# A Complete Equational Theory for Real-Clifford+CH Quantum Circuits

## Alexandre Clément

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Alexandre Clément. A Complete Equational Theory for Real-Clifford+CH Quantum Circuits. 2026. [hal-05487235]

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# A Complete Equational Theory for Real-Clifford+CH Quantum Circuits

Alexandre Clément

Université Paris-Saclay, ENS Paris-Saclay, CNRS, Inria, LMF,
91190, Gif-sur-Yvette, France
alexandre.clement@inria.fr

## Abstract

We introduce a complete equational theory for the fragment of quantum circuits generated by the real Clifford gates plus the two-qubit controlled-Hadamard gate. That is, we give a simple set of equalities between circuits of this fragment, and prove that any other true equation can be derived from these. This is the first such completeness result for a finitely-generated, universal fragment of quantum circuits, with no parameterized gates and no need for ancillas.

## 1 Introduction

Quantum circuits form the basic language which is essentially at the core of quantum computing. The role that they play in practice is sometimes compared to that of an assembly language in classical computing. But formally, they are in fact the quantum equivalent of *boolean circuits*. Like boolean circuits, they consist of a sequence of *gates*, which are applied to some quantum data which is most often in the form of *qubits*, the quantum equivalent of bits.

Quantum computing can *a priori* be realized on a variety of physical platforms with various characteristics, and for now it is essentially impossible to predict what physical implementation will turn out to be the best, neither is it clear whether one technology will completely supplant the others; therefore, various different classes of quantum circuits are *a priori* relevant to formalize and study. Even staying inside the most standard model of quantum computing, where the data is encoded in qubits, several *fragments* of the language of quantum circuits, corresponding to particular sets of elementary gates with which to build the circuits, are relevant to consider. Some of them have received particular attention, such as the well-known Clifford+T and Toffoli-Hadamard fragments. An essential characteristic for a gate set is to be *computationally universal*, that is, to make it possible in principle to implement any arbitrary quantum program, up to arbitrarily small error. One reason that makes Clifford+T and Toffoli-Hadamard interesting is that they are universal while being generated by a very small and simple set of gates – in addition to being possibly relevant for promising implementations such as superconducting circuits.

A central task in quantum computing is to manipulate quantum circuits, and in particular to transform circuits into equivalent ones that are better according to some criterion: this has various practical applications such as optimization, satisfaction of hardware constraints, error correction, or verification. A promising approach towards manipulation of quantum circuits is via graphical rewriting. In the recent years, several *graphical languages* for formal manipulation of quantum circuits (and of other aspects of quantum processes) have been introduced, such as the ZX-, ZW-, ZH-, and ZXW-calculi, among others [2, 12, 16, 19, 29]. A central question

in the formalization of such languages is the completeness of their equational theory, that is, the fact that the set of rewriting rules with which they are equipped be powerful enough so that any semantically meaningful transformation can be done. A related important question is to make the complete equational theory as simple as possible, and if possible, to be finally able to prove that it is minimal in the sense that all redundancies have been simplified out. There are now various complete equational theories for graphical languages [20, 23–25, 35], some of which being minimal [14] or near-minimal [3, 4, 32–35].

The language of quantum circuits itself can be viewed as a graphical language, and finding complete equational theories for it has its own added value compared to relying on external auxiliary languages like the ZX-calculus. For instance, applying arbitrary ZX transformations to a quantum circuit can produce a ZX-diagram from which it may be hard to get back a quantum circuit [13, 16]. Although the question of finding a complete equational theory for quantum circuits had been formulated in the early 2000’s [22, 26], the first complete equational theory was found only a few years ago [11], followed by simplifications and a proof of minimality [9, 10]. Those results are for the full language of quantum circuits in the qubit setting. That is, the gate set is very simple but still contains a parameterized gate, and the minimal equational theory is relatively simple but involves non-trivial computations on the parameters.

In this paper, we consider a finitely generated fragment of quantum circuits, without parameterized gates, namely the fragment generated by real Clifford gates – the Hadamard gate $H$, the Pauli $Z$ gate, and the controlled-$Z$ gate $CZ$ – together with the controlled Hadamard gate $CH$. We call this fragment *Real-Clifford+CH*. This is a sub-fragment of Clifford+T, which contains Toffoli-Hadamard<fnref n="1" /> and therefore is universal. To our knowledge, this is the first completeness result for a fragment of quantum circuits generated by a finite gate set, without parameterized gates.

Similarly to [11], our result is based on a back-and-forth translation into an auxiliary language for which a completeness result is known, namely a presentation of the matrix group $U_n(\mathbb{Z}[\frac{1}{\sqrt{2}}])$ generated by one- and two-level matrices [17]. Those one- and two-level matrices have a similar structure to the optical circuits used in [11].

*Related works.* In addition to the graphical languages mentioned, other languages for quantum computing are equipped with complete equational theories [15, 17, 21]. The main difference is that those languages are equipped with a control constructor (or an

<footnote n="1">¹Except in the case of 3-qubit circuits without ancillas.</footnote>
$$ \begin{align} \text{id}_k \circ C &= C &= C \circ \text{id}_k & \tag{a} \\ \boxed{C} \quad \boxed{\phantom{C}} \quad \boxed{C} &= \boxed{C} &= \boxed{\phantom{C}C} \end{align} $$

$$ \begin{align} (C_3 \circ C_2) \circ C_1 &= C_3 \circ (C_2 \circ C_1) & \tag{b} \\ \boxed{C_1} \quad \boxed{\phantom{C_1}} \quad \boxed{C_2} \quad \boxed{\phantom{C_2}} \quad \boxed{C_3} &= \boxed{C_1} \quad \boxed{\phantom{C_1}} \quad \boxed{C_2} \quad \boxed{\phantom{C_2}} \quad \boxed{C_3} \end{align} $$

$$ \boxed{\phantom{C}} \otimes C = C = C \otimes \boxed{\phantom{C}} & \tag{c} $$

$$ \boxed{C} \quad \boxed{\phantom{C}} = \boxed{C} = \boxed{\phantom{C}} $$

$$ \sigma_k \circ (C \otimes \boxed{\phantom{C}}) = (\boxed{\phantom{C}} \otimes C) \circ \sigma_k & \tag{f} \\ \boxed{C} \quad \boxed{\phantom{C}} = \boxed{C} $$

$$ \begin{align} (C_1 \otimes C_2) \otimes C_3 &= C_1 \otimes (C_2 \otimes C_3) & \tag{d} \\ \boxed{C_1} \quad \boxed{\phantom{C_1}} \quad \boxed{C_2} &= \boxed{C_1} \quad \boxed{\phantom{C_1}} \\ \boxed{C_2} \quad \boxed{\phantom{C_2}} \quad \boxed{C_3} &= \boxed{C_2} \quad \boxed{\phantom{C_2}} \\ \boxed{C_3} \quad \boxed{\phantom{C_3}} \quad \boxed{C_4} &= \boxed{C_3} \quad \boxed{\phantom{C_4}} \end{align} $$

$$ \begin{align} (C_2 \circ C_1) \otimes (C_4 \circ C_3) &= (C_2 \otimes C_4) \circ (C_1 \otimes C_3) & \tag{e} \\ \boxed{C_1} \quad \boxed{\phantom{C_1}} \quad \boxed{C_2} &= \boxed{C_1} \quad \boxed{\phantom{C_1}} \quad \boxed{C_2} \\ \boxed{C_3} \quad \boxed{\phantom{C_3}} \quad \boxed{C_4} &= \boxed{C_3} \quad \boxed{\phantom{C_4}} \quad \boxed{C_4} \end{align} $$

$$ \begin{align} \Box C \circ \Box C &= \boxed{\phantom{C}} \otimes \boxed{\phantom{C}} & \tag{g} \\ \Box \Box C &= \boxed{\phantom{C}} \end{align} $$

where $id_0 = \boxed{\phantom{C}}$ and $id_{k+1} = id_k \otimes \boxed{\phantom{C}}$, and $\sigma_0 := \boxed{\phantom{C}}$, $\sigma_{k+1} := (\Box C \otimes id_k) \circ (\Box C \otimes \sigma_k)$.

<figure><img src="image_2.png" /><figcaption>Figure 2: Alternative equations to Equation (f).</figcaption></figure>

$U_n(Z[\frac{1}{\sqrt{2}}])$. However, one has

$$ P \otimes P = \frac{1}{2\sqrt{2}} \begin{pmatrix} \sqrt{2} + 1 & 1 & 1 & \sqrt{2} - 1 \\ 1 & -\sqrt{2} - 1 & \sqrt{2} - 1 & -1 \\ 1 & \sqrt{2} - 1 & -\sqrt{2} - 1 & -1 \\ \sqrt{2} - 1 & -1 & -1 & \sqrt{2} + 1 \end{pmatrix} $$

which is in that group, and can be represented by a real-Clifford+CH circuit. We draw this shortcut with a slight dashed line in order to keep track of the fact that it is actually defined using two-qubit gates:

$$ \begin{array}{c} -P- \\ \rightarrow \\ -P- \end{array} := \begin{array}{c} \rightarrow \\ \boxed{H} \\ \rightarrow \\ \boxed{H} \end{array} \quad \begin{array}{c} \rightarrow \\ \boxed{H} \\ \rightarrow \\ \boxed{H} \end{array} \quad \begin{array}{c} \rightarrow \\ \boxed{H} \\ \rightarrow \\ \boxed{H} \end{array} \quad \begin{array}{c} \rightarrow \\ \boxed{H} \\ \rightarrow \\ \boxed{H} \end{array} $$

It will also be useful to represent a pair of *P* gates acting on non-adjacent qubits:

<figure><img src="image_3.png" /></figure>

Finally, similarly to [11], another particularly useful class of shortcuts is *multi-controlled gates*.

On the one hand, we will use multi-controlled *Z* and *H* gates. Note that for $n \ge 3$, one cannot represent an *n*-qubit multi-controlled

<figure><img src="image_4.png" /><figcaption>Figure 3: Standard one- and two-qubit gates, defined as shortcuts.</figcaption></figure>

*Z* or *H* gate as an *n*-qubit real-Clifford+CH circuit.<fnref n="3" /> However, one can represent an (*n*−1)-qubit multi-controlled *Z* or *H* gate tensored with the 1-qubit identity.

On the other hand, we will also use multi-controlled *ZX* and *XZ* gates (where *ZX* and *XZ* are just the product of the *Z* and *X* gates in both order), which do not have such a restriction.

<footnote n="3">³This is due to the fact that it has determinant −1 and that we are only working with real matrices. See also Section 4.</footnote>
$$(-1)_{[a]}^2 \approx \epsilon \qquad (a1) \qquad H_{[a,b]}^2 \approx \epsilon \qquad (a3)$$

$$X_{[a,b]}^2 \approx \epsilon \qquad (a2) \qquad (-1)_{[a]} H_{[b,c]} \approx H_{[b,c]} (-1)_{[a]} \qquad (b4)$$

$$X_{[a,b]} H_{[c,d]} \approx H_{[c,d]} X_{[a,b]} \qquad (b5)$$

$$H_{[a,b]} H_{[c,d]} \approx H_{[c,d]} H_{[a,b]} \qquad (b6)$$

$$H_{[b,c]} X_{[a,b]} \approx X_{[a,b]} H_{[a,c]} \qquad (c4)$$

$$H_{[a,c]} X_{[b,c]} \approx X_{[b,c]} H_{[a,c]} \qquad (c5)$$

$$(-1)_{[a]}(-1)_{[b]} H_{[a,b]} \approx H_{[a,b]}(-1)_{[a]}(-1)_{[b]} \qquad (d1)$$

$$(-1)_{[b]} H_{[a,b]} \approx H_{[a,b]} X_{[a,b]} \qquad (d2)$$

$$(H_{[c,d]} H_{[a,c]} H_{[b,d]})^4 \approx H_{[a,b]} H_{[c,d]} \qquad (d3)$$

$$(H_{[a,c]} H_{[b,d]} H_{[a,b]} H_{[c,d]} H_{[b,d]} X_{[c,e]} X_{[d,f]})^3 \approx H_{[c,e]} H_{[d,f]} H_{[e,f]} H_{[c,e]} H_{[d,f]} X_{[c,e]} X_{[d,f]} \qquad (d4)$$

$$X_{[c,d]} \approx X_{[b,c]} \qquad (e1) \qquad H_{[c,d]} \approx X_{[b,c]} H_{[b,c]} X_{[b,c]} \qquad (e2)$$

<caption>Figure 6: The complete equational theory given in [17]. In each equation, all the indices must be distinct. In Equations (e1) and (e2), we require $b < c$, and in all other equations, the two indices of each $H$ or $X$ must be in increasing order.</caption>

$$(-1)_{[0]}^2 \approx \epsilon \qquad (a1^*) \qquad H_{[a,b]}^2 \approx \epsilon \qquad (a3)$$

$$X_{[a,b]}^2 \approx \epsilon \qquad (a2) \qquad (-1)_{[1]} H_{[0,2]} \approx H_{[0,2]} (-1)_{[1]} \qquad (b4^*)$$

$$H_{[0,1]} H_{[2,3]} \approx H_{[2,3]} H_{[0,1]} \qquad (b6^*)$$

$$H_{[a,c]} X_{[b,c]} \approx X_{[b,c]} H_{[a,c]} \qquad (c5)$$

$$(-1)_{[b]} H_{[a,b]} \approx H_{[a,b]} X_{[a,b]} \qquad (d2)$$

$$H_{[0,1]} H_{[0,2]} H_{[1,3]} H_{[0,1]} (-1)_{[0]} (-1)_{[1]} H_{[0,2]} H_{[1,3]} \approx H_{[0,2]} H_{[1,3]} H_{[0,1]} (-1)_{[0]} (-1)_{[1]} H_{[0,2]} H_{[1,3]} H_{[0,1]} \qquad (d3^*)$$

$$H_{[0,1]} H_{[0,2]} H_{[1,3]} H_{[0,4]} H_{[1,5]} H_{[0,1]} (-1)_{[0]} (-1)_{[1]} H_{[0,4]} H_{[1,5]} H_{[0,2]} H_{[1,3]} \approx H_{[0,2]} H_{[1,3]} H_{[0,4]} H_{[1,5]} H_{[0,1]} (-1)_{[0]} (-1)_{[1]} H_{[0,4]} H_{[1,5]} H_{[0,2]} H_{[1,3]} H_{[0,1]} \qquad (d4^*)$$

$$H_{[c,d]} X_{[b,c]} \approx X_{[b,c]} H_{[b,c]} \qquad (e2^*)$$

<caption>Figure 7: Simpler equational theory, equivalent to that of Figure 6. In each equation, all the indices must be distinct but are not required to be in a particular order.</caption>

## 4.1 Expressiveness of Real-Clifford+CH Circuits

First, recall the relevant generating set for $U_N(\mathbb{Z}[\frac{1}{\sqrt{2}}])$, made of what are called 1- and 2-level matrices:

**Definition 4.1 ([1, 17]).** Let $M$ be an $m \times m$ matrix, let $m < N$, and let $0 \le a_1, ..., a_m \le N-1$. An $m$-level matrix of type $M$ is the $N \times N$ matrix $M_{[a_1, ..., a_m]}$ defined by

$$(M_{[a_1, ..., a_m]})_{i,j} = \begin{cases} M_{i', j'} & \text{if } i = a_{i'} \text{ and } j = a_{j'} \\ \delta_{i,j} & \text{otherwise.} \end{cases}$$

**Proposition 4.2 ([1]).** For $N \ge 1$, a matrix belongs to the group $U_N(\mathbb{Z}[\frac{1}{\sqrt{2}}])$ if and only if it can be expressed as a product of generators from

$$\mathcal{G}_N := \{(-1)_{[a]}, X_{[a,b]}, H_{[a,b]} \mid a, b \in \{0, ..., N-1\}, a \neq b\}.$$

We can now state the completeness result of [17]:

**Definition 4.3.** The free monoid generated by $\mathcal{G}_N$ (i.e. the monoid of words over $\mathcal{G}_N$) is denoted by $\mathcal{G}_N^*$. The empty word is denoted by $\epsilon$. The matrix associated with a word $w = G_1G_2...G_k \in \mathcal{G}_N^*$ is called its semantics, and denoted by $[[w]] := G_1 \cdot G_2 \cdot ... \cdot G_k$.
We denote by $\approx$ the smallest congruence satisfying the equations of Figure 6. That is, for any two words $w_1, w_2 \in \mathcal{G}_N^*$, one has $w_1 \approx w_2$ iff $w_1$ and $w_2$ can be transformed into one another using the equations of Figure 6.

**Theorem 4.4 (Soundness and Completeness of $\approx$ for $\mathcal{G}_N^*$ [17]).**
Let $w_1, w_2 \in \mathcal{G}_N^*$. Then $[w_1] = [w_2]$ if and only if $w_1 \approx w_2$.

Note that contrary to [17], we have not a priori required that $a < b$ in $X_{[a,b]}$ or $H_{[a,b]}$, hence the additional conditions in the caption of Figure 6, and the fact that Equations (e1) and (e2) are here considered as axioms, whereas in [17] they were just shortcut definitions.

We can now express the parity conditions needed to characterize the matrix group spanned by real-Clifford+CH circuits:

**Lemma 4.5.** Given any $A \in U_N(\mathbb{Z}[\frac{1}{\sqrt{2}}])$, the parity of the number of generators of the form $H_{[a,b]}$ (resp. of the form $(-1)_{[a]}$ or $X_{[a,b]}$) in an arbitrary decomposition of A as a product of generators from $\mathcal{G}_N$ only depends on A.

*PROOF.* By Theorem 4.4, any two decompositions of A as a product of generators from $\mathcal{G}_N$ can be transformed into one another using the equations of Figure 6. One can check that all those equations preserve both the parity of the number of generators of the form $H_{[a,b]}$, and the parity of the number of generators of the form $(-1)_{[a]}$ or $X_{[a,b]}$. $\square$

In the following, the parity of the number of $H_{[a,b]}$ in an arbitrary decomposition of A will be called the *H-parity* of A, and the parity of the total number of $(-1)_{[a]}$ and $X_{[a,b]}$ in such a decomposition will be called the *ZX-parity* of A.

**Proposition 4.6.** For $N \ge 1$, a matrix belongs to the group $U_N(\mathbb{Z}[\frac{1}{\sqrt{2}}])$ and has even ZX- and H-parities if and only if it can be expressed as a product of generators from

$$ \mathcal{P}_N := \{ ((-1)_{[a]})(-1)_{[c]}), ((-1)_{[a]}X_{[c,d]}), (X_{[a,b]}X_{[c,d]}), (H_{[a,b]}H_{[c,d]}) \mid a, b, c, d \in \{0, \dots, n-1\}, a \neq b \land c \neq d \} $$

*PROOF.* The algorithm given in [17] for decomposing a given element of $U_N(\mathbb{Z}[\frac{1}{\sqrt{2}}])$ into a product of generators of $\mathcal{P}_N$ can be slightly modified so that its output just consists of a sequence of $H_{[a,b]}$ followed by a sequence of $(-1)_{[a]}$ and $X_{[a,b]}$ (the modified algorithm is described in Appendix A.1). By Lemma 4.5, these two sequences are of even length, so we can group the generators by pairs in each. Finally, it remains to get rid of the pairs of the form $(X_{[c,d]}(-1)_{[a]})$ by noticing that this is semantically equivalent to $((-1)_{[a]}X_{[c,d]})$ if $a \notin \{c, d\}$, to $((-1)_{[d]}X_{[c,d]})$ if $a = c$, and to $((-1)_{[c]}X_{[c,d]})$ if $a = d$. $\square$

**Lemma 4.7.** For $n = 2$, the set of matrices that can be represented by an n-qubit real-Clifford+CH circuit is exactly $U_{2n}(\mathbb{Z}[\frac{1}{\sqrt{2}}])$. For $n \ge 3$, it is the subset of $U_{2n}(\mathbb{Z}[\frac{1}{\sqrt{2}}])$ consisting of the matrices with even ZX- and H-parities.

*PROOF.* First, it is clear from the definition of the semantics that any real-Clifford+CH circuit represents an element of $U_{2n}(\mathbb{Z}[\frac{1}{\sqrt{2}}])$.

For $n = 2$, we need to show that any element of $U_4(\mathbb{Z}[\frac{1}{\sqrt{2}}])$ can be represented by a real-Clifford+CH circuit. To do so, it suffices to give for every generator of $\mathcal{G}_N$ a circuit with corresponding semantics. This is done in Definition 7.3.

For $n \ge 3$, similarly, it suffices to express each generator of $\mathcal{P}_N$ as a real-Clifford+CH circuit. This is done in Definition 8.3. $\square$

## 4.2 A Complete Equational Theory for $\mathcal{P}_N^*$

Thus, for $n \ge 3$, the matrices represented by n-qubit real-Clifford+CH circuits form a subgroup of $U_{2n}(\mathbb{Z}[\frac{1}{\sqrt{2}}])$ of index 4.

Although we are talking about groups, note that in the following we will essentially only work with the monoid structure. Indeed, this does not fundamentally change the presentations since all inverses are easy to express from the original generators, and makes more sense from a quantum circuit point of view.

There is a standard technique for obtaining a complete equational theory for a subgroup when one is known for the whole group, based on the *Reidemeister-Schreier theorem* [30, 31]. This theorem has been adapted to monoids in [6] and used for obtaining complete equational theories for 2-qubit Clifford+T [6] and 3-qubit Clifford+CS [7] quantum circuits. One drawback of this technique is that it often leads to a combinatorial explosion that makes it computationally costly, even in relatively simple cases. This is the reason that the completeness result of [7] is limited to 3 qubit, whereas the presentation of the underlying matrix group is a priori valid in arbitrary dimension.

Rather than trying to obtain a complete equational theory for real-Clifford+CH quantum circuits directly by applying the Reidemeister-Schreier theorem to the equational theory of [17], we adopt an alternative approach in two steps:

* first, we use the Reidemeister-Schreier theorem to obtain a presentation of the submonoid of $U_N(\mathbb{Z}[\frac{1}{\sqrt{2}}])$ generated by $\mathcal{P}_N$

* then we use a back-and-forth encoding similar to that of [11] in order to turn this presentation into a complete equational theory for real-Clifford+CH circuits.

Additionally, we perform some simplifications of the equational theories before and after each step. In particular, we start by slightly simplifying the equational theory of [17]; the simplified complete equational theory is shown in Figure 7.

**Proposition 4.8 (Soundness and Completeness of the Equational Theory of Figure 7 w.r.t. $\approx$).** For any $w_1, w_2 \in \mathcal{G}_N^*$, one has $w_1 \approx w_2$ if and only if $w_1$ and $w_2$ can be transformed into one another using the equations of Figure 7. In other words, $\approx$ is the smallest congruence satisfying the relations of Figure 7.

*PROOF.* One can check that the equations of Figure 7 are semantically sound. Hence, by Theorem 4.4, they can be derived from those of Figure 6. Conversely, it remains to prove that the equations of Figure 6 can be derived from those of Figure 7. This is done in Appendix A.2. $\square$

Details about how we use the Reidemeister-Schreier theorem are given in Appendix A.3.1. The raw complete equational theory that we obtain by applying the Reidemeister-Schreier theorem is shown in Appendix A.3.2. Note that it contains many equations, which are
not necessarily well-suited for the next step which is to translate them using the decoding function defined in Section 8; hence, before proceeding to the back-and-forth encoding, we simplify this equational theory into a smaller one, shown in Figure 8. The equational theory of Figure 8 is actually not the simplest possible from the point of view of a human reader (although much simpler than that of Appendix A.3.2), but is optimized so as to ease the decoding step.

This gives us the following completeness result:

**Definition 4.9.** The free monoid generated by $\mathcal{P}_N$ (i.e. the monoid of words over $\mathcal{P}_N$) is denoted by $\mathcal{P}_N^*$. The empty word is denoted by $\epsilon$. The matrix associated with a word $w = P_1P_2...P_k \in \mathcal{P}_N^*$ is called its semantics, and denoted by $[w] := P_1 \cdot P_2 \cdot ... \cdot P_k$.

We denote by $\approx$ the smallest congruence satisfying the equations of Figure 8. That is, for any two words $w_1, w_2 \in \mathcal{P}_N^*$, one has $w_1 \approx w_2$ iff $w_1$ and $w_2$ can be transformed into one another using the equations of Figure 8.

**Theorem 4.10 (Soundness and Completeness of $\approx$ for $\mathcal{P}_N^*$).** Let $w_1, w_2 \in \mathcal{P}_N^*$ with $N \ge 8$. Then $[w_1] = [w_2]$ if and only if $w_1 \approx w_2$.

**PROOF.** To prove the soundness, it suffices to check that all the equations of Figure 8 preserve the semantics. The proof of completeness is given in Appendices A.3 and A.4. $\square$

# 5 Auxiliary Equations/Properties of Multi-Controlled Gates

Similarly to [11], the complete equational theory for Real-Clifford+CH circuits that we would obtain naively by the back-and-forth encoding technique described in Sections 7 and 8 consists mainly of equations involving a large (possibly unbounded) number of multi-controlled gates. One of the main contributions of this paper is to derive those equations from the much simpler equational theory shown in Figure 4. To do so, we will rely on auxiliary equations, that we derive from those of Figure 4 in Appendices B and D. Note that like in [11], a key role is played by basic properties of multi-controlled gates.

One can in particular recognize the main auxiliary properties used in [11], namely that the different controls of a single multi-controlled gate play the same role and therefore can be swapped (Equations (307), (313) and (353)); that a positively-controlled gate can be merged with the negatively-controlled version of the same gate (Equations (309), (310) and (354)); and that a positively-controlled gate always commute with a gate which is negatively-controlled by the same qubit (Equations (335) to (352)). Equations (311), (312), (317) and (355) can be seen as the analogues of the additivity and periodicity properties of [11].

Other notable auxiliary equations, more specific to the real-Clifford+CH fragment, include in particular the fact that the dashed box of the multi-controlled Z and H gates indeed behaves as just an identity wire (Equations (274) and (277), together with Lemmas D.9 and D.11 and their various particular cases proven beforehand in Lemma D.8).

**Remark 4.** One can show that Equation (19) can be derived in particular from Equation (272) or from Equation (274). Thus, Equations (272) and (274) are two alternative axiom schemas that could each have

been put instead of Equation (19) in the equational theory of Figure 4, without altering its completeness.

**Remark 5.** In particular, the 4-qubit case of Equation (272) is trivial since it follows from the topological rules. Hence, the 4-qubit case of Equation (19) can be derived from the other equations (see Equation (154) in Appendix D.3). This partly explains the maybe surprising fact that the equational theory of Figure 4 does not contain any axiom on 4 qubits.

Note that in the proof of completeness, we will essentially prove the completeness for n-qubit circuits, for each n separately. Therefore, we can instead reason by induction on n (the base cases being given by Lemmas 7.1 and 7.6), which does not change the way that the proof of completeness itself is done, but is useful for proving auxiliary equations. This can be reformulated as the fact that we can use the following lemma:

**Lemma 5.1 (Induction Hypothesis for the Proof of Completeness).** For any real-Clifford+CH circuits $C_1, C_2$, if $[[C_1]] = [[C_2]]$ then $QC \vdash (\bigcirc \otimes C_1) = (\bigcirc \otimes C_2)$ and $QC \vdash (C_1 \otimes \bigcirc) = (C_2 \otimes \bigcirc)$.

# 6 The Category of Sequentialized Circuits

As we have said, the proof of completeness will be based on a back-and-forth translation, similar to that of [11], between the PROP of real-Clifford+CH circuits and the monoids $\mathcal{G}_N^*$ and $\mathcal{P}_N^*$. However, real-Clifford+CH circuits have a richer structure: they form not only a monoid (for every particular number of qubits) – which corresponds to Equations (a) and (b) of Figure 1 – but they also satisfy the other coherence relations of a PROP. Hence, we cannot directly encode circuits into words since those relations would not be preserved and therefore the encoding would be ill-defined. Instead, we will use what we call *sequentialized circuits*, that is, elements of the category $QC_{seq}$ of raw circuits quotiented only by Equations (a) and (b).

The PROP QC of usual real-Clifford+CH circuits is recovered by quotienting by the remaining topological rules, namely Equations (c) to (g).<fnref n="6" /> Given a sequentialized circuit $C \in QC_{seq}$, we denote by $|C|$ its image by the canonical projection onto QC. Conversely, any circuit $C \in QC$ has several representatives $C'$ in $QC_{seq}$, satisfying $|C'| = C$.

We are now ready to prove the main result of the paper. We proceed as follows: first, we prove the completeness for 1-qubit circuits, which is easy to do by using a normal form, in Section 7.1; then we prove the completeness for 2-qubit circuits by means of a back-and-forth encoding into $\mathcal{G}_4^*$, in Section 7.2. Finally, we prove the completeness for circuits on $n \ge 3$ qubits by a back-and-forth encoding into $\mathcal{P}_{2n}^*$, in Section 8. Note that while the encoding function needs to be defined on sequentialized circuits, the decoding function can be defined directly into the PROP QC.

<footnote n="6">⁶More precisely, by quotienting by the variants of Equations (c) to (g) obtained by projecting the two sides of each equation according to the canonical projection $QC_{raw} \rightarrow QC_{raw}/\{(a), (b)\}$.</footnote>
$$ ((-1)_{[a]}(-1)_{[a]}) \approx \epsilon \qquad (20) $$

$$ ((-1)_{[a]}(-1)_{[b]}) \approx ((-1)_{[b]}(-1)_{[a]}) \qquad (21) $$

$$ ((-1)_{[a]}(-1)_{[b]}) ((-1)_{[b]}(-1)_{[c]}) \approx ((-1)_{[a]}(-1)_{[c]}) \qquad (22) $$

$$ ((-1)_{[a]}(-1)_{[a+1]}) \approx ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[a]}X_{[a,a+1]}) \qquad (23) $$

$$ ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a+1,a+2]}) ((-1)_{[a]}X_{[a,a+1]}) \approx ((-1)_{[a+1]}X_{[a+1,a+2]}) ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a+1,a+2]}) \qquad (24) $$

$$ ((-1)_{[a]}X_{[a,c]}) \approx ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a+1,c]}) ((-1)_{[a]}X_{[a,a+1]}) \qquad a+1<c, \ c-a \text{ odd} \qquad (25) $$

$$ ((-1)_{[c]}X_{[a,c]}) \approx ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[c]}X_{[a+1,c]}) ((-1)_{[a]}X_{[a,a+1]}) \qquad a+1<c, \ c-a \text{ odd} \qquad (26) $$

$$ ((-1)_{[a]}X_{[a,c]}) \approx ((-1)_{[c-1]}X_{[c-1,c]}) ((-1)_{[a]}X_{[a,c-1]}) ((-1)_{[c]}X_{[c-1,c]}) \qquad a+1<c, \ c-a \text{ even} \qquad (27) $$

$$ ((-1)_{[c]}X_{[a,c]}) \approx ((-1)_{[c-1]}X_{[c-1,c]}) ((-1)_{[c-1]}X_{[a,c-1]}) ((-1)_{[c]}X_{[c-1,c]}) \qquad a+1<c, \ c-a \text{ even} \qquad (28) $$

$$ ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a,a+1]}) \approx \epsilon \qquad (29) $$

$$ ((-1)_{[c]}X_{[a,b]}) \approx ((-1)_{[c]}(-1)_{[a]}) ((-1)_{[a]}X_{[a,b]}) \qquad c \neq a, b \qquad (30) $$

$$ ((-1)_{[a+1]}(-1)_{[c]}) ((-1)_{[a]}X_{[a,a+1]}) \approx ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[a]}(-1)_{[c]}) \qquad c \neq a, a+1 \qquad (31) $$

$$ ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[b]}X_{[b,b+1]}) \approx ((-1)_{[b]}X_{[b,b+1]}) ((-1)_{[a]}X_{[a,a+1]}) \qquad a+1 < b \qquad (32) $$

$$ ((-1)_{[b]}X_{[a,b]}) \approx ((-1)_{[b]}X_{[b,a]}) \qquad (33) $$

$$ (X_{[a,b]}X_{[c,d]}) \approx ((-1)_{[a]}X_{[a,b]}) ((-1)_{[b]}X_{[c,d]}) \qquad (34) $$

$$ (H_{[a,b]}H_{[c,d]}) \approx \Sigma_{a,b,c,d} \circ (H_{[0,1]}H_{[3,2]}) \circ \Sigma'_{a,b,c,d} \qquad \text{see caption} \qquad (35) $$

$$ (H_{[a,b]}H_{[c,d]}) \approx W_1 \qquad \text{see caption} \qquad (36) $$

$$ (H_{[a,b]}H_{[c,d]}) \approx W_2 \qquad \text{see caption} \qquad (37) $$

$$ ((-1)_{[a]}X_{[a,a+1]}) (H_{[0,1]}H_{[3,2]}) \approx (H_{[0,1]}H_{[3,2]}) ((-1)_{[a]}X_{[a,a+1]}) \qquad a \geq 4 \qquad (38) $$

$$ ((-1)_{[0]}(-1)_{[1]}) (H_{[0,1]}H_{[3,2]}) \approx (H_{[0,1]}H_{[3,2]}) ((-1)_{[0]}(-1)_{[1]}) \qquad (39) $$

$$ ((-1)_{[3]}(-1)_{[4]}) (H_{[0,1]}H_{[3,2]}) \approx (H_{[0,1]}H_{[3,2]}) ((-1)_{[3]}(-1)_{[4]}) ((-1)_{[2]}X_{[2,3]}) \qquad (40) $$

$$ (H_{[0,1]}H_{[3,2]}) (H_{[0,1]}H_{[3,2]}) \approx \epsilon \qquad (41) $$

$$ (H_{[a,b]}H_{[e,f]}) (H_{[e,f]}H_{[c,d]}) \approx (H_{[a,b]}H_{[c,d]}) \qquad \text{see caption} \qquad (42) $$

$$ (H_{[0,1]}H_{[3,2]}) (H_{[3,2]}H_{[4,5]}) \approx (X_{[0,3]}X_{[1,2]}) (H_{[3,2]}H_{[4,5]}) (X_{[0,3]}X_{[1,2]}) \qquad (43) $$

$$ (X_{[0,3]}X_{[1,2]}) (H_{[0,1]}H_{[3,2]}) \approx (H_{[0,1]}H_{[3,2]}) (X_{[0,3]}X_{[1,2]}) \qquad (44) $$

$$ (H_{[0,1]}H_{[7,6]}) (H_{[0,3]}H_{[1,2]}) (H_{[0,1]}H_{[7,6]}) ((-1)_{[0]}(-1)_{[1]}) (H_{[0,3]}H_{[1,2]}) \approx \qquad (45) $$

$$ (H_{[0,3]}H_{[1,2]}) ((-1)_{[0]}(-1)_{[1]}) (H_{[0,1]}H_{[7,6]}) (H_{[0,3]}H_{[1,2]}) (H_{[0,1]}H_{[7,6]}) \qquad \text{see caption} \qquad (46) $$

$$ (H_{[0,1]}H_{[7,6]}) (H_{[0,3]}H_{[1,2]}) (H_{[3,4]}H_{[2,5]}) (H_{[3,2]}H_{[4,5]}) (H_{[7,6]}H_{[4,5]}) ((-1)_{[2]}(-1)_{[3]}) (H_{[3,4]}H_{[2,5]}) (H_{[0,3]}H_{[1,2]}) \approx \qquad (46) $$

$$ (H_{[0,3]}H_{[1,2]}) (H_{[3,4]}H_{[2,5]}) ((-1)_{[2]}(-1)_{[3]}) (H_{[7,6]}H_{[4,5]}) (H_{[3,2]}H_{[4,5]}) (H_{[3,4]}H_{[2,5]}) (H_{[0,3]}H_{[1,2]}) (H_{[0,1]}H_{[7,6]}) \qquad \text{see caption} \qquad (47) $$

**Figure 8:** Simpler equational theory, equivalent to that of Appendix A.3.2 (and therefore complete) when the dimension is at least 8. In Equation (36), we require that the tuple $(G_n(a), G_n(b), G_n(c), G_n(d))$ be of the form $(x0y0z, x1y0z, x0y1z, x1y1z)$ for some bitstrings $x, y, z$; in Equation (37), we require that it be of the form $(x0y0z, x0y1z, x1y0z, x1y1z)$; and in Equation (35), we require that it be in neither of these two forms. $\Sigma_{a,b,c,d}$ and $\Sigma'_{a,b,c,d}$ are defined in Definition 8.3. In Equation (42), we require that $|\{a, b, c, d\}| \le 3$ and that $e < f$ be the two smallest elements of $\{0, ..., 2^n - 1\} \setminus \{a, b, c, d\}$. The right-hand sides of Equations (36) and (37) are words specifically tailored to behave well with respect to decoding (see Definitions 8.3 and 2.4 and Appendix E.5); they are defined in Appendix A.4.1.

