}%{r'$}
	\AxiomC{$\tmfive \partobind {n_2} \tmfive'$}
	\BinaryInfC{$t = \tmfour}%{r \tmfive \partobindlong {n_1 + n_2} \tmfour}%{r' \tmfive' = s$}
	\DisplayProof\]
	(with $n = n_1 + n_2$). We examine the conclusion.
	 If $\tmfour}%{r \tmfive \partonotstrat \tmfour}%{r' \tmfive' $ the claim holds.
	 Otherwise, we have $\tmfour}%{r \tmfive \partostratind {d} \tmfour}%{r' \tmfive'$ with   $d=\deg{\tmfour}%{r \tmfive}$. Let us write explicitly all the depths:

		\[\AxiomC{$\tmfour}%{r \partostratind {d_{\tmfour}%{r}} \tmfour}%{r'$}
	\AxiomC{$\tmfive \partostratind{d_{\tmfive}} \tmfive'$}
	\BinaryInfC{$t = \tmfour}%{r \tmfive \partostratind d \tmfour}%{r' \tmfive' = s$}
	\DisplayProof\]
We first \blue{prove that  if $d=\deg{\tmfour}%{r \tmfive}$ then  either $\deg \tmfour}%{r = d_{\tmfour}%{r}$ or $\deg \tmfive = d_{\tmfive}$}. We then use the {\textit{i.h.}}\xspace\ on the opportune premiss.
\begin{itemize}
	\item Observe that
	$\deg  {\tmfour}%{r \tmfive}\leq \deg  {\tmfour}%{r } $ and $\deg  {\tmfour}%{r \tmfive}\leq \deg  {\tmfive}+1 $  hold in both cases defining $\deg{\tmfour}%{r \tmfive}$.
	By definition, $d=\min\{ d_{\tmfour}%{r}, d_{\tmfive}+1  \}$.
	\begin{enumerate}
		\item[(a.)]  If  $d=d_{\tmfour}%{r}$, then $d=\deg  {\tmfour}%{r \tmfive}\leq \deg \tmfour}%{r \leq
	d_{\tmfour}%{r}=d $, hence $\deg \tmfour}%{r =
	d_{\tmfour}%{r}$.
\item[(b.)]	If  $d=d_{\tmfive}+1$, then $d=\deg  {\tmfour}%{r \tmfive}\leq \deg \tmfive +1 \leq	  d_{\tmfive}+1 =d $, hence $\deg \tmfive =
	d_{\tmfive}$.
	\end{enumerate}

	\item If (a.) holds, we apply the {\textit{i.h.}}\xspace\ to $\tmfour}%{r \partobind {n_1} \tmfour}%{r'$, and obtain that $n_1>0$ and $\tmfour}%{r \Rew{x} \tmthree}%{u_1 \partobind {n_1-1} \tmfour}%{r'$, hence $n>0 $ and  $\tmfour}%{r\tmfive  \Rew{x} \tmthree}%{u_1\tmfive$ and $\tmthree}%{u_1 \tmfive\partobind {n_1-1+n_2} \tmfour}%{r'\tmfive'$.  If (b.) holds, we apply the {\textit{i.h.}}\xspace\ to $\tmfive \partobind {n_2} \tmfive'$.
\end{itemize}

	\item \emph{$\beta$ step}:
	\[\AxiomC{$u \partobind {n_1} u'$}
	\AxiomC{$r \partobind {n_2} r'$}
	\BinaryInfC{$t = (\lax u)r \partobindlong {n_1 + \sizep{u'}x \cdot n_2 +1} u'\isub x {r'} = s$}
	\DisplayProof\]
	With $n = n_1 + \sizep{u'}x \cdot n_2 +1 > 0$. We have $t = (\lax u)r \Rew{x} u\isub\varr$ and by substitutivity of $\partobind n$ (\reflemma{partobind-subs}) $u\isub\varr \partobindlong{n_1 + \sizep{u'}x \cdot n_2} u' \isubx{r'} = s$.
	\qedhere
\end{itemize}

\begin{corollary}[Split]
	If $t \parRew{\beta} s$ then there exists $u$ such that $t \Rew{x}^* u \partonotstrats$.
\end{corollary}
\proof
If $t \parRew{\beta} s$ then $t \partobind n s$ for some $n$. We prove the statement by induction on $n$. By Indexed Split, there are two cases:
\begin{itemize}
	\item \emph{$\blue{u} \partonotstrats$}. This is an instance of the statement (take $u = t$).

	\item \emph{$n>0$ and there exists $r$ such that $t \Rew{x} r \partobind {n-1} s$}. By {\textit{i.h.}}\xspace, there exists $u$ such that $r \Rew{x}^* u \partonotstrat s$, and so $t \Rew{x}^* u \partonotstrats$.\qed
\end{itemize}

\CF
	la prova sopra e' cut-and-paste di \refcoro{e-i-split}.  Io direi semplicemente: \\
\blue{The proof is identical to that of \refcoro{e-i-split}, where  $\Rew{x}$ e $\partonotstrat$ respectively replace $\Rew{h}$ and $\parRew{\neg h}$.}}

\newpage
\subsection{Domanda sulle altre prove di indexed split }
\CF{Mi domando se anche nelle altre prove di indexed split, non sia piu' semplice (mentalmente) usare per l'applicazione lo schema:
	\begin{enumerate}
		\item if $\tmfour}%{r \tmfive \Rightarrow_{\neg e} \tmfour}%{r'\tmfive'$ the claim holds
		\item otherwise, we use the {\textit{i.h.}}\xspace
	\end{enumerate}
}

\subsubsection{Head: la prova seguente di  Indexed Split basta?}
\CF{E' corretto nell'applicazione sostituire i tre sottocasi con la parte blu seguente?}

\begin{proposition}[Indexed split]
	If $t \partobind n s$ then $t \partonoths$, or $n>0$ and there exists $u$ such that $t \Rew{h} u \partobind {n-1} s$.
\end{proposition}

\proof
By induction on $t \partobind n s$. We use freely the fact that if $t \partobind n s$ then $t \parRew{\beta} s$. Cases:
\begin{itemize}
	\item \emph{Variable}: $t = x \partobind 0 x = s$. Then $t = x \partonothx = s$.

	\item \emph{Abstraction}: $t = \la \vart' \partobind n  \la x s' = s$ because $t' \partobind n s'$. It follows by the {\textit{i.h.}}\xspace.

	\item \emph{Application}:
	\[\AxiomC{$r \partobind {n_1} r'$}
	\AxiomC{$p \partobind {n_2} p'$}
	\BinaryInfC{$t = r p \partobindlong {n_1 + n_2} r' p' = s$}
	\DisplayProof\]
	With $n = n_1 + n_2$. \blue{We examine the conclusion.
	\begin{itemize}
		\item 	If $r p \parRew{\neg h} r' p'$, the claim is holds.
\item 	Otherwise, observe that the  left premiss is not of the form $r \parRew{\neg h} r'$ (otherwise,  $r p \parRew{\neg h} r' p'$ ). By the {\textit{i.h.}}\xspace,
	\emph{$n_1>0$ and there exists $r''$ such that $r \Rew{h} r'' \partobind {n_1-1} r'$}. Then $t = r p \Rew{h} r'' p$ and
	\[\AxiomC{$r'' \partobindlong {n_1-1} r'$}
	\AxiomC{$p \partobind {n_2} p'$}
	\BinaryInfC{$r'' p \partobindlong {n_1-1 + n_2} r' p' = s$}
	\DisplayProof\]
	\end{itemize}
	}

	\item \emph{$\beta$ step}:
	\[\AxiomC{$u \partobind {n_1} u'$}
	\AxiomC{$r \partobind {n_2} r''$}
	\BinaryInfC{$t = (\lax u)r \partobindlong {n_1 + \sizep{u'}x \cdot n_2 +1} u'\isub x {r'} = s$}
	\DisplayProof\]
	With $n = n_1 + \sizep{u'}x \cdot n_2 +1 > 0$. We have $t = (\lax u)r \Rew{h} u\isub\varr$ and by substitutivity of $\partobind n$ (\reflemma{partobind-subs}) $u\isub\varr \partobindlong{n_1 + \sizep{u'}x \cdot n_2} u' \isubx{r'} = s$.
	\qedhere
\end{itemize}

\subsubsection{Leftmost: indexed split}
\CF{Tento di implementare anche  nel caso della leftmost, ma qui vado per puro pattern-matching, perche' LO e' una riduzione che io non capisco}

\begin{proposition}[Indexed split]
	\label{prop:quant-out-in-split-left}
	If $t \partobind n s$ then $t \parRew{\neg \losym} s$ or $n>0$ and there exists $u$ such that $t \Rew{\losym} u \partobind {n-1} s$.
\end{proposition}

\begin{proof}
	By induction on $t \partobind n s$. We use freely the fact that if $t \partobind n s$ then $t \parRew{\beta} s$. Cases:
	\begin{itemize}
		\item \emph{Variable}: $t = x \partobind 0 x = s$. Then $t = x \parRew{\neg \losym} x = s$.

