Q&\mapsto&Q^{\prime}\end{array}\right.$

The induction hypothesis give that the order type of $P^{\prime}(y)$ is less
than $\omega^{\omega^{\omega(\operatorname{NR}(y)+1)}}$. Then

$\omega^{\omega^{\omega\operatorname{NR}(x)}}\leq\mu<\omega^{\omega^{\omega(\operatorname{NR}(y)+1)}}\leq\omega^{\omega^{\omega\operatorname{NR}(x)}}$

and we get the contradiction.

This completes the proof.

∎

######  Corollary 6.2.

If $\operatorname{NR}(x)<\lambda$ then $\nu(\partial x)<\lambda$

######  Proposition 6.2.

For all $x\in\textnormal{No}$, let $\alpha$ the minimum ordinal such that
$\kappa_{-\alpha}\prec^{K}t$ for all log-atomic $t$ such that there is some
path $P\in\mathcal{P}_{\mathbb{L}}(x)$ and some index $k\in\mathbb{N}$ such
that $P(k)=t$. Then, for all path $P$,

$\operatorname{NR}(\partial P)\leq k(\operatorname{NR}(x)+1)+\omega(\alpha+1)$

and $\operatorname{NR}(\partial
x)\leq\omega(\operatorname{NR}(x)+\alpha+2)\otimes\nu(\partial
x)\leq\omega^{\omega^{\omega(\operatorname{NR}(x)+1)}+\alpha}$

###### Proof.

Let $P$ be a path of such that $\partial P\neq 0$. Then there is some
$k\in\mathbb{N}$ such that $\partial P=P(0)\cdots
P(k-1)\partial_{\mathbb{L}}P(k)$. With Corollary 6.2.2, we get

| $\operatorname{NR}(\partial P)$ | $\ \leq\ $ | $\underset{i=0}{\overset{k-1}{\sum}}\operatorname{NR}(P(i))+\operatorname{NR}(\partial_{\mathbb{L}}P(k))+k$ |
---|---|---|---|---
|  | $\ \leq\ $ | $k\operatorname{NR}(x)+\operatorname{NR}(\partial_{\mathbb{L}}P(k))+k$ | (Lemma 6.3)
|  | $\ \leq\ $ | $k\operatorname{NR}(x)+k+\operatorname{NR}\left(\exp\left(-\underset{\kappa_{-\beta}\succeq^{K}P(k)}{\overset{}{\sum}}\underset{n\geq 1}{\overset{}{\sum}}\ln_{n}\kappa_{-\beta}+\underset{n\geq 1}{\overset{}{\sum}}\ln_{n}P(k)\right)\right)$ |
|  | $\ \leq\ $ | $k\operatorname{NR}(x)+k+\operatorname{NR}\left(-\underset{\kappa_{-\beta}\succeq^{K}P(k)}{\overset{}{\sum}}\underset{n\geq 1}{\overset{}{\sum}}\ln_{n}\kappa_{-\beta}+\underset{n\geq 1}{\overset{}{\sum}}\ln_{n}P(k)\right)$ |
|  |  |  | (Proposition 6.2.2)
|  | $\ \leq\ $ | $k\operatorname{NR}(x)+k+(\omega\otimes(\alpha\oplus 1))$ | (Lemma 6.2.2)
|  | $\ \leq\ $ | $\omega(\operatorname{NR}(x)+1)+\omega(\alpha+1)$ |

This bound does not depend on $P$. Then applying Proposition 1 and Lemma 6.2.2
we get

| $\operatorname{NR}(\partial x)$ | $\ \leq\ $ | $(\omega(\operatorname{NR}(x)+\alpha+2))\otimes\nu(\partial x)$
---|---|---|---
|  | $\ <\ $ | $(\omega(\operatorname{NR}(x)+\alpha+2))\otimes\omega^{\omega^{\omega(\operatorname{NR}(x)+1)}}$
|  | $\ \leq\ $ | $\omega^{\omega^{\omega(\operatorname{NR}(x)+1)}+\alpha}$

∎

###  6.5 Anti-derivative of a surreal number

Berarducci and Mantova provided a derivation, $\partial$. An other strong
property of this derivation is that is as a compositional inverse, an anti-
derivation. The first thing to prove it is to prove that there is an
asymptotic anti derivation.

######  Proposition 6.2 ([BM18b, Berarducci and Mantova, Proposition 7.4]).

There is a class function
$A:\textnormal{No}^{*}\to\mathbb{R}\omega^{\textnormal{No}^{*}}$ such that

$\forall x\in\textnormal{No}^{*}\qquad x\sim\partial A(x)$

Basically, $A(x)$ is the leading term of $x\mathchoice{\dfrac{xu/\partial
u}{\partial(xu/\partial u)}}{\dfrac{xu/\partial u}{\partial(xu/\partial
u)}}{\frac{xu/\partial u}{\partial(xu/\partial u)}}{\frac{xu/\partial
u}{\partial(xu/\partial u)}}$ where $u=\kappa_{\alpha}$ for $\alpha$ a
sufficiently large ordinal. Actually we can be even more precise and give a
more explicit formula for Berarducci and Mantova’s asymptotic anti-derivation
[BM18b].

######  Lemma 6.2.

Let $u=\ln_{n}\kappa_{-\alpha}$ for some $n\in\mathbb{N}$ and some ordinal
$\alpha$. Let $x=\partial u\exp\varepsilon$. If $\varepsilon\succ\ln u$, then

$\partial\left(\mathchoice{\dfrac{x}{\partial\varepsilon}}{\dfrac{x}{\partial\varepsilon}}{\frac{x}{\partial\varepsilon}}{\frac{x}{\partial\varepsilon}}\right)\sim
x$

###### Proof.

Let
$y=\mathchoice{\dfrac{x}{\partial\varepsilon}}{\dfrac{x}{\partial\varepsilon}}{\frac{x}{\partial\varepsilon}}{\frac{x}{\partial\varepsilon}}=\mathchoice{\dfrac{\partial
u}{\partial\varepsilon}}{\dfrac{\partial
u}{\partial\varepsilon}}{\frac{\partial
u}{\partial\varepsilon}}{\frac{\partial
u}{\partial\varepsilon}}\exp(\varepsilon)$. Since $\varepsilon\succ\ln u$,
Proposition 6.3 ensures that
$\partial\varepsilon\succ\mathchoice{\dfrac{\partial u}{u}}{\dfrac{\partial
u}{u}}{\frac{\partial u}{u}}{\frac{\partial u}{u}}$. Then,
$\mathchoice{\dfrac{\partial u}{\partial\varepsilon}}{\dfrac{\partial
u}{\partial\varepsilon}}{\frac{\partial
u}{\partial\varepsilon}}{\frac{\partial u}{\partial\varepsilon}}\prec
u\not\asymp 1$

| $\partial y$ | $\ =\ $ | $\mathchoice{\dfrac{\partial u}{\partial\varepsilon}}{\dfrac{\partial u}{\partial\varepsilon}}{\frac{\partial u}{\partial\varepsilon}}{\frac{\partial u}{\partial\varepsilon}}\partial\varepsilon\exp(\varepsilon)+\partial\left(\mathchoice{\dfrac{\partial u}{\partial\varepsilon}}{\dfrac{\partial u}{\partial\varepsilon}}{\frac{\partial u}{\partial\varepsilon}}{\frac{\partial u}{\partial\varepsilon}}\right)\exp(\varepsilon)$
---|---|---|---
|  | $\ =\ $ | $x+\partial\left(\mathchoice{\dfrac{\partial u}{\partial\varepsilon}}{\dfrac{\partial u}{\partial\varepsilon}}{\frac{\partial u}{\partial\varepsilon}}{\frac{\partial u}{\partial\varepsilon}}\right)\exp(\varepsilon)$

Proposition 6.3 gives that $\partial\left(\mathchoice{\dfrac{\partial
u}{\partial\varepsilon}}{\dfrac{\partial
u}{\partial\varepsilon}}{\frac{\partial
u}{\partial\varepsilon}}{\frac{\partial
u}{\partial\varepsilon}}\right)\prec\partial u$. Then

$\partial y\sim x$

∎

######  Lemma 6.2.

Let $u=\ln_{n}\kappa_{-\alpha}$ for some $n\in\mathbb{N}$ and some ordinal
$\alpha$. Let $x=\partial u\exp(\varepsilon)$. If $\varepsilon\sim r\ln u$ for
some $r\in\mathbb{R}\setminus\\{0,-1\\}$, then

$\partial\left(\mathchoice{\dfrac{1}{r+1}}{\dfrac{1}{r+1}}{\frac{1}{r+1}}{\frac{1}{r+1}}\mathchoice{\dfrac{ux}{\partial
u}}{\dfrac{ux}{\partial u}}{\frac{ux}{\partial u}}{\frac{ux}{\partial
u}}\right)\sim x$

###### Proof.

Let us compute the above derivative.

$\partial\left(\mathchoice{\dfrac{1}{r+1}}{\dfrac{1}{r+1}}{\frac{1}{r+1}}{\frac{1}{r+1}}\mathchoice{\dfrac{ux}{\partial
u}}{\dfrac{ux}{\partial u}}{\frac{ux}{\partial u}}{\frac{ux}{\partial
u}}\right)=\partial\left(\mathchoice{\dfrac{u\exp(\varepsilon)}{r+1}}{\dfrac{u\exp(\varepsilon)}{r+1}}{\frac{u\exp(\varepsilon)}{r+1}}{\frac{u\exp(\varepsilon)}{r+1}}\right)=\mathchoice{\dfrac{x}{r+1}}{\dfrac{x}{r+1}}{\frac{x}{r+1}}{\frac{x}{r+1}}+\mathchoice{\dfrac{u\partial\varepsilon\exp(\varepsilon)}{r+1}}{\dfrac{u\partial\varepsilon\exp(\varepsilon)}{r+1}}{\frac{u\partial\varepsilon\exp(\varepsilon)}{r+1}}{\frac{u\partial\varepsilon\exp(\varepsilon)}{r+1}}$

Using Proposition 6.3, we get that $\partial\varepsilon\sim\partial(r\ln
u)=r\mathchoice{\dfrac{\partial u}{u}}{\dfrac{\partial u}{u}}{\frac{\partial
u}{u}}{\frac{\partial u}{u}}$. Then, since $r\neq-1$, we get that

$\partial\left(\mathchoice{\dfrac{1}{r+1}}{\dfrac{1}{r+1}}{\frac{1}{r+1}}{\frac{1}{r+1}}\mathchoice{\dfrac{ux}{\partial
u}}{\dfrac{ux}{\partial u}}{\frac{ux}{\partial u}}{\frac{ux}{\partial
u}}\right)\sim x$

∎

######  Lemma 6.2.

Let $u=\ln_{n}\kappa_{-\alpha}$ for some $n\in\mathbb{N}$ and some ordinal
$\alpha$. Let $x=\partial u\exp(\varepsilon)$. If $\varepsilon\prec\ln u$,
then

$\partial\left(\mathchoice{\dfrac{ux}{\partial u}}{\dfrac{ux}{\partial
u}}{\frac{ux}{\partial u}}{\frac{ux}{\partial u}}\right)\sim x$

###### Proof.

Let us compute the above derivative.

$\partial\left(\mathchoice{\dfrac{ux}{\partial u}}{\dfrac{ux}{\partial
u}}{\frac{ux}{\partial u}}{\frac{ux}{\partial
u}}\right)=\partial\left(u\exp(\varepsilon)\right)=x+u\partial\varepsilon\exp(\varepsilon)$

Using Proposition 6.3, we get that $\partial\varepsilon\prec\partial\ln
u=\mathchoice{\dfrac{\partial u}{u}}{\dfrac{\partial u}{u}}{\frac{\partial
u}{u}}{\frac{\partial u}{u}}$. Then,
$u\partial\varepsilon\exp(\varepsilon)\prec x$ and we get that

$\partial\left(\mathchoice{\dfrac{ux}{\partial u}}{\dfrac{ux}{\partial
u}}{\frac{ux}{\partial u}}{\frac{ux}{\partial u}}\right)\sim x$

∎

######  Theorem 6.3.

Let $x$ be a term. Write $|x|=\partial u\exp(\varepsilon)$ with
$u=\ln_{n}\kappa_{-\alpha}=\lambda_{-\omega\alpha-n}$ with $\omega\alpha+n$
such minimal that $\varepsilon\not\sim-\ln u$. Then,

$A(x)\sim\left\\{\begin{array}[]{@{}r@{\quad}l@{}}\mathchoice{\dfrac{x}{\partial\varepsilon}}{\dfrac{x}{\partial\varepsilon}}{\frac{x}{\partial\varepsilon}}{\frac{x}{\partial\varepsilon}}&\varepsilon\succ\ln
u\\\ \mathchoice{\dfrac{ux}{(r+1)\partial u}}{\dfrac{ux}{(r+1)\partial
u}}{\frac{ux}{(r+1)\partial u}}{\frac{ux}{(r+1)\partial u}}&\varepsilon\sim
r\ln u\quad r\neq 0,-1\\\ \mathchoice{\dfrac{ux}{\partial
u}}{\dfrac{ux}{\partial u}}{\frac{ux}{\partial u}}{\frac{ux}{\partial
u}}&\varepsilon\prec\ln u\end{array}\right.$

In this theorem, the quantities $\kappa_{a}$ and $\lambda_{a}$ are defined in
Definitions 6.1 and 6.1.

###### Proof.

Since $A(x)=-A(-x)$, we may assume that $x>0$. Then, we just need to apply
Lemmas 6.5, 6.5, and 6.5. ∎

######  Corollary 6.3.

Let $x$ be a non-zero surreal number. Write $|x|=\partial u\exp(\varepsilon)$
with $u=\ln_{n}\kappa_{-\alpha}=\lambda_{-\omega\alpha-n}$ with
$\omega\alpha+n$ such minimal that $\varepsilon\not\sim-\ln u$. Then,

$A(x)=\left\\{\begin{array}[]{@{}r@{\quad}l@{}}\mathchoice{\dfrac{t}{s}}{\dfrac{t}{s}}{\frac{t}{s}}{\frac{t}{s}}&\varepsilon\succ\ln
u\\\\[11.38092pt] \mathchoice{\dfrac{ut}{(r+1)\partial
u}}{\dfrac{ut}{(r+1)\partial u}}{\frac{ut}{(r+1)\partial
u}}{\frac{ut}{(r+1)\partial u}}&\varepsilon=r\ln u+\eta\quad
r\neq-1,\eta\prec\ln u\end{array}\right.$

where $t$ is the leading term of $x$ and $s$ the leading term of
$\partial\varepsilon$.

###### Proof.

Just use Theorem 6.3 and the definition of $A$. ∎

We are now ready to build the anti-derivation for surreal numbers. We start
with a useful lemma due to Aschenbrenner, van den Dries and van der Hoeven. We
give it in a form that matches our notations.

######  Definition 6.3.

A function $\Phi$ is strongly linear is for all summable family
$\left\\{x_{i}\right\\}_{i\in I}$,

$\Phi\left(\underset{i\in I}{\overset{}{\sum}}x_{i}\right)=\underset{i\in
I}{\overset{}{\sum}}\Phi(x_{i})$

######  Lemma 6.3 ([AvdDvdH05, Aschenbrenner, van den Dries, van der Hoeven,
Corollary 1.4]).

Let $\Phi$ a strongly linear map defined over a field $\mathbb{K}$ of surreal
numbers. Assume that for any monomial $\omega^{a}\in\mathbb{K}$, we have
$\Phi(\omega^{a})\prec\omega^{a}$. Then
$\underset{n\in\mathbb{N}}{\overset{}{\sum}}\Phi^{n}(x)$ makes sense as a
surreal number (i.e $\left\\{\Phi^{n}(x)\right\\}_{n\in\mathbb{N}}$ is
summable) and if it belongs to $\mathbb{K}$ for all $x$, we have

$(\operatorname{id}-\Phi)^{-1}=\underset{n\in\mathbb{N}}{\overset{}{\sum}}\Phi^{n}$

######  Definition 6.3.

We define an extension of $A$, denoted $\mathcal{A}$, to all surreal numbers
by

$\mathcal{A}\left(\underset{i<\nu}{\overset{}{\sum}}r_{i}\omega^{a_{i}}\right)=\underset{i<\nu}{\overset{}{\sum}}r_{i}A(\omega^{a_{i}})$

We also introduce the function
$\Phi=\operatorname{id}-\partial\circ\mathcal{A}$.

Proposition 6.3 ensures that the function $\mathcal{A}$ is well defined.
Moreover, this function is obviously strongly linear. We now consider, given a
surreal number $x$, the sequence

$\left\\{\begin{array}[]{@{}r@{\quad}l@{}}&x_{0}=x\\\
&x_{n+1}=x_{n}-\partial\mathcal{A}(x_{n})=\Phi(x_{n})\end{array}\right.$

Note that if $\omega^{a}=\partial u\exp\varepsilon$ with
$u=\lambda_{-\omega\otimes\alpha-n}=\ln_{n}\kappa_{-\alpha}$ and
$\varepsilon\prec\ln\lambda_{-\omega\otimes\beta-m}$ for
$\omega\otimes\beta+m<\omega\otimes\alpha+n$, and $\omega\otimes\alpha+n$
maximum for that property, we have

$\Phi(\omega^{a})=\left\\{\begin{array}[]{@{}r@{\quad}l@{}}\left(1-\mathchoice{\dfrac{\partial\varepsilon}{s}}{\dfrac{\partial\varepsilon}{s}}{\frac{\partial\varepsilon}{s}}{\frac{\partial\varepsilon}{s}}\right)\omega^{a}-\partial\left(\mathchoice{\dfrac{\partial
u}{s}}{\dfrac{\partial u}{s}}{\frac{\partial u}{s}}{\frac{\partial
u}{s}}\right)\exp\varepsilon&\varepsilon\succ\ln u\quad s\text{ dominant term
of }\partial\varepsilon\\\
\mathchoice{\dfrac{\omega^{a}}{r+1}}{\dfrac{\omega^{a}}{r+1}}{\frac{\omega^{a}}{r+1}}{\frac{\omega^{a}}{r+1}}\partial\eta\mathchoice{\dfrac{u}{\partial
u}}{\dfrac{u}{\partial u}}{\frac{u}{\partial u}}{\frac{u}{\partial
u}}&\varepsilon=r\ln u+\eta\quad r\neq-1\end{array}\right.$

######  Corollary 6.3.

The operator $\operatorname{id}-\Phi$ is invertible with inverse
$\underset{i\in\mathbb{N}}{\overset{}{\sum}}\Phi^{i}$. Moreover
$\mathcal{A}\circ\underset{i\in\mathbb{N}}{\overset{}{\sum}}\Phi^{i}$ is an
operator that sends every $x$ to some anti-derivative of $x$.