# 7 One- and Two-Qubit Completeness

## 7.1 One-Qubit Completeness

**Lemma 7.1 (1-Qubit Completeness).** For any 1-qubit quantum circuits $C_1, C_2$ over $\{-\otimes, \bullet, \bullet\}$, if $\llbracket C_1 \rrbracket = \llbracket C_2 \rrbracket$ then QC $\vdash$ $C_1 = C_2$.

**PROOF.** We say that a 1-qubit circuit is in normal form if it is of the form

$$ (\otimes_{\otimes-\bullet})^k \quad \text{or} \quad \otimes_{\otimes-\bullet} \circ (\otimes_{\otimes-\bullet})^k $$

where $k \in \{0, ..., 7\}$.
One can check by direct computation that if two circuits in normal form have the same semantics then they are equal. Thus, it remains to prove that any 1-qubit circuit can be put in normal form.

Note that a 1-qubit circuit is just a sequence of $\otimes_H$- and $\bullet$-gates. By repeatedly applying Equations (1) and (2), we can ensure that it does not contain two consecutive identical gates. That is, it alternates between $\otimes_H$- and $\bullet$, or in other words, it is of one of the following three forms:

$$ (\otimes_H \bullet)^k \qquad \otimes_H \bullet \circ (\otimes_H \bullet)^k \qquad (\otimes_H \bullet)^k \circ \bullet $$

By applying Equation (3) $\lfloor \frac{k}{8} \rfloor$ times, we can ensure that $k \in \{0, ..., 7\}$. Thus, if the circuit is in one of the first two forms, then it is in normal form. If it is of the third form, then we do the following transformations:

$$ \begin{align*} (\otimes_H \bullet)^k \circ \bullet & \stackrel{(3)}{=} (\otimes_H \bullet)^k \circ \bullet \circ (\otimes_H \bullet)^8 \\ &= \bullet \circ (\bullet \otimes_H \bullet)^k \circ (\otimes_H \bullet)^8 \\ & \stackrel{(2)(1)}{=} \bullet \circ (\otimes_H \bullet)^{8-k} \\ & \stackrel{(2)}{=} \otimes_H \bullet \circ (\otimes_H \bullet)^{8-k-1} \end{align*} $$

which gives us a circuit in normal form. □

## 7.2 Two-Qubit Completeness

**Definition 7.2 (2-Qubit Encoding).** Let $E: \text{QC}_{\text{seq}}^{(2)} \to \mathcal{G}_4^*$ be defined as follows: for any 2-qubit sequentialized circuit C, let $E(C) = E_{0,0}(C)$, where $E_{k,l}$ is defined on any m-qubit sequentialized circuit satisfying $k + m + l = 2$, inductively as follows:

* $E_{k,l}(C_1 \otimes C_2) = E_{k+n_1,l}(C_2) \circ E_{k,l+n_2}(C_1)$, where $C_1$ (resp. $C_2$) is acting on $n_1$ (resp. $n_2$) qubits;

* $E_{k,l}(C_2 \circ C_1) = E_{k,l}(C_2) \circ E_{k,l}(C_1)$;

and for the generators:

* $E_{k,l}(\bullet) = \epsilon$,

* $E_{k,l}(\text{---}) = \epsilon$,

* $E_{0,0}(\text{---}) = X_{[1,2]}$,

* $E_{1,0}(\otimes_H \bullet) = H_{[0,1]}H_{[2,3]}$,

* $E_{0,1}(\otimes_H \bullet) = H_{[0,2]}H_{[1,3]}$,

* $E_{1,0}(\bullet) = (-1)_{[1]}(-1)_{[3]}$,

* $E_{0,1}(\bullet) = (-1)_{[2]}(-1)_{[3]}$,

* $E_{0,0}(\text{---}) = (-1)_{[3]}$,

* $E_{0,0}(\text{---}) = H_{[2,3]}$.

**Definition 7.3 (2-Qubit Decoding).** Let $D: \mathcal{G}_4^* \to \text{QC}^{(2)}$ be the monoid homomorphism defined as follows on the generators:

* $D((-1)_{[0]}) = \text{---}$,

* $D((-1)_{[1]}) = \text{---}$,

* $D((-1)_{[2]}) = \text{---}$,

* $D((-1)_{[3]}) = \text{---}$,

* $D(X_{[2,3]}) = D(X_{[3,2]}) = \text{---}$,

* $D(X_{[0,1]}) = D(X_{[1,0]}) = \text{---}$,

* $D(X_{[0,2]}) = D(X_{[2,0]}) = \text{---}$,

* $D(X_{[1,3]}) = D(X_{[3,1]}) = \text{---}$,

* $D(X_{[1,2]}) = D(X_{[2,1]}) = \text{---}$,

* $D(X_{[0,3]}) = D(X_{[3,0]}) = \text{---}$,

* $D(H_{[2,3]}) = \text{---}$,

* $D(H_{[0,1]}) = \text{---}$,

* $D(H_{[0,2]}) = \text{---}$,

* $D(H_{[1,3]}) = \text{---}$,

* $D(H_{[2,1]}) = \text{---}$,

* $D(H_{[0,3]}) = \text{---}$,

* $D(H_{[3,2]}) = \text{---}$,

* $D(H_{[1,0]}) = \text{---}$,

* $D(H_{[2,0]}) = \text{---}$,

* $D(H_{[3,1]}) = \text{---}$,

* $D(H_{[1,2]}) = \text{---}$,

* $D(H_{[3,0]}) = \text{---}$,

**Lemma 7.4.** For any 2-qubit circuit C, and any sequentialized circuit C' such that $|C'| = C$,

$$ \mathrm{QC} \vdash \mathrm{D}(E(C')) = C. $$

**PROOF.** The proof is given in Appendix C.1. □

**Lemma 7.5.** For any words $w_1, w_2 \in \mathcal{G}_4^*$, if $w_1 \approx w_2$ then $\mathrm{QC} \vdash \mathrm{D}(w_1) = \mathrm{D}(w_2)$. □

**PROOF.** The proof is given in Appendix C.2. □

**Lemma 7.6 (2-Qubit Completeness).** For any 2-qubit real-Clifford+CH quantum circuits $C_1, C_2$,

$$ \text{if } \llbracket C_1 \rrbracket = \llbracket C_2 \rrbracket \text{ then } \mathrm{QC} \vdash C_1 = C_2. $$

**PROOF.** Let $C'_1, C'_2$ be representatives in $\text{QC}_{\text{seq}}^{(2)}$ of respectively $C_1$ and $C_2$, so that $|C'_1| = C_1$ and $|C'_2| = C_2$. Since $\llbracket C_1 \rrbracket = \llbracket C_2 \rrbracket$ and the encoding preserves the semantics, one has $\llbracket E(C'_1) \rrbracket = \llbracket E(C'_2) \rrbracket$. Hence, by Theorem 4.4, we have $E(C'_1) \approx E(C'_2)$. By Lemma 7.5, this implies that $\mathrm{QC} \vdash \mathrm{D}(E(C'_1)) = \mathrm{D}(E(C'_2))$. In turn, by Lemma 7.4, this implies that $\mathrm{QC} \vdash C_1 = C_2$. □

## 8 Completeness in the General Case

To define the back-and-forth encoding between n-qubits circuits and the elements of $\mathcal{P}_{2n}$ which represent $2^n$-dimensional matrices, it will be handful, like in [11], to encode the integers $0, ..., 2^n - 1$ as n-length bitstrings, by using not the usual binary encoding but rather the *Gray code*, which has the advantage that the encodings of two consecutive integers always differ by exactly one bit:

**Definition 8.1 (Gray Code [11]).** The n-bit Gray code of an integer $k \in \{0, ..., 2^n - 1\}$ is the word $G_n(k) \in \{0, 1\}^n$, inductively defined by $G_0(0) = \epsilon$ and

$$ G_n(k) = \begin{cases} 0 & G_{n-1}(k) \text{ if } k < 2^{n-1}, \\ 1 & G_{n-1}(2^n - 1 - k) \text{ if } k \ge 2^{n-1}. \end{cases} $$
One can easily check that $G_n: \{0, ..., 2^n - 1\} \rightarrow \{0, 1\}^n$ is a bijection.

**Definition 8.2 (Encoding).** For any $n \ge 3$, let $E: \text{QC}_{\text{seq}} \rightarrow \mathcal{P}_{2n}^*$ be defined as follows: for any n-qubit sequentialized circuit C, let $E(C) = E_{0,0}(C)$, where $E_{k,l}$ is defined on any m-qubit sequentialized circuit satisfying $k + m + l = n$, inductively as follows (where the order of the products is left unspecified since it does not matter in the proof):

* $E_{k,l}(C_1 \otimes C_2) = E_{k+n_1,l}(C_2) \circ E_{k,l+n_2}(C_1)$, where $C_1$ (resp. $C_2$) is acting on $n_1$ (resp. $n_2$) qubits;

* $E_{k,l}(C_2 \circ C_1) = E_{k,l}(C_2) \circ E_{k,l}(C_1)$;

* $E_{k,l}(\overline{\phantom{C_2}}) = \epsilon$;

* $E_{k,l}(\underline{\phantom{C_2}}) = \epsilon$;

if $\ell \ge 1$, then

* $E_{k,l}(\text{--}) = \prod_{\substack{x \in \{0,1\}^k \\ y \in \{0,1\}^{\ell-1}}} ((-1)_{[G_n^{-1}(x10y)]} (-1)_{[G_n^{-1}(x11y)]})$;

* $E_{k,l}(\text{+}) = \prod_{\substack{x \in \{0,1\}^k \\ y \in \{0,1\}^{\ell-1}}} ((-1)_{[G_n^{-1}(x110y)]} (-1)_{[G_n^{-1}(x111y)]})$;

* $E_{k,l}(\text{--H}) = \prod_{\substack{x \in \{0,1\}^k \\ y \in \{0,1\}^{\ell-1}}} (H_{[G_n^{-1}(x00y),G_n^{-1}(x10y)]} H_{[G_n^{-1}(x01y),G_n^{-1}(x11y)]})$;

* $E_{k,l}(\text{+H}) = \prod_{\substack{x \in \{0,1\}^k \\ y \in \{0,1\}^{\ell-1}}} (H_{[G_n^{-1}(x100y),G_n^{-1}(x110y)]} H_{[G_n^{-1}(x101y),G_n^{-1}(x111y)]})$;

and if $\ell = 0$ (note that because of the condition $k + m + \ell = n$, we necessarily have $k \ge 1$ in this case), then

* $E_{k,0}(\text{--}) = \prod_{x \in \{0,1\}^{k-1}} ((-1)_{[G_n^{-1}(0x1)]} (-1)_{[G_n^{-1}(1x1)]})$;

* $E_{k,0}(\text{+}) = \prod_{x \in \{0,1\}^{k-1}} ((-1)_{[G_n^{-1}(0x11)]} (-1)_{[G_n^{-1}(1x11)]})$;

* $E_{k,0}(\text{--H}) = \prod_{x \in \{0,1\}^{k-1}} (H_{[G_n^{-1}(0x0),G_n^{-1}(0x1)]} H_{[G_n^{-1}(1x0),G_n^{-1}(1x1)]})$;

* $E_{k,0}(\text{+H}) = \prod_{x \in \{0,1\}^{k-1}} (H_{[G_n^{-1}(0x10),G_n^{-1}(0x11)]} H_{[G_n^{-1}(1x10),G_n^{-1}(1x11)]})$;

finally, for any $\ell \ge 0$,

* $E_{k,l}(\text{--C}) = E_{k,l}(\text{+C}) \circ$

$$ \prod_{\substack{x \in \{0,1\}^k \\ y \in \{0,1\}^{\ell}}} ((-1)_{[G_n^{-1}(x11y)]} X_{[G_n^{-1}(x10y),G_n^{-1}(x11y)]}) \circ $$

$$ \prod_{\substack{x \in \{0,1\}^k \\ y \in \{0,1\}^{\ell}}} ((-1)_{[G_n^{-1}(x01y)]} X_{[G_n^{-1}(x01y),G_n^{-1}(x11y)]}) \circ $$

$$ \prod_{\substack{x \in \{0,1\}^k \\ y \in \{0,1\}^{\ell}}} ((-1)_{[G_n^{-1}(x11y)]} X_{[G_n^{-1}(x10y),G_n^{-1}(x11y)]}) . $$

**Definition 8.3 (Decoding).** For any $n \ge 3$, let $D: \mathcal{P}_{2n}^* \rightarrow \text{QC}$ be the monoid homomorphism defined as follows on the generators:

**Case of** $((-1)_{[a]})(-1)_{[b]}$.

* For any $a \in \{0, ..., 2^n - 1\}$, we define

$$ D((-1)_{[a]})(-1)_{[a]} := (\text{---})^{\otimes n} . $$

* For any $a \in \{0, ..., 2^n - 2\}$, we define

$$ D((-1)_{[a]})(-1)_{[a+1]} := \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] \node (A) at (0,0) {$\Lambda^x$}; \node (B) at (0,1) {$\Lambda^y$}; \draw (A) -- (B); \draw (A) -- (-.5,0); \draw (B) -- (-.5,0); \draw (-.5,0) -- (0,0); \node at (0,0) {---}; \node at (0,1) {---}; \end{tikzpicture} $$

where $\beta \in \{0, 1\}$, $x \in \{0, 1\}^k$, and $y \in \{0, 1\}^{n-k-1}$ are such that $G_n(a) = x\beta y$ and $G_n(a+1) = x\beta y$.⁷

* If $a \le b$, then we define⁸

$$ D((-1)_{[a]})(-1)_{[b]} := \overset{b-1}{\underset{k=a}{\bigwedge}} D((-1)_{[k]})(-1)_{[k+1]} $$

where we use the notations $\prod_{k=n}^{m} C_k := C_m \circ \dots \circ C_n$ and $\prod_{k=n}^{m} C_k := C_n \circ \dots \circ C_m$, with by convention the products being the identity if they are empty (i.e. if $m < n$).

* Finally, if $a > b$, we define $D((-1)_{[a]})(-1)_{[b]} := D((-1)_{[b]})(-1)_{[a]}$.

**Case of** $((-1)_{[c]})(X_{[a,b]})$.

* For any $a \in \{0, ..., 2^n - 2\}$, we define

$$ D((-1)_{[a]}X_{[a,a+1]}) := \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] \node (A) at (0,0) {$\Lambda^x$}; \node (B) at (0,1) {$\Lambda^y$}; \draw (A) -- (B); \draw (A) -- (-.5,0); \draw (B) -- (-.5,0); \draw (-.5,0) -- (0,0); \node at (0,0) {---}; \node at (0,1) {---}; \end{tikzpicture} , \quad D((-1)_{[a+1]}X_{[a,a+1]}) := \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}] \node (A) at (0,0) {$\Lambda^x$}; \node (B) at (0,1) {$\Lambda^y$}; \draw (A) -- (B); \draw (A) -- (-.5,0); \draw (B) -- (-.5,0); \draw (-.5,0) -- (0,0); \node at (0,0) {---}; \node at (0,1) {---}; \end{tikzpicture} $$

where $\beta \in \{0, 1\}$, $x \in \{0, 1\}^k$, and $y \in \{0, 1\}^{n-k-1}$ are such that $G_n(a) = x\beta y$ and $G_n(a+1) = x\beta y$.

* If $a < b$, then we define inductively on $b - a$ (using the previous case as base case):

- if $b - a$ is odd, $D((-1)_{[a]}X_{[a,b]}) =$

$$ D((-1)_{[a+1]}X_{[a,a+1]}) \circ D((-1)_{[a+1]}X_{[a+1,b]}) \circ D((-1)_{[a]}X_{[a,a+1]}) $$

and $D((-1)_{[b]}X_{[a,b]}) =$

$$ D((-1)_{[a+1]}X_{[a,a+1]}) \circ D((-1)_{[b]}X_{[a+1,b]}) \circ D((-1)_{[a]}X_{[a,a+1]}) ; $$

- if $b - a$ is even, $D((-1)_{[a]}X_{[a,b]}) =$

$$ D((-1)_{[b-1]}X_{[b-1,b]}) \circ D((-1)_{[a]}X_{[a,b-1]}) \circ D((-1)_{[b]}X_{[b-1,b]}) $$

and $D((-1)_{[b]}X_{[a,b]}) =$

$$ D((-1)_{[b-1]}X_{[b-1,b]}) \circ D((-1)_{[b-1]}X_{[a,b-1]}) \circ D((-1)_{[b]}X_{[b-1,b]}) . $$

* If $a > b$, then we define

$$ D((-1)_{[a]}X_{[a,b]}) := D((-1)_{[a]}X_{[b,a]}), $$

$$ D((-1)_{[b]}X_{[a,b]}) := D((-1)_{[b]}X_{[b,a]}). $$

⁷$\beta, x, y$ necessarily exist by the properties of the Gray code.

⁸Note that the definition is consistent in the case where $b = a + 1$.
# 9 Conclusion and Perspectives

We have introduced a complete equational theory for unitary quantum circuits over the gate set $\{\overline{H}, \bullet, \overline{H}, \bullet\}$. To our knowledge, this is the second universal fragment of unitary quantum circuits to be given a complete axiomatization, and the first which is finitely generated. One can note the relative similarity between our complete equational theory and that of [9], which is proved to be minimal: both equational theories are composed of a few simple equations on at most 3 qubits, plus an equation schema consisting of one equation for each number of qubits; in both cases, the equation schema has a trivial semantic interpretation, and its meaning comes from the fact that the multi-controlled gate is expressed as a macro built from 1- and 2- qubit gates. A difference between our equational theory and that of [9] is that since we are working in a discrete setting, ours does not require working with continuous parameters.

An obvious follow-up question for future work is whether the unbounded equation schema of Equation (19) is actually necessary, or can be derived from a finite number of equations. In the second case, one would have to find a derivation of it; in the first case, one would have to find an argument to show its necessity. Note that the necessity argument used in [9] does not apply to our case.

Our result is for unitary circuits, without ancillas. One could proceed in the same way as in [10] to extend the equational theory and its completeness to circuits with qubit initializations, ancillas, and qubit discarding (i.e. trace-out). If Equation (19) turned out to be necessary in the unitary case, then another open question would be whether it can be derived in the presence of ancillas, similarly to what is done in [9, 10].

Another open problem is to find complete equational theories for other relevant fragments of quantum circuits. In particular, one could a priori proceed in the same way as we do in this paper in order to obtain complete equational theories for the Toffoli-Hadamard and Clifford+CS fragments, relying on the corresponding matrix group presentations given respectively in [27] and [5]. Doing the same for Clifford+T would require first to extend the presentation of [18] to arbitrary dimension.

# Acknowledgments

This work has been partially funded by the French National Research Agency (ANR) within the framework of "Plan France 2030", under the research projects EPIQ ANR-22-PETQ-0007, HQI-Acquisition ANR-22-PNCQ-0001 and HQI-R&D ANR-22-PNCQ-0002.

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# Appendix Contents

<table><tr><td>A</td><td>Auxiliary Language</td><td>15</td></tr><tr><td>A.1</td><td>Modified Exact Synthesis Algorithm</td><td>15</td></tr><tr><td>A.2</td><td>Slight Simplification of the Complete Equational Theory of [17]</td><td>15</td></tr><tr><td>A.3</td><td>Applying the Reidemeister-Schreier Theorem</td><td>21</td></tr><tr><td>A.3.1</td><td>Elements of the Reidemeister-Schreier Theorem for Monoids</td><td>21</td></tr><tr><td>A.3.2</td><td>Raw Output of the Reidemeister-Schreier Theorem</td><td>23</td></tr><tr><td>A.4</td><td>Simplifying the Output of the Reidemeister-Schreier Theorem</td><td>27</td></tr><tr><td>A.4.1</td><td>Details about Equations (36) and (37)</td><td>27</td></tr><tr><td>A.4.2</td><td>Proof of Completeness of ≈ for P<sub>N</sub><sup>*</sup></td><td>28</td></tr><tr><td>B</td><td>Some auxiliary Equations and Useful Properties</td><td>55</td></tr><tr><td>B.1</td><td>A Few Useful General Properties of Circuits</td><td>55</td></tr><tr><td>B.2</td><td>Auxiliary Equations on 1 and 2 Qubits</td><td>56</td></tr><tr><td>C</td><td>Two-Qubit Completeness</td><td>62</td></tr><tr><td>C.1</td><td>Proof of Lemma 7.4</td><td>62</td></tr><tr><td>C.2</td><td>Proof of Lemma 7.5</td><td>62</td></tr><tr><td>D</td><td>Other Auxiliary Equations and Properties of Multi-Controlled gates</td><td>66</td></tr><tr><td>D.1</td><td>Additional Auxiliary Equations on 2 Qubits</td><td>66</td></tr><tr><td>D.2</td><td>Auxiliary Equations on 3 Qubits</td><td>66</td></tr><tr><td>D.3</td><td>Auxiliary Equations on 4 Qubits</td><td>79</td></tr><tr><td>D.4</td><td>Auxiliary Equations on 5 Qubits</td><td>136</td></tr><tr><td>D.5</td><td>Basic Auxiliary Equations on n Qubits and Properties of Multi-Controlled Gates</td><td>153</td></tr><tr><td>D.6</td><td>Further Auxiliary Equations on n Qubits and Properties of Multi-Controlled Gates</td><td>183</td></tr><tr><td>D.7</td><td>Commutation Properties of Multi-Controlled Gates</td><td>189</td></tr><tr><td>D.8</td><td>Properties of Multi-Controlled ZX and XZ Gates</td><td>234</td></tr><tr><td>D.9</td><td>Additional Useful Auxiliary Equations on n Qubits</td><td>237</td></tr><tr><td>E</td><td>Completeness in the General Case</td><td>244</td></tr><tr><td>E.1</td><td>Definition of Σ<sub>a,b,c,d</sub> and Σ'<sub>a,b,c,d</sub></td><td>244</td></tr><tr><td>E.2</td><td>Proof of Lemma 8.4</td><td>245</td></tr><tr><td>E.3</td><td>Proof of Lemma 8.5</td><td>247</td></tr><tr><td>E.4</td><td>Proof of Lemma 8.6</td><td>250</td></tr><tr><td>E.5</td><td>Proof of Lemma 8.8</td><td>252</td></tr></table>

# A Auxiliary Language

## A.1 Modified Exact Synthesis Algorithm

## A.2 Slight Simplification of the Complete Equational Theory of [17]

**Lemma A.1.** The equations of Figure 6 can be derived from those of Figure 7.

**PROOF.** Strictly speaking, except for Equations (e1) and (e2), we only need to consider the cases where the two indices of each generator are in increasing order. However, for more simplicity, we prove each equation in full generality without this restriction (note that the two indices of each generator still need to be distinct).

*Proof of Equation (a1):*

* If $a = 0$, then Equation (a1) is just Equation (a1*).

* If $a \neq 0$:

$$(-1)^2_{[a]} \stackrel{(a2)}{\approx} X^2_{[0,a]} (-1)^2_{[a]}$$

$$\stackrel{(c1)}{\approx} X_{[0,a]} (-1)^2_{[0]} X_{[0,a]}$$

$$\stackrel{(a1')}{\approx} X^2_{[0,a]}$$

$$\stackrel{(a2)}{\approx} \epsilon$$
**Algorithm 1** Exact synthesis for $U_N(\mathbb{Z}[\frac{1}{\sqrt{2}}])$ (cf. [17]).

**Input:** An element $A$ of $U_N(\mathbb{Z}[\frac{1}{\sqrt{2}}])$

**Output:** A sequence $W_1, \dots, W_t$ of words over $\mathcal{G}_N$ such that $W_1 \cdots W_t A = I$

1: $B \leftarrow A$

2: **for** $j \leftarrow N-1, N-2, \dots, 0$ **do**

3: $k \leftarrow \text{lde}(Be_j)$

4: **while** $k > 0$ **do**

5: &nbsp;&nbsp;&nbsp;&nbsp;Let $i_1$ be the least integer such that $\sqrt{2}^k B[i_1, j] \equiv 1 \pmod{2}$ or $\equiv 1 + \sqrt{2} \pmod{2}$

6: &nbsp;&nbsp;&nbsp;&nbsp;Let $i_2$ be the least integer $> i_1$ such that $\sqrt{2}^k B[i_2, j] \equiv \sqrt{2}^k B[i_1, j] \pmod{2}$

7: &nbsp;&nbsp;&nbsp;&nbsp;W $\leftarrow H_{[i_1, i_2]}$

8: &nbsp;&nbsp;&nbsp;&nbsp;Output W

9: $B \leftarrow WB, k \leftarrow \text{lde}(Be_j)$

10: **end while**

11: **end for**

12: **for** $j \leftarrow N-1, N-2, \dots, 0$ **do**

13: $v \leftarrow Be_j$

14: **if** $v \neq e_j$ **then**

15: &nbsp;&nbsp;&nbsp;&nbsp;There are $a \le j$ and $\tau \in \{0, 1\}$ such that $v = (-1)^\tau e_a$.

16: **if** $a = j$ **then** W $\leftarrow (-1)^\tau$$_{[a]}$

17: **if** $a < j$ **then** W $\leftarrow X_{[a,j]}(-1)^\tau$$_{[a]}$

18: &nbsp;&nbsp;&nbsp;&nbsp;Output W

19: $B \leftarrow WB$

20: **end if**

21: **end for**

► See [17] for the definition of lde

► See [1] for the existence of $i_1$ and $i_2$

► lde$(Be_j) = 0$ for all $j$, therefore $B$ is now a permutation matrix

**Proof of Equation (e1):**

$$X_{[b,c]} \stackrel{(a2)}{\approx} X_{[b,c]}X_{[b,c]}X_{[b,c]}$$

$$\stackrel{(a3)}{\approx} X_{[b,c]}H_{[b,c]}H_{[b,c]}X_{[b,c]}X_{[b,c]}$$

$$\stackrel{(d2)}{\approx} X_{[b,c]}H_{[b,c]}(-1)_{[c]}H_{[b,c]}X_{[b,c]}$$

$$\stackrel{(e2^*)}{\approx} H_{[c,b]}X_{[b,c]}(-1)_{[c]}H_{[b,c]}X_{[b,c]}$$

$$\stackrel{(c1)}{\approx} H_{[c,b]}(-1)_{[b]}X_{[b,c]}H_{[b,c]}X_{[b,c]}$$

$$\stackrel{(e2^*)}{\approx} H_{[c,b]}(-1)_{[b]}H_{[c,b]}X_{[b,c]}X_{[b,c]}$$

$$\stackrel{(a2)}{\approx} H_{[c,b]}(-1)_{[b]}H_{[c,b]}$$

$$\stackrel{(d2)}{\approx} H_{[c,b]}H_{[c,b]}X_{[c,b]}$$

$$\stackrel{(a3)}{\approx} X_{[c,b]}$$

**Proof of Equation (e2):**

$$H_{[c,b]} \stackrel{(a2)}{\approx} H_{[c,b]}X_{[b,c]}X_{[b,c]}$$

$$\stackrel{(e2^*)}{\approx} X_{[b,c]}H_{[b,c]}X_{[b,c]}$$

**Proof of Equation (e3):**

$$X_{[b,c]}X_{[a,b]} \stackrel{(a3)}{\approx} X_{[b,c]}H_{[a,b]}H_{[a,b]}X_{[a,b]}$$

$$\stackrel{(d2)}{\approx} X_{[b,c]}H_{[a,b]}(-1)_{[b]}H_{[a,b]}$$
$$ (c5) \quad \approx H_{[a,c]}X_{[b,c]}(-1)_{[b]}H_{[a,b]} $$

$$ (a2) \quad \approx H_{[a,c]}X_{[b,c]}(-1)_{[b]}X_{[b,c]}X_{[b,c]}H_{[a,b]} $$

$$ (c1) \quad \approx H_{[a,c]}X_{[b,c]}X_{[b,c]}(-1)_{[c]}X_{[b,c]}H_{[a,b]} $$

$$ (a2) \quad \approx H_{[a,c]}(-1)_{[c]}X_{[b,c]}H_{[a,b]} $$

$$ (c5) \quad \approx H_{[a,c]}(-1)_{[c]}H_{[a,c]}X_{[b,c]} $$

$$ (d2) \quad \approx H_{[a,c]}H_{[a,c]}X_{[a,c]}X_{[b,c]} $$

$$ (a3) \quad \approx X_{[a,c]}X_{[b,c]} $$

### Proof of Equation (c2):

$$ X_{[a,b]}X_{[a,c]} \quad \stackrel{(e1)}{\approx} \quad X_{[a,b]}X_{[c,a]} $$

$$ \stackrel{(c3)}{\approx} X_{[c,b]}X_{[a,b]} $$

$$ \stackrel{(e1)}{\approx} X_{[b,c]}X_{[a,b]} $$

### Proof of Equation (c4):

$$ X_{[a,b]}H_{[a,c]} \quad \stackrel{(a2)}{\approx} \quad X_{[a,b]}H_{[a,c]}X_{[c,a]}X_{[c,a]} $$

$$ \stackrel{(e2')}{\approx} X_{[a,b]}X_{[c,a]}H_{[c,a]}X_{[c,a]} $$

$$ \stackrel{(c3)}{\approx} X_{[c,b]}X_{[a,b]}H_{[c,a]}X_{[c,a]} $$

$$ \stackrel{(c5)}{\approx} X_{[c,b]}H_{[c,b]}X_{[a,b]}X_{[c,a]} $$

$$ \stackrel{(c3)}{\approx} X_{[c,b]}H_{[c,b]}X_{[c,b]}X_{[a,b]} $$

$$ \stackrel{(e2')}{\approx} H_{[b,c]}X_{[c,b]}X_{[c,b]}X_{[a,b]} $$

$$ \stackrel{(a2)}{\approx} H_{[b,c]}X_{[a,b]} $$

The following variants of Equations (c2), (c4), (c5) and (d2) will be useful to prove the remaining equations:

$$ X_{[a,b]}X_{[b,c]} \approx X_{[a,c]}X_{[a,b]} \qquad (c2^\dagger) $$

$$ X_{[a,b]}H_{[b,c]} \approx H_{[a,c]}X_{[a,b]} \qquad (c4^\dagger) $$

$$ X_{[b,c]}H_{[a,c]} \approx H_{[a,b]}X_{[b,c]} \qquad (c5^\dagger) $$

$$ H_{[a,b]}(-1)_{[b]} \approx X_{[a,b]}H_{[a,b]} \qquad (d2^\dagger) $$

### Proof of Equation (c2$^\dagger$):

$$ X_{[a,c]}X_{[a,b]} \quad \stackrel{(a2)}{\approx} \quad X_{[a,b]}X_{[a,b]}X_{[a,c]}X_{[a,b]} $$

$$ \stackrel{(c2)}{\approx} X_{[a,b]}X_{[b,c]}X_{[a,b]}X_{[a,b]} $$

$$ \stackrel{(a2)}{\approx} X_{[a,b]}X_{[b,c]} $$

### Proof of Equation (c4$^\dagger$):

$$ H_{[a,c]}X_{[a,b]} \quad \stackrel{(a2)}{\approx} \quad X_{[a,b]}X_{[a,b]}H_{[a,c]}X_{[a,b]} $$

$$ \stackrel{(c4)}{\approx} X_{[a,b]}H_{[b,c]}X_{[a,b]}X_{[a,b]} $$

$$ \stackrel{(a2)}{\approx} X_{[a,b]}H_{[b,c]} $$
Proof of Equation (c5†):

$$ 
\begin{align*}
H_{[a,b]} X_{[b,c]} & \stackrel{(a2)}{\approx} X_{[b,c]} X_{[b,c]} H_{[a,b]} X_{[b,c]} \\
& \stackrel{(c5)}{\approx} X_{[b,c]} H_{[a,c]} X_{[b,c]} X_{[b,c]} \\
& \stackrel{(a2)}{\approx} X_{[b,c]} H_{[a,c]}
\end{align*}
$$

Proof of Equation (d2†):

$$ 
\begin{align*}
H_{[a,b]} (-1)_{[b]} & \stackrel{(a3)(a2)}{\approx} H_{[a,b]} (-1)_{[b]} H_{[a,b]} X_{[a,b]} X_{[a,b]} H_{[a,b]} \\
& \stackrel{(d2)}{\approx} H_{[a,b]} (-1)_{[b]} (-1)_{[b]} H_{[a,b]} X_{[a,b]} H_{[a,b]} \\
& \stackrel{(a1)(a3)}{\approx} X_{[a,b]} H_{[a,b]}
\end{align*}
$$

Proof of Equation (b2):<fnref n="10" />

$$ 
\begin{align*}
(-1)_{[c]} X_{[a,b]} & \stackrel{(a2)}{\approx} X_{[a,c]} X_{[a,c]} (-1)_{[c]} X_{[a,b]} \\
& \stackrel{(c1)}{\approx} X_{[a,c]} (-1)_{[a]} X_{[a,c]} X_{[a,b]} \\
& \stackrel{(c2\dagger)}{\approx} X_{[a,c]} (-1)_{[a]} X_{[a,b]} X_{[b,c]} \\
& \stackrel{(c1)}{\approx} X_{[a,c]} X_{[a,b]} (-1)_{[b]} X_{[b,c]} \\
& \stackrel{(c1)}{\approx} X_{[a,c]} X_{[a,b]} X_{[b,c]} (-1)_{[c]} \\
& \stackrel{(c2\dagger)}{\approx} X_{[a,c]} X_{[a,c]} X_{[a,b]} (-1)_{[c]} \\
& \stackrel{(a2)}{\approx} X_{[a,b]} (-1)_{[c]}
\end{align*}
$$

Proof of Equation (b5):

$$ 
\begin{align*}
H_{[c,d]} X_{[a,b]} & \stackrel{(a2)}{\approx} X_{[a,c]} X_{[a,c]} H_{[c,d]} X_{[a,b]} \\
& \stackrel{(c4\dagger)}{\approx} X_{[a,c]} H_{[a,d]} X_{[a,c]} X_{[a,b]} \\
& \stackrel{(c2\dagger)}{\approx} X_{[a,c]} H_{[a,d]} X_{[a,b]} X_{[b,c]} \\
& \stackrel{(c4\dagger)}{\approx} X_{[a,c]} X_{[a,b]} H_{[b,d]} X_{[b,c]} \\
& \stackrel{(c4\dagger)}{\approx} X_{[a,c]} X_{[a,b]} X_{[b,c]} H_{[c,d]} \\
& \stackrel{(c2\dagger)}{\approx} X_{[a,c]} X_{[a,c]} X_{[a,b]} H_{[c,d]} \\
& \stackrel{(a2)}{\approx} X_{[a,b]} H_{[c,d]}
\end{align*}
$$

Proof of Equation (b3):<fnref n="11" />

$$ 
\begin{align*}
X_{[c,d]} X_{[a,b]} & \stackrel{(a2)}{\approx} X_{[a,c]} X_{[a,c]} X_{[c,d]} X_{[a,b]} \\
& \stackrel{(c2\dagger)}{\approx} X_{[a,c]} X_{[a,d]} X_{[a,c]} X_{[a,b]} \\
& \stackrel{(c2\dagger)}{\approx} X_{[a,c]} X_{[a,d]} X_{[a,b]} X_{[b,c]}
\end{align*}
$$

<footnote n="10">¹⁰We have renamed the indices in order to emphasize the similarity with the proofs of Equations (b5) and (b3).</footnote>

<footnote n="11">¹¹Note that the proof is identical to that of Equation (b5), with *H* replaced by *X* and Equation (c4†) replaced by Equation (c2†).</footnote>
$$ (c2^\dagger) \quad \approx \quad X_{[a,c]}X_{[a,b]}X_{[b,d]}X_{[b,c]} $$

$$ (c2^\dagger) \quad \approx \quad X_{[a,c]}X_{[a,b]}X_{[b,c]}X_{[c,d]} $$

$$ (c2^\dagger) \quad \approx \quad X_{[a,c]}X_{[a,c]}X_{[a,b]}X_{[c,d]} $$

$$ \begin{align*} (a2) & \approx X_{[a,b]}X_{[c,d]} \\ & \approx (-1)_{[b]}(-1)_{[0]} \end{align*} $$

### Proof of Equation (b1):

* If $\{a, b\} = \{0, 1\}$, then Equation (b1) is just Equation (b1*).