		\item \emph{Abstraction}: $t = \la \vart' \partobind n  \la x s' = s$ because $t' \partobind n s'$. It follows by the {\textit{i.h.}}\xspace.

		\item \emph{Application}:
		\[\AxiomC{$r \partobind {n_1} r'$}
		\AxiomC{$p \partobind {n_2} p'$}
		\BinaryInfC{$t = r p \partobindlong {n_1 + n_2} r' p' = s$}
		\DisplayProof\]
		With $n = n_1 + n_2$.\blue{We examine the conclusion.
		\begin{itemize}
			\item  If $r p \parRew{\neg \losym} r' p'$, the claim holds.
			\item  Otherwise, observe that $r$ is not neutral (otherwise $r p \parRew{\neg \losym} r' p'$ by  the 5th rule) and the left premiss is not of the form
		$r \parRew{\neg \losym} r'$ (otherwise $r p \parRew{\neg \losym} r' p'$ by the 2nd rule). By the {\textit{i.h.}}\xspace,
		\emph{$n_1>0$ and there exists $r''$ such that $r \Rew{\losym} r'' \partobind {n_1-1} r'$}. Then $t = r p \Rew{\losym} r'' p$ and
		\[\AxiomC{$r'' \partobindlong {n_1-1} r'$}
		\AxiomC{$p \partobind {n_2} p'$}
		\BinaryInfC{$r'' p \partobindlong {n_1-1 + n_2} r' p' = s$}
		\DisplayProof\]
		\end{itemize}
		}

		\item \emph{$\beta$ step}:
		\[\AxiomC{$u \partobind {n_1} u'$}
		\AxiomC{$r \partobind {n_2} r''$}
		\BinaryInfC{$t = (\lax u)r \partobindlong {n_1 + \sizep{s}x \cdot n_2 +1} u'\isub x {r'} = s$}
		\DisplayProof\]
		With $n = n_1 + \sizepsx \cdot n_2 +1 > 0$. We have $t = (\lax u)r \Rew{\losym} u\isub\varr$ and by substitutivity of $\partobind n$ (\reflemmap{partobind-subs}{ind-par}) $u\isub\varr \partobindlong{n_1 + \sizepsx \cdot n_2} u' \isubx{r'} = s$.
	\end{itemize}
\end{proof}

\subsection{Omitted proofs of \refsect{cbv-calculus} (\texorpdfstring{{CbV}\xspace $\l$-calculus}{{CbV}\xspace lambda-calculus})}
\setcounter{lemmaAppendix}{\value{l:tobv-subs}}
\begin{lemmaAppendix}[Substitutivity of $\partobvind n$]
	\label{lappendix:tobv-subs}
	\NoteState{l:tobv-subs}
 If $t \partobvind n t'$ and $v \partobvind m v'$ then $t \isubxv \partobvind{k} t' \isubx{v'}$ where $k = n + \sizep{t'}x\cdot m$.
\end{lemmaAppendix}

\begin{proof}
By induction on $t \partobind n t'$. It follows the exact same pattern of the proof of substitutivity of $\partobind n$. Cases:
\begin{itemize}
	\item \emph{Variable}: two sub-cases
	\begin{itemize}
	\item $t = x$: then $t = x \partobvind 0 x = t'$ then $t \isubxv = x\isubxv = v \partobvind m v' = x \isubx{v'} = t' \isubx{v'}$ that satisfies the statement because $n + \sizep{t'}x\cdot m = 0+1\cdot m = m$.
	 \item $t = y$: then $t = y \partobvind 0 y = t'$ then $t \isubxv = y\isubxv = y \partobvind 0 y = y \isubx{v'} = t' \isubx{v'}$ that satisfies the statement because $n + \sizep{t'}x\cdot m = n +0\cdot m=n$.
	 \end{itemize}

	\item \emph{Abstraction}, {\em i.e.}\xspace $t = \la{y}{u} \partobvind{n} \la{y}{u'} = t'$ because $u \partobvind{n} u'$; we can suppose without loss of generality that $y \neq x$, hence $\sizep{u'}{x} = \sizep{t'}{x}$ and $t\isub{x}{v} = \la{y}(u\isub{x}{v})$ and $t'\isub{x}{v'} = \la{y}(u'\isub{x}{v'})$.
		By {\textit{i.h.}}\xspace, $u\isub{x}{v} \partobvindlong{n +  \sizep{u'}{x} \cdot m} u'\isub{x}{v'}$, thus
		\begin{align*}
		\AxiomC{$u\isub{x}{v} \partobvindlong{n +  \sizep{u'}{x} \cdot m} u'\isub{x}{v'}$}

		\UnaryInfC{$t\isub{x}{v} = \la {y}u\isub{x}{v} \partobvindlong{n +  \sizep{t'}{x} \cdot m} \la{y}u'\isub{x}{v'} = t'\isub{x}{v'}$}
		\DisplayProof\,.
		\end{align*}

	\item \emph{Application}, {\em i.e.}\xspace
	\[\AxiomC{$u \partobvind {n_u} u'$}
	\AxiomC{$r \partobvind {n_r} r'$}

	\BinaryInfC{$t = u r \partobvindlong {n_u + n_r} u' r'= t'$}
	\DisplayProof\]
	with $n = n_u + n_r$. By {\textit{i.h.}}\xspace, $u\isubxv \partobvindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{v'}$ and $r\isubxv \partobvindlong {n_r +\sizep{r'}x \cdot m} r'\isubx{v'}$. Then
	\[\AxiomC{$u\isubxv \partobvindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{v'}$}
	\AxiomC{$r\isubxv \partobvindlong {n_r +\sizep{r'}x \cdot m} r'\isubx{v'}$}

	\BinaryInfC{$t\isubxv = u\isubxv r\isubxv \partobvind{k} u'\isubx{v'} r'\isubx{v'}= t'\isubx{v'}$}
	\DisplayProof\]
	with $k = n_u +\sizep{u'}x \cdot m +n_r +\sizep{r'}x \cdot m$, which proves the statement because $$k = n_u +\sizep{u'}x \cdot m +n_r +\sizep{r'}x \cdot m = n + (\sizep{u'}x + \sizep{r'}x)\cdot m = n + \sizep{t'}x \cdot m.$$

\item \emph{$\beta_\val$-step}, {\em i.e.}\xspace ($w$ and $\wal'$ are values)
	\[\AxiomC{$u \partobvind {n_u} u'$}
	\AxiomC{$w \partobvind {n_{w}} \wal'$}
\BinaryInfC{$t = (\lay u)w \partobvindlong {n_u + \sizep{u'}y \cdot n_{w} +1} u'\isub y {\wal'} = t'$}
\DisplayProof\]
with $n = n_u + \sizep{u'}y \cdot n_{w} +1$. We can assume without loss of generality that $y\neq x$,
		hence, $\sizep{t'}{x} = \sizep{u'\isuby {\wal'}}x = \sizep{u'}{x} + \sizep{u'}{y}\cdot\sizep{\wal'}{x} $ and $t\isubxv = (\la\vartwou\isubxv) (w\isubxv)$ and $t'\isubx{v'} = u'\isubx{v'} \isuby{\wal'\isubx{v'}}$.