###### Proof.

Lemma 6.5 ensure that $\operatorname{id}-\Phi$ has a inverse expressed by
$\underset{i\in\mathbb{N}}{\overset{}{\sum}}\Phi^{i}$. We also have that
$\operatorname{id}-\Phi=\partial\circ\mathcal{A}$. Then,

$\partial\circ\left(\mathcal{A}\circ\underset{i\in\mathbb{N}}{\overset{}{\sum}}\Phi^{i}\right)=(\partial\circ\mathcal{A})\circ\left(\partial\circ\mathcal{A}\right)^{-1}=\operatorname{id}$

In particular, for all $x$,
$\left(\mathcal{A}\circ\underset{i\in\mathbb{N}}{\overset{}{\sum}}\Phi^{i}\right)(x)$
is a anti-derivative of $x$. ∎

##  7 Field stable under exponential, logarithm, derivation and anti-
derivation

###  7.1 Length of the series of a derivative of a monomial

####  7.1.1 Case $\varepsilon\preceq\ln u$

######  Lemma 7.0.

Assume $x=\omega^{a}=\partial u\exp\varepsilon$ with $\varepsilon=r\ln u+\eta$
and $u=\ln_{n}\kappa_{-\alpha}$. Let
$b\in\operatorname{supp}\Phi(\omega^{a})$. Then, there is a path
$P\in\mathcal{P}_{\mathbb{L}}(\eta)$ such that

| $\omega^{b}$ | $\ \asymp\ $ | $\partial u\exp\left(r\ln u+\eta-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}\kappa_{-\alpha}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\right)$
---|---|---|---

###### Proof.

It is just a calculation. First notice that
$\mathchoice{\dfrac{\omega^{a}}{r+1}}{\dfrac{\omega^{a}}{r+1}}{\frac{\omega^{a}}{r+1}}{\frac{\omega^{a}}{r+1}}\mathchoice{\dfrac{u}{\partial
u}}{\dfrac{u}{\partial u}}{\frac{u}{\partial u}}{\frac{u}{\partial u}}$ is a
term as a product of terms. Then, let
$b\in\operatorname{supp}\Phi(\omega^{a})$. There is path $P$ of $\eta$ such
that

$\omega^{b}\asymp\omega^{a}\mathchoice{\dfrac{u}{\partial
u}}{\dfrac{u}{\partial u}}{\frac{u}{\partial u}}{\frac{u}{\partial u}}\partial
P=u\partial P\exp(r\ln u+\eta)$

write $\partial P=P(0)\cdots P(k_{P}-1)\partial_{\mathbb{L}}P(k_{P})$

Since $P(0)$ is a term of $\eta\prec\ln u$, we also have $P(0)\prec\ln u$.
Moreover since $\eta$ consists in purely infinite term, so is $P(0)$ and then
$\ln|P(0)|\prec P(0)$. Since $P(1)$ is a purely infinite term of $\ln|P(0)|$,
we get that $P(1)\prec P(0)$. By induction, for all $i$, $P(i+1)\preceq
P(i)\leq P(0)$. In particular, $P(k_{P})\preceq
P(0)\preceq^{k}\kappa_{-\alpha}$. Then, $\kappa_{-\alpha}\succeq^{K}P(k_{P})$.
That leads to

| $\partial_{\mathbb{L}}(P(k_{P}))$ | $\ =\ $ | $\exp\left(-\underset{\beta\leq\alpha,\ m\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{m=1}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}P(k_{P})\right)$
---|---|---|---
| $\partial_{\mathbb{L}}(P(k_{P}))$ | $\ =\ $ | $\partial u\exp\left(-\underset{m=n+1}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}{\kappa_{-\alpha}}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{m=1}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}P(k_{P})\right)$
Since $P(k_{P})\in\mathbb{L}$, |  |  |
| $\partial_{\mathbb{L}}(P(k_{P}))$ | $\ =\ $ | $\partial u\exp\left(-\underset{m=n+1}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}{\kappa_{-\alpha}}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=k_{P}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\right)$
Then | $\partial P$ | $\ =\ $ | $\partial u\exp\left(-\underset{m=n+1}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}{\kappa_{-\alpha}}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\right)$

Finally,

| $\omega^{b}$ | $\ \asymp\ $ | $\partial u\exp(r\ln u+\eta)u$
---|---|---|---
|  |  | $\qquad\times\exp\left(-\underset{m=n+1}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}{\kappa_{-\alpha}}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\right)$
| $\omega^{b}$ | $\ \asymp\ $ | $\partial u\exp\left(r\ln u+\eta-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}{\kappa_{-\alpha}}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\right)$

∎

######  Proposition 7.0.

Assume $x=\omega^{a}=\partial u\exp\varepsilon$ with $\varepsilon=r\ln u+\eta$
and $u=\ln_{n}\kappa_{-\alpha}$. We denote for
$P_{0},\dots,P_{k}\in\mathcal{P}_{\mathbb{L}}(\eta)$ and
$i_{1},\dots,i_{k}\in\mathbb{N}^{*}$,

| $e\left(\begin{array}[]{c}P_{0},\dots,P_{k}\\\ i_{1},\dots,i_{k}\end{array}\right)$ | $\ =\ $ | $-(k+1)\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}{\kappa_{-\alpha}}$
---|---|---|---
|  |  | $\qquad-\underset{j=0}{\overset{k}{\sum}}\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P_{j}(k_{P_{j}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{j=0}{\overset{k}{\sum}}\underset{i=i_{j}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{j}(i)|$

with $i_{0}=0$. For $k\in\mathbb{N}$ define $E_{1,k}$ by:

| $e\left(\begin{array}[]{c}P_{0},\dots,P_{k}\\\ i_{1},\dots,i_{k}\end{array}\right)\in E_{1,k}$ | $\ \Leftrightarrow\ $ | $P_{0},\dots,P_{k}\in\mathcal{P}_{\mathbb{L}}(\eta)\quad\wedge\quad i_{1},\dots,i_{k}\in\mathbb{N}^{*}$
---|---|---|---
|  |  | $\quad\wedge\quad\forall j\in\left\llbracket\,1\ ;\ k\,\right\rrbracket\quad\exists j^{\prime}\in\left\llbracket\,0\ ;\ j-1\,\right\rrbracket$
|  |  | $\qquad\qquad\forall i\in\left\llbracket\,0\ ;\ i_{j}-1\,\right\rrbracket\qquad P_{j^{\prime}}(i)=P_{j}(i)$
|  |  | $\quad\wedge\quad\forall j\in\left\llbracket\,1\ ;\ k\,\right\rrbracket$
|  |  | $\qquad\quad\operatorname{supp}P_{j}(i_{j})\subseteq\operatorname{supp}e\left(\begin{array}[]{c}P_{0},\dots,P_{k-1}\\\ i_{1},\dots,i_{k-1}\end{array}\right)\par$

| $E_{1}$ | $\ =\ $ | $\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k}$
---|---|---|---
| $E_{2}$ | $\ =\ $ | $\left\\{\left.\begin{array}[]{c}-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}\kappa_{-\alpha}\\\ -\underset{\gamma<\beta,\ m\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{m}\kappa_{-\gamma}-\underset{m=1}{\overset{p}{\sum}}\ln_{m}\kappa_{-\beta}\end{array}\ \vphantom{\begin{array}[]{c}\beta>\alpha\\\ \exists P\in\mathcal{P}_{\mathbb{L}}(\eta)\quad\kappa_{-\beta}\succeq^{K}P(k_{P})\\\ p\in\mathbb{N}\end{array}}\right|\ \begin{array}[]{c}\beta>\alpha\\\ \exists P\in\mathcal{P}_{\mathbb{L}}(\eta)\quad\kappa_{-\beta}\succeq^{K}P(k_{P})\\\ p\in\mathbb{N}\end{array}\right\\}$
| $E_{3}$ | $\ =\ $ | $\left\\{\left.-\underset{m=n+2}{\overset{p}{\sum}}\ln_{m}{\kappa_{-\alpha}}\ \vphantom{p\geq n+2}\right|\ p\geq n+2\right\\}$
| $E$ | $\ =\ $ | $E_{1}\cup E_{2}\cup E_{3}$

and $\left\langle E\right\rangle$ be the monoid it generates. Let
$b\in\underset{\ell=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\bigcup}}\operatorname{supp}\Phi^{\ell}(\omega^{a})$.
Then, there is $y\in\left\langle E\right\rangle$ such that

$\omega^{b}\asymp\partial u\exp(r\ln u+\eta+y)$

###### Proof.

We prove it by induction on $\ell$.

  * •

If $b\in\operatorname{supp}\omega^{a}$, then $y=0$ works.

  * •

Assume the property for $\ell\in\mathbb{N}$ and let
$b\in\sup\Phi^{\ell+1}(\omega^{a})$. Then there is
$c\in\operatorname{supp}\Phi^{\ell}(\omega^{a})$ such that
$b\in\operatorname{supp}\Phi(\omega^{c})$. Apply the induction hypothesis on
$c$ and on $y$ associated to $c$. Since any element $e\in E$ is such that
$e\prec\ln u$, we have $y\prec\ln u$ then Apply Lemma 7.1.1 to get that there
is $P\in\mathcal{P}_{\mathbb{L}}(\eta+y)$ such that

| $\omega^{b}$ | $\ \asymp\ $ | $\partial u\exp\left(r\ln u+\eta+y-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}\kappa_{-\alpha}\vphantom{\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}}\right.$
---|---|---|---
|  |  | $\qquad\qquad\left.-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\right)$

If $P(0)$ a term of $\eta$, up to some real factor, then there is a real
number $s$ and some $e\in E_{1,0}$ such that

$\exp\left(-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}\kappa_{-\alpha}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
\kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\right)=s\exp
e$

Then $y+e\in\left\langle E\right\rangle$ and $\omega^{b}\asymp\partial
u\exp(r\ln u+\eta+y+e)$. If not, then $P(0)$ is a term of $y$ (not up to a
real factor, an actual term). Hence, we have the following cases :

    * ➢

$P(0)=s\ln_{p}\kappa_{-\alpha}$ for some $s\in\mathbb{R}^{*}_{-}$ and $p\geq
n+2$. Then,

| $-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}\kappa_{-\alpha}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|$ | $\ =\ $ | $\ln|s|-\underset{m=n+2}{\overset{p}{\sum}}\ln_{m}\kappa_{-\alpha}$
---|---|---|---
|  | $\ \in\ $ | $\ln|s|+E_{3}$

Then,

$y-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}\kappa_{-\alpha}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
\kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\in\mathbb{R}+\left\langle
E\right\rangle$

    * ➢

$P(0)=s\ln_{p}\kappa_{-\beta}$ with $\beta>\alpha$ and $p\in\mathbb{N}^{*}$
such that there is some path $Q\in\mathcal{P}_{\mathbb{L}}(\eta)$ such that
$\kappa_{-\beta}\succeq^{K}Q(k_{Q})$. Then

$-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}\kappa_{-\alpha}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
\kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\in\ln|s|+E_{2}$

Then,

$y-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}\kappa_{-\alpha}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
\kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\in\mathbb{R}+\left\langle
E\right\rangle$

    * ➢

There are some paths $P_{0},\dots,P_{k}\in\mathcal{P}_{\mathbb{L}}(\eta)$ and
some non-zero integers $i_{1},\dots,i_{k}$ such that

$\forall j\in\left\llbracket\,1\ ;\ k\,\right\rrbracket\ \exists
j^{\prime}\in\left\llbracket\,0\ ;\ j-1\,\right\rrbracket\ \forall
i\in\left\llbracket\,0\ ;\ i_{j}-1\,\right\rrbracket\quad
P_{j^{\prime}}(i)=P_{j}(i)$

and

| $\exists y^{\prime}\in\left\langle E\right\rangle\quad y$ | $\ =\ $ | $y^{\prime}-(k+1)\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}{\kappa_{-\alpha}}$
---|---|---|---
|  |  | $\quad-\underset{j=0}{\overset{k}{\sum}}\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P_{j}(k_{P_{j}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{j=0}{\overset{k}{\sum}}\underset{i=i_{j}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{j}(i)|$

and such that $P(0)\in\mathbb{R}z$ for some $z$ a term of some
$\ln|P_{j}({i_{k+1}}^{\prime})|$ with $j\in\left\llbracket\,0\ ;\
k\,\right\rrbracket$ and ${i_{k+1}}^{\prime}\geq i_{j}$. Let $P_{k+1}$ be the
following path :

$P_{k+1}(i)=\left\\{\begin{array}[]{@{}r@{\quad}l@{}}P_{j}(i)&i\leq{i_{k+1}}^{\prime}\\\
z&i={i_{k+1}}^{\prime}+1\\\
P(i-{i_{k+1}}^{\prime}-1)&i>{i_{k+1}}^{\prime}+1\end{array}\right.$

Then, $P_{k+1}\in\mathcal{P}(\eta)$. Moreover, $\partial
P_{k+1}=\underbrace{P_{j}(0)\cdots P_{j}({i_{k+1}}^{\prime})}_{\neq
0}\underbrace{\partial P}_{\neq 0}$. Then
$P_{k+1}\in\mathcal{P}_{\mathbb{L}}(\eta)$. Note also that for all $\beta$,

$\kappa_{-\beta}\succeq^{K}P_{k+1}(k_{P_{k+1}})\Longleftrightarrow\kappa_{-\beta}\succeq^{K}P(k_{P})$

Finally,

|  | $-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}\kappa_{-\alpha}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|$
---|---|---
| $\ =\ $ | $-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}\kappa_{-\alpha}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P_{k+1}(k_{P_{k+1}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i={i_{k+1}}^{\prime}+1}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{k+1}(i)|+\ln\underbrace{\left|\mathchoice{\dfrac{P(0)}{z}}{\dfrac{P(0)}{z}}{\frac{P(0)}{z}}{\frac{P(0)}{z}}\right|}_{\in\mathbb{R}^{*}_{+}}$

From that we derive that

|  | $y-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}\kappa_{-\alpha}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|$
---|---|---
| $\ =\ $ | $y^{\prime}-(k+2)\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}{\kappa_{-\alpha}}$
|  | $\quad-\underset{j=0}{\overset{k+1}{\sum}}\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P_{j}(k_{P_{j}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{j=0}{\overset{k+1}{\sum}}\underset{i=i_{j}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{j}(i)|+\ln\left|\mathchoice{\dfrac{P(0)}{z}}{\dfrac{P(0)}{z}}{\frac{P(0)}{z}}{\frac{P(0)}{z}}\right|$
| $\ \in\ $ | $\mathbb{R}+\left\langle E\right\rangle$

where $i_{k+1}={i_{k+1}}^{\prime}+1$ and $P_{k+1}(i_{k})=z$ has indeed its
support (which is reduced to a singleton) included in the one of
$e\left(\begin{array}[]{c}P_{0},\dots,P_{k}\\\
i_{1},\dots,i_{k}\end{array}\right)$.

Then there is a real number $s$, and $e\in\left\langle E\right\rangle$ such
that

$\omega^{b}\asymp\partial u\exp(r\ln u+\eta+e+s)\asymp\partial u\exp(r\ln
u+\eta+e)$

Then we get the property at rank $\ell+1$.

By the induction principle, we conclude that the proposition is true for any
$\ell\in\mathbb{N}$. ∎

######  Corollary 7.0.

Let $x=\underset{i<\nu}{\overset{}{\sum}}r_{i}\omega^{a_{i}}$ such that

$\exists u=\ln_{n}\kappa_{-\alpha}\ \exists r\in\mathbb{R}\ \forall
a\in\operatorname{supp}x\ \exists\eta\prec\ln
u\quad\omega^{a}=\partial(u)\exp(r\ln u+\eta)$

We denote for $P_{0},\dots,P_{k}\in\mathcal{P}_{\mathbb{L}}(x)$ and
$i_{1},\dots,i_{k}\in\mathbb{N}\setminus\\{0,1\\}$,

| $e\left(\begin{array}[]{c}P_{0},\dots,P_{k}\\\ i_{1},\dots,i_{k}\end{array}\right)$ | $\ =\ $ | $-(k+1)\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}{\kappa_{-\alpha}}$
---|---|---|---
|  |  | $\quad-\underset{j=0}{\overset{k}{\sum}}\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P_{j}(k_{P_{j}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{j=0}{\overset{k}{\sum}}\underset{i=i_{j}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{j}(i)|$

with $i_{0}=0$. For $k\in\mathbb{N}$ define $E_{1,k}$ by:

| $e\left(\begin{array}[]{c}P_{0},\dots,P_{k}\\\ i_{1},\dots,i_{k}\end{array}\right)\in E_{1,k}$ | $\ \Leftrightarrow\ $ | $P_{0},\dots,P_{k}\in\mathcal{P}_{\mathbb{L}}(\eta)\quad\wedge\quad i_{1},\dots,i_{k}\in\mathbb{N}\setminus\\{0,1\\}$
---|---|---|---
|  |  | $\quad\wedge\quad\forall i\in\left\llbracket\,0\ ;\ k\,\right\rrbracket\qquad P_{i}(1)\prec\ln u$
|  |  | $\quad\wedge\quad\forall j\in\left\llbracket\,1\ ;\ k\,\right\rrbracket\quad\exists j^{\prime}\in\left\llbracket\,0\ ;\ j-1\,\right\rrbracket$
|  |  | $\qquad\qquad\forall i\in\left\llbracket\,0\ ;\ i_{j}-1\,\right\rrbracket\qquad P_{j^{\prime}}(i)=P_{j}(i)$
|  |  | $\quad\wedge\quad\forall j\in\left\llbracket\,1\ ;\ k\,\right\rrbracket$
|  |  | $\qquad\quad\operatorname{supp}P_{j}(i_{j})\subseteq\operatorname{supp}e\left(\begin{array}[]{c}P_{0},\dots,P_{k-1}\\\ i_{1},\dots,i_{k-1}\end{array}\right)\par$

Let also

| $E_{1}$ | $\ =\ $ | $\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k}$
---|---|---|---
| $E_{2}$ | $\ =\ $ | $\left\\{\left.\begin{array}[]{c}-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}\kappa_{-\alpha}-\underset{\gamma<\beta,\ m\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{m}\kappa_{-\gamma}\\\ \quad-\underset{m=1}{\overset{p}{\sum}}\ln_{m}\kappa_{-\beta}\end{array}\\!\\!\ \vphantom{\\!\\!\begin{array}[]{c}\beta>\alpha\\\ \exists P\in\mathcal{P}_{\mathbb{L}}(x)\ \kappa_{-\beta}\succeq^{K}P(k_{P})\\\ p\in\mathbb{N}\end{array}}\right|\ \\!\\!\begin{array}[]{c}\beta>\alpha\\\ \exists P\in\mathcal{P}_{\mathbb{L}}(x)\ \kappa_{-\beta}\succeq^{K}P(k_{P})\\\ p\in\mathbb{N}\end{array}\right\\}$
| $E_{3}$ | $\ =\ $ | $\left\\{\left.-\underset{m=n+2}{\overset{p}{\sum}}\ln_{m}{\kappa_{-\alpha}}\ \vphantom{p\geq n+2}\right|\ p\geq n+2\right\\}$
| $E$ | $\ =\ $ | $E_{1}\cup E_{2}\cup E_{3}$

and $\left\langle E\right\rangle$ be the monoid it generates. Let
$b\in\underset{\ell=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\bigcup}}\operatorname{supp}\Phi^{\ell}(x)$.
Then, there is $y\in\left\langle E\right\rangle$ such that

$\omega^{b}\asymp\exp(y)$

###### Proof.

Since $\Phi$ is strongly linear, we just need to apply Proposition 7.1.1 to
each term of $x$. For each term, $\partial u\exp(r\ln u+\eta)$ is term we add
at the beginning $P_{0}$. Each path involved is shifted one rank. ∎

######  Proposition 7.0.

Let $x=\underset{i<\nu}{\overset{}{\sum}}r_{i}\omega^{a_{i}}$ such that

$\exists u=\ln_{n}\kappa_{-\alpha}\ \exists r\in\mathbb{R}\ \forall
a\in\operatorname{supp}x\ \exists\eta\prec\ln
u\quad\omega^{a}=\partial(u)\exp(r\ln u+\eta)$

Consider $E_{1}$, $E_{2}$ and $E_{3}$ as defined Corollary 7.1.1. Let $\gamma$
be the smallest ordinal such that $\kappa_{-\gamma}\prec^{K}P(k_{P})$ for all
path $P\in\mathcal{P}_{\mathbb{L}}(\eta)$. Let $\lambda$ the least
$\varepsilon$-number greater than $\operatorname{NR}(x)$ and $\gamma$. Then
$E=E_{1}\cup E_{2}\cup E_{3}$ is reverse well-ordered with order type at most
$2\lambda+\omega(\gamma+1)$.