* If $a = 0$ and $b \neq 0, 1$:

$$ (-1)_{[0]}(-1)_{[b]} \quad \begin{align*} (a2) & \approx X_{[1,b]}X_{[1,b]}(-1)_{[0]}(-1)_{[b]} \\ (b2) & \approx X_{[1,b]}(-1)_{[0]}X_{[1,b]}(-1)_{[b]} \end{align*} $$

$$ (c1) \quad \approx X_{[1,b]}(-1)_{[0]}(-1)_{[1]}X_{[1,b]} $$

$$ (b1^*) \quad \approx X_{[1,b]}(-1)_{[1]}(-1)_{[0]}X_{[1,b]} $$

$$ (c1) \quad \approx X_{[1,b]}X_{[1,b]}(-1)_{[b]}(-1)_{[0]} $$

* If $b = 0$ and $a \neq 0, 1$, or if $a = 1$ and $b \neq 0, 1$, or if $b = 1$ and $a \neq 0, 1$, then the proof is identical, with $a$ replaced by $b$ and/or $0$ replaced by $1$.

* If $\{a, b\} \cap \{0, 1\} = \emptyset$:

$$ (-1)_{[a]}(-1)_{[b]} \quad \begin{align*} (a2) & \approx X_{[0,a]}X_{[0,a]}(-1)_{[a]}(-1)_{[b]} \\ (a2) & \approx X_{[0,a]}X_{[1,b]}X_{[1,b]}X_{[0,a]}(-1)_{[a]}(-1)_{[b]} \end{align*} $$

$$ (c1) \quad \approx X_{[0,a]}X_{[1,b]}X_{[1,b]}(-1)_{[0]}X_{[0,a]}(-1)_{[b]} $$

$$ (b2) \quad \approx X_{[0,a]}X_{[1,b]}X_{[1,b]}(-1)_{[0]}(-1)_{[b]}X_{[0,a]} $$

$$ (c1^*) \quad \approx X_{[0,a]}X_{[1,b]}(-1)_{[0]}(-1)_{[1]}X_{[1,b]}X_{[0,a]} $$

$$ (b2) \quad \approx X_{[0,a]}X_{[1,b]}(-1)_{[1]}X_{[1,b]}(-1)_{[0]}X_{[0,a]} $$

$$ (c1) \quad \approx X_{[0,a]}X_{[1,b]}X_{[1,b]}(-1)_{[b]}X_{[0,a]}(-1)_{[a]} $$

$$ (a2) \quad \approx (-1)_{[b]}(-1)_{[a]} $$

The proofs of Equations (b4) and (b6) follow the same principle as the proof of Equation (b1): using Equation (a2) followed by Equations (c1), (b2), (c4), (c5) and (b5) (and maybe (e1)), we can arbitrarily rename the indices up to conjugating by some $X$ generators, so that we can conclude using Equation (b4*) (resp. Equation (b6*)).
Proof of Equation (d1):

$$(-1)_{[a]}(-1)_{[b]}H_{[a,b]} \stackrel{(d2)}{\approx} (-1)_{[a]}H_{[a,b]}X_{[a,b]}$$

$$\stackrel{(e1)}{\approx} (-1)_{[a]}H_{[a,b]}X_{[b,a]}$$

$$\stackrel{(e2^*)}{\approx} (-1)_{[a]}X_{[b,a]}H_{[b,a]}$$

$$\stackrel{(a3)}{\approx} (-1)_{[a]}H_{[b,a]}H_{[b,a]}X_{[b,a]}H_{[b,a]}$$

$$\stackrel{(d2)}{\approx} H_{[b,a]}X_{[b,a]}H_{[b,a]}X_{[b,a]}H_{[b,a]}$$

$$\stackrel{(d2)}{\approx} H_{[b,a]}X_{[b,a]}(-1)_{[a]}H_{[b,a]}H_{[b,a]}$$

$$\stackrel{(a3)}{\approx} H_{[b,a]}X_{[b,a]}(-1)_{[a]}$$

$$\stackrel{(e1)}{\approx} H_{[b,a]}X_{[a,b]}(-1)_{[a]}$$

$$\stackrel{(e2^*)}{\approx} X_{[a,b]}H_{[a,b]}(-1)_{[a]}$$

$$\stackrel{(a3)}{\approx} H_{[a,b]}H_{[a,b]}X_{[a,b]}H_{[a,b]}(-1)_{[a]}$$

$$\stackrel{(d2)}{\approx} H_{[a,b]}(-1)_{[b]}H_{[a,b]}H_{[a,b]}(-1)_{[a]}$$

$$\stackrel{(a3)}{\approx} H_{[a,b]}(-1)_{[b]}(-1)_{[a]}$$

$$\stackrel{(b1)}{\approx} H_{[a,b]}(-1)_{[a]}(-1)_{[b]}$$

□

Proof of Equation (d3): By proceeding similarly to the proofs of Equations (b1) to (b6), we can rename the indices arbitrarily, therefore it suffices to prove the following equation:

$$(H_{[2,3]}H_{[0,2]}H_{[1,3]})^4 \approx H_{[0,1]}H_{[2,3]}$$

which we do as follows:

$$
\begin{align*}
& (H_{[2,3]}H_{[0,2]}H_{[1,3]})^4 \\
&= H_{[2,3]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]} \\
&\stackrel{(a1)(b1)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]} \\
&\stackrel{(b4)(d2)(b5)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}X_{[0,2]}X_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]} \\
&\stackrel{(c5^*)(c4^*)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}X_{[0,2]}X_{[1,3]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]} \\
&\stackrel{(b5)(d2^*)(b4)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}(-1)_{[2]}(-1)_{[3]}H_{[2,3]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]} \\
&\stackrel{(d1)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]} \\
&\stackrel{(b4)(d2)(b5)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]}X_{[0,2]}X_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]} \\
&\stackrel{(c5^*)(c4^*)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}X_{[0,2]}X_{[1,3]}H_{[0,2]}H_{[1,3]} \\
&\stackrel{(b5)(d2^*)(b4)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}(-1)_{[2]}(-1)_{[3]} \\
&\stackrel{(a1)(b1)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}(-1)_{[0]}(-1)_{[1]}(-1)_{[2]}(-1)_{[3]}(-1)_{[2]}(-1)_{[3]}(-1)_{[0]}(-1)_{[1]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}(-1)_{[2]}(-1)_{[3]} \\
&\stackrel{(b1)(b4)(d1)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}(-1)_{[0]}(-1)_{[1]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]}(-1)_{[2]}(-1)_{[3]}(-1)_{[0]}(-1)_{[1]}H_{[0,1]}H_{[0,2]}H_{[1,3]}(-1)_{[2]}(-1)_{[3]} \\
&\stackrel{(b4)(d2)(b5)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}(-1)_{[0]}(-1)_{[1]}H_{[0,2]}H_{[1,3]}X_{[0,2]}X_{[1,3]}H_{[2,3]}H_{[0,2]}H_{[1,3]}(-1)_{[2]}(-1)_{[3]}(-1)_{[0]}(-1)_{[1]}H_{[0,1]}H_{[0,2]}H_{[1,3]}(-1)_{[2]}(-1)_{[3]} \\
&\stackrel{(c5^*)(c4^*)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}(-1)_{[0]}(-1)_{[1]}H_{[0,2]}H_{[1,3]}H_{[0,1]}X_{[0,2]}X_{[1,3]}H_{[0,2]}H_{[1,3]}(-1)_{[2]}(-1)_{[3]}(-1)_{[0]}(-1)_{[1]}H_{[0,1]}H_{[0,2]}H_{[1,3]}(-1)_{[2]}(-1)_{[3]} \\
&\stackrel{(b5)(d2^*)(b4)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}(-1)_{[0]}(-1)_{[1]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}(-1)_{[2]}(-1)_{[3]}(-1)_{[0]}(-1)_{[1]}H_{[0,1]}H_{[0,2]}H_{[1,3]}(-1)_{[2]}(-1)_{[3]} \\
&\stackrel{(b1)(a1)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,2]}H_{[1,3]}H_{[0,1]}(-1)_{[0]}(-1)_{[1]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}(-1)_{[0]}(-1)_{[1]}H_{[0,1]}H_{[0,2]}H_{[1,3]}(-1)_{[2]}(-1)_{[3]} \\
&\stackrel{(d3^*)}{\approx} H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}H_{[0,1]}(-1)_{[0]}(-1)_{[1]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}(-1)_{[2]}(-1)_{[3]}(-1)_{[0]}(-1)_{[1]}H_{[0,1]}H_{[0,2]}H_{[1,3]}(-1)_{[2]}(-1)_{[3]}
\end{align*}
$$
$$ (b1)(b6)(a3)(a1) \approx H_{[2,3]}(-1)_{[2]}(-1)_{[3]}H_{[0,1]}(-1)_{[2]}(-1)_{[3]} $$

$$ (b4)(b6)(b1)(a1) \approx H_{[0,1]}H_{[2,3]} $$

Proof of Equation (d4): By proceeding similarly to the proofs of Equations (b1) to (b6), we can rename the indices arbitrarily, therefore it suffices to prove the following equation:

$$ (H_{[0,4]}H_{[1,5]}H_{[0,1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]})^3 \approx H_{[4,2]}H_{[5,3]}H_{[2,3]}H_{[4,2]}H_{[5,3]}X_{[4,2]}X_{[5,3]} $$

which we do as follows:

$$ \begin{align*} & (H_{[0,4]}H_{[1,5]}H_{[0,1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]})^3 \\ &= H_{[0,4]}H_{[1,5]}H_{[0,1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]}H_{[0,4]}H_{[1,5]}H_{[0,1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]} \\ & \quad (b5)(c5) \approx H_{[0,4]}H_{[1,5]}H_{[0,1]}H_{[0,4]}H_{[1,5]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}X_{[4,2]}X_{[5,3]}X_{[4,2]}X_{[5,3]}H_{[0,4]}H_{[1,5]}H_{[0,1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]} \\ & \quad (b3)(a2) \approx H_{[0,4]}H_{[1,5]}H_{[0,1]}H_{[0,4]}H_{[1,5]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}H_{[0,4]}H_{[1,5]}H_{[0,1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]} \\ & \quad (a1)(b1) \approx H_{[0,4]}H_{[1,5]}H_{[0,1]}(-1)_{[0]}(-1)_{[1]}(-1)_{[4]}(-1)_{[5]}(-1)_{[4]}(-1)_{[5]}(-1)_{[0]}(-1)_{[1]}H_{[0,4]}H_{[1,5]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}H_{[0,4]}H_{[1,5]}H_{[0,1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]} \\ & \quad (d1)(b4)(b1) \approx (-1)_{[0]}(-1)_{[1]}(-1)_{[4]}(-1)_{[5]}H_{[0,4]}H_{[1,5]}(-1)_{[4]}(-1)_{[5]}H_{[0,1]}(-1)_{[0]}(-1)_{[1]}H_{[0,4]}H_{[1,5]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}H_{[0,4]}H_{[1,5]}H_{[0,1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]} \\ & \quad (b4)(d2*)^3(b5) \approx (-1)_{[0]}(-1)_{[1]}(-1)_{[4]}(-1)_{[5]}X_{[0,4]}X_{[1,5]}H_{[0,4]}H_{[1,5]}H_{[0,1]}(-1)_{[0]}(-1)_{[1]}H_{[0,4]}H_{[1,5]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}H_{[0,4]}H_{[1,5]}H_{[0,1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]} \\ & \quad (a3)(b6) \approx (-1)_{[0]}(-1)_{[1]}(-1)_{[4]}(-1)_{[5]}X_{[0,4]}X_{[1,5]}H_{[0,4]}H_{[1,5]}H_{[0,1]}(-1)_{[0]}(-1)_{[1]}H_{[0,4]}H_{[1,5]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}H_{[0,4]}H_{[1,5]}H_{[0,1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]} \\ & \quad (d4*) \approx (-1)_{[0]}(-1)_{[1]}(-1)_{[4]}(-1)_{[5]}X_{[0,4]}X_{[1,5]}H_{[0,4]}H_{[1,5]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}H_{[0,4]}H_{[1,5]}H_{[0,1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]} \\ & \quad (b6)(a3) \approx (-1)_{[0]}(-1)_{[1]}(-1)_{[4]}(-1)_{[5]}X_{[0,4]}X_{[1,5]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}H_{[0,4]}H_{[1,5]}H_{[0,1]}(-1)_{[0]}(-1)_{[1]}H_{[0,1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]} \\ & \quad (d1)(a3) \approx (-1)_{[0]}(-1)_{[1]}(-1)_{[4]}(-1)_{[5]}X_{[0,4]}X_{[1,5]}H_{[0,2]}H_{[1,3]}H_{[0,1]}H_{[0,2]}H_{[1,3]}H_{[0,4]}H_{[1,5]}(-1)_{[0]}(-1)_{[1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]} \\ & \quad (c4)(c5) \approx (-1)_{[0]}(-1)_{[1]}(-1)_{[4]}(-1)_{[5]}H_{[4,2]}H_{[5,3]}H_{[4,5]}H_{[4,2]}H_{[5,3]}X_{[0,4]}X_{[1,5]}H_{[0,4]}H_{[1,5]}(-1)_{[0]}(-1)_{[1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]} \\ & \quad (b5)(d2*)^3(b4) \approx (-1)_{[0]}(-1)_{[1]}(-1)_{[4]}(-1)_{[5]}H_{[4,2]}H_{[5,3]}H_{[4,5]}H_{[4,2]}H_{[5,3]}H_{[0,4]}H_{[1,5]}(-1)_{[4]}(-1)_{[5]}(-1)_{[0]}(-1)_{[1]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]} \\ & \quad (b1)(b4)(d1) \approx (-1)_{[0]}(-1)_{[1]}(-1)_{[4]}(-1)_{[5]}H_{[4,2]}H_{[5,3]}H_{[4,5]}H_{[4,2]}H_{[5,3]}(-1)_{[4]}(-1)_{[5]}(-1)_{[0]}(-1)_{[1]}H_{[0,4]}H_{[1,5]}H_{[0,4]}H_{[1,5]}X_{[4,2]}X_{[5,3]} \\ & \quad (b6)(a3) \approx (-1)_{[0]}(-1)_{[1]}(-1)_{[4]}(-1)_{[5]}H_{[4,2]}H_{[5,3]}H_{[4,5]}H_{[4,2]}H_{[5,3]}(-1)_{[4]}(-1)_{[5]}(-1)_{[0]}(-1)_{[1]}X_{[4,2]}X_{[5,3]} \\ & \quad (b1)(b4)(a1) \approx (-1)_{[4]}(-1)_{[5]}H_{[4,2]}H_{[5,3]}H_{[4,5]}H_{[4,2]}H_{[5,3]}(-1)_{[4]}(-1)_{[5]}X_{[4,2]}X_{[5,3]} \\ & \quad (a1)(b1) \approx (-1)_{[4]}(-1)_{[5]}(-1)_{[2]}(-1)_{[3]}(-1)_{[2]}(-1)_{[3]}H_{[4,2]}H_{[5,3]}H_{[4,5]}H_{[4,2]}H_{[5,3]}(-1)_{[4]}(-1)_{[5]}X_{[4,2]}X_{[5,3]} \\ & \quad (b4)(d2)(b5) \approx (-1)_{[4]}(-1)_{[5]}(-1)_{[2]}(-1)_{[3]}H_{[4,2]}H_{[5,3]}X_{[4,2]}X_{[5,3]}H_{[4,5]}H_{[4,2]}H_{[5,3]}(-1)_{[4]}(-1)_{[5]}X_{[4,2]}X_{[5,3]} \\ & \quad (c4)(c5) \approx (-1)_{[4]}(-1)_{[5]}(-1)_{[2]}(-1)_{[3]}H_{[4,2]}H_{[5,3]}H_{[2,3]}X_{[4,2]}X_{[5,3]}H_{[4,2]}H_{[5,3]}(-1)_{[4]}(-1)_{[5]}X_{[4,2]}X_{[5,3]} \\ & \quad (b5)(d2*)^3(b4) \approx (-1)_{[4]}(-1)_{[5]}(-1)_{[2]}(-1)_{[3]}H_{[4,2]}H_{[5,3]}H_{[2,3]}H_{[4,2]}H_{[5,3]}(-1)_{[2]}(-1)_{[3]}(-1)_{[4]}(-1)_{[5]}X_{[4,2]}X_{[5,3]} \\ & \quad (b1)(b4)(d1)(a1) \approx H_{[4,2]}H_{[5,3]}H_{[2,3]}H_{[4,2]}H_{[5,3]}X_{[4,2]}X_{[5,3]} \end{align*} $$

## A.3 Applying the Reidemeister-Schreier Theorem

### A.3.1 Elements of the Reidemeister-Schreier Theorem for Monoids
We denote by $U_N^{\text{even}}(\mathbb{Z}[\frac{1}{\sqrt{2}}])$ the submonoid (or subgroup) of $U_N(\mathbb{Z}[\frac{1}{\sqrt{2}}])$ consisting of the matrices with even $H$- and $ZX$-parities.

In order to apply the Reidemeister-Schreier theorem for monoids described in [6] (Section 4.2), it suffices to provide the following elements:

* a set $C$, consisting of a representative of each of the four cosets of $U_N^{\text{even}}(\mathbb{Z}[\frac{1}{\sqrt{2}}])$ as a subgroup of $U_N(\mathbb{Z}[\frac{1}{\sqrt{2}}])$, the representative of $U_N^{\text{even}}(\mathbb{Z}[\frac{1}{\sqrt{2}}])$ itself being necessarily the identity $I$

* a function $f: \mathcal{P}_N \to \mathcal{G}_N^*$ (where $\mathcal{G}_N^*$ denotes the free monoid generated by $\mathcal{G}_N$) preserving the semantics

* a function $h: C \times \mathcal{G}_N \to \mathcal{P}_N^* \times C$ such that for any $(c, g)$, if $(p, c') = h(c, g)$ then $[cg] = [pc']$.

Then, let $h^{**}: C \times \mathcal{G}_N^* \to \mathcal{P}_N^* \times C$ be the function defined by $h^{**}(c_0, x_1...x_k) = (w_1...w_k, c_k)$, where $h(c_{i-1}, x_i) = (w_i, c_i)$ for all $i = 1, ..., k$. The Reidemeister-Schreier theorem for monoids stated in [6] tells us that the equational theory consisting of the following equations is complete:
(a) for every $x \in \mathcal{P}_N$, an equation $x \approx v$, where $v$ is such that $h^{**}(I, f(x)) = (v, c)$ (note that in this case, the preservation of the semantics implies that $c = I$)

(b) for every equation of Figure 7, of the form $w \approx w'$, and every $c \in C$, an equation $v \approx v'$, where $v, v'$ are such that $h^{**}(c, w) = (v, c')$ and $h^{**}(c, w') = (v', c'')$ (note that in this case, the preservation of the semantics implies that $c' = c''$).

The set $C$ and the functions $f$ and $h$ are defined in Definition A.2 below (by case distinction on the output of $f$ and $h$). The equational theory built as just described is given in Appendix A.3.2.

**Definition A.2 (Elements of the Reidemeister-Schreier theorem for monoids [6]).**

$$C = \{\epsilon, (-1)_{[1]}, H_{[0,1]}, H_{[0,1]}(-1)_{[1]}\}$$

$$f: \begin{array}{rcl} ((-1)_{[a]}(-1)_{[b]}) & \mapsto & (-1)_{[a]}(-1)_{[b]} \\ ((-1)_{[a]}X_{[c,d]}) & \mapsto & (-1)_{[a]}X_{[c,d]} \\ (X_{[a,b]}X_{[c,d]}) & \mapsto & X_{[a,b]}X_{[c,d]} \\ (H_{[a,b]}H_{[c,d]}) & \mapsto & H_{[a,b]}H_{[c,d]} \end{array}$$

$$h: \begin{array}{rcl} (\epsilon, (-1)_{[a]}) & \mapsto & (((-1)_{[1]}(-1)_{[a]}), (-1)_{[1]}) \\ (\epsilon, X_{[a,b]}) & \mapsto & \begin{cases} (((-1)_{[ab]}X_{[a,b]}), (-1)_{[1]}) & \text{if } a = 1 \text{ or } b = 1 \\ (((-1)_{[1]}X_{[a,b]}), (-1)_{[1]}) & \text{if } a, b \neq 1 \end{cases} \\ (\epsilon, H_{[a,b]}) & \mapsto & ((H_{[a,b]}H_{[0,1]}), H_{[0,1]}) \end{array}$$

$$(((-1)_{[1]}, (-1)_{[a]}) \mapsto (((-1)_{[1]}(-1)_{[a]}), \epsilon)$$

$$(((-1)_{[1]}, X_{[a,b]}) \mapsto (((-1)_{[1]}X_{[a,b]}), \epsilon)$$

$$(((-1)_{[1]}, H_{[a,b]}) \mapsto \begin{cases} (((-1)_{[b]}X_{[a,b]})(H_{[a,b]}H_{[0,1]}), H_{[0,1]}(-1)_{[1]}) & \text{if } a = 1 \text{ or } b = 1 \\ (H_{[a,b]}H_{[0,1]}), H_{[0,1]}(-1)_{[1]} & \text{if } a, b \neq 1 \end{cases})$$

$$(H_{[0,1]}, (-1)_{[a]}) \mapsto \begin{cases} (((-1)_{[1]}(-1)_{[a]}), H_{[0,1]}(-1)_{[1]}) & \text{if } a \in \{0, 1\} \\ (((-1)_{[a]}X_{[0,1]}), H_{[0,1]}(-1)_{[1]}) & \text{if } a \notin \{0, 1\} \end{cases}$$

$$(H_{[0,1]}, X_{[a,b]}) \mapsto \begin{cases} (((-1)_{[1]}X_{[0,1]}), H_{[0,1]}(-1)_{[1]}) & \text{if } \{a, b\} = \{0, 1\} \\ (X_{[0,1]}X_{[a,b]})(H_{[a+b,1]}H_{[0,1]}), H_{[0,1]}(-1)_{[1]} & \text{if } \{a, b\} \cap \{0, 1\} = \{0\} \\ (((-1)_{[ab]}X_{[a,b]})(H_{[0,ab]}H_{[0,1]}), H_{[0,1]}(-1)_{[1]}) & \text{if } \{a, b\} \cap \{0, 1\} = \{1\} \\ (X_{[a,b]}X_{[0,1]}), H_{[0,1]}(-1)_{[1]} & \text{if } \{a, b\} \cap \{0, 1\} = \emptyset \end{cases}$$

$$(H_{[0,1]}, H_{[a,b]}) \mapsto ((H_{[0,1]}H_{[a,b]}), \epsilon)$$

$$(H_{[0,1]}(-1)_{[1]}, (-1)_{[a]}) \mapsto \begin{cases} (((-1)_{[1]}(-1)_{[a]}), H_{[0,1]}) & \text{if } a \in \{0, 1\} \\ (((-1)_{[a]}X_{[0,1]}), H_{[0,1]}) & \text{if } a \notin \{0, 1\} \end{cases}$$
$$ (H_{[0,1]}(-1)_{[1]}, X_{[a,b]}) \mapsto \begin{cases} \begin{pmatrix} ((-1)_{[0]}X_{[0,1]}), H_{[0,1]} \end{pmatrix} & \text{if } \{a, b\} = \{0, 1\} \\ \begin{pmatrix} (X_{[0,1]}X_{[a,b]})(H_{[a+b,1]}H_{[0,1]}), H_{[0,1]} \end{pmatrix} & \text{if } \{a, b\} \cap \{0, 1\} = \{0\} \\ \begin{pmatrix} (X_{[0,1]}X_{[a,b]})(H_{[0,ab]}H_{[0,1]}), H_{[0,1]} \end{pmatrix} & \text{if } \{a, b\} \cap \{0, 1\} = \{1\} \\ \begin{pmatrix} (X_{[a,b]}X_{[0,1]}), H_{[0,1]} \end{pmatrix} & \text{if } \{a, b\} \cap \{0, 1\} = \emptyset \end{cases} \\ \\ (H_{[0,1]}(-1)_{[1]}, H_{[a,b]}) \mapsto \begin{cases} \begin{pmatrix} (H_{[0,1]}H_{[a,b]})((-1)_{[a]}X_{[a,b]}), (-1)_{[1]} \end{pmatrix} & \text{if } a=1 \text{ or } b=1 \\ \begin{pmatrix} (H_{[0,1]}H_{[a,b]}), (-1)_{[1]} \end{pmatrix} & \text{if } a, b \neq 1 \end{cases} $$

### A.3.2 Raw Output of the Reidemeister-Schreier Theorem.
We follow the following naming convention for the equations:

* The equations coming from Item (a) are called (DE-***) or (DE-**-k), where ** is either ZZ, ZX, XX, or HH depending on whether x is $((-1)_{[a]}(-1)_{[b]})$, $((-1)_{[a]}X_{[c,d]}), (X_{[a,b]}X_{[c,d]}), or (H_{[ab]}H_{[c,d]})$ respectively.

* The equations coming from Item (b) are called (**-††) or (**-††-k), where ** is either $\epsilon$, Z, H, or HZ depending on whether c is $\epsilon$, $(-1)_{[1]}, H_{[0,1]}, or H_{[0,1]}(-1)_{[1]}$ respectively, and †† is the label of the original equation in Figure 7.

In both cases, the additional index k is used when we need to split cases depending on the values of the parameters.

The equational theory is the following:

$$ \begin{align} ((-1)_{[a]}(-1)_{[b]}) &\approx ((-1)_{[1]}(-1)_{[a]}) ((-1)_{[1]}(-1)_{[b]}) \tag{DE-ZZ} \\ ((-1)_{[a]}X_{[c,d]}) &\approx ((-1)_{[1]}(-1)_{[a]}) ((-1)_{[1]}X_{[c,d]}) \tag{DE-ZX} \\ (X_{[a,b]}X_{[c,d]}) &\approx ((-1)_{[ab]}X_{[a,b]}) ((-1)_{[1]}X_{[c,d]}) \tag{a = 1 or b = 1} \tag{DE-XX-1} \\ (X_{[a,b]}X_{[c,d]}) &\approx ((-1)_{[1]}X_{[a,b]}) ((-1)_{[1]}X_{[c,d]}) \tag{a, b \neq 1} \tag{DE-XX-2} \\ (H_{[a,b]}H_{[c,d]}) &\approx (H_{[a,b]}H_{[0,1]}) (H_{[0,1]}H_{[c,d]}) \tag{DE-HH} \end{align} $$

$$ ((-1)_{[1]}(-1)_{[0]})^2 \approx \epsilon \tag{\epsilon-a1} $$

$$ ((-1)_{[1]}(-1)_{[0]})^2 \approx \epsilon \tag{Z-a1} $$

$$ ((-1)_{[1]}(-1)_{[0]})^2 \approx \epsilon \tag{H-a1} $$

$$ ((-1)_{[1]}(-1)_{[0]})^2 \approx \epsilon \tag{HZ-a1} $$

$$ (H_{[j,k]}H_{[0,1]}) (H_{[0,1]}H_{[j,k]}) \approx \epsilon \tag{\epsilon-a3} $$

$$ ((-1)_{[k]}X_{[j,k]}) (H_{[j,k]}H_{[0,1]}) (H_{[0,1]}H_{[j,k]}) ((-1)_{[j]}X_{[j,k]}) \approx \epsilon \tag{j = 1 or k = 1} \tag{Z-a3-1} $$

$$ (H_{[j,k]}H_{[0,1]}) (H_{[0,1]}H_{[j,k]}) \approx \epsilon \tag{j, k \neq 1} \tag{Z-a3-2} $$

$$ (H_{[0,1]}H_{[j,k]}) (H_{[j,k]}H_{[0,1]}) \approx \epsilon \tag{H-a3} $$

$$ (H_{[0,1]}H_{[j,k]}) ((-1)_{[j]}X_{[j,k]}) ((-1)_{[k]}X_{[j,k]}) (H_{[j,k]}H_{[0,1]}) \approx \epsilon \tag{j = 1 or k = 1} \tag{HZ-a3-1} $$

$$ (H_{[0,1]}H_{[j,k]}) (H_{[j,k]}H_{[0,1]}) \approx \epsilon \tag{j, k \neq 1} \tag{HZ-a3-2} $$

$$ ((-1)_{[jk]}X_{[j,k]}) ((-1)_{[1]}X_{[j,k]}) \approx \epsilon \tag{j = 1 or k = 1} \tag{\epsilon-a2-1} $$

$$ ((-1)_{[1]}X_{[j,k]}) ((-1)_{[1]}X_{[j,k]}) \approx \epsilon \tag{j, k \neq 1} \tag{\epsilon-a2-2} $$

$$ ((-1)_{[1]}X_{[j,k]}) ((-1)_{[jk]}X_{[j,k]}) \approx \epsilon \tag{j = 1 or k = 1} \tag{Z-a2-1} $$

$$ ((-1)_{[1]}X_{[j,k]}) ((-1)_{[1]}X_{[j,k]}) \approx \epsilon \tag{j, k \neq 1} \tag{Z-a2-2} $$

$$ ((-1)_{[1]}X_{[0,1]}) ((-1)_{[0]}X_{[0,1]}) \approx \epsilon \tag{j, k} = \{0, 1\} \tag{H-a2-1} $$

$$ (X_{[0,1]}X_{[j,k]}) (H_{[j+k,1]}H_{[0,1]}) (X_{[0,1]}X_{[j,k]}) (H_{[j+k,1]}H_{[0,1]}) \approx \epsilon \tag{j, k} \cap \{0, 1\} = \{0\} \tag{H-a2-2} $$

$$ ((-1)_{[jk]}X_{[j,k]}) (H_{[0,jk]}H_{[0,1]}) (X_{[0,1]}X_{[j,k]}) (H_{[0,jk]}H_{[0,1]}) \approx \epsilon \tag{j, k} \cap \{0, 1\} = \{1\} \tag{H-a2-3} $$

$$ (X_{[j,k]}X_{[0,1]}) (X_{[j,k]}X_{[0,1]}) \approx \epsilon \tag{j, k} \cap \{0, 1\} = \emptyset \tag{H-a2-4} $$
$$ ((-1)_{[0]} X_{[0,1]} ) ((-1)_{[1]} X_{[0,1]} ) \approx \epsilon \qquad \{j, k\} = \{0, 1\} \qquad (\text{HZ-a2-1}) $$

$$ (X_{[0,1]} X_{[j,k]} ) (H_{[j+k,1]} H_{[0,1]} ) (X_{[0,1]} X_{[j,k]} ) (H_{[j+k,1]} H_{[0,1]} ) \approx \epsilon \qquad \{j, k\} \cap \{0, 1\} = \{0\} \qquad (\text{HZ-a2-2}) $$