		By {\textit{i.h.}}\xspace, $u\isubxv \partobvindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{v'}$ and $w\isubxv \partobvindlong {n_{w} +\sizep{\wal'}x \cdot m} \wal'\isubx{v'}$.
		Then
	\[
	\def4pt{1pt}
	\AxiomC{$u\isubxv \partobvindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{v'}$}
	\AxiomC{$w\isubxv \partobvindlong {n_{w} +\sizep{\wal'}x \cdot m} \wal'\isubx{v'}$}

	\BinaryInfC{$t\isubxv = (\la\vartwou\isubxv) (w\isubxv) \partobvind{k} u'\isubx{v'} \isuby{\wal'\isubx{v'}}= t'\isubx{v'}$}
	\DisplayProof
	\def4pt{4pt}
	\]
where $k = n_u +\sizep{u'}x \cdot m +\sizep{u'}y\cdot(n_{w} +\sizep{\wal'}x \cdot m)+1 = n_u +\sizep{u'}x \cdot m +\sizep{u'}y\cdot(n_{w} +\sizep{\wal'}x \cdot m)+1
=
n_u + \sizep{u'}y\cdot n_{w} + 1 + \sizep{u'}x \cdot m + \sizep{u'}y\cdot \sizep{\wal'}x \cdot m
= n + \sizep{t'}x\cdot m$.
\qedhere
\end{itemize}
\end{proof}

\subsection{Omitted proofs  of \refsect{weak-cbv} (weak {CbV}\xspace)}

\setcounter{propositionAppendix}{\value{prop:macro-cbv}}
\begin{propositionAppendix}[Weak {CbV}\xspace macro-step system]
	\label{propappendix:macro-cbv}
	\NoteState{prop:macro-cbv}
\begin{enumerate}
 \item \label{pappendix:macro-cbv-merge} \emph{Merge}: if $t \parRew{\neg \wsym}\cdot \Rew{\wsym} u$ then $t \parRew{\betav} u$.
 \item \label{pappendix:macro-cbv-indexed-split} \emph{Indexed split}: if $t \partobvind n s$ then $t \partonotws$, or $n>0$ and $t \Rew{\wsym} \cdot \partobvind {n-1} s$.
 \item \label{pappendix:macro-cbv-split} \emph{Split}: if $t \parRew{\betav} s$ then $t \Rew{\wsym}^* \cdot \partonotws$.
\end{enumerate}
That is, $(\Lambda, \set{\Rew{\wsym}, \Rew{\neg \wsym}})$ is a macro-step system with respect to $\parRew{\betav}$ and $\parRew{\neg \wsym}$.
\end{propositionAppendix}

\begin{proof}
\hfill
\begin{enumerate}
 \item \emph{Merge}: by induction on $t \partonotws$. Note that the cases in which $s=x$ or $s=\l x. \tmtwo}%{s'$ are not possible.  Hence $ s=r'p' $  and $t \partonotws$ is derived as follows
	\begin{align*}
	\AxiomC{$r \parRew{\neg \wsym} r'$}
	\AxiomC{$p \parRew{\neg \wsym} p'$}
	\BinaryInfC{$t = rp \parRew{\neg \wsym} r'p' =s$}
	\DisplayProof
	\end{align*}
\begin{enumerate}
\item If $ r' \Rew{\wsym} r''$ then $s = r'' p'$. The {\textit{i.h.}}\xspace gives $r \parRew{\betav} r''$, and $t \parRew{\betav} s$ is derived as follows (remember that $\parRew{\neg \wsym} \subseteq \parRew{\betav}$):
\begin{align*}
	\AxiomC{$r \parRew{\betav} r''$}
	\AxiomC{$p \parRew{\betav} p'$}
	\BinaryInfC{$t = rp \parRew{\betav} r''p' =s$}
	\DisplayProof
	\end{align*}

	\item If $ p' \Rew{\wsym} p''$ it is analogous to the previous case.

\item If 	$ s \Rew{\wsym} u$ by a top $\beta_\val$ step then $r' = \la x \tmsix}%{q'$. Now, by definition of $\parRew{\neg \wsym}$ the step $r \parRew{\neg \wsym} r'$ necessarily has the form $r = \la x \tmsix}%{q \parRew{\neg \wsym} \la x \tmsix}%{q' = r'$ for some $\tmsix}%{q$ such that $\tmsix}%{q\parRew{\betav} \tmsix}%{q'$. Then the hypothesis is $t = (\la x \tmsix}%{q) p \parRew{\neg \wsym} (\la x \tmsix}%{q') p' \Rew{\wsym} \tmsix}%{q'\isub{x}{p'} = u$ and  $t \parRew{\betav} s$ is derived as follows (remember that $\parRew{\neg \wsym} \subseteq \parRew{\betav}$):
\begin{align*}
\AxiomC{$\tmsix}%{q\parRew{\betav} \tmsix}%{q'$}
\AxiomC{$p \parRew{\betav} p'$}
\BinaryInfC{$t = (\la x \tmsix}%{q) p \parRew{\betav}  \tmsix}%{q'\isub{x}{p'} =s $}
\DisplayProof
\end{align*}
\end{enumerate}

\item \emph{Indexed split}: by induction on the definition of $t \partobvind n s$.
	We freely use the fact that if $t \partobvind n s$ then $t \parRew{\betav} s$. Cases:
	\begin{itemize}
		\item \emph{Variable}: $t = x \partobvind 0 x = s$.
		Then, $t = x \partonotwx = s$.

		\item \emph{Abstraction}: $t = \la \vart' \partobvind n  \la x s' = s$ because $t' \partobvind n s'$.
		Then,
		\begin{align*}
		\AxiomC{$t' \parRew{\betav} s'$}
		\UnaryInfC{$t = \la{x}t' \parRew{\neg \wsym} \la{x}s' = s$}
		\DisplayProof
		\,.
		\end{align*}

	\item \emph{Application}:
	\[\AxiomC{$r \partobvind {n_1} r'$}
	\AxiomC{$p \partobvind {n_2} p'$}
	\BinaryInfC{$t = r p \partobvindlong {n_1 + n_2} r' p' = s$}
	\DisplayProof\]
	with $n = n_1 + n_2$. There are only two cases:
	\begin{itemize}
		\item either $r p \parRew{\neg \wsym} r' p'$, and then the claim holds.
		\item or $r p \not\parRew{\neg \wsym} r' p'$, and then $r \not\parRew{\neg \wsym} r'$ or $p \not\parRew{\neg \wsym} p'$ (otherwise $r p \parRew{\neg \wsym} r' p'$).
		Suppose $r \not\parRew{\neg \wsym} r'$ (the other case is analogous).
		By {\textit{i.h.}}\xspace applied to $r \partobvind{n_1} r'$,
		$n_1>0$ and there is $r''$ such that $r \Rew{\wsym} r'' \partobvind {n_1-1} r'$.
		So, $t = r p \Rew{\wsym} r'' p$ and
		\[\AxiomC{$r'' \partobvindlong {n_1-1} r'$}
		\AxiomC{$p \partobvind {n_2} p'$}
		\BinaryInfC{$r'' p \partobvindlong {n_1-1 + n_2} r' p' = s$}
		\DisplayProof
		\ .
		\]
	\end{itemize}

		\item \emph{$\beta_\val$ step}:
		\[\AxiomC{$p \partobvind {n_1} p'$}
		\AxiomC{$r$ is a value}
		\AxiomC{$r \partobvind {n_2} r'$}
		\TrinaryInfC{$t = (\lax p)r \partobvindlong {n_1 + \sizep{p'}x \cdot n_2 +1} p'\isub x {r'} = s$}
		\DisplayProof\]
		with $n = n_1 + \sizep{p'}x \cdot n_2 +1 > 0$.
		We have $t = (\lax p)r \Rew{\wsym} p\isub\varr := u$ and substitutivity of $\partobvind n$ (\reflemma{tobv-subs}) gives $u = p\isub\varr \partobvindlong{n_1 + \sizep{p'}x \cdot n_2} p' \isubx{r'} = s$.
	\end{itemize}

	\item \emph{Split}: exactly as in the head case (\refpropp{macro-head}{split}), using the Indexed Split property for weak {CbV}\xspace (\refpoint{macro-cbv-indexed-split} above).
	\qedhere
	\end{enumerate}
\end{proof}

\setcounter{propositionAppendix}{\value{essential-weak}}
\begin{propositionAppendix}[Weak {CbV}\xspace essential system]
	\label{propappendix:essential-weak}
	\NoteState{prop:essential-weak}
	\begin{enumerate}
		\item\label{pappendix:essential-weak-persistence} \emph{Persistence}: if $t \Rew{\wsym} s$ and $t \Rew{\neg \wsym} u$ then $u \Rew{\wsym} r$ for some $r$.
		\item\label{pappendix:essential-weak-diamond} \emph{Diamond}: if $s\lRew{\wsym} t \towu$ with $s \neq u$ then $s \Rew{\wsym} r \lRew{\wsym} u$ for some $r$.
	\end{enumerate}
	Then, $(\Lambda, \set{ \Rew{\wsym}, \Rew{\neg \wsym}})$ is an essential system.
\end{propositionAppendix}

\begin{proof}\hfill
	\begin{enumerate}
		\item \emph{Persistence}: By induciton on the definition of $t \Rew{\neg \wsym} s$.
		Cases:
		\begin{itemize}
			\item \emph{Abstraction}: $t = \lax' \Rew{\neg \wsym} \lax''$ because $p \Rew{\betav} p'$. This case is impossible because $t$ is $\Rew{\wsym}$ normal, against the hypothesis that $t \Rew{\wsym} u$.