###### Proof.

First notice that $E_{3}$ is reverse well-ordered with order type $\omega$.
$E_{2}$ is also reverse well-ordered with order at most
$\omega+\omega\otimes\gamma+n\leq\omega\otimes(\gamma+1)$. We then focus on
$E_{1}$. We denote again

| $e\left(\begin{array}[]{c}P_{0},\dots,P_{k}\\\ i_{1},\dots,i_{k}\end{array}\right)$ | $\ =\ $ | $-(k+1)\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}{\kappa_{-\alpha}}$
---|---|---|---
|  |  | $\qquad-\underset{j=0}{\overset{k}{\sum}}\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P_{j}(k_{P_{j}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{j=0}{\overset{k}{\sum}}\underset{i=i_{j}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{j}(i)|$

  1. (i)

We first claim that for all $i\geq 3$ and all path $P\in\mathcal{P}(x)$ such
that $P(1)\prec\ln u$, $P(i)\prec P(2)\preceq\ln_{2}u$. Let
$P\in\mathcal{P}(x)$ such that $P(1)\prec\ln u$. Assume $P(2)\succ\ln_{2}u$.
Then, since $P(2)$ is a term of $\ln|P(1)|$, we also have
$\ln|P(1)|\succ\ln_{2}(u)$. Then, either $\ln|P(1)|<-m\ln_{2}u$ for all
$m\in\mathbb{N}$, or $\ln|P(1)|>m\ln_{2}(u)$ for all $m\in\mathbb{N}$. By
definition, $P(1)$ is purely infinite. In particular, $\ln|P(1)|$ cannot be
negative. Then,

$\forall m\in\mathbb{N}\qquad\ln|P(1)|>m\ln_{2}u$

and $\forall m\in\mathbb{N}\qquad|P(1)|>\left(\ln u\right)^{m}$ ($\exp$ is
increasing)

which is a contradiction with $P(1)\prec\ln u$, since $\ln u$ is infinitely
large. Since, for $i\geq 2$, $P(i)$ is infinitely large, $\ln|P(i)|\prec
P(i)$, and since $P(i+1)\preceq\ln|P(i)|$, we have for all $i\geq 1$,
$P(i+1)\prec P(i)$. By induction, we get

$\forall i\geq 3\qquad P(i)\prec P(2)\preceq\ln_{2}u$

  2. (ii)

We claim that for all path $P\in\mathcal{P}(x)$ such that $P(1)\prec\ln u$, if
$P(2)\asymp\ln_{2}u$, then, denoting $r$ the real number such that $P(2)\sim
r\ln_{2}u$, we have $0<r\leq 1$ . Let $P\in\mathcal{P}(x)$ such that
$P(1)\prec\ln u$ and assume $P(2)\asymp\ln_{2}u$. Since $P(2)$ is a term there
is a non-zero real number $r$ such that $P(1)=r\ln_{2}u$. From (i), we know
that $P(2)$ is the dominant term of $\ln|P(1)|$ so that

$\ln|P(1)|\sim r\ln_{2}u$

If $r<0$, Proposition 5.1 ensures that $|P(1)|\prec 1$ what is impossible
since $P(1)$ is infinite. Then $r>0$. If now $r>1$ then again with Proposition
5.1, $|P(1)|\succ\ln u$ what is not true. Then, $0<r\leq 1$.

  3. (iii)

For all $j$ and $i\geq 2$, $\ln|P_{j}(i)|\preceq\ln_{3}u\prec\ln_{2}u$.
Indeed, using (i), we know that $P_{j}(i)\preceq\ln_{2}u$. Then, there is a
natural number $m\geq 1$ such that $|P_{j}(i)|\leq m\ln_{2}u$. Using the fact
that $\ln$ is increasing,

$\ln|P_{j}(i)|\leq\ln_{3}u+\ln m\preceq\ln_{3}u\prec\ln_{2}u$

  4. (iv)

We now claim that $E_{1,k}>E_{1,k+2}$. Indeed, using (ii) and (iii) if
$e_{1}\in E_{1,k}$, then there is
$s\in\mathchoice{\left[\,-(k+1)\,;-k\,\right]}{\left[\,-(k+1)\,;-k\,\right]}{\left[\,-(k+1)\,;\,-k\,\right]}{\left[\,-(k+1)\,;\,-k\,\right]}$
such that $e_{1}\sim s\ln_{2}u$. Similarly, for $e_{2}\in E_{1,k+2}$, there is
$s^{\prime}\in\mathchoice{\left[\,-(k+3)\,;-(k+2)\,\right]}{\left[\,-(k+3)\,;-(k+2)\,\right]}{\left[\,-(k+3)\,;\,-(k+2)\,\right]}{\left[\,-(k+3)\,;\,-(k+2)\,\right]}$
such that $e_{2}\sim s^{\prime}\ln_{2}u$.

  5. (v)

We define the following sequence :

     * •

$a_{0}=\omega^{\omega^{\omega(\operatorname{NR}(x)+1)}}$

     * •

$a_{k+1}=\omega^{\omega^{\omega(\omega(\operatorname{NR}(x)+\gamma+4)a_{k}+1)}}$

We show that $E_{1,k}$ is reverse well-ordered with order type less than
$a_{k}$. We also claim that the equivalence classes of $E_{1,k}/{\asymp}$ are
finite and that

$\operatorname{NR}\left(\underset{t\in E_{1,k}}{\overset{}{\sum}}\exp
t\right)\leq\omega(\operatorname{NR}(x)+\gamma+4)a_{k}$

We show it by induction on $k\in\mathbb{N}$.

     * •

For $k=0$, let $t\in E_{1,0}$. Take $P\in\mathcal{P}_{\mathbb{L}}(x)$, minimal
for $<_{lex}$ such that $P(1)\prec\ln u$ and $t=e(P;)$. Then

| $\partial(\ln u)\exp t$ | $\ =\ $ | $|P(0)\cdots P(k_{P}-1)|\exp\left(-\underset{\footnotesize\begin{array}[]{c}\beta\ |\ \kappa_{-\beta}\succeq^{K}P(k_{P})\\\ m\in\mathbb{N}^{*}\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}\right.$
---|---|---|---
|  |  | $\hskip 135.00023pt\left.\vphantom{\underset{\footnotesize\begin{array}[]{c}\beta\ |\ \kappa_{-\beta}\succeq^{K}P(k_{P})\\\ m\in\mathbb{N}^{*}\end{array}}{\overset{}{\sum}}}+\underset{i=k_{P}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\right)$
|  | $\ =\ $ | $|\partial P|$

Since there are finitely many paths $Q\in\mathcal{P}_{\mathbb{L}}(x)$ such
that $\partial P\asymp\partial Q$, there are finitely many $t^{\prime}\in
E_{1,0}$ such that

$\partial(\ln u)\exp t\asymp\partial(\ln u)\exp t^{\prime}$

Since $\exp$ is an increasing function and $\partial(\ln u)>0$, we get , using
Proposition 1, that $E_{1,0}$ is reverse well-ordered with order type less
than
$\omega\otimes\omega^{\omega^{\omega(\operatorname{NR}(x)+1)}}=\omega^{\omega^{\omega(\operatorname{NR}(x)+1)}}=a_{0}$.
Finally, it remains to compute the nested rank of $\underset{t\in
E_{1,0}}{\overset{}{\sum}}\exp t$. Write

$t=-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}{\kappa_{-\alpha}}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
\kappa_{-\beta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|$

| $\operatorname{NR}\left(t\right)$ | $\ =\ $ | $\operatorname{NR}\left(-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}{\kappa_{-\alpha}}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)$ |
---|---|---|---|---
|  | $\ \leq\ $ | $\operatorname{NR}\left(-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}{\kappa_{-\alpha}}-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=k_{P_{0}}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)$ |
|  |  | $+\underset{i=0}{\overset{k_{P_{0}}-1}{\sum}}\operatorname{NR}\left(\ln|P_{0}(i)|\right)+k_{P_{0}}$ | (Lemma 6.2.2)
|  | $\ \leq\ $ | $(\omega\oplus\omega\otimes\gamma\oplus\omega)+\underset{i=0}{\overset{k_{P_{0}}-1}{\sum}}\operatorname{NR}\left(\ln|P_{0}(i)|\right)+k_{P_{0}}$ | (Lemma 6.2.2)
|  | $\ \leq\ $ | $(\omega\oplus\omega\otimes\gamma\oplus\omega)+k_{P_{0}}(\operatorname{NR}(x)+1)$ | (using Proposition 6.2.2)
|  | $\ \leq\ $ | $\omega(\operatorname{NR}(x)+\gamma+4)$ |

Then, since the equivalence classes of $E_{1,0}/{\asymp}$ are finite,

$\operatorname{NR}\left(\underset{t\in E_{1,0}}{\overset{}{\sum}}\exp
t\right)\leq\omega(\operatorname{NR}(x)+\gamma+4)a_{0}$

     * •

Assume the property for some $k\in\mathbb{N}$. Let $t\in E_{1,k+1}$. Let
$(P_{0},0),\dots,(P_{k+1},i_{k+1})$ minimal for the order $(<_{lex},<)_{lex}$
such that $t=e\left(\begin{array}[]{c}P_{0},\dots,P_{k+1}\\\
i_{1},\dots,i_{k+1}\end{array}\right)$. Then,

| $t$ | $\ =\ $ | $e\left(\begin{array}[]{c}P_{0},\dots,P_{k}\\\ i_{1},\dots,i_{k}\end{array}\right)-\underset{m=n+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{m}{\kappa_{-\alpha}}$
---|---|---|---
|  |  | $\qquad-\underset{\footnotesize\begin{array}[]{c}\beta>\alpha,m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P_{k+1}(k_{P_{k+1}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=i_{k+1}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{k+1}(i)|$

Write $s=e\left(\begin{array}[]{c}P_{0},\dots,P_{k}\\\
i_{1},\dots,i_{k}\end{array}\right)$. We then have,

| $\partial(\ln u)\exp t$ | $\ =\ $ | $\exp(s)\exp\left(-\underset{\footnotesize\begin{array}[]{c}\beta\ |\ \kappa_{-\beta}\succeq^{K}P_{k+1}(k_{P_{k+1}})\\\ m\in\mathbb{N}^{*}\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}\right.$
---|---|---|---
|  |  | $\hskip 108.00018pt\left.+\underset{i=i_{k+1}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{k+1}(i)|\vphantom{\underset{\footnotesize\begin{array}[]{c}\beta\ |\ \kappa_{-\beta}\succeq^{K}P_{k+1}(k_{P_{k+1}})\\\ m\in\mathbb{N}^{*}\end{array}}{\overset{}{\sum}}}\right)$

Consider the following path:

$\left\\{\begin{array}[]{@{}r@{\quad}l@{}}R(0)=\exp s\\\
R(i)=P_{k+1}(i-1+i_{k+1})&i>0\end{array}\right.$

It is indeed a path since, by definition of $E_{1,k+1}$,
$\operatorname{supp}P_{k+1}(i_{k+1})$ must be contained in
$\operatorname{supp}s$. Then,

$\partial(\ln u)\exp t=\partial R$

Moreover, $R\in\mathcal{P}_{\mathbb{L}}\left(\underset{s\in
E_{1,k}}{\overset{}{\sum}}\exp s\right)$. By induction hypothesis and
Proposition 1, $E_{1,k+1}$ has order type less than

$\omega^{\omega^{\omega(\omega(\operatorname{NR}(x)+\gamma+4)a_{k}+1)}}=a_{k+1}$

Since the equivalences classes of
$\mathcal{P}_{\mathbb{L}}\left(\underset{s\in E_{1,k}}{\overset{}{\sum}}\exp
s\right)/{\asymp}$ are finite, the ones of $E_{1,k+1}/{\asymp}$ are also
finite. Finally, using Lemmas 6.2.2 and 6.2.2,

| $\operatorname{NR}(t)$ | $\ \leq\ $ | $(\omega\oplus\omega\otimes\gamma\oplus\omega)+\underset{j=0}{\overset{k+1}{\sum}}\underset{i=i_{j}}{\overset{k_{P_{j}}-1}{\sum}}\operatorname{NR}\left(\ln|P_{j}(i)|\right)+\underset{j=0}{\overset{k+1}{\sum}}\max(0,k_{P_{j}}-i_{j})$
---|---|---|---
|  | $\ \leq\ $ | $\omega(\operatorname{NR}(x)+\gamma+4)$

Then, $\operatorname{NR}\left(\underset{t^{\prime}\in
E_{1,k+1}}{\overset{}{\sum}}\exp
t^{\prime}\right)\leq\omega(\operatorname{NR}(x)+\gamma+4)a_{k+1}$

We conclude thanks to the induction principle.

  6. (vi)

By easy induction, for all $k\in\mathbb{N}$, $a_{k}<\lambda$.

  7. (vii)

Using (iv), we get that for all $N\in\mathbb{N}$,
$\underset{k=0}{\overset{N}{\bigcup}}E_{1,2k}$ is an initial segment of
$\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,2k}$. We also have that
$\underset{k=0}{\overset{N}{\bigcup}}E_{1,2k+1}$ is an initial segment of
$\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,2k+1}$. Using (v), we get
that $\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,2k}$ has order type
at most

$\sup\left\\{\left.\underset{k=0}{\overset{N}{\oplus}}a_{2k}\
\vphantom{N\in\mathbb{N}}\right|\
N\in\mathbb{N}\right\\}=\sup\left\\{\left.a_{2N}\
\vphantom{N\in\mathbb{N}}\right|\ N\in\mathbb{N}\right\\}\underset{\text{by
(vi)}}{\leq}\lambda$

Similarly, $\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,2k+1}$ has
order type at most $\lambda$. Using Proposition 2, we conclude that $E_{1}$
has order type at most $2\lambda$.

Using again proposition 2, point (vii) above and the properties of $E_{2}$ and
$E_{3}$ mentioned in the beginning of this proof, we get that $E$ is reverse
well-ordered with order type at most $2\lambda+\omega(\gamma+1)$. ∎

######  Corollary 7.1.

Let $x=\underset{i<\nu}{\overset{}{\sum}}r_{i}\omega^{a_{i}}$ such that

$\exists u=\ln_{n}\kappa_{-\alpha}\quad\exists r\in\mathbb{R}\quad\forall
a\in\operatorname{supp}x\quad\exists\eta\prec\ln u\qquad\omega^{a}=\partial
u\exp(r\ln u+\eta)$

Let $\gamma$ be the smallest ordinal such that
$\kappa_{-\gamma}\prec^{K}P(k_{P})$ for all path
$P\in\mathcal{P}_{\mathbb{L}}(\eta)$. Let $\lambda$ the least
$\varepsilon$-number greater than $\operatorname{NR}(x)$ and $\gamma$. Then
$\underset{\ell=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\bigcup}}\operatorname{supp}\Phi^{\ell}(x)$
is reverse well-ordered with order type less at most
$\omega^{\omega\left(2\lambda+\omega(\gamma+1)+1\right)}$

###### Proof.

Just use Propositions 7.1.1, 7.1.1 and 2. ∎

####  7.1.2 Case $\varepsilon\succ\ln u$

######  Lemma 7.2.

Let $x$ be a surreal number. Let $P$ be the dominant path of $x$ and
$Q\in\mathcal{P}_{\mathbb{L}}(x)$. Then, $P(k_{P})\succeq^{K}Q(k_{Q})$. In
particular, for all ordinal $\beta$, if $\kappa_{-\beta}\succeq^{K}P(k_{P})$,
then $\kappa_{-\beta}\succeq^{K}Q(k_{Q})$.

###### Proof.

  1. (i)

We first claim that for all $i\in\mathbb{N}$, $P(i)\succeq Q(i)$. We prove it
by induction.