$$ (X_{[0,1]} X_{[j,k]} ) (H_{[0,jk]} H_{[0,1]} ) ((-1)_{[j,k]} X_{[j,k]} ) (H_{[0,jk]} H_{[0,1]} ) \approx \epsilon \qquad \{j, k\} \cap \{0, 1\} = \{1\} \qquad (\text{HZ-a2-3}) $$

$$ (X_{[j,k]} X_{[0,1]} ) (X_{[j,k]} X_{[0,1]} ) \approx \epsilon \qquad \{j, k\} \cap \{0, 1\} = \emptyset \qquad (\text{HZ-a2-4}) $$

$$ ((-1)_{[1]} (-1)_{[0]} ) ((-1)_{[1]} (-1)_{[1]} ) \approx ((-1)_{[1]} (-1)_{[1]} ) ((-1)_{[1]} (-1)_{[0]} ) \qquad (\epsilon \text{-b1}^*) $$

$$ ((-1)_{[1]} (-1)_{[0]} ) ((-1)_{[1]} (-1)_{[1]} ) \approx ((-1)_{[1]} (-1)_{[1]} ) ((-1)_{[1]} (-1)_{[0]} ) \qquad (\text{Z-b1}^*) $$

$$ ((-1)_{[1]} (-1)_{[0]} ) ((-1)_{[1]} (-1)_{[1]} ) \approx ((-1)_{[1]} (-1)_{[1]} ) ((-1)_{[1]} (-1)_{[0]} ) \qquad (\text{H-b1}^*) $$

$$ ((-1)_{[1]} (-1)_{[0]} ) ((-1)_{[1]} (-1)_{[1]} ) \approx ((-1)_{[1]} (-1)_{[1]} ) ((-1)_{[1]} (-1)_{[0]} ) \qquad (\text{HZ-b1}^*) $$

$$ ((-1)_{[1]} (-1)_{[1]} ) (H_{[0,2]} H_{[0,1]} ) \approx (H_{[0,2]} H_{[0,1]} ) ((-1)_{[1]} (-1)_{[1]} ) \qquad (\epsilon \text{-b4}^*) $$

$$ ((-1)_{[1]} (-1)_{[1]} ) (H_{[0,2]} H_{[0,1]} ) \approx (H_{[0,2]} H_{[0,1]} ) ((-1)_{[1]} (-1)_{[1]} ) \qquad (\text{Z-b4}^*) $$

$$ ((-1)_{[1]} (-1)_{[1]} ) (H_{[0,1]} H_{[0,2]} ) \approx (H_{[0,1]} H_{[0,2]} ) ((-1)_{[1]} (-1)_{[1]} ) \qquad (\text{H-b4}^*) $$

$$ ((-1)_{[1]} (-1)_{[1]} ) (H_{[0,1]} H_{[0,2]} ) \approx (H_{[0,1]} H_{[0,2]} ) ((-1)_{[1]} (-1)_{[1]} ) \qquad (\text{HZ-b4}^*) $$

$$ (H_{[0,1]} H_{[0,1]} ) (H_{[0,1]} H_{[2,3]} ) \approx (H_{[2,3]} H_{[0,1]} ) (H_{[0,1]} H_{[0,1]} ) \qquad (\epsilon \text{-b6}^*) $$

$$ ((-1)_{[1]} X_{[0,1]} ) (H_{[0,1]} H_{[0,1]} ) (H_{[0,1]} H_{[2,3]} ) \approx (H_{[2,3]} H_{[0,1]} ) (H_{[0,1]} H_{[0,1]} ) ((-1)_{[0]} X_{[0,1]} ) \qquad (\text{Z-b6}^*) $$

$$ (H_{[0,1]} H_{[0,1]} ) (H_{[2,3]} H_{[0,1]} ) \approx (H_{[0,1]} H_{[2,3]} ) (H_{[0,1]} H_{[0,1]} ) \qquad (\text{H-b6}^*) $$

$$ (H_{[0,1]} H_{[0,1]} ) ((-1)_{[0]} X_{[0,1]} ) (H_{[2,3]} H_{[0,1]} ) \approx (H_{[0,1]} H_{[2,3]} ) ((-1)_{[1]} X_{[0,1]} ) (H_{[0,1]} H_{[0,1]} ) \qquad (\text{HZ-b6}^*) $$

$$ ((-1)_{[jk]} X_{[j,k]} ) ((-1)_{[1]} (-1)_{[k]} ) \approx ((-1)_{[1]} (-1)_{[j]} ) ((-1)_{[1]} X_{[j,k]} ) \qquad (j = 1 \text{ or } k = 1) \qquad (\epsilon \text{-c1-1}) $$

$$ ((-1)_{[1]} X_{[j,k]} ) ((-1)_{[1]} (-1)_{[k]} ) \approx ((-1)_{[1]} (-1)_{[j]} ) ((-1)_{[1]} X_{[j,k]} ) \qquad j, k \neq 1 \qquad (\epsilon \text{-c1-2}) $$

$$ ((-1)_{[1]} X_{[j,k]} ) ((-1)_{[1]} (-1)_{[k]} ) \approx ((-1)_{[1]} (-1)_{[j]} ) ((-1)_{[jk]} X_{[j,k]} ) \qquad (j = 1 \text{ or } k = 1) \qquad (\text{Z-c1-1}) $$

$$ ((-1)_{[1]} X_{[j,k]} ) ((-1)_{[1]} (-1)_{[k]} ) \approx ((-1)_{[1]} (-1)_{[j]} ) ((-1)_{[1]} X_{[j,k]} ) \qquad j, k \neq 1 \qquad (\text{Z-c1-2}) $$

$$ ((-1)_{[1]} X_{[0,1]} ) ((-1)_{[1]} (-1)_{[k]} ) \approx ((-1)_{[1]} (-1)_{[j]} ) ((-1)_{[0]} X_{[0,1]} ) \qquad \{j, k\} = \{0, 1\} \qquad (\text{H-c1-1}) $$

$$ (X_{[0,1]} X_{[0,1]} ) (H_{[k,1]} H_{[0,1]} ) ((-1)_{[k]} X_{[0,1]} ) \approx ((-1)_{[1]} (-1)_{[0]} ) (X_{[0,1]} X_{[0,k]} ) (H_{[k,1]} H_{[0,1]} ) \qquad j = 0, k \notin \{0, 1\} \qquad (\text{H-c1-2}) $$

$$ ((-1)_{[k]} X_{[1,k]} ) (H_{[0,k]} H_{[0,1]} ) ((-1)_{[k]} X_{[0,1]} ) \approx ((-1)_{[1]} (-1)_{[1]} ) (X_{[0,1]} X_{[1,k]} ) (H_{[0,k]} H_{[0,1]} ) \qquad j = 1, k \notin \{0, 1\} \qquad (\text{H-c1-3}) $$

$$ (X_{[0,1]} X_{[j,0]} ) (H_{[j,1]} H_{[0,1]} ) ((-1)_{[1]} (-1)_{[0]} ) \approx ((-1)_{[j]} X_{[0,1]} ) (X_{[0,1]} X_{[j,0]} ) (H_{[j,1]} H_{[0,1]} ) \qquad k = 0, j \notin \{0, 1\} \qquad (\text{H-c1-4}) $$

$$ ((-1)_{[j]} X_{[j,1]} ) (H_{[0,j]} H_{[0,1]} ) ((-1)_{[1]} (-1)_{[1]} ) \approx ((-1)_{[j]} X_{[0,1]} ) (X_{[0,1]} X_{[j,1]} ) (H_{[0,j]} H_{[0,1]} ) \qquad k = 1, j \notin \{0, 1\} \qquad (\text{H-c1-5}) $$

$$ (X_{[j,k]} X_{[0,1]} ) ((-1)_{[k]} X_{[0,1]} ) \approx ((-1)_{[j]} X_{[0,1]} ) (X_{[j,k]} X_{[0,1]} ) \qquad \{j, k\} \cap \{0, 1\} = \emptyset \qquad (\text{H-c1-6}) $$

$$ ((-1)_{[0]} X_{[0,1]} ) ((-1)_{[1]} (-1)_{[k]} ) \approx ((-1)_{[1]} (-1)_{[j]} ) ((-1)_{[1]} X_{[0,1]} ) \qquad \{j, k\} = \{0, 1\} \qquad (\text{HZ-c1-1}) $$

$$ (X_{[0,1]} X_{[0,1]} ) (H_{[k,1]} H_{[0,1]} ) ((-1)_{[k]} X_{[0,1]} ) \approx ((-1)_{[1]} (-1)_{[0]} ) (X_{[0,1]} X_{[0,k]} ) (H_{[k,1]} H_{[0,1]} ) \qquad j = 0, k \notin \{0, 1\} \qquad (\text{HZ-c1-2}) $$

$$ (X_{[0,1]} X_{[1,k]} ) (H_{[0,k]} H_{[0,1]} ) ((-1)_{[k]} X_{[0,1]} ) \approx ((-1)_{[1]} (-1)_{[1]} ) ((-1)_{[k]} X_{[1,k]} ) (H_{[0,k]} H_{[0,1]} ) \qquad j = 1, k \notin \{0, 1\} \qquad (\text{HZ-c1-3}) $$

$$ (X_{[0,1]} X_{[j,0]} ) (H_{[j,1]} H_{[0,1]} ) ((-1)_{[1]} (-1)_{[0]} ) \approx ((-1)_{[j]} X_{[0,1]} ) (X_{[0,1]} X_{[j,0]} ) (H_{[j,1]} H_{[0,1]} ) \qquad k = 0, j \notin \{0, 1\} \qquad (\text{HZ-c1-4}) $$

$$ (X_{[0,1]} X_{[j,1]} ) (H_{[0,j]} H_{[0,1]} ) ((-1)_{[1]} (-1)_{[1]} ) \approx ((-1)_{[j]} X_{[0,1]} ) ((-1)_{[j]} X_{[j,1]} ) (H_{[0,j]} H_{[0,1]} ) \qquad k = 1, j \notin \{0, 1\} \qquad (\text{HZ-c1-5}) $$

$$ (X_{[j,k]} X_{[0,1]} ) ((-1)_{[k]} X_{[0,1]} ) \approx ((-1)_{[j]} X_{[0,1]} ) (X_{[j,k]} X_{[0,1]} ) \qquad \{j, k\} \cap \{0, 1\} = \emptyset \qquad (\text{HZ-c1-6}) $$

$$ ((-1)_{[1]} X_{[j,k]} ) ((-1)_{[j]} X_{[1,j]} ) (H_{[1,j]} H_{[0,1]} ) \approx (H_{[1,k]} H_{[0,1]} ) (X_{[0,1]} X_{[j,k]} ) (H_{[j+k,1]} H_{[0,1]} ) \qquad l = 1, (j = 0 \text{ or } k = 0) \qquad (\epsilon \text{-c5-1}) $$

$$ ((-1)_{[1]} X_{[j,k]} ) ((-1)_{[j]} X_{[1,j]} ) (H_{[1,j]} H_{[0,1]} ) \approx (H_{[1,k]} H_{[0,1]} ) (X_{[j,k]} X_{[0,1]} ) \qquad l = 1, j, k \notin \{0, 1\} \qquad (\epsilon \text{-c5-2}) $$

$$ ((-1)_{[0]} X_{[1,0]} ) ((-1)_{[1]} X_{[1,1]} ) (H_{[1,1]} H_{[0,1]} ) \approx (H_{[l,0]} H_{[0,1]} ) ((-1)_{[1]} X_{[0,1]} ) \qquad j = 1, k = 0 \qquad (\epsilon \text{-c5-3}) $$

$$ ((-1)_{[k]} X_{[1,k]} ) ((-1)_{[1]} X_{[l,1]} ) (H_{[l,1]} H_{[0,1]} ) \approx (H_{[l,k]} H_{[0,1]} ) ((-1)_{[k]} X_{[1,k]} ) (H_{[0,k]} H_{[0,1]} ) \qquad j = 1, k \notin \{0, 1\} \qquad (\epsilon \text{-c5-4}) $$
$$
\begin{align*}
& ((-1)_{[0]} X_{[0,1]})(H_{[l,0]} H_{[0,1]}) \approx (H_{[l,1]} H_{[0,1]}) ((-1)_{[1]} X_{[0,1]}) && k=1, j=0 \quad (\epsilon\text{-c5-5}) \\
& ((-1)_{[j]} X_{[j,1]})(H_{[l,j]} H_{[0,1]}) \approx (H_{[l,1]} H_{[0,1]}) ((-1)_{[j]} X_{[j,1]})(H_{[0,j]} H_{[0,1]}) && k=1, j \notin \{0,1\} \quad (\epsilon\text{-c5-6}) \\
& ((-1)_{[1]} X_{[j,k]})(H_{[l,j]} H_{[0,1]}) \approx (H_{[l,k]} H_{[0,1]}) (X_{[0,1]} X_{[j,k]})(H_{[j+k,1]} H_{[0,1]}) && l \neq 1, \{j,k\} \cap \{0,1\} = \{0\} \quad (\epsilon\text{-c5-7}) \\
& ((-1)_{[1]} X_{[j,k]})(H_{[l,j]} H_{[0,1]}) \approx (H_{[l,k]} H_{[0,1]}) (X_{[j,k]} X_{[0,1]}) && l \neq 1, \{j,k\} \cap \{0,1\} = \emptyset \quad (\epsilon\text{-c5-8}) \\
& ((-1)_{[1]} X_{[j,k]})(H_{[l,j]} H_{[0,1]}) \approx ((-1)_{[k]} X_{[1,k]})(H_{[1,k]} H_{[0,1]}) (X_{[0,1]} X_{[j,k]})(H_{[j+k,1]} H_{[0,1]}) && l=1, \quad (j=0 \text{ or } k=0) \quad (\text{Z-c5-1}) \\
& ((-1)_{[1]} X_{[j,k]})(H_{[l,j]} H_{[0,1]}) \approx ((-1)_{[k]} X_{[1,k]})(H_{[1,k]} H_{[0,1]}) (X_{[j,k]} X_{[0,1]}) && l=1, \quad j,k \notin \{0,1\} \quad (\text{Z-c5-2}) \\
& ((-1)_{[1]} X_{[0,1]})(H_{[l,0]} H_{[0,1]}) \approx ((-1)_{[1]} X_{[l,1]})(H_{[l,1]} H_{[0,1]}) ((-1)_{[0]} X_{[0,1]}) && k=1, j=0 \quad (\text{Z-c5-3}) \\
& ((-1)_{[1]} X_{[j,1]})(H_{[l,j]} H_{[0,1]}) \approx ((-1)_{[1]} X_{[l,1]})(H_{[l,1]} H_{[0,1]}) (X_{[0,1]} X_{[j,1]})(H_{[0,j]} H_{[0,1]}) && k=1, j \notin \{0,1\} \quad (\text{Z-c5-4}) \\
& ((-1)_{[1]} X_{[1,0]})(H_{[l,1]} H_{[0,1]}) \approx (H_{[l,0]} H_{[0,1]}) ((-1)_{[0]} X_{[0,1]}) && j=1, k=0 \quad (\text{Z-c5-5}) \\
& ((-1)_{[1]} X_{[1,k]})(H_{[l,1]} H_{[0,1]}) \approx (H_{[l,k]} H_{[0,1]}) (X_{[0,1]} X_{[1,k]})(H_{[0,k]} H_{[0,1]}) && j=1, k \notin \{0,1\} \quad (\text{Z-c5-6}) \\
& ((-1)_{[1]} X_{[j,k]})(H_{[l,j]} H_{[0,1]}) \approx (H_{[l,k]} H_{[0,1]}) (X_{[0,1]} X_{[j,k]})(H_{[j+k,1]} H_{[0,1]}) && l \neq 1, \{j,k\} \cap \{0,1\} = \{0\} \quad (\text{Z-c5-7}) \\
& ((-1)_{[1]} X_{[j,k]})(H_{[l,j]} H_{[0,1]}) \approx (H_{[l,k]} H_{[0,1]}) (X_{[j,k]} X_{[0,1]}) && l \neq 1, \{j,k\} \cap \{0,1\} = \emptyset \quad (\text{Z-c5-8}) \\
& (X_{[0,1]} X_{[j,k]})(H_{[j+k,1]} H_{[0,1]}) (H_{[0,1]} H_{[1,j]}) ((-1)_{[1]} X_{[1,j]}) \approx (H_{[0,1]} H_{[1,k]}) ((-1)_{[1]} X_{[j,k]}) && l=1, \quad (j=0 \text{ or } k=0) \quad (\text{H-c5-1}) \\
& (X_{[j,k]} X_{[0,1]})(H_{[0,1]} H_{[1,j]}) ((-1)_{[1]} X_{[1,j]}) \approx (H_{[0,1]} H_{[1,k]}) ((-1)_{[1]} X_{[j,k]}) && l=1, \quad j,k \notin \{0,1\} \quad (\text{H-c5-2}) \\
& ((-1)_{[1]} X_{[0,1]})(H_{[0,1]} H_{[l,1]}) ((-1)_{[1]} X_{[l,1]}) \approx (H_{[0,1]} H_{[l,0]}) ((-1)_{[0]} X_{[1,0]}) && j=1, k=0 \quad (\text{H-c5-3}) \\
& ((-1)_{[k]} X_{[1,k]})(H_{[0,k]} H_{[0,1]}) (H_{[0,1]} H_{[l,1]}) ((-1)_{[1]} X_{[l,1]}) \approx (H_{[0,1]} H_{[l,k]}) ((-1)_{[k]} X_{[1,k]}) && j=1, k \notin \{0,1\} \quad (\text{H-c5-4}) \\
& ((-1)_{[1]} X_{[0,1]})(H_{[0,1]} H_{[l,0]}) \approx (H_{[0,1]} H_{[l,1]}) ((-1)_{[0]} X_{[0,1]}) && k=1, j=0 \quad (\text{H-c5-5}) \\
& ((-1)_{[j]} X_{[j,1]})(H_{[0,j]} H_{[0,1]}) (H_{[0,1]} H_{[j,l]}) \approx (H_{[0,1]} H_{[l,1]}) ((-1)_{[j]} X_{[j,1]}) && k=1, j \notin \{0,1\} \quad (\text{H-c5-6}) \\
& (X_{[0,1]} X_{[j,k]})(H_{[j+k,1]} H_{[0,1]}) (H_{[0,1]} H_{[l,j]}) \approx (H_{[0,1]} H_{[l,k]}) ((-1)_{[1]} X_{[j,k]}) && l \neq 1, \{j,k\} \cap \{0,1\} = \{0\} \quad (\text{H-c5-7}) \\
& (X_{[j,k]} X_{[0,1]})(H_{[0,1]} H_{[l,j]}) \approx (H_{[0,1]} H_{[l,k]}) ((-1)_{[1]} X_{[j,k]}) && l \neq 1, \{j,k\} \cap \{0,1\} = \emptyset \quad (\text{H-c5-8}) \\
& (X_{[0,1]} X_{[j,k]})(H_{[j+k,1]} H_{[0,1]}) (H_{[0,1]} H_{[1,j]}) \approx (H_{[0,1]} H_{[1,k]}) ((-1)_{[1]} X_{[1,k]}) ((-1)_{[1]} X_{[j,k]}) && l=1, \quad (j=0 \text{ or } k=0) \quad (\text{HZ-c5-1}) \\
& (X_{[j,k]} X_{[0,1]})(H_{[0,1]} H_{[1,j]}) \approx (H_{[0,1]} H_{[1,k]}) ((-1)_{[1]} X_{[1,k]}) ((-1)_{[1]} X_{[j,k]}) && l=1, \quad j,k \notin \{0,1\} \quad (\text{HZ-c5-2}) \\
& ((-1)_{[0]} X_{[0,1]})(H_{[0,1]} H_{[l,0]}) \approx (H_{[0,1]} H_{[l,1]}) ((-1)_{[l]} X_{[1,1]}) ((-1)_{[1]} X_{[0,1]}) && k=1, j=0 \quad (\text{HZ-c5-3}) \\
& (X_{[0,1]} X_{[j,1]})(H_{[0,j]} H_{[0,1]}) (H_{[0,1]} H_{[l,j]}) \approx (H_{[0,1]} H_{[l,1]}) ((-1)_{[l]} X_{[1,1]}) ((-1)_{[1]} X_{[j,1]}) && k=1, j \notin \{0,1\} \quad (\text{HZ-c5-4}) \\
& ((-1)_{[0]} X_{[0,1]})(H_{[0,1]} H_{[l,1]}) \approx (H_{[0,1]} H_{[l,0]}) ((-1)_{[1]} X_{[1,0]}) && j=1, k=0 \quad (\text{HZ-c5-5}) \\
& (X_{[0,1]} X_{[1,k]})(H_{[0,k]} H_{[0,1]}) (H_{[0,1]} H_{[l,1]}) \approx (H_{[0,1]} H_{[l,k]}) ((-1)_{[1]} X_{[1,k]}) && j=1, k \notin \{0,1\} \quad (\text{HZ-c5-6}) \\
& (X_{[0,1]} X_{[j,k]})(H_{[j+k,1]} H_{[0,1]}) (H_{[0,1]} H_{[l,j]}) \approx (H_{[0,1]} H_{[l,k]}) ((-1)_{[1]} X_{[j,k]}) && l \neq 1, \{j,k\} \cap \{0,1\} = \{0\} \quad (\text{HZ-c5-7}) \\
& (X_{[j,k]} X_{[0,1]})(H_{[0,1]} H_{[l,j]}) \approx (H_{[0,1]} H_{[l,k]}) ((-1)_{[1]} X_{[j,k]}) && l \neq 1, \{j,k\} \cap \{0,1\} = \emptyset \quad (\text{HZ-c5-8})
\end{align*}
$$

$$
\begin{align*}
& ((-1)_{[0]} X_{[j,k]})((-1)_{[k]} X_{[j,k]})(H_{[j,k]} H_{[0,1]}) \approx (H_{[k,j]} H_{[0,1]})((-1)_{[1]} X_{[0,1]}) && \{j,k\} = \{0,1\} \quad (\epsilon\text{-e2}^*-1) \\
& ((-1)_{[j,k]} X_{[j,k]})((-1)_{[k]} X_{[j,k]})(H_{[j,k]} H_{[0,1]}) \approx (H_{[k,j,k]} H_{[0,1]})((-1)_{[jk]} X_{[j,k]}) (H_{[0,j,k]} H_{[0,1]}) && \{j,k\} \cap \{0,1\} = \{1\} \quad (\epsilon\text{-e2}^*-2) \\
& ((-1)_{[1]} X_{[j,k]})(H_{[j,k]} H_{[0,1]}) \approx (H_{[k,j,k]} H_{[0,1]}) (X_{[0,1]} X_{[j,k]})(H_{[j+k,1]} H_{[0,1]}) && \{j,k\} \cap \{0,1\} = \{0\} \quad (\epsilon\text{-e2}^*-3) \\
& ((-1)_{[1]} X_{[j,k]})(H_{[j,k]} H_{[0,1]}) \approx (H_{[k,j,k]} H_{[0,1]}) (X_{[j,k]} X_{[0,1]}) && \{j,k\} \cap \{0,1\} = \emptyset \quad (\epsilon\text{-e2}^*-4) \\
& ((-1)_{[1]} X_{[j,k]})(H_{[j,k]} H_{[0,1]}) \approx ((-1)_{[j]} X_{[k,j,k]})(H_{[k,j,k]} H_{[0,1]}) ((-1)_{[0]} X_{[0,1]}) && \{j,k\} = \{0,1\} \quad (\text{Z-e2}^*-1) \\
& ((-1)_{[1]} X_{[j,k]})(H_{[j,k]} H_{[0,1]}) \approx ((-1)_{[l,j]} X_{[k,j,k]})(H_{[k,j,k]} H_{[0,1]}) (X_{[0,1]} X_{[j,k]})(H_{[0,j,k]} H_{[0,1]}) && \{j,k\} \cap \{0,1\} = \{1\} \quad (\text{Z-e2}^*-2) \\
& ((-1)_{[1]} X_{[j,k]})(H_{[j,k]} H_{[0,1]}) \approx (H_{[k,j,k]} H_{[0,1]}) (X_{[0,1]} X_{[j,k]})(H_{[j+k,1]} H_{[0,1]}) && \{j,k\} \cap \{0,1\} = \{0\} \quad (\text{Z-e2}^*-3) \\
& ((-1)_{[1]} X_{[j,k]})(H_{[j,k]} H_{[0,1]}) \approx (H_{[k,j,k]} H_{[0,1]}) (X_{[j,k]} X_{[0,1]}) && \{j,k\} \cap \{0,1\} = \emptyset \quad (\text{Z-e2}^*-4) \\
& ((-1)_{[1]} X_{[0,1]})(H_{[0,1]} H_{[k,j,k]}) ((-1)_{[j]} X_{[j,k]}) \approx (H_{[0,1]} H_{[k,j,k]})((-1)_{[0]} X_{[j,k]}) && \{j,k\} = \{0,1\} \quad (\text{H-e2}^*-1) \\
& ((-1)_{[l,j,k]} X_{[j,k]}) (H_{[0,j,k]} H_{[0,1]}) (H_{[0,1]} H_{[j,k]}) ((-1)_{[l]} X_{[l,j,k]}) \approx (H_{[0,1]} H_{[k,j,k]})((-1)_{[l]} X_{[j,k]}) && \{j,k\} \cap \{0,1\} = \{1\} \quad (\text{H-e2}^*-2) \\
& (X_{[0,1]} X_{[j,k]})(H_{[j+k,1]} H_{[0,1]}) (H_{[0,1]} H_{[j,k]}) \approx (H_{[0,1]} H_{[k,j,k]})((-1)_{[1]} X_{[j,k]}) && \{j,k\} \cap \{0,1\} = \{0\} \quad (\text{H-e2}^*-3) \\
& (X_{[j,k]} X_{[0,1]})(H_{[0,1]} H_{[j,k]}) \approx (H_{[0,1]} H_{[k,j,k]})((-1)_{[1]} X_{[j,k]}) && \{j,k\} \cap \{0,1\} = \emptyset \quad (\text{H-e2}^*-4) \\
& ((-1)_{[0]} X_{[0,1]})(H_{[0,1]} H_{[j,k]}) \approx (H_{[0,1]} H_{[k,j,k]})((-1)_{[k]} X_{[k,j,k]})(-1)_{[1]} X_{[j,k]} && \{j,k\} = \{0,1\} \quad (\text{HZ-e2}^*-1)
\end{align*}
$$
$$
\begin{align*}
(X_{[0,1]}X_{[j,k]})(H_{[0,jk]}H_{[0,1]})(H_{[0,1]}H_{[j,k]}) &\approx (H_{[0,1]}H_{[k,j]})((-1)_{[k]}X_{[k,j]})(-(-1)_{[1]}X_{[j,k]}) & \{j,k\} \cap \{0,1\} &= \{1\} & \text{(HZ-e2*-2)} \\
(X_{[0,1]}X_{[j,k]})(H_{[j+k,1]}H_{[0,1]})(H_{[0,1]}H_{[j,k]}) &\approx (H_{[0,1]}H_{[k,j]})((-1)_{[1]}X_{[j,k]}) & \{j,k\} \cap \{0,1\} &= \{0\} & \text{(HZ-e2*-3)} \\
(X_{[j,k]}X_{[0,1]})(H_{[0,1]}H_{[j,k]}) &\approx (H_{[0,1]}H_{[k,j]})((-1)_{[1]}X_{[j,k]}) & \{j,k\} \cap \{0,1\} &= \emptyset & \text{(HZ-e2*-4)}
\end{align*}
$$

$$
\begin{align*}
(H_{[j,k]}H_{[0,1]})((-1)_{[1]}X_{[0,1]}) &\approx ((-1)_{[1]}(-1)_{[k]})((-1)_{[k]}X_{[j,k]}) (H_{[j,k]}H_{[0,1]}) & \{j,k\} &= \{0,1\} & (\epsilon\text{-d2-1}) \\
(H_{[j,k]}H_{[0,1]})((-1)_{[j,k]}X_{[j,k]}) (H_{[0,jk]}H_{[0,1]}) &\approx ((-1)_{[1]}(-1)_{[k]})((-1)_{[k]}X_{[j,k]}) (H_{[j,k]}H_{[0,1]}) & \{j,k\} \cap \{0,1\} &= \{1\} & (\epsilon\text{-d2-2}) \\
(H_{[j,k]}H_{[0,1]}) (X_{[0,1]}X_{[j,k]}) (H_{[j+k,1]}H_{[0,1]}) &\approx ((-1)_{[1]}(-1)_{[k]}) (H_{[j,k]}H_{[0,1]}) & \{j,k\} \cap \{0,1\} &= \{0\} & (\epsilon\text{-d2-3}) \\
(H_{[j,k]}H_{[0,1]}) (X_{[j,k]}X_{[0,1]}) &\approx ((-1)_{[1]}(-1)_{[k]}) (H_{[j,k]}H_{[0,1]}) & \{j,k\} \cap \{0,1\} &= \emptyset & (\epsilon\text{-d2-4}) \\
((-1)_{[k]}X_{[j,k]}) (H_{[j,k]}H_{[0,1]})((-1)_{[0]}X_{[0,1]}) &\approx ((-1)_{[1]}(-1)_{[k]}) (H_{[j,k]}H_{[0,1]}) & \{j,k\} &= \{0,1\} & (\text{Z}\text{-d2-1}) \\
((-1)_{[k]}X_{[j,k]}) (H_{[j,k]}H_{[0,1]}) (X_{[0,1]}X_{[j,k]}) (H_{[0,jk]}H_{[0,1]}) &\approx ((-1)_{[1]}(-1)_{[k]}) (H_{[j,k]}H_{[0,1]}) & \{j,k\} \cap \{0,1\} &= \{1\} & (\text{Z}\text{-d2-2}) \\
(H_{[j,k]}H_{[0,1]}) (X_{[0,1]}X_{[j,k]}) (H_{[j+k,1]}H_{[0,1]}) &\approx ((-1)_{[1]}(-1)_{[k]}) (H_{[j,k]}H_{[0,1]}) & \{j,k\} \cap \{0,1\} &= \{0\} & (\text{Z}\text{-d2-3}) \\
(H_{[j,k]}H_{[0,1]}) (X_{[j,k]}X_{[0,1]}) &\approx ((-1)_{[1]}(-1)_{[k]}) (H_{[j,k]}H_{[0,1]}) & \{j,k\} \cap \{0,1\} &= \emptyset & (\text{Z}\text{-d2-4}) \\
(H_{[0,1]}H_{[j,1]})((-1)_{[j]}X_{[j,1]}) &\approx ((-1)_{[1]}(-1)_{[1]}) (H_{[0,1]}H_{[j,1]})((-1)_{[j]}X_{[j,1]}) & k &= 1 & (\text{H}\text{-d2-1}) \\
(H_{[0,1]}H_{[1,0]})((-1)_{[0]}X_{[1,0]}) &\approx ((-1)_{[1]}(-1)_{[0]}) (H_{[0,1]}H_{[1,0]})((-1)_{[1]}X_{[1,0]}) & j &= 1, k = 0 & (\text{H}\text{-d2-2}) \\
(H_{[0,1]}H_{[1,k]})((-1)_{[k]}X_{[1,k]}) &\approx ((-1)_{[k]}X_{[0,1]}) (H_{[0,1]}H_{[1,k]})((-1)_{[1]}X_{[1,k]}) & j &= 1, k \notin \{0,1\} & (\text{H}\text{-d2-3}) \\
(H_{[0,1]}H_{[j,0]})((-1)_{[1]}X_{[j,0]}) &\approx ((-1)_{[1]}(-1)_{[0]}) (H_{[0,1]}H_{[j,0]}) & j & \neq 1, k = 0 & (\text{H}\text{-d2-4}) \\
(H_{[0,1]}H_{[j,k]})((-1)_{[1]}X_{[j,k]}) &\approx ((-1)_{[k]}X_{[0,1]}) (H_{[0,1]}H_{[j,k]}) & j & \neq 1, k \notin \{0,1\} & (\text{H}\text{-d2-5}) \\
(H_{[0,1]}H_{[j,1]})((-1)_{[j]}X_{[j,1]})((-1)_{[1]}X_{[j,1]}) &\approx ((-1)_{[1]}(-1)_{[1]}) (H_{[0,1]}H_{[j,1]}) & k &= 1 & (\text{HZ}\text{-d2-1}) \\
(H_{[0,1]}H_{[1,0]})((-1)_{[1]}X_{[1,0]})((-1)_{[1]}X_{[1,0]}) &\approx ((-1)_{[1]}(-1)_{[0]}) (H_{[0,1]}H_{[1,0]}) & j &= 1, k = 0 & (\text{HZ}\text{-d2-2}) \\
(H_{[0,1]}H_{[1,k]})((-1)_{[1]}X_{[1,k]})((-1)_{[1]}X_{[1,k]}) &\approx ((-1)_{[k]}X_{[0,1]}) (H_{[0,1]}H_{[1,k]}) & j &= 1, k \notin \{0,1\} & (\text{HZ}\text{-d2-3}) \\
(H_{[0,1]}H_{[j,0]})((-1)_{[1]}X_{[j,0]}) &\approx ((-1)_{[1]}(-1)_{[0]}) (H_{[0,1]}H_{[j,0]}) & j & \neq 1, k = 0 & (\text{HZ}\text{-d2-4}) \\
(H_{[0,1]}H_{[j,k]})((-1)_{[1]}X_{[j,k]}) &\approx ((-1)_{[k]}X_{[0,1]}) (H_{[0,1]}H_{[j,k]}) & j & \neq 1, k \notin \{0,1\} & (\text{HZ}\text{-d2-5})
\end{align*}
$$

$$
\begin{align*}
& (H_{[0,1]}H_{[0,1]}) (H_{[0,1]}H_{[0,2]}) (H_{[1,3]}H_{[0,1]}) (H_{[0,1]}H_{[0,1]}) \\
& ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) \approx & (\epsilon\text{-d3}^*) \\
& (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) (H_{[0,1]}H_{[0,1]}) ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) \\
& (H_{[0,1]}H_{[0,2]}) (H_{[1,3]}H_{[0,1]}) (H_{[0,1]}H_{[0,1]})
\end{align*}
$$

$$
\begin{align*}
& ((-1)_{[1]}X_{[0,1]}) (H_{[0,1]}H_{[0,1]}) (H_{[0,1]}H_{[0,2]}) ((-1)_{[3]}X_{[1,3]}) (H_{[1,3]}H_{[0,1]}) (H_{[0,1]}H_{[0,1]}) ((-1)_{[0]}X_{[0,1]}) \\
& ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) ((-1)_{[1]}X_{[1,3]}) \approx & \text{(Z-d3*)} \\
& (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) ((-1)_{[1]}X_{[1,3]}) ((-1)_{[0]}X_{[0,1]}) \\
& (H_{[0,1]}H_{[0,2]}) ((-1)_{[3]}X_{[1,3]}) (H_{[1,3]}H_{[0,1]}) (H_{[0,1]}H_{[0,1]}) ((-1)_{[0]}X_{[0,1]})
\end{align*}
$$