			\item \emph{Application left}: $t = pq \Rew{\neg \wsym} p_{\neg{\mathtt w}}q = s$ because $p \Rew{\neg \wsym} p_{\neg{\mathtt w}}$.
			According to the definition of $\Rew{\wsym} $, there are the following sub-cases:
			\begin{enumerate}
				\item $t = pq \Rew{\wsym} p_\wsymq = u$ with $p \Rew{\wsym} p_{\mathtt w}$;
				by {\textit{i.h.}}\xspace applied to $p$, we have $p_{\neg{\mathtt w}} \Rew{\wsym} r'$  for some term $r'$, and so
				\begin{align*}
				\AxiomC{$p_{\neg{\mathtt w}} \Rew{\wsym} r'$}
				\UnaryInfC{$s = p_{\neg{\mathtt w}}q \Rew{\wsym} r'q =: r$}
				\DisplayProof
				\,;
				\end{align*}
				\item $t = pq \Rew{\wsym} pq' = u$ with  $q \Rew{\wsym} q'$;
				hence
				\begin{align*}
				\AxiomC{$q \Rew{\wsym} q'$}
				\UnaryInfC{$s = p_{\neg{\mathtt w}}q \Rew{\wsym} p_{\neg{\mathtt w}}q' =`: r$}
				\DisplayProof
				\,;
				\end{align*}

				\item $t = (\la{x}{p'})q \Rew{\wsym} p'\isub{x}{q} = u$ where $p = \la{x}{p'}$ and $q$ is a value.
				According to the definition of $t \Rew{\neg \wsym} s$,
				\begin{align*}
				\AxiomC{$p \Rew{\betav} p_{\neg{\mathtt w}}$}
				\UnaryInfC{$\la{x}p' \Rew{\neg \wsym} \la{x}p_{\neg{\mathtt w}}'$}
				\UnaryInfC{$t = (\la{x}p')q \Rew{\neg \wsym} (\la{x}{p_{\neg{\mathtt w}}'})q = s$}
				\DisplayProof
				\end{align*}
				where $p_{\neg{\mathtt w}} = \la{x}{p_{\neg{\mathtt w}}'}$; therefore
				\begin{align*}
				\AxiomC{}
				\UnaryInfC{$s = (\la{x}p_{\neg{\mathtt w}}')q \Rew{\wsym} p_{\neg{\mathtt w}}'\isub{x}{q} =: r$}
				\DisplayProof
				\,.
				\end{align*}
			\end{enumerate}

			\item \emph{Application right}: analogous to the previous point.
		\end{itemize}

		\item \emph{Diamond}:
		The idea of the proof is that, when $(\la{x}t)v \Rew{\wsym} t\isub{x}{v}$, the $\beta_\val$-redexes in $v$ (which is a value) can be duplicated in $t\isub{x}{v}$ but they are under an abstraction and $\Rew{\wsym}$ does not reduce  under abstractions.

		Formally, the proof is by induction on $t$.
		Note that $t$ is not a value, because values are $\Rew{\wsym}$-normal, since $\Rew{\wsym}$ does not reduce under abstractions.
		Therefore, $t = t_0t_1$.
		The case where $t_0 = \la{x}{t'}$ and $t_1$ is a value is impossible, because $t_0$ and $t_1$ would be $\Rew{\wsym}$-normal and so from $s \lRew{\wsym} t \Rew{\wsym} u$ it would follow $s = t'\isub{x}{t_1} = u$, which contradicts the hypothesis.
		The remaining cases for $t = t_0t_1$ are:
		\begin{itemize}
			\item $s = t_0s_1 \lRew{\wsym} t = t_0t_1 \Rew{\wsym} u_0t_1 = u$ with $t_0 \Rew{\wsym} u_0$,  and $t_1 \Rew{\wsym}  s_1$.
			Then, $s \Rew{\wsym} u_0s_1 \lRew{\wsym} u$.

			\item $s = s_0 t_1 \lRew{\wsym} t = t_0t_1 \Rew{\wsym} u_0t_1 = u$ with $s_0 \lRew{\wsym} t_0 \Rew{\wsym} u_0$.
			By {\textit{i.h.}}\xspace, there is $r_0$ such that $s_0 \Rew{\wsym} r_0 \lRew{\wsym} u_0$.
			Thus, $s \Rew{\wsym} r_0t_1 \lRew{\wsym} u$.

			\item $s = t_0s_1  \lRew{\wsym} t = t_0t_1 \Rew{\wsym} t_0u_0 = u$ with $s_1 \lRew{\wsym} t_1 \Rew{\wsym} u_1$.
			Analogous to the previous case.
			\qedhere
		\end{itemize}
	\end{enumerate}
\end{proof}

\subsection{Omitted proofs  of \refsect{cbv-calculus} ({CbV}\xspace $\l$-calculus)}
\setcounter{lemmaAppendix}{\value{l:tobv-subs}}
\begin{lemmaAppendix}[Substitutivity of $\partobvind n$]
	\label{lappendix:tobv-subs}
	\NoteState{l:tobv-subs}
 If $t \partobvind n t'$ and $v \partobvind m v'$ then $t \isubxv \partobvindlong {n + \sizep{t'}x\cdot m} t' \isubx{v'}$.
\end{lemmaAppendix}

\begin{proof}
By induction on $t \partobind n t'$. It follows the exact same pattern of the proof of substitutivity of $\partobind n$. Cases:
\begin{itemize}
	\item \emph{Variable}: two sub-cases
	\begin{itemize}
	\item $t = x$: then $t = x \partobvind 0 x = t'$ then $t \isubxv = x\isubxv = v \partobvind m v' = x \isubx{v'} = t' \isubx{v'}$ that satisfies the statement because $n + \sizep{t'}x\cdot m = 0+1\cdot m = m$.
	 \item $t = y$: then $t = y \partobvind 0 y = t'$ then $t \isubxv = y\isubxv = y \partobvind 0 y = y \isubx{v'} = t' \isubx{v'}$ that satisfies the statement because $n + \sizep{t'}x\cdot m = n +0\cdot m=n$.
	 \end{itemize}

	\item \emph{Abstraction}, {\em i.e.}\xspace $t = \la{y}{u} \partobvind{n} \la{y}{u'} = t'$ because $u \partobvind{n} u'$; we can suppose without loss of generality that $y \neq x$, hence $\sizep{u'}{x} = \sizep{t'}{x}$ and $t\isub{x}{v} = \la{y}(u\isub{x}{v})$ and $t'\isub{x}{v'} = \la{y}(u'\isub{x}{v'})$.
		By {\textit{i.h.}}\xspace, $u\isub{x}{v} \partobvindlong{n +  \sizep{u'}{x} \cdot m} u'\isub{x}{v'}$, thus
		\begin{align*}
		\AxiomC{$u\isub{x}{v} \partobvindlong{n +  \sizep{u'}{x} \cdot m} u'\isub{x}{v'}$}