     * •

For $i=0$, $P(0)$ is the leading term of $x$ and $Q(0)$ is some term of $x$.
Therefore, $P(0)\succeq Q(0)$.

     * •

Assume $P(i)\succeq Q(i)$. $P(i+1)$ is the leading term of $\ln|P(i)|$. $P(i)$
and $Q(i)$ are both infinitely large. Then $\ln|P(i)|$ and $\ln|Q(i)|$ are
both positive infinitely large. If $Q(i+1)\succ P(i+1)$ then, in particular,
$\ln|Q(i)|\succ\ln|P(i)|$ what is impossible since $P(i)\succeq Q(i)$. Then
$P(i+1)\succeq Q(i+1)$.

We conclude thanks to induction principle.

  2. (ii)

Take $k=\max(k_{P},k_{Q})$. Using (i), we have :

$P(k_{P})\asymp^{K}P(k)\succeq Q(k)\asymp^{K}Q(k_{Q})$

Hence, $P(k_{P})\succeq^{K}Q(k_{Q})$.

∎

######  Lemma 7.3.

Assume $x=\omega^{a}=\partial u\exp\varepsilon$ with $\varepsilon\succ\ln u$
and $u=\ln_{n}\kappa_{-\alpha}$. Let
$b\in\operatorname{supp}\Phi(\omega^{a})$. Then, we have one of theses cases :

  * •

there is a path $P\in\mathcal{P}(\eta)$ and $i\in\mathbb{N}$ such that

$\omega^{b}\asymp\partial
u\exp\left(\varepsilon-\underset{\footnotesize\begin{array}[]{c}\beta\geq\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
P_{0}(k_{P_{0}})\succ^{K}\kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{j=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln\left|\mathchoice{\dfrac{P(i+j)}{P_{0}(j)}}{\dfrac{P(i+j)}{P_{0}(j)}}{\frac{P(i+j)}{P_{0}(j)}}{\frac{P(i+j)}{P_{0}(j)}}\right|\right)$

and $\forall j\in\left\llbracket\,0\ ;\ i-1\,\right\rrbracket\qquad
P(j)=P_{0}(j)$

  * •

There is some $(\beta,m)<_{lex}(\alpha,n)$ such that there is some
$\eta\prec\ln_{m}\kappa_{-\beta}$ such that

$\omega^{b}\asymp\partial(\ln_{m}\kappa_{-\beta})\exp\eta$

where $\eta=\varepsilon+\eta^{\prime}$ and $\eta^{\prime}$ only depends on
$\alpha,\beta,n,m$ and $P_{0}$, the dominant path of $\varepsilon$ :

$\eta^{\prime}=\underset{\footnotesize\begin{array}[]{c}(\zeta,p)>_{lex}(\beta,m)\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{(\beta,m)<_{lex}(\zeta,p)<_{lex}(\alpha,n)}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|$

or
$\eta^{\prime}=\underset{\footnotesize\begin{array}[]{c}(\zeta,p)\geq_{lex}(\alpha,n)\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|$

###### Proof.

We have
$\Phi(\omega^{a})=\left(1-\mathchoice{\dfrac{\partial\varepsilon}{s}}{\dfrac{\partial\varepsilon}{s}}{\frac{\partial\varepsilon}{s}}{\frac{\partial\varepsilon}{s}}\right)\omega^{a}-\partial\left(\mathchoice{\dfrac{\partial
u}{s}}{\dfrac{\partial u}{s}}{\frac{\partial u}{s}}{\frac{\partial
u}{s}}\right)\exp\varepsilon$

Let $b\in\operatorname{supp}\Phi(\omega^{a})$. Then either

$b\in\operatorname{supp}\left(\left(1-\mathchoice{\dfrac{\partial\varepsilon}{s}}{\dfrac{\partial\varepsilon}{s}}{\frac{\partial\varepsilon}{s}}{\frac{\partial\varepsilon}{s}}\right)\omega^{a}\right)$

or $b\in\operatorname{supp}\left(\partial\left(\mathchoice{\dfrac{\partial
u}{s}}{\dfrac{\partial u}{s}}{\frac{\partial u}{s}}{\frac{\partial
u}{s}}\right)\exp\varepsilon\right)$

  * •

First case :
$b\in\operatorname{supp}\left(\left(1-\mathchoice{\dfrac{\partial\varepsilon}{s}}{\dfrac{\partial\varepsilon}{s}}{\frac{\partial\varepsilon}{s}}{\frac{\partial\varepsilon}{s}}\right)\omega^{a}\right)$.
Then there is a path $P$, which is not the dominant path, such that

$\omega^{b}\asymp\mathchoice{\dfrac{\partial P}{s}}{\dfrac{\partial
P}{s}}{\frac{\partial P}{s}}{\frac{\partial
P}{s}}\omega^{a}\asymp\mathchoice{\dfrac{\exp\left(-\underset{\footnotesize\begin{array}[]{c}\beta\geq\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
\kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}\beta\geq\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
\kappa_{-\beta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)}}{\dfrac{\exp\left(-\underset{\footnotesize\begin{array}[]{c}\beta\geq\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
\kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}\beta\geq\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
\kappa_{-\beta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)}}{\frac{\exp\left(-\underset{\footnotesize\begin{array}[]{c}\beta\geq\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
\kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}\beta\geq\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
\kappa_{-\beta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)}}{\frac{\exp\left(-\underset{\footnotesize\begin{array}[]{c}\beta\geq\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
\kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}\beta\geq\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
\kappa_{-\beta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)}}\omega^{a}$

where $P_{0}$ is the dominant path of $\varepsilon$. Using Lemma 7.2, we get

$\omega^{b}\asymp\omega^{a}\exp\left(-\underset{\footnotesize\begin{array}[]{c}\beta\geq\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
P_{0}(k_{P_{0}})\succ^{K}\kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln\left|\mathchoice{\dfrac{P(i)}{P_{0}(i)}}{\dfrac{P(i)}{P_{0}(i)}}{\frac{P(i)}{P_{0}(i)}}{\frac{P(i)}{P_{0}(i)}}\right|\right)$

  * •

Second case :
$b\in\operatorname{supp}\left(\partial\left(\mathchoice{\dfrac{\partial
u}{s}}{\dfrac{\partial u}{s}}{\frac{\partial u}{s}}{\frac{\partial
u}{s}}\right)\exp\varepsilon\right)$. First notice that $\partial\partial
u=S_{u}\partial u$ where

$S_{u}=-\underset{\beta<\alpha\
m\in\mathbb{N}^{*}}{\overset{}{\sum}}\exp\left(-\underset{\zeta<\beta\
p\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{p=1}{\overset{m-1}{\sum}}\ln_{p}\kappa_{-\beta}\right)-\underset{m=1}{\overset{n-1}{\sum}}\exp\left(-\underset{\beta<\alpha\
p\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{p}\kappa_{-\beta}-\underset{p=1}{\overset{m-1}{\sum}}\ln_{p}\kappa_{-\alpha}\right)$

Hence, if $b\in\operatorname{supp}\left(\mathchoice{\dfrac{\partial\partial
u}{s}}{\dfrac{\partial\partial u}{s}}{\frac{\partial\partial
u}{s}}{\frac{\partial\partial u}{s}}\exp\varepsilon\right)$, there is some
$(\beta,m)<_{lex}(\alpha,n)$ such that

$\omega^{b}\asymp\omega^{a}\mathchoice{\dfrac{\exp\left(-\underset{\zeta<\beta\
p\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{p=1}{\overset{m-1}{\sum}}\ln_{p}\kappa_{-\beta}\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)}}{\dfrac{\exp\left(-\underset{\zeta<\beta\
p\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{p=1}{\overset{m-1}{\sum}}\ln_{p}\kappa_{-\beta}\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)}}{\frac{\exp\left(-\underset{\zeta<\beta\
p\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{p=1}{\overset{m-1}{\sum}}\ln_{p}\kappa_{-\beta}\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)}}{\frac{\exp\left(-\underset{\zeta<\beta\
p\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{p=1}{\overset{m-1}{\sum}}\ln_{p}\kappa_{-\beta}\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)}}$

Therefor,
$\omega^{b}\asymp\omega^{a}\exp\left(\underset{\footnotesize\begin{array}[]{c}(\zeta,p)\geq_{lex}(\beta,m)\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)$

Notice that
$\underset{\footnotesize\begin{array}[]{c}(\zeta,p)\geq_{lex}(\beta,m)\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}\sim\ln_{m}\kappa_{-\beta}\succ
P_{0}(0)$

and then

| $\omega^{a}$ | $\ \asymp\ $ | $\partial(\ln_{m}\kappa_{-\beta})\exp\left(\varepsilon+\underset{\footnotesize\begin{array}[]{c}(\zeta,p)>_{lex}(\beta,m)\\\ \zeta\ |\ \kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}\right.$
---|---|---|---
|  |  | $\left.\qquad\vphantom{\underset{\footnotesize\begin{array}[]{c}(\zeta,p)>_{lex}(\beta,m)\\\ \zeta\ |\ \kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}}-\underset{(\beta,m)<_{lex}(\zeta,p)<_{lex}(\alpha,n)}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)$

Since
$\varepsilon-\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\sim\varepsilon\prec\ln_{m}\kappa_{-\beta}$

and $\underset{\footnotesize\begin{array}[]{c}(\zeta,p)>_{lex}(\beta,m)\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{(\beta,m)<_{lex}(\zeta,p)<_{lex}(\alpha,n)}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}\prec\ln_{m}\kappa_{-\beta}$

Moreover,

$\operatorname{NR}\left(\ln\partial(\ln_{m}\kappa_{-\beta})+\varepsilon-\underset{(\beta,m)<_{lex}(\zeta,p)<_{lex}(\alpha,n)}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}\right)\leq\operatorname{NR}(x)$

and using Proposition 6.4,

| $\operatorname{NR}\left(\underset{\footnotesize\begin{array}[]{c}(\zeta,p)>_{lex}(\beta,m)\\\ \zeta\ |\ \kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)$ | $\ \leq\ $ | $\operatorname{NR}(\partial P_{0})$
---|---|---|---
|  | $\ \leq\ $ | $k_{P_{0}}(\operatorname{NR}(x)+1)+\omega(\gamma+1)$

We then conclude that there is some $\eta\prec\ln_{m}\kappa_{-\beta}$ such
that

$\omega^{b}=\partial(\ln_{m}\kappa_{-\beta})\exp\eta$

and by Corollary 6.2.2,

$\operatorname{NR}(\omega^{b})\leq(k_{P_{0}}+1)(\operatorname{NR}(x)+1)+\omega(\gamma+1)$

Now assume $b\in\operatorname{supp}\left(\mathchoice{\dfrac{\partial
s}{s^{2}}}{\dfrac{\partial s}{s^{2}}}{\frac{\partial s}{s^{2}}}{\frac{\partial
s}{s^{2}}}\omega^{a}\right)$. Notice that

| $\partial s$ | $\ =\ $ | $s\left(-\underset{\footnotesize\begin{array}[]{c}m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\partial\ln_{m}\kappa_{-\beta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\partial\ln|P_{0}(i)|\right)$
---|---|---|---
|  | $\ =\ $ | $s\left(-\underset{\footnotesize\begin{array}[]{c}m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\exp\left(-\underset{\zeta<\beta\ p\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{p=1}{\overset{m-1}{\sum}}\ln_{p}\kappa_{-\beta}\right)\right.$
|  |  | $\left.\vphantom{\underset{\footnotesize\begin{array}[]{c}m\in\mathbb{N}^{*}\\\ \beta\ |\ \kappa_{-\beta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}}\qquad+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\partial\ln|P_{0}(i)|\right)$

We then have the following sub-cases :

    * ➢

There is some $m\in\mathbb{N}^{*}$ and some ordinal $\beta$ such that
$\kappa_{-\beta}\succeq^{K}P_{0}(k_{P_{0}})$ such that

| $\omega^{b}$ | $\ \asymp\ $ | $\mathchoice{\dfrac{\exp\left(-\underset{\zeta<\beta\ p\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{p=1}{\overset{m-1}{\sum}}\ln_{p}\kappa_{-\beta}\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\ \zeta\ |\ \kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)}}{\dfrac{\exp\left(-\underset{\zeta<\beta\ p\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{p=1}{\overset{m-1}{\sum}}\ln_{p}\kappa_{-\beta}\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\ \zeta\ |\ \kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)}}{\frac{\exp\left(-\underset{\zeta<\beta\ p\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{p=1}{\overset{m-1}{\sum}}\ln_{p}\kappa_{-\beta}\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\ \zeta\ |\ \kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)}}{\frac{\exp\left(-\underset{\zeta<\beta\ p\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{p=1}{\overset{m-1}{\sum}}\ln_{p}\kappa_{-\beta}\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\ \zeta\ |\ \kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)}}\omega^{a}$
---|---|---|---
|  | $\ \asymp\ $ | $\partial(\ln_{m}\kappa_{-\beta})\exp\left(\varepsilon+\underset{\footnotesize\begin{array}[]{c}(\zeta,p)\geq_{lex}(\alpha,n)\\\ \zeta\ |\ \kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right)$

with

$\varepsilon-\underset{\footnotesize\begin{array}[]{c}\zeta\geq\alpha\
p\in\mathbb{N}^{*}\\\ \zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\sim\varepsilon\prec\ln_{m}\kappa_{-\beta}$

We then conclude that there is some $\eta\prec\ln_{m}\kappa_{-\beta}$ such
that

$\omega^{b}=\partial(\ln_{m}\kappa_{-\beta})\exp\eta$

and by Corollary 6.2.2,

$\operatorname{NR}(\omega^{b})\leq(k_{P_{0}}+1)(\operatorname{NR}(x)+1)+\omega(\gamma+1)$

    * ➢

There is some path $P\in\mathcal{P}_{\mathbb{L}}(\varepsilon)$ and some $i\geq
1$ such that for all $j<i$, $P(j)=P_{0}(j)$ and

$\omega^{b}\asymp\mathchoice{\dfrac{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{j=i}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(j)|\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{j=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(j)|\right)}}{\dfrac{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{j=i}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(j)|\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{j=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(j)|\right)}}{\frac{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{j=i}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(j)|\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{j=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(j)|\right)}}{\frac{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{j=i}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(j)|\right)}{\exp\left(-\underset{\footnotesize\begin{array}[]{c}p\in\mathbb{N}^{*}\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}+\underset{j=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(j)|\right)}}\omega^{a}$

As in the first case, we get

$\omega^{b}\asymp\omega^{a}\exp\left(-\underset{\footnotesize\begin{array}[]{c}\beta\geq\alpha,m\in\mathbb{N}^{*}\\\
\beta\ |\
P_{0}(k_{P_{0}})\succ^{K}\kappa_{-\beta}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{j=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln\left|\mathchoice{\dfrac{P(i+j)}{P_{0}(j)}}{\dfrac{P(i+j)}{P_{0}(j)}}{\frac{P(i+j)}{P_{0}(j)}}{\frac{P(i+j)}{P_{0}(j)}}\right|\right)$

∎

######  Proposition 7.4.

Assume $x=\omega^{a}=\partial u\exp\varepsilon$ with $\varepsilon\succ\ln u$
and $u=\ln_{n}\kappa_{-\alpha}$. Let $P_{0}$ be the dominant path of
$\varepsilon$. We denote for
$P_{1},\dots,P_{k+k^{\prime}}\in\mathcal{P}_{\mathbb{L}}(\varepsilon)$,
$i_{1},\dots,i_{k+k^{\prime}}\in\mathbb{N}^{*}$ and
$(\beta,m)\leq_{lex}(\alpha,n)$,

| $e^{(\beta,m)}\left(\begin{matrix}P_{1},\dots,P_{k}\\\ P_{k+1},\dots,P_{k+k^{\prime}}\\\ i_{1},\dots,i_{k+k^{\prime}}\end{matrix}\right)$ | $\ =\ $ | $-k\underset{i=0}{\overset{\infty}{\sum}}\ln|P_{0}(i)|+\underset{j=1}{\overset{k}{\sum}}\underset{i=i_{j}}{\overset{\infty}{\sum}}\ln|P_{j}(i)|$
---|---|---|---
|  |  | $-\underset{j=1}{\overset{k}{\sum}}\underset{\footnotesize\begin{array}[]{c}\gamma\geq\alpha,\ell\in\mathbb{N}^{*}\\\ \gamma\ |\ P_{0}(k_{P_{0}})\succ^{K}\kappa_{-\gamma}\succeq^{K}P_{j}(k_{P_{j}})\end{array}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma}$
|  |  | $-k^{\prime}\underset{\ell=m+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{\ell}{\kappa_{-\beta}}+\underset{j=k+1}{\overset{k+k^{\prime}}{\sum}}\underset{i=i_{j}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{j}(i)|$
|  |  | $-\underset{j=k+1}{\overset{k+k^{\prime}}{\sum}}\underset{\footnotesize\begin{array}[]{c}\gamma>\beta,\ell\in\mathbb{N}^{*}\\\ \gamma\ |\ \kappa_{-\gamma}\succeq^{K}P_{j}(k_{P_{j}})\end{array}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma}\par$

We now define $E_{1,k,k^{\prime}}^{(\beta,m)}$ as follows:

|  | $e^{(\beta,m)}\left(\begin{matrix}P_{1},\dots,P_{k}\\\ P_{k+1},\dots,P_{k+k^{\prime}}\\\ i_{1},\dots,i_{k+k^{\prime}}\end{matrix}\right)\in E_{1,k,k^{\prime}}^{(\beta,m)}$
---|---|---
| $\ \Leftrightarrow\ $ | $P_{1},\dots,P_{k}\in\mathcal{P}_{\mathbb{L}}(\varepsilon)\setminus\\{P_{0}\\}$
|  | $\wedge\quad P_{k+1},\dots,P_{k+k^{\prime}}\in\mathcal{P}_{\mathbb{L}}(\varepsilon)$
|  | $\wedge\quad i_{1},\dots,i_{k}\in\mathbb{N}$
|  | $\wedge\quad i_{k+1},\dots,i_{k+k^{\prime}}\in\mathbb{N}^{*}$
|  | $\wedge\quad\forall j\in\left\llbracket\,1\ ;\ k+k^{\prime}\,\right\rrbracket\ \exists j^{\prime}\in\left\llbracket\,0\ ;\ j-1\,\right\rrbracket$
|  | $\qquad\qquad\forall i\in\left\llbracket\,0\ ;\ i_{j}-1\,\right\rrbracket\quad P_{j^{\prime}}(i)=P_{j}(i)$
|  | $\wedge\quad\forall j\in\left\llbracket\,k+1\ ;\ k+k^{\prime}\,\right\rrbracket$
|  | $\qquad\qquad\operatorname{supp}P_{j}(i_{j})\subseteq\operatorname{supp}e^{(\beta,m)}\left(\begin{matrix}P_{1},\dots,P_{k}\\\ P_{k+1},\dots,P_{j}\\\ i_{1},\dots,i_{j}\end{matrix}\right)$

$E_{1}^{(\beta,m)}=\left\\{\begin{array}[]{@{}r@{\quad}l@{}}\underset{k\in\mathbb{N},\
k^{\prime}\in\mathbb{N}^{*}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}}^{(\beta,m)}&(\beta,m)\neq(\alpha,n)\\\
\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,0}^{(\beta,m)}&(\beta,m)=(\alpha,n)\end{array}\right.$

Define sets $E_{2}^{(\beta,m)}$ as follows:

  * •

If $(\beta,m)\neq(\alpha,n)$, then

$-\underset{\ell=m+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{\ell}\kappa_{-\beta}-\underset{\gamma^{\prime}<\gamma,\
\ell\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma^{\prime}}-\underset{\ell=1}{\overset{p}{\sum}}\ln_{\ell}\kappa_{-\gamma}\in
E_{2}^{(\beta,m)}$

iff $\gamma>\beta$, $p\in\mathbb{N}$ and there is some
$P\in\mathcal{P}_{\mathbb{L}}(\varepsilon)$ such that
$\kappa_{-\gamma}\succeq^{K}P(k_{P})$

  * •

If $(\beta,m)=(\alpha,n)$, then

$-\underset{j=0}{\overset{\infty}{\sum}}\ln|P_{0}(j)|-\underset{\gamma>\zeta>\alpha,\
\ell\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\zeta}-\underset{\ell=1}{\overset{p}{\sum}}\ln_{\ell}\kappa_{-\gamma}\in
E_{2}^{(\beta,m)}$

iff $\gamma>\alpha$, $p\in\mathbb{N}$ and there is some
$P\in\mathcal{P}_{\mathbb{L}}(\varepsilon)$ such that
$\kappa_{-\gamma}\succeq^{K}P(k_{P})$.