$$
\begin{align*}
& (H_{[0,1]}H_{[0,1]}) (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) (H_{[0,1]}H_{[0,1]}) \\
& ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) (H_{[0,1]}H_{[0,2]}) (H_{[1,3]}H_{[0,1]}) \approx & \text{(H-d3*)} \\
& (H_{[0,1]}H_{[0,2]}) (H_{[1,3]}H_{[0,1]}) (H_{[0,1]}H_{[0,1]}) ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) \\
& (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) (H_{[0,1]}H_{[0,1]})
\end{align*}
$$

$$
\begin{align*}
& (H_{[0,1]}H_{[0,1]})((-1)_{[0]}X_{[0,1]}) (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) ((-1)_{[1]}X_{[1,3]}) ((-1)_{[1]}X_{[0,1]}) (H_{[0,1]}H_{[0,1]}) \\
& ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) (H_{[0,1]}H_{[0,2]}) ((-1)_{[3]}X_{[1,3]}) (H_{[1,3]}H_{[0,1]}) \approx & \text{(HZ-d3*)} \\
& (H_{[0,1]}H_{[0,2]})((-1)_{[3]}X_{[1,3]}) (H_{[1,3]}H_{[0,1]}) (H_{[0,1]}H_{[0,1]}) ((-1)_{[0]}X_{[0,1]}) ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) \\
& (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) ((-1)_{[1]}X_{[1,3]}) ((-1)_{[1]}X_{[0,1]}) (H_{[0,1]}H_{[0,1]})
\end{align*}
$$
$$ (H_{[0,1]}H_{[0,1]}) (H_{[0,1]}H_{[0,2]}) (H_{[1,3]}H_{[0,1]}) (H_{[0,1]}H_{[0,4]}) (H_{[1,5]}H_{[0,1]}) (H_{[0,1]}H_{[0,1]}) \\ ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) (H_{[0,4]}H_{[0,1]}) (H_{[0,1]}H_{[1,5]}) (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) \approx (\epsilon \text{-d4}^*) \\ (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) (H_{[0,4]}H_{[0,1]}) (H_{[0,1]}H_{[1,5]}) (H_{[0,1]}H_{[0,1]}) ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) \\ (H_{[0,1]}H_{[0,4]}) (H_{[1,5]}H_{[0,1]}) (H_{[0,1]}H_{[0,2]}) (H_{[1,3]}H_{[0,1]}) (H_{[0,1]}H_{[0,1]}) $$

$$ ((-1)_{[1]}X_{[0,1]}) (H_{[0,1]}H_{[0,1]}) (H_{[0,1]}H_{[0,2]}) ((-1)_{[3]}X_{[1,3]}) (H_{[1,3]}H_{[0,1]}) (H_{[0,1]}H_{[0,4]}) ((-1)_{[5]}X_{[1,5]}) (H_{[1,5]}H_{[0,1]}) (H_{[0,1]}H_{[0,1]}) ((-1)_{[0]}X_{[0,1]}) \\ ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) (H_{[0,4]}H_{[0,1]}) (H_{[0,1]}H_{[1,5]}) ((-1)_{[1]}X_{[1,5]}) (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) ((-1)_{[1]}X_{[1,3]}) \approx (\text{Z-d4}^*) \\ (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) ((-1)_{[1]}X_{[1,3]}) (H_{[0,4]}H_{[0,1]}) (H_{[0,1]}H_{[1,5]}) ((-1)_{[1]}X_{[1,5]}) ((-1)_{[1]}X_{[0,1]}) (H_{[0,1]}H_{[0,1]}) ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) \\ (H_{[0,1]}H_{[0,4]}) ((-1)_{[5]}X_{[1,5]}) (H_{[1,5]}H_{[0,1]}) (H_{[0,1]}H_{[0,2]}) ((-1)_{[3]}X_{[1,3]}) (H_{[1,3]}H_{[0,1]}) (H_{[0,1]}H_{[0,1]}) ((-1)_{[0]}X_{[0,1]}) $$

$$ (H_{[0,1]}H_{[0,1]}) (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) (H_{[0,4]}H_{[0,1]}) (H_{[0,1]}H_{[1,5]}) (H_{[0,1]}H_{[0,1]}) \\ ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) (H_{[0,1]}H_{[0,4]}) (H_{[1,5]}H_{[0,1]}) (H_{[0,1]}H_{[0,2]}) (H_{[1,3]}H_{[0,1]}) \approx (\text{H-d4}^*) \\ (H_{[0,1]}H_{[0,2]}) (H_{[1,3]}H_{[0,1]}) (H_{[0,1]}H_{[0,4]}) (H_{[1,5]}H_{[0,1]}) (H_{[0,1]}H_{[0,1]}) ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) \\ (H_{[0,4]}H_{[0,1]}) (H_{[0,1]}H_{[1,5]}) (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) (H_{[0,1]}H_{[0,1]}) $$

$$ (H_{[0,1]}H_{[0,1]}) ((-1)_{[0]}X_{[0,1]}) (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) ((-1)_{[1]}X_{[1,3]}) (H_{[0,4]}H_{[0,1]}) (H_{[0,1]}H_{[1,5]}) ((-1)_{[1]}X_{[1,5]}) ((-1)_{[1]}X_{[0,1]}) (H_{[0,1]}H_{[0,1]}) \\ ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) (H_{[0,1]}H_{[0,4]}) ((-1)_{[5]}X_{[1,5]}) (H_{[1,5]}H_{[0,1]}) (H_{[0,1]}H_{[0,2]}) ((-1)_{[3]}X_{[1,3]}) (H_{[1,3]}H_{[0,1]}) \approx (\text{HZ-d4}^*) \\ (H_{[0,1]}H_{[0,2]}) ((-1)_{[3]}X_{[1,3]}) (H_{[1,3]}H_{[0,1]}) (H_{[0,1]}H_{[0,4]}) ((-1)_{[5]}X_{[1,5]}) (H_{[1,5]}H_{[0,1]}) (H_{[0,1]}H_{[0,1]}) ((-1)_{[0]}X_{[0,1]}) ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[1]}(-1)_{[1]}) \\ (H_{[0,4]}H_{[0,1]}) (H_{[0,1]}H_{[1,5]}) ((-1)_{[1]}X_{[1,5]}) (H_{[0,2]}H_{[0,1]}) (H_{[0,1]}H_{[1,3]}) ((-1)_{[1]}X_{[1,3]}) ((-1)_{[1]}X_{[0,1]}) (H_{[0,1]}H_{[0,1]}) $$

## A.4 Simplifying the Output of the Reidemeister-Schreier Theorem

A.4.1 Details about Equations (36) and (37). The right-hand sides of Equations (36) and (37) are defined respectively as follows:

$$ \begin{aligned} W_1 := & \left( \overset{m+\ell}{\underset{j=m+1}{\bigwedge}} E_{n-j-2,j}(\bigcirc) \right) \circ \left( \overset{m-1}{\underset{j=0}{\bigwedge}} E_{n-j-2,j}(\bigcirc) \circ E_{n-j-3,j+1}(\bigcirc) \right) \circ E_{n-2,0}(\bigcirc) \circ \\ & \left( \overset{n-3}{\underset{j=k+\ell}{\bigwedge}} \mathcal{E}_{j,n-j-2}^{1-z_j}(-\oplus) \right) \circ \left( \overset{k+\ell-1}{\underset{j=k}{\bigwedge}} \mathcal{E}_{j,n-j-2}^{1-y_j}(-\oplus) \right) \circ \left( \overset{k-1}{\underset{j=0}{\bigwedge}} \mathcal{E}_{j,n-j-2}^{1-x_j}(-\oplus) \right) \circ \\ & (H_{[0,1]}H_{[3,2]}) \circ \left( \overset{k-1}{\underset{j=0}{\bigwedge}} \mathcal{E}_{j,n-j-2}^{1-x_j}(-\oplus) \right) \circ \left( \overset{k+\ell-1}{\underset{j=k}{\bigwedge}} \mathcal{E}_{j,n-j-2}^{1-y_j}(-\oplus) \right) \circ \left( \overset{n-3}{\underset{j=k+\ell}{\bigwedge}} \mathcal{E}_{j,n-j-2}^{1-z_j}(-\oplus) \right) \circ \\ & E_{n-2,0}(\bigcirc) \circ \left( \overset{m-1}{\underset{j=0}{\bigwedge}} E_{n-j-3,j+1}(\bigcirc) \circ E_{n-j-2,j}(\bigcirc) \right) \circ \left( \overset{m+\ell}{\underset{j=m+1}{\bigwedge}} E_{n-j-2,j}(\bigcirc) \right) \end{aligned} $$

and

$$ W_2 := \left( \overset{m+\ell}{\underset{j=m+1}{\bigwedge}} E_{n-j-2,j}(\bigcirc) \right) \circ \left( \overset{m-1}{\underset{j=0}{\bigwedge}} E_{n-j-2,j}(\bigcirc) \circ E_{n-j-3,j+1}(\bigcirc) \right) \circ $$
$$
\begin{align*}
& \left( \left( \overleftarrow{\prod}_{j=k+\ell}^{n-3} \mathcal{E}_{j,n-j-2}^{1-z_j} (-\oplus) \right) \circ \left( \overleftarrow{\prod}_{j=k}^{k+\ell-1} \mathcal{E}_{j,n-j-2}^{1-y_j} (-\oplus) \right) \circ \left( \overleftarrow{\prod}_{j=0}^{k-1} \mathcal{E}_{j,n-j-2}^{1-x_j} (-\oplus) \right) \circ \right. \\
& \qquad \left. (H_{[0,1]} H_{[3,2]}) \circ \right. \\
& \qquad \left. \left( \overleftarrow{\prod}_{j=0}^{k-1} \mathcal{E}_{j,n-j-2}^{1-x_j} (-\oplus) \right) \circ \left( \overleftarrow{\prod}_{j=k}^{k+\ell-1} \mathcal{E}_{j,n-j-2}^{1-y_j} (-\oplus) \right) \circ \left( \overleftarrow{\prod}_{j=k+\ell}^{n-3} \mathcal{E}_{j,n-j-2}^{1-z_j} (-\oplus) \right) \circ \right. \\
& \qquad \left. \left( \overleftarrow{\prod}_{j=0}^{m-1} E_{n-j-3,j+1} (\bigcirc) \circ E_{n-j-2,j} (\bigcirc) \right) \circ \left( \overleftarrow{\prod}_{j=m+1}^{m+\ell} E_{n-j-2,j} (\bigcirc) \right) \right)
\end{align*}
$$

where the notations $\overleftarrow{\prod}$ and $\overrightarrow{\prod}$ are introduced in Definition 8.3, and $\mathcal{E}_{k,l}^{1-x}(-\oplus)$ is a shortcut defined as follows (where the order of the product is left unspecified):

$$
\mathcal{E}_{k,l}^{1-x}(-\oplus) := \begin{cases} \epsilon & \text{if } x=1 \\ E_{k,l}(-\bullet) \circ \prod_{\substack{x \in \{0,1\}^k \\ y \in \{0,1\}^l}} ((-1)_{[G_n^{-1}(x1y)]} X_{[G_n^{-1}(x0y),G_n^{-1}(x1y)]}) & \text{if } x=0. \end{cases}
$$

A.4.2 Proof of Completeness of ≈ for $\mathcal{P}_N^*$. The goal of this appendix is to prove the completeness part of Theorem 4.10 (stated as Corollary A.12 below). For this purpose, we prove increasingly strong partial completeness results, together with some auxiliary equations, so as to be finally able to derive the equations of Appendix A.3.2 with minimum effort.

**Lemma A.3.** *The equations of Figure 9 are consequences of the equations of Figure 8.*

PROOF. <fnref n="12" />

*Proof of Equation (47):*

$$
\begin{align*}
& ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a]} X_{[a,a+1]}) \stackrel{(20)(22)(21)}{\approx} ((-1)_{[a]} (-1)_{[a+1]})^2 ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a]} X_{[a,a+1]}) \\
& \stackrel{(23)}{\approx} ((-1)_{[a]} X_{[a,a+1]})^4 ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a]} X_{[a,a+1]}) \\
& \stackrel{(29)}{\approx} ((-1)_{[a]} X_{[a,a+1]})^4 \\
& \stackrel{(23)(21)(22)(20)}{\approx} \epsilon
\end{align*}
$$

*Proof of Equation (48):*

$$
\begin{align*}
& ((-1)_{[a+1]} X_{[a,a+1]}) \stackrel{(20)(22)(21)}{\approx} ((-1)_{[a]} (-1)_{[a+1]})^2 ((-1)_{[a+1]} X_{[a,a+1]}) \\
& \stackrel{(23)}{\approx} ((-1)_{[a]} X_{[a,a+1]})^4 ((-1)_{[a+1]} X_{[a,a+1]}) \\
& \stackrel{(29)}{\approx} ((-1)_{[a]} X_{[a,a+1]})^3
\end{align*}
$$

*Proof of Equation (49):* If $c-a$ is odd, then Equation (49) is just Equation (25), therefore we only have to consider the case where $c-a$ is even. We proceed by induction on $c-a$. If $c-a=2$, then we have

$$
\begin{align*}
& ((-1)_{[a]} X_{[a,a+2]}) \stackrel{(27)}{\approx} ((-1)_{[a+1]} X_{[a+1,a+2]}) ((-1)_{[a]} X_{[a,a+1]}) ((-1)_{[a+2]} X_{[a+1,a+2]}) \\
& \stackrel{(48)}{\approx} ((-1)_{[a+1]} X_{[a+1,a+2]}) ((-1)_{[a]} X_{[a,a+1]}) ((-1)_{[a+1]} X_{[a+1,a+2]})^3 \\
& \stackrel{(24)}{\approx} ((-1)_{[a]} X_{[a,a+1]})^3 ((-1)_{[a+1]} X_{[a+1,a+2]}) ((-1)_{[a]} X_{[a,a+1]})
\end{align*}
$$

<footnote n="12">¹²Note that throughout all the paper, whenever we write a derivation to prove an equation to which parameter conditions are attached, we implicitly assume the conditions of the equation that we are proving.</footnote>
$$ ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a]} X_{[a,a+1]} \approx \epsilon \qquad (47) $$

$$ ((-1)_{[a+1]} X_{[a,a+1]}) \approx ((-1)_{[a]} X_{[a,a+1]})^3 \qquad (48) $$

$$ ((-1)_{[a]} X_{[a,c]}) \approx ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+1]} X_{[a+1,c]}) ((-1)_{[a]} X_{[a,a+1]}) \quad a+1 < c \qquad (49) $$

$$ ((-1)_{[c]} X_{[a,c]}) \approx ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[c]} X_{[a+1,c]}) ((-1)_{[a]} X_{[a,a+1]}) \quad a+1 < c \qquad (50) $$

$$ ((-1)_{[a]} X_{[a,b]}) ((-1)_{[b]} X_{[a,b]}) \approx \epsilon \qquad (51) $$

$$ ((-1)_{[a]}(-1)_{[b]}) \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[a]} X_{[a,b]}) \quad a \neq b \qquad (52) $$

$$ ((-1)_{[c]} X_{[a,b]}) \approx ((-1)_{[c]}(-1)_{[a]}) ((-1)_{[a]} X_{[a,b]}) \qquad (53) $$

$$ ((-1)_{[a]} X_{[a,c]}) \approx ((-1)_{[a]} X_{[a,a+1]})^3 ((-1)_{[a+1]} X_{[a+1,c]}) ((-1)_{[a]} X_{[a,a+1]}) \quad a+1 < c \qquad (54) $$

$$ ((-1)_{[c]}(-1)_{[d]}) ((-1)_{[a]} X_{[a,a+1]}) \approx ((-1)_{[a]} X_{[a,a+1]}) ((-1)_{[c]}(-1)_{[d]}) \quad \{a, a+1\} \cap \{c, d\} = \emptyset \qquad (55) $$

$$ ((-1)_{[a]}(-1)_{[b]}) ((-1)_{[a]} X_{[a,b]}) \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[a]}(-1)_{[b]}) \qquad (56) $$

$$ ((-1)_{[b]}(-1)_{[a]}) ((-1)_{[a]} X_{[a,b]}) \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[b]}(-1)_{[a]}) \qquad (57) $$

$$ ((-1)_{[a]}(-1)_{[c]}) ((-1)_{[a]} X_{[a,b]}) \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[b]}(-1)_{[c]}) \quad c \neq a, b \qquad (58) $$

$$ ((-1)_{[c]}(-1)_{[a]}) ((-1)_{[a]} X_{[a,b]}) \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[c]}(-1)_{[b]}) \quad c \neq a, b \qquad (59) $$

$$ ((-1)_{[b]}(-1)_{[c]}) ((-1)_{[a]} X_{[a,b]}) \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[a]}(-1)_{[c]}) \quad c \neq a, b \qquad (60) $$

$$ ((-1)_{[c]}(-1)_{[b]}) ((-1)_{[a]} X_{[a,b]}) \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[c]}(-1)_{[a]}) \quad c \neq a, b \qquad (61) $$

$$ ((-1)_{[c]}(-1)_{[d]}) ((-1)_{[a]} X_{[a,b]}) \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[c]}(-1)_{[d]}) \quad \{a, b\} \cap \{c, d\} = \emptyset \qquad (62) $$

$$ ((-1)_{[a]}(-1)_{[b]}) ((-1)_{[c]}(-1)_{[d]}) \approx ((-1)_{[a]}(-1)_{[c]}) ((-1)_{[b]}(-1)_{[d]}) \qquad (63) $$

$$ ((-1)_{[a]} X_{[a,b]}) ((-1)_{[c]} X_{[c,d]}) \approx ((-1)_{[c]} X_{[c,d]}) ((-1)_{[a]} X_{[a,b]}) \quad \{a, b\} \cap \{c, d\} = \emptyset \qquad (64) $$

<caption>Figure 9: Useful auxiliary equations.</caption>

$$ \approx^{(48)} ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+1]} X_{[a+1,a+2]}) ((-1)_{[a]} X_{[a,a+1]}) $$

And if $c - a \ge 4$, then we have

$$ ((-1)_{[a]} X_{[a,c]}) $$

$$ \approx^{(27)} ((-1)_{[c-1]} X_{[c-1,c]}) ((-1)_{[a]} X_{[a,c-1]}) ((-1)_{[c]} X_{[c-1,c]}) $$

$$ \approx^{(25)} ((-1)_{[c-1]} X_{[c-1,c]}) ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+1]} X_{[a+1,c-1]}) ((-1)_{[a]} X_{[a,a+1]}) ((-1)_{[c]} X_{[c-1,c]}) $$
induction hypothesis
$$ \approx^{(48)(32)} ((-1)_{[c-1]} X_{[c-1,c]}) ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+2]} X_{[a+1,a+2]}) ((-1)_{[a+2]} X_{[a+2,c-1]}) ((-1)_{[a+1]} X_{[a+1,a+2]}) ((-1)_{[a]} X_{[a,a+1]}) ((-1)_{[c]} X_{[c-1,c]}) $$

$$ \approx^{(48)(32)} ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+2]} X_{[a+1,a+2]}) ((-1)_{[c-1]} X_{[c-1,c]}) ((-1)_{[a+2]} X_{[a+2,c-1]}) ((-1)_{[c]} X_{[c-1,c]}) ((-1)_{[a+1]} X_{[a+1,a+2]}) ((-1)_{[a]} X_{[a,a+1]}) $$

$$ \approx^{(27)} ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+2]} X_{[a+1,a+2]}) ((-1)_{[a+2]} X_{[a+2,c]}) ((-1)_{[a+1]} X_{[a+1,a+2]}) ((-1)_{[a]} X_{[a,a+1]}) $$

$$ \approx^{(25)} ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+1]} X_{[a+1,c]}) ((-1)_{[a]} X_{[a,a+1]}) $$

*Proof of Equation (50):* If $c-a$ is odd, then Equation (50) is just Equation (26), therefore we only have to consider the case where $c-a$ is even. We proceed by induction on $c-a$. If $c-a=2$, then we have

$$ ((-1)_{[a+2]} X_{[a,a+2]}) \approx^{(28)} ((-1)_{[a+1]} X_{[a+1,a+2]}) ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+2]} X_{[a+1,a+2]}) $$

$$ \approx^{(47)(29)} ((-1)_{[a+1]} X_{[a+1,a+2]}) ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+2]} X_{[a+1,a+2]}) \\ ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+1]} X_{[a+1,a+2]}) ((-1)_{[a]} X_{[a,a+1]}) \\ ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+2]} X_{[a+1,a+2]}) ((-1)_{[a]} X_{[a,a+1]}) $$

$$ \approx^{(49)} ((-1)_{[a+1]} X_{[a+1,a+2]}) ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+2]} X_{[a+1,a+2]}) \\ ((-1)_{[a]} X_{[a,a+2]}) $$
$$ ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a+2]} X_{[a+1,a+2]})(-1)_{[a]} X_{[a,a+1]} ) $$

$$ \begin{align*} \stackrel{(27)}{\approx} & ((-1)_{[a+1]} X_{[a+1,a+2]})(-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a+2]} X_{[a+1,a+2]} \\ & ((-1)_{[a+1]} X_{[a+1,a+2]})(-1)_{[a]} X_{[a,a+1]})(-1)_{[a+2]} X_{[a+1,a+2]} \\ & ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a+2]} X_{[a+1,a+2]})(-1)_{[a]} X_{[a,a+1]} ) \end{align*} $$

$$ \stackrel{(47)(29)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a+2]} X_{[a+1,a+2]})(-1)_{[a]} X_{[a,a+1]} ) $$

And if $c - a \ge 4$, then we have

$$ ((-1)_{[c]} X_{[a,c]} ) $$

$$ \stackrel{(28)}{\approx} ((-1)_{[c-1]} X_{[c-1,c]})(-1)_{[c-1]} X_{[a,c-1]})(-1)_{[c]} X_{[c-1,c]} ) $$

$$ \stackrel{(26)}{\approx} ((-1)_{[c-1]} X_{[c-1,c]})(-1)_{[a+1]} X_{[a,a+1]})(-1)_{[c-1]} X_{[a+1,c-1]})(-1)_{[a]} X_{[a,a+1]})(-1)_{[c]} X_{[c-1,c]} ) $$

induction hypothesis
$\approx$

$$ \stackrel{(48)(32)}{\approx} ((-1)_{[c-1]} X_{[c-1,c]})(-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a+2]} X_{[a+1,a+2]})(-1)_{[c-1]} X_{[a+2,c-1]})(-1)_{[a+1]} X_{[a+1,a+2]})(-1)_{[a]} X_{[a,a+1]})(-1)_{[c]} X_{[c-1,c]} ) $$

$\approx$

$$ \stackrel{(28)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a+2]} X_{[a+1,a+2]})(-1)_{[c-1]} X_{[c-1,c]})(-1)_{[c-1]} X_{[a+2,c-1]})(-1)_{[c]} X_{[c-1,c]})(-1)_{[a+1]} X_{[a+1,a+2]})(-1)_{[a]} X_{[a,a+1]} ) $$

$$ \stackrel{(26)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a+2]} X_{[a+1,a+2]})(-1)_{[c]} X_{[a+2,c]})(-1)_{[a+1]} X_{[a+1,a+2]})(-1)_{[a]} X_{[a,a+1]} ) $$

$$ \stackrel{(26)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[c]} X_{[a+1,c]})(-1)_{[a]} X_{[a,a+1]} ) $$

*Proof of Equations (51) and (52):* First, we prove both equations in the case $a < b$ by decreasing finite induction on $a$.

If $a = b - 1$, then Equation (51) is an instance of Equation (29). Assuming Equation (51) for $a + 1$, we prove it for $a$, as follows:

$$ ((-1)_{[a]} X_{[a,b]})(-1)_{[b]} X_{[a,b]} ) $$

$$ \stackrel{(49)(50)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a+1]} X_{[a+1,b]})(-1)_{[a]} X_{[a,a+1]})(-1)_{[a+1]} X_{[a,a+1]})(-1)_{[b]} X_{[a+1,b]})(-1)_{[a]} X_{[a,a+1]} ) $$

$$ \stackrel{(29)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a+1]} X_{[a+1,b]})(-1)_{[b]} X_{[a+1,b]})(-1)_{[a]} X_{[a,a+1]} ) $$

induction hypothesis
$\approx$

$$ \stackrel{(20)(22)(21)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a]} X_{[a,a+1]} ) $$

$$ \stackrel{(20)(22)(21)}{\approx} ((-1)_{[a]}(-1)_{[a+1]})^2 ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a]} X_{[a,a+1]} ) $$

$$ \stackrel{(23)}{\approx} ((-1)_{[a]} X_{[a,a+1]})^4 ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a]} X_{[a,a+1]} ) $$

$$ \stackrel{(29)}{\approx} ((-1)_{[a]} X_{[a,a+1]})^4 ) $$

$\stackrel{(23)(21)(22)(20)}{\approx}$

$\epsilon$

If $a = b - 1$, then Equation (52) is an instance of Equation (23). Assuming Equation (52) for $a + 1$, we prove it for $a$, as follows:

$$ ((-1)_{[a]} X_{[a,b]})(-1)_{[a]} X_{[a,b]} ) $$

$$ \stackrel{(49)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a+1]} X_{[a+1,b]})(-1)_{[a]} X_{[a,a+1]})(-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a+1]} X_{[a+1,b]})(-1)_{[a]} X_{[a,a+1]} ) $$

$$ \stackrel{(29)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a+1]} X_{[a+1,b]})(-1)_{[a+1]} X_{[a+1,b]})(-1)_{[a]} X_{[a,a+1]} ) $$

induction hypothesis
$\approx$

$$ \stackrel{(31)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a+1]}(-1)_{[b]})(-1)_{[a]} X_{[a,a+1]} ) $$

$$ \stackrel{(20)(22)(21)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a]} X_{[a,a+1]})(-1)_{[a]}(-1)_{[b]} ) $$

$$ \stackrel{(23)}{\approx} ((-1)_{[a]}(-1)_{[a+1]})^2 ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a]} X_{[a,a+1]})(-1)_{[a]}(-1)_{[b]} ) $$

$$ \stackrel{(29)}{\approx} ((-1)_{[a]} X_{[a,a+1]})^4 ((-1)_{[a+1]} X_{[a,a+1]})(-1)_{[a]} X_{[a,a+1]})(-1)_{[a]}(-1)_{[b]} ) $$

$$ \stackrel{(29)}{\approx} ((-1)_{[a]} X_{[a,a+1]})^4 ((-1)_{[a]}(-1)_{[b]} ) $$

$\stackrel{(23)(21)(22)(20)}{\approx}$

$$ \stackrel{(23)(21)(22)(20)}{\approx} ((-1)_{[a]}(-1)_{[b]} ) $$
It remains to prove Equations (51) and (52) in the case $b < a$. By Equation (33), this is equivalent to proving the following equations for $a < b$:

$$ ((-1)_{[b]}X_{[a,b]}) ((-1)_{[a]}X_{[a,b]}) \approx \epsilon \\ ((-1)_{[b]}X_{[a,b]}) ((-1)_{[b]}X_{[a,b]}) \approx ((-1)_{[b]})(-1)_{[a]} $$

The proof of the first one is as follows:

$$ \begin{array}{l} ((-1)_{[b]}X_{[a,b]}) ((-1)_{[a]}X_{[a,b]}) \\ \stackrel{(20)(22)(21)}{\approx} ((-1)_{[a]}(-1)_{[b]})^2 ((-1)_{[b]}X_{[a,b]}) ((-1)_{[a]}X_{[a,b]}) \\ \text{case } a < b \\ \text{of } (52) \\ \stackrel{\approx}{\approx} ((-1)_{[a]}X_{[a,b]})^4 ((-1)_{[b]}X_{[a,b]}) ((-1)_{[a]}X_{[a,b]}) \\ \text{case } a < b \\ \text{of } (51) \\ \stackrel{\approx}{\approx} ((-1)_{[a]}X_{[a,b]})^3 ((-1)_{[a]}X_{[a,b]}) \\ \text{case } a < b \\ \text{of } (52) \\ \stackrel{\approx}{\approx} ((-1)_{[a]}(-1)_{[b]})^2 \\ \stackrel{(21)(22)(20)}{\approx} \epsilon \end{array} $$

and the proof of the second one is as follows:

$$ \begin{array}{l} ((-1)_{[b]}X_{[a,b]}) ((-1)_{[b]}X_{[a,b]}) \\ \stackrel{(20)(22)}{\approx} ((-1)_{[b]}(-1)_{[a]}) ((-1)_{[a]}(-1)_{[b]}) ((-1)_{[b]}X_{[a,b]}) ((-1)_{[b]}X_{[a,b]}) \\ \text{case } a < b \\ \text{of } (52) \\ \stackrel{\approx}{\approx} ((-1)_{[b]}(-1)_{[a]}) ((-1)_{[a]}X_{[a,b]}) ((-1)_{[a]}X_{[a,b]}) ((-1)_{[b]}X_{[a,b]}) ((-1)_{[b]}X_{[a,b]}) \\ \text{case } a < b \\ \text{of } (51) \\ \stackrel{\approx}{\approx} ((-1)_{[b]}(-1)_{[a]}) \end{array} $$

*Proof of Equation (53):* If $c \neq a, b$, then the equation is just Equation (30). If $c = a$, then the result follows from Equation (20). Therefore, it only remains to treat the case where $c = b$:

$$ \begin{array}{l} ((-1)_{[b]}X_{[a,b]}) \\ \stackrel{(51)(33)}{\approx} ((-1)_{[b]}X_{[a,b]}) ((-1)_{[b]}X_{[a,b]}) ((-1)_{[a]}X_{[a,b]}) \\ \stackrel{(33)(52)}{\approx} ((-1)_{[b]}(-1)_{[a]}) ((-1)_{[a]}X_{[a,b]}) \end{array} $$

*Proof of Equation (54):*

$$ \begin{align*} ((-1)_{[a]}X_{[a,c]}) & \stackrel{(49)}{\approx} ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a+1,c]}) ((-1)_{[a]}X_{[a,a+1]}) \\ & \stackrel{(53)}{\approx} ((-1)_{[a+1]}(-1)_{[a]}) ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a+1,c]}) ((-1)_{[a]}X_{[a,a+1]}) \\ & \stackrel{(21)}{\approx} ((-1)_{[a]}(-1)_{[a+1]}) ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a+1,c]}) ((-1)_{[a]}X_{[a,a+1]}) \\ & \stackrel{(23)}{\approx} ((-1)_{[a]}X_{[a,a+1]})^3 ((-1)_{[a+1]}X_{[a+1,c]}) ((-1)_{[a]}X_{[a,a+1]}) \end{align*} $$

*Proof of Equation (55):*

$$ \begin{align*} ((-1)_{[c]}(-1)_{[d]}) ((-1)_{[a]}X_{[a,a+1]}) & \stackrel{(22)(21)}{\approx} ((-1)_{[a+1]}(-1)_{[c]}) ((-1)_{[a+1]}(-1)_{[d]}) ((-1)_{[a]}X_{[a,a+1]}) \\ & \stackrel{(31)}{\approx} ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[a]}(-1)_{[c]}) ((-1)_{[a]}(-1)_{[d]}) \\ & \stackrel{(21)(22)}{\approx} ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[c]}(-1)_{[d]}) \end{align*} $$

*Proof of Equation (56):*

$$ \begin{align*} ((-1)_{[a]}(-1)_{[b]}) ((-1)_{[a]}X_{[a,b]}) & \stackrel{(52)}{\approx} ((-1)_{[a]}(-1)_{[b]})^3 \\ & \stackrel{(52)}{\approx} ((-1)_{[a]}X_{[a,b]}) ((-1)_{[a]}(-1)_{[b]}) \end{align*} $$
**Proof of Equation (57):**

$$ ((-1)_{[b]}(-1)_{[a]}) ((-1)_{[a]}X_{[a,b]}) \stackrel{(21)}{\approx} ((-1)_{[a]}(-1)_{[b]}) ((-1)_{[a]}X_{[a,b]}) \\ \stackrel{(56)}{\approx} ((-1)_{[a]}X_{[a,b]}) ((-1)_{[a]}(-1)_{[b]}) \\ \stackrel{(21)}{\approx} ((-1)_{[a]}X_{[a,b]}) ((-1)_{[b]}(-1)_{[a]}) $$

**Proof of Equation (58):** First, we prove Equation (58) in the case where $a < b$ and $c \notin [a, b]$ (that is, either $c < a < b$ or $a < b < c$) by decreasing finite induction on $a$.

If $a = b - 1$, then we have

$$ ((-1)_{[a]}(-1)_{[c]}) ((-1)_{[a]}X_{[a,a+1]}) \stackrel{(22)}{\approx} ((-1)_{[a]}(-1)_{[a+1]}) ((-1)_{[a+1]}(-1)_{[c]}) ((-1)_{[a]}X_{[a,a+1]}) \\ \stackrel{(31)}{\approx} ((-1)_{[a]}(-1)_{[a+1]}) ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[a]}(-1)_{[c]}) \\ \stackrel{(23)}{\approx} ((-1)_{[a]}X_{[a,a+1]})^3 ((-1)_{[a]}(-1)_{[c]}) \\ \stackrel{(23)}{\approx} ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[a]}(-1)_{[a+1]}) ((-1)_{[a]}(-1)_{[c]}) \\ \stackrel{(21)(22)}{\approx} ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[a+1]}(-1)_{[c]}) $$

Assuming $c \notin [a, b]$ and Equation (58) for $a + 1$, we prove Equation (58) for $a$, as follows:

$$ ((-1)_{[a]}(-1)_{[c]}) ((-1)_{[a]}X_{[a,b]}) \\ \stackrel{(49)}{\approx} ((-1)_{[a]}(-1)_{[c]}) ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a+1,b]}) ((-1)_{[a]}X_{[a,a+1]}) \\ \stackrel{(51)(33)}{\approx} ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[a]}(-1)_{[c]}) ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a+1,b]}) ((-1)_{[a]}X_{[a,a+1]}) \\ \stackrel{(31)}{\approx} ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[a+1]}(-1)_{[c]}) ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a+1,b]}) ((-1)_{[a]}X_{[a,a+1]}) \\ \stackrel{(29)}{\approx} ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[a+1]}(-1)_{[c]}) ((-1)_{[a+1]}X_{[a+1,b]}) ((-1)_{[a]}X_{[a,a+1]}) \\ \text{induction hypothesis} \stackrel{(55)}{\approx} ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a+1,b]}) ((-1)_{[b]}(-1)_{[c]}) ((-1)_{[a]}X_{[a,a+1]}) \\ \stackrel{(49)}{\approx} ((-1)_{[a]}X_{[a,b]}) ((-1)_{[b]}(-1)_{[c]}) $$

Then, we prove Equation (58) in the case $a < c < b$, again by decreasing finite induction on $a$.