		\UnaryInfC{$t\isub{x}{v} = \la {y}u\isub{x}{v} \partobvindlong{n +  \sizep{t'}{x} \cdot m} \la{y}u'\isub{x}{v'} = t'\isub{x}{v'}$}
		\DisplayProof\,.
		\end{align*}

	\item \emph{Application}, {\em i.e.}\xspace
	\[\AxiomC{$u \partobvind {n_u} u'$}
	\AxiomC{$r \partobvind {n_r} r'$}

	\BinaryInfC{$t = u r \partobvindlong {n_u + n_r} u' r'= t'$}
	\DisplayProof\]
	with $n = n_u + n_r$. By {\textit{i.h.}}\xspace, $u\isubxv \partobvindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{v'}$ and $r\isubxv \partobvindlong {n_r +\sizep{r'}x \cdot m} r'\isubx{v'}$. Then
	\[\AxiomC{$u\isubxv \partobvindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{v'}$}
	\AxiomC{$r\isubxv \partobvindlong {n_r +\sizep{r'}x \cdot m} r'\isubx{v'}$}

	\BinaryInfC{$t\isubxv = u\isubxv r\isubxv \partobvindlong {n_u +\sizep{u'}x \cdot m +n_r +\sizep{r'}x \cdot m} u'\isubx{v'} r'\isubx{v'}= t'\isubx{v'}$}
	\DisplayProof\]
that proves the statement because $$n_u +\sizep{u'}x \cdot m +n_r +\sizep{r'}x \cdot m = n + (\sizep{u'}x + \sizep{r'}x)\cdot m = n + \sizep{t'}x \cdot m$$

\item $\beta$, {\em i.e.}\xspace
	\[\AxiomC{$u \partobvind {n_u} u'$}
	\AxiomC{$\val'' \partobvind {n_{\val''}} \val'''$}
\BinaryInfC{$t = (\lay u)\val'' \partobvindlong {n_u + \sizep{u'}y \cdot n_{\val''} +1} u'\isub y {\val'''} = t'$}
\DisplayProof\]
with $n = n_u + \sizep{u'}y \cdot n_{\val''} +1$. We can assume without loss of generality that $y\neq x$,
		hence, $\sizep{t'}{x} = \sizep{u'\isuby {\val'''}}x = \sizep{u'}{x} + \sizep{u'}{y}\cdot\sizep{\val'''}{x} $ and $t\isubxv = (\la\vartwou\isubxv) (\val''\isubxv)$ and $t'\isubx{v'} = u'\isubx{v'} \isuby{\val'''\isubx{v'}}$.

		By {\textit{i.h.}}\xspace, $u\isubxv \partobvindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{v'}$ and $\val''\isubxv \partobvindlong {n_{\val''} +\sizep{\val'''}x \cdot m} \val'''\isubx{v'}$. Then
	\[\AxiomC{$u\isubxv \partobvindlong {n_u +\sizep{u'}x \cdot m} u'\isubx{v'}$}
	\AxiomC{$\val''\isubxv \partobvindlong {n_{\val''} +\sizep{\val'''}x \cdot m} \val'''\isubx{v'}$}

	\BinaryInfC{$t\isubxv = (\la\vartwou\isubxv) (\val''\isubxv) \partobvindlong {n_u +\sizep{u'}x \cdot m +\sizep{u'}y\cdot(n_{\val''} +\sizep{\val'''}x \cdot m)+1} u'\isubx{v'} \isuby{\val'''\isubx{v'}}= t'\isubx{v'}$}
\DisplayProof\]

The number on the $\parRew{\betav}$ arrow satisfies
$$n_u +\sizep{u'}x \cdot m +\sizep{u'}y\cdot(n_{\val''} +\sizep{\val'''}x \cdot m)+1
=
\underbrace{n_u + \sizep{u'}y\cdot n_{\val''} + 1}_n + \underbrace{\sizep{u'}x \cdot m + \sizep{u'}y\cdot \sizep{\val'''}x \cdot m}_{\sizep{t'}x\cdot m}$$
\end{itemize}
\end{proof}

\subsection{Omitted proofs  of \refsect{weak-cbv} (weak {CbV}\xspace)}

\setcounter{propositionAppendix}{\value{prop:macro-cbv}}
\begin{propositionAppendix}[Weak {CbV}\xspace macro-step system]
	\label{propappendix:macro-cbv}
	\NoteState{prop:macro-cbv}
\begin{enumerate}
 \item \label{pappendix:macro-cbv-merge} \emph{Merge}: if $t \parRew{\neg \wsym}\cdot \Rew{\wsym} u$ then $t \parRew{\betav} u$.
 \item \label{pappendix:macro-cbv-indexed-split} \emph{Indexed split}: if $t \partobvind n s$ then $t \partonotws$, or $n>0$ and $t \Rew{\wsym} \cdot \partobvind {n-1} s$.
 \item \label{pappendix:macro-cbv-split} \emph{Split}: if $t \parRew{\betav} s$ then $t \Rew{\wsym}^* \cdot \partonotws$.
\end{enumerate}
That is, $(\Lambda, \set{\Rew{\wsym}, \Rew{\neg \wsym}})$ is a macro-step system with respect to $\parRew{\betav}$ and $\parRew{\neg \wsym}$.
\end{propositionAppendix}

\begin{proof}
\hfill
\begin{enumerate}
 \item \emph{Merge}: by induction on $t \partonotws$. Note that the cases in which $s=x$ or $s=\l x. \tmtwo}%{s'$ are not possible.  Hence $ s=r'p' $  and $t \partonotws$ is derived as follows
	\begin{align*}
	\AxiomC{$r \parRew{\neg \wsym} r'$}
	\AxiomC{$p \parRew{\neg \wsym} p'$}
	\BinaryInfC{$t = rp \parRew{\neg \wsym} r'p' =s$}
	\DisplayProof
	\end{align*}
\begin{enumerate}
\item If $ r' \Rew{\wsym} r''$ then $s = r'' p'$. The {\textit{i.h.}}\xspace gives $r \parRew{\betav} r''$, and $t \parRew{\betav} s$ is derived as follows (remember that $\parRew{\neg \wsym} \subseteq \parRew{\betav}$):
\begin{align*}
	\AxiomC{$r \parRew{\betav} r''$}
	\AxiomC{$p \parRew{\betav} p'$}
	\BinaryInfC{$t = rp \parRew{\betav} r''p' =s$}
	\DisplayProof
	\end{align*}

	\item If $ p' \Rew{\wsym} p''$ it is analogous to the previous case.

\item If 	$ s \Rew{\wsym} u$ by a top $\beta_\val$ step then $r' = \la x \tmsix}%{q'$. Now, by definition of $\parRew{\neg \wsym}$ the step $r \parRew{\neg \wsym} r'$ necessarily has the form $r = \la x \tmsix}%{q \parRew{\neg \wsym} \la x \tmsix}%{q' = r'$ for some $\tmsix}%{q$ such that $\tmsix}%{q\parRew{\betav} \tmsix}%{q'$. Then the hypothesis is $t = (\la x \tmsix}%{q) p \parRew{\neg \wsym} (\la x \tmsix}%{q') p' \Rew{\wsym} \tmsix}%{q'\isub{x}{p'} = u$ and  $t \parRew{\betav} s$ is derived as follows (remember that $\parRew{\neg \wsym} \subseteq \parRew{\betav}$):
\begin{align*}
\AxiomC{$\tmsix}%{q\parRew{\beta} \tmsix}%{q'$}
\AxiomC{$p \parRew{\beta} p'$}
\BinaryInfC{$t = (\la x \tmsix}%{q) p \parRew{\beta}  \tmsix}%{q'\isub{x}{p'} =s $}
\DisplayProof
\end{align*}
\end{enumerate}

\item \emph{Indexed split}: by induction on the definition of $t \partobvind n s$.
	We freely use the fact that if $t \partobvind n s$ then $t \parRew{\betav} s$. Cases:
	\begin{itemize}
		\item \emph{Variable}: $t = x \partobvind 0 x = s$.
		Then, $t = x \partonotwx = s$.