Let also

| $E_{3}^{(\beta,m)}$ | $\ =\ $ | $\left\\{\begin{array}[]{@{}r<EMAIL_ADDRESS>\vphantom{p\geq m+2}\right|\ p\geq m+2\right\\}&(\beta,m)\neq(\alpha,n)\\\ \varnothing&(\beta,m)=(\alpha,n)\end{array}\right.$
---|---|---|---
| $E^{(\beta,m)}$ | $\ =\ $ | $E_{1}^{(\beta,m)}\cup E_{2}^{(\beta,m)}\cup E_{3}^{(\beta,m)}$

and $\left\langle E^{(\beta,m)}\right\rangle$ be the monoid it generates.
Finally, let $H^{(\beta,m)}$ defined by cases as follows:

$\left\\{\begin{array}[]{@{}r@{\quad}l@{}}\left\\{\underset{\footnotesize\begin{array}[]{c}(\zeta,p)>_{lex}(\beta,m)\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}\hskip
64.00012pt\right.\\\
-\underset{(\beta,m)<_{lex}(\zeta,p)<_{lex}(\alpha,n)}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|,\\\
\left.\underset{\footnotesize\begin{array}[]{c}(\zeta,p)\geq_{lex}(\alpha,n)\\\
\zeta\ |\
\kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\right\\}&(\beta,m)\neq(\alpha,n)\\\
\\{0\\}&(\beta,m)=(\alpha,n)\end{array}\right.$

Let
$b\in\underset{q=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\bigcup}}\operatorname{supp}\Phi^{q}(\omega^{a})$.
Then, there are $\eta\in H^{(\beta,m)}$ and $y\in\left\langle
E^{(\beta,m)}\right\rangle$ such that

$\omega^{b}\asymp\partial(\ln_{m}\kappa_{-\beta})\exp(\varepsilon+\eta+y)$

###### Proof.

We prove it by induction on $q$.

  * •

If $b\in\operatorname{supp}\Phi^{0}(\omega^{a})$, then $b=a$ and $y=0$ with
$(\beta,m)=(\alpha,n)$ and $\eta=0$ works.

  * •

Assume the property for some $q\in\mathbb{N}$. Let
$b\in\operatorname{supp}\Phi^{q+1}(\omega^{b})$. Then there is
$c\in\operatorname{supp}\Phi^{q}(\omega^{a})$ such that
$b\in\operatorname{supp}\Phi(\omega^{c})$. Apply the induction hypothesis on
$c$. Take $(\beta,m)$, $\eta\in H^{(\beta,m)}$ and $y\in\left\langle
E^{(\beta,m)}\right\rangle$ such that

$\omega^{c}\asymp\partial(\ln_{m}\beta)\exp(\varepsilon+\eta+y)$

    * ➢

If $(\beta,m)<_{lex}(\alpha,n)$, then
$y,\varepsilon\prec\ln_{n+1}\kappa_{-\beta}$. Hence, using Lemma 7.1.1, we get
that there is $P\in\mathcal{P}_{\mathbb{L}}(\varepsilon+\eta+y)$ such that

| $\omega^{b}$ | $\ \asymp\ $ | $\partial(\ln_{m}\kappa_{-\beta})\exp\left(\varepsilon+\eta+y-\underset{\ell=m+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{\ell}\kappa_{-\beta}\vphantom{\underset{\footnotesize\begin{array}[]{c}\gamma>\beta,\ell\in\mathbb{N}^{*}\\\ \gamma\ |\ \kappa_{-\gamma}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}}\right.$
---|---|---|---
|  |  | $\left.\qquad-\underset{\footnotesize\begin{array}[]{c}\gamma>\beta,\ell\in\mathbb{N}^{*}\\\ \gamma\ |\ \kappa_{-\gamma}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\right)$

If $P(0)$ a term of $\varepsilon$, up to some real factor, then there is a
real number $s$ and some $e\in E_{1,0,1}^{(\beta,m)}$ such that

| $s\exp e$ | $\ =\ $ | $\exp\left(-\underset{\ell=m+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{\ell}\kappa_{-\beta}-\underset{\footnotesize\begin{array}[]{c}\gamma>\beta,\ell\in\mathbb{N}^{*}\\\ \gamma\ |\ \kappa_{-\gamma}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\right)$
---|---|---|---

Then $y+e\in\left\langle E^{(\beta,m)}\right\rangle$ and
$\omega^{b}\asymp\partial(\ln_{m}\kappa_{-\beta})\exp(\varepsilon+y+e)$. If
not, then $P(0)$ is a term of $\eta+y$. Hence, we have the following cases:

      * $\because$

$P(0)=s\ln_{p}\kappa_{-\beta}$ for some $s\in\mathbb{R}^{*}_{-}$ and $p\geq
m+2$. Then,

|  | $-\underset{\ell=m+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{\ell}\kappa_{-\beta}-\underset{\footnotesize\begin{array}[]{c}\gamma>\beta,\ell\in\mathbb{N}^{*}\\\ \gamma\ |\ \kappa_{-\gamma}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|$
---|---|---
| $\ =\ $ | $\ln|s|-\underset{\ell=m+2}{\overset{p}{\sum}}\ln_{\ell}\kappa_{-\beta}\in\ln|s|+E_{3}^{(\beta,m)}$

Then,

$y-\underset{\ell=m+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{\ell}\kappa_{-\beta}-\underset{\footnotesize\begin{array}[]{c}\gamma>\beta,\ell\in\mathbb{N}^{*}\\\
\gamma\ |\
\kappa_{-\gamma}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\in\mathbb{R}+\left\langle
E^{(\beta,m)}\right\rangle$

      * $\because$

$P(0)=s\ln_{p}\kappa_{-\gamma}$ with $\gamma>\beta$ and $p\in\mathbb{N}^{*}$
such that there is some path $Q\in\mathcal{P}_{\mathbb{L}}(\varepsilon)$ such
that $\kappa_{-\beta}\succeq^{K}Q(k_{Q})$. Then

$-\underset{\ell=m+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{\ell}\kappa_{-\beta}-\underset{\footnotesize\begin{array}[]{c}\gamma>\beta,\ell\in\mathbb{N}^{*}\\\
\gamma\ |\
\kappa_{-\gamma}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\in\ln|s|+E_{2}^{(\beta,m)}$

Then,

$y-\underset{\ell=m+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{\ell}\kappa_{-\beta}-\underset{\footnotesize\begin{array}[]{c}\gamma>\beta,\ell\in\mathbb{N}^{*}\\\
\gamma\ |\
\kappa_{-\gamma}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|\in\mathbb{R}+\left\langle
E^{(\beta,m)}\right\rangle$

      * $\because$

There are some paths
$P_{1},\dots,P_{k+k^{\prime}}\in\mathcal{P}_{\mathbb{L}}(\varepsilon)$ and
some integers $i_{1},\dots,i_{k+k^{\prime}}$ such that

$e^{(\beta,m)}\left(\begin{matrix}P_{1},\dots,P_{k}\\\
P_{k+1},\dots,P_{k+k^{\prime}}\\\
i_{1},\dots,i_{k+k^{\prime}}\end{matrix}\right)\in
E_{1,k,k^{\prime}}^{(\beta,m)}$

and $\exists y^{\prime}\in\left\langle E\right\rangle\qquad
y=y^{\prime}+e^{(\beta,m)}\left(\begin{matrix}P_{1},\dots,P_{k}\\\
P_{k+1},\dots,P_{k+k^{\prime}}\\\
i_{1},\dots,i_{k+k^{\prime}}\end{matrix}\right)$

and finally such that $P(0)\in\mathbb{R}z$ for some $z$ a term of some
$\ln|P_{j}({i_{k+k^{\prime}+1}}^{\prime})|$ with $j\in\left\llbracket\,0\ ;\
k+k^{\prime}\,\right\rrbracket$ and ${i_{k+k^{\prime}+1}}^{\prime}\geq i_{j}$.
Let $P_{k+k^{\prime}+1}$ be the following path :

$P_{k+k^{\prime}+1}(i)=\left\\{\begin{array}[]{@{}r@{\quad}l@{}}P_{j}(i)&i\leq{i_{k+1}}^{\prime}\\\
z&i={i_{k+1}}^{\prime}+1\\\
P(i-{i_{k+1}}^{\prime}-1)&i>{i_{k+1}}^{\prime}+1\end{array}\right.$

Then, $P_{k+k^{\prime}+1}\in\mathcal{P}(\varepsilon)$. Moreover,

$\partial P_{k+k^{\prime}+1}=\underbrace{P_{j}(0)\cdots
P_{j}({i_{k+1}}^{\prime})}_{\neq 0}\underbrace{\partial P}_{\neq 0}$

Then $P_{k+k^{\prime}+1}\in\mathcal{P}_{\mathbb{L}}(\varepsilon)$. Note also
that for all $\beta$,

$\kappa_{-\beta}\succeq^{K}P_{k+k^{\prime}+1}(k_{P_{k+k^{\prime}+1}})\Longleftrightarrow\kappa_{-\beta}\succeq^{K}P(k_{P})$

Finally,

|  | $-\underset{\ell=m+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{\ell}\kappa_{-\beta}-\underset{\footnotesize\begin{array}[]{c}\gamma>\beta,\ell\in\mathbb{N}^{*}\\\ \gamma\ |\ \kappa_{-\gamma}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|$
---|---|---
| $\ =\ $ | $-\underset{\ell=m+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{\ell}\kappa_{-\beta}-\underset{\footnotesize\begin{array}[]{c}\gamma>\beta,\ell\in\mathbb{N}^{*}\\\ \gamma\ |\ \kappa_{-\gamma}\succeq^{K}P_{k+k^{\prime}+1}(k_{P_{k+k^{\prime}+1}})\end{array}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma}$
|  | $\qquad\qquad+\underset{i={i_{k+1}}^{\prime}+1}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{k+k^{\prime}+1}(i)|+\ln\underbrace{\left|\mathchoice{\dfrac{P(0)}{z}}{\dfrac{P(0)}{z}}{\frac{P(0)}{z}}{\frac{P(0)}{z}}\right|}_{\in\mathbb{R}^{*}_{+}}$

From that we derive that

|  | $y-\underset{\ell=m+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{\ell}\kappa_{-\beta}-\underset{\footnotesize\begin{array}[]{c}\gamma>\beta,\ell\in\mathbb{N}^{*}\\\ \gamma\ |\ \kappa_{-\gamma}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma}+\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P(i)|$
---|---|---
| $\ =\ $ | $y^{\prime}+e^{(\beta,m)}\left(\begin{matrix}P_{1},\dots,P_{k}\\\ P_{k+1},\dots,P_{k+k^{\prime}+1}\\\ i_{1},\dots,i_{k+k^{\prime}+1}\end{matrix}\right)+\ln\left|\mathchoice{\dfrac{P(0)}{z}}{\dfrac{P(0)}{z}}{\frac{P(0)}{z}}{\frac{P(0)}{z}}\right|\in\mathbb{R}+\left\langle E^{(\beta,m)}\right\rangle$

where $i_{k+k^{\prime}+1}={i_{k+k^{\prime}+1}}^{\prime}+1$ and
$P_{k+k^{\prime}+1}(i_{k+k^{\prime}})=z$ has indeed its support (which is
reduced to a singleton) included in the one of
$e^{(\beta,m)}\left(\begin{matrix}P_{1},\dots,P_{k}\\\
P_{k+k^{\prime}},\dots,P_{k+k^{\prime}}\\\
i_{1},\dots,i_{k+k^{\prime}}\end{matrix}\right)$.

Then there is a real number $s$, and $e\in\left\langle
E^{(\beta,m)}\right\rangle$ such that

$\omega^{b}\asymp\partial(\ln_{m}\beta)\exp(\varepsilon+\eta+e+s)\asymp\partial(\ln_{m}\beta)\exp(\varepsilon+\eta+e)$

Then we get the property at rank $q+1$.

    * ➢

If $(\beta,m)=(\alpha,n)$, we have $\eta=0$ and write

$y=y^{\prime}+e^{(\alpha,n)}\left(\begin{matrix}P_{1},\dots,P_{k}\\\
\varnothing\\\ i_{1},\dots,i_{k+k^{\prime}}\end{matrix}\right)$

with, $y^{\prime}\in\left\langle E^{(\beta,m)}\right\rangle$ and,
$k,k^{\prime}\in\mathbb{N}$. Using Lemma 7.3, we have

      * $\because$

First case : $\omega^{b}\asymp\partial u\exp(\varepsilon+y+e)$

where

$e=-\underset{\footnotesize\begin{array}[]{c}\gamma\geq\alpha,\ell\in\mathbb{N}^{*}\\\
\gamma\ |\
P_{0}(k_{P_{0}})\succ^{K}\kappa_{-\gamma}\succeq^{K}P(k_{P})\end{array}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma}+\underset{j=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln\left|\mathchoice{\dfrac{P(i+j)}{P_{0}(j)}}{\dfrac{P(i+j)}{P_{0}(j)}}{\frac{P(i+j)}{P_{0}(j)}}{\frac{P(i+j)}{P_{0}(j)}}\right|$

for some path $P\in\mathcal{P}_{\mathbb{L}}(\varepsilon+y)$ and some
$i\in\mathbb{N}$ such that

$\forall j\in\left\llbracket\,0\ ;\ i-1\,\right\rrbracket\qquad P(j)=P_{0}(j)$

Indeed, $y\in\left\langle E^{(\alpha,n)}\right\rangle$. In particular,
$y\prec\varepsilon$ and then $\varepsilon+y\sim\varepsilon$ so that $P_{0}$ is
also the dominant path of $\varepsilon+y$.

        * ·

If $P(0)$ is, up to a real factor, a term of $\varepsilon$, then we get that
there is some path $Q\in\mathcal{P}_{\mathbb{L}}(\varepsilon)$ and a real
number $s$ such that

$y+e=y^{\prime}+e^{(\beta,m)}\left(\begin{matrix}P_{1},\dots,P_{k},Q\\\
\varnothing\\\ i_{1},\dots,i_{k},i\end{matrix}\right)+s$

Since $y\prec\varepsilon$, and $P\neq P_{0}$, we also have $Q\neq P_{0}$. Then
$y+e\in\left\langle
E^{(\beta,m)}\right\rangle+E_{1,k+1,k^{\prime}}^{(\beta,m)}+s$. Let

$y^{\prime\prime}=y+e-s\in\left\langle E^{(\beta,m)}\right\rangle$

then, $\omega^{b}\asymp\partial u\exp(\varepsilon+y^{\prime\prime})$

In particular, $y^{\prime\prime}\prec\varepsilon$.

        * ·

If $P(0)$ is a term of $y$, and more precisely if it can be written as
$P(0)=s\ln_{p}\kappa_{-\gamma}$ for $s\in\mathbb{R}$, $p\in\mathbb{N}$ and
$\gamma\geq\alpha$ such that

$P_{0}(k_{P_{0}})\succ^{K}\kappa_{-\gamma}\succeq^{K}Q(k_{Q})$

for some path $Q\in\mathcal{P}_{\mathbb{L}}(\varepsilon)\setminus\\{P_{0}\\}$.
Then,

$e=-\underset{j=0}{\overset{\infty}{\sum}}\ln|P_{0}(j)|-\underset{\gamma>\zeta>\alpha,\
\ell\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\zeta}-\underset{\ell=1}{\overset{p+i}{\sum}}\ln_{\ell}\kappa_{-\gamma}+\mathds{1}_{i=0}\ln|s|\in
E_{2}^{(\beta,m)}+\mathbb{R}$

Then $y+e-\ln|s|\in\left\langle E^{(\beta,m)}\right\rangle$ and since
$e\prec\varepsilon$, $y+e-s\prec\varepsilon$ and

$\omega^{b}\asymp\partial u\exp(\varepsilon+y+e-\ln|s|)$

        * ·

If $P(0)$ is a term of $y$, and more precisely if it can be written as
$P(0)=s\ln|P_{\ell}(j)|$ for some $s\in\mathbb{R}$ and some
$\ell\in\left\llbracket\,0\ ;\ k+k^{\prime}\,\right\rrbracket$ (actually it is
true if we have chosen well the $y^{\prime}$ in the beginning, but up to a
renaming, it is true). Consider the following path

$Q(p)=\left\\{\begin{array}[]{@{}r@{\quad}l@{}}P_{\ell}(p)&p\leq j\\\
P(p-j)&p>j\end{array}\right.$

We have $Q\in\mathcal{P}_{\mathbb{L}}(\varepsilon)$ and

$y+e=y^{\prime}+e^{(\beta,m)}\left(\begin{matrix}P_{1},\dots,P_{k},Q\\\
\varnothing\\\ i_{1},\dots,i_{k},j\end{matrix}\right)+\ln|s|$

Then $y+e-\ln|s|\in\left\langle E^{(\beta,m)}\right\rangle$ and since
$e\prec\varepsilon$, $y+e-s\prec\varepsilon$ and

$\omega^{b}\asymp\partial u\exp(\varepsilon+y+e-\ln|s|)$

This concludes the first case.