If $a = c - 1$, then since $b \neq c$, we necessarily have $b > a + 1$. We have

$$ ((-1)_{[a]}(-1)_{[a+1]}) ((-1)_{[a]}X_{[a,b]}) \\ \stackrel{(49)}{\approx} ((-1)_{[a]}(-1)_{[a+1]}) ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a+1,b]}) ((-1)_{[a]}X_{[a,a+1]}) \\ \stackrel{(21)(52)(33)}{\approx} ((-1)_{[a+1]}X_{[a,a+1]})^3 ((-1)_{[a+1]}X_{[a+1,b]}) ((-1)_{[a]}X_{[a,a+1]}) \\ \stackrel{(33)(52)}{\approx} ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[a+1]}(-1)_{[a]}) ((-1)_{[a+1]}X_{[a+1,b]}) ((-1)_{[a]}X_{[a,a+1]}) \\ \text{previous case} \stackrel{(21)}{\approx} ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a+1,b]}) ((-1)_{[b]}(-1)_{[a]}) ((-1)_{[a]}X_{[a,a+1]}) \\ \stackrel{(21)}{\approx} ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a+1,b]}) ((-1)_{[a]}(-1)_{[b]}) ((-1)_{[a]}X_{[a,a+1]}) \\ \text{previous case} \stackrel{(49)}{\approx} ((-1)_{[a]}X_{[a,b]}) ((-1)_{[a+1]}(-1)_{[b]}) \\ \stackrel{(21)}{\approx} ((-1)_{[a]}X_{[a,b]}) ((-1)_{[b]}(-1)_{[a+1]}) $$

Assuming $a + 1 < c < b$ and Equation (58) for $a + 1$, we prove Equation (58) for $a$, in the same way as in the previous case:

$$ ((-1)_{[a]}(-1)_{[c]}) ((-1)_{[a]}X_{[a,b]}) \\ \stackrel{(49)}{\approx} ((-1)_{[a]}(-1)_{[c]}) ((-1)_{[a+1]}X_{[a,a+1]}) ((-1)_{[a+1]}X_{[a+1,b]}) ((-1)_{[a]}X_{[a,a+1]}) $$
$$^{(51)(33)} \approx ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a]} X_{[a,a+1]}) ((-1)_{[a]} (-1)_{[c]}) ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+1]} X_{[a+1,b]}) ((-1)_{[a]} X_{[a,a+1]})$$

$$^{(31)} \approx ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+1]} (-1)_{[c]}) ((-1)_{[a]} X_{[a,a+1]}) ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+1]} X_{[a+1,b]}) ((-1)_{[a]} X_{[a,a+1]})$$

$$^{(29)} \approx ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+1]} (-1)_{[c]}) ((-1)_{[a+1]} X_{[a+1,b]}) ((-1)_{[a]} X_{[a,a+1]})$$

$$\begin{align*} \text{induction} \\ \text{hypothesis} \\ \approx \quad ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+1]} X_{[a+1,b]}) ((-1)_{[b]} (-1)_{[c]}) ((-1)_{[a]} X_{[a,a+1]}) \end{align*}$$

$$^{(55)} \approx ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a+1]} X_{[a+1,b]}) ((-1)_{[a]} X_{[a,a+1]}) ((-1)_{[b]} (-1)_{[c]})$$

$$^{(49)} \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[b]} (-1)_{[c]})$$

Finally, it remains to prove Equation (58) in the case $b < a$:

$$((-1)_{[a]} (-1)_{[c]}) ((-1)_{[a]} X_{[a,b]})$$

$$^{(33)} \approx ((-1)_{[a]} (-1)_{[c]}) ((-1)_{[a]} X_{[b,a]})$$

$$^{(53)} \approx ((-1)_{[a]} (-1)_{[c]}) ((-1)_{[a]} (-1)_{[b]}) ((-1)_{[b]} X_{[b,a]})$$

$$^{(21)(22)} \approx ((-1)_{[b]} (-1)_{[c]}) ((-1)_{[b]} X_{[b,a]})$$

$$\begin{align*} \text{previous} \\ \text{cases} \\ \approx \quad ((-1)_{[b]} X_{[b,a]}) ((-1)_{[a]} (-1)_{[c]}) \end{align*}$$

$$^{(22)} \approx ((-1)_{[b]} X_{[b,a]}) ((-1)_{[a]} (-1)_{[b]}) ((-1)_{[b]} (-1)_{[c]})$$

$$^{(52)} \approx ((-1)_{[b]} X_{[b,a]}) ((-1)_{[a]} X_{[a,b]}) ((-1)_{[a]} X_{[a,b]}) ((-1)_{[b]} (-1)_{[c]})$$

$$^{(33)(51)} \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[b]} (-1)_{[c]})$$

Proof of Equation (59):

$$((-1)_{[c]} (-1)_{[a]}) ((-1)_{[a]} X_{[a,b]}) \stackrel{(21)}{\approx} ((-1)_{[a]} (-1)_{[c]}) ((-1)_{[a]} X_{[a,b]})$$

$$^{(58)} \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[b]} (-1)_{[c]})$$

$$^{(21)} \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[c]} (-1)_{[b]})$$

Proof of Equation (60):

$$((-1)_{[b]} (-1)_{[c]}) ((-1)_{[a]} X_{[a,b]}) \stackrel{(22)}{\approx} ((-1)_{[b]} (-1)_{[a]}) ((-1)_{[a]} (-1)_{[c]}) ((-1)_{[a]} X_{[a,b]})$$

$$^{(58)(57)} \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[b]} (-1)_{[a]}) ((-1)_{[b]} (-1)_{[c]})$$

$$^{(21)(22)} \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[a]} (-1)_{[c]})$$

Proof of Equation (61):

$$((-1)_{[c]} (-1)_{[b]}) ((-1)_{[a]} X_{[a,b]}) \stackrel{(21)}{\approx} ((-1)_{[b]} (-1)_{[c]}) ((-1)_{[a]} X_{[a,b]})$$

$$^{(60)} \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[a]} (-1)_{[c]})$$

$$^{(21)} \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[c]} (-1)_{[a]})$$

Proof of Equation (62):

$$((-1)_{[c]} (-1)_{[d]}) ((-1)_{[a]} X_{[a,b]}) \stackrel{(22)}{\approx} ((-1)_{[c]} (-1)_{[a]}) ((-1)_{[a]} (-1)_{[d]}) ((-1)_{[a]} X_{[a,b]})$$

$$^{(58)(59)} \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[c]} (-1)_{[b]}) ((-1)_{[b]} (-1)_{[d]})$$

$$^{(22)} \approx ((-1)_{[a]} X_{[a,b]}) ((-1)_{[c]} (-1)_{[d]})$$

Proof of Equation (63):

$$(((-1)_{[a]} (-1)_{[b]}) ((-1)_{[c]} (-1)_{[d]}) \stackrel{(20)}{\approx} ((-1)_{[a]} (-1)_{[b]}) ((-1)_{[b]} (-1)_{[b]}) ((-1)_{[c]} (-1)_{[d]})$$

$$^{(22)} \approx ((-1)_{[a]} (-1)_{[b]}) ((-1)_{[b]} (-1)_{[c]}) ((-1)_{[c]} (-1)_{[b]}) ((-1)_{[c]} (-1)_{[d]})$$
$$ \stackrel{(21)}{\approx} ((-1)_{[a]}(-1)_{[b]}) ((-1)_{[b]}(-1)_{[c]}) ((-1)_{[c]}(-1)_{[a]}) ((-1)_{[c]}(-1)_{[d]}) $$

$$ \stackrel{(22)}{\approx} ((-1)_{[a]}(-1)_{[c]}) ((-1)_{[b]}(-1)_{[d]}) $$

**Proof of Equation (64):** First, we consider the case where either $a < b < c < d$ or $c < d < a < b$. We proceed by induction on $b-a$. If $b-a=1$, then we proceed by induction on $d-c$. If $d-c=1$, then Equation (64) is just an instance of Equation (32). If $d-c > 1$, then we have

$$ ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[c]}X_{[c,d]}) $$

$$ \stackrel{(54)}{\approx} ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[c]}X_{[c,c+1]})^3 ((-1)_{[c+1]}X_{[c+1,d]}) ((-1)_{[c]}X_{[c,c+1]}) $$

induction hypothesis
$\approx$

$$ ((-1)_{[c]}X_{[c,c+1]})^3 ((-1)_{[c+1]}X_{[c+1,d]}) ((-1)_{[c]}X_{[c,c+1]}) ((-1)_{[a]}X_{[a,a+1]}) $$

$$ \stackrel{(54)}{\approx} ((-1)_{[c]}X_{[c,d]}) ((-1)_{[a]}X_{[a,a+1]}) $$

Now, if $b-a > 1$, we have

$$ ((-1)_{[a]}X_{[a,b]}) ((-1)_{[c]}X_{[c,d]}) $$

$$ \stackrel{(54)}{\approx} ((-1)_{[a]}X_{[a,a+1]})^3 ((-1)_{[a+1]}X_{[a+1,c]}) ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[c]}X_{[c,d]}) $$

induction hypothesis
$\approx$

$$ ((-1)_{[c]}X_{[c,d]}) ((-1)_{[a]}X_{[a,a+1]})^3 ((-1)_{[a+1]}X_{[a+1,c]}) ((-1)_{[a]}X_{[a,a+1]}) $$

$$ \stackrel{(54)}{\approx} ((-1)_{[c]}X_{[c,d]}) ((-1)_{[a]}X_{[a,b]}) $$

Then we consider the case where $c < a < b < d$. Again, we proceed by induction on $b-a$. If $b-a=1$, then we proceed by induction on $a-c$. If $a-c=1$, then we have

$$ ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,d]}) $$

$$ \stackrel{(49)}{\approx} ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c+1]}X_{[c,c+1]}) ((-1)_{[c+1]}X_{[c+1,d]}) ((-1)_{[c]}X_{[c,c+1]}) $$

$$ \stackrel{(49)}{\approx} ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c+1]}X_{[c,c+1]}) ((-1)_{[c+2]}X_{[c+1,c+2]}) ((-1)_{[c+2]}X_{[c+2,d]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) $$

$$ \stackrel{(53)}{\approx} ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c+1]}(-1)_{[c]}) ((-1)_{[c]}X_{[c,c+1]}) ((-1)_{[c+2]}(-1)_{[c+1]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c+2]}X_{[c+2,d]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) $$

$$ \stackrel{(59)}{\approx} ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c+1]}(-1)_{[c]}) ((-1)_{[c+2]}(-1)_{[c]}) ((-1)_{[c]}X_{[c,c+1]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c+2]}X_{[c+2,d]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) $$

$$ \stackrel{(60)(58)}{\approx} ((-1)_{[c+2]}(-1)_{[c]}) ((-1)_{[c+1]}(-1)_{[c]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c+2]}X_{[c+2,d]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) $$

$$ \stackrel{(24)}{\approx} ((-1)_{[c+2]}(-1)_{[c]}) ((-1)_{[c+1]}(-1)_{[c]}) ((-1)_{[c]}X_{[c,c+1]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) ((-1)_{[c+2]}X_{[c+2,d]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) $$

$$ \stackrel{(53)}{\approx} ((-1)_{[c+2]}(-1)_{[c]}) ((-1)_{[c+1]}X_{[c,c+1]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) ((-1)_{[c+2]}X_{[c+2,d]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) $$

$$ \stackrel{(33)(61)}{\approx} ((-1)_{[c+1]}X_{[c,c+1]}) ((-1)_{[c+2]}(-1)_{[c+1]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) ((-1)_{[c+2]}X_{[c+2,d]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) $$

$$ \stackrel{(53)}{\approx} ((-1)_{[c+1]}X_{[c,c+1]}) ((-1)_{[c+2]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) ((-1)_{[c+2]}X_{[c+2,d]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) $$

$$ \stackrel{\text{previous case}}{\approx} ((-1)_{[c+1]}X_{[c,c+1]}) ((-1)_{[c+2]}X_{[c+1,c+2]}) ((-1)_{[c+2]}X_{[c+2,d]}) ((-1)_{[c]}X_{[c,c+1]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) $$

$$ \stackrel{(24)}{\approx} ((-1)_{[c+1]}X_{[c,c+1]}) ((-1)_{[c+2]}X_{[c+1,c+2]}) ((-1)_{[c+2]}X_{[c+2,d]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) ((-1)_{[c]}X_{[c,c+1]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) $$

$$ \stackrel{(49)}{\approx} ((-1)_{[c]}X_{[c,d]}) ((-1)_{[c+1]}X_{[c+1,c+2]}) $$

And if $a-c > 1$, we have

$$ ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[c]}X_{[c,d]}) $$

$$ \stackrel{(54)}{\approx} ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[c]}X_{[c,c+1]})^3 ((-1)_{[c+1]}X_{[c+1,d]}) ((-1)_{[c]}X_{[c,c+1]}) $$

$$ \stackrel{(32)}{\approx} ((-1)_{[c]}X_{[c,c+1]})^3 ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[c+1]}X_{[c+1,d]}) ((-1)_{[c]}X_{[c,c+1]}) $$

induction hypothesis
$\approx$

$$ ((-1)_{[c]}X_{[c,c+1]})^3 ((-1)_{[c+1]}X_{[c+1,d]}) ((-1)_{[a]}X_{[a,a+1]}) ((-1)_{[c]}X_{[c,c+1]}) $$

$$ \stackrel{(32)}{\approx} ((-1)_{[c]}X_{[c,c+1]})^3 ((-1)_{[c+1]}X_{[c+1,d]}) ((-1)_{[c]}X_{[c,c+1]}) ((-1)_{[a]}X_{[a,a+1]}) $$

$$ \stackrel{(54)}{\approx} ((-1)_{[c]}X_{[c,d]}) ((-1)_{[a]}X_{[a,a+1]}) $$

Now, if $b-a > 1$, we have

$$ ((-1)_{[a]}X_{[a,b]}) ((-1)_{[c]}X_{[c,d]}) $$
$$ \begin{align*} & \stackrel{(54)}{\approx} \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right)^3 \left(\left(-1\right)_{[a+1]} X_{[a+1,c]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[c]} X_{[c,d]}\right) \\ & \text{induction} \\ & \stackrel{\text{hypothesis}}{\approx} \left(\left(-1\right)_{[c]} X_{[c,d]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right)^3 \left(\left(-1\right)_{[a+1]} X_{[a+1,c]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \\ & \stackrel{(54)}{\approx} \left(\left(-1\right)_{[c]} X_{[c,d]}\right) \left(\left(-1\right)_{[a]} X_{[a,b]}\right) \end{align*} $$

Then we consider the case where $a < c < b < d$. We proceed by induction on $b-c$. If $b-c=1$, then we proceed by induction on $c-a$. If $c-a=1$, then we have

$$ \begin{align*} & \stackrel{(49)}{\approx} \left(\left(-1\right)_{[a]} X_{[a,a+2]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,d]}\right) \\ & \stackrel{(49)}{\approx} \left(\left(-1\right)_{[a+1]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+2,d]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \\ & \stackrel{(53)}{\approx} \left(\left(-1\right)_{[a+1]} \left(-1\right)_{[a]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+2,d]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \\ & \stackrel{(24)}{\approx} \left(\left(-1\right)_{[a+1]} \left(-1\right)_{[a]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+2,d]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \\ & \stackrel{(29)}{\approx} \left(\left(-1\right)_{[a+1]} \left(-1\right)_{[a]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+2,d]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \\ & \stackrel{\text{first case}}{\approx} \left(\left(-1\right)_{[a+1]} \left(-1\right)_{[a]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+2,d]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \\ & \stackrel{(29)}{\approx} \left(\left(-1\right)_{[a+1]} \left(-1\right)_{[a]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+2,d]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \\ & \stackrel{(53)}{\approx} \left(\left(-1\right)_{[a+1]} \left(-1\right)_{[a]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+2,d]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} \left(-1\right)_{[a+1]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \\ & \stackrel{(24)}{\approx} \left(\left(-1\right)_{[a+1]} \left(-1\right)_{[a]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+2,d]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} \left(-1\right)_{[a+1]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \\ & \stackrel{(22)(21)(53)}{\approx} \left(\left(-1\right)_{[a+1]} \left(-1\right)_{[a]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+2,d]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} \left(-1\right)_{[a]}\right) \left(\left(-1\right)_{[a+1]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \\ & \stackrel{(58)(62)(60)}{\approx} \left(\left(-1\right)_{[a+1]} \left(-1\right)_{[a]}\right) \left(\left(-1\right)_{[a+2]} \left(-1\right)_{[a]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+2,d]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+1]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \\ & \stackrel{(21)(22)}{\approx} \left(\left(-1\right)_{[a+2]} \left(-1\right)_{[a+1]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+2,d]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+1]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \\ & \stackrel{(53)}{\approx} \left(\left(-1\right)_{[a+2]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+2]} X_{[a+2,d]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a+1]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[a+1]} X_{[a+1,a+2]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \\ & \stackrel{(49)}{\approx} \left(\left(-1\right)_{[a+1]} X_{[a+1,d]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+2]}\right) \end{align*} $$

And if $c-a > 1$, we have

$$ \begin{align*} & \stackrel{(54)}{\approx} \left(\left(-1\right)_{[a]} X_{[a,c+1]}\right) \left(\left(-1\right)_{[c]} X_{[c,d]}\right) \\ & \text{first case} \\ & \stackrel{\text{induction}}{\approx} \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right)^3 \left(\left(-1\right)_{[a+1]} X_{[a+1,c+1]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[c]} X_{[c,d]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \\ & \text{hypothesis} \\ & \stackrel{\text{first case}}{\approx} \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right)^3 \left(\left(-1\right)_{[c]} X_{[c,d]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \left(\left(-1\right)_{[c]} X_{[c,d]}\right) \left(\left(-1\right)_{[a]} X_{[a,a+1]}\right) \\ & \stackrel{(54)}{\approx} \left(\left(-1\right)_{[c]} X_{[c,d]}\right) \left(\left(-1\right)_{[a]} X_{[a,c+1]}\right) \end{align*} $$

Now, if $b-c > 1$, we have

$$ \begin{align*} & \stackrel{(54)}{\approx} \left(\left(-1\right)_{[a]} X_{[a,b]}\right) \left(\left(-1\right)_{[c]} X_{[c,d]}\right) \\ & \text{previous case} \\ & \stackrel{\text{induction}}{\approx} \left(\left(-1\right)_{[c]} X_{[c,c+1]}\right)^3 \left(\left(-1\right)_{[a]} X_{[a,b]}\right) \left(\left(-1\right)_{[c+1]} X_{[c+1,d]}\right) \left(\left(-1\right)_{[c]} X_{[c,c+1]}\right) \\ & \text{hypothesis} \\ & \stackrel{\text{previous case}}{\approx} \left(\left(-1\right)_{[c]} X_{[c,c+1]}\right)^3 \left(\left(-1\right)_{[c+1]} X_{[c+1,d]}\right) \left(\left(-1\right)_{[a]} X_{[a,b]}\right) \left(\left(-1\right)_{[c]} X_{[c,c+1]}\right) \\ & \stackrel{(54)}{\approx} \left(\left(-1\right)_{[c]} X_{[c,d]}\right) \left(\left(-1\right)_{[a]} X_{[a,b]}\right) \end{align*} $$

By symmetry, the case $a < c < d < b$ is similar to the case $c < a < b < d$, and the case $c < a < d < b$ is similar to the case $a < c < b < d$. Thus, we have treated all the cases where $a < b$ and $c < d$.

Finally, if $b < a$ and $c < d$, we have

$$ \begin{align*} & \stackrel{(51)}{\approx} \left(\left(-1\right)_{[a]} X_{[a,b]}\right) \left(\left(-1\right)_{[c]} X_{[c,d]}\right) \\ & \stackrel{(51)}{\approx} \left(\left(-1\right)_{[a]} X_{[a,b]}\right) \left(\left(-1\right)_{[c]} X_{[c,d]}\right) \left(\left(-1\right)_{[b]} X_{[b,a]}\right) \left(\left(-1\right)_{[a]} X_{[b,a]}\right) \end{align*} $$
$$
\begin{align*}
\begin{array}{c} \text{previous cases} \\ \approx \end{array}
& \quad ((-1)_{[a]} X_{[a,b]}) ((-1)_{[b]} X_{[b,a]}) ((-1)_{[c]} X_{[c,d]}) ((-1)_{[a]} X_{[b,a]}) \\
(33)(51) \quad \approx
& \quad ((-1)_{[c]} X_{[c,d]}) ((-1)_{[a]} X_{[a,b]})
\end{align*}
$$

The case where $a < b$ and $d < c$ follows by symmetry. Finally, the case where $b < a$ and $d < c$ follows by exactly the same derivation:

$$
\begin{align*}
& ((-1)_{[a]} X_{[a,b]}) ((-1)_{[c]} X_{[c,d]}) \\
& \stackrel{(51)}{\approx} ((-1)_{[a]} X_{[a,b]}) ((-1)_{[c]} X_{[c,d]}) ((-1)_{[b]} X_{[b,a]}) ((-1)_{[a]} X_{[b,a]}) \\
\begin{array}{c} \text{previous cases} \\ \approx \end{array}
& ((-1)_{[a]} X_{[a,b]}) ((-1)_{[b]} X_{[b,a]}) ((-1)_{[c]} X_{[c,d]}) ((-1)_{[a]} X_{[b,a]}) \\
(33)(51) \quad \approx
& ((-1)_{[c]} X_{[c,d]}) ((-1)_{[a]} X_{[a,b]})
\end{align*}
$$

□

**Lemma A.4.** Any word over the generators of $\mathcal{P}_N$ containing no generators of the form $(H_{[a,b]}H_{[c,d]})$ is $\approx$-equivalent to a unique word, called its normal form, of the form

$$
(((-1)_{[a_1]}(-1)_{[b_1]}) \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) \prod_{i=1}^{N-1} \prod_{j=d_i}^{N-i-1} (((-1)_{[j]}X_{[j,i+1]}) \quad (B))
$$

with $k \ge 0$; $a_1 < b_1 < \cdots < a_k < b_k$; and for every $i \in \{1, \dots, N\}$, $d_i \in \{0, \dots, N-i\}$ (with the empty products being the identity). Moreover, two words with the same semantics have the same normal form.

**Remark 6.** In the following, when we talk about words of the form (B) or being put in the form (B), we implicitly also require that they satisfy the conditions on the parameters $k, a_i, b_i, d_i$ given in Lemma A.4.

PROOF OF LEMMA A.4.

Uniqueness given the semantics: Given a word of the form (B), its semantics is the product of a permutation matrix with a diagonal matrix whose diagonal elements are all $\pm 1$. It is easy to see that the $d_i$'s are uniquely determined by the permutation, as the product $\prod_{i=1}^{N-1} (\overrightarrow{\prod}_{j=d_i}^{N-i-1} X_{[j,j+1]})$ corresponds to a well-known canonical decomposition of the permutation as a product of transpositions. Then, the $d_i$'s being known, $k$ together with the $a_i$'s and the $b_i$'s are uniquely determined by the elements of the diagonal matrix.

Existence: First, all generators of $\mathcal{P}_N$, except those of the form $(H_{[a,b]}H_{[c,d]})$, can be written by means of the equations of Figure 8 as a product of generators of the form $((-1)_{[j]}X_{[j,j+1]})$. Indeed: first, by Equations (34) and (53), any of these generators can be written as a product of generators of the form $((-1)_{[a]}(-1)_{[b]})$ and $((-1)_{[a]}X_{[a,b]})$; then by Equations (33), (53), (20) and (21), one can ensure that $a < b$ in all generators of the form $((-1)_{[a]}(-1)_{[b]})$ and $((-1)_{[a]}X_{[a,b]})$; then by Equation (52), one can write $((-1)_{[a]}(-1)_{[b]})$ as a product of two $((-1)_{[a]}X_{[a,b]})$; finally, when $a < b$, Equation (54) allows us to write $((-1)_{[a]}X_{[a,b]})$, as a product of generators of the form $((-1)_{[j]}X_{[j,j+1]})$ by direct induction on $b-a$.

Therefore, it suffices to show that any product of $((-1)_{[j]}X_{[j,j+1]})$ can be put in the form (B). We proceed by induction: The identity is of the form (B), namely with $k=0$ and $d_i=N-i$ for all $i$. Then it remains to show that for any word $P$ of the form (B) and any $a \in \{0, \dots, N-2\}$, the word $P((-1)_{[a]}X_{[a,a+1]})$ can be put in the form (B):

$$
\begin{align*}
& ((-1)_{[a_1]}(-1)_{[b_1]})(-1)_{[a_k]}(-1)_{[b_k]} \prod_{i=1}^{N-1} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \circ ((-1)_{[a]} X_{[a,a+1]}) \\
& \stackrel{(64)}{\approx} ((-1)_{[a_1]}(-1)_{[b_1]})(-1)_{[a_k]}(-1)_{[b_k]} \prod_{i=1}^{N-a} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \circ ((-1)_{[a]} X_{[a,a+1]}) \circ \prod_{i=N-a+1}^{N-1} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \\
& = ((-1)_{[a_1]}(-1)_{[b_1]})(-1)_{[a_k]}(-1)_{[b_k]} \prod_{i=1}^{N-a-2} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \circ \prod_{j=d_{N-a-1}+1}^{a} ((-1)_{[j]} X_{[j,j+1]}) \circ \prod_{j=d_{N-a}+1}^{a-1} ((-1)_{[j]} X_{[j,j+1]}) \circ ((-1)_{[a]} X_{[a,a+1]}) \circ \prod_{i=N-a+1}^{N-1} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right)
\end{align*}
$$
* If $d_{N-a-1} = d_{N-a} + 1$, then by Equation (64), this can be rewritten into

$$ \begin{align*} & ((-1)_{[a_1]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \circ \prod_{j=d_{N-a-1}}^{a} ((-1)_{[j]} X_{[j,j+1]})((-1)_{[j-1]} X_{[j-1,j]}) \circ ((-1)_{[a]} X_{[a,a+1]}) \circ \prod_{i=N-a+1}^{N-1} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \\ & \stackrel{(24)}{\approx} ((-1)_{[a_1]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \circ ((-1)_{[d_{N-a-1}-1]} X_{[d_{N-a-1}-1,d_{N-a-1}]} ) \circ \prod_{j=d_{N-a-1}}^{a} ((-1)_{[j]} X_{[j,j+1]})((-1)_{[j-1]} X_{[j-1,j]}) \circ \prod_{i=N-a+1}^{N-1} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \\ & \stackrel{(64)}{\approx} ((-1)_{[a_1]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \circ ((-1)_{[d_{N-a-1}-1]} X_{[d_{N-a-1}-1,d_{N-a-1}]} ) \circ \prod_{j=d_{N-a-1}}^{a} ((-1)_{[j]} X_{[j,j+1]}) \circ \prod_{i=N-a+1}^{N-1} ((-1)_{[j]} X_{[j,j+1]}) \circ \prod_{i=N-a+1}^{N-1} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \\ & = ((-1)_{[a_1]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \circ \prod_{j=d_{N-a-1}}^{a} ((-1)_{[j]} X_{[j,j+1]}) \circ \prod_{i=N-a-1}^{a-1} ((-1)_{[j]} X_{[j,j+1]}) \circ \prod_{i=N-a+1}^{N-1} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \end{align*} $$

which is in the form (B).

* If $d_{N-a-1} > d_{N-a} + 1$, then this is equal to

$$ \begin{align*} & ((-1)_{[a_1]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \circ \prod_{j=d_{N-a-1}}^{a} ((-1)_{[j]} X_{[j,j+1]})(-1)_{[a]} \circ \prod_{j=d_{N-a-1}}^{a} ((-1)_{[j]} X_{[j,j+1]})(-1)_{[a]} \circ \prod_{j=d_{N-a-1}}^{a} ((-1)_{[j]} X_{[j,j+1]})(-1)_{[a]} \circ \prod_{i=N-a+1}^{N-1} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \\ & \stackrel{(64)}{=} ((-1)_{[a_1]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \circ \prod_{j=d_{N-a-1}}^{a} ((-1)_{[j]} X_{[j,j+1]})(-1)_{[a]} \circ \prod_{j=d_{N-a-1}}^{a} ((-1)_{[j]} X_{[j,j+1]})(-1)_{[a]} \circ \prod_{i=N-a+1}^{N-1} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \\ & \stackrel{(64)}{=} ((-1)_{[a_1]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \circ \prod_{j=d_{N-a-1}}^{a} ((-1)_{[j]} X_{[j,j+1]})(-1)_{[a]} \circ \prod_{j=d_{N-a-1}}^{a} ((-1)_{[j]} X_{[j,j+1]})(-1)_{[a]} \circ \prod_{i=N-a+1}^{N-1} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \\ & = ((-1)_{[a_1]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \circ \prod_{j=d_{N-a-1}}^{a} ((-1)_{[j]} X_{[j,j+1]})(-1)_{[a]} \circ \prod_{j=d_{N-a-1}}^{a-1} ((-1)_{[j]} X_{[j,j+1]})(-1)_{[a]} \circ \prod_{i=N-a+1}^{N-1} \left( \prod_{j=d_i}^{N-i-1} ((-1)_{[j]} X_{[j,j+1]}) \right) \end{align*} $$

which is in the form (B).
• And if $d_{N-a-1} < d_{N-a} + 1$, then this is equal to

$$ ((-1)_{[a]}(-1)_{[b_1]} \cdots (-1)_{[a_k]}(-1)_{[b_k]} \prod_{i=1}^{N-a-2} \left( \prod_{j=\text{d}_{N-a-1}}^{N-1} (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a-1}}^{d_{N-a}} \left( \prod_{j=\text{d}_{N-a-1}}^{a} (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a}}^{a-1} \left( \prod_{j=\text{d}_{N-a}}^{(-1)[j]} X_{[j,j+1]} \right) \circ \left( \prod_{j=\text{d}_{N-a}}^{(-1)[a]} X_{[a,a+1]} \right) \circ \prod_{i=\text{N}-a-1}^{N-1} \left( \prod_{j=\text{d}_i}^{(-1)[j]} X_{[j,j+1]} \right) $$

$$ \stackrel{(64)}{\approx} ((-1)_{[a]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=\text{d}_i}^{N-1} (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a-1}}^{d_{N-a}} \left( \prod_{j=\text{d}_{N-a-1}}^{a} (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a-1}}^{a} \left( \prod_{j=\text{d}_{N-a}}^{(-1)[j]} X_{[j-1,j]} \right) \circ \left( (-1)_{[a]} X_{[a,a+1]} \right) \circ \prod_{i=\text{N}-a+1}^{N-1} \left( \prod_{j=\text{d}_i}^{(-1)[j]} X_{[j,j+1]} \right) $$

$$ \stackrel{(24)}{\approx} ((-1)_{[a]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=\text{d}_i}^{N-1} (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a-1}}^{d_{N-a}} \left( \prod_{j=\text{d}_{N-a-1}}^{(-1)[j]} X_{[j,j+1]} \right) \circ \left( (-1)_{[d_{N-a}]} X_{[d_{N-a},d_{N-a+1}]} \right) \circ \prod_{j=\text{d}_{N-a+1}}^{a} \left( (-1)_{[j]} X_{[j,j+1]} \right) \circ \left( (-1)_{[j-1]} X_{[j-1,j]} \right) \circ \prod_{i=\text{N}-a+1}^{N-1} \left( \prod_{j=\text{d}_i}^{(-1)[j]} X_{[j,j+1]} \right) $$

$$ \stackrel{(23)}{\approx} ((-1)_{[a]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=\text{d}_{N-a-1}}^{N-1} (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a-1}}^{d_{N-a-1}} \left( \prod_{j=\text{d}_{N-a-1}}^{(-1)[j]} X_{[j,j+1]} \right) \circ \left( (-1)_{[d_{N-a}]} (-1)_{[d_{N-a+1}]} \right) \circ \prod_{j=\text{d}_{N-a+1}}^{a} \left( (-1)_{[j]} X_{[j,j+1]} \right) \circ \left( (-1)_{[j-1]} X_{[j-1,j]} \right) \circ \prod_{i=\text{N}-a+1}^{N-1} \left( \prod_{j=\text{d}_i}^{(-1)[j]} X_{[j,j+1]} \right) $$

$$ \stackrel{(56) \text{ to } (62)}{=} ((-1)_{[a]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) ((-1)_{[a]}(-1)_{[\beta_1]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=\text{d}_i}^{N-1} (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a-1}}^{d_{N-a-1}} \left( (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a-1}}^{a} \left( (-1)_{[j]} X_{[j,j+1]} \right) \circ \left( (-1)_{[j-1]} X_{[j-1,j]} \right) \circ \prod_{i=\text{N}-a-1}^{N-1} \left( \prod_{j=\text{d}_i}^{(-1)[j]} X_{[j,j+1]} \right) \quad \text{for some } \alpha, \beta \in \{0, \dots, N-1\} $$

$$ \stackrel{(64)}{\approx} ((-1)_{[a]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=\text{d}_i}^{N-1} (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a-1}}^{d_{N-a-1}} \left( (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a-1}}^{a} \left( (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a}}^{a-1} \left( (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{i=\text{N}-a+1}^{N-1} \left( \prod_{j=\text{d}_i}^{(-1)[j]} X_{[j,j+1]} \right) $$

$$ \stackrel{(64)}{\approx} ((-1)_{[a]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) ((-1)_{[a]}(-1)_{[\beta_1]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=\text{d}_i}^{N-1} (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a-1}}^{a} \left( (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a-1}}^{d_{N-a-1}} \left( (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a}}^{a-1} \left( (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{i=\text{N}-a+1}^{N-1} \left( \prod_{j=\text{d}_i}^{(-1)[j]} X_{[j,j+1]} \right) $$

$$ = ((-1)_{[a]}(-1)_{[b_1]} \cdots ((-1)_{[a_k]}(-1)_{[b_k]}) ((-1)_{[a]}(-1)_{[\beta_1]}) \prod_{i=1}^{N-a-2} \left( \prod_{j=\text{d}_i}^{N-1} (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a-1}}^{a} \left( (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{j=\text{d}_{N-a-1}}^{d_{N-a-1}} \left( (-1)_{[j]} X_{[j,j+1]} \right) \circ \prod_{i=\text{N}-a+1}^{N-1} \left( \prod_{j=\text{d}_i}^{(-1)[j]} X_{[j,j+1]} \right) $$

finally, to put this in the form (B), it suffices to put the left part composed of $(-1)$'s in the appropriate form, which can be done using Equations (20) to (22) and (63). $\square$

**Corollary A.5.** Given any two words $w, w'$ over the generators of $\mathcal{P}_N$ containing no generators of the form $(H_{[a,b]}H_{[c,d]})$, if $\llbracket w \rrbracket = \llbracket w' \rrbracket$ then $w \approx w'$.