		\item \emph{Abstraction}: $t = \la \vart' \partobvind n  \la x s' = s$ because $t' \partobvind n s'$.
		Then,
		\begin{align*}
		\AxiomC{$t' \parRew{\betav} s'$}
		\UnaryInfC{$t = \la{x}t' \parRew{\neg \wsym} \la{x}s' = s$}
		\DisplayProof
		\,.
		\end{align*}

	\item \emph{Application}:
	\[\AxiomC{$r \partobvind {n_1} r'$}
	\AxiomC{$p \partobvind {n_2} p'$}
	\BinaryInfC{$t = r p \partobvindlong {n_1 + n_2} r' p' = s$}
	\DisplayProof\]
	with $n = n_1 + n_2$. There are only two cases:
	\begin{itemize}
		\item either $r p \parRew{\neg \wsym} r' p'$, and then the claim holds.
		\item or $r p \not\parRew{\neg \wsym} r' p'$, and then $r \not\parRew{\neg \wsym} r'$ or $p \not\parRew{\neg \wsym} p'$ (otherwise $r p \parRew{\neg \wsym} r' p'$).
		Suppose $r \not\parRew{\neg \wsym} r'$ (the other case is analogous).
		By {\textit{i.h.}}\xspace applied to $r \partobvind{n_1} r'$,
		$n_1>0$ and there is $r''$ such that $r \Rew{\wsym} r'' \partobvind {n_1-1} r'$.
		So, $t = r p \Rew{\wsym} r'' p$ and
		\[\AxiomC{$r'' \partobvindlong {n_1-1} r'$}
		\AxiomC{$p \partobvind {n_2} p'$}
		\BinaryInfC{$r'' p \partobvindlong {n_1-1 + n_2} r' p' = s$}
		\DisplayProof
		\ .
		\]
	\end{itemize}

		\item \emph{$\beta_\val$ step}:
		\[\AxiomC{$p \partobvind {n_1} p'$}
		\AxiomC{$r$ is a value}
		\AxiomC{$r \partobvind {n_2} r'$}
		\TrinaryInfC{$t = (\lax p)r \partobvindlong {n_1 + \sizep{p'}x \cdot n_2 +1} p'\isub x {r'} = s$}
		\DisplayProof\]
		with $n = n_1 + \sizep{p'}x \cdot n_2 +1 > 0$.
		We have $t = (\lax p)r \Rew{\wsym} p\isub\varr := u$ and substitutivity of $\partobvind n$ (\reflemma{tobv-subs}) gives $u = p\isub\varr \partobvindlong{n_1 + \sizep{p'}x \cdot n_2} p' \isubx{r'} = s$.
	\end{itemize}

	\item \emph{Split}: exactly as in the head case (\refpropp{macro-head}{split}), using the Indexed Split property for weak {CbV}\xspace (\refpoint{macro-cbv-indexed-split} above).
	\qedhere
	\end{enumerate}
\end{proof}
\newpage
\setcounter{propositionAppendix}{\value{essential-weak}}
\begin{propositionAppendix}[Weak {CbV}\xspace essential system]
	\label{propappendix:essential-weak}
	\NoteState{prop:essential-weak}
	\begin{enumerate}
		\item\label{pappendix:essential-weak-persistence} \emph{Persistence}: if $t \Rew{\wsym} s$ and $t \Rew{\neg \wsym} u$ then $u \Rew{\wsym} r$ for some $r$.
		\item\label{pappendix:essential-weak-diamond} \emph{Diamond}: if $s\lRew{\wsym} t \towu$ with $s \neq u$ then $s \Rew{\wsym} r \lRew{\wsym} u$ for some $r$.
	\end{enumerate}
	Then, $(\Lambda, \set{ \Rew{\wsym}, \Rew{\neg \wsym}})$ is an essential system.
\end{propositionAppendix}

\blue{\begin{proof}We prove that if $t \Rew{\neg \wsym} \tmtwo}%{s$ and $\tmtwo}%{s$ is $\Rew{\wsym}$-normal, then $t$ is $\Rew{\wsym}$-normal, by induction on the term $\tmtwo}%{s$. Cases.
	\begin{enumerate}
		\item $\tmtwo}%{s=x$ does not apply, because no $\Rew{\neg \wsym}$-step is possible.
		\item $\tmtwo}%{s=\la x \tmfour}%{r$. We have $t=\la x \tmfour}%{r \Rew{\neg \wsym} \la x \tmfour}%{r'=\tmtwo}%{s$,
		and $\tmfour}%{r \Rew{\neg \wsym}  \tmfour}%{r'$,  with $\tmfour}%{r'$ $\Rew{\wsym}$-normal.  By the {\textit{i.h.}}\xspace, $\tmfour}%{r $ is $\Rew{\wsym}$-normal, hence  so is $t=\la x \tmfour}%{r$.

	\item $\tmtwo}%{s= \tmsix}%{q\tmfour}%{r$. Then $t=\tmsix}%{q_1\tmfour}%{r_1$, with either $\tmsix}%{q_1\Rew{\neg \wsym}\tmsix}%{q$ (and $\tmfour}%{r_1=\tmfour}%{r$),
	or  $\tmfour}%{r_1\Rew{\neg \wsym}\tmfour}%{r$ (and $\tmsix}%{q_1=\tmsix}%{q$).
	\begin{enumerate}
		\item By the {\textit{i.h.}}\xspace., $\tmsix}%{q_1$ and $\tmfour}%{r_1$ are both $\Rew{\wsym}$-normal.
		\item  The $\beta$-rule does not apply, because   if $ \tmsix}%{q_1 $ is  an abstraction and $\tmfour}%{r_1$ is  a value, then $ \tmsix}%{q$ is  an abstraction and $\tmfour}%{r$ is  a value,
		against the assumption that $\tmtwo}%{s$ is $\Rew{\wsym}$-normal.
	\end{enumerate}
	Hence, no rule for $\Rew{\wsym}$ applies, and   $\tmfour}%{r=\tmsix}%{q_1\tmfour}%{r_1$ is $\Rew{\wsym}$-normal

	\end{enumerate}

\end{proof}}

\begin{proof}\hfill
	\begin{enumerate}
		\item \emph{Persistence}: By induction on the definition of $t \Rew{\neg \wsym} s$.
		Cases:
		\begin{itemize}
			\item \emph{Abstraction}: $t = \lax' \Rew{\neg \wsym} \lax''$ because $p \Rew{\betav} p'$. This case is impossible because $t$ is $\Rew{\wsym}$ normal, against the hypothesis that $t \Rew{\wsym} u$.

			\item \emph{Application left}: $t = pq \Rew{\neg \wsym} p_{\neg{\mathtt w}}q = s$ because $p \Rew{\neg \wsym} p_{\neg{\mathtt w}}$.
			According to the definition of $\Rew{\wsym} $, there are the following sub-cases:
			\begin{enumerate}
				\item $t = pq \Rew{\wsym} p_\wsymq = u$ with $p \Rew{\wsym} p_{\mathtt w}$;
				by {\textit{i.h.}}\xspace applied to $p$, we have $p_{\neg{\mathtt w}} \Rew{\wsym} r'$  for some term $r'$, and so
				\begin{align*}
				\AxiomC{$p_{\neg{\mathtt w}} \Rew{\wsym} r'$}
				\UnaryInfC{$s = p_{\neg{\mathtt w}}q \Rew{\wsym} r'q =: r$}
				\DisplayProof
				\,;
				\end{align*}
				\item $t = pq \Rew{\wsym} pq' = u$ with  $q \Rew{\wsym} q'$;
				hence
				\begin{align*}
				\AxiomC{$q \Rew{\wsym} q'$}
				\UnaryInfC{$s = p_{\neg{\mathtt w}}q \Rew{\wsym} p_{\neg{\mathtt w}}q' =`: r$}
				\DisplayProof
				\,;
				\end{align*}

				\item $t = (\la{x}{p'})q \Rew{\wsym} p'\isub{x}{q} = u$ where $p = \la{x}{p'}$ and $q$ is a value.
				According to the definition of $t \Rew{\neg \wsym} s$,
				\begin{align*}
				\AxiomC{$p \Rew{\betav} p_{\neg{\mathtt w}}$}
				\UnaryInfC{$\la{x}p' \Rew{\neg \wsym} \la{x}p_{\neg{\mathtt w}}'$}
				\UnaryInfC{$t = (\la{x}p')q \Rew{\neg \wsym} (\la{x}{p_{\neg{\mathtt w}}'})q = s$}
				\DisplayProof
				\end{align*}
				where $p_{\neg{\mathtt w}} = \la{x}{p_{\neg{\mathtt w}}'}$; therefore
				\begin{align*}
				\AxiomC{}
				\UnaryInfC{$s = (\la{x}p_{\neg{\mathtt w}}')q \Rew{\wsym} p_{\neg{\mathtt w}}'\isub{x}{q} =: r$}
				\DisplayProof
				\,.
				\end{align*}
			\end{enumerate}

			\item \emph{Application right}: analogous to the previous point.
		\end{itemize}

		\item \emph{Diamond}:
		The idea of the proof is that, when $(\la{x}t)v \Rew{\wsym} t\isub{x}{v}$, the $\beta_\val$-redexes in $v$ (which is a value) can be duplicated in $t\isub{x}{v}$ but they are under an abstraction and $\Rew{\wsym}$ does not reduce  under abstractions.