      * $\because$

Second case : There are $(\beta^{\prime},m^{\prime})<_{lex}(\alpha,n)$ and
$\eta^{\prime}\in H^{(\beta,m)}$ such that
$\omega^{b}\asymp\partial(\ln_{m^{\prime}}\kappa_{-\beta^{\prime}})\exp(\varepsilon+\eta^{\prime}+y)$.
This immediately conclude the second case.

We then have the property at rank $q+1$.

Thanks to the induction principle, we conclude that the property holds for any
$q\in\mathbb{N}$. ∎

######  Corollary 7.0.

Let $x$ be a surreal number such that

$\exists u=\ln_{n}\kappa_{-\alpha}\quad\exists
r_{0}\in\mathbb{R}^{*}\quad\exists a_{0}\in\textnormal{No}\quad\forall
a\in\operatorname{supp}x\quad\exists\varepsilon\sim r\omega^{a_{0}}\succ\ln
u\qquad\omega^{a}\asymp\partial u\exp\varepsilon$

Let $\mathcal{P}_{0}(x)=\left\\{\left.P\in\mathcal{P}_{\mathbb{L}}(x)\
\vphantom{\begin{array}[]{c}P(1)=r\omega^{a_{0}}\\\ \forall i\geq 1\quad
P(i+1)\sim\ln|P(i)\end{array}}\right|\
\begin{array}[]{c}P(1)=r\omega^{a_{0}}\\\ \forall i\geq 1\quad
P(i+1)\sim\ln|P(i)\end{array}\right\\}$

It is the set of all the possible dominant paths of the epsilon to which we
add the corresponding term of $x$ at the beginning. We denote for
$P_{0}\in\mathcal{P}_{0}(x)$,
$P_{1},\dots,P_{k+k^{\prime}}\in\mathcal{P}_{\mathbb{L}}(x)$,
$i_{1},\dots,i_{k+k^{\prime}}\in\mathbb{N}^{*}$ and
$(\beta,m)\leq_{lex}(\alpha,n)$,

| $e^{(\beta,m)}\left(\begin{matrix}P_{0};P_{1},\dots,P_{k}\\\ P_{k+1},\dots,P_{k+k^{\prime}}\\\ i_{1},\dots,i_{k+k^{\prime}}\end{matrix}\right)$ | $\ =\ $ | $-k\underset{i=1}{\overset{\infty}{\sum}}\ln|P_{0}(i)|-k^{\prime}\underset{\ell=m+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{\ell}{\kappa_{-\beta}}$
---|---|---|---
|  |  | $-\underset{j=1}{\overset{k}{\sum}}\underset{\scriptsize\begin{array}[]{c}\gamma\geq\alpha,\ell\in\mathbb{N}^{*}\\\ \gamma\ |\ P_{0}(k_{P_{0}})\succ^{K}\kappa_{-\gamma}\succeq^{K}P_{j}(k_{P_{j}})\end{array}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma}$
|  |  | $+\underset{j=1}{\overset{k}{\sum}}\underset{i=i_{j}}{\overset{\infty}{\sum}}\ln|P_{j}(i)|$
|  |  | $-\underset{j=k+1}{\overset{k+k^{\prime}}{\sum}}\underset{\footnotesize\begin{array}[]{c}\gamma>\beta,\ell\in\mathbb{N}^{*}\\\ \gamma\ |\ \kappa_{-\gamma}\succeq^{K}P_{j}(k_{P_{j}})\end{array}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma}$
|  |  | $+\underset{j=k+1}{\overset{k+k^{\prime}}{\sum}}\underset{i=i_{j}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{j}(i)|$

We now define $E_{1,k,k^{\prime}}^{(\beta,m)}$ as follows:

|  | $e^{(\beta,m)}\left(\begin{matrix}P_{0};P_{1},\dots,P_{k}\\\ P_{k+1},\dots,P_{k+k^{\prime}}\\\ i_{1},\dots,i_{k+k^{\prime}}\end{matrix}\right)\in E_{1,k,k^{\prime}}^{(\beta,m)}$
---|---|---
| $\ \Leftrightarrow\ $ | $P_{0}\in\mathcal{P}_{0}(x)\wedge P_{1},\dots,P_{k}\in\mathcal{P}_{\mathbb{L}}(\varepsilon)\setminus\\{P_{0}\\}$
|  | $\wedge\quad P_{k+1},\dots,P_{k+k^{\prime}}\in\mathcal{P}_{\mathbb{L}}(x)$
|  | $\wedge\quad i_{1},\dots,i_{k}\in\mathbb{N}^{*}$
|  | $\wedge\quad i_{k+1},\dots,i_{k+k^{\prime}}\in\mathbb{N}\setminus\\{0,1\\}$
|  | $\wedge\quad\forall j\in\left\llbracket\,1\ ;\ k+k^{\prime}\,\right\rrbracket\ \exists j^{\prime}\in\left\llbracket\,0\ ;\ j-1\,\right\rrbracket$
|  | $\qquad\qquad\forall i\in\left\llbracket\,0\ ;\ i_{j}-1\,\right\rrbracket\quad P_{j^{\prime}}(i)=P_{j}(i)$
|  | $\wedge\quad\forall j\in\left\llbracket\,k+1\ ;\ k+k^{\prime}\,\right\rrbracket$
|  | $\qquad\qquad\operatorname{supp}P_{j}(i_{j})\subseteq\operatorname{supp}e^{(\beta,m)}\left(\begin{matrix}P_{0};P_{1},\dots,P_{k}\\\ P_{k+1},\dots,P_{j}\\\ i_{1},\dots,i_{j}\end{matrix}\right)$

$E_{1}^{(\beta,m)}=\left\\{\begin{array}[]{@{}r@{\quad}l@{}}\underset{k\in\mathbb{N},\
k^{\prime}\in\mathbb{N}^{*}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}}^{(\beta,m)}&(\beta,m)\neq(\alpha,n)\\\
\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,0}^{(\beta,m)}&(\beta,m)=(\alpha,n)\end{array}\right.$

Define sets $E_{2}^{(\beta,m)}$ as follows:

  * •

If $(\beta,m)\neq(\alpha,n)$, then

$-\underset{\ell=m+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{\ell}\kappa_{-\beta}-\underset{\gamma^{\prime}<\gamma,\
\ell\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\gamma^{\prime}}-\underset{\ell=1}{\overset{p}{\sum}}\ln_{\ell}\kappa_{-\gamma}\in
E_{2}^{(\beta,m)}$

iff $\gamma>\beta$, $p\in\mathbb{N}$ and there is some
$P\in\mathcal{P}_{\mathbb{L}}(\varepsilon)$ such that
$\kappa_{-\gamma}\succeq^{K}P(k_{P})$

  * •

If $(\beta,m)=(\alpha,n)$, then

$-\underset{j=0}{\overset{\infty}{\sum}}\ln|P_{0}(j)|-\underset{\gamma>\zeta>\alpha,\
\ell\in\mathbb{N}^{*}}{\overset{}{\sum}}\ln_{\ell}\kappa_{-\zeta}-\underset{\ell=1}{\overset{p}{\sum}}\ln_{\ell}\kappa_{-\gamma}\in
E_{2}^{(\beta,m)}$

iff $\gamma>\alpha$, $p\in\mathbb{N}$, $P_{0}\in\mathcal{P}_{0}(x)$,
$P_{0}(k_{P_{0}})\succ^{K}\kappa_{-\gamma}$ and there is some
$P\in\mathcal{P}_{\mathbb{L}}(\varepsilon)$ such that
$\kappa_{-\gamma}\succeq^{K}P(k_{P})$.

Let also

| $E_{3}^{(\beta,m)}$ | $\ =\ $ | $\left\\{\begin{array}[]{@{}r<EMAIL_ADDRESS>\vphantom{p\geq m+2}\right|\ p\geq m+2\right\\}&(\beta,m)\neq(\alpha,n)\\\ \varnothing&(\beta,m)=(\alpha,n)\end{array}\right.$
---|---|---|---
| $E^{(\beta,m)}$ | $\ =\ $ | $E_{1}^{(\beta,m)}\cup E_{2}^{(\beta,m)}\cup E_{3}^{(\beta,m)}$

and $\left\langle E^{(\beta,m)}\right\rangle$ be the monoid it generates.
Finally, let $H^{(\beta,m)}$ defined by cases as follows:

  * •

If $(\beta,m)\neq(\alpha,n)$, then

| $H^{(\beta,m)}$ | $\ =\ $ | $\left\\{\left.\begin{array}[]{r}\underset{\footnotesize\begin{array}[]{c}(\zeta,p)>_{lex}(\beta,m)\\\ \zeta\ |\ \kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{(\beta,m)<_{lex}(\zeta,p)<_{lex}(\alpha,n)}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}\\\ -\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\end{array}\ \vphantom{P_{0}\in\mathcal{P}_{0}(x)}\right|\ P_{0}\in\mathcal{P}_{0}(x)\right\\}$
---|---|---|---
|  |  | $\qquad\bigcup\left\\{\left.\underset{\footnotesize\begin{array}[]{c}(\zeta,p)\geq_{lex}(\alpha,n)\\\ \zeta\ |\ \kappa_{-\zeta}\succeq^{K}P_{0}(k_{P_{0}})\end{array}}{\overset{}{\sum}}\ln_{p}\kappa_{-\zeta}-\underset{i=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|\ \vphantom{P_{0}\in\mathcal{P}_{0}(x)}\right|\ P_{0}\in\mathcal{P}_{0}(x)\right\\}$

  * •

If $(\beta,m)=(\alpha,n)$, then

$H^{(\beta,m)}=\left\\{\left.-\ln|P_{0}(x)|\
\vphantom{P_{0}\in\mathcal{P}_{0}(x)}\right|\
P_{0}\in\mathcal{P}_{0}(x)\right\\}$

Let
$b\in\underset{q=0}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\bigcup}}\operatorname{supp}\Phi^{q}(x)$.
Then, there are $\eta\in H^{(\beta,m)}$ and $y\in\left\langle
E^{(\beta,m)}\right\rangle$ such that

$\omega^{b}\asymp\mathchoice{\dfrac{\partial\ln_{m}\kappa_{-\beta}}{\partial
u}}{\dfrac{\partial\ln_{m}\kappa_{-\beta}}{\partial
u}}{\frac{\partial\ln_{m}\kappa_{-\beta}}{\partial
u}}{\frac{\partial\ln_{m}\kappa_{-\beta}}{\partial u}}\exp(\eta+y)$

###### Proof.

Since $\Phi$ is strongly linear, we just need to apply Proposition 7.4 to each
term of $x$. Each path of $\mathcal{P}_{0}(x)$ involved is shifted one rank.
In $H^{(\beta,m)}$ we the add $\ln|P_{0}(0)|$ compare to Proposition 7.4. Then
$\exp(\eta)$ gives also $|\partial u\exp\varepsilon|$. We just remove it so
that it does not appear twice. ∎

######  Proposition 7.5.

Let $x$ be a surreal number such that there $u$ of the form
$u=\ln_{n}\kappa_{-\alpha}$, $r\in\mathbb{R}^{*}$ and
$a_{0}\in\textnormal{No}$ such that $r\omega^{a_{0}}\succ\ln u$ and

$\forall a\in\operatorname{supp}x\quad\exists\varepsilon\sim
r\omega^{a_{0}}\qquad\omega^{a}\asymp\partial u\exp\varepsilon$

Let $\mathcal{P}_{0}(x)=\left\\{\left.P\in\mathcal{P}_{\mathbb{L}}(x)\
\vphantom{\begin{array}[]{c}P(1)=r\omega^{a_{0}}\\\ \forall i\geq 1\quad
P(i+1)\sim\ln|P(i)\end{array}}\right|\
\begin{array}[]{c}P(1)=r\omega^{a_{0}}\\\ \forall i\geq 1\quad
P(i+1)\sim\ln|P(i)\end{array}\right\\}$

Consider $E_{1}^{(\beta,m)}$, $E_{2}^{(\beta,m)}$ and $E_{3}^{(\beta,m)}$ as
defined in Corollary 7.1.2. Let $\xi$ be the smallest ordinal such that
$\kappa_{-\xi}\prec^{K}P(k_{P})$ for all path
$P\in\mathcal{P}_{\mathbb{L}}(x)$. Let $\lambda$ the least
$\varepsilon$-number greater than $\operatorname{NR}(x)$ and $\xi$. Then
$E^{(\beta,m)}=E_{1}^{(\beta,m)}\cup E_{2}^{(\beta,m)}\cup E_{3}^{(\beta,m)}$
is reverse well-ordered with order type at most $2\lambda+\omega(\xi+1)$.

###### Proof.

First notice that $E_{3}^{(\beta,m)}$ is reverse well-ordered with order type
at most $\omega$. $E_{2}^{(\beta,m)}$ is also reverse well-ordered with order
at most $\omega\otimes\xi$. We then focus on $E_{1}^{(\beta,m)}$. For the
moment, we will assume $(\beta,m)<_{lex}(\alpha,n)$.

  1. (i)

We first claim that for all $i\geq 3$ and all path $P\in\mathcal{P}(x)$,
$P(i)\prec P(2)\preceq\ln_{m+2}\kappa_{-\beta}$. It is indeed the same proof
as the point (i) of the proof of Proposition 7.1.1.

  2. (ii)

We claim that for all path $P\in\mathcal{P}(x)$, if
$P(2)\asymp\ln_{m+2}\kappa_{-\beta}$, then, if $r$ is the real number such
that $P(2)\sim r\ln_{m+2}\kappa_{-\beta}$, we have $0<r\leq 1$. It is indeed
the same proof as the point (ii) of the proof of Proposition 7.1.1.

  3. (iii)

For all $j$ and $i\geq 2$,
$\ln|P_{j}(i)|\preceq\ln_{m+3}\kappa_{-\beta}\prec\ln_{m+2}\kappa_{-\beta}$.
Indeed, using (i), we know that $P_{j}(i)\preceq\ln_{m+2}\kappa_{-\beta}$.
Then, there is a natural number $m\geq 1$ such that $|P_{j}(i)|\leq
m\ln_{m+2}\kappa_{-\beta}$. Using the fact that $\ln$ is increasing,

$\ln|P_{j}(i)|\leq\ln_{m+3}\kappa_{-\beta}+\ln
m\preceq\ln_{m+3}\kappa_{-\beta}\prec\ln_{m+2}\kappa_{-\beta}$

  4. (iv)

We now claim that
$\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}}^{(\beta,m)}>\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}+2}^{(\beta,m)}$.
Indeed, using (ii) and (iii) if
$e_{1}\in\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}}^{(\beta,m)}$,
then there is
$s\in\mathchoice{\left[\,-(k+1)\,;-k\,\right]}{\left[\,-(k+1)\,;-k\,\right]}{\left[\,-(k+1)\,;\,-k\,\right]}{\left[\,-(k+1)\,;\,-k\,\right]}$
such that $e_{1}\sim s\ln_{m+2}\kappa_{-\beta}$. Similarly, for
$e_{2}\in\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}+2}^{(\beta,m)}$,
there is
$s^{\prime}\in\mathchoice{\left[\,-(k+3)\,;-(k+2)\,\right]}{\left[\,-(k+3)\,;-(k+2)\,\right]}{\left[\,-(k+3)\,;\,-(k+2)\,\right]}{\left[\,-(k+3)\,;\,-(k+2)\,\right]}$
such that $e_{2}\sim s^{\prime}\ln_{m+2}\kappa_{-\beta}$.

  5. (v)

We define the following sequence :

     * •

$a_{0}=1$

     * •

$a_{k+1}=\omega^{\omega^{\omega(\omega(\operatorname{NR}(x)+\xi+1)a_{k}+1)}}$

We show that $E_{1,k,0}^{(\beta,m)}$ is reverse well-ordered with order type
less than $a_{k}$. We also claim that the equivalence classes of
$E_{1,k,0}^{(\beta,m)}/{\asymp}$ are finite and that

$\operatorname{NR}\left(\underset{t\in
E_{1,k,0}^{(\beta,m)}}{\overset{}{\sum}}\exp
t\right)\leq\omega(\operatorname{NR}(x)+\xi+1)a_{k}$

We show it by induction on $k\in\mathbb{N}$.