**PROOF.** Since $w$ and $w'$ are semantically equivalent, by Lemma A.4 they are $\approx$-equivalent to the same normal form (of the form (B)), which implies that they are $\approx$-equivalent to each other. $\square$

**Lemma A.6.** For any word $w$ over the generators of $\mathcal{P}_N$ containing no generators of the form $(H_{[a,b]}H_{[c,d]})$, if $\llbracket w (H_{[0,1]}H_{[3,2]}) \rrbracket = \llbracket (H_{[0,1]}H_{[3,2]}) w \rrbracket$, then $w$ is $\approx$-equivalent to a word of the form

$$ ((-1)_{[0]}(-1)_{[1]})^{\alpha} ((-1)_{[3]}(-1)_{[2]})^{\beta} (X_{[0,3]}X_{[1,2]})^{\gamma} Q $$

where $Q$ is a product of generators of the form $(-1)_{[a]}X_{[a,a+1]}$ with $a \ge 4$, and $\alpha, \beta, \gamma \in \{0, 1\}$.

**PROOF.** By analyzing the effect of $\llbracket w (H_{[0,1]}H_{[3,2]}) \rrbracket$ and $\llbracket (H_{[0,1]}H_{[3,2]}) w \rrbracket$ on the vectors of the canonical basis of $\mathbb{C}^N$, noting in particular that $\llbracket w \rrbracket$ is a just permutation matrix up to the signs of its entries, one can show that $w$ must be semantically equivalent to a word of the indicated form. Then the result follows by Corollary A.5. $\square$

**Corollary A.7.** For any words $A, A', B, B'$ over the generators of $\mathcal{P}_N$ containing no generators of the form $(H_{[a,b]}H_{[c,d]})$, if $\llbracket AA' \rrbracket = \llbracket BB' \rrbracket = I$ and $\llbracket A (H_{[0,1]}H_{[3,2]}) A' \rrbracket = \llbracket B (H_{[0,1]}H_{[3,2]}) B' \rrbracket$, then $A (H_{[0,1]}H_{[3,2]}) A' \approx B (H_{[0,1]}H_{[3,2]}) B'$.

**PROOF.** One has

$$ \llbracket B' A (H_{[0,1]} H_{[3,2]}) \rrbracket = \llbracket B' A (H_{[0,1]} H_{[3,2]}) A' \rrbracket \\ = \llbracket B' B (H_{[0,1]} H_{[3,2]}) B' A \rrbracket \\ = \llbracket (H_{[0,1]} H_{[3,2]}) B' A \rrbracket. $$

Therefore, by Lemma A.6, $B'A$ is $\approx$-equivalent to a word of the form

$$ ((-1)_{[0]}(-1)_{[1]})^{\alpha} ((-1)_{[3]}(-1)_{[2]})^{\beta} (X_{[0,3]}X_{[1,2]})^{\gamma} Q $$

where $Q$ is a product of generators of the form $(-1)_{[a]}X_{[a,a+1]}$ with $a \ge 4$, and $\alpha, \beta, \gamma \in \{0, 1\}$. Then one has

$$ \begin{align*} A (H_{[0,1]} H_{[3,2]}) A' &\stackrel{\text{Corollary A.5}}{\approx} BB' A (H_{[0,1]} H_{[3,2]}) A' \\ &\approx B ((-1)_{[0]} (-1)_{[1]})^{\alpha} ((-1)_{[3]} (-1)_{[2]})^{\beta} (X_{[0,3]} X_{[1,2]})^{\gamma} Q (H_{[0,1]} H_{[3,2]}) A' \\ &\stackrel{(38)}{\approx} B ((-1)_{[0]} (-1)_{[1]})^{\alpha} ((-1)_{[3]} (-1)_{[2]})^{\beta} (X_{[0,3]} X_{[1,2]})^{\gamma} (H_{[0,1]} H_{[3,2]}) QA' \end{align*} $$
$$ (H_{[a,b]}H_{[c,d]}) \approx \Sigma_{a,b,c,d} \circ (H_{[0,1]}H_{[3,2]}) \circ \Sigma'_{a,b,c,d} \qquad \{a, b\} \cap \{c, d\} = \emptyset \qquad (65) $$

$$ (H_{[a,b]}H_{[c,d]}) \approx (H_{[c,d]}H_{[a,b]}) \qquad \{a, b\} \cap \{c, d\} = \emptyset \qquad (66) $$

$$ (H_{[b,a]}H_{[c,d]}) \approx (H_{[a,b]}H_{[c,d]}) ((-1)_{[b]}X_{[a,b]}) \qquad \{a, b\} \cap \{c, d\} = \emptyset \qquad (67) $$

$$ ((-1)_{[a]}X_{[a,a+1]}) (H_{[a,b]}H_{[c,d]}) \approx (H_{[a+1,b]}H_{[c,d]}) ((-1)_{[a]}X_{[a,a+1]}) \qquad |\{a, a+1, b, c, d\}| = 5 \qquad (68) $$

$$ ((-1)_{[a]}X_{[a,a+1]}) (H_{[a+1,b]}H_{[c,d]}) \approx (H_{[a,b]}H_{[c,d]}) ((-1)_{[a]}X_{[a,b]}) ((-1)_{[a]}X_{[a,a+1]}) \qquad |\{a, a+1, b, c, d\}| = 5 \qquad (69) $$

$$ (H_{[a,b]}H_{[a,b]}) \approx \epsilon \qquad (70) $$

$$ (H_{[a,b]}H_{[c,d]}) (H_{[c,d]}H_{[e,f]}) \approx (H_{[a,b]}H_{[e,f]}) \qquad (71) $$

$$ (H_{[a,b]}H_{[b,a]}) \approx ((-1)_{[b]}X_{[a,b]}) \qquad \{c, d\} \cap \{e, f\} = \emptyset \qquad (72) $$

$$ (H_{[a,b]}H_{[c,d]}) (H_{[e,f]}H_{[g,h]}) \approx (H_{[a,b]}H_{[e,f]}) (H_{[c,d]}H_{[g,h]}) \qquad \{c, d\} \cap \{e, f\} = \emptyset \qquad (73) $$

$$ ((-1)_{[a]}X_{[a,b]}) (H_{[c,d]}H_{[e,f]}) \approx (H_{[c,d]}H_{[e,f]}) ((-1)_{[a]}X_{[a,b]}) \qquad \{a, b\} \cap \{c, d, e, f\} = \emptyset \qquad (74) $$

$$ ((-1)_{[a]}X_{[a,b]}) (H_{[a,b]}H_{[c,d]}) \approx (H_{[a,b]}H_{[c,d]}) ((-1)_{[b]}X_{[a,b]}) \qquad \{a, b\} \cap \{c, d\} = \emptyset \qquad (75) $$

$$ ((-1)_{[a]}X_{[a,e]}) (H_{[a,b]}H_{[c,d]}) \approx (H_{[e,b]}H_{[c,d]}) ((-1)_{[a]}X_{[a,e]}) \qquad |\{a, e, b, c, d\}| = 5 \qquad (76) $$

$$ ((-1)_{[a]}X_{[a,e]}) (H_{[e,b]}H_{[c,d]}) \approx (H_{[a,b]}H_{[c,d]}) ((-1)_{[a]}X_{[a,e]}) ((-1)_{[a]}X_{[a,e]}) \qquad |\{a, e, b, c, d\}| = 5 \qquad (77) $$

**Figure 10: More useful auxiliary equations. Equation (65) is the same as Equation (35) but without the restriction on the tuple $(G_n(a), G_n(b), G_n(c), G_n(d))$**

$$ \text{Corollary A.5} \qquad \approx \quad B ((-1)_{[0]}(-1)_{[1]})^{\alpha} (X_{[0,3]}X_{[1,2]})^{\gamma} ((-1)_{[0]}(-1)_{[1]})^{\beta} (H_{[0,1]}H_{[3,2]}) QA' $$

$$ \approx \quad B ((-1)_{[0]}(-1)_{[1]})^{\alpha} (X_{[0,3]}X_{[1,2]})^{\gamma} (H_{[0,1]}H_{[3,2]}) ((-1)_{[0]}(-1)_{[1]})^{\beta} QA' $$

$$ \approx \quad B ((-1)_{[0]}(-1)_{[1]})^{\alpha} (H_{[0,1]}H_{[3,2]}) (X_{[0,3]}X_{[1,2]})^{\gamma} ((-1)_{[0]}(-1)_{[1]})^{\beta} QA' $$

$$ \approx \quad B (H_{[0,1]}H_{[3,2]}) ((-1)_{[0]}(-1)_{[1]})^{\alpha} (X_{[0,3]}X_{[1,2]})^{\gamma} ((-1)_{[0]}(-1)_{[1]})^{\beta} QA' $$

$$ \text{Corollary A.5} \qquad \approx \quad B (H_{[0,1]}H_{[3,2]}) ((-1)_{[0]}(-1)_{[1]})^{\alpha} ((-1)_{[3]}(-1)_{[2]})^{\beta} (X_{[0,3]}X_{[1,2]})^{\gamma} QA' $$

$$ \approx \quad B (H_{[0,1]}H_{[3,2]}) B'AA' $$

$$ \text{Corollary A.5} \qquad \approx \quad B (H_{[0,1]}H_{[3,2]}) B' $$

**Corollary A.8. The equations of Figure 10 are consequences of the equations of Figure 8.**

PROOF.

*Proof of Equation (65): Given Equations (35), (36) and (37) and their soundness, and the fact that Equation (65) is sound for all values of a, b, c, d, the result follows directly from Corollary A.7.*

Proof of Equation (66):

$$ (H_{[a,b]}H_{[c,d]}) \qquad \stackrel{(65)}{\approx} \quad \Sigma_{a,b,c,d} \circ (H_{[0,1]}H_{[3,2]}) \circ \Sigma'_{a,b,c,d} $$

$$ \text{Corollary A.7} \qquad \approx \quad \Sigma_{c,d,a,b} \circ (H_{[0,1]}H_{[3,2]}) \circ \Sigma'_{c,d,a,b} $$

$$ \stackrel{(65)}{\approx} \quad (H_{[c,d]}H_{[a,b]}) $$

Proof of Equation (67):

$$ (H_{[b,a]}H_{[c,d]}) \qquad \stackrel{(65)}{\approx} \quad \Sigma_{b,a,c,d} \circ (H_{[0,1]}H_{[3,2]}) \circ \Sigma'_{b,a,c,d} $$

$$ \text{Corollary A.7} \qquad \approx \quad \Sigma_{c,d,a,b} \circ ((-1)_{[3]}(-1)_{[4]}) (H_{[0,1]}H_{[3,2]}) ((-1)_{[2]}(-1)_{[4]}) \circ \Sigma'_{c,d,a,b} $$

$$ \stackrel{(40)}{\approx} \quad \Sigma_{c,d,a,b} \circ (H_{[0,1]}H_{[3,2]}) ((-1)_{[3]}(-1)_{[4]}) ((-1)_{[2]}X_{[2,3]}) ((-1)_{[2]}(-1)_{[4]}) \circ \Sigma'_{c,d,a,b} $$

$$ \text{Corollary A.5} \qquad \approx \quad \Sigma_{c,d,a,b} \circ (H_{[0,1]}H_{[3,2]}) \circ \Sigma'_{c,d,a,b} \circ ((-1)_{[b]}X_{[a,b]}) $$
$$
\begin{align*}
\text{Corollary A.7} &\approx \sum_{a,b,c,d} \circ (H_{[0,1]} H_{[3,2]}) \circ \Sigma'_{a,b,c,d} \circ ((-1)_{[b]} X_{[a,b]}) \\
&\approx (H_{[a,b]} H_{[c,d]}) ((-1)_{[b]} X_{[a,b]})
\end{align*}
$$

Proof of Equation (68):

$$
\begin{align*}
\text{Corollary A.5} &\approx ((-1)_{[a]} X_{[a,a+1]}) (H_{[a,b]} H_{[c,d]}) \\
&\approx ((-1)_{[a]} X_{[a,a+1]}) (H_{[a,b]} H_{[c,d]}) ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a]} X_{[a,a+1]}) \\
&\approx ((-1)_{[a]} X_{[a,a+1]}) \circ \Sigma_{a,b,c,d} \circ (H_{[0,1]} H_{[3,2]}) \circ \Sigma'_{a,b,c,d} \circ ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a]} X_{[a,a+1]}) \\
\text{Corollary A.7} &\approx \Sigma_{a+1,b,c,d} \circ (H_{[0,1]} H_{[3,2]}) \circ \Sigma'_{a+1,b,c,d} \circ ((-1)_{[a]} X_{[a,a+1]}) \\
&\approx (H_{[a+1,b]} H_{[c,d]}) ((-1)_{[a]} X_{[a,a+1]})
\end{align*}
$$

Proof of Equation (69):

$$
\begin{align*}
\text{Corollary A.5} &\approx ((-1)_{[a]} X_{[a,a+1]}) (H_{[a+1,b]} H_{[c,d]}) ((-1)_{[a]} X_{[a,a+1]}) ((-1)_{[a]} X_{[a,a+1]}) \\
&\approx ((-1)_{[a]} X_{[a,a+1]}) \circ \Sigma_{a+1,b,c,d} \circ (H_{[0,1]} H_{[3,2]}) \circ \Sigma'_{a+1,b,c,d} \circ ((-1)_{[a+1]} X_{[a,a+1]}) ((-1)_{[a]} X_{[a,a+1]}) \\
\text{Corollary A.7} &\approx \Sigma_{c,d,a,b} \circ ((-1)_{[3]}(-1)_{[4]}) (H_{[0,1]} H_{[3,2]}) ((-1)_{[3]}(-1)_{[4]}) \circ \Sigma'_{c,d,a,b} \circ ((-1)_{[a]} X_{[a,a+1]}) \\
&\approx \Sigma_{c,d,a,b} \circ (H_{[0,1]} H_{[3,2]}) ((-1)_{[3]}(-1)_{[4]}) ((-1)_{[2]} X_{[2,3]}) ((-1)_{[3]}(-1)_{[4]}) \circ \Sigma'_{c,d,a,b} \circ ((-1)_{[a]} X_{[a,a+1]}) \\
\text{Corollary A.5} &\approx \Sigma_{c,d,a,b} \circ (H_{[0,1]} H_{[3,2]}) \circ \Sigma'_{c,d,a,b} \circ ((-1)_{[a]} X_{[a,b]}) \circ ((-1)_{[a]} X_{[a,a+1]}) \\
\text{Corollary A.7} &\approx \Sigma_{a,b,c,d} \circ (H_{[0,1]} H_{[3,2]}) \circ \Sigma'_{a,b,c,d} \circ ((-1)_{[a]} X_{[a,b]}) \circ ((-1)_{[a]} X_{[a,a+1]}) \\
&\approx (H_{[a,b]} H_{[c,d]}) ((-1)_{[a]} X_{[a,b]}) \circ ((-1)_{[a]} X_{[a,a+1]})
\end{align*}
$$

Proof of Equation (70):

$$
\begin{align*}
(H_{[a,b]} H_{[a,b]}) &\stackrel{(42)}{\approx} (H_{[a,b]} H_{[e,f]}) (H_{[e,f]} H_{[a,b]}) \\
(H_{[a,b]} H_{[e,f]}) &\stackrel{(66)}{\approx} (H_{[a,b]} H_{[e,f]}) (H_{[a,b]} H_{[e,f]}) \\
(H_{[a,b]} H_{[e,f]}) &\stackrel{(65)}{\approx} \Sigma_{a,b,e,f} \circ (H_{[0,1]} H_{[3,2]}) \circ \Sigma'_{a,b,e,f} \circ \Sigma_{a,b,e,f} \circ (H_{[0,1]} H_{[3,2]}) \circ \Sigma'_{a,b,e,f} \\
\text{Corollary A.5} &\stackrel{(41)}{\approx} \Sigma_{a,b,e,f} \circ \Sigma'_{a,b,e,f} \\
\text{Corollary A.5} &\stackrel{\epsilon}{\approx}
\end{align*}
$$

where e, f are defined in Figure 8.

Proof of Equation (71): First, we consider the case where $a, b, c, d, e, f$ are all distinct. Let $\mathcal{T}_{a,b,c,d,e,f}$ and $\mathcal{T}'_{a,b,c,d,e,f}$ be words over the generators of $\mathcal{P}_N$ containing no generators of the form $(H_{[g,h]} H_{[i,j]})$, such that $[\mathcal{T}'_{a,b,c,d,e,f}] = [\mathcal{T}_{a,b,c,d,e,f}]^{-1}$, and either $[\mathcal{T}_{a,b,c,d,e,f}] = \tau$ or $[\mathcal{T}_{a,b,c,d,e,f}] = \tau \circ [(-1)_{[v]}]$, where $v \notin \{0, 1, 3, 2, 4, 5\}$ and $\tau$ is a permutation matrix mapping respectively $|0\rangle, |1\rangle, |3\rangle, |2\rangle, |4\rangle, |5\rangle$ to $|a\rangle, |b\rangle, |c\rangle, |d\rangle, |e\rangle, |f\rangle$.<fnref n="13" /> We have

$$
\begin{align*}
(H_{[a,b]} H_{[c,d]}) (H_{[c,d]} H_{[e,f]}) & \\
\stackrel{(65)}{\approx} & \Sigma_{a,b,c,d} \circ (H_{[0,1]} H_{[3,2]}) \circ \Sigma'_{a,b,c,d} \circ \Sigma_{c,d,e,f} \circ (H_{[0,1]} H_{[3,2]}) \circ \Sigma'_{c,d,e,f} \\
\text{Corollary A.7} & \approx \mathcal{T}_{a,b,c,d,e,f} \circ (H_{[0,1]} H_{[3,2]}) \circ \mathcal{T}'_{a,b,c,d,e,f} \circ \mathcal{T}_{a,b,c,d,e,f} \circ \Sigma_{3,2,4,5} \circ (H_{[0,1]} H_{[3,2]}) \circ \Sigma'_{3,2,4,5} \circ \mathcal{T}'_{a,b,c,d,e,f} \\
\stackrel{(65)}{\approx} & \mathcal{T}_{a,b,c,d,e,f} \circ (H_{[0,1]} H_{[3,2]}) \circ \mathcal{T}'_{a,b,c,d,e,f} \circ \mathcal{T}_{a,b,c,d,e,f} \circ (H_{[3,2]} H_{[4,5]}) \circ \mathcal{T}'_{a,b,c,d,e,f}
\end{align*}
$$

<footnote n="13">¹³ $\mathcal{T}_{a,b,c,d,e,f}$ and $\mathcal{T}'_{a,b,c,d,e,f}$ (as well as $\Xi_{i,j|c,d}^{(a,b,e,f)}$ and $\Xi_{i,j|c,d}^{(a,b,e,f)'}$ below) can be constructed in a similar way to $\Sigma_{a,b,c,d}$ and $\Sigma_{a,b,c,d}$ (see Appendix E.1).</footnote>
$$
\begin{align*}
\text{Corollary A.5} &\approx \mathcal{T}_{a,b,c,d,e,f} \circ (H_{[0,1]}H_{[3,2]}) (H_{[3,2]}H_{[4,5]}) \circ \mathcal{T}'_{a,b,c,d,e,f} \\
(43) &\approx \mathcal{T}_{a,b,c,d,e,f} \circ (X_{[0,3]}X_{[1,2]}) (H_{[3,2]}H_{[4,5]}) (X_{[0,3]}X_{[1,2]}) \circ \mathcal{T}'_{a,b,c,d,e,f} \\
(65) &\approx \mathcal{T}_{a,b,c,d,e,f} \circ (X_{[0,3]}X_{[1,2]}) \circ \Sigma_{3,2,4,5} \circ (H_{[0,1]}H_{[3,2]}) \circ \Sigma'_{3,2,4,5} \circ (X_{[0,3]}X_{[1,2]}) \circ \mathcal{T}'_{a,b,c,d,e,f} \\
\text{Corollary A.7} &\approx \Sigma_{a,b,e,f} \circ (H_{[0,1]}H_{[3,2]}) \circ \Sigma'_{a,b,e,f} \\
(65) &\approx (H_{[a,b]}H_{[e,f]})
\end{align*}
$$

Then, we consider the case where $|\{a, b, e, f\}| \le 3$ and $\{c, d\} \cap \{a, b, e, f\} = \emptyset$. Let $\Xi_{i,j|c,d}^{(a,b,e,f)}$ and $\Xi_{i,j|c,d}^{(a,b,e,f)'}$ be words over the generators of $\mathcal{P}_N$ containing no generators of the form $(H_{[g,h]}H_{[k,l]})$, such that $[\Xi_{i,j|c,d}^{(a,b,e,f)'}] = [\Xi_{i,j|c,d}^{(a,b,e,f)}]^{-1}$, and either $[\Xi_{i,j|c,d}^{(a,b,e,f)}] = \xi$ or $[\Xi_{i,j|c,d}^{(a,b,e,f)}] = \xi \circ [(-1)_{[\rho]}]$, where$^{14} \rho \notin \{a, b, c, d, e, f, i, j\}$ and $\xi$ is a permutation matrix mapping respectively $|a\rangle, |b\rangle, |e\rangle, |f\rangle, |i\rangle, |j\rangle$ to $|a\rangle, |b\rangle, |e\rangle, |f\rangle, |c\rangle, |d\rangle$. We have

$$
\begin{align*}
(H_{[a,b]} H_{[c,d]})(H_{[c,d]} H_{[e,f]}) & \tag{65} \\
\Sigma_{a,b,c,d} \circ (H_{[0,1]} H_{[3,2]}) \circ \Sigma'_{a,b,c,d} \circ \Sigma_{c,d,e,f} \circ (H_{[0,1]} H_{[3,2]}) \circ \Sigma'_{c,d,e,f} &
\end{align*}
$$

$$
\text{Corollary A.7} \quad \approx \quad \Xi_{i,j|c,d}^{(a,b,e,f)} \circ \Sigma_{a,b,i,j} \circ (H_{[0,1]}H_{[3,2]}) \circ \Sigma'_{a,b,i,j} \circ \Xi_{i,j|c,d}^{(a,b,e,f)'} \circ \Xi_{i,j|c,d}^{(a,b,e,f)} \circ \Sigma_{i,j,e,f} \circ (H_{[0,1]}H_{[3,2]}) \circ \Sigma'_{i,j,e,f} \circ \Xi_{i,j|c,d}^{(a,b,e,f)'}
$$

$$
(65) \quad \approx \quad \Xi_{i,j|c,d}^{(a,b,e,f)} \circ (H_{[a,b]}H_{[i,j]}) \circ \Xi_{i,j|c,d}^{(a,b,e,f)'} \circ \Xi_{i,j|c,d}^{(a,b,e,f)} \circ (H_{[i,j]}H_{[e,f]}) \circ \Xi_{i,j|c,d}^{(a,b,e,f)'}
$$

$$
\text{Corollary A.5} \quad \approx \quad \Xi_{i,j|c,d}^{(a,b,e,f)} \circ (H_{[a,b]}H_{[i,j]}) (H_{[i,j]}H_{[e,f]}) \circ \Xi_{i,j|c,d}^{(a,b,e,f)'}
$$

$$
(42) \quad \approx \quad \Xi_{i,j|c,d}^{(a,b,e,f)} \circ (H_{[a,b]}H_{[e,f]}) \circ \Xi_{i,j|c,d}^{(a,b,e,f)'}
$$

$$
(65) \quad \approx \quad \Xi_{i,j|c,d}^{(a,b,e,f)} \circ \Sigma_{a,b,e,f} \circ (H_{[0,1]}H_{[3,2]}) \circ \Sigma'_{a,b,e,f} \circ \Xi_{i,j|c,d}^{(a,b,e,f)'}
$$

$$
\text{Corollary A.7} \quad \approx \quad \Sigma_{a,b,e,f} \circ (H_{[0,1]}H_{[3,2]}) \circ \Sigma'_{a,b,e,f}
$$

$$
(65) \quad \approx \quad (H_{[a,b]} H_{[e,f]})
$$

We have treated all the cases where $\{c, d\} \cap \{a, b, e, f\} = \emptyset$, therefore it remains to treat the case where $\{c, d\} \cap \{a, b, e, f\} \neq \emptyset$. In this case, let $i, j$, with $i < j$, be the two smallest elements of $\{0, ..., N-1\} \setminus \{a, b, c, d, e, f\}$. We have

$$
\begin{align*}
(H_{[a,b]} H_{[c,d]})(H_{[c,d]} H_{[e,f]}) & \\
\text{previous cases} & \approx (H_{[a,b]} H_{[i,j]}) (H_{[i,j]} H_{[c,d]}) (H_{[c,d]} H_{[i,j]}) (H_{[i,j]} H_{[e,f]}) \\
\text{previous cases} & \approx (H_{[a,b]} H_{[i,j]}) (H_{[i,j]} H_{[i,j]}) (H_{[i,j]} H_{[e,f]}) \\
(70) & \approx (H_{[a,b]} H_{[i,j]}) (H_{[i,j]} H_{[e,f]}) \\
\text{previous cases} & \approx (H_{[a,b]} H_{[e,f]})
\end{align*}
$$

Proof of Equation (72): Let $c, d$ be such that $c \neq d$ and $\{a, b\} \cap \{c, d\} = \emptyset$. Such indices exist since $N \ge 4$. We have

$$
(H_{[a,b]} H_{[b,a]}) \stackrel{(71)}{\approx} (H_{[a,b]} H_{[c,d]})(H_{[c,d]} H_{[b,a]})
$$

$$
(66) \quad \approx \quad (H_{[a,b]} H_{[c,d]})(H_{[b,a]} H_{[c,d]})
$$

$$
(67) \quad \approx \quad (H_{[a,b]} H_{[c,d]})(H_{[a,b]} H_{[c,d]})(-1)_{[b]} X_{[a,b]}
$$

$$
(66) \quad \approx \quad (H_{[a,b]} H_{[c,d]})(H_{[c,d]} H_{[a,b]})(-1)_{[b]} X_{[a,b]}
$$

$$
(71) \quad \approx \quad (H_{[a,b]} H_{[a,b]})(-1)_{[b]} X_{[a,b]}
$$

$$
(70) \quad \approx \quad (-1)_{[b]} X_{[a,b]}
$$

<footnote>$^{14}$ $\rho$ exists since $N \ge 8$, and $|\{a, b, e, f\}| \le 3$ implies $|\{a, b, c, d, e, f, i, j\}| < 8$.</footnote>
Proof of Equation (73):

$$ \begin{align*} (H_{[a,b]}H_{[c,d]})(H_{[e,f]}H_{[g,h]}) & \stackrel{(70)}{\approx} (H_{[a,b]}H_{[c,d]})(H_{[c,d]}H_{[c,d]})(H_{[e,f]}H_{[g,h]}) \\ & \stackrel{(71)}{\approx} (H_{[a,b]}H_{[c,d]})(H_{[c,d]}H_{[e,f]})(H_{[e,f]}H_{[c,d]})(H_{[e,f]}H_{[g,h]}) \\ & \stackrel{(66)}{\approx} (H_{[a,b]}H_{[c,d]})(H_{[c,d]}H_{[e,f]})(H_{[c,d]}H_{[e,f]})(H_{[e,f]}H_{[g,h]}) \\ & \stackrel{(71)}{\approx} (H_{[a,b]}H_{[e,f]})(H_{[c,d]}H_{[g,h]}) \end{align*} $$

Proof of Equation (74):

$$ \begin{align*} ((-1)_{[a]}X_{[a,b]})(H_{[c,d]}H_{[e,f]}) & \stackrel{(33)}{\approx} ((-1)_{[a]}X_{[b,a]})(H_{[c,d]}H_{[e,f]}) \\ & \stackrel{(72)}{\approx} (H_{[b,a]}H_{[a,b]})(H_{[c,d]}H_{[e,f]}) \\ & \stackrel{(73)}{\approx} (H_{[b,a]}H_{[c,d]})(H_{[a,b]}H_{[e,f]}) \\ & \stackrel{(66)}{\approx} (H_{[c,d]}H_{[b,a]})(H_{[e,f]}H_{[a,b]}) \\ & \stackrel{(73)}{\approx} (H_{[c,d]}H_{[e,f]})(H_{[b,a]}H_{[a,b]}) \\ & \stackrel{(72)}{\approx} (H_{[c,d]}H_{[e,f]})((-1)_{[a]}X_{[b,a]}) \\ & \stackrel{(33)}{\approx} (H_{[c,d]}H_{[e,f]})((-1)_{[a]}X_{[a,b]}) \end{align*} $$

Proof of Equation (75):

$$ \begin{align*} ((-1)_{[a]}X_{[a,b]})(H_{[a,b]}H_{[c,d]}) & \stackrel{(33)(72)}{\approx} (H_{[b,a]}H_{[a,b]})(H_{[a,b]}H_{[c,d]}) \\ & \stackrel{(71)}{\approx} (H_{[b,a]}H_{[c,d]}) \\ & \stackrel{(67)}{\approx} (H_{[a,b]}H_{[c,d]})((-1)_{[b]}X_{[a,b]}) \end{align*} $$

Proof of Equations (76) and (77):

• If $e - a = 1$, then Equations (76) and (77) are just instances of Equations (68) and (69) respectively.