		Formally, the proof is by induction on $t$.
		Note that $t$ is not a value, because values are $\Rew{\wsym}$-normal, since $\Rew{\wsym}$ does not reduce under abstractions.
		Therefore, $t = t_0t_1$.
		The case where $t_0 = \la{x}{t'}$ and $t_1$ is a value is impossible, because $t_0$ and $t_1$ would be $\Rew{\wsym}$-normal and so from $s \lRew{\wsym} t \Rew{\wsym} u$ it would follow $s = t'\isub{x}{t_1} = u$, which contradicts the hypothesis.
		The remaining cases are:
		\begin{itemize}
			\item $s = t_0s_1 \lRew{\wsym} t = t_0t_1 \Rew{\wsym} u_0t_1 = u$ with $t_0 \Rew{\wsym} u_0$,  and $t_1 \Rew{\wsym}  s_1$.
			Then, $s \Rew{\wsym} u_0s_1 \lRew{\wsym} u$.

			\item $s = s_0 t_1 \lRew{\wsym} t = t_0t_1 \Rew{\wsym} u_0t_1 = u$ with $s_0 \lRew{\wsym} t_0 \Rew{\wsym} u_0$.
			By {\textit{i.h.}}\xspace, there is $r_0$ such that $s_0 \Rew{\wsym} r_0 \lRew{\wsym} u_0$.
			Thus, $s \Rew{\wsym} r_0t_1 \lRew{\wsym} u$.

			\item $s = t_0s_1  \lRew{\wsym} t = t_0t_1 \Rew{\wsym} t_0u_0 = u$ with $s_1 \lRew{\wsym} t_1 \Rew{\wsym} u_1$.
			Analogous to the previous case.
			\qedhere
		\end{itemize}
	\end{enumerate}
\end{proof}

\subsection{Omitted proofs  of \refsect{weak-cbv}}
\setcounter{lemmaAppendix}{\value{l:tobv-subs}}
\begin{lemmaAppendix}[Substitutivity of $\partobvind n$]
	\label{lappendix:tobv-subs}
	\NoteState{l:tobv-subs}
 If $t \partobvind n t'$ and $s \partobvind m s'$ where $s$ is a value,  then $t \isub\vars \partobvindlong {n + \sizep{t'}x\cdot m} t' \isubx{s'}$.
\end{lemmaAppendix}

\begin{proof}

		By induction on the definition of $t \partobvind{n} t'$. Cases:
	\begin{itemize}
		\item \emph{Variable}, {\em i.e.}\xspace $t = x \partobvind{0} x = t'$. Then, $\sizep{t'}x = 1$ and $t\isub{x}{s} = s \partobvind{m} s' = t'\isub{x}{s'}$, where $m = 0 + \sizep{t'}{x} \cdot m$.
		\item \emph{Abstraction}, {\em i.e.}\xspace $t = \la{y}{u} \partobvind{n} \la{y}{u'} = t'$ because $u \partobvind{n} u'$; we can suppose without loss of generality that $y \notin \fv{s} \cup \{x\} \supseteq \fv{s'} \cup \{x\}$, hence $\sizep{u'}{x} = \sizep{t'}{x}$ and $t\isub{x}{s} = \la{y}(u\isub{x}{s})$ and $t'\isub{x}{s'} = \la{y}(u'\isub{x}{s'})$.
		By {\textit{i.h.}}\xspace, $u\isub{x}{s} \partobvindlong{n +  \sizep{u'}{x} \cdot m} u'\isub{x}{s'}$, thus
		\begin{align*}
		\AxiomC{$u\isub{x}{s} \partobvindlong{n +  \sizep{u'}{x} \cdot m} u'\isub{x}{s'}$}

		\UnaryInfC{$t\isub{x}{s} = \la {y}u\isub{x}{s} \partobvindlong{n +  \sizep{t'}{x} \cdot m} \la{y}u'\isub{x}{s'} = t'\isub{x}{s'}$}
		\DisplayProof\,.
		\end{align*}
		\item \emph{Application}, {\em i.e.}\xspace $t = {u}r \partobvind{n_1 + n_2} {u'}r' = t'$ because $u \partobvind{n_1} u'$ and $r \partobvind{n_2} r'$.
		By {\textit{i.h.}}\xspace, $u\isub{x}{s} \partobvindlong{n_1 +  \sizep{u'}{x} \cdot m} u'\isub{x}{s'}$ and $r\isub{x}{s} \partobvindlong{n_2 +  \sizep{r'}{x} \cdot m} r'\isub{x}{s'}$, hence
		\begin{align*}
		\AxiomC{$u\isub{x}{s} \partobvindlong{n_1 +  \sizep{u'}{x} \cdot m} u'\isub{x}{s'}$}
		\AxiomC{$r\isub{x}{s} \partobvindlong{n_2 +  \sizep{r'}{x} \cdot m} r'\isub{x}{s'}$}

		\BinaryInfC{$t\isub{x}{s} = u\isub{x}{s} r\isub{x}{s} \partobvindlong{n +  \sizep{t'}{x} \cdot m} u'\isub{x}{s'} r'\isub{x}{s'} = t'\isub{x}{s'}$}
		\DisplayProof
		\end{align*}
		where $\sizep{t'}{x} = \sizep{u'}{x} + \sizep{r'}{x}$ and $n = n_1 + n_2$.

		\item \emph{$\beta_\val$ step}:
		\[\AxiomC{$u \partobvind {n_1} u'$}
		\AxiomC{$r$ is a value}
		\AxiomC{$r \partobvind {n_2} r'$}
		\TrinaryInfC{$t = (\lay u)r \partobvind{n} u'\isub y {r'} = t'$}
		\DisplayProof\]
		where $n = n_1 + \sizep{u'}{x} \cdot n_2 +1$.
		We can assume without loss of generality that $y \notin \fv{s} \cup \{x\} \supseteq \fv{s'} \cup \{x\}$;
		hence, $\sizep{t'}{x} = \sizep{u'}{x} + \sizep{u'}{y}\sizep{r'}{x} $ and $t\isub{x}{r} = (\la{y}{u}\isub{x}{s})r\isub{x}{s}$ and $t'\isub{x}{r'} = {u'}\isub{y}{r'}\isub{x}{s'} = u'\isub{y}{s'}\isub{x}{r'\isub{y}{s'}}$.
		By {\textit{i.h.}}\xspace, $u\isub{x}{s} \partobvindlong{n_1 + m \cdot \sizep{u'}{x}} u'\isub{x}{s'}$ and $r\isub{x}{s} \allowbreak\partobvindlong{n_2 + m \cdot \sizep{r'}{x}} r'\isub{x}{s'}$.
		Therefore, since $r\isub{x}{s}$ is a value,
		\[\AxiomC{$u\isub{x}{s} \partobvindlong{n_1 + m \cdot \sizep{u'}{x}} u'\isub{x}{s'}$}
		\AxiomC{$r\isub{x}{r} \partobvindlong{n_2 + m \cdot \sizep{r'}{x}} r'\isub{x}{s'}$}
		\BinaryInfC{$t \isub{x}{s} = (\lay u\isub{x}{s})r\isub{x}{s} \partobvindlong{n_1 + m \cdot \sizep{u'}{x} + (n_2 + m \cdot \sizep{r'}{x})\sizep{u'}{y} + 1} u'\isub y {r'}\isub{x}{s'} = t'\isub{x}{s'}$}
		\DisplayProof\]
		where $n_1 + m \cdot \sizep{u'}{x} + (n_2 + m \cdot \sizep{r'}{x})\sizep{u'}{y} + 1 = n + m(\sizep{u'}{x} + \sizep{u'}{y}\cdot \sizep{r'}{x}) = n + m \cdot \sizep{t'}{x}$.
	\end{itemize}