     * •

For $k=0$, $E_{1,0,0}^{(\beta,m)}=\\{0\\}$. Then it is reverse well-ordered
with order type $1$. We also have

$\operatorname{NR}\left(\underset{t\in
E_{1,0,0}^{(\beta,m)}}{\overset{}{\sum}}\exp
t\right)=\operatorname{NR}(1)=1\leq\omega(\operatorname{NR}(x)+\xi+1)$

     * •

Assume the property for some $k\in\mathbb{N}$. Let $t\in
E_{1,k+1,0}^{(\beta,m)}$. Let
$(P_{0},0),(P_{1},i_{1}),\dots,(P_{k+1},i_{k+1})$ minimal for the order
$(<_{lex},<)_{lex}$ such that

$t=e^{(\beta,m)}\left(\begin{matrix}P_{0};P_{1},\dots,P_{k+1}\\\
\varnothing\\\ i_{1},\dots,i_{k+1}\end{matrix}\right)$

Then,

| $t$ | $\ =\ $ | $e^{(\beta,m)}\left(\begin{matrix}P_{0};P_{1}\dots,P_{k}\\\ \varnothing\\\ i_{1},\dots,i_{k}\end{matrix}\right)-\underset{i=1}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{0}(i)|$
---|---|---|---
|  |  | $\qquad-\underset{\footnotesize\begin{array}[]{c}\gamma\geq\alpha,m\in\mathbb{N}^{*}\\\ \gamma\ |\ P_{0}(k_{P_{0}})\succ^{K}\kappa_{-\gamma}\succeq^{K}P_{k+1}(k_{P_{k+1}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\gamma}+\underset{i=i_{k+1}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{k+1}(i)|$

Write $s=e^{(\beta,m)}\left(\begin{matrix}P_{0};P_{1},\dots,P_{k}\\\
\varnothing\\\ i_{1},\dots,i_{k}\end{matrix}\right)$ and consider the
following path:

$\left\\{\begin{array}[]{@{}r@{\quad}l@{}}R(0)=\exp s\\\
R(i)=P_{k+1}(i-1+i_{k+1})&i>0\end{array}\right.$

It is indeed a path since, by definition of $E_{1,k+1,0}^{(\beta,m)}$,
$\operatorname{supp}P_{k+1}(i_{k+1})$ must be contained in
$\operatorname{supp}s$. We then have,

$\exp t=\mathchoice{\dfrac{\partial R}{\partial P_{0}[1:]}}{\dfrac{\partial
R}{\partial P_{0}[1:]}}{\frac{\partial R}{\partial P_{0}[1:]}}{\frac{\partial
R}{\partial P_{0}[1:]}}$

Moreover, $R\in\mathcal{P}_{\mathbb{L}}\left(\underset{s\in
E_{1,k,0}^{(\beta,m)}}{\overset{}{\sum}}\exp s\right)$. By assumption on $x$,
the set $\left\\{\left.P_{0}[1:]\
\vphantom{P_{0}\in\mathcal{P}_{0}(x)}\right|\
P_{0}\in\mathcal{P}_{0}(x)\right\\}$ is a singleton. Therefore, so is the set
$\left\\{\left.\partial P_{0}[1:]\
\vphantom{P_{0}\in\mathcal{P}_{0}(x)}\right|\
P_{0}\in\mathcal{P}_{0}(x)\right\\}$. By induction hypothesis and Proposition
1, $E_{1,k+1,0}^{(\beta,m)}$ has order type less than

$\omega^{\omega^{\omega(\omega(\operatorname{NR}(x)+\xi+1)a_{k}+1)}}=a_{k+1}$

Since the equivalences classes of
$\mathcal{P}_{\mathbb{L}}\left(\underset{s\in
E_{1,k,0}^{(\beta,m)}}{\overset{}{\sum}}\exp s\right)/{\asymp}$ are finite,
the ones of $E_{1,k+1,0}^{(\beta,m)}/{\asymp}$ are also finite. Finally, using
Lemmas 6.2.2 and 6.2.2,

| $\operatorname{NR}(t)$ | $\ \leq\ $ | $(\omega\otimes\xi)+\underset{i=1}{\overset{k_{P_{0}}-1}{\sum}}\operatorname{NR}(\ln|P_{0}(i)|)+k_{P_{0}}$
---|---|---|---
|  |  | $\qquad+\underset{j=1}{\overset{k+1}{\sum}}\underset{i=i_{j}}{\overset{k_{P_{j}}-1}{\sum}}\operatorname{NR}\left(\ln|P_{j}(i)|\right)+\underset{j=0}{\overset{k+1}{\sum}}\max(0,k_{P_{j}}-i_{j})+4$
|  | $\ \leq\ $ | $\omega(\operatorname{NR}(x)+\xi+1)$

Then, $\operatorname{NR}\left(\underset{t^{\prime}\in
E_{1,k+1}^{(\beta,m)}}{\overset{}{\sum}}\exp
t^{\prime}\right)\leq\omega(\operatorname{NR}(x)+\xi+1)a_{k+1}$

We conclude thanks to the induction principle.

  6. (vi)

We have
$\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,0}^{(\beta,m)}\subseteq\left\langle
E_{1,1,0}^{(\beta,m)}\right\rangle$. Then, using (v) and applying Proposition
2, it has order type at most $\omega^{\widehat{a_{1}}}\leq\omega^{\omega
a_{1}}$.

  7. (vii)

We define the following sequence :

     * •

$b_{0}=\omega^{\widehat{a_{1}}}$

     * •

$b_{k^{\prime}+1}=\omega^{\omega^{\omega(\omega(\operatorname{NR}(x)+\xi+4)b_{k^{\prime}}+1)}}$

We show that
$\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}}^{(\beta,m)}$
is reverse well-ordered with order type less than $b_{k^{\prime}}$. We also
claim that the equivalence classes of
$\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}}^{(\beta,m)}/{\asymp}$
are finite and that

$\operatorname{NR}\left(\underset{t\in\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}}^{(\beta,m)}}{\overset{}{\sum}}\exp
t\right)\leq\omega(\operatorname{NR}(x)+\xi+4)b_{k^{\prime}}$

We show it by induction on $k^{\prime}\in\mathbb{N}$.

     * •

For $k^{\prime}=0$, we just apply (vi).

     * •

Assume the property for some $k^{\prime}\in\mathbb{N}$. Let
$t\in\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}+1}^{(\beta,m)}$.
Let $(P_{0},0)(P_{1},i_{1}),\dots,(P_{k+k^{\prime}+1},i_{k+k^{\prime}+1})$
minimal for the order $(<_{lex},<)_{lex}$ such that
$t=e^{(\beta,m)}\left(\begin{matrix}P_{0};P_{1},\dots,P_{k}\\\
P_{k+1},\dots,P_{k+k^{\prime}+1}\\\
i_{1},\dots,i_{k+k^{\prime}+1}\end{matrix}\right)$. Then,

| $t$ | $\ =\ $ | $e^{(\beta,m)}\left(\begin{matrix}P_{0};P_{1},\dots,P_{k}\\\ P_{k+1},\dots,P_{k+k^{\prime}}\\\ i_{1},\dots,i_{k+k^{\prime}}\end{matrix}\right)-\underset{\ell=m+2}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln_{\ell}{\kappa_{-\beta}}$
---|---|---|---
|  |  | $\qquad-\underset{\footnotesize\begin{array}[]{c}\gamma>\beta,\ell\in\mathbb{N}^{*}\\\ \gamma\ |\ \kappa_{-\gamma}\succeq^{K}P_{k+k^{\prime}+1}(k_{P_{k+k^{\prime}+1}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}+\underset{i=i_{k+1}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{k+1}(i)|$

Write $s=e^{(\beta,m)}\left(\begin{matrix}P_{0};P_{1},\dots,P_{k}\\\
P_{k+1},\dots,P_{k+k^{\prime}}\\\
i_{1},\dots,i_{k+k^{\prime}}\end{matrix}\right)$. We then have,

| $\partial(\ln_{m+1}\kappa_{-\beta})\exp t$ | $\ =\ $ | $\exp(s)\exp\left(\underset{i=i_{k+1}}{\overset{{\mbox{\raisebox{0.56905pt}{\tiny{$+$}}}\infty}}{\sum}}\ln|P_{k+1}(i)|\right.$
---|---|---|---
|  |  | $\left.\qquad-\underset{\footnotesize\begin{array}[]{c}\ell\in\mathbb{N}^{*}\\\ \gamma\ |\ \kappa_{-\gamma}\succeq^{K}P_{k+k^{\prime}+1}(k_{P_{k+k^{\prime}+1}})\end{array}}{\overset{}{\sum}}\ln_{m}\kappa_{-\beta}\right)$

Consider the following path:

$\left\\{\begin{array}[]{@{}r@{\quad}l@{}}R(0)=\exp s\\\
R(i)=P_{k+k^{\prime}+1}(i-1+i_{k+1})&i>0\end{array}\right.$

It is indeed a path since, by definition of
$E_{1,k,k^{\prime}+1}^{(\beta,m)}$,
$\operatorname{supp}P_{k+k^{\prime}+1}(i_{k+k^{\prime}+1})$ must be contained
in $\operatorname{supp}s$. Then,

$\partial(\ln_{m+1}\kappa_{-\beta})\exp t=\partial R$

Moreover,
$R\in\mathcal{P}_{\mathbb{L}}\left(\underset{s\in\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}}^{(\beta,m)}}{\overset{}{\sum}}\exp
s\right)$. By induction hypothesis and Proposition 1,
$\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}+1}^{(\beta,m)}$
has order type less than

$\omega^{\omega^{\omega(\omega(\operatorname{NR}(x)+\xi+4)b_{k^{\prime}}+1)}}=b_{k^{\prime}+1}$

Since the equivalences classes of
$\mathcal{P}_{\mathbb{L}}\left(\underset{s\in\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}}^{(\beta,m)}}{\overset{}{\sum}}\exp
s\right)/{\asymp}$ are finite, the ones of
$\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}+1}^{(\beta,m)}/{\asymp}$
are also finite. Finally, using Lemmas 6.2.2 and 6.2.2,

| $\operatorname{NR}(t)$ | $\ \leq\ $ | $(\omega\oplus\omega\otimes\xi\oplus\omega)+\underset{j=0}{\overset{k+1}{\sum}}\underset{i=i_{j}}{\overset{k_{P_{j}}-1}{\sum}}\operatorname{NR}\left(\ln|P_{j}(i)|\right)+\underset{j=0}{\overset{k+1}{\sum}}\max(0,k_{P_{j}}-i_{j})$
---|---|---|---
|  | $\ \leq\ $ | $\omega(\operatorname{NR}(x)+\xi+4)$

Then,
$\operatorname{NR}\left(\underset{t^{\prime}\in\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,k^{\prime}+1}^{(\beta,m)}}{\overset{}{\sum}}\exp
t^{\prime}\right)\leq\omega(\operatorname{NR}(x)+\xi+4)b_{k^{\prime}+1}$

We conclude thanks to the induction principle.

  8. (viii)

By easy induction, for all $k\in\mathbb{N}$, $b_{k^{\prime}}<\lambda$.

  9. (ix)

Using (iv), we get that for all $N\in\mathbb{N}$,
$\underset{k^{\prime}=0}{\overset{N}{\bigcup}}\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,,k,2k^{\prime}}^{(\beta,m)}$
is an initial segment of
$\underset{k^{\prime}\in\mathbb{N}}{\overset{}{\bigcup}}\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,2k^{\prime}}^{(\beta,m)}$.
We also have that
$\underset{k^{\prime}=0}{\overset{N}{\bigcup}}\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,2k^{\prime}+1}^{(\beta,m)}$
is an initial segment of
$\underset{k^{\prime}\in\mathbb{N}}{\overset{}{\bigcup}}\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,2k^{\prime}+1}^{(\beta,m)}$.
Using (vii), we get that
$\underset{k^{\prime}\in\mathbb{N}}{\overset{}{\bigcup}}\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,2k^{\prime}}^{(\beta,m)}$
has order type at most

$\sup\left\\{\left.\underset{k=0}{\overset{N}{\oplus}}b_{2k^{\prime}}\
\vphantom{N\in\mathbb{N}}\right|\
N\in\mathbb{N}\right\\}=\sup\left\\{\left.b_{2N}\
\vphantom{N\in\mathbb{N}}\right|\ N\in\mathbb{N}\right\\}\underset{\text{by
(viii)}}{\leq}\lambda$

Similarly,
$\underset{k^{\prime}\in\mathbb{N}}{\overset{}{\bigcup}}\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,2k^{\prime}+1}^{(\beta,m)}$
has order type at most $\lambda$. Using Proposition 2, we conclude that
$E_{1}^{(\beta,m)}$ has order type at most $2\lambda$.

Now we deal with the case $(\beta,m)=(\alpha,n)$. A close looking at point (v)
above reveals that the property it shows does not depend on $(\beta,m)$. Then
we have, using a similar argument as in points (viii) and (ix), that
$\underset{k\in\mathbb{N}}{\overset{}{\bigcup}}E_{1,k,0}^{(\alpha,n)}$ has
order type at most $2\lambda$. Then, for any $(\beta,m)\leq_{lex}(\alpha,n)$,
$E_{1}^{(\beta,m)}$ is reverse well-ordered with order type at most
$2\lambda$. Using again Proposition 2 and the properties of
$E_{2}^{(\beta,m)}$ and $E_{3}^{(\beta,m)}$ mentioned in the beginning of this
proof, we get that $E^{(\beta,m)}$ is reverse well-ordered with order type at
most $2\lambda+\omega(\xi+1)$. ∎

###  7.2 Length of the series of the anti-derivative of an arbitrary surreal
number

See 1

###### Proof.

Let $\alpha<\gamma$ and $n\in\mathbb{N}$. Write
$x=\underset{a\in\operatorname{supp}x}{\overset{}{\sum}}r_{a}\omega^{a}$. For
any ordinal $\alpha<\gamma$, $n\in\mathbb{N}$,
$r\in\mathbb{R}\setminus\\{-1\\}$ and any term $s\omega^{a_{0}}$, define
$S_{\alpha,n,1,s\omega^{a_{0}}}$ to be the set

$\left\\{\left.a\in\operatorname{supp}x\
\vphantom{\exists\varepsilon\in\textnormal{No}_{\infty}\
\forall(\beta,m)<_{lex}(\alpha,n)\quad\left\\{\begin{array}[]{@{}r@{\quad}l@{}}\ln_{n}\kappa_{-\alpha}\prec\varepsilon\\\
\varepsilon\sim s\omega^{a_{0}}\\\ \varepsilon\ln_{m}\kappa_{-\beta}\\\
\omega^{a}\asymp\partial(\ln_{n}\kappa_{-\alpha})\exp\varepsilon\end{array}\right.}\right|\
\exists\varepsilon\in\textnormal{No}_{\infty}\
\forall(\beta,m)<_{lex}(\alpha,n)\quad\left\\{\begin{array}[]{@{}r@{\quad}l@{}}\ln_{n}\kappa_{-\alpha}\prec\varepsilon\\\
\varepsilon\sim s\omega^{a_{0}}\\\ \varepsilon\ln_{m}\kappa_{-\beta}\\\
\omega^{a}\asymp\partial(\ln_{n}\kappa_{-\alpha})\exp\varepsilon\end{array}\right.\right\\}$

and consider also

$S_{\alpha,n,2,r}=\left\\{\left.a\in\operatorname{supp}x\
\vphantom{\exists\varepsilon\in\textnormal{No}_{\infty}\quad\varepsilon\sim
r\ln_{n}\kappa_{-\alpha}\wedge\omega^{a}\asymp\partial(\ln_{n}\kappa_{-\alpha})\exp\varepsilon}\right|\
\exists\varepsilon\in\textnormal{No}_{\infty}\quad\varepsilon\sim
r\ln_{n}\kappa_{-\alpha}\wedge\omega^{a}\asymp\partial(\ln_{n}\kappa_{-\alpha})\exp\varepsilon\right\\}$

$x_{\alpha,n,1,s\omega^{a_{0}}}=\underset{a\in
S_{\alpha,n,1,s\omega^{a_{0}}}}{\overset{}{\sum}}r_{a}\omega^{a}\qquad\text{and}\qquad
x_{\alpha,n,2,r}=\underset{a\in
S_{\alpha,n,2,r}}{\overset{}{\sum}}r_{a}\omega^{a}$

All theses surreal numbers have disjoint supports and

$x=\underset{s\omega^{a_{0}}\in\mathbb{R}\omega^{\textnormal{No}}}{\overset{}{\sum}}\underset{\alpha<\gamma}{\overset{}{\sum}}\underset{n\in\mathbb{N}}{\overset{}{\sum}}x_{\alpha,n,1,s\omega^{a_{0}}}+\underset{r\in\mathbb{R}\setminus\\{-1\\}}{\overset{}{\sum}}\underset{\alpha<\gamma}{\overset{}{\sum}}\underset{n\in\mathbb{N}}{\overset{}{\sum}}x_{\alpha,n,2,r}$

We then study both sums of the above equality.

  * •

The set $\left\\{\left.r\in\mathbb{R}\setminus\\{-1\\}\
\vphantom{S_{\alpha,n,2,r}\neq\varnothing}\right|\
S_{\alpha,n,2,r}\neq\varnothing\right\\}$ is reverse well-ordered with order
type at most $\nu(x)$. Let

$S_{0}=\underset{i\in\mathbb{N}}{\overset{}{\bigcup}}\operatorname{supp}\Phi^{i}\left(\underset{r\in\mathbb{R}\setminus\\{-1\\}}{\overset{}{\sum}}\underset{\alpha<\gamma}{\overset{}{\sum}}\underset{n\in\mathbb{N}}{\overset{}{\sum}}x_{\alpha,n,2,r}\right)$

Since $\Phi$ is strongly linear,

$S_{0}\subseteq\underset{r\in\mathbb{R}\setminus\\{-1\\}}{\overset{}{\bigcup}}\underset{\alpha<\gamma}{\overset{}{\bigcup}}\underset{n\in\mathbb{N}}{\overset{}{\bigcup}}\underset{i\in\mathbb{N}}{\overset{}{\bigcup}}\operatorname{supp}\Phi^{i}(x_{\alpha,n,2,r})$

Using Corollary 7.1,
$\underset{i\in\mathbb{N}}{\overset{}{\bigcup}}\operatorname{supp}\Phi^{i}(x_{\alpha,n,2,r})$
is reverse well-ordered with order type at most
$\omega^{\omega(2\lambda+\omega(\gamma+1)+1)}$. Moreover, Lemma 7.1.1 ensure
that is $(\alpha,n,r)>_{lex}(\alpha^{\prime},n^{\prime},r^{\prime})$, then
$\underset{i\in\mathbb{N}}{\overset{}{\bigcup}}\operatorname{supp}\Phi^{i}(x_{\alpha,n,2,r})<\underset{i\in\mathbb{N}}{\overset{}{\bigcup}}\operatorname{supp}\Phi^{i}(x_{\alpha^{\prime},n^{\prime},2,r^{\prime}})$.
We end up with the fact that
$\underset{i\in\mathbb{N}}{\overset{}{\bigcup}}\operatorname{supp}\Phi^{i}\left(\underset{r\in\mathbb{R}\setminus\\{-1\\}}{\overset{}{\sum}}\underset{\alpha<\gamma}{\overset{}{\sum}}\underset{n\in\mathbb{N}}{\overset{}{\sum}}x_{\alpha,n,2,r}\right)$
is reverse well-ordered with order type at most
$\omega^{\omega(2\lambda+\omega(\gamma+1)+1)}\nu(x)\gamma$.