• If $a - e = 1$, then we proceed as follows.<fnref n="15" /> For Equation (76):

$$ \begin{align*} & ((-1)_{[a]}X_{[a,a-1]}) (H_{[a,b]}H_{[c,d]}) \\ \text{Corollary A.5} \quad & \stackrel{(69)}{\approx} ((-1)_{[a-1]}X_{[a-1,a]})^3 (H_{[a,b]}H_{[c,d]}) \\ & \stackrel{(68)}{\approx} ((-1)_{[a-1]}X_{[a-1,a]})^2 (H_{[a-1,b]}H_{[c,d]}) ((-1)_{[a-1]}X_{[a-1,b]}) ((-1)_{[a-1]}X_{[a-1,a]}) \\ & \stackrel{(69)}{\approx} (H_{[a-1,b]}H_{[c,d]}) ((-1)_{[a-1]}X_{[a-1,b]}) ((-1)_{[a-1]}X_{[a-1,a]}) ((-1)_{[a-1]}X_{[a-1,b]}) ((-1)_{[a-1]}X_{[a-1,a]}) \\ \text{Corollary A.5} \quad & \stackrel{(68)}{\approx} (H_{[a-1,b]}H_{[c,d]}) ((-1)_{[a-1]}X_{[a-1,a]}) \end{align*} $$

and for Equation (77):

$$ \begin{align*} & ((-1)_{[a]}X_{[a,a-1]}) (H_{[a-1,b]}H_{[c,d]}) \\ \text{Corollary A.5} \quad & \stackrel{(68)}{\approx} ((-1)_{[a-1]}X_{[a-1,a]})^3 (H_{[a-1,b]}H_{[c,d]}) \\ & \stackrel{(69)}{\approx} ((-1)_{[a-1]}X_{[a-1,a]})^2 (H_{[a,b]}H_{[c,d]}) ((-1)_{[a-1]}X_{[a-1,a]}) \\ & \stackrel{(68)}{\approx} (H_{[a-1,b]}H_{[c,d]}) ((-1)_{[a-1]}X_{[a-1,b]}) ((-1)_{[a-1]}X_{[a-1,a]})^2 \\ \text{Corollary A.5} \quad & \stackrel{(69)}{\approx} (H_{[a,b]}H_{[c,d]}) ((-1)_{[a-1]}X_{[a-1,a]}) ((-1)_{[a-1]}X_{[a-1,b]}) ((-1)_{[a-1]}X_{[a-1,a]})^2 \\ & \stackrel{(68)}{\approx} (H_{[a,b]}H_{[c,d]}) ((-1)_{[a-1]}X_{[a-1,a]}) ((-1)_{[a-1]}X_{[a-1,b]}) ((-1)_{[a-1]}X_{[a-1,a]})^2 \end{align*} $$

<footnote n="15">¹⁵Notice that the derivations are identical to the case below where $e < a$ and $\{a, ..., e\} \cap \{b, c, d\} = \emptyset$</footnote>
• If $a < e$ and $\{a, ..., e\} \cap \{b, c, d\} = \emptyset$, then we proceed by finite induction on $e - a$. The base case has already been treated. The induction step for Equation (76) is as follows:

$$ ((-1)_{[a]} X_{[a,e]} (H_{[a,b]} H_{[c,d]}) \stackrel{(49)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]} ((-1)_{[a+1]} X_{[a+1,e]} ((-1)_{[a]} X_{[a,a+1]} (H_{[a,b]} H_{[c,d]} \\ \text{induction hypothesis} \approx ((-1)_{[a+1]} X_{[a,a+1]} ((-1)_{[a+1]} X_{[a+1,e]} (H_{[a+1,b]} H_{[c,d]} ((-1)_{[a]} X_{[a,a+1]} \\ \text{induction hypothesis} \approx ((-1)_{[a+1]} X_{[a,a+1]} (H_{[e,b]} H_{[c,d]} ((-1)_{[a+1]} X_{[a+1,e]} ((-1)_{[a]} X_{[a,a+1]} \\ \stackrel{(74)}{\approx} (H_{[e,b]} H_{[c,d]} ((-1)_{[a+1]} X_{[a,a+1]} ((-1)_{[a+1]} X_{[a+1,e]} ((-1)_{[a]} X_{[a,a+1]} \\ \stackrel{(49)}{\approx} (H_{[e,b]} H_{[c,d]} ((-1)_{[a]} X_{[a,e]} \\ \text{induction hypothesis} \approx ((-1)_{[a]} X_{[a,e]} (H_{[e,b]} H_{[c,d]} \\ \stackrel{(49)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]} ((-1)_{[a+1]} X_{[a+1,e]} ((-1)_{[a]} X_{[a,a+1]} (H_{[e,b]} H_{[c,d]} ((-1)_{[a]} X_{[a,a+1]} \\ \stackrel{(74)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]} ((-1)_{[a+1]} X_{[a+1,e]} (H_{[e,b]} H_{[c,d]} ((-1)_{[a]} X_{[a,a+1]} \\ \stackrel{(49)}{\approx} (H_{[e,b]} H_{[c,d]} ((-1)_{[a]} X_{[a,b]} ((-1)_{[a]} X_{[a,e]} \\ \text{case } a-e=1 \\ \text{for } (76)} \approx (H_{[a,b]} H_{[c,d]} ((-1)_{[a+1]} X_{[a,a+1]} ((-1)_{[a+1]} X_{[a+1,b]} ((-1)_{[a+1]} X_{[a+1,e]} ((-1)_{[a]} X_{[a,a+1]} \\ \stackrel{(51)}{\approx} (H_{[a,b]} H_{[c,d]} ((-1)_{[a+1]} X_{[a,a+1]} ((-1)_{[a+1]} X_{[a+1,b]} ((-1)_{[a]} X_{[a,a+1]} ((-1)_{[a+1]} X_{[a,a+1]} ((-1)_{[a]} X_{[a,a+1]} \\ \stackrel{(49)}{\approx} (H_{[e,b]} H_{[c,d]} ((-1)_{[a]} X_{[a,b]} ((-1)_{[a]} X_{[a,e]} \\ \text{If } e < a \text{ and } \{e, ..., a\} \cap \{b, c, d\} = \emptyset, \text{ then we deduce the result from the previous case, as follows. For Equation (76):} \\ \text{Corollary A.5} \stackrel{\text{Corollary A.5}}{\approx} ((-1)_{[a]} X_{[a,e]} (H_{[a,b]} H_{[c,d]} \\ \text{case } a < e \\ \text{of } (77)} \approx ((-1)_{[e]} X_{[e,a]})^3 (H_{[a,b]} H_{[c,d]} \\ \text{case } a < e \\ \text{of } (76)} \approx ((-1)_{[e]} X_{[e,a]})^2 (H_{[e,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{case } a < e \\ \text{of } (77)} \approx ((-1)_{[e]} X_{[e,a]} (H_{[a,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,a]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{case } a < e \\ \text{of } (77)} \approx (H_{[e,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} ((-1)_{[e]} X_{[e,a]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{Corollary A.5} \stackrel{\text{Corollary A.5}}{\approx} (H_{[e,b]} H_{[c,d]} ((-1)_{[a]} X_{[a,e]} \\ \text{and for Equation (77):} \\ ((-1)_{[a]} X_{[a,e]} (H_{[e,b]} H_{[c,d]} \\ \text{Corollary A.5} \stackrel{\text{Corollary A.5}}{\approx} ((-1)_{[e]} X_{[e,a]})^3 (H_{[e,b]} H_{[c,d]} \\ \text{case } a < e \\ \text{of } (76)} \approx ((-1)_{[e]} X_{[e,a]})^2 (H_{[a,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,b]} \\ \text{case } a < e \\ \text{of } (77)} \approx ((-1)_{[e]} X_{[e,a]} (H_{[e,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{case } a < e \\ \text{of } (76)} \approx (H_{[a,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,a]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{Corollary A.5} \stackrel{\text{Corollary A.5}}{\approx} (H_{[a,b]} H_{[c,d]} ((-1)_{[a]} X_{[a,b]} ((-1)_{[a]} X_{[a,e]} \\ \text{To treat the general case, it will be useful to prove the following auxiliary equation:} \\ ((-1)_{[a]} X_{[a,c]} (H_{[a,b]} H_{[c,d]}) \approx (H_{[c,b]} H_{[a,d]} ((-1)_{[a]} X_{[a,d]} ((-1)_{[a]} X_{[a,c]} \\ \text{where } [a, c] \text{ denotes the interval } \{a, ..., c\} \text{ if } a < c, \text{ or } \{c, ..., a\} \text{ if } c < a.} \\ |\{a, b, c, d\}| = 4, \ [a, c] \cap \{b, d\} = \emptyset, \ |a - c| \le N - 4 \quad (78) \end{aligned} $$

and the induction step for Equation (77) is as follows:

$$ ((-1)_{[a]} X_{[a,e]} (H_{[e,b]} H_{[c,d]}) \\ \stackrel{(49)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]} ((-1)_{[a+1]} X_{[a+1,e]} ((-1)_{[a]} X_{[a,a+1]} (H_{[e,b]} H_{[c,d]} \\ \stackrel{(74)}{\approx} ((-1)_{[a+1]} X_{[a,a+1]} ((-1)_{[a+1]} X_{[a+1,e]} (H_{[e,b]} H_{[c,d]} ((-1)_{[a]} X_{[a,a+1]} \\ \text{induction hypothesis} \approx ((-1)_{[a+1]} X_{[a,a+1]} (H_{[a+1,b]} H_{[c,d]} ((-1)_{[a+1]} X_{[a+1,b]} ((-1)_{[a+1]} X_{[a+1,e]} ((-1)_{[a]} X_{[a,a+1]} \\ \text{case } a-e=1 \\ \text{for } (76)} \approx (H_{[a,b]} H_{[c,d]} ((-1)_{[a+1]} X_{[a,a+1]} ((-1)_{[a+1]} X_{[a+1,b]} ((-1)_{[a+1]} X_{[a+1,e]} ((-1)_{[a]} X_{[a,a+1]} \\ \stackrel{(51)}{\approx} (H_{[a,b]} H_{[c,d]} ((-1)_{[a+1]} X_{[a,a+1]} ((-1)_{[a+1]} X_{[a+1,b]} ((-1)_{[a]} X_{[a,a+1]} ((-1)_{[a+1]} X_{[a,a+1]} ((-1)_{[a]} X_{[a,a+1]} \\ \stackrel{(49)}{\approx} (H_{[e,b]} H_{[c,d]} ((-1)_{[a]} X_{[a,b]} ((-1)_{[a]} X_{[a,e]} \\ \text{If } e < a \text{ and } \{e, ..., a\} \cap \{b, c, d\} = \emptyset, \text{ then we deduce the result from the previous case, as follows. For Equation (76):} \\ \text{Corollary A.5} \stackrel{\text{Corollary A.5}}{\approx} ((-1)_{[a]} X_{[a,e]} (H_{[a,b]} H_{[c,d]} \\ \text{case } a < e \\ \text{of } (77)} \approx ((-1)_{[e]} X_{[e,a]})^3 (H_{[a,b]} H_{[c,d]} \\ \text{case } a < e \\ \text{of } (76)} \approx ((-1)_{[e]} X_{[e,a]})^2 (H_{[e,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{case } a < e \\ \text{of } (77)} \approx ((-1)_{[e]} X_{[e,a]} (H_{[a,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,a]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{case } a < e \\ \text{of } (77)} \approx (H_{[a,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,a]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{Corollary A.5} \stackrel{\text{Corollary A.5}}{\approx} (H_{[e,b]} H_{[c,d]} ((-1)_{[a]} X_{[a,e]} \\ \text{and for Equation (77):} \\ ((-1)_{[a]} X_{[a,e]} (H_{[e,b]} H_{[c,d]} \\ \text{Corollary A.5} \stackrel{\text{Corollary A.5}}{\approx} ((-1)_{[e]} X_{[e,a]})^3 (H_{[e,b]} H_{[c,d]} \\ \text{case } a < e \\ \text{of } (76)} \approx ((-1)_{[e]} X_{[e,a]})^2 (H_{[a,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,b]} \\ \text{case } a < e \\ \text{of } (77)} \approx ((-1)_{[e]} X_{[e,a]} (H_{[e,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{case } a < e \\ \text{of } (76)} \approx (H_{[a,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,a]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{Corollary A.5} \stackrel{\text{Corollary A.5}}{\approx} (H_{[a,b]} H_{[c,d]} ((-1)_{[a]} X_{[a,b]} ((-1)_{[a]} X_{[a,e]} \\ \text{To treat the general case, it will be useful to prove the following auxiliary equation:} \\ ((-1)_{[a]} X_{[a,c]} (H_{[a,b]} H_{[c,d]}) \approx (H_{[c,b]} H_{[a,d]} ((-1)_{[a]} X_{[a,d]} ((-1)_{[a]} X_{[a,c]} \\ \text{where } [a, c] \text{ denotes the interval } \{a, ..., c\} \text{ if } a < c, \text{ or } \{c, ..., a\} \text{ if } c < a.} \\ |\{a, b, c, d\}| = 4, \ [a, c] \cap \{b, d\} = \emptyset, \ |a - c| \le N - 4 \quad (78) \end{aligned} $$

$$ ((-1)_{[a]} X_{[a,e]} (H_{[a,b]} H_{[c,d]}) \\ \stackrel{\text{Corollary A.5}}{\approx} ((-1)_{[e]} X_{[e,a]})^3 (H_{[a,b]} H_{[c,d]}) \\ \text{case } a < e \\ \text{of } (77)} \approx ((-1)_{[e]} X_{[e,a]})^2 (H_{[e,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{case } a < e \\ \text{of } (76)} \approx ((-1)_{[e]} X_{[e,a]} (H_{[a,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,a]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{case } a < e \\ \text{of } (77)} \approx (H_{[e,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} ((-1)_{[e]} X_{[e,a]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{Corollary A.5} \stackrel{\text{Corollary A.5}}{\approx} (H_{[e,b]} H_{[c,d]} ((-1)_{[a]} X_{[a,e]} \\ \text{and for Equation (77):} \\ ((-1)_{[a]} X_{[a,e]} (H_{[e,b]} H_{[c,d]}) \\ \text{Corollary A.5} \stackrel{\text{Corollary A.5}}{\approx} ((-1)_{[e]} X_{[e,a]})^3 (H_{[e,b]} H_{[c,d]} \\ \text{case } a < e \\ \text{of } (76)} \approx ((-1)_{[e]} X_{[e,a]})^2 (H_{[a,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,b]} \\ \text{case } a < e \\ \text{of } (77)} \approx ((-1)_{[e]} X_{[e,a]} (H_{[a,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,a]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{case } a < e \\ \text{of } (76)} \approx (H_{[a,b]} H_{[c,d]} ((-1)_{[e]} X_{[e,a]} ((-1)_{[e]} X_{[e,b]} ((-1)_{[e]} X_{[e,a]} \\ \text{Corollary A.5} \stackrel{\text{Corollary A.5}}{\approx} (H_{[a,b]} H_{[c,d]} ((-1)_{[a]} X_{[a,b]} ((-1)_{[a]} X_{[a,e]} \\ \text{To treat the general case, it will be useful to prove the following auxiliary equation:} \\ ((-1)_{[a]} X_{[a,c]} (H_{[a,b]} H_{[c,d]}) \approx (H_{[c,b]} H_{[a,d]} ((-1)_{[a]} X_{[a,d]} ((-1)_{[a]} X_{[a,c]} \\ \text{where } [a, c] \text{ denotes the interval } \{a, ..., c\} \text{ if } a < c, \text{ or } \{c, ..., a\} \text{ if } c < a.} \\ |\{a, b, c, d\}| = 4, \ [a, c] \cap \{b, d\} = \emptyset, \ |a - c| \le N - 4 \quad (78) \end{aligned} $$
To prove Equation (78), let $f_b, g_b, f_d, g_d \in \{0, ..., N-1\} \setminus [a, c]$ be such that $f_b \neq g_b, f_d \neq g_d, f_b, g_b \neq d$ and $f_d, g_d \neq b$. Such indices exist thanks to the condition $|a-c| \le N-4$. One has the following derivation (where the references to Equations (76) and (77) refer to their restricted versions proven above):

$$ \begin{align*} & ((-1)_{[a]} X_{[a,c]})(H_{[a,b]} H_{[c,d]}) \\ & \stackrel{(71)}{\approx} ((-1)_{[a]} X_{[a,c]})(H_{[a,b]} H_{[fd,g_d]})(H_{[fd,g_d]} H_{[fb,g_b]})(H_{[fb,g_b]} H_{[c,d]}) \\ & \stackrel{(66)}{\approx} ((-1)_{[a]} X_{[a,c]})(H_{[a,b]} H_{[fd,g_d]})(H_{[fd,g_d]} H_{[fb,g_b]})(H_{[c,d]} H_{[fb,g_b]}) \\ & \stackrel{(76)}{\approx} (H_{[c,b]} H_{[fd,g_d]})((-1)_{[a]} X_{[a,c]})(H_{[fd,g_d]} H_{[fb,g_b]})(H_{[c,d]} H_{[fb,g_b]}) \\ & \stackrel{(74)}{\approx} (H_{[c,b]} H_{[fd,g_d]})(H_{[fd,g_d]} H_{[fb,g_b]})((-1)_{[a]} X_{[a,c]})(H_{[c,d]} H_{[fb,g_b]}) \\ & \stackrel{(77)}{\approx} (H_{[c,b]} H_{[fd,g_d]})(H_{[fd,g_d]} H_{[fb,g_b]})(H_{[a,d]} H_{[fb,g_b]})((-1)_{[a]} X_{[a,d]})((-1)_{[a]} X_{[a,c]}) \\ & \stackrel{(66)}{\approx} (H_{[c,b]} H_{[fd,g_d]})(H_{[fd,g_d]} H_{[fb,g_b]})(H_{[fb,g_b]} H_{[a,d]})((-1)_{[a]} X_{[a,d]})((-1)_{[a]} X_{[a,c]}) \\ & \stackrel{(71)}{\approx} (H_{[c,b]} H_{[a,d]})((-1)_{[a]} X_{[a,d]})((-1)_{[a]} X_{[a,c]}) \end{align*} $$

Now, to prove Equations (76) and (77) in the general case, let $i_0 < \dots < i_k$ be such that $1 \le k \le 4$, $\{i_0, i_k\} = \{a, e\}$ and $\{i_1, i_2, \dots, i_{k-1}\} = [a, e] \cap \{b, c, d\}$ (namely, $i_1, \dots, i_{k-1}$ are the indices among $b, c, d$ that are between $a$ and $e$). By Corollary A.5, one has

$$ ((-1)_{[a]} X_{[a,e]}) \approx ((-1)_{[i_1]} X_{[i_0, i_1]}) \cdots ((-1)_{[i_{k-1}]} X_{[i_{k-2}, i_{k-1}]}) \circ ((-1)_{[i_{k-1}]} X_{[i_{k-1}, i_k]}) \circ ((-1)_{[i_{k-2}]} X_{[i_{k-2}, i_{k-1}]}) \cdots ((-1)_{[i_0]} X_{[i_0, i_1]}) \quad (C) \\ \text{and for every } j \in \{0, \dots, k-2\}, $$

$$ ((-1)_{[i_{j+1}]} X_{[i_j, i_{j+1}]}) \approx ((-1)_{[i_j]} X_{[i_j, i_{j+1}]})^3 \quad (D) $$

Every pair of indices $i_j, i_{j+1}$ is such that the other three indices are outside of the interval $[i_j, i_{j+1}]$, which also implies that $|i_j - i_{j+1}| \le N-4$. Therefore, starting from $((-1)_{[a]} X_{[a,e]})(H_{[a,b]} H_{[c,d]})$ (resp. $((-1)_{[a]} X_{[a,e]})(H_{[e,b]} H_{[c,d]})$) and after decomposing $((-1)_{[a]} X_{[a,e]})$ according to Equations (C) and (D), we can use the already proven cases of Equations (76) and (77) together with Equation (78) and Equation (75) (all four being potentially used in combination with Equations (66) and (67)) to push all the $((-1)_{[i_j]} X_{[i_j, i_{j+1}]})$ to the right. One can check that we get $(H_{[e,b]} H_{[c,d]})$ (resp. $(H_{[a,b]} H_{[c,d]})$), multiplied on its right by a product of generators of the form $((-1)_{[f]} X_{[f,g]})$ and $((-1)_{[g]} X_{[f,g]})$. By Corollary A.5, we can rewrite this product into $((-1)_{[a]} X_{[a,e]})$ (resp. into $((-1)_{[a]} X_{[a,b]})((-1)_{[a]} X_{[a,e]})$), which finishes the proof. $\square$

**Corollary A.9.** Any word over the generators of $\mathcal{P}_N$ containing at most one generator of the form $(H_{[a,b]} H_{[c,d]})$ is $\approx$-equivalent to a unique word, called its normal form, of the form

$$ P \quad \text{or} \quad (H_{[a,b]} H_{[c,d]}) P $$

with $a < b; c < d; \{a, b\} \neq \{c, d\}$; if $\{a, b\} \cap \{c, d\} = \emptyset$ then $a < c$; and $P$ being of the form (B). Moreover, two words with the same semantics have the same normal form.

**PROOF.**

*Uniqueness given the semantics:* By analysis of the effect of $[[P]]$ and $[[((H_{[a,b]} H_{[c,d]}) P]]$ on the canonical basis of $\mathbb{C}^N$, noting that $[[P]]$ is a permutation matrix up to the signs of its entries.

*Existence:* If the word does not contain any $(H_{[a,b]} H_{[c,d]})$, then the result is just a consequence of Lemma A.4.

If the word contains a single $(H_{[a,b]} H_{[c,d]})$, then it is of the form $w = p_1 (H_{[a,b]} H_{[c,d]}) p_2$ where $p_1, p_2$ are two words containing no generators of the form $(H_{[e,f]} H_{[g,h]})$. As we have shown in the proof of Lemma A.4, the word $p_1$ can be written as a product of generators of the form $((-1)_{[j]} X_{[j,j+1]})$, therefore it suffices to prove the result for $p_1 = ((-1)_{[j]} X_{[j,j+1]})$. For practical purposes, we will prove the result in a slightly more general case, namely for $p_1 = ((-1)_{[j]} X_{[j,k]}$).

Let $e, f \in \{0, ..., N-1\}$ such that $e \neq f$ and $e, f \notin \{a, b, c, d, j, k\}$. Such indices exist since $N \ge 8$. One has

$$ \begin{align*} ((-1)_{[j]} X_{[j,k]})(H_{[a,b]} H_{[c,d]}) &\stackrel{(71)}{\approx} ((-1)_{[j]} X_{[j,k]})(H_{[a,b]} H_{[e,f]})(H_{[e,f]} H_{[c,d]}) \\ &\stackrel{(66)}{\approx} ((-1)_{[j]} X_{[j,k]})(H_{[a,b]} H_{[e,f]})(H_{[c,d]} H_{[e,f]}) \end{align*} $$

Let us first consider $((-1)_{[j]} X_{[j,k]})(H_{[a,b]} H_{[e,f]})$. There are seven cases:
* If $a = j$ and $b = k$, then

$$ ((-1)_{[j]} X_{[j,k]})(H_{[a,b]} H_{[e,f]}) \stackrel{(75)}{\approx} (H_{[a,b]} H_{[e,f]})((-1)_{[k]} X_{[j,k]}) $$

$$ \stackrel{(33)}{\approx} (H_{[a,b]} H_{[e,f]})((-1)_{[k]} X_{[k,j]}) $$

* If $a = k$ and $b = j$, then

$$ ((-1)_{[j]} X_{[j,k]})(H_{[a,b]} H_{[e,f]}) $$

$$ = ((-1)_{[j]} X_{[j,k]})(H_{[k,j]} H_{[e,f]}) $$

$$ \stackrel{(67)}{\approx} ((-1)_{[j]} X_{[j,k]})(H_{[j,k]} H_{[e,f]})((-1)_{[k]} X_{[j,k]}) $$

$$ \stackrel{(75)}{\approx} (H_{[j,k]} H_{[e,f]})((-1)_{[k]} X_{[j,k]})((-1)_{[k]} X_{[j,k]}) $$

$$ \stackrel{(53)(21)}{\approx} (H_{[j,k]} H_{[e,f]})((-1)_{[j]}(-1)_{[k]})((-1)_{[j]} X_{[j,k]})((-1)_{[k]}(-1)_{[j]})((-1)_{[j]} X_{[j,k]}) $$

$$ \stackrel{(56)}{\approx} (H_{[j,k]} H_{[e,f]})((-1)_{[j]} X_{[j,k]})((-1)_{[j]}(-1)_{[k]})((-1)_{[k]}(-1)_{[j]})((-1)_{[j]} X_{[j,k]}) $$

$$ \stackrel{(22)(20)}{\approx} (H_{[j,k]} H_{[e,f]})((-1)_{[j]} X_{[j,k]})^2 $$

* If $a = j$ and $b \neq k$, then

$$ ((-1)_{[j]} X_{[j,k]})(H_{[a,b]} H_{[e,f]}) = ((-1)_{[j]} X_{[j,k]})(H_{[j,b]} H_{[e,f]}) $$

$$ \stackrel{(76)}{\approx} (H_{[k,b]} H_{[e,f]})((-1)_{[j]} X_{[j,k]}) $$

* If $a = k$ and $b \neq j$, then

$$ ((-1)_{[j]} X_{[j,k]})(H_{[a,b]} H_{[e,f]}) = ((-1)_{[j]} X_{[j,k]})(H_{[k,b]} H_{[e,f]}) $$

$$ \stackrel{(77)}{\approx} (H_{[j,b]} H_{[e,f]})((-1)_{[j]} X_{[j,b]})((-1)_{[j]} X_{[j,k]}) $$

* If $b = j$ and $a \neq k$, then

$$ ((-1)_{[j]} X_{[j,k]})(H_{[a,b]} H_{[e,f]}) $$

$$ = ((-1)_{[j]} X_{[j,k]})(H_{[a,j]} H_{[e,f]}) $$

$$ \stackrel{(67)}{\approx} ((-1)_{[j]} X_{[j,k]})(H_{[j,a]} H_{[e,f]})((-1)_{[a]} X_{[j,a]}) $$

$$ \stackrel{(76)}{\approx} (H_{[k,a]} H_{[e,f]})((-1)_{[j]} X_{[j,k]})((-1)_{[a]} X_{[j,a]}) $$

$$ \stackrel{(67)}{\approx} (H_{[a,k]} H_{[e,f]})((-1)_{[k]} X_{[a,k]})((-1)_{[j]} X_{[j,k]})((-1)_{[a]} X_{[j,a]}) $$

$$ \stackrel{(33)}{\approx} (H_{[a,k]} H_{[e,f]})((-1)_{[k]} X_{[k,a]})((-1)_{[j]} X_{[j,k]})((-1)_{[a]} X_{[j,a]}) $$

$$ \stackrel{(51)}{\approx} (H_{[a,k]} H_{[e,f]})((-1)_{[j]} X_{[j,k]})((-1)_{[k]} X_{[j,k]})((-1)_{[k]} X_{[k,a]})((-1)_{[j]} X_{[j,k]})((-1)_{[a]} X_{[j,a]}) $$

$$ \text{Corollary A.5} \qquad \stackrel{\approx}{(H_{[a,k]} H_{[e,f]})((-1)_{[j]} X_{[j,k]})((-1)_{[j]} X_{[j,a]})((-1)_{[a]} X_{[j,a]})} $$
$$ \stackrel{(51)}{\approx} (H_{[a,k]} H_{[e,f]}) ((-1)_{[j]} X_{[j,k]}) $$

• If $b = k$ and $a \neq j$, then

$$ \begin{align*} & ((-1)_{[j]} X_{[j,k]}) (H_{[a,b]} H_{[e,f]}) \\ &= ((-1)_{[j]} X_{[j,k]}) (H_{[a,k]} H_{[e,f]}) \\ &\stackrel{(67)}{\approx} ((-1)_{[j]} X_{[j,k]}) (H_{[k,a]} H_{[e,f]}) ((-1)_{[a]} X_{[k,a]}) \\ &\stackrel{(77)}{\approx} (H_{[j,a]} H_{[e,f]}) ((-1)_{[j]} X_{[j,a]}) ((-1)_{[j]} X_{[j,k]}) ((-1)_{[a]} X_{[k,a]}) \\ &\stackrel{(33)}{\approx} (H_{[j,a]} H_{[e,f]}) ((-1)_{[j]} X_{[j,a]}) ((-1)_{[j]} X_{[j,k]}) ((-1)_{[a]} X_{[a,k]}) \end{align*} $$

• If $\{a, b\} \cap \{j, k\} = \emptyset$, then

$$ ((-1)_{[j]} X_{[j,k]}) (H_{[a,b]} H_{[e,f]}) \stackrel{(74)}{\approx} (H_{[a,b]} H_{[e,f]}) ((-1)_{[j]} X_{[j,k]}) $$

In all seven cases, we have transformed $(( -1 )_{[j]} X_{[j,k]} ) ( H_{[a,b]} H_{[e,f]} )$ into some $(H_{[a',b']}) H_{[e,f]}$) $q$, where $\{a', b'\} \cap \{e, f\} = \emptyset$, and $q$ is the product of between one and three generators of the form $(( -1 )_{[j']}) X_{[j',k']})$, which are such that $j', k'$ are distinct from $e, f$. Then we can apply the same reasoning to transform $q$ $(H_{[c,d]} H_{[e,f]})$ into some $(H_{[c',d']}) H_{[e,f]}$) $p$, where $\{c', d'\} \cap \{e, f\} = \emptyset$, and $p$ is the product of between one and nine generators of the form $(( -1 )_{[j']}) X_{[j',k']})$.

This shows that the word $w = p_1 (H_{[a,b]} H_{[c,d]}) p_2$ can be put in the form $(H_{[a',b']}) H_{[e,f]})(H_{[c',d']}) H_{[e,f]}) p_3$, where $p_3$ is a product of generators of the form $(( -1 )_{[j']}) X_{[j',k']})$. The next step is to ensure that the indices $a', b, c', d'$ satisfy the required conditions.

If $a' < b'$, then we proceed as follows

$$ \begin{align*} (H_{[a',b']} H_{[e,f]}) (H_{[c',d']} H_{[e,f]}) p_3 &\stackrel{(67)}{\approx} (H_{[b',a']} H_{[e,f]}) ((-1)_{[a']} X_{[b',a']}) (H_{[c',d']} H_{[e,f]}) p_3 \\ &\stackrel{(33)}{\approx} (H_{[b',a']} H_{[e,f]}) ((-1)_{[a']} X_{[a',b']}) (H_{[c',d']} H_{[e,f]}) p_3 \end{align*} $$

and we apply the same reasoning as above to obtain $(H_{[b',a']} H_{[e,f]}) (H_{[c'',d'')} H_{[e,f]}) p_3'$, where $p_3'$ is a product of generators of the form $(( -1 )_{[j']}) X_{[j',k']})$. Thus, we have got $(H_{[a'',b'')} H_{[e,f]}) (H_{[c'',d'')} H_{[e,f]}) p_3'$, where $p_3'$ is a product of generators of the form $(( -1 )_{[j']}) X_{[j',k']})$, and $a'' < b''$. Then, if $c'' < d''$, we apply Equations (67) and (33) one more time to get $(H_{[a'',b'']} H_{[e,f]}) (H_{[d'',c'']} H_{[e,f]}) p_3''$, where $p_3''$ is still a product of generators of the form $(( -1 )_{[j']}) X_{[j',k']})$. Thus, we have got $(H_{[a'',b'']} H_{[e,f]}) (H_{[c'',d'']} H_{[e,f]}) p_3''$, where $p_3''$ is a product of generators of the form $(( -1 )_{[j']}) X_{[j',k']})$, and where $a'' < b''$ and $c'' < d''$.

Finally, we have

$$ \begin{align*} (H_{[a'',b'']} H_{[e,f]}) (H_{[c'',d'']} H_{[e,f]}) p_3'' &\stackrel{(66)}{\approx} (H_{[a'',b'']} H_{[e,f]}) (H_{[e,f]} H_{[c'',d'']}) p_3'' \\ &\stackrel{(71)}{\approx} (H_{[a'',b'']} H_{[c'',d'']}) p_3'' \end{align*} $$

Up to applying Equation (70) in the case where $\{a'', b''\} = \{c'', d''\}$, or Equation (66) in the case where $\{a'', b''\} \cap \{c'', d''\} = \emptyset$ and $a'' > c''$, and to applying Lemma A.4 to $p_3''$, we get the desired form. $\square$

**Corollary A.10.** Given any two words $w, w'$ over the generators of $\mathcal{P}_N$, such that one of the two contains at most one generator of the form $(H_{[a,b]} H_{[c,d]})$, and the other contains at most two generators of this form, if $[[w]] = [[w']]$ then $w \approx w'$.

**PROOF.** By applying Corollary A.9 as many times as the number of generators of the form $(H_{[a,b]} H_{[c,d]})$, we can transform $w$ (resp. $w'$) into a word $\tilde{w}$ (resp. $\tilde{w}'$) containing at most as many such generators as $w$ (resp. $w'$) and where all such generators are on the left. Then by analysis of the effect of $[[\tilde{w}]]$ and $[[\tilde{w}']]]$ on the canonical basis of $\mathbb{C}^N$, one can show that if $\tilde{w}$ or $\tilde{w}'$ contains two generators of the form $(H_{[a,b]} H_{[c,d]})$, then one can necessarily make them cancel out by means of Equations (66), (70), (71) and (73). Then the result follows by applying Corollary A.9 again, thanks to the uniqueness of the normal form. $\square$

**Corollary A.11.** The equations of Figure 11 are consequences of the equations of Figure 8.

**PROOF.**
$$ (H_{[0,1]}H_{[0,2]}) (H_{[1,3]}H_{[0,1]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,2]}H_{[1,3]}) \approx \qquad (79) \\ (H_{[0,2]}H_{[1,3]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,1]}H_{[0,2]}) (H_{[1,3]}H_{[0,1]}) $$

$$ (H_{[0,1]}H_{[0,2]}) (H_{[1,3]}H_{[0,4]}) (H_{[1,5]}H_{[0,1]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,4]}H_{[1,5]}) (H_{[0,2]}H_{[1,3]}) \approx \qquad (80) \\ (H_{[0,2]}H_{[1,3]}) (H_{[0,4]}H_{[1,5]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,1]}H_{[0,4]}) (H_{[1,5]}H_{[0,2]}) (H_{[1,3]}H_{[0,1]}) $$

<caption>Figure 11: Simpler versions of Equations (45) and (46), more complicated to decode but simpler to use in derivations.</caption>

Proof of Equation (79):

$$ (H_{[0,1]}H_{[0,2]}) (H_{[1,3]}H_{[0,1]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,2]}H_{[1,3]}) $$

$$ \text{Corollary A.5} \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) ((-1)_{[4]}X_{[2,3]}) (H_{[0,1]}H_{[0,2]}) (H_{[1,3]}H_{[0,1]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,2]}H_{[1,3]}) $$

$$ \text{Corollary A.10} \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) (H_{[0,1]}H_{[0,3]}) ((-1)_{[4]}X_{[2,3]}) (H_{[1,3]}H_{[0,1]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,2]}H_{[1,3]}) $$

$$ \text{Corollary A.10} \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) (H_{[0,1]}H_{[0,3]}) (H_{[1,2]}H_{[0,1]}) ((-1)_{[4]}X_{[2,3]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,2]}H_{[1,3]}) $$

$$ \text{Corollary A.5} \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) (H_{[0,1]}H_{[0,3]}) (H_{[1,2]}H_{[0,1]}) ((-1)_{[1]}(-1)_{[0]}) ((-1)_{[4]}X_{[2,3]}) (H_{[0,2]}H_{[1,3]}) $$

$$ \text{Corollary A.10} \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) (H_{[0,1]}H_{[0,3]}) (H_{[1,2]}H_{[0,1]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,3]}H_{[1,2]}) ((-1)_{[4]}X_{[2,3]}) $$

$$ (70) \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) (H_{[0,1]}H_{[0,3]}) (H_{[1,2]}H_{[0,1]}) (H_{[7,6]}H_{[7,6]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,3]}H_{[1,2]}) ((-1)_{[4]}X_{[2,3]}) $$

$$ (73) \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) (H_{[0,1]}H_{[0,3]}) (H_{[1,2]}H_{[7,6]}) (H_{[0,1]}H_{[7,6]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,3]}H_{[1,2]}) ((-1)_{[4]}X_{[2,3]}) $$

$$ (66) \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) (H_{[0,1]}H_{[0,3]}) (H_{[7,6]}H_{[1,2]}) (H_{[0,1]}H_{[7,6]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,3]}H_{[1,2]}) ((-1)_{[4]}X_{[2,3]}) $$

$$ (73) \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) (H_{[0,1]}H_{[7,6]}) (H_{[0,3]}H_{[1,2]}) (H_{[0,1]}H_{[7,6]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,3]}H_{[1,2]}) ((-1)_{[4]}X_{[2,3]}) $$

$$ (21) \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) (H_{[0,1]}H_{[7,6]}) (H_{[0,3]}H_{[1,2]}) (H_{[0,1]}H_{[7,6]}) ((-1)_{[0]}(-1)_{[1]}) (H_{[0,3]}H_{[1,2]}) ((-1)_{[4]}X_{[2,3]}) $$

$$ (45) \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) (H_{[0,3]}H_{[1,2]}) ((-1)_{[0]}(-1)_{[1]}) (H_{[0,1]}H_{[7,6]}) (H_{[0,3]}H_{[1,2]}) (H_{[0,1]}H_{[7,6]}) ((-1)_{[4]}X_{[2,3]}) $$

$$ (21) \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) (H_{[0,3]}H_{[1,2]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,1]}H_{[7,6]}) (H_{[0,3]}H_{[1,2]}) (H_{[0,1]}H_{[7,6]}) ((-1)_{[4]}X_{[2,3]}) $$

$$ (73) \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) (H_{[0,3]}H_{[1,2]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,1]}H_{[0,3]}) (H_{[7,6]}H_{[1,2]}) (H_{[0,1]}H_{[7,6]}) ((-1)_{[4]}X_{[2,3]}) $$

$$ (66) \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) (H_{[0,3]}H_{[1,2]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,1]}H_{[0,3]}) (H_{[1,2]}H_{[7,6]}) (H_{[0,1]}H_{[7,6]}) ((-1)_{[4]}X_{[2,3]}) $$

$$ (73) \qquad \approx \qquad ((-1)_{[4]}X_{[2,3]}) (H_{[0,3]}H_{[1,2]}) ((-1)_{[1]}(-1)_{[0]}) (H_{[0,1]}H_{[0,3]}) (H_{[1,2]}H_{[0,1]}) (H_{[7,6]}H_{[7,6]}) ((-1)_{[4]}X_{[2,3]}) $$
$$ \stackrel{(70)}{\approx} ((-1)_{[4]} X_{[2,3]} (H_{[0,3]} H_{[1,2]} ((-1)_{[1]} (-1)_{[0]})) (H_{[0,1]} H_{[0,3]} (H_{[1,2]} H_{[0,1]})) ((-1)_{[4]} X_{[2,3]}) $$

$$ \text{Corollary A.10} \quad \stackrel{\sim}{\approx} (H_{[0,2]} H_{[1,3]}) ((-1)_{[1]} (-1)_{[0]}) (H_{[0,1]} H_{[0,2]}) (H_{[1,3]} H_{[0,1]}) $$