\end{proof}

\setcounter{propositionAppendix}{\value{prop:i-e-merge-cbv}}
\begin{propositionAppendix}[Merge]
\label{propappendix:i-e-merge-cbv}
\NoteState{prop:i-e-merge-cbv}
	If $t \partonotws \Rew{\wsym} u$ then $t \parRew{\betav} u$.
\end{propositionAppendix}

\begin{proof}
	By induction on the form of  $ s $.
	Cases $s=x$ or $s=\l x. \tmtwo}%{s'$ are not possible.  Hence $ s=r'p' $  and $t \partonotws$ is derived as follows
	\begin{align*}
	\AxiomC{$r \parRew{\neg \wsym} r'$}
	\AxiomC{$p \parRew{\neg \wsym} p'$}
	\BinaryInfC{$t = rp \parRew{\neg \wsym} r'p' $}
	\DisplayProof
	\end{align*}
\begin{enumerate}
	\item If 	$ s \Rew{\wsym} u$ by a $\beta_\val$-rule (let $r=\la x \tmsix}%{q'$, $p$ value) then $s= (\la x \tmsix}%{q') p' \Rew{\wsym} \tmsix}%{q'\isub{x}{p'} $ and  we conclude by the following derivation
\begin{align*}
\AxiomC{$\tmsix}%{q\parRew{\beta} \tmsix}%{q'$}
\AxiomC{$p \parRew{\beta} p'$}
\BinaryInfC{$  (\la x \tmsix}%{q) p \parRew{\beta}  \tmsix}%{q'\isub{x}{p'}  $}
\DisplayProof
\end{align*}

\item Otherwise, either $ r' \Rew{\wsym} r''$ or $p' \Rew{\wsym} p''$, and  we apply the {\textit{i.h.}}\xspace\ to the concerned premiss, concluding by application rule.
\end{enumerate}
\end{proof}

\setcounter{propositionAppendix}{\value{prop:quant-out-in-split-cbv}}
\begin{propositionAppendix}[Indexed split]
	\label{propappendix:quant-out-in-split-cbv}
	\NoteState{propappendix:quant-out-in-split-cbv}
	If $t \partobvind n s$ then either $t \partonotws$ or $n>0$ and there exists $u$ such that $t \Rew{\wsym} u \partobvind {n-1} s$.
\end{propositionAppendix}

\begin{proof}
	By induction on the definition of $t \partobvind n s$.
	We use freely the fact that if $t \partobvind n s$ then $t \parRew{\betav} s$. Cases:
	\begin{itemize}
		\item \emph{Variable}: $t = x \partobvind 0 x = s$. Then $t = x \partonotwx = s$.

		\item \emph{Abstraction}: $t = \la \vart' \partobvind n  \la x s' = s$ because $t' \partobvind n s'$.
		Then,
		\begin{align*}
		\AxiomC{$t' \parRew{\betav} s'$}
		\UnaryInfC{$t = \la{x}t' \parRew{\neg \wsym} \la{x}s' = s$}
		\DisplayProof
		\end{align*}

		\item \emph{Application}:
		\[\AxiomC{$r \partobvind {n} r'$}
		\AxiomC{$p \partobvind {m} p'$}
		\BinaryInfC{$t = r p \partobvindlong {n + m} r' p' = s$}
		\DisplayProof\]
		By {\textit{i.h.}}\xspace applied to $r \partobvind{n} r'$ and $p \partobvind{m} p'$, there are three sub-cases:
		\begin{enumerate}
			\item either $r \parRew{\neg \wsym} r'$ and $p \parRew{\neg \wsym} p'$, and then
			\[\AxiomC{$r \parRew{\neg \wsym} r'$}
			\AxiomC{$p \parRew{\neg \wsym} p'$}
			\BinaryInfC{$t = r p \parRew{\neg \wsym} r' p' = s$}
			\DisplayProof\,;\]
			\item or $n > 0$ and there exists $r''$ such that $r \Rew{\wsym} r'' \partobvind {n-1} r'$;
			then, $t = r p \Rew{\wsym} r'' p := u$ and
			\[\AxiomC{$r'' \partobvindlong {n-1} r'$}
			\AxiomC{$p \partobvind {m} p'$}
			\BinaryInfC{$u = r'' p \partobvindlong {n-1 + m} r' p' = s$}
			\DisplayProof \,;\]
			\item or $m > 0$ and there exists $p''$ such that $p \Rew{\wsym} p'' \partobvind {m-1} p'$;
			then, $t = r p \Rew{\wsym} r p'' := u$ and
			\[\AxiomC{$r \partobvindlong {n} r'$}
			\AxiomC{$p'' \partobvind {m-1} p'$}
			\BinaryInfC{$u = rp'' \partobvindlong {n + m - 1} r' p' = s$}
			\DisplayProof \,.\]
		\end{enumerate}

		\item \emph{$\beta_\val$ step}:
		\[\AxiomC{$p \partobvind {n_1} p'$}
		\AxiomC{$r$ is a value}
		\AxiomC{$r \partobvind {n_2} r'$}
		\TrinaryInfC{$t = (\lax p)r \partobvindlong {n_1 + \sizep{p'}x \cdot n_2 +1} p'\isub x {r'} = s$}
		\DisplayProof\]
		with $n = n_1 + \sizep{p'}x \cdot n_2 +1 > 0$. We have $t = (\lax p)r \Rew{\wsym} p\isub\varr := u$ and by \reflemmap{tobv-subs}{ind-par} $u = p\isub\varr \partobvindlong{n_1 + \sizep{p'}x \cdot n_2} p' \isubx{r'} = s$.
		\qedhere
	\end{itemize}
\end{proof}

\blue{\paragraph{On the definition of Internal Parallel Step.}
	A parallel steps  is internal, written
	$M\parRew{\neg \wsym} N$,  if
	$M\parRew{} N$ and $M\Rew{\neg \wsym}^* N$ (obviously,  $\parRew{\neg \wsym} \,\subseteq\, \parRew{}$).  We give explicitly the inference rules. We prove that the inference rules characterize the definition of parallel internal step.

\begin{lemma}\label{l:basicpar}If $t \parRew{\betav} \tmtwo}%{s$ then $t \to^* \tmtwo}%{s$. Moreover, if $\parRew{\betav}$ is obtained via the $\parallel\beta_\val$-rule, then $t \Rew{\betav}^*t'\Rew{\betav} \tmtwo}%{s$ where  the last step is a $\beta_\val$-rule.
\end{lemma}
\begin{proof}
	Induction on the derivation of $t \parRew{} \tmtwo}%{s$
\end{proof}

\begin{lemma} The following are equivalent:
	\begin{enumerate}
		\item $t \parRew{\betav} \tmtwo}%{s$ and $t \Rew{\neg \wsym}^*\tmtwo}%{s$
		\item $t \parRew{\neg \wsym}\tmtwo}%{s$
	\end{enumerate}
\end{lemma}

\begin{proof}
\begin{itemize}
	\item $1\rightarrow 2$ by induction on $t$. We observe that if $t=\tmfour}%{r\tmtwo}%{s$,  $t \parRew{\betav} \tmtwo}%{s$ can only be conclusion of a
	$\parallel$App-rule, and not of a $\parallel$$\beta_\val$-rule, because $t \Rew{\neg \wsym}^*\tmtwo}%{s$ and \reflemma{basicpar}.
	\item $2\rightarrow 1$ is immediate
\end{itemize}
\end{proof}}
\section{Technical appendix: omitted proofs and lemmas}
\label{sect:proofs}

The enumeration of propositions, theorems, lemmas already stated in the body of the article is unchanged.

\input{./proofs/proofs-head}

\input{./proofs/proofs-weak-cbv}

\input{./proofs/proofs-leftmost}

\input{./proofs/proofs-stratified-cbn2}

redpajama_set_name: RedPajamaArXiv