  * •

Let
$S_{1}=\underset{i\in\mathbb{N}}{\overset{}{\bigcup}}\operatorname{supp}\Phi^{i}\left(\underset{s\omega^{a_{0}}\in\mathbb{R}\omega^{\textnormal{No}}}{\overset{}{\sum}}\underset{\alpha<\gamma}{\overset{}{\sum}}\underset{n\in\mathbb{N}}{\overset{}{\sum}}x_{\alpha,n,1,s\omega^{a_{0}}}\right)$

Since $\Phi$ is strongly linear,

$S_{1}\subseteq\underset{s\omega^{a_{0}}\in\mathbb{R}\omega^{\textnormal{No}}}{\overset{}{\bigcup}}\underset{\alpha<\gamma}{\overset{}{\bigcup}}\underset{n\in\mathbb{N}}{\overset{}{\bigcup}}\underset{i\in\mathbb{N}}{\overset{}{\bigcup}}\operatorname{supp}\Phi^{i}(x_{\alpha,n,1,s\omega^{a_{0}}})$

Denote $H^{(\beta,m)}(x_{\alpha,n,1,s\omega^{a_{0}}})$,
$E^{(\beta,m)}(x_{\alpha,n,1,s\omega^{a_{0}}})$ the sets defined as in
Corollary 7.1.2 for $x_{\alpha,n,1,s\omega^{a_{0}}}$. Then, using this
corollary, $S_{1}$ is contained in the set

$\underset{\tiny\begin{array}[]{c}\beta<\gamma\\\ m\in\mathbb{N}\\\
s\omega^{a_{0}}\in\mathbb{R}\omega^{\textnormal{No}}\end{array}}{\overset{}{\bigcup}}\underset{\tiny\begin{array}[]{c}\alpha,n\
|\ (\beta,m)\leq_{lex}(\alpha,n)\\\
\alpha<\gamma\end{array}}{\overset{}{\bigcup}}\underset{\tiny\begin{array}[]{c}\eta\in
H^{(\beta,m)}(x_{\alpha,n,1,s\omega^{a_{0}}})\\\ y\in\left\langle
E^{(\beta,m)}(x_{\alpha,n,1,s\omega^{a_{0}}})\right\rangle\end{array}}{\overset{}{\bigcup}}\operatorname{supp}\left(\mathchoice{\dfrac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}{\dfrac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}{\frac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}{\frac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}\exp(\eta+y)\right)$

We also know that
$(\alpha,n,s\omega^{a_{0}})>_{lex}(\alpha^{\prime},n^{\prime},s^{\prime}\omega^{a_{0}^{\prime}})$,
then the set

$\underset{\tiny\begin{array}[]{c}\eta\in
H^{(\beta,m)}(x_{\alpha,n,1,s\omega^{a_{0}}})\\\ y\in\left\langle
E^{(\beta,m)}(x_{\alpha,n,1,s\omega^{a_{0}}})\right\rangle\end{array}}{\overset{}{\bigcup}}\operatorname{supp}\left(\mathchoice{\dfrac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}{\dfrac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}{\frac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}{\frac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}\exp(\eta+y)\right)$

is contained in the set

$\underset{\tiny\begin{array}[]{c}\eta\in
H^{(\beta,m)}\left(x_{\alpha^{\prime},n^{\prime},1,s^{\prime}\omega^{a_{0}^{\prime}}}\right)\\\
y\in\left\langle
E^{(\beta,m)}\left(x_{\alpha^{\prime},n^{\prime},1,s^{\prime}\omega^{a_{0}^{\prime}}}\right)\right\rangle\end{array}}{\overset{}{\bigcup}}\operatorname{supp}\left(\mathchoice{\dfrac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}{\dfrac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}{\frac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}{\frac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}\exp(\eta+y)\right)$

Propositions 7.5 and 2 guarantee that all of theses sets are reverse well-
ordered with order type less than $\omega^{2\omega\lambda}$. Let

$S_{\beta,m}=\underset{\tiny\begin{array}[]{c}\alpha,n\ |\
(\beta,m)\leq_{lex}(\alpha,n)\\\ \alpha<\gamma\\\
s\omega^{a_{0}}\in\mathbb{R}\omega^{\textnormal{No}}\end{array}}{\overset{}{\bigcup}}\underset{\tiny\begin{array}[]{c}\eta\in
H^{(\beta,m)}(x_{\alpha,n,1,s\omega^{a_{0}}})\\\ y\in\left\langle
E^{(\beta,m)}(x_{\alpha,n,1,s\omega^{a_{0}}})\right\rangle\end{array}}{\overset{}{\bigcup}}\operatorname{supp}\left(\mathchoice{\dfrac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}{\dfrac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}{\frac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}{\frac{\partial\ln_{m}\kappa_{-\beta}}{\partial\ln_{n}\kappa_{-\alpha}}}\exp(\eta+y)\right)$

The set of possible $s\omega^{a_{0}}$ is reverse well-ordered with order type
at most $\nu(x)$. Moreover, $\alpha$ and $n$ are determined from
$s\omega^{a_{0}}$. Then $S_{\beta,m}$ is reverse well-ordered with order type
at most $\omega^{2\omega^{\lambda+1}}\nu(x)$. Finally, if
$(\beta,m)>_{lex}(\beta^{\prime},m^{\prime})$, then $S_{\beta,m}\subsetneq
S_{\beta^{\prime},m^{\prime}}$ and there are at most $\omega\gamma$ such
couples. Then, $S_{1}$ is reverse well-ordered with order type at most
$\omega^{2\omega\lambda+1}\nu(x)\gamma$.

Both sets have order type less than $\omega^{\omega^{\lambda+2}}$, which is a
multiplicative ordinal. Using Proposition 2,
$\underset{i\in\mathbb{N}}{\overset{}{\bigcup}}\operatorname{supp}\Phi^{i}(x)$
has order type less than $\omega^{\omega^{\lambda+2}}$.

∎

###  7.3 Stability of some surreal fields

We are ready to exhibit a surreal field that is stable under
$\exp,\ln,\partial$ and anti-derivation and that is not No itself. We actually
have a lot of such fields.

To get a field stable under derivation and anti-derivation, it sufficient that
the field is stable under $\exp$ and $\ln$, and, if for all $\beta<\alpha$,
$\kappa_{-\beta}$ is in the field, and if $\omega\otimes\alpha$ is less than
the authorized length of a serie, then $\kappa_{-\alpha}$ must also be in the
field.

See 1

###### Proof.

Let
$\mathbb{K}=\underset{\beta<\alpha}{\overset{}{\bigcup}}\mathbb{R}_{\varepsilon_{\beta}}^{\Gamma_{\beta}^{\uparrow\varepsilon_{\beta}}}$
As an increasing union of fields, $\mathbb{K}$ is indeed a field.

  1. (i)

Using Theorem 1.2, each field
$\mathbb{R}_{\varepsilon_{\beta}}^{\Gamma_{\beta}^{\uparrow\varepsilon_{\beta}}}$
is stable under $\exp$ and $\ln$, then so is $\mathbb{K}$.

  2. (ii)

Write
$\Gamma_{\beta}^{\uparrow\varepsilon_{\beta}}=\left(\Gamma_{i,\beta}\right)_{i<\gamma_{\varepsilon_{\beta}}}$.
We use the notation introduce in the beginning Definition 1. We prove by
induction on $i<\gamma_{\varepsilon_{\beta}}$ that for $x\in\Gamma_{i,\beta}$,
$\operatorname{NR}(\omega^{x})<\eta_{\beta}e_{i}$.

     * •

For $i=0$ we have $e_{0}=1$ and $\Gamma_{0,\beta}=\Gamma_{\beta}$. By
assumption on $\Gamma_{\beta}$, for all $x\in\Gamma_{0,\beta}$,
$\operatorname{NR}(\omega^{x})<\eta_{\beta}=\eta_{\beta}e_{0}$.

     * •

Assume the property for some ordinal $i$. Then let $x\in\Gamma_{i+1,\beta}$.
Write $x=u+v+\underset{k=1}{\overset{p}{\sum}}h(w_{k})$ with
$u\in\Gamma_{i,\beta}$,
$v\in\mathbb{R}_{e_{i}}^{g\left((\Gamma_{i,\beta})_{+}^{*}\right)}$ and
$w_{k}$ such $r\omega^{w_{k}}$ is a term of some element
$y_{k}\in\Gamma_{i,\beta}$, for some $r\in\mathbb{R}$. Using Corollary 6.2.2,

$\operatorname{NR}(\omega^{x})\leq\operatorname{NR}(\omega^{u})+\operatorname{NR}(\omega^{v})+\underset{k=1}{\overset{p}{\sum}}\operatorname{NR}(\omega^{h(w_{k})})+p+1$

From induction hypothesis,

$\operatorname{NR}(\omega^{u})<\eta_{\beta}e_{i}$

Write $v=\underset{j<\nu}{\overset{}{\sum}}r_{j}\omega^{g(a_{j})}$. Then
$\omega^{v}=\exp\left(\underset{j<\nu}{\overset{}{\sum}}r_{j}\omega^{a_{j}}\right)$.
From induction hypothesis,
$\operatorname{NR}(\omega^{a_{j}})<\eta_{\beta}e_{i}$. Then
$\operatorname{NR}(r_{j}\omega^{a_{j}})<\eta_{\beta}e_{i}+1$. Then

$\operatorname{NR}(\omega^{v})=\operatorname{NR}\left(\underset{j<\nu}{\overset{}{\sum}}r_{j}\omega^{a_{j}}\right)\leq(\eta_{\beta}e_{i}+1)\otimes\nu\leq(\eta_{\beta}e_{i}+1)\otimes
e_{i}\leq\eta_{\beta}e_{i}^{2}$

We also have

$\operatorname{NR}(\omega^{h(w_{k})})=\operatorname{NR}(\omega^{\omega^{w_{k}}})\leq\operatorname{NR}(\omega^{y_{k}})<\eta_{\beta}e_{i}$

Finally,
$\operatorname{NR}(\omega^{x})\leq(p+1)(\eta_{\beta}e_{i}+1)+\eta_{\beta}e_{i}^{2}<\eta_{\beta}e_{i+1}$

     * •

Assume $i$ is a limit ordinal. Then by definition of $\Gamma_{i,\beta}$ for
any $x\in\Gamma_{i,\beta}$ there is some $j<i$ such that
$x\in\Gamma_{j,\beta}$. Then induction hypothesis concludes.

  3. (iii)

Let $x\in\mathbb{K}$ and $\beta<\alpha$ such that
$x\in\mathbb{R}_{\varepsilon_{\beta}}^{\Gamma_{\beta}^{\uparrow\varepsilon_{\beta}}}$.
Using (ii), there is $i<\gamma_{\varepsilon_{\beta}}$ such that

$\operatorname{NR}(x)\leq(\eta_{\beta}e_{i}+1)\otimes\nu(x)<\eta_{\beta}\otimes\varepsilon_{\beta}=\varepsilon_{\beta}$

Since $\eta_{\beta}\otimes\varepsilon_{\beta}$ is a limit ordinal, then we
also have
$\operatorname{NR}(x)+1<\eta_{\beta}\otimes\varepsilon_{\beta}=\varepsilon_{\beta}$.

  4. (iv)

Let $x\in\mathbb{K}$ and $\beta<\alpha$ such that
$x\in\mathbb{R}_{\varepsilon_{\beta}}^{\Gamma_{\beta}^{\uparrow\varepsilon_{\beta}}}$.
Using (iii) and Proposition 1, $\nu(\partial
x)<\omega^{\omega^{\omega(\operatorname{NR}(x)+1)}}<\varepsilon_{\beta}$.
Using Corollary 1 and (i), we also have for all $P\in\mathcal{P}(x)$,
$\partial
P\in\mathbb{R}\omega^{\Gamma_{\beta}^{\uparrow\varepsilon_{\beta}}}$. Then,

$\partial
x\in\mathbb{R}_{\varepsilon_{\beta}}^{\Gamma_{\beta}^{\uparrow\varepsilon_{\beta}}}\subseteq\mathbb{K}$

Then $\mathbb{K}$ is stable under $\partial$.

  5. (v)

Let $x\in\mathbb{K}$ and $\beta<\alpha$ such that
$x\in\mathbb{R}_{\varepsilon_{\beta}}^{\Gamma_{\beta}^{\uparrow\varepsilon_{\beta}}}$.
Using Proposition 1 and the definition of $\mathcal{A}$,

$\nu\left(\mathcal{A}\circ\left(\underset{i\in\mathbb{N}}{\overset{}{\sum}}\Phi^{i}\right)(x)\right)<\omega^{\omega^{\lambda+2}}$

where $\lambda$ is least $\varepsilon$-number greater $\operatorname{NR}(x)$
and such that

$\forall
P\in\mathcal{P}_{\mathbb{L}}(x)\qquad\kappa_{-\lambda}\prec^{K}P(k_{P})$

Using (iii), $\operatorname{NR}(x)<\varepsilon_{\beta}$. Let
$P\in\mathcal{P}_{\mathbb{L}}(x)$. Using (i),
$\mathbb{R}_{\varepsilon_{\beta}}^{\Gamma_{\beta}^{\uparrow\varepsilon_{\beta}}}$
is stable under $\exp$ and $ln$. Since $P(i+1)$ is a term of $\ln|P(i)|$, if
$P(i)\in\omega^{\Gamma_{\beta}}$, then $P(i+1)\in\omega^{\Gamma_{\beta}}$. By
induction, $P(k_{P})\in\omega^{\Gamma_{\beta}}$. Since $P(k_{P})$ is
infinitely large, $P(k_{P})\in\omega^{\left(\Gamma_{\beta}\right)^{*}_{+}}$.
By assumption on $\Gamma_{\beta}$,
$P(k_{P})\succ^{K}\kappa_{-\varepsilon_{\beta}}$. Finally,
$\lambda\leq\varepsilon_{\beta}$ and

$\nu\left(\mathcal{A}\circ\left(\underset{i\in\mathbb{N}}{\overset{}{\sum}}\Phi^{i}\right)(x)\right)<\omega^{\omega^{\varepsilon_{\beta}+2}}<\varepsilon_{\beta+1}$

Propositions 7.1.1 and 7.4 and the third assumption about $\Gamma_{\beta}$
ensure that each term of
$\mathcal{A}\circ\left(\underset{i\in\mathbb{N}}{\overset{}{\sum}}\Phi^{i}\right)(x)$
is in $\omega^{\Gamma_{\beta}}\subseteq\omega^{\Gamma_{\beta+1}}$. Then

$\mathcal{A}\circ\left(\underset{i\in\mathbb{N}}{\overset{}{\sum}}\Phi^{i}\right)(x)\in\mathbb{R}_{\varepsilon_{\beta+1}}^{\Gamma_{\beta+1}^{\uparrow\varepsilon_{\beta+1}}}$

Application of Corollary 6.5 gives that $\mathbb{K}$ is stable under anti-
derivation.

∎

The previous theorem may seem have a lot of strong hypothesis but we can
actually give a non-trivial application.

Take $\alpha=\omega$ and for $n<\omega$,
$\Gamma_{n}=\left\\{\left.x\in\textnormal{No}_{\varepsilon_{n}}\
\vphantom{\operatorname{NR}(\omega^{x})<\varepsilon_{n-1}}\right|\
\operatorname{NR}(\omega^{x})<\varepsilon_{n-1}\right\\}$, with
$\varepsilon_{-1}:=\omega$. We first recall that from Lemma 6.1, for any
ordinal $\alpha$,

$\kappa_{-\alpha}=\omega^{\omega^{-\omega\otimes\alpha}}$

in particular
$\kappa_{-\varepsilon_{n}}=\omega^{\omega^{-\omega\otimes\varepsilon_{n}}}=\omega^{\omega^{-\varepsilon_{n}}}=\omega^{\mathchoice{\dfrac{1}{\varepsilon_{n}}}{\dfrac{1}{\varepsilon_{n}}}{\frac{1}{\varepsilon_{n}}}{\frac{1}{\varepsilon_{n}}}}$

From Theorem 3.7, we know that the sign sequence of
$\omega^{-\omega\otimes\alpha}$ is $(+)(-)^{\omega\otimes\alpha}$, which has
length $1\oplus\omega\otimes\alpha$.

  * •

Since $\varepsilon_{n}$ is an $\varepsilon$-number, hence an additive ordinal,
for any $n\in\mathbb{N}$, $\Gamma_{n}$ is an abelian group.

  * •

Of course for any $n\leq m$, $\Gamma_{n}\subseteq\Gamma_{m}$.

  * •

Since
$\left|\omega^{-\varepsilon_{n}}\right|_{+-}=1\oplus\omega\otimes\varepsilon_{n}=\varepsilon_{n}$,
we have $\kappa_{-\varepsilon_{n}}\notin\omega^{\Gamma_{\beta}}$. However, for
$\alpha<\varepsilon_{n}$,
$\left|\omega^{-\omega\otimes\alpha}\right|_{+-}=1\oplus\omega\otimes\alpha<\varepsilon_{n}$,
and, since $\kappa_{-\alpha}\in\mathbb{L}$,
$\operatorname{NR}(\kappa_{-\alpha})=0<\varepsilon_{n-1}$. Therefor, we have
$\kappa_{-\alpha}\in\omega^{\Gamma_{\beta}}$. Since,
$\kappa_{-\varepsilon_{\beta}}\in\omega^{\textnormal{No}}$, is
$x\in\omega^{\Gamma_{\beta}}$ is such that
$x\preceq^{K}\kappa_{-\varepsilon_{\beta}}$ then
$\kappa_{-\varepsilon_{\beta}}$ is a prefix of $x$ and
$\left|x\right|_{+-}\geq\varepsilon_{\beta}$ what is impossible from Lemma
5.4.

  * •

We can take $\eta_{\beta}=\varepsilon_{\beta-1}<\varepsilon_{\eta}$.

Theorem 1 applies and
$\underset{n\in\mathbb{N}}{\overset{}{\bigcup}}\mathbb{R}_{\varepsilon_{n}}^{\Gamma_{n}^{\uparrow\varepsilon_{n}}}$
is stable under $\exp$, $\ln$, $\partial$ and anti-derivation. As a final
note, we can notice that

$\underset{n\in\mathbb{N}}{\overset{}{\bigcup}}\mathbb{R}_{\varepsilon_{n}}^{\Gamma_{n}^{\uparrow\varepsilon_{n}}}=\underset{n\in\mathbb{N}}{\overset{}{\bigcup}}\mathbb{R}_{\varepsilon_{n}}^{\textnormal{No}_{\varepsilon_{n}}^{\uparrow\varepsilon_{n}}}$